PHOTOPLASTICITY

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• Release Decision: RIPPUB
• Original Classification: K
• Document Page Count: 202
• Document Creation Date: January 4, 2017
• Document Release Date: June 18, 2014
• Sequence Number: 1
• Publication Date: August 1, 1959
• Content Type: REPORT

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TH:111SIBTIOR
PHOTOPLASTICITY
By. S. I. Gubkin, S. 1. Dobrovolfsky, B. B. Boyko
August 1959, 194 pages
Al
STAT
PREPARED BY
LIAISON OFFICE �
TECHNICAL INFORMATION CENTER
HMO
WRICHT-PATTERSON AIR FORCE BASE. OHIO
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Aloademiya Newk MS*
Fiziko�Tekhnicheskiy Institut
Es. IQubkin1
S. I. Dobrovorsky, B. B. Boyko
POTOPLASTICHNOST1
Izdatellstvo Akademii Nauk Belorusskoy SSR
Minsk 1957
167 pages
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SITICIARY
This monograph presents the basic principl s of a new
experimental method for studying the processes of plastic
deformation and stresses by translucence in polarized light
optically sensitive materials which
171 'VS Los. -
subjected to a
permanent deformation. This is known as the method. of photo
plasticity.
The initial presentation of the new method is given in
the present �momograph in terms of viscous flow.
�
The results of this work may be used in studying models
of the various processes of plastic deformation.
This book is intended for use by engineers and scientific
personnel.
�
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INTRODUCTION
One of the authors of this monograph, S. I. Gubkin, organized.
in 1949, at the PhrimaI.TAnhni nal Institute of the Academy of
Sciences of BSSR, a laboratory for the purpose of developing the
method of photoplasticity.
The initial investigations in this laboratory were conducted
by S. I. Oubkin and S. I. Dobrovoltsky. Some results of these
investigations were published in 221..E1 USSR [10, 11]. In
1952; the staff in the laboratory was joined by B. B. BoYko.
3y the end of 1954; it was Possible to conclude that as
taN
result of the laboratory investigations one of the basic problems
of photoplasticity was resolved in its basic form -- determining
the stress state by methods of photoplasticity under viscous flaw
conditions. The solution of this problem which depicts the basic
characteristics of this method permits us to consider that the
problem of photoplasticity has been solved in principle and estab-
lished as an independent method of investigation.
In order to permit the most rapid possible development and
application of this useful method; the Scientific Council of the
Physical-Technical Institute of AN BSSR recommended that the
laboratory publish this monograph.
The present monoLvraph Rnmmpry.i.700 the wmt-r4-' -bt-411-imuLA
the investigations conducted during the development of the method
of photoplasticity at the PTI [Physical-Technical Institute]
AN BSSR under the direction and with participation of the Member of
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�
� AN BSS, S. I. Gubkin.
The work of preparing the m^7,^rrraph for publication was divided
among the authors as follows: S. I. Gubkin prepared an outline of
the work and prepared the first and sixth chapters and also edited
the manuscript; B. B. Boyko prepared the fourth chapter and the
'second sFction of the fifth chapter, and S. I. Dobrovol'sky prepared
the second and third chapters and the first parvA third sections of
the fifth chapter.
This monograph was designed for readers who are familiar with
the method of photoelasticity.
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.CHAPTEI I
IOTHOD OF PHOTOPIASTICITY
1, Photmelasticity
At the present time there is no field of teciamology in which one does not
have to deal with stresses and strains, both elastic and plastic. The study
of distribution of stresses in an elastically deformed body and the determina�
tion of these stresses is one of the most important problems of modern mechanics.
An analytical solution of these problems based on theory of elasticity i�
usually associated with a solution of partial differential equations with
partial derivatives and with difficulties in finding boundary conditions.
luc
A4.P-.P4elet.11-4-4.fte,
are particularly troublesome
mi. a a
problems involving irregular
autllnes and a complex distribution of applied loads. The solution of
problems which are of the greatest practical interest are those associated for
instance with stress concentratons in machine parts subjected to repetitive
loads of alternating sign, and in a number of cases problems of this kind pre�
sent the greatest difficulties.
The theory of double refraction which appeared iu the forties of the
last century in regard to "compressed and nonuniformly l'entatl noncrystalline
bodies" which was derived by F. Neuman from the conditions of static equilib�
rium aniwhich.was subsequently developed. by Maxwell, lay the basis for the
creation of an experimental method of measurement of stresses in an elastic
medium.
The French scientist Leger conducted at the end of the last century
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extensive investigations leading to the initial development of the experimental
method of measurement in an elastic medium which subsequently was named the
method of photoelasticity (i.e., a methnd for obtaining a pictorial represen-
tation of the state of stress for an elastically deformed medium). This work
was highly praised by our_compatriot, the founder of metallography, D. K.
Chernov, in his communication of 10 March 1884 on the topic, "Some Generali-
zations on New Observations in the Working of Steel" /1/.
In 1912, under the direction of Prof. V.
VirpichaVi there was con-
structed and establ4heri'lly A. K. Zaytsev the first Russian laboratory for
studying stresses in transparent models by the method of photoelasticity.
This installation was constructed for the laboratory of apilied mechanics of
the Present Leningrad Polytechnic
Institutee
A detail historical"
Q
reView of
this method is given in "Proceedings of the Conference on the Optical Method
of Measuring Stresses" /2/.
The essence of the method of photoelasticity consists of the following.
An optically isotropic plate prepared from an optically sensitive transparent
material having been placed in a state of nonuniform elastic straiii becomes
optically anisotropic as long as the applied load causing the elastic defor-
mation is maintained. Therefore, polarized white or monochromatic light dis-
plays double refraction in passing through the loaded plate. This effect con-
sists in the splitting of a light ray into two rays which vibrate in two
mutually perpendicular planes and which propagate in the medium with different
velocities. Because of the difference in the velocities, we observe a differ-
ence between the propagations of the two rays defined by the formula
cd(cL� 00,
� where R is the difference in the paths,
d is the thickness of the plate;
c is a constant known as the optical coefficient of stress which depends
vormissiNiMA
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on the properties of the material;
61 and 62 are the principal stresses at a given point.
It must be noted that the maximum shearing stress is given by
�
max 2
at a2
�
In the case of circularly polarized monochromatic light subjected to double
refraction in passing through a loaded place in the field of the polariscope,
we obtain on the photographic plate a syatemof
dark.
affly�Ar
fringes which are called
isochromatics (Fig, 1). Each of these fringes represents the locus of the
points in which the shearing stress has the same value. Iu this instance,
the magnitude of the shearing stress is determined by the value assigned to a
fringe and to its order. The order n of the fringe is determined by
ference in the path expressed in wavelength
is given by the expression
of light. The value of
In order to determine the fringe value there are appropriate methods.
16,14V
fringe
(2)
Figure 1. A system of dark fringes (isochromatics) produced by polarized light
in elastic optically sensitive materials*
In the case where white light is used, the fringes corresponding to equal
shearing stresses have different colors which deprnd upon the value of the
maximum shearing stress (Figure. 2, see insert between pages 46 - 11.7 ).
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Experience has shown that for quantitative studies it is more convenient to
use morochromatic light, while white light which gives a color picture of the
distribution of stresses is more useful for graphical demonstration.
When we use plane polarized light, we observe on the general fringe pat-
tern certain supplementary dark lines or regions which are called isoclinics.
ISoclinics represent the locus of the points in which the principal stresses
have the same direction, one of which coincides with the orientation of the
�
plane of polarizatio-
^f light. Having the isoclinics; we can construct a
grid .of trajectories of the principal stresses.
Knowing at each point the magnitude of the maximum shearing
the directions of the principal stresses as well as the boundary
stress and
conditions,
it is possible to determine the sensor of the stresses at each point of the
elastically deformed -body using the methods of the
The photoelastic method
toSAG%.04.47 160.1. IiiiiKarvymme=
has undergoneextensive development during the
�
recent years. The work done over a period of many years in a number of scien-
tific schools in the development of this method has led to greater perfection
of this method. Improvement in the experimental technique has enabled us to
obtain an undistorted picture of the stresses along the edges of the model.
This has presented great possibilities for studying the concentration of
stresses. The considerable material accumulated in recent years
^vs
the nnnf.
ficients of. stress concentrations has clarified +11n nausea of fatigue failures
and has shown the way in a number of cases to eliminating these failures.
These data were obtained primarily as a result of investigations carried out
by means of the.photoelastic method.
.AU Ult= loslursal t.""a�
AL Li
i-ha'sa in no other method which permits us to determine
so completely the state of stress with considerable precision and with rela-
tively small expenditure of time and effort.
The recent developments in three-dimensional photoelasticity offer even
more promising 'perspectives.
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Of particular significance in the development of this method are'th.V.
works of the Soviet physicist A. V. Stepanov with the following results
(a) The peculiarities of stress distribution in elastic anisotropic
media compared with isotropic media under the same conditions have been de-
termined /3/. An analysis of these peculiarities leads to the conclusion
that even in so-called quasi-isotropic media, among which all polycrystalline
metals may be included, the distribution of stresses may sometimes differ sig-
nificantly from the distribution in isotropicmedia. Therefore, 'ellen we de-
termine the distribution of stresses in metal parts by means of models of
isotropic material,, we. do not always obtain a precise picture of the actual'
Acamt.rillntion of stresses in these parts;
�(b)
A group of anisotropic crystalline substances was discovered whose
mechanical properties and crystalline structure are-similar to the mechanical
properties and structure of metallic crystals /4/. Among
Utivs.ww
alillatances are
haloid salts of silver and thallium and their various alloys. These substan-
ces rere discovered by A. V. Stepanov about 1935, and, at the July session of
Academy of Sciences of USSR in 1944, he reported a new optical method of
determining stresses in an elastic mediums differing from the usual photo-
�
elastic method, in which models prepared from a transparent material of crys-
talline structure were utilized. A. V. Stepanov proposed that the "trans-
parent metals" be used as materials for such models. Among these he named
chloric silvers and other haloid salts of silver and thallium' andtheir
various alloys.
A. V. Stepanov's proposal permits us to determine by means of elastically
loaded models of "transparent metals" a much more precise picture of the dis-
tribution of stresses than by means of models made of similarly loaded iso-
tropic material. Acceptance of this proposal will lead to considerable fur-
ther development of the existing method of photoelasticity.
* A. V. Stepanov, Author's certificate No 47829y 30 June 1936.
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At the present time the engineering profession is already aware of the
fact that the photoelastic method is a reliable and perfected tool in the
hands of investigators and designers which yields In many cases a rapid and
accurate solution of very difficult practical problems and. which at the same
time offers-the means of confirming solutions by mathematical theory of elas-
ticity, thus increasing the value of the theory itself. Therefore, during the
recent years, there have appeared many important papers on the subject of photo
elasticity, many new laboratories have been established for experimental deter-
mination of states of stress by the photoelastic method,and numerous studies
are in progress to perfect still further this visual method of stress analysis
which in the words of A. K. Zaytsev "makes the invisible visible" /5/.
2. The Need for Development of an Experimental Method of Studying the State.
of Stress with Plastic Deformations.
When we determine the state of stress in plastically deformed bodies by
analytical methods, we must use the mathematical tools which are now available
to us In the mathematical theory of plasticity and those data on the mechani-
cal behavior of bodies which we can obtain by modern laboratory experimental
techniques. The solution of problems In the theory of plasticity as well as
.in the theory of elasticity involve the use of very difficult differential
equations with partial derivatives and the difficulties associated with formu-
lation of boundary conditions.
However, the difficulties which arise in solving problems in the theory
of plasticity, taking into account the present state of the science of deforma-
tions, are infinitely more difficult than the difficulties arising in the prob-
lems of the theory of elasticity.
This can be explained by the following circumstances:
1. The mathematical theory of plasticity is not nearly as well developed
as the mathematical theory of elasticity. This is explained first of all by
the considerably greater scientific maturity of the theory of elasticity as
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compared with the theory of plasticity, which is a relatively recent offshoot
of science, and also by the inadequate study of the physical nature of the
plastic process.
2. In the general case considerably greater difficulties arise in for.
mulating the boundary conditions for problems in plasticity than for those in
elasticity.
Particularly great difficulties arise in the solution of problems in
which plastic deformation is accompanied by surface friction /V. The latter
has exceptionally great egfeot
Ark 1/11
la' 64
the character of stress distribution
in the deformed body and on the magnitude of the deforming forces.
In a
number of cases this magnitude is affected not so much by the mechanical pro-
perties of metal (which determine its resistance to plastic deformation)
as the
conditions of contact friction. As is shown by experiments, the deforming
force may be reduced by a factor of more than five in certain cases by changing
the conditions of contact friction.
410 In spite of the great significance of the conditions of contact, we do
not as yet know the laws governing the contact friction during plastic de-
formation. In many practical cases of deformation the specific frictional
force is given by the complex. function:
VI, 44,
tst F (N ,
where' C is a specific frictional force;
m
N is
the normal pressure;
f%l
16,0,1
1.-;sn is the yield point in shear of the surface layer situated near
the contact surface;
W is the velocity of slip of the particles of the material in the
surface layer;
tn is the temperature of the contact layer (it may differ from the
�
temperature of the body being deformed).
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Experimental determinOion of function (3) is at times extremely
3. Consider-.. Consider greater difficulties arise in describing the mechanical
behavior of a plastically deformed body than in describing the behavior of an
elastic body.
The description of the mechanical behavior of an isotropic elastic body
is completely determined by two constants -- Young's modulus and Poisson's
ratio. However, the mechanical behavior of a plastically deformed body may.
described in the g!enaral case only by a complex function of the following.
typet:;,-
aw-ama (to, q, v, a, x, c).
Here 6sq is the resistance of the material to deformation;
t is the temperature, o,
is the uag.L-ee of deformation;
is tha rate of deformation;
is the average stress;
is the chemical composition;
(4)
w is the structure of the material being deformed which zs a rule
changes during the pr less of deformation*
Thus, instead of completely determined constants which describe the
elastic behavior of the body, in describing the plastic behavior of the same
body we must deal with a complex function which considerably complicates the
solution of the problem. At the same time, the laboratory procedure by means
of which we can determine precisely the constants describing the behavior of
an elastic body is not yet sufficiently developed for a precise determination
of the functions of type (4).
4. The nature of the plastic process has not yet been adequately studied.
This
akes it impossible to take into account certain phenomena which accompany
�
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the process of plastic deformation and affect th
thus lessening the precision,.
*stresses.,
Therefore, the science of plasticity needs the creation of an experimental
method of measuring the stresses by a method similar to the photoelastic method
even to a greater degree than the science of elasticity. This is necessary
to conserve equipment and time in solving many important problems in plasticity
as well as to assure
mathematical theory.
further progress of the science of plasticity and its
The development of the experimental method of measuring
stresses in plastically deformed bodies will permit us to verify the existing
methods of mathematical theory of plasticity, as well as to perfect and develop
them. Such a method in addition will broaden our concept of the nature of
plastic deformation 4Tolalfo
The basis of the method of photoelasticity is the effect of double re-
fraction caused by elastic deformation of the medium. However, plastic de-
formation-can occur and develop onlz.1.11_832.2122.Ils2222Albsped medium. This
postulate is a most important law of plastic deformation. From it we can draw
a corollary regarding the possibility of observing the effect of double re-
fraction in a plastically deformed body. Thus,
of creating an experimental method of measuring
deformation. The basis of the proposed methodt
there arises the possibility
stresses caused by plastic
just as in the pt.otoelastic
method, is the effect of double refraction caused by elastic deformation, i.e.,
by a reversible displacement of kinetic units of a substance (atoms, molecules,
macromolecules), which also accompany every process of plastic change in form.
Therefore, the given method may be called the method of photoplasticity as
being analogous with the method of photoelasticity (the method of obtaining a
pictorial representation of the states of stress for a plastically deformed
medium).
The method of photoplasticity, while based on the same physical phenote-
non as the method of photoelasticity, differs from it in certain principles,
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and, therefore,
more fully below.
The method of photoelasticity utilizes the process of elastic
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y be considered quite Independent. We shall discuss this
Aefeirmsatinn
and solves the problem of stress distribution by supposing that the stresses
in the model do not exceed the elastic limit. Therefore, the mechanical be-
havior of models of isotropic material is described by Hooke's law with optly
two constants being involved
AMP Allik
Young's modulus and Poisson's ratio.
The method of photoplasticity is the model analysis of the process of
plastic deformation and solves the problem of states of stress by supposing
that the stresses in the model exceed the elastic limit. The mechanical be-
havior of the models in this case is described by the laws of plasticity,
which have a variable form depending upon the nature of the material being
_ mec
deformed and the conditioas of deformatiori. relme the hanical behavior of
the models in the case of the method of photoplasticity ma be different.
Therefore, every method must be developed by solving typical problems corres-
ponding to the classification of the rheological behavior of solid bodies.
3. Classification of the Rheolo ical Behavior of Solid.Bodies.
The character of the relationship between resistance to shear and such
parameters as rate of deformation, the degree of deformation and average
stress, is well as the nature of failure of the substance yield an adequate
'picture of the rheological behavior of the substance at a given temperature.
In this instance all solid substances may be distinguished by their character-
istic deformations and they may be represented by several rheological badies
among which are:
L. Brittle Body. An ideal brittle body fails at elastic deformations
of negligible value without indicating any yielding whatsoever. By deforma-
tions of negligible value we mean deformations less than 0.001% (one of the
specified values' of the elastic limit). In the Case of presence of any symp-
toms of yielding or 'failure it small elastic deformations of the order of
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0.001-1%, the body is brittle if the residual deformations are of the same
magnitude as the elastic ones.
2. Elastic Body. An ideal elastic body permits unlimited elastic de-
formations without any indications of yielding or failure. The body which
fails without any symptoms of plastic deformation, but at large values of
elastic deformations is an elastic body. By large elastic deformations we
mean deformations which have minimum values measured in tens of percent. The
relationshiv between the stresses and deformations may be either linear or
nonlinear. An elastic body may be combined with a brittle ore. The, we may
have elastic-brittle bodies. However, if the failure takes place as a result
of large elastic deformations, then sucha failure cannot be defined as a
brittle failure. This, then, becomes a special type
associate with elastic bodies.
of failure Wa
3. Viscous Bodx. The basic characteristics of a viscous body are the
following:
(a) The resistance of a viscous body to shear, depends upon the rate of
deformation and does not depend Upon its .magnitude;
(b) during the flow process
f
a viscous body there is an absence of re-
sidual changes in its structure and properties;
(c) before a certain rate of deformation determined by the nature of the
given viscous body and temperature is attained, this body shows infinitely large
.plastic deformations without any symptoms of discontinuities.
In is well-known work P. P. Kobeko cites a description of interesting
experiments by Kornfeld and Ryvkin in which they subjected a stream of liquid
to rapid impact applied at right angles to the direction of its flow and
photographed the stream at the moment of impact /7/. For an impact applied
at 19 m/sec the stream was deflected plastically as a unit, while for an impact
applied at 23 m/seo the stream is shattered as brittle glass and is fragmented
in the manner shown clearly in Figure 3.
=
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A. Nadai shows that one of the compounds of silica (silicone) prepared
in the form of a sphere flows under its own weight, but when this sphere is
�
permitted to fall 1-2 m, it rebounds as an ideally elastic material /8/.
However, the elastic state is apparently only a transitional olL,: and if the
rate of deformation is increased, the sphere becomes a brittle body. Accord-
ing to the observations of the authors, certain gels behave in precisely the
same manner for very slow rates as elasto-viscous bodies, and at rates of de-
frrpmafinn
f.arf-min vall)aca they behave an elastic Ones.
The resistance to shear of a viscous body for a given rate of deforma-
tion depends notably upon the hydrostatic pressure. This relationship may
be expressed by the formula
e
tn.
where Cris the average stress;
Tr, is the resistance to shear for cr= 0;
7'6 is the resistance to shear at a mean stress equal to
a is a function dependent upon the molecular weight.
;
Figure 3. a the form of a stream of liquid with a viscosity of 5000 poises
subjected to lateral Impact at a velocity of 19 m/sec; b the same for an im-
pact with a velocity of 23 misec (from Kornfeld and Ryvkin).
Formula (5) is analogous to the well-known formula which gives the rela-
tionship between the coefficient of viscosity and the hydrostatic pressure.
It is useful to consider two types of viscous bodies: a liquid viscous
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A
body and a solid one. The liquid viscous body has a low yield point and as a
00 result assumes under its own weight after a certain length of time the shape
of its container. A solid viscous body has such a yield point that under the
influence of its own weight it cannot assume the form of its container.
The rheological curves of a viscous body are shown in Figure 4.
4. Plastic Bodz. A plastic body possesses the following characteristics:
(a) its resistance to shear does not depalad iitr,,,za the rate
f deformation
but may depend upon its magnitude;
(b)
during the process of plastic deformation there occurs a
the structure and properties of the body.
change in
It is necessary to distinguish between a nonfailing plastic body and a
failinr one. A nonfailing plastic body produces infinitely large plastic de-
formation
without ="Y SYMPtOME Of
discontinuities, while
embrittled by plastic deformation and
fails when the deformation attains a
certain value. A plastically em-
brittled body (nonplastic body) must
be differentiated from a brittle one.
The former may fail at very small
elastic deformations without any
noticeable geometric symptons of
plastic deformation just as a brittle
body, but it differs from a brittle
body in that failure may occur in it
a, failing body becomes
Figure 4. Rheological curves of a vis-
cous body: /r is the stress; V is the
rate of deformation; E is the extent
of defernation,---. indicates an infi-
nitely large plastic deformation.
only as a result of a preliminary plastic deformation, while a brittle body
fails without it. An enbrittled body and a 1.-,rttle one also differ in the
character of their failures. The failure of an entrittled body is usually
termed a viscous failure while that of a brittle body is called a brittle
failure.
17
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The mean value of the stress has practically no effect on the resistance
of the plastic nonfailing body to shear. In the case of a plastic failing
body, the effect of the mean stress becomes more pronounced as the body be-
comes more embrittled in the process of plastic deformation.
Figure 5 shows rheological curves characteristic of a plaetic body.
Figure 5a shows
stage we observe the
of deformation
the rheological curve of a nonfalling body. In the initial
greatest increase in the strength of the material. As the
irnPPAMAA, t strength wain decreases and finally ceaseei
entirely, as a result of which the rheological curve becomes a straight line
Parallel to the. axis of abscissa. The straight line in Figure 5b is character-
istic either for a material already strengthened by means of preliminary plas-
tic deformation or for the very first stage of deformation of certain materials
having a sharply defined yield point and which are undergoing cold working. A
material with such a rheological behavior is often called an ideal plastic
body /8/. The curve in Figure 5c is typical of plastic materials which fail
as a result of embrittlement in the process of plastic deformation.
mi
.r!k
1 sa
straight line in Figure 5d may in certain cases represent with sufficient
accuracy that portion of the actual rheological curve corresponding to a given
stare of deformation. The curve in Figure Se is characteristic for materials
with a sharply defined embrittlement occurring during the process of deformation.
In most cases, this curve indicates that under given conditions of change of
fora the-predeftinant mechanism of flow is an intercrystalline deformation which
takes place with progressive breakdown of the bonds between crystals.
It is quite obvious that there are in nature combinations of different
rheological bodies. Therefore, it is proper to describe the behavior of the
following combined bodies: (1) plastic-viscous; (2) plastic-brittle; (3)
elastic-plastic; (4) elastic-viscous; (5) elastic-brittle; (6) viscous-
plastic-elastic; (7) elastic-viscous-plastic; (8) elactic-plastic-brittle;
(9) elastic-viscous-plastic-brittle.
18
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v: 0
Oft
7.1
k
Figure 5. Rheological curves of a
plastic body:--4 indicates infinitely
large plastic deformation; I indicates
failure.
���������
Figure 6. Rheological curves for
combined bodies.
The combined bodies possess respectively the combination of characteris-
tics of the basic Theological bodies which form the combination. The rheo-
logical curves of the combined bodies also reflect the combinations of the
curves ^f the haai'- bodies (zee Firlre 6).
It must be noted that elasticity to some degree is present in all natural
bodies. In addition to that; the process of flow can only occur in an elas-
tically deformed body. However, in many cases it is expedient to neglect the
elasticity to simplify the calculations and the description of the behavior
of the bodies. For instance, it may be neglected in describing the behavior
of a largo number of viscous bodies. Whether or not the elasticity of a vis-
cous body may be neglected is even considered one of the most important cri-
teria for deciding whether a body is solid or liquid. For instance, in the
experiments of Kornfeld and Ryvkin mentioned above,
the .experimenters investi-
gated the behavior of a stream of solution of rosin in mineral oil. This sub-
stance having a viscosity
3
= 5�10 poises may be considered only as a liquid
body as nudged by the aggregate of its properties. The authors also dealt
with a rosin plasticized .ith a rosin ^41 but in different proportions thorn
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those used in the above experiments. The viscosity of this substance wol..s
= 2.5�108; it was not possible to determine the yield point by ordinary
TI
means because of its extremely small value. The rheological behavior of this
body corresponds to the graph shown in Figure 38. At the same time, according
to external characteristics, this solid body which can be deformed plastically
with ease under a slow action of applied forces, under impact shatters into
small fragments as a brittle body. By-means of pressure this body may acquire
r.941
form, whinft chances only after a considerable period measured in months.
Thus, according to all symptoms this body approaches a solid viscous body. A
body possessing elastic properties to such an extent that they cannot be neg-
lected must
10.I. MIL
VC'
classified as a solid body, and in this instance we must des-
cribe its 'theological behavior by the combined rheological characteristic
shown in Figure 6a, Even a body whose elastic-deformation can be neglected
but which has a yield point which cannot be neglected must be classificsd as
a solid body.
In considering large plastic deformations of a plastic body we can neglect
elastic deformations, but in the case of plastic deformations which are com-
mensurate with elastic ones we must consider that we are dealing with an
elastic plastic body whose description has great practical significance.
Rheological characteristics of this body are shown in Figure 6b.
-Rheological characteristics determined by the relationship between re-
sistance to shear and the magnitude of deformation are the same for both the
plastic-brittle body and the plastic body which fails in the process of defor-
mation (an embrittled body). However, these bodies are distinguished by some-
what different rheological behavior because of the completely different cha-
racter of failure. In addition to that, the average stress has a considerably
greater effect on the resistance to shear of a plastic-brittle body than on
the resistance to shear of a plastic failing body. Figure 6c shows the rheo-
logical characteristic of a plastic-brittle body with average stress taken into
20
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1
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account, and in this instance this characteristic is given not by a single
curve, but by A family of curves. Red aandstone and marble (Figure 7) indicate
a behavior analogous to that described in Figure 6c. The average stress exerts
so great effect on the behavior of the
logioNow4
that by increasing.the absolute
value of the negative mean stress it is possible in a number of cases to trans-
form a body from a brittle-plastic state into a plastic one and even to obtain
" et. 4- 4 "
/leinfAiling body in some cases.
POO
1 1
IL"?'
# % � 1 .
oil
eel
'qet-65q11'"'i
.
/Al
.. ..2�15
0 I
r _ries
4041 VIII
1
,,,,
s
-
-7%
62 �
6.2 2 NVOr
- a T
FA
,,-...',...,,,..,,�
�
etwo
..,
1
1 I
1 1 i .i I 1 1 1I
1
, I i
347011
0 1 2 3
a
7
VIO. widow
Figure 7. Rheological curves for marble (a) and sandstone (b) under triaxial
compression (from Karman).
Legend: a) failure
rlastic-viscous bodies which are widespread in nature present consider-
able interest to us. Among them we cah consider first of all the great ma-
jority of polycrystalline metallic alloys. The metallic substance in a tem-
perature range below the temperature of recrystallization shows symptoms of a
plastic body to the rreatest degree, particularly for the lower temperatures.
The lower the temperature of deformation, the greater the Increase in strength
and the less the effect of the rate of deformation on the resistance to shear.
However, the behavior of metallic alloys becomes different at temperatures
exceeding the temperature of recrystallization. Then we begin to notice symp-
toms of a viscous body and these sgymptons are more pronounced at higher
temperatures.
Thus, if at the low temperatures symptoms of a plastic body nredominata,
then at temperatures approaching the temperature of fusion symptoms of a
21
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viscous body predominate in many cases. At high temperatures we may observe
that the extent of deformation has no effect on resistance to shear In spite
of the presence of structural transformations occurring during the process of
deformation, and the rheological behavior is described by characteristics
associated with a viscous body (Figure 4). However, the physical significance
of the rheological characterilic in this case is quite different from the
characteristic shown in Figure 4. In the case of a plastic-viscous body the
characteristic indicates that the gain in strength occurring in the process
,,,asint-ic deformation is completely eliminated by the weakening process
which accompanies the deformation, whlle in the case of a viicous body the
deformation Proceeds without gain in strength and residual structural changes.
The mechanical properties of the substance reflect the nature of the in-
termolecular forces. can draw certain conclusions regarding the structure.
of the substance in a series of cases from the values of the mechanical
properties.
A study of the rheological behavior of the substance along with an ana-
lysis of the mechanical properties and mechanism of flow improves our under-
standing of the structure of the substance and the structural changes caused
by,its flow. Conversely, the study of the structure and the structural chan-
ges of a substance aids us in gaining a better understanding of its 'theologi-
cal behavior. It is known that the character of the changes of many mecha-
nical properties of an alloy depends upon its chemical composition and the
type of its composition diagram. Besides that, many mechanical properties
and the rheological behavior of bodies depend on the peculiarities of their
structure and molecular weight. Thus, the viscosity and rheological behavior
of bodies cannot be considered as being unrelated to their molecular weight.
In the case of a small molecular weight we are dealing with mobile liquids.
As the molecular weight of polymers increases, the mobile fluid is transformed
successively into a viscous, then an elastic-viscous, body, and, finally, into
22'
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an elastic (highly elastic) body. Mechanisms of plasticity also depend on
%.Lalw
.1+.0
411. IA No � � IA.+ '41
-t4-t^rillAm 11^flic-im are nnnneiated with unordered,
mechanisms of plasticity. In such bodies the predominant mechanism is a
simple amorphous mechanism of plasticity. In plastic bodies we observe
orderly mechanisms of plasticity such as intercrystalline slip and twinning.
When we consider not only the orderliness of the structure itself but the
actual order as is shown well by a diagram taken from P. P. Kobeko /7/ (see
Figure 8), then we may derive considerable useful material for clarifying
the theological peculiarities of A substance which depend upon the degree
of orderliness of its structure.
The deformation characteristics of rheological bodies described above
(based on generalizations of materials derived from physical and physical-
tOndaminal
4vIvectignt4rinm) �sm fn pnsrm
the question of classifying the
rheological behavior of solid bodies with respect to its chemical and struc-
tural peculiarities. The creation of such a classification would aid in
obtaining a solution of a series of most important problems in various fields
of science involving the questions of deformations and chemical compositions.
Figure 8. A schematic twoi-dimensional diagram of crystalline structure
(left) and amorphous structure (right) of quartz.
The deformation characteristics which describe the behavior of basic
rheological bodies as proposed here are not yet sufficiently developed for
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the proposed classificatiou. Both the deformation characteristics and the
terminology need to be defined more accurately. At the same time our concept
of the symptoms of rheological behavior must be broadened in order to estab-
lish a more intimate relationship with chemical and structural properties
of the substance.
However, in order to solve the particular problem related to the typical
problems of photoplasticity, the classification of rheological behavior of
solid bodies proposed here R.ay be utilized successfully in its precont form.
On the basis of this classificati
draw the conclusion that the basic
typical problems of the method of photoplasticity are: (1) viscous, (2)
plastic, and (3) viscous-plastic. For ere.ch of these problems there is a
r' '4�
a
40 .an=010 �11,
attack.
Ilite&11
loavy...s.^...N. 4.11.v rw�G.A..0 MC%
an analytical method of
In the present monograph there is given a solution of the problem of
viscous photoplasticity, and methods for solutions of other problems in
photoplasticity are indicated. The problem is posed in this fashion since
the viscous problem is the simplest one both with respect to experimental
technique and theoretical treatment. In the process of solving this problem
a series of important questions of methods of attack which have a great sig-.
nificance for all problems of photoplasticity will be elarified; this solu-
tion will facilitate the solution of more complex problems among which are
the problems of plastic and viscous-plastic behavior.
4. Basic. Problems of Photo plasticity.
A
Ars �
V. Stepanov showed that the transparent crysta]s which he discovered
behave under pressure in a manner analagous to metals: they become stronger,
acquire a metallic texture, accumulate potential ener i
gy n the form of re-
sidual stresses and indicate the possibility of a large plastic deformation
by means of the very same mechanisms which operate in producing plastic de-
rmation in a metallic substance. Basing his reaooning on the Indicated
24 �
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properties of transparent crystals, he considers it possible to apply an
optical method for studying the states of stress which arise as a result of
plastic deformation in specimens having a grainy structure while they undergo
a change in crystalline structure. In A. V. Stepanov's opinion, investiga-
tions of this kind may have great significance in the study of various prob-
lems in metallurgy: geology and the theory of elasticity of a homogeneous and
quasi-isotropic body, the interaction between discrete particles of a semi-
crystal, the behavior of an individual grain and its boundaries, which will
aid
in the development of our concepts of such phenomena as elastic hysteresis,
creep, fatigue and relaxation. Similar investigations may also be undertaken
for the study of the process of cutting of metals and the forming of metal
by pressure. By studying the stresses present in transparent crystalline
substances in polarized light, it is possible to reproduce in a model the
complex technological process of a combination of mechanical and thermal
va,GCLUAll=Lito.
A. V. Stepanov showed experimentally that in plastic tensioning
of a flat specimen of polycrystalline chloric silver the 2-rains being deformed
become outlined fairly clearly (see Figure 9). In addition to that, he indi-
cated the possibility of studying the nature of residual stresses by means of
"transparent metals" investigated in polarized light*
S. 0. Tsobkallo and B. A. Kuznetzov have shown that it is possible to
study the nature of fatigue Li by means of "transparent rptnlm" under polar�
ized light. D. B. Gogoberidze and I. D. Kirvalidze drew the sane conclusion
having studied by means of an optical method various fatigue phenomena in
monocrystals of rock salt /9/.
Si I. Gubkin and S. I. D,Dbrovol'sky suggested in their work (devoted to
the application of transparent -models to the study of forming of parts by
means of pressure) that during the plastic deformation of certain slowly
deformed gels the observed pattern of isochromatics indicates the geometric
locus of maximum shearing stresses of the same magnitude jg. The same
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authors have established in another work that the very same well-defined
pattern of isochromatice may be observed in plastic deformations of certain
transparent resins gg. They also established that during plastic deforma-
tion of transparent optically sensitive materials it is possible to observe
both an ordered and disordered isochromatic pattern.
Figure 9. View of a specimen of chloric silver in terasioa Qvio&aU41-
field of a plane polariscope (from Stepanov).
An isochromatic grid is called an elyviaireA ono
WUCILIL
406
4n the
reveals a certain
isochromatic pattern characteristic for the given type of loading (Figure
10a). A disordered isochromatic grid is one which reveals a field of va-
riously colored polygons with a real or apparent absence of any system of
coloring of these polygons (Figure 10b).
'4
Flaira3,1
LIN - A �
424/114,4 r
ti,4141k .
*-
t..!:4;12L:411L. � .**4
* 4.
Figure 10. Isochromatic pattern: a an ordered one; b -- a disordered ones
An ordered pattern of isochromatics is observed in homogeneous media
such as certain plastic polymers (certain synthetic and natural resins) and
also in semi-crystals whose grains are so fine that they approach in their
STAT
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behavior a homogeneous substance. A disordered pattern of isochromatics is
observed in nonhomogeneous media such as in polycrystals whose grains have a
different coloring. The finer the grain structure, the greater the likeli-
hood that the disordered isochromatic
pattern will become an ordered one.
In Figure 10b there is shown a
polycrystal of chloric silver with a
surface plastically deformed with a
die. In view of the fact that the
grain structure is coarse, we have a
disordered isochromatic pattara
sisting of variously colored
F44.111"., 11_ A detorted view of iso-
chromatics (a die forced into a speci-
men of chloric silver).
As the grain structure becomes finer, the isochromatic pattern becomes ordered
although the iringv Tollittern is distorted (Figure 11).
Both an ordered and a disordered isochromatic pattern may find its
special application. For a quantitative determination of distribution of
stresses an ordered fringe pattern is necessary. For the study of various
phenomena which accompany the process of plastic formations in many cases
a disordered isochromatic pattern has a predominant significance since it
permits us to study the mechanisms of these phenomena and to observe their
kinetics. A disordered isochromatic pattern may be called a structural iso-
chromatic pattern inasmuch as it reveals the structure of the model under-
going deformation. An ordered isochromatic pattern which reveals the dis-
tribution of forces in the volume of the model undergoing deformation may be
called an isochromatic pattern of macroforces.
S. I. Gubkin and S. I. Dobrovol'sky established that in addition to an
isochromatic pattern it is possible to obtain a pattern of isoclinics from
transparent optically sensitive material undergoing plastic deformation. The
latter makes it possible to obtain a grid of normal principal stresses and
27
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110
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maximum shearing stresses (stress grids). Such grids (Figures 112-113)
present a qualitative, picture of the distribution of stresses. If, however,
we succeed in evaluating the value of a fringe of a transparent optically
sensitive material subjected to plastic deformation, we shall obtain all the
necessary information for a quantitative solution of the problem of distribu-
tion of stresSes in a plastically deformed material analagous to the problem
which is solved by the method of photoelasticity (127.
Thus, along with the method of photoelasticity there arises the method
of-photoplastic.ity which is based on the same phenomenon of double refraction
as the mAtheA of photoelasticity, but which differs from the latter in a
whole series of specific peculiarities which put the method of photoplasticity
into a category of completely independent methods having considerably greater
range of applicability than the method of photoelasticity.
At the present time, the theory of elasticity serves a great variety of
sciences and areas of technology.
a
ra result, this method is applicable to
a large variety of problems which can be solved by this theory. The theory
of plasticity is called upon in solving extremely complex problems in the
field of geology, geophysics and mechanics of mineralogy; it serves as a basis
for development of the theory of shaping metals by means of pressure and cut-
ting processes, the formation of cermets and welding; it acquires an increasing
significance in the development of the theory of both metallic and nonmetallic
alloys; it assists in advancing the calculation of strength of structures and
in the various areas of applied mechanics; it has great significance in the
development of branches of chemical technology (for instance, that concerned
with pressure molding of plastics, and the production of synthetic fibers and
'paints), Even 'Certain medical sciences have recently shown an interest4-1,21%
�Ls
various applications of the theory of plasticity.
Hence, photoplasticity as a method of model analysis of plastic deforma-
tion processes observed in various natural phenomena and encountered in various
�
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41Ir
fields of technology is called upon to solve a great variety of problems.
All the problems of modellinE the processes of plastic deformation may be
classified as problems having the following objectives:
(1) an analysis of stress distributions in plastically deformed bodies;
(2) a study of physical phenomena accompanying plastic flow.
The first group Of problems in its turn maybe divided into two subgroups.
In the analysis of stress distributions in plastically deformed bodies the
problems may be divided into qualitative and quantitative ones the same as
in a chemical analysis. In this instance the question Is not one of pre-
cision of measurements but of the requirements of the problem. In case of
many problems
WA0A.A%0M.4.Mos....0
certain processes
f
frgwilesfrirstl ninti crtxrsyNln-v=irAl nhpnnmena and also for
shaping metals by means of pressure and cutting, the
basic requirement is that of establishing the general picture of distribution
of stresses for the purpose of revealing the peculiarities of the phenomenon
or process and of establishing the character of its development. In such
cases, it is not necessary to obtain the absolute values of the stresses.
The isochromatic .pattern of the stress distributions without any quantitative
analysis
71,1M4".4.4%
gm44-ea
sufficient
for establishing the peculiari-
ties of the phenomenon or process under study. Thus, the establishment of
+110 isochromatic pattern of distribution of stresses in the last instant of
the filling of the mold in a stamping process reveals the characteristic
features of the process being studied without any quantitative analysis of
the fringe pattern. The photograph in Figure. 73 shows that during the last
instant of the stamping process there occurs thraughout the casting a uniform
state of stress and that only in the vicinity of the slit do we observe any
nonuniform distribution of stresses. This conclusion assists us in forming
a correct concept of
4,Lie
mechanism of the stamping process
In solving problems similar to this our attention must be directed
primarily to realizing those conditions of similarity which have a derisive
29
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influence on the peculiarities under investigation and the character of the
process being studied by means of the model.
The second subgroup of problems which are analagous to the problem of
photoelasticity are
of entirely different nature, since the basic requirement
in this case is the determination of the values of stresses at each point of
the body being deformed. In solving these problems the basic requirement is
that of determining the precise value of a fringe and determination of the
directions of principal stresses by means of deciphering of the isoclinics.
Particular attention most be directed to the possibility of realizing the
conditions of a plane state of stress.
The second group of problems of photoplasticity has as its objective the
study of physical phenomena which accompany
defor-tiewl.
Nany of
these phenomena have great practical significance. Among the basic problems
in this group may be enumerated the study by means of models of processes of
plastic deformation designed to increase our knowledge of:
(1) the mechanism of flow and fractures;
(2) the nature of residual streF3P-e5;
(3) the nature of fatigue, relaxation, creep, elastic aftereffect and
contact friction.
TIva
4.)14c2
grOup Of: prOb1euITV47#Ar�
�ammw�&.0
.��� AM.
����� 1����
�
"it �
����11...
firstthe groups we may deal
with both qualitative and quantitative problems.
5. Basic Characteristics of the Eethod of PhotoplaELlsall.
In nni:Era of
the
fsar,f 4.1hAf
the.
gtmma1-1
forms the basis of btv,
the photoelastic and photoplastic methods, these methods differ from each
other in substance.
Among the charLcteristic peculiarities of the method of photoplasticity
we may enumerate the following:
1. By means of the method of photoplasticity we may study In models
not only the distribution of stresses in plastically deformed bodies but also
1
IIII
I
w,
I
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0,1
LL.... structural changes occurring during nIastic deformation, using for this
purpose optically sensitive models of crystalline materials (haloid salts of
thallium and their alloys).
The discovery by A. V. Stepanov of optically sensitive crystalline ma-
terials permxts us an the one hand to significantly improve the method of
photoelasticity, and on the other hand to create a new area of studies by
means of a method of photoplasticity, since there is a possibility of uti-
lizing the structural isochromatic pattern for the study of mechanisms of
plasticity and failure and also to study the nature of the phenomena accom-
panying plastic deformation.
Thant the ranee of problems which can be solved by the method of
yulouv�
p4.0.0,-Lc44,
is considerably greater than the range of problems which can be
solved by the method of photoelasticity. While the method of photoelasticity
solves the problems of distribution of at.veaaaa in elastically deformed
VtimA4Asat
the method of photoplasticity in addition to these problems also solves prob.
lems concerned with the nature of plastic deformation.
2. The area of qualitative problems on distribution of stresses which
can be solved by the method of photoplasticity is considerably wider than the
arPn nf Pimilar prnblpm= pnlirpti by thP rfhr nf phntnalaptinitv. At the name
time, certain of the qualitative problems of the method of photoplasticity may
be utilized in studying by means of models the geological and reophysical phe-
nomena of nature involving plastic deformation.
3. The requirements imposed on the materials used for models in the
methods of photoelasticity and photoplasticity are different, hence the model
materials themselves are different.
4. The behavior of materials In the models being studied by the methods
of photoelasticity and photoplasticity is different. Therefore, the nlechani-
cal properties of the model materials which must be determined are also dif-
ferent. In the method of photoelasticity we must know such properties of the
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model material as Youag's mod-1-- and Poisson's ratio, while in the method of
photoplasticity we must know the relationship between the shearing stresses
and either the degree or the rate of deformation. Sometimes it is necessary
to know both of these relationships and also the relationship between the
shearing stress and the average applied stress.
5. The experimental procedures in the methods of photoelasticity and
photoplasticity are basically different. In the method of photoplasticity,
deformation in the direction of incident light must be avoided, otherwise
the surface being illuminated becomes uneven. and the pattern of isochromatics
�
becomes sharply distorted. Deformation in the direction of the incident light
.1r^iAA4 by means of glass plates of required thickness which restrain this
deformation by exerting the necessary forces. This leads to formation of
certain frictional forces on the surface of the glass. Sometimes these for-
ces may be neglected. This is possible in those cases where the frictional
forces are considerably reduced by applying a lubricant. The effect of the
frictional forces is noticeable only at a slight distance from the surface of
the model. Therefore, as is shown by experiment the frictional forces exerted
by the glass may be neglected if the thickness of the model in the direction
of light is taken sufficiently large so that the isochromatic pattern being
observed is essentially in a state of,plane strain. The avoidance of defor-
mation in the direction or incident light, reduction of the frictional
ir
�11.� �al
* 4.
Ill Ns
the surface of restraining glass plates and the use of proper thickness of
motlel are specific features of the experimental procedure in the method
of photoplasticity.
6. The procedure for determination of the fringe
value fnlft the model
material in the method of photoplasticity has distinctive features which
differ from the procedure used in the method of photoelasticity.
Thus, the methods of photoelasticity and photoplasticity are essentially
different methods both with respect to the experimental procedures and with
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respect to the materials used for the models, as well as the behavior of
these materials in models and the determination of the properties which des�
cribe their behavior.
a
��g�
�
to
�
��
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CHAPTER II
MATERIALS UTILIZED IN l'HE METHOD CF PHOTOPLASTICITY
1.�yecifications for Materials Used in the Photoelastic Method.
The materials used in the method of photoelasticity must have certain
optical and mechanical properties. Among these properties are the following:
1. An adequate transparency.
� 2. Mechanical and optical homogeneity.
3. Sufficient hardness.
4. Ease
of mar.hininm.
� 5. A high value of piezo-optical constants characterizing the ability
of the substance to become doubly refractive under the action of mechanical
forces.
6. A linear relationship between the stresses and strains, and also
between stresses and the difference between the paths of the rays.
7. Absence of initial refraction of light.
8. Adequate constancy of optical and mechanical properties for small
changes in temperature.
9. Absence of "edge effect" following machining.
10. Absence of noticeable optical and mechanical creep and also absence
of elastic aftereffect and plastic flow.
We must add still other special requirements to those enumerated for
the study of ths stressed state in three-dimensional models. Thus, for the
study of distribution of stresses in three-dimensional models by the method
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of "freezing," the material must have the ability of "freezing" the stress
pattern. In utilizing the method of scattering of light the model being de-
formed must have the required sitilT4iiiiityvil
At the present time we still do not have materials which would satisfy
completely all the indicated requirements. The majority of optically sensi-
tive materials utilized successfully in the photoelastic iethod both in our
country and abroad satisfy only the most basic of the requirements.
2. S ecifications for Materials Used in the Method of Photoplasticity.
The materials utilized in the method of photoplasticity must meet
essentially the same requirements as in the method of photoelasticity (except
for the requirements given under 6 and 10), and in addition to that, they
settles+
mx.a+
1101,10,115,M
�(a.)
special requirements which may be reduced to the following:
+ha
material must produce large plastic deformations;
(b) the rheological behavior of the material must correspond to the
rheological behavior
0 f
the prototype.
For the purpose of photoplasticity we may use both amorphous and crys-
talline materials.
In certain cases the materials must be capable of serving as model
materials for fatigue processes, creep and elastic aftereffect, while in
other cases it must permit measuring the residual stresses of the first,
second and third kind indicated, as well as the effect of the structure on
the plastic behavior of crystalline bodies.
Amorphous bodies are substances with a disordered disposition of kinetic
units (atoms, molecules and macromolecules). These bodies.may be classified
into two basic groups. The first group consists of simple amorphous sub-
stances containing molecules of sEall dimensions (small molecular weight);
the second group includes compounds consisting of macromolecules. Each
macromolecule is a complex of chemically combined simple molecules. Such
amorphous bodies (polymers) have a distinctive characteristic of relatively
1
;S TAT
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1111-----,y
^
36
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��
high strength.
Up to the present time, the method of photoelasticity has utilized
_materials which belong primarily to the second group of amorphous bodies.
Recently, the work of the authors has indicated that among simple amor-
phous substances there are transparent optically sensitive materials which
are quite plastic at the same time, and that these may be successfully utilized
for the purposes of photoplasticity.
Thane,
mnitat.inla
are characterized by the following peculiarities:
mechanical and optical homogeneity; linear relationship between the difference
in the optical paths
nnA
the magnitude of the maximum shearing stresses (see
formula (2)); absence of an initial double refraction following processing
and machining; absence of any irreversible changes
of structure and proper-
ties during the process of flow; very large plastic deformations without any
symptoms of failure at definite rates of deformation and temperature; depend-
ence of the resistance to shear upon the rate of deformation and the average
stress (hydrostatic pressure) and independence of the resistance to shear on
the extent of defOrmation;. noticeable effect of temperature changes on the
optical sensitivity and viscosity, and reduction in the fringe value of the
material with an increase ia.temperature. In addition to that, the rheo-
logical behavior of these materials corresponds to the
lotar.4fttAVF,
tiVA4.10-va.va
of sa viscous
solid body (see Chapter I, Section 3). Therefore, such materials may be
successfully applied in studying states of stress under conditions of viscous
flow.
As was already indicated, the problem of studying processes of plastic
deformation by means of models is an analysis of the state of stress in the
body undergoing deformation and the study of phenomena accompanying plastic
flow.. The solution of these problems is associated with crystalline materials
such as metals and alloys. Naturally, more complete solution of the indicated
problems by the method of photoplasticity may be attained by acing models of
STAT t
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materials which either coikpletely or to a large extent simulate the proper�
ties of metals and alloys.
The materials utilized in the photoplastic method must also have the
following distinctive characteristics in addition. to such properties as a
high optical sensitivity and transparency:
1. Plasticity. Resistance of such materials to shear must depend upon
the degree of deformation and be independent of its rate. In the process of
plastic deformation there must occur a change in structure and properties.
We must be able to vary the grain size and obtain the proper structure in
the. mmtariml by
remthivting-Aatfnrmnfinn and therma treatmente
2. Absence of an initial double refraction. When double refraction is
present,
we must
daudLe to
select
giOlgelWALA
thermalo
treatment as will assure
complete removal of residual stresses or reduce them to an acceptable magnitude.
3. Similarity of rheological curves to the corresponding curves of the
material in the prototype. The material of the model must be capable of sim-
ulating creep, elastic aftereffect and plastic flow.
�
4. Possibility of obtaining transparent; optically sensitive alloys
with different structures.
5. The appearance during the process of defornation of residual stresses
of the first,
and third kind.
3. Classification of Materials.
In view of the fact that the materials being utilized for the method of
photonlasticity have been studied relatively little, it is not possible at the
present time to enumerate the materials whosfl, rhPillng''f.121
Ileb1115.tr4ow.nro
ftevaaf.alliAWA
corresponds
not only to the classes but to the sub-classes of the classification proposed
in Chapter I. It appears that the most expedient canner of classifying all
materials being used for the optical methods of analysis of states of stress
is to divide them into three groups: (l) elastic; (2) viscous; (3) plastic.
Such a division corresponds to the basic classes of the proposed classification.
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materials which either completely or to a large extent simulate the proper�
ties of metals and alloys.
The materials utilized in the photoplastic method must also have the
following distinctive characteristics in addition to such properties as a
high optical sensitivity and transparency:
1. Plasticity. Resistance of such materials to shear must depend upon
the degree of deformation and be independent of its rate. In the process of
plastic deformation there must occur a change in structure and properties*
We must be able to vary the grain size and obtain the proper structure in
the material by combining deformation and thermal treatment.
2. Absence of an initial double refraction. When double refraction is
present, we must be able to select such thermal treatment as will assure
complete removal of residual stresses or reduce them to an acceptable magnitude.
3. Similarity of rheological curves to the corresponding curves of the
material in the prototype. The material of the model must be capable of sim-
ulating creep, elastic aftereffect and plastic flow.
�
4. Possibility of obtaining transparent, optically sensitive alloys
with different structures*
5. The appearance during the process of deformation of residual stresses
of the first, second and third kind.
3. Classification of Materials.
In view of the fact that the materials being utilized for the method of
photoplasticity have been studied relatively little, it is not possible at the
present time to enumerate the materials whose rheological behavior corresponds
not only to the classes but to the sub-classes of the classification proposed
in Chapter I. It appears that the most expedient manner of classifying All
materials being used for the optical methods of analysis of states of stress
is to divide them into three groups: (1) elastic; (2) viscous; (3) plastic.
Such a division corresponds to the basic classes of the proposed classification.
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Among the materials to be enumerated in the first group we must list
glass, celluloid, bakelite, phenolite, trolon, viskhomlit, plexiglass, wells
and others, in the second group we find synthetic and natural resins; in the
third group
we find haloid salts of silver, thallium and their alloys ("trans-
parent metals"). Certain materials which we have enumerated in one group may
be successfully utilized for solving problems which specifically apply to
another group. Thus, there are data which indicate the possibility of uti-
lizing elastic materials
OW MO
celluloid Ziff and plexiglass
411.11
for the study of
a state of stress accompanied by plastic deformation. In testing these ma-
terials at high temperatures, it is possible to simulate in models processes
4ellynlvinr viscous flow.
We shall give a brief characteristic of the basic properties of elastic,
viscous and plastic materials used in the photoelastic and photoplastic
procedures.
Glass. Glass satisfies the majority of the requirements for materials
used in the method of photoelasticity. This serves as the basis for its
application as the very first material for solution of practical problems.
Glass is the most transparent material, it has a high modulus of elas-
ticity, it is rigid; isotrovic, insensitive to temperature variations, free
as not received any
of creep and is comparatively inexpensive. However, itwide application in the photoelastic method of analysis in view of its low 111
h
tt:
optical sensitivity and the difficulties involved in its fabrication. Cue of
its basic shortcomings is the formation of nonhomogeneous and anisotropic
structure when models are prepared from glass by casting process. The
ration of complex models from separate glass prisms by cementing involves
considerable expenditure of money.
Celluloid. This material is an organic substance t 6N3�11)' which is
obtained from a solution of nitrocellulose iu a mixture of alcohol and ether
after this solution is treated with camphor. It is sufficiently transparent
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and isotropic. Its optical sensitivity is greater than that of glass by a
factor of five. The basic advantage of celluloid is its good cementability
by means of acetone, butyl acetate, and ethyl acetate. This permits us to
prepare from this material complex models of large dimensions. The Properties
of celluloid improve and become stabilized with time. After being aged for a
period of several years, it has a very slight creep.
The transparency of celluloid decreases considerably as the thickness
of the plate increases. The transparency decreases 'still further after it is
exposed to ultraviolet light. Celluloid has comparatively low mechanical
strength.
The stress-strain diagram of celluloid in tension shows Lis that it is
capable of producing small elastic-plastic deformations in the range of
a.t."1"0
ses of 200-500 kg/cm2. At the same time, the difference in the optical paths
is proportional to the stresses beyond the elastic limit approximately up to
500 kg/cm2. This permits us to utilize celluloid for the study of stresses
�
at elastic-plastic deformations.
Bakelite. (BT-61-893). This material is a phenol-formaldehyde plastic;
bakelite also satisfies the majority of the basic requirements specified for
mutat-in-1s used in the photoelastis method.
Bakelite is transparent and colorless, is sufficiently hard and is not
excessively brittle; this permits us to machine it with ordinary tools with-
out difficulties. Being isotropic, it shows a linear relationship between
stresses and strains up to 420 kg/cm2 and a linear relationship between the
stresses and fringe order up to 500 kg/cm2 Ei.?.T Bakelite has considerable
tensile strength. Its properties at room temperature are practically constant.
Its high optical sensitivity permits us to obtain a high order of fringes, thus
increasing considerably the precision of the exIeriments.
Bakelite is a two-phase system whose components have different meltiTz
points jg. A change of temperature within the limits of 15-30�C practically
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has no effect upon the fringe value (sensitivity), its modulus of elasticity
and Poisson's ratio. At temperatures of 60 to 1100c the fringe value and the
modulus of elasticity decrease rapidly. At a temperature of approximately
llee the change in the properties of bakelite practically ceases. At high
temperatures it is a completely elastic material since the optical effect and
the deformations caused by the load disappear completely upon removal
%pa.
U1-1W
load. The relationships between stresses and deformations, and between
atiessoa and the order of the fringes are linear. At room temperature, the
material is hard and its deformations are elastic. As the temperature is
increased there _nclIrom A
failure by flow (viscous) which corresponds to its
transition to a liquid state. As the temperature is changed, the load is
gradually transferred
LUI
solid wrid. The value of qr decreases
0 ����
sharply.
This process continues up to a temperature of 1100.
As the heated and deformed bakelite is cooled, there occurs solidifica-
tion of the liquid component phase which leads to "freezing" of the deforma-
tions of its primary grid. This prevents the disappearance of the grid as
the load is removed. The fringes of the "frozen" pattern coincide completely
In all
with the configuration of the fringes obtained at
room temperature.
other respects the bakelite behaves just as the material of the type known
as viskhomlit.
Material I14-44 (Type of Viskhomlit). An optically sensitive material
designated as I14-44 prepared in the Institute of Machine Design, Academy of
Sciences USSR, satisfies the majority of the requirements specified for ma.
terials used in
ea-
FAL0.0%.wa.masuic method.
Preparation of the material is carried out in three stages. In the first
a resin 443 obtained by condensation of phenol with formaldehyde in the
2
presence of oxalic acid as a catylyst; in the second stage the alcohol in the
lacquer obtained during the condensation is evaporated and the material is
cast into molds;. in the third stage polymerization of the product occurs as
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the temperature is increased Zig.
The product obtained after evaporation is quite plastic and soluble
(resin of stage A). In the polymerization process the resin is transformed
into the final state (into an insoluble and nonfusible resin of stage C).,
Its structure consists of an insoluble three-dimens1_41na1 grid filled with a
soluble and fusible product. Arrangement of the molecules of the nonfusible
three-dimensional frame is random and is not subject to any crystallographic
law. Therefore, the material is isotropic in all its properties. The mecha-
nical and optical properties of viskhomlit depend upon the purity of the raw
materials, the degree and method of polymerization, and the temperature and
duration of material under load. At temperatures up to 50�C the phenol.
formaldehyde plastic remains an elastic material irg, in the range of tem-
peratures WA
it",
.Pm.n1.1%0 V ULte material is elastic-viscous, and at temperatures
from 80-/10� to 180�C (depending upon the grade) it remains an elastic ma-
terial while its properties are but slightly sensitive 'to changes in tempera-
ture. At room temperature we observe a mechanical and optical creep, but
their values can be decreased if the measurements are made 10-15 minutes
after the model is loaded (within the limits of porportionality).
The material 114744 finds successful application in the study of stresses
in three-dimensional models. If the model is loaded at a temperature of
80-110�C and is then cooled to room temperature, the fringe pattern obtained
under load is "frozen." Then the model may be cut into any number of parts
while the optical pattern observed under load is not destroyed. The differ-
ence in the optical paths at a given point of the model being deformed is
proportional to the maximum shearing stress.
The material 1M-44 has certain residual stresses and consequently a
certain degree of double refraction inlmediately after beillg formed. This is
caused by nonuniform shrinkage of the material during its polymerization.
In addition to internal stresses distributed throughout its volume, there is
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11i
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produced along the contours a special layer with a considerable degree of
double refractivity. The appearance of the edge effect is explained by
physical-chemical processes (primarily by evaporation or absorption of
moisture). The aging of this material over a period of several years im-
proves its quality. The magnitude of the edge effect increases with time.
The sign of the internal stresses caused by the edge effect depends upon the
method of the preliminary machining of the material.
Material, of type IN-44 has characteristics which are equal to those of
bakelite of grade BT-61-8931 which is widely used in the United States for
the photoelastic method. This can be
seen from data given in Tame
�
Table 1
Indexes
Bakelite BT-61-893
_A
.2%
at 20�
Fringe value
in kg/cm2
Modulus of elasticity E in
longitudinal direction '
In kg/cm2
Poisson's ratio
Proportionttl limitcrpr
in. kg/cm'
Tensile stgength Ctir
in kg/cm'
6,0 ,
0.9 7.7
1
140-42�103 200 43.24103 1 86
0.37 0.42 o. 365 I
, 1 0.5
50a 1
1300 128.
0.286
10
25
492
1200
11.3
2-316
psonic glass (plexiglass). This material is a plasticized polymer of
methyl ether of metacryllic acid EiY. It is the most transparent plastic
of all. An increase in the model thickness does not appreciably reduce its
transparency. Ultraviolet light has no effect on the transparency of the
organic glass for practical purposes. Plexiglass is low in optical sensi-
tivity. In order to obtain isochromatic lines of several orders it is neces-
sary to apply considerable force. It has a desirable property of being able
to deform plastically. Kodels of large dimensions may be prepared from this
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material by cutting from a whole piece or by cementing separate parts.
Plexiglass can be used to measure distribution of stresses at elevated tem-
peratures (up to 1200), whereupon its optical sensitivity increases consider-
ably.
�lea
Gelatin-Glycerine Optically Sensitive Material. This material is pre-
pared from photo-gelatin (with molecular weight of the order of 900), glyce-.
rine, ethyl alcohol, sodium chloride, and iq-naphthol...The gelatin-glycerine
material is transparent, has a very high coefficient of optical sensitivity
(several hundred times as large as that of viskhomlit and bakelite). This
makes it applicable in studying stresses caused by three-dimensional forces.
Sensitivity of the material depends
in Mairly
..L'GAPIPC-Utel. VI/ ULL W UW.1.1.1.0CLIA.AM'w
tion of photo-gelatin, concentration of glycerine, salt and temperature.
Being elastic and weak, the gelatin-glycerine material (after being
formed) is destroyed under appreciable load. However; if it is aged for a
long period at room temperature or if it is subjected to a lengthy steam
bath, it becomes more dense, stronger and slightly darker, thus losing some
of its optical sensitivity. However, after such treatment, the material
becomes suitable both for a short-term as well as a sustained test and it is
able to resist considerable loads.
In a short-term load test, gelatin-glycerine .models undergo only elastic
deformations. The pattern of isochromatic lines observed under load vanishes
Immediately upon removal of the load. In the case of a load maintained for
long time (of the order of several days), plastic deformation is observed
and the model changes its form permanently.
Figure 12 shows three stages of flow of a model from a container under
a 'ustained load test. rhoton-aph a indicates the form and dimensions of the
model prior to the test; b shows the beginning of an irreversible change of
form, c shows the form of the model after flow of a considerable amount of
material through a die. After the model was removed from the container; its
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dimensions changed somewhat due to the removal of the elastic component of deformation,
but the residual stresses remained in the material.
The character of distribution of the isochromatics and their order in the model
under test may be seen in Figure 13, which was obtained by illuminating with circular-
ly polarized light (see insert between pages 46.47). This pattern remained in the
model over a considerable period of time (more than three years).
,
� e-4-�
.;
. , 4
, -.Pak � -
?
a
Figure 12. Flaw of gelatin-glycerine material from a container: a--model
before test; b....beginning of an irreversible change in form; c--model
after a sustained test.
Figure 14 shows photographs obtained with such a model after a certain period.
Comparison of Figures 13 and 14 a shows that no significant changes in the isochro-
matic pattern have occurred. Figure 14 b shows a photograph made after the model was
mit Into two parts along the axis of flaw.
esnn
ha '64rIttha iang-hrnrelt4^s were
preserved in the two separate parts of the model. This shows that there develops
in the model during the process of plastic deformation a texture indicated by the
double refraction shown in the photograph.
Figure 15 shows a color photograph which indicates the disposition of the 4.�
chromatics and their order in the model deformed by
(see insert between pages 46.47).
1.
Thfl a II e.1 ^
WINC6), /.rom. l% re^,4,4 4 "
iqiiNOAA rfc.i /r4.1 maw3.11.-
AA1
AATTIACKWAAWII.
0Wwv.A.MG1110LLI
a flat container
prepared in the indicated manner,
we can study the character of the distribution of the residual stresses in models
subjected to finite plastic deformations.
44
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44.6. +Wiz ����
�
,
..e.444
r'
�
Figure 14. An isochromatic pattern in a specimen of gelatin-glycerine ma-
terial obtained three years after deformation: a -- before the model was
cut; b -- after the model was cut into two parts along the axis of flow.
Resins. Rosin (a mixture of resinous acids) in its solid state possesses
optical sensitivity. The rosin acquires different colors and degrees of trans-
parency depending upon the degree of oxidation of it componants.
sinous acids which are oxidized to lesser extent (C20H3002) yield lighter
grades of rosin. The latter are most suitable for preparation of plastic
optically sensitive materials. Since the rosin itself is very brittle, it
cannot be used for the photoelastic method, and certainly not the photo-
plastic method. However, by plasticizing we can obtain from the hard rosin
materials of the required properties. In order to plasticize this
we can use various mineral and vegetable oils; however, some of them when
mixed with the rosin do not yield materials of the required properties. We
must use as a plasticizer such oils as when mixed with the rosin yield a
material which resists oxidation.
In order to obtain an optically sensitive, transparent plastic material
which meets the basic requirements specified for materials to be used in the
photoplastic method, it is best to use a rosin of high grade and a pure rosin
oil. A rosin oil of the highest fraction is obtained by distilling rosin of
light grades under vacuum (distilling temperature 3000C, pressure 5 mm Hg).
A mixture of rosin with rosin oil* in certain proportions yields a material
* The rosin oil was obtained in the Laboratory of Technical Catalysis and
V4w. 4�ins2 Arsidemv of Sciences BSSR
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of different hardness and plasticity. In mixing rosin and rosin oil in the
ratios of 2:1 and 3:1 we obtain a very plastic material in which the pattern
of isochromatics under load is preserved only a'very short time. In this
case it is difficult to fix the pattern of isochromatics and isoclinic& which
accompany the process of plastic deformation. 3y mixing the components in the
ratios of 5:1 and 6:1 we obtain a hard elastic material in which the iso-
chromatic pattern due to loading is preserved for a considerable length of
time. However, models prepared from such a material are likely to be destroyed
even for relatively srlall deformations.
ML
1111:7
mixture most suitable for the method of photoplasticity proved to be
one consisting of four parts of rosin and one part of rosin oil. The material
fst itionnaplav�dart+, halierk4�%reilletw
eme4.1nor.
"hdrelft,a0,11.11
Atioftsam,
1,0i*W,A.WLW,AVY
eh
S.6.6414.16
gs,,A is41r4A
matic pattern and permits large plastic deformations at high rates of deforma-
tion. It proved to be Possible to remelt the material repeatedly without any
..�&&J
appreciable loss of transparency (darkening). In order to guard against pos-
sible contamination during the repeated use of the same batch of material it
may be filtered through gauze in a molten state. Our experiments were conducted
in the main with material of this composition.
The optical sensitivity of
mentally Em-uarmualeu fringe.value
1
% of oil in mixture 1,
the resin described is quite high. The experi-
for this Material at a stabilized process of
flow for the yellow line of mercury
and a temperature of 20�C is 1%
2.20 + 0.015 kg/cm. For a sufficiently
Figure 16. variation of the index
refraction of the rosin mixed with
rosin oil with the oil content.
in the photograph up to 40 fringes
Tests of the resin at various
Declassified in Part - Sanitized Copy Approved for Release
����
small dimension of the monochromatic
source of light and well regulated
of optical apparatus and the use of a
light filter, it is possible to
(Figure 46c).
obtain.
temperatures show that its sensitivity
146
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� ���
..1 �
.:;;W:r1C17%Ii71
'
a
Figure 2. Colored fringe pattern obtained:
a) with resin; b) with gelatin-glycerine material.
Figure 13. Isochroktatics retained by the model of
gelatine-glycerine material after test.
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Figure 15. Isochromatics retained by the model of
gelatine-glycerine material after a compresaive test.
Figure 17. Isochromatic pattern observed during the flow of
rosinwbutadiene hydrocarbon dioxide mixture.
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increases with temperature. The sensitivity of the material changes also
withlthe proportions of its components. The smaller the amount of resin in
the mixture, the greater the sensitivity. The index of refraction varies with
the composition linearly (Figure 16).
Physical tests of cylindrical specimens (d = 10 alms h = 14 mm) of the
resin of indicated composition show that the static deformation at various
speeds of the testing machine (10, 20, 40 and 80 ram/min) Indicates, as might
be expected, that the material has no defined flow limit. The rate of flow
of the resin, and consequently the rate of its deformation at room temperature
within wide limits (from 1.19 to 33.7 kg/cm2) is proportional to the shearing
stress. The relationship between the coefficient of internal friction and the
temperature for a temperature range of 12 to 270C is shown in Table 2, For a
temperature change of 1�C the viscosity of the resin decreases a.,1,...�Imataly
by a factor of 1.4.
When we study the process of plastic deformation by means of resin models,
we must keep in mind the possibility of large reduction of viscosity at points
of stress concentration associated with temperature rise. This circuzstance
may lead to a redistribution of stresses in the model under test.
Table 2
toC
12
15
18
21
24
27
7
lo. /1 355
205
81
G-7
8.1 1 3.45
The result of physical tests at room temperature show that the resin is
the most suitaole material for models in the study of the process of rlastic
deformation of solid bodies under coiaditforls of viscous flow.
Dioxide of Butadiene --drocarbon (C._e0 H222 0 )
oxidizing hydrocarbon C__H
.eu 22 in vacuum.
color.
It iF
� The dioxide is obtained by
visccs fluid of
4-
The higher the fraction of the material, the denser the dioxide. When
Obtained in the Laboratory for Organic Chemistry of the Academy of Sciences
of BSSR.
47
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I
1 I
I
1
I
mmiimmomiTAT
�
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mixed with a resin it yields a transparent optically sensitive material of
varying plasticity (depending upon the proportions of the components).
The most suitable material was obtained from a mixture of rosin and the
dioxide fraction obtained at a temperature of 204-218�C and a pressure of 1.5
mm Hg for a component ratio of 1:2. This material is transparent and has a
light-yellow color. The isochromatic pattern has bright and vivid colors
(see insert, Figure 17). For large rates of loading the material is destroyed,
while for low rates it deforms plastically. It permits repeated use by re-
melting without exceeding the temperature .of the melting point. When this
temperature is exceeded the material gradually darkens and loses its trans-
parency, thus becoming unsuitable
for further work.
The mixture of dioxide of buLadieile hydrocarboa with rosin of higher
grades yields isochromatics
IN9
high orders.
Abletinic Acid (CH 0_)* This material is a crystalline substance of
2U 30 e
monoclinic structure obtained by isomerization of primary resin acids. It
has a melting point of 172-173�C. In its molten state it is transparent;
after cooling this material becomes amorphous and after a certain length of
2.0.614.21W
fa01.0
e. 1 1 4 nos
4f.
���� on 'um am.= 4.� id
Abietinic acid mixed with rosin oil in the ratio of 3:1 yields a material
of light-yellow color of high optical sensitivity. The isochromatic pattern
is bright but disappears rapidly, and the colors are vivid. For a component
ratio of 4:1 we obtain a material which has the properties indicated above,
is
and
retains its isochromatic pattern for a long time. When the
model is loaded until a fringe of the tenth order Is obtained, the fringe
pattern is retained in excess of
minute's (at room temperature).
ON 51
unstable material, abietinic acid oxidizes rapidly when heated (for repeated
melts), darkens and becomes unsuitable for further use.
repeated use of this material is quite limited.
T
view of this,
s' Obtained in the Laboratory of Chemistry of Forest Products, Academy of
Sciences, BSSR.
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�,
Canadian Balsaill. When this material is evaporated to an almost solid
state, it is adequately transparent, plastic and optically sensitive. When
models made of this material are deformed, we observe bright and vivid iso-
chromatics. Canadian balsam may be used repeatedly upon being remelted.
Silver-Fir Balsam. In the solid state this material possesses optical
sensitivity. In order to make this material plastic (viscous) it must be
boiled with xylol (or some other solvent). Varying the quantities of the
components, we can obtain materials of various plasticity and transparency.
When models made of such a material are deformed they produce bright and vivid
isochromatics. This material permits repeated melts without any noticeable
loss of transparency.
Materials obtained with Canadian or silver-fir balsam as a base possess
high optical sensitivity, plasticity and other required properties, but they
adhere tenaciously to instruments. Besides that, they are quite expensive
and in short supply.
Chloric Silver. This material has a crystalline structure /V. Its
crystals have a simple cubic grid of the type of EaCl and have a period
a = 5.54 kx. By combining mechanical and thermal treatments of a casting, it
is possible to obtain t'ransparent and almost colorless specimens of any di-
mension and form. The volume of the specimen (casting) under a given set of
conditions may be filled with a single or several grains, while under another
set of conditions it may contain hundreds and even millions of grains. The
grains may have different dimensions, form and orientation in one and the same
casting, just as is the case with ordinary metals and alloys. We may obtain
textured specimens and also specimens with an almost homogeneous grain struc-
ture, with grains of uniform size. Specimens of chloric silver may have the
structure of cast products, recrystallized metal, etc. In order to obtain the
required structure, we utilize ordinary methods of treatment of metal.
Haloid salts of silver and thallium have mechanical properties at room
49
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� or...
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temperature which are quite different from the ordinary properties of the
component masses of salts and minerals. The mechanical properties of chloric
silver are such that it can be truly called "transparent metal." It can be
treated at room temperature by all
the types of treatment applicable to
metals. Chloric silver may be forged,
rolled, stamped. Dressed, etc. At
room temperature it is approximately
ten times weaker than copper, like
lead, it may be scratched by nail,
it can
el�
easily flexed by hand, and
6 kginl?
3
2
I 1 .
t I
I-
J,..
.-'
...
...._
i i�---
1
1
:
-i
, ji.
Hi_ I
II
1
111E11
11
11
ah
Figure 18. Curve of true stresses ob-
tained for a static compression of spe-
cimens of chloric silver.
it has a metallic ring. The mechanical properties of chloric silver both by
itself and in alloyed form, depend upon the magnitude of deformation. In the
process of plastic deformation it becomes less plastic. In this respect,
chloric silver shows its ability to be strengthened by cold working at room
temperature.
The test by the method of indentation with a cone shows that the hard-
ness of chloric silver increases considerably after it is deformild
q.mcs^i_
Amilmw
mens which are compressed 75% have their hardness increased approximately by
A factor of 1.75 compared with undeformed specimens. Thus, if the hardness
of the undeformed specimens is 10.5 kg/cm2, specimens compressed 60 and 75%
have 'hardness values of 16.2 and 18.5 kg/cm2 respectively.
Curves of the true stresses .obtained from the diagrams of static com-
pression for specimens of chloric silver having a cylindrical form (8 mm in
diameter and 12 mm long) prepared from a casting which had previously been
deformed 60% in compression and then heat treated for 5 cycles of 5 hours
each at a temperature of 1500C, are quite similar to analogous curves for a
"cold worked" metal, The curve of true stresses for the indicated case is
shown in Figure 18.
EllimmimmimmiimismilININ Declassified in Part - Sanitized Copy Approved
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The thermal treatment of cold worked specimens of chloric silver brings
about the removal of stresses, which may be either relaxation or complete
recrystallization depending upon the temperature of the heat treatment. As
a result of thermal treatment the cold worked specimen is returned to its .
initial soft and plastic state as may be seen in Figure 19. For purpose of
comparison there are given in Figure 20 stress-strain diagrams of polycrys-
talline specimens of copper.
The character of the flow of chloric silver and also the nature of frac-
tured and compressed specimens are similar to the corresponding phenomena
observed for metals. Fracture occurs with a formation of a necked-down
section. For heat treated specimens the fractured section degenerates into
a point (Figure 21). The entire length of the
Anintimmilsktinvt of residual
c.LAJLAWAU.LV144.
Ariociymeto%
vNart4^4iNa+eft
4T .A.V0.0.20 10 110,
47!I
During a tensile test of cold worked
we observe formation of a necked-down section and in addition to
that we obAiarve
the formation of the surface of fracture. Specimens compressed
to a high degree do not show any symptoms of fracture (Figure 22).
t 40
go 30
t 20
Ca.
10
tts
to 20 30
Li Ydnutfemue 6 Z
Figure 19. Stress.:strain
polycrystalline specimens
silver in tension: 1 -- h
2 cold worked (from A�,
Legend: a) stress in kg/mm4
b) elongation in
Crystsls of chloric silver being in
diagrams of
of chloric
eat treated;
V. Stepanov)
0 I 10 ZO 30 40
rtf5 .Z
in
r
Figure 20. Stress-strain diagrams ot
polycrystalline specimens of copper
in tension: 1 -- heat treated; 2
cold worked (from Muller).
Legend: same as Figure 19
tropic. If a specimen of
forces, it becomes doubly
lab MP
the cubic system are optically iso-
this material is subjected to external mechanical
refractive.
A characteristic peculiarity of the
51
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piezo-optical properties of chloric silver is its high optical sensitivity.
That places it among the best contemporary substances having high optical
sensitivities (viskhomlit, bakslite, resin).
Figure 21. A specimen of chloric Figure 22. A cylindrical specimen of
silver tested in tension (from A. V. chloric silver compressed 60%.
Stepanov).
Up to certain small values of loads chloric silver behaves as elastic
material. For larger loads the material develops permanent double refractivity.
This indicates that plastic deformation brings about residual stresses in this
material. In a deformed specimen of chloric silver one can observe residual
stresses of the first, second and third types. Residual stresses brought
about as a result of plastic deformation may be removed by means of special
heat treatment.
Optically sensitive chloric silver has high piezo-optical coefficients
which are of the same order of magnitude as for viskhomlit and bakelite. The
transparency of chloric silver is reduced considerably under the action of
light. However, this is not a serious obstacle in working with this material.
The process of decomposition of this material is not very intensive. Appli-
cation of light filters which block the short-wave portion of the spectrum
reduces considerably the decomposition of chloric silver in light.
A very desirable property of chloric silver is its ability to regenerate
and its ability to be used several times over for experimental work.
4. Effect of the Nature of the Material Bei Deformed on the Character o
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Stress Distribution.
111 Study of the effect of the nature (structure) of materials and their
�
mechanical properties on the stress distributions of a deformed body or a
body undergoing deformation is one of the basic problems of the method of
photoplasticity. Depending on whether the material is amorphous or crystal-
line, the model undergoing deformation under given conditions (type of load-
ing, geometry of the model and type of instrument) will yield a completely
definite picture of
the distribution of stresses.
The process of plastic deformation is accompanied by formation of re-
sidual stresses. In homogeneous bodies one can observe only residual stresses
of the fitst type, while in crystalline bodies residual stresses of first,
second and third type may be observed. Thus, the distributions of stress in
crystalline and amorphous bodies subjected to identical conditions of defor
mation differ in this essential respect.
In order to confirm the stated postulate we shall compare fringe patterns
observed for a load of the same type in models prepared from elastic, vis-
cous, and plastic material.
For the purpose of comparative tests the authors utilized a flat model
loaded with a die.
Figures 231 ,% and 25 snow the fringe pattern in the
entire field of models loaded apIroximately to the same extent.
In the case of elastic material the fringes are continuous and form
circles which touch the corners of the die. This confirms the well-known
postulate that for the loading under consideration fringes are circular and
pass through the edges of the die. ue. As the load was removed, the fringe pat-
tern accompanying the load was completely removed:
In the case of viscous material the fringes are continuous nnri well
defined. They form ovals elongated in the direction of movement of the die.
As the model is unloaded, the fringes persist for a long time, and gradually
disappear. The fact that these lines are continuous indicates that the
53
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Figure 23. Fringes observed upon de- Figure 24. Fringes observed upon de-
forming a Viscous material (resin), forming an elastic material (viskhomlit).
residual stresses being observed are
Figures 2A
of the first type.
cinA
b.. 11 %A
Dr7�
c_ro.
show fringe patterns immediately after
models of various dimensions are com-
pletely unloaded. Figure 27b shows
the fringes several minutes after the
Figure 25. Fringes observed upon de- model is unloaded, the model corres-
forming of a plastic polycrystalline
material with a fine grain structure ponding to Figure 27a.
(chloric silver).
In the case of polycrystalline
material with a sufficiently fine grain structure we note considerable dis-
tortion of the fringes even in the first stages of finite plastic deformation.
However, even in this case the fringes have an oval shape. As the model is
loaded, the order of the fringes increases (Figure 116). After the model is
unloaded, a portion of the fringes corresponding to elastic deformation dis-
appears immediately. However, it retains fringes of a certain order. The
fringe pattern retained by the model corresponds to the residual stresses of
the first, second, and third type. In Figure 117 we may consider the two
�
1
well defined fringes in the right hand side of the model as being caused by
IP residual stresses of the first type. In the remaining portion of the model,
the fringes are considerably distorted and do not give us any clear picture
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MINIIMI11111111
411
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of the orderly distribution of residual
essence residual stresses of the second
the model.
Figure 10b which shows the fringe
pattern obtained with a model of con-
siderable dimensions made of plastic
polycrystalline material with a
coarse structure (1-2 mm) is even
more characteristic in this respect.
In this view it is impossible to see
any well defined continuous fringes.
stresses. These stresses are in
and third type in this portion of
Figure 26.
a model of
Fringe pattern retained by
resin after removal of the
load.
Each grain behaves in a distinct manner, yielding i+s own individual
nf ic
nnffialny,
Th4 ic
1,3..1.441t. Vi 111C
grains in polarized light. The individual properties and the different dis-
position of the discrete grains give their own optical pattern, departing
from the orderly fringe pattern which we would have obtained in the presence
f residual stresses of the first type only. In the given case the residual
stresses of the second and third type are larger in magnitude than the resi-
dual stresses of the first type. Here we are dealing with a structural iso-
chromatic picture.
Figure 27. Fringe patterns obsel-ved in models of resin: a -- immediately
after removal of load; b several minutes after removal of load.
For a fine and homogeneous grain structure the distorting influence of
the residual stresses of the second and third type is not as strongly indicated
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and in this Instance one can observe a certain orderliness in the fringe
pattern AM OW a pattern which shows the distribution of forces throughout the
volume of the deformed model.
���
41110.
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410
1.
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CHAPTER III
SPECIAL FEATURLS OF EXPERIMEkITAL TECHNIQUE
22tical Installation and Apparatus.
Optical installations for the photoplastic method as distinct from those
used in a photoelastic method must have primarily devices for assuring con-
stant rates of loading over a broad range of loads and rates. Therefore,
such installations must have a mechanical device meeting thls
vsgsrmirAMent
and they must have a recorder for recording the load-deformation curves.
Such an apparatus was installed in the FTI and AN BS SR Ziff. Its ge.wal
view is shown in Figure 28.
Figure 28. Optical installation of the Physical-Technical Institute of AN
BSSR.
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The optical installation consists of an optical and loading portion.
arrangement of the optical portion does not differ in principle from the
ordinary apparatus applied in the photoelastic method (Figure 29).
Figure 29. Diagram of the optical part of the installation: 1 source of
light; 2 condenser; 3 -- filter; 4 model; 5 polaroids; 6 OM NI objec-
tive; 7 -- "quarter-wave" plates.
Figure 30. Position of the container with the model in the field of the
optical installation.
In investigating various processes of plastic deformation, the model is
placed in the container between polaroids (Figure 30). Loading of the model
is achieved according to various schemes by the loading portion of the equip-
ment. The loading mechanism -permits recording of loads from 100 g to 180 kg.
The necessity of obtaining an even application of load and very low rates of
loading with highly optically sensitive plastic materials dictated the require-
ment that the range of speeds of the loading crosshead be maintained from 0.3
to 300 mm per minute. The load being applied to the model is measured
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automatically by a lever-pendulum mechanism consisting of several levers, a
pendulum indicator and a load scale. The record of "load deformation" is
obtained by a recording mechanism.
The study of the processes of de-
formation under conditions of plane
strain is carried out in universal
devices -- flat containers shown in
Figures 31 and 32.
In the container shown in
Figure 31, the openings in the front and rear walls are covered with quartz
plates 25 mm thick. The space between the plates contains steel inserts of
required form and thickness. The model under investigation is placed between
the inserts and is loaded by them.
The container shown in Figure 32 may be used for testing of models of
40 various thicknesses. The portions of the container with the die opening are
made of steel plates of various thicknesses (Figure 33). Optically flat
Figure 31. Container in assembled form.
plates of the required thickness are attached to the interior portions of the
container by means of side bars of rectangular section and bolts. The width
of the working portion of the container (in which the model is placed) may
be adjusted by the width of the loading die.
Figure 32. Container in assembled
form.
Figure 33. One of the interior. plates of
the container shown in Figure 3c...
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IIweight) is thorousehly stirred with a glass rod and after air bubbles are ex
pelled, it is poured into molds made of synthetic rubber. Other materials
-
e not suitable for this purpose, since the highly adhesive resin sticks to
ar
them and it is impossible to strip the molds without spoiling tae model. The
II
II
II
11
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For the study of the stamping process, stamps of the required form are
fabricated and placed in the containers.
In the first container (Figure 31) the diameter of the openings in the
front and rear walls is equal to r&LI- mm, which permits the testing of 60x60x16
mm models; in the second, the diameter of the openings is 52 mm, which permits
the testing of 38x38x50 mm models. The construction of these devices permits
the casting of models directly into the almost assembled container. After
test, the containers are taken apart and tne models are removed from them
without difficulty.
2. Fabrication and Machining of Modela.
Ustol..law npsine
n-
The procedure used in preparing models of resin differs substance
from the corresponding procedure used for the elastic method. The laboratory
luz-mAAlls ol meet. uy pressure
� �
���I ���
T:VTIT A'tt TIAC,Tn 1.t.tr
611= �br
its own procedure for preparation of materials and preparations of "0'1=1' of
resin.
In view of the fact that models of the required form and thickness can-
not be cut or machined from a slab or piece of resin because of its brittle-
ness and to the cutting tools., the following procedure is used.
The mixture (consisting of the components taken in proper proportions by
molds have the dimensions and the form of the required model. The molten
resin is cast with some excess since as it cools the material shrinks. After
the model cools,
the form or its Darts are easily stripped fro the model
without disturbing it. The excess of the model thickness is re:noved by rub-
bing on a stretched strip of gauze with application of suitable alcohol. The
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alcohol dissolves the resin quite easily and it is only necessary to expend
several minutes to remove a thickness of material of about 1 mm. The other
dimensions of the model are similarly adjusted by polishing with gauze
....
CL. 4,CU
with alcohol. By means of this treatment, it is possible to prepare a model
of any form and dimensions. The finished model is coated lightly with a pure
transformer or vegetable Oil which increases its transparency.
The model prepared in this manner is placed in a container and is tested
or aged for the necessary length of time.
When imperfections in the fit of the model are noted, it is loaded
lightly prior to the test. This is done in the first place in order to obtain
intimate contact between the model and the various parts of the container and
in so doing to eliminate Imperfections on the model surface; in the second
place, this is done to remove from the field of the model those fine lines
which are due t.T the presence of droplets of lubricating oil. However, when
the model is loaded, stresses are induced in it. In order to remove these
s-tresses, the model is sometimes unloaded, while in other cases 2-3 minutes
are allowed to elapse until the optical pattern disappears and the mnr1P1
comes quite clear and transparent.
0111.4.0.1V.1
preliminary nnprntinnR are completed v the test of the model
is initiated.
The method of preparation of models of resin described above is not the
only possible one. The model may be cast into the container -which is almost
completely assembled. In order to do this, the molten resin is cast (taking
into account the thickness ot the model and tne excess needed to account for
shrinkage) directly into the mold formed by the walls of the instrument and
inserts of synthetic rubber. After tie model cools, it is polished until tne
required thickness is obtained on one surface only (the front,.. Then the
front part of the container or the face plate is placed, the parts are bolted
together and the model may be tested. Such a 1,rocedure eliminates the operation
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required to fit the dimensions and shape of the model to the container.
Models of Chloric Silver.
Optically sensitive chloric silver is prepared in accordance with the
procedure developed by A. V. Stepanov L127.
The powder of chloric silver is obtained from a 33-percent solution of
silver nitrate as a precipitate produced with a 15-percent solution of sodium
chloride. In order to obtain the precipitate the component materials must be
chemically pure. _Before the solutions are mixed, they are heated to a tem-
perature of 600. This accelerates the coagulation of the particles of chlo-
ric silver.
After the precipitate is obtained from the reaction which is accompanied
by a vigorous stirring of the solutions, the mixture is permitted to stand
for 8-12 hours to permit the sedimentation of .tie flakes of the precipitated
chloric.silver. After this, half of the portion of tAe clear solution which
is above the chloric silver at the bottom of the vessel, is siphoned out and
is replaced by pre distilled water. During the following washings and de-
cantations the precipitate of chloric. silver is separated from the soluble -
products of the reaction. In practice this operation is repeated 14-18
times. This assures the presence in the final solution of soluble products
in an amount not exceeding 1/16,000 - 1/250,000 parts of the initial amount.
After washing, the powder of chloric silver is dried at a thermostatical-
ly controlled temperature of 60-80�C. In this process of drying, the powder
becomes finer.
The entire process of preparation of the product (AgC1) is carried out
in a dark-red light..
In order to obtain a transparent optically sensitive material it is
necessary to melt the powder of chloric silver (melting temperature 457.5o)
and having caot the material into a crucible of thin glass it must be slowly
cooled. However, the material obtained decomposes rapidly and darkens (in
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view of liberation of metallid silver). The darkened chloric silver is
restored to transparency by refinement with chlorine. In order to do this,
dry chlorine obtained during the reaction of Potassium permanganate with
hydrochloric acid is passed through the melt at a temperature of 500-5250
The refinement takes from 10-30 minutes depending upon the degree of darken-
ing of the material. The transparent chloric silver with an oily sheen ob-
tained after this process may now be utilized in the investigation.
However, there are still certain impurities in the chloric silver thus
obtained which may cause the material to decompose rapidly while exposed to
light during the test. The removal of these impurities is carried out by
means of crystallization. As the result of formation of a crystalline grid
during the slow cooling of the molten chloric silver, all the foreign bodies
and metallic silver are displaced to the end of the casting and may be removed
when the model is machined. This is achieved by virtue of the fact that the
11 .
cooling of the melt proceeds from one end and the boundary between solid and
liquid chloric silver slowly moves from the lower part of the crucible to
the top. During this process, the impurities are gradually displaced into
the upper portion of the casting and form a dark layer at the top.
1
.1
After the melt of chloric silver is refined, it is cast in a crucible
of molybdenum glass and is then placed in a vertical crystallizer. In order
to obtain a. completely transparent casting without any defects it is neces-
sary to obtain certain conditions in the crystallizer furnace. After the
crucible with the molten material is maintained for a certain length of time
in the upper part of the furnace of the crystallizer at a ter]pera ure not
exceeding 5250, a device is started which moves the crucible downward at a
speed of 1-6 cm/hour. The fineness of the grain structure will depend upon
+ crucible
the .rate of movement
crucible passes through a ring diaphragm (furnace)
r
-y
th temperature
is maintained within the limits of 495-500�C and tr,e melt begins to crystallize.
63
1
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After the entire crucible passes through the ring diaphragm and the crucible
cools slowly at the bottom part of the furnace of the crystallizer to a
temperature of 100-1200, the crucible with the casting is removed. Because
of the large difference in the coefficients of thermal expansion of chloric
silver (32.94 x 10 -) and molybdenum glass (5., x10-6), during cooling the
casting is in a condition corresponding to three-dimensional tens4on. there-
fore, in order, to prevent cracking wi ain the casting, the walls of tie cru-
cible are made so thin (0.2-0.3 mm) that the glass is destroyed by the com-
pressive stress before the casting can be damaged by the tensile stresses.
The casting must then ki 1_
e hgtni-
treated at a temperature of 360-370oC for
a period of 6 hours with a consequent slow cooling in the furnace during a
lengthy period (10 hours).
Chloric silver is very weakly soluble in water and the majority of other
substances. In order to show the magnitude and boundaries of the grains, it
is necessary to etch the casting. For this purpose, one may use solutions of
ammoniac, hyposulphate or potassium cyanide, which are all good solvents of
chloric silver.
Various methods of mechanical and thermal treatment of the material are
used depending upon the function to be served by the modls _repared from
Plastic optically sensitive
In order to obtain a fine and sufficiently homogeneous grain structure
upon recrystallization, it is necessary to realize a preliminary Plastic de-
formation of sufficiently high magnitude under conditions of absolute homo-
geneity. In this instance the thermal treatment must be carried out at low
temperatures. When such a procedure of mechanical and thermal tratrInt is
used, we succeed in obtaining specimens with grains of the order of C.1-0.05 mm.
Specimens in the shape of discs(cylinders) about 24-30 mm high, are cut
from the casting obtained by crystallization of the melt and heat treatment
at an elevated temperature . Thr, height of the cylinders must not be greater
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than twice the diameter and the ends must be perpendicular to the axis. The
cylinders are compressed between flat polished plates of plexiglass or chrome
plated steel plates in a hydraulic press. As a lubricant one must use a pure
transformer oil, liquid vaseline, or grease. The degree of deformation is
from 90-92, i.e., the cylinder with an initial height of 26 mm is transformed
a plate with a thickness of 2.4-2.6 mm. The plates thus obtained are
polished to the required dimension after which models of required dimensions
are cut from the plates. Then finer polishing of the mnaizi surfaces
ried out on frosted glass plates with abrasives of different fineness and a
solution of hyposulphate. The models are polished on felt, wool cloth, or
chamois and are then placed on flat and parallel thick glass plates to be
heat treated in a tnermostatically controlled furnace.
As was already shown, in order to obtain a sufficiently fine grain
structure, the temperature for the heat treatment must not be high. Thus,
for a deformation of 92-95% it must not exceed 100�C LT87, while for a de-
formation of 90 percent it must not exceed 150�C. Naturally, for a temrera-
ture of the heat treatment of 100�C the time of the treatment must be rather
long (of the order of 10-12 hours); for 1500C it may be considerably reduced
to approximately 2-5 hours.
In order to obtain a more uniform grain structure it is necessary that
the coarser grains in the original casting be sufficiently uniform. If in
the original structure there are significant inhomogeneities, then the re-
crystallized grains will be quite inhomos.eneous. The large and fine grains
Will 1-IP
cnnnr1 side by side in the heat treated specimen. Therefore, if in
the oril--;inal casting there is present a significant inhomogenity of grains,
it
reccmrPended that the castin7 be subjected to a preliminary treatment
consisting of compression of the order of 40-50 percent with subsequent heat
treatment. After such a preliminary treatment the material is subjected to
treatment indicated above.
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Adequate results are obtained in preparation of plates in the following
manner. The disk is compressed from 13 to 7.6 mm. In doing so, the defor-
mation attains a value of 42 percent. After that the casting is heat treated
at a temyerature of 170�C over a period of 4 hours and is then compressed to
2.2 mm. During this operation the deformation attains a value of 71 percent.
The subsequent heat treatment is again performed at a temperature of 1700C
over a period of 3-4 hours. The models are prepared from the plate obtained
in this fashion. Figure 115 shows the isochromatic pattern obtained in a
model prepared in the indicated manner for the case where the load is applied
ln 'A
wiull a u_ Le mm wiuLe.
The ����� 0�� 11. ���� 4- 4
linesare continuous.
The
lerie,Azal
Auvut
behaves mechanically and optically as a continuous and homogeneous body.
During this
*.rtellr..171
1.16 Nur. A.
01^A.011ne.
IrfNal,..,1.4.1. imp
"
.L
averaging process of the optical pheno-
mena in the model bfquasi-isotropic fine grain structure, and the optical
isoclinic coincides with the elastic one. In view of the fact that the
�
averaged difference in optical paths varies continuously, there will be ob-
served a system of continuous isochromatics.
In models having a thickness of 2 mm, one of which is indicated in Figure
10b, there are approximately 20-40 grains in the Path of the polarized ray of
light. This number is sufficient to obtain in the field of view of the de-
formed model a system of continuous isochromatics and isoclinics.
The heat treatment of the models must be performed in air. It is still
better to use paraffin for this purpose by virtue of the greater heat capacity
which facilitates the regulation of temperature in tne heat treatment.
The development of chloric silver of finely rry=tn11i.ne
it possible to obtain
out the volume of the
a macro-pattern of the distribution Qf stresses
mnir-mc,
through-
specimen (model). However, in order to solve a series
of problems, sL'ecimens with a coarser grain
in investigatinp: certain fatigue properties
dividual grains occupy the entire width and
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66
structure are needed.
nrm
nmmimg-1
For example
In which
thickness cf the specimens. It
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becomes important to study the state of stress in individual grains and also
at the boundaries of the grains. For this purpose specimens are needed which
are prepared in such a manner that the path of the ray of light contains only
a single grain.
The procedure for preparation of polycrystalline specimens of chloriC
silver (monocrystalline in the direction of thickness) was developed by S. 0.
Tzobkallo /197 and applied by him in the study of the nature of fatigue fail-
ures by optical means. This method consisted of carrying out
ses of crystallization and mechanical and thermal treatments.
In order to obtain specimens monocrystalline in the direction of thick-
ness, the crystallization of chloric silver is carried out in a horizontal
crystallizer with an application of flat glass inserts. In this process we
obtain plates with elongated grains with a magnitude up to 20 mm. Crystals
obtained in this fashion have an uneven surface and must be ground and then
polished. The best results are obtained by recrystallization of strips of
chloric silver obtained by pressing. In strips having a cross-section of
lx10 mm pressed in a container having a diameter of 15 mm with a deformation
of 94 percent it is possible to obtain grains having .dimensions of 1.5-2.0 mm
by application of the high temperature heat treatment (340�C) over a period of
24 hours. The graph in Figure 34 shows the relationship between the grain
size and the temrierature of recrystallization.
In certain special cases (such as determination of piezo-optical constant)
it is necessary to have specimens containing but a sins-le grain both in direc-
AperiAl prnnoec,
nf th-inknt.Piz nrIA
=NM
r1
the ,q-;rection of the width. This may be achieved in
the following manner /22/. We start with a strip having a cross-section of
lx10 mm and a grain size of 1.5-2.0 mm obtained by pressing in a container.
These strips, which receive a 1)reliminary cold wor1,--:7
by collprescion
tension, which is worse) up to the critical deformation of 3 percent, are
then heat treated in the following manner. The specimens are maintained in a
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I
II
II
11 pression. .
11 : 4
11 t
II By utilizing a very low rate of crystallization of molten chloric silver
z a
10 .
=nei other 1,=1^4,1
..........., ..,... w�............, ........
14.� ^4. 4-1-tift1141tm and
11
., a �
4Z;
CIla
t
?
4.
4U
t'
V
0 2 4 6 a m- L72-1 and 2,5 reported in their work the
11Ra
their alloys, it is possible to obtain
a monocrystals of considerable dimensions
n f
V. N. Krasnov and A. V. Stepanov
anicnfrnpin nlafac nf 4-ha allny
II CMeneMb derztoaplauuu. Z
b
II
I
I
Limmimmommramm
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furnace for 24 hours at a temperature of 3100C after which the temperature
is raised to 340�C over a long period of time (10 hours) and the specimens
are maintained at that temperature for 20 hours. Then the temperature is
raised to 360�C in a period of 4 hours and the specimens are maintained at
that ter:-oerature for 2 hours. After such a treatment the specimens must be
slowly cooled. Such a procedure makes it possible to obtain specimens having
a grain thickness of 1 mm and a width of 8 mm. In Figure 35 there is shown
the relationship between grain size and the extent of deformation in com-
6
z
4 17.
r-r
X 0
16:1/ AV 400
b Tetinepamypa pexpucmannu3a4uu
Figure 34. Relationship between the grain size and the temperature of re-
crystallization.
a) Average grain size, mm.
b) Temperature of recrystallization.
Figure
35. Relationship between the
grain size and the 2a6-nitude of plas-
tic deformation in compression.
a) Grain size, rm.
b) Magnitude of deformation,
%
68
�
of bromic and iodic thallium (T1Br
60 percent, T1T_
Mg. OM.
COO
40 percent) having
dimensions of 3.28x37.00x35 mm in which
they .st-Inciieri the
tai--e of stress under
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the action of a concentrated load. The plates were heat treated prior to the
test at a temperature of 150�C for a period of 6 hours. Following the heat
treatment, only slight residual stresses, which had no significant effect on
the results of the experiments, were observed. In the second case [2-27 mono-
crystalline plates of a fluoric casting having dimensions of 20.87x22.55x3.28
mm and others were utilized for
4-1-1, 4- A- -r 4 *4-'
134.11LAJ ;.%4. .1:n16:Lat.-Ion of fracture.
In this
instance the heat treatment was carried out at A temperature of 7000C over a
norinti of in hmIrm.,
Thus, when we use a proper method of mechanical and thermal treatment we
&-=n obtain
from the salts of silver and thallium uoun monocr%r_5tals
dimensions and polycrystalline materials with a very fine grain structure.
3. Experimental Technique.
In order to obtain quantitative results by the photoplastic method it
is necessary to utilize models of considerable thickness, as will be shown
below. An increase in the thickness of the model is associated with a series
of experimental difficulties. The transparency of
model decreases notice-
ably and the intensity of the transmitted polarized light is correspondingly
reduced. This leads to considerable increase in the exposure time in photo-
graphing the optical pattern. However, the more serious difficulties are
those related to the problem of obtaining a sharp outling of the model, which
is associated with the volumetric effect formation of shadows around the
contour of the model. Then the outline is poorly visible, the determination
of the true values of stress at the edges as well as at other points of the
model is considerably more difficult.
The shadows which are observed along the edges of the model are associ-
ated
improper
experimental procedure: improper installation of the model
and poor machining of tile surfaces of the model and atiaratus, failure to
secure a rectangular section and rounding of the edges (Figure 36).
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However, the volumetric effect may also occur for other reasons. If the
model and tne apparatus are well machined and the model is correctly installed
on the optical bench but the optical
arrangement does not assure parallel-
of the polarized rays of light,
then in investigating models of con-
siderable dimensions
we observe sha-
dows along its edges (Figure 37).
In the given case, the necessary con-
dition
fnr a sharp image of the edge
of the model is that'the polarized raz.-s of light in the working
the installation must be rigorously parallel.
The Indicated difficulties =ay be largely Avoided by observing a number
of conditions.
1. Use of a'point source of light of high intensity. Mercury lamps of
intense luminosity, such as lamps SVDSh-250 and SVDSh-l000, may be used.
Figure 36. Causes of poor visibility
of the model edges: a -- properly
machined but improperly installed model;
improperly machined edge surface;
rounding of edges during improper
polishing of the model.
portion of
r^,
2. Use of highly transparent materials
this purpose the models must be machined
terials having a transparency of high order.
Fi_grnrP
for preparation of the models.
from originally very pure ma-
Mdge effect caused by lack of parallelism
light.
3. Carefully adjusted position
of the rays of polarized
of the components of the optical system
and in particular the condenser (in order to
obtain a rigorously parallel beam
of light). For this it is necessary that the ratio of the diameter of the
light source to the focal length of the condenser be a minimum.
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4. Careful fit of the model dimensions to the contour and thickness of
the holders
5. Assurance of constant thickness of the model over its entire surface.
When the model is placed between the polaroids, its surfaces must be rigorous-
ly perpendicular to the optical axis of the system and its center must coincide
with the axis. Failure to secure a rectangular section of the model and its
improper orientation with respect to the direction of polarized light becomes
evident first of all in the magnitude of the volumetric effect. This effect
may cause exherimental errors in tilA determination of the order of the frin-
ges by as much as 10 percent.
In
1-11,1^4-^rplmnI,1,4vItl.
the optical pattern one must use photographic objec-
tives of high caplcity, highly sensitive film of high contrast, developers
which tend to increase the contrast in the negatives, use. of thin negatives
and the proper type of photosensitive paper.
In testing models of resin in a flat container with glass sidewalls the
.observed ortical pattern reveals the fact that the resin adheres to the glass
plates of the container. If a lubricant is used, it is displaced at the con-
tact points of the resin model and the glass in the process of deformation.
The glass surfaces in contact with the resin are acted upon during the flow
of the material. In view of this tnere are created considerable shearing
stresses and this fact causes the appearance of a certain optical effect in
the glass plates themselves. This optical effect, which affects the optical
pattern, must be taken into account in planning the experiment.
This indicated effect may be
teal
to a minimum by usin in the.con-
tainer glass plates with a low optical sehsitivity. Then we shall not observe
any appreciable change in the optical paths within the glass and in certain
cases the effect of double refractivity in tne glass plates may be neglected.
Thus, it follows from the above that the experinental procedure in the
photoplastic L.etnod has its characteristic features which distinguish it
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basically from the photoelastic method.
4.Photographing of Isochromatics and Isoclinics.
The photographing of optical effects observed with linearly and circular-
ly polarized light witn deformed models undergoing viscous flow has its
si,ecial characteristics.
While in the testing of models by the photoelastic method we photograph
a static fringe pattern, in the tests of models of resin we obtain a pattern
whinA varies with time. Therefore, all the experimental procedures must be
rigorously planned so that the photographing is carried out at those stages
of the experiment which are of greatest interest to the experimenter. There-
fore, it is necessary to use photographic apparatus of high quality which is
convenient to operate, and photographic materials of high sensitivity so as
to redtwe the exposure to a
For this purpose-the universal photographic camera (UFK) is quite suit-
able; it has an "Industar-.13" lens with a focal length of 300 mm and receives
9x12 cm film. Even more convenient is a camera with a lens having a focal
length of 110 mm. In this instance a removable plate holder is used for
photographing with 6x9 cm film.
Distortion of the form of the model and the fringe pattern cannot be
tolerated in the polariscope. In order to eliminate this distortion, a pro-
per optical system is used. In order to check the presence of distortions on
the ground glass of the camera, a rectangular grid is engraved upon the ground
glass which aids in controlling the appearance of distortions.
In using narrow film the most convenient camera is one of the reflex
type, especially a "Kino-Exacta" with a "Biotar" (1:2) or "Zenith" lens. In
order that the film accommodate the entire optical pattern, it is necessary
to make the proper adjustment in magnification.
In order to obtain high contrast negatives with the cameras as described,
one must use highly sensitive aerial photographic film, high contrast,
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"Iso-ortho" plates, 6x9 "Isopan" film, or fine grained motion picture film l4.24
In order to obtain color negatives, one must use film of type B. In this
instance the light source must be a vertical projection lamp of 500 watts.
This yields bright and vivid color photographs, both of the isochromatics
and isoclinics.
Good results are obtained by photographing isoclinics with ordinary
photographic material (MZ) using a monochromatic source of light. The use
of a filter and a slight overexposure aids in increasing the sharmess of the,
isoclinics in the general fringe pattern.
Isoclinics may be sketched directly on a sheet of paver by utilizing a
ground glass camera back or by using a special attachment for projecting the
lines on �a horizontal surface. In particularly important cases, one must
obtain both a photographic record and a recorded sketch. Each record supple-
ments the other and aids in locating with precision the location of isoclinics
in various parts of the model. In such cases one must also take into account
the particular properties of these lines along free boundaries or the contours
in which shear 1nnaR are Absent.
Development of the negatives in high contrast developers improves the
contrast quality of the negatives. However, even with correctly exposed and
developed negatives it is possible to improve tie contrast by thinning out
the negatives or certain parts of it. The use of different grades of paper,
partial exposure, equalization during exposure, partial development of selected
portions of tile positive, various grades of developer, retouching, etc, may
successfully aid in improving the quality of the positive.
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1
STAT
, . 1
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�
�
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�
CHAPTER IV
VISCOUS PHOTOPLASTICITY
An irreversible change in the form of a body occurs by virtue of viscous
and plastic flow. Each of these forms of flow has its specific features
�
vivA46.16..su
require different modelling procedures. In this chapter we shall consider the
basic principles involved in modelling of processes of viscous flow, and we
...U.11 A mip....4c,
faLloo wiewyy
rtmliwe-4,1%,11.ftwb
1,4P.A.v.s.uout
atte-tion .0
elastic and photoplastic methods.
cs="ndatfti-c
which
differ in the photo-
1. Viscous Flow.
As was previously indicated, the basic distinction of viscous flow is
the dependence of the deviational portion of the stress tensor upon the rate
of deformation and its independence of the magnitude of deformation. This
form of flow is common to all amorphous bodies, which include among them the
Majority of natural and synthetic plastics. It must be remembered, however,
that viscous flow also occurs in bodies of crystalline
obAu%,uur.
Min11110.
11401.41.1
formations which follow the law of viscous flow are observed in the case of
tectonic flow of rock formations and in the creep of metals. Apparently,
deformation in the form of viscous flow is common
la.f
all bodies in nature.
In crystalline bodies this deformation is as a rule accompanied by structural
changes, a fact which complicates considerably the observed relationships.
The kinetic-molecular theory of viscous flow troposed by Eiring (2-7
and further developed by Frenkel, establishes the relation between the shear-
ing stresses and the rate of shearing deformation:
lilimmmim...p. Declassified in Part - Sanitized Copy Approved for Release
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A
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ma.
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where
t==ci Arsh (c21),
is the shearing stress;
is the rate of shearing deformation;
(6)
C1, C2 are constants which characterize the properties of the sub-
stance at a given temperature.
For small shearing stresses, when c27S1 formula (6) approaches asymptotically
the form of
ln(2c,i)
== co + �
where C0 = Cl in (2C2).
if cm/sec2
0.2 03 a4 0.5
Relationships of this type between
stresses and the rate of deformi.t4^n
are observed in stabilized processes
of creep in metals.
In considering the problems of in-
vestigation by means of models, we need
Figure 38. Relationship between the not be concerned with the mechanism of
shearing stress and the rate of inden-
tation of the ball in Hepler's consist-different types of change in the form
ency apparatus.
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of the body. In this instance we are only concerned with the relationship
between the tensors of stresses, deformations, and the rates of deformations.
These relationships are always determined experimentally.
In the following material (Je shall consider tne flow of material having
_7
a viscosity in excess of 10' poises. In this case, the internal frictional
forces exceed the body forces
most processes which are of Interest to us,
and consequently the forces due to mass and inertia may be neglected. Under
these hypotheses, the relationship between the stress components and the rates
of flow-may be represented by the following equations up to Certain magnitudes
f shearing strem=am:*x=--* +
Ox
OR),
,=+2'
o,
av: (7)
z
�
hq, ov.
�- _r a
a
,b
__ re (avY 4- ...t r-1 e
.,,,.., dz dy e
. r
1 o ' D g si_ ali _ II it,
'r'' == 1 t d x -1- di P j
where cf
is the average stress equal to
3 '
ri is the coefficient of the internal friction.
Together with the differential equations of equilibrium
ass '
7 acxv.iL 611:_rt =4), a
dx 1 ay 1- dz
. �3Y 4.
' +A., = Os b
; 7F� ox T.14.
,
63, ; dt
"dz c
9
and the condition of invariability of volume
avY + aux �0
dx sly az
(8)
(9)
These equations describe completely the process of flow under the indicated
conditions.
If we confine ourselves to consideration of the cases in which the
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coefficient of internal friction 7? may be assumed to be constant, then
substituting the values of the components of the tensor of the stresses (7)
in (8), and taking into account equation (9), we shall obtain:
a _
riAvx=0, a
d3 -FiAvy==0, b
dy
ax
c
(10)
In these equations, the operator has the usual value
al 4. aa 4. di
ex2 .d,z3
Differentiating equations (10a) with respect to x, (10b) with respect to yv
(10c) with respect to z, adding, and taking into account condition (9), we
obtain an equation for () given oy:
Ac =0.
(11)
In the system of equations (10) it is also easy to eliminate 6 and to
obtain a system of equations only for tae rates of flow. Viscous materials
whose flow is described by equations (10). have the property of adhering com-
pletely to the instrument causing deformation. Thus, the rate of flow of the
material undergoing deformation at the surface of the instrument equals its
velocity. To consider a particular case, the rate of flow for a stationary
instrument surface is equal to zero. This property yields a boundary con-
dition for tne rate of flow. The consequence of this i tie fact that the
ormal stresses on the surface of the illstrument are equal to the mean stress
In order to prove this assertion, let us consider the stresses at an ar-
bitrary point on the surface of tile instrument. .ithout imposinp: any limita-
tions on the general nature of tne proof, we can assume that the pla..e tangent
to the surface of the instrument at this point is parallel to the plane XOY.
Because of the complete
r
adherenceiVia _ - = 01 and on the basis of (9)
OIX ay
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_
e9z
= 0 also, whence we obtain on the basis of (7c) that
Equations (9), (10) and (11) are linear, and their use is well developed;
this permits us to apply them to the calculations and analysis of many pro-
cesses of flow.
In spite of this, in those cases where the deformation is characterized
by a geometrically complex plastic pattern, the calculations are either too
time consuminc or impossible to solve in practice.
� It must be borne in mind, however, that in realitY the coefficient of
internal friction Y1 is a function of the averaze stress (5 and tae tem-
11 perature. As was already shown, the dependence of the coefficient of internal
friction on the mean stress hs the form
MI
(12)
II
material
two points in the 'resin material whose
According to the authors' experiments to be described below, for resin
flow process is under study the diffe-
rence between average stresses is equal to 100 kg/cm, then the viscosity of
the material at these two points will differ by a factor of more than three.
It is quite apparent that an assumption that the coefficient of internal fric-
has a value of the order of
2 11
0.012 cm /kg. Consequently, if at
these conditions.' If, however, this
assumes the form of (12), we shall then obtain
tion is constant is not applicable under
coefficient YI in equation (7)
nonlinear differential equations the solutions of which lead to great mathe-
"ttif'n1 difficulties even for the simplest processes. In view of this, the
study of many processes of viscous flow at the present time is only possible
in an experimental form.
plastic method.
particularly when we wish to
11
One of the experimental methods may be the photo-
similitude are of paramount im-
STA
MEMMEMMEMMiiiiI111111
_ In modelling various physical processes,
obtain quantitative results, the questions of
pertale
A� .2
1S
usual in such cases, we shall consider as similar those
8
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processes which possess a similarity in tnose fields of physical variables
which are of interest to us in the process under study. Similarity of the
fields of the physical magnitudes is present in the case where one field may
be obtained from the other by changing the scales of measurement. In study-
ing process of viscous flow we are primarily interested in the fields
representing rates of flow and stresses. In the following; we shall consider
two processes as being similar if the rates of flow and the stresses at cor-
responding points are proportional. (The coefficients of proportionality
"
for the rates of flow and the stresses are
; general AAffe,/,14-
%A %. ',.�age �
The basic theorem of the theory of sif,ilarity, the
of Kirpicheva
Gukhman, states that: "Two phenomena are similar if they are described by
one and the same system of differential equations and if they have similar
conditions which determine the sign" L27. In the case under consideration,
among
L
1011e
conditions of similarity which assure constancy of sign we must
Include geometric similarity and similarity of boundary conditions. The
necessity of geometric similarity is obvious and needs no special extlanation.
Under conditions of surface stickini.-1 similarity of boundary conditions is al-
ways present. Let us establish those conditions of similarity wnich make it
possible to describe the processes by a single system of differential equations.
As was already shown, in the study of many processes of viscous flow it
is necessary to take Into account the dependence of the coefficient of inter-
nal friction lq upon the average stress. Let us derive the conditions of
similarity taking this dependence into account. When the relationsAip bet-
ween ji and u -s of the fore-. _:iven in
(12), the flow process of a very
viscous incompressible mediuz is described by equations (7), (8), (9) and
(12). Substituting the value of the coefficient of internal friction (12)
in (7), we obtain:
ik denotes a
when I
4.
aht = Tkie
w t1-1
I AV
1"�'I
dx,,
components equal to 1, wnen i = k, and
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(13)
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where i k assume the values of x, ys z
The medium undergoing deformation is characterized by two coefficients:
n(0), the viscosity in the case where the average stress is zero, and
()C, the parameter indicating the sensitivity of t.rle internal friction to
dhanges in the mean value of stress. The media which we are considering may
have various values of 77(0) and CX �
Let us reduce equation (13) to a dimensionless fcr. In doing this,
let us change the scale rvr the quantities which enter into the equation in
such a manner that dimensional values may be canceled in the equation. Let
tit et
%SAW
representesz%representekmatu.La. dimensional tiLMULJ-Lj .LU LUC equation as a product of a
constant dimensional value which may be regarded as a new unit of measure-
ment and a dimensionless variable quantity-
-
Let
a he =-- Cipr, Vi=Vp Vt; .11(0) -
�9 --et � 1.
1
(14)
Here all the letters with the subscript p denote constant dimensional
values, and the capital letters denote variable dimensionless quantities.
Substituting (14) in (13), we obtain
aPa
fa PE p
cip,,ix=ta �E�rii.+Ne
X
av, oti,
k OX, OX,).
Dimensional factors in equation (15) may be eliminated provided that
7Ip
VP
In addition to that, since the exponent of e is dimensionless
Here
lr
(15)
(3.6)
(17)
a dimensionless constant. It can be easily demonstrated that equa-
tions (8) and (9) do not impose any conditions on flich of
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Let us consider two processes whichwe shall in the following call the
10 1st and 2nd processes. The quantities which appear in each of these pro-
cesses will be designated by the appropriate subscript.
The conditions for the 1st process are given by
and for the 2nd, given by
opi 71", xpa
a1 o1 K
a
(18)
U!,,C14 =IC
(19)
Both processes. are described by the same equations but with different units
of mea5urement. Therefore, the processes are similar
Let us divide the expressions (18a) by (19a), and (18b) by (19b):
pi 1P2� � _!E_.
a
7ii)2 Xpi
OP!
Ctipi
A
12122
= 1 .
(20)
The ratios of the analogous values which characterize the 1st and 2nd process
are called multipliers of similarity.
Let
a..
Ca ; - C;; ;
apt
0p2 vp=
-aft
(21)
By means cf these designations the conditions of similarity may be written
in the form:
Cri � C,
CGI.
Ca 3.
1.
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(22)
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Let us suppose that we must study by means of a model the flow process
of a material characterized by the constants 77(0) and Q . Vie have at
our disposal a material for preparation of a model whose constants are )7 (0)1
The scale of the model is usually selected by
0
ex=
perimental means and the means of machining the models and also by taking into
account the available equipment. Thus, Ca
be preassigned.
Using. equation (22b), we determine
On the basis of (22-)
Since C. is the ratio
mal����
01.111���
f'
"
'wt.%
of the velocities of flow at the corresponding
%41+1
Cs
and Cx may be considered to
( 23 )
points of the model and the actual process, it is equal to the ratio of the
speeds of the machines causing the loads. At the same time on the basis of
(23) all the stresses in the corresponding points of the model will be reduced
by a factor of Coc In an analogous manner we can obtain the ratios between
the loads in the model and the actual process which assures similarity in the
distribution of stresses. It is useful to note that, as follows from (23)
and (24)1 varying only the speed of the machine and utilizing one material
for the models we can determine =tress distributions similar to those which
exist in the actual material undergoing a process of flow with 7/ an
assuming arbitrary values. If we can neglect in the processes under consider-
ation the effect of the mean va)ue of stress on the coefficient of
4-te-n-1
friction, then the condition (22b) vanishes. The distribution of stresses in
wrir.h
processes is always similar and the ratio of stresses at corresponding
points is determined by the expression (22a). We must note that in deriving
the conditions of similarity we did not take into account the heating of the
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material in the flow process and the reduction of the viscosity associated
lb� with the increase in temperature. In a number of cases this factor must be
�
taken into account.
2. Optical Anisotropy in Conditions of Viscous Flow.
In Chapter II it was indicated that during viscous flow of certain
bodies the initially isotropic medium acquires optical anisotropy. This
phenomenon is observed among resins, plastics, glasses and fluids. The
effect of double refraction in viscous flow was studied primarily in fluids
-2
having a viscosity of the order of 10 - 10 poises. Ordinarily this effect
is called the Maxwell effect. It is established that for molecular fluids
and with very small rates of shear the difference between the indexes of
refraction of the ordinary and extraordinary rays is proportional to the rate
of shear, and the directions of the principal optical .,.= form nn angle of
45� with the plane of shear. In the case where the fluid is a colloidal sus-
pension or a solution of polymers, the relationships become more complicated.
Thus, the angle between the major optical axis and the plane of shear de-
creases with an increasing rate of shear approaching a certain constant. The
relationship between the difference of the indexes of refraction and the rate
of shear in this case proves to be more complex and is different for different
substances. These data yield information on the form of molecules of the
polyters and on the degree of their polymerization. Considerable work in this
field was accomplished by Tsvetkov and his associates.
Modern molecular theories of Maxwell's effect indicate several possible
ex-aanations of the forced anisotropy caused by the presence of the velocity
gradient LT.g. First of all, this phenomenon may occur if the medium contains
molecules which may be polarized anisotropically and have in addition an
elongated form. In the absence of deformation in the fluid, these molecules
are oriented at random and the fluid is isotropic as a whole. If we now
create in the fluid a velocity gradient, it aids in orientation of the
83'
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elongated molecules along certain directions. The directions of orientation
of the molecules cease to be equally probable, directions of predominant
orientation appear and the fluid acquires optical anisotropy. While the
velocity qradient introduces order in the orientation, the thermal movernent
of the molecules destroys this order continuously. The simultaneous effect
of these opposing factors leads to the state where -a certain degree of order-
liness is established in the liquid, depending on the relationship of the
indicated factors. Calculations show that in the cases where the equilibrium
state of orderliness is small, the directions of predominant orientation of
the major axes of the molecules form an angle of 450 with the direction of
t" velocity gradient. These directions then become the directions of the
major optical axes. Such a mechanism of anisotropy presupposes saturation,
since there is a limit value which is attained upon complete orientation.
such a case. th infinT, n the
difference in the indexes of refrac-
tion and the velocity gradient may be nearly linear only as long as the degree
of orderliness is not great.
The forced anisotropy May also ,be the consequence of deformation of
molecules under the action of stresses. This mechanism may be the predomi-
nant one for deformation of polymers.
In addition to that, if the medium contains elongated particles for
which the index of refraction differs from the mean index of refraction of
the medium, then in the case of ordered orientation of these particles we
also obtain anisotropy. We must keep in mind the fact that the latter mechan-
ism of optical anisotropy may exist only in those cases where the particles
have dimensions exceeding the wavelength of light.
The theories which consider 1-/I�
,..4cated mechanisms of Maxwell's effect
yield results which only agree qualitatively with the experimental results.
In order to utilize double refraction for the purpose of study of pro-
cesses of viscous flow we must know those relationships which exist between
.81+
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1.1
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the observed optical effect and the various parameters of the process. Of
greatest interest to us is the relationship between the magnitude of forced
anisotropy, rate of deformation, and the state of stress.
The authors have made measurements of the difference in the paths of
ordinary and extraordinary rays passing through deformed resin consisting of
80 percent rosin and 20 percent of rosin oil. Viscosity of such a material
at t = 200 is 71(0) = 2.5x108 poises. Data which characterize the rheological
behavior of this resin are shown in Figure 38. The linear relationship bet-
ween the rate of deformation and the maximum shear stress indicates that in
the given case we are dealing with a viscous flow described by equatiJns (7).
Those methods of deforming bodies which are used in similar tests of
materials in the method
11.
Photoelasticity are naturally not applicable
�
'I In
a.ddida
the case of flow of the medium. In a given case we must have a plane station-
ary process of flow in which the state of stress and the rates of deformation
are known with precision. We have obtained deformation by means of concentric
1110
shear, and the substance of this method will become clear from examnation of
the diagram shown in Figure 39. This method of deforwing a body is ordinarily
employed also in the study of Maxwell's effect in fluids.
Figure 39. Diagram illustrating
deformation by the
tric displacement.
mAm+Unti ^f
^^1,1^^11.1
'4*.
Figure 40.
The essential parts of the instrument were two concentric rings, the
clearance between which was filled with the material under investigation.
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�
After hardening the material adhered tenaciously to the walls of the rings.
Figure 40 shows tne rings with the specimen. The outer ring was fixed while
the inner ring was subjected to a given torque. During rotation of the inner
ring each point of tne -ecimen was deformed by the method of applying simple
shear. For any isotropic tnaterial subjected to such a deformation, the tra-
jectories of maximum shearing stresses are concentric circles, while trajec-
tories of principal stresses form two families of logarithmic spirals of the
following form:
r =.Ae-
� +Cp
(Figure 39). At the same time,
but have opposite signs.
LUG
principal stresses are equal in magnitude
The maximum shearing stress is determined by the expression
Al
tootax == 'CAT m= �
2mr2d
where M is the torque;
(ql
r is the radius of the point under consideration;
d is the thickness of the specimen.
At the same time the. distribution of velocities of flow is established
in the specimen by the following equation Z777:
er =0;
==w
r (r r11)
T12 (r112
(26)
Here vr and v
are respectively the radial and tangential components
of the velocity;
Lii the angular velocity of rotation of the
r, is the radius of the inner ring;
r2 is the radius of the outer ring;
is tne radius of the point in the specimen
Consequently, the rate of shear is
avy � 1),r �2 �2 r , � 2
2
dr 1- _i1._,kr2
.86
inner ring;
being considered.
(27)
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cope.
The apparatus with the specimen was
A .
i
.e.s-cat,Ou n the
The illumination was directed along the axis of
diagram of the experimental setup was shown previously
a given torque was applied to the inner ring, it began
.r4 ehl A
-,e.
e tV10 poinTkist.
the rings. The optical
in Figure 29. After
to rotate. The field
of view of the specimen became covered with fringes. At some time each fringe
shifted from the center to the periphery and the radii of the fringes increased.
However, after a certain length of time the fringes became stable and their
location was dependent on the magnitude of the applied t^rglim.
After this the
optical pattern was photographed and the radius was determined from the nega-
tive taking into account the change
of scale in the photographic process.
The corresponding shearing stress was computed by formula (25).
In the given case, just as is done in the photoelastic method, it is
convenient to measure the difference in the paths of the rays In
fati.vft nf tha
wavelengths of the light being used. In this instance the difference in the
paths at the point of the model under consideration is equal to the fringe
order. In Figure 41, there is shown a graph illustrating the relationship of
the fringe order to the magnitude of the shearing stress for a specimen 1 cm
thick. These data were obtained with the yellow line of mercury (A=
5770/90 R) at a temperature of 20�C.
The experimental results indicate
Trnax'Wera2
25
20
45
10
5
2
6 8 10 12 fl
that a linear relationship
ween the difference in the
light and the magnitude
exists bet-
paths of
of the maximum
shearing stress up to a limiting value
of at least 26 kg/cm2. Since in the
given case the rate
n
shear is por-
rticnal tc t:rse=, shearing stress, we
Figure 41. Relationship between the may speak of proportionality between the
maximum shearing stress and the fringe
order for defor:Lation in resin.
Declassified in
difrrence -In the paths
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of the
rifal.re.2
de. 11..7
minA
S.A.41141,10116
STAT
,.,,g0;������111;;;Ce
�
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the rate of shear. The linear relationship thus obtained makes it possible to
characterize
terial, just
the material
the optical sensitivity in terms of the fringe value of the ma-
as in the photoelastic method. As we know, the fringe value for
is measured by the magnitude of tne
maximum shearing shress which
produces a difference of one wavelength of given light, for a path of light
traversing a medium 1 cm thick.
�
According to the definition; the frinze values are calculated according
to the formula
to-tax
Ft
where t is the fringe value of the material,
0
n. is the order of the frincp_
Cr, aa applied to the experiments described above,
=
9Tren
The fringe value of the material under consideration
20 specimens for t
yellow.line of
MIND
42.
20oC prcved to be equal to
In all the experiments
(28)
(29)
as obtained from
2.2 4- 0.015 kg/cm for a
the isoclinics were situated
along the radii forming an angle of 45� with the plane of polarization irres-
pective of the magnitude and stage of loading. Thus, the optical isoclinic&
coincide with mechanical isoclinics.
In order to obtain a more detailed explanation of certain peculiarities
of the increase in the path difference with time while the
a constant load, analagous tests were conducted at a lower
reduction of
4-amneArsatIlres
was accompanied by an increase in
specimen is under
temperature. The
viscosity, and as
a result the stabilization process of the optical pattern became drawn out.
At the same time, the following was observed. At the instant the load is
applied, the difference in the paths of the rays attains a certain value in a
single increment and then gradually increases under conditions of creep for a
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sustained load. At first the rate of increase in the difference of the paths
of the rays is large, although it has
develops, the rate of increase of the
path difference decreases and ap-
proaches zero.- When the load is
suddenly removed, a certain part of
the path difference disappears to-
gether with the load, and the re-
mainder decreases over a period of
several seconds, and even scores of
a finite value. As the flow process
20 40 t. sec.
Figure 42. Change in
ence for rays passing
undergoing flow under
(t = 0 at the Instant
-
the path differ-
through resin
constant load
of application
seconds. If the direction of the load is reversed after it has been 'applied
for a considerable length of time (say, by
torque)
the path difference is reduced to
creases having changed its sign.
reversing the direction of applied
7prn in 1..2AirmenTIAms ;Inn +Ilan in�
In Figure 42 there is shown a graph illustrating the increase in the
difference of the paths of rays with time under a constant load.
This experi-
ment was conducted at 14�C. It can be seen from the graph that the final path
difference is approximately twice as large as the value corresponding to the
instant of. application of the load. This must be taken into account in experi-
ments involving models. All the measurements and photographing must be don,
only after the pattern under observation becomes stabilized.
The indicated peculiarity in the change of difference in the paths of the
rays during the flow process is definite evidence
given case optical anisotropy is due basically to
orientation of molecules comes about
of the fact that in the
tfte fact that an orderly
= rARIlit of defcr=ation.
tion with this, we must consider the question of the possible
=echanical anisotropy which occurs in the flow
tation of molecules must assuredly
In connec-
magnitude of
Process, since an orderly orien-
quantity.
bring about mechanical anisctropy. In par-
ammommimm...mmmommimmAl
ticularl the coefficient of internal friction 77 must
89
become a
fd:Irt=^1".
���� FN. Ow, %A, a.
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As was already indicated, the- anisotropy caused by orientation of mole-
cules has a limit which is attained when complete orderliness is achieved.
The curve showing the variation of the difference between Indexes of refrac-
tion Has the share of a saturation curve which approaches asymptotically its
limiting value. In this case the proportionality between the difference of
the two indexes of refraction and the shearing stress under conditions of
viscous flow may exist only as long as the distribution of molecule orienta-
tions approaches a uniform distribution. However, according to experimental da-
ta this proportionality is well observed up to Tmelex.26 kg/cm2 as a limit. This
forms a basis for a hypothesis that for shearing stresses within thAnp, limitRt
the degree of orderliness in orientation is quite small. We shall also con-
sider how large is the relative difference between the indexes.of refraction
of the ordinary and extraordinary rays which is observed in analogous experi-
ments. As can be easily shown, the difference between the indexes of refrac-
tion is related to the Magnitude of the maximum shearing stress as follows:
�
n I --
�
e Ilitta.V
Even in the case where the shearing stress I = 100 kg/cm2, when
ow
1
we utilize the previously obtained values of A. and To we arrive at the
fact that the difference of the indexes of refractiono
- 0.0026,
which constitutes only 0.175 percent of the mean value of the index of re-
fraction of the material under Atudy. There are reasons to believe that the
magnitude of mechanical anisotropy is of the order of the optical one. All
this gives us grounds to consider resin as being mechanically an isotropic
material under the Indicated conditions of deformation (that is, we can con-
sider the coefficient of internal friction as being a scalar quantity).
It proved to be possible to conduct the exi-eriment with concentric shear-
ing displacements only up to 26 kg/cm2 as a limiting stress, since the speci-
mens fail at stresses exceeding this value. Usually the failure occurred
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1 ;
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along spiral lines whicil coincided fairly well with the trajectories of
1� principal compressive stresses. A photograph of a specimen after failure
(%
is shown in Figure 43. It is obvious that the material is destroyed in ten-
sion under the action of a tensile stress having a magnitude of tne order of
26 kg/cm2, which then imposes A limit upon the shearing stress that can be
used in the method of concentric shearing displacements.
One of the special attributes of
the deforming process by means of con-
centric displacements is that the mean
stress is equal to zero. On -FhP other
hand, the mean stress generally attains
considerable values in models which
represent various processes of
formation. Thus, there arose the Figure 43. View of a speci;en which
failed att-max = 26 kg/cm".
necessity of verifying the observed
fringe value for a specimen subjected to hydrostatic pressure varying within
wide limits. This proved to be possible to achieve experimentally by forcing
the material under investigation through a channel of square cross section.
The general view of the apparatus and the diagram of the experimental set-up
are shown in Figures 44 and 45.
A slit 5 mm wide was cut through a steel plate 5 mm thick. Optically
flat glass plates were bolted to both sides of this plate by means of two
other plates and six bolts. The slit, covered on both sides with glass
plates, formed a channel. One end of the channel was sealed with an accurate-
ly machined aluminum washer and then the channel was filled with the cast
material being studied. A die which exerted pressure through the washer had
a cross section somewhat smaller than the channel so as to minimize resistance
to its movement. The optical arrangement in the experiment was analogous to
that shown in Figure 29. Photographs of the optical patterns obtained in
91
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1111111111111111111111.11111111&
Figure
ratus used
�
*la
General view u.L une appa-
for forcing resin through
a channel.
Figure
set-up
4411$
45. Diagram of the experimental
in forcing the material through
a channel.
Nof
I
11
lii 1111.
a b
II
1.11111111MMMOMMI
Figure 46. Fringe pattern in forcing resin through channel: a --, at a load
Imodm.
of 120 kg; b at a load of 150 kg; c
92 _ - fringes near the loading washer.
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linearly polarized light are shown
9 '-
Figure 46. The dark line along the
axis of symmetry is the neutral line, or the line of zero order. Near the
ends, it passes over into an Isoclinic with a parameter equal to zero.
When a specimen is forced through the channel, the pressure upon the
material from the washer end is equalized by the tangential forces on the
sidewalls of the channel. Thus, we have an equality
k a
D,A rr dr_ lir
�
-a -,gy ������ 110,01.0
fir do (30)
where P is the force on the rasher;
h is the height of the stecimerl;
a is the side of the square section;
defines the stress in the plane of the wall of the channel.
zy - -
' fy 1 1- . � 1 � 1 � .2 I
a.flCvultaber_i_cm.& cimer invest.iga;.ion auneres 60 AllriPse.47!A or the
instrument, the latter are the
imytrfne.,
where the maximum shearing stresses
exist (this question will be considered in greater detail in the following
pages). Consequently, on the surfaces of the channel (-
into account the fact that
� IF11
C
dx =--
Tmax dx = n(z),
equation (25) may be rewritten in the form
p=4 ato dz
zy = 1rnaf
Taking
(31)
i.i
I
1
1 I
where n(z) is the fringe order at the wall of the channel at a section z. 11
The experimentally determined -pressure forces P, are compared in Table
11
11
STAT
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3 with the forces F. confuted from onallatiov, f71N
The integration in formula
(31) was replaced by a summation, and the variation of n between adjacent
exIerimental points was ass=ed to be linear. The fringe value used in the
calculations was that observed In the exileriments involving a concentric
sharing displacement.
.�..
AM!.
AW-
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1+2
.62
P -P,
e p
1
40 1 2
4.3
38
39
59
56
57
62
3
3
6
5
76 9
78 7
85 . ft 4
75 lo
1 119
11
115
15
130 I 11.4
1 177
)
16
4
Table 3
p P P
% aver.
Pe Pe
4.8
9.5
7.2
4.9
907
8.1
0.0
iLhO
8.2
4.7
11.8
8,5
11.5
12.3
62
6
5.7
8.8
The tabulated data show that the computed Isressure, as a rule, is some-
what smaller tn-- that antlied to the pressure washer. This can apparently
be explained by the fact that a part of tne force is resisted by the pressure
washer, a fact which we d
�
111.1 Jr, 4.
4-
in4-^ Ar.ennnt.-
In sT.ite of a certain
amount of scatter In the computed values of the force, t'nese results enable
us to conclude that the fringe value does not depend upon the mean value of
stress.
Let us now consider the optical pattern obtained for the field of view
of the cdannel. At the same time, let us confine our attention to the flow
process as it proceeds at sections sufficiently far removed from the ends
of the specimen. Experiments have shown that the distorting influence of the
ends is only appreciable at distances.from the ends not exceeding the width
of the channel. When we take into account the effect of the hydrostatic
pressure on the viscosity, the flow process is described by equations (7),
(A);
(9) and (12) (the medium is assumed to be incompressible). As a boun-
dary condition, we assume that the velocity of flow along the side walls of
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the channel is equal to zero.
Let us rake two simplifying assumptions. First, let us assume that at
all points the flow velocity has a direction parallel to the axis of the
channel. Consequently, Vx = 01
V
0. Whence, utilizing (9),
= 0 or Vz = Vz(x,y)
Thus, this condition for an incompressible
medium is
we obtain
tantamount to the
assumption that distribution of flow velocities is one and the same for all
transverse sections of the cflannel.
Secondly, we shall Assume that the average stress cr is governed only
6= 6(z)
by the relationship z 4
It .must be noted that when n
are exact.
LUW=VVA, in
�
,
is independent of 0 , these hypotheses
II
the g...=... WO ADP considering-, both of these conditions II
and (12) into (8), and taking into account
, 11
IIwe shall obtain
can be satisfied only approximately. It is impossible to obtain exact solu-
tions on '" '--4s of the indicated hypotheses.
Substituting expressions (7)
(9) and the indicated 'hypotheses,
-- 004
4-10) e AVAx. A = 0.
dz (32)
Let us rewrite equation (32) in the form
esci(z) dam
Vz(x, y)
laq dz (33)
In this extression, the left hand side contains only x and y, while II
the right hand side contains only .z. Equality is Tossible only when each of
II
II
them is equal tc one and the same constaht, which we shall denote by B. We
shall obtain to equations:
VAx, =B;
(34)
STAT
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14
,d111111k
Clair
111b
.71sw
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?NZ) 1144
� ==
Tim dz
( 35)
Equation (34) describes the distribution of flow velocities at a trans-
verse section and constitutes Poisson's equation with a constant right hand
side. By virtue of the adherence of the medium to the apparatus and utilizing
the notation of Figure 45, the boundary conditions may be written as follows:
the
V(c),Y) =v ( =V
= -a, Y) (x, -a/2) (lc, a/2 =
)
V 0
The solution of equation (34) for the indicated boundary conditions has
following
form gg:
B 4a2
m�O
cht901._ Y sin(2fft
,
� �
a
(1 + 2m)8 � c /rpm, +
2
1
1
x(a � x) I
2
1
(36)
The series contained in this expression can be differentiated and in-
tegrated term by term, and as a consequence, by utilizing (7) we can obtain
the magnitude of the shearing stress er
. Yz
tyz 71 v i,2
es
x
-
_ 113a sh(2m + 1)i-. sin (2m+1)?: -
41 y a a
(1 + 2tn)2c h (2m +1)-f-
2
1-zr7N
Taking into account the fact that , 0 , and utilizing
dx d31 az
(7a, 7b, 7c), we obtain 6 -
X G(Y = CIL ='r Cf , consequently, the trajectories
of the quasi-principal stresses* at any section x = const (planes -cermendicular
_
to the direction of incident light) form an angle of 45� with the axis z and
* quasi-principal stresses are cor7uted as principal stress from
the components
in a selected plane, for instance in a plane Perpendicular to the direction of
incident light. Thus, in the case under nnnirig.rafio, they are computed by the
formula_
(ci.atIr +az � I
� lir+ 4Tlyr
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-ryz is a quasi-maximum shearing stress. This conclusion is verified ex-
114 perimentally: the entire field of view of the channel, except for small regions
near the ends, is occupied by an isoclinic having a parameter equal to 450.
Thus, the fringe order is given by the expression
a
1
�
Substituting into (38) the value �firfrom (37), we obtain
Yz
n
RB-nas sh(2m + 1)1 -1-a
Li
(2m + 1)s ch(2m + 1)
0
( 3 8 )
(39)
Let us establish the relationship of the fringe order to z. In order
to do this, let us integrate equation (35), and as a result we obtain
aq4
"4(U)
C Bz.
The constant of Integration C we shall determine on the basis of the
following boundary conditions: Cr = 0, at the exit from the channel, with
O. Thus
But since
actal
it follows that
1
ae4
1
� � � Bz
Crrio
no)
--21911i(OZ
( 0)
(41)
Substituting the value of from (41) into (39), we obtain finally the
equation of the fringes in the field of view of the channel:
n(y4)
s h (2m + 1)7c --t-V-
8BaN (0) a
To(1 � B an (0)z) rn�a i, Lir% 4 � It
kLra 1)-Crikr71
z-r ij
fl!-0 2
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(42)
111
I
i
SI
II
II ;
MI
MO at a given point to the order of the
IIfringe at the edge of the cnannel for
I'
the same section based on expression
(42). The same graph also shows the
values of this ratio obtained experi-
mentally. The experimental points were
IIor exceeded 5 mm. The overall lenth
II
of the
11 of 71,c).
of the
values.
--,,
II
II Am
II wr
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As regards the constant 13, its sign and magnitude may be obtained from
(36). It is convenient to express B in terms of the volume of the material
passing through the transverse section of the channel in a unit of time. In
this case, we havt on the basis of (31)
where U
284b
(43)
the voluthe of the medium passing through the transverse section of
the channel in a unit of time;
4 rt
1.11.
a is the side of the square cross section.
Let us compare the exi:erimentally obtained distribution of fringe order
the field of view of the channel with. that based on expression (42).
Figure 47 shows a graph of the fringe order distribution at a transverse
section of the channel. The solid line shows the ratio of the fringe order
obtained for various sections whose
distances from the ends were equal to
Rs
n9
0
I
1
ill
r-
1
11, �
,
1 Z
I
�
42
L14
45
ge
2v
a
Figure 47. Relative order of the fringe
at a transverse section of a square
channel.
specimens in this instance was 30-35 mm. It follows from examination
graph that the distribution of the fringe orders at a transverse section
specimen in its middle portion agrees satisfactorily with the co.a....putd
Let us consider the distribution of the order of frr,es in the field of
the view of the channel along its longitudinal sections. Since the distribu-
tions of the order of fringes at various transverse sections proved to be
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1
1 1
I!
I
II
I.
STAY
1
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similar, in the given case we may confine ourselves to a comparison of com-
puted and experimental values along a single arbitrarily selected section.
In view of the fact that the order of the fringes is highest at the edges
nf
the channel, the results we obtain are most accurate when we examine the dis-
tribution of the order of the fringes precisely along the edges of the speci-
men. For this purpose, let us solve the expression (42) for z and present
it in the form
here
NZ)
(44)
1
zo ;
(45)
Borro)
' il:(2.m4- � 114:r-
11
8a2 422
. ., z
NI
A------ .
4o
(2m + 1)' (46)
11
crizor =
m -o
,2_,_
Thus, zo is a constant of the given experiment "".1.A.:" depends upon the
flow velocity, and A is a general constant of all experiments. The values
III
of zo and A may be computed by the method of least squares from the experi-
mental values of z and n . The values of A determined in this fashion
(z)
from the data of 10 experiments proved to be concordant. If we substitute
II
the computed values of zo and A in (44), we shall find that the values of z
computed by this formula agree well with the experimental values. Figure
11
48 shows the results of such a comparison.
As can be seen from the graph, a systematic diverence of experimental II
and computed results is first observed in tLe vicinity of t e pressure washer
11
at a distance from it equal approximately to the width of the channel. This
11
is quite natural -,'nc,, 4---_- cnlculations cited do not +--,- into account those
distortions in the flow process which are introduced by the proximity of the
pressure washer.
99
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P.5
a.03 ja Z o
Z
Figure 43. Ratio of the observed or-
der of tne frin,,-e in the field of
view of tile c:kannel to that computed
by formula (44). Point D corresponds
to the location of the washer; CD is
equal to the length of the side of the
square section of the channel.
The constant A may be computed
more simply by means of two experimental
points. Knowing the order of fringes
at two points n1 and n2 and the dis-
tance z2 - z1 between them, we obtain
on the basis of (44)
A
am,
1,2
(z2 - z1)n1n2
n2 - n1
(47)
Utilizing the experimental values
and A, and taking into account (46)
we can compute the value of a,. According to the data of the series of ex-
periments where rosin oil was used as a plasticizer, a = 0.012 + 0.0005
0
cm'/kg. The thermodynamic theory of viscosity gives the value of a In the
form of
V ...
---- /V that. If we assume hat the molecular weight of the rosin
-RT
oil is 3.t2 aid its density is .1.1 rNnv.v, we
obtain from this theory for a tempera-
ture of 20�C, a value of CX of 0.0127 cm2, /x..g. Agreement with the experi-
mentally obtained value may be considered to be good. This result confirms
anew the correctness of our calculations, and also confirms the validity of
a previous deduction 1-..gArdin, the
-771AIMInekVIellet^
the
ek.1-1 c 4 * 4 4 4-vm.
of the resin from the hydrostatic pressure and the related change in viscosity.
On the whole, the results of our experiments with pressing of resin
through a channel furnished convincing evidence that it is possible to conduct
a quantitative study of trie state of stress of a material undergoing viscol:s
flow by the method cf photoplasticity.
3. Certain Secial Features of the Problem of Plane :-:odels UnderfroinT
-TieN*.7
�
Results of investigations cited in the previous paragraphs show that the
basic relationships between the state of stress and the optical anisotropy
in conditions of viscous flow of resin, are analogous to the corresponding
100
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. �
I
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relationships for the case of elastic deformation in amorphous bodies. Thus,
many of the experimental procedures and methods of reduction of data which
are used in the photoelastic nethods may also be used in the models designed
for the study of viscous flow.
However, the use of models under conditions of viscous flow also has a
number of substantial special features. The process of plane flow, In particu-
gives rise to a number of considerable difficulties.
Let us consider this question in more detail. A plane process, as we
know, is one in which the variable 'Qhysical quantities which characterize it
are independent of one of its coordinates (in our case these variables are
flow velocities and the stresses). In the use of models in the photo=
elastic method a plane state of stress is realized relatively simply. For
this purpose we select a model having a thickness which is small compared
with its other dimensions and apply the load only
along Lnle periphery
of the
model. In accordance with Saint Venant's principle, the variation of stresses
across the thickness decreases rapidly with the distance from the loaded peri-
phery and the state of stress approaches a state of plane stress. It may be
noted that tne state of plane stress is approximated more closely as the
thickness of the model decreases.
The basic difference in the modelling technique for processes involving
an irreversible change in the form of the bodies is the fact that in the great
majority of such processes it is impossible to confine the api.lication of
11 surfaces of tne apparatus touching tne specimen which
forces to the periphery of t-:e specimen. (In the following discussion all
are nnrAllel to the
11
11
direction of incident light will be terned as peripheral surfaces, and those
which are po-r-c.nr:-;c-1----
'
2-rec . .
on
14 1--k-
Ji
will be termed side surfaces.)
Let us consider this sT,ecial feature of the modelling of a process in-
volving plane plastic deformation by means of a model representing- extrusinn.,
The diagram illustrating the 1--rocess is shown in Figure 49. The specimen is
101
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illuminated in the direction of axis of x. The unavoidable presence of side
surfaces 1 and 2 and the frictional forces on them, leads to the creation of
shearing stresses in tne deformled body. From symmetry it is clear that
zx
the shearinc stresses at correstomdinE: points or --�n' are equal in
.
magnitude but are opposite in sign, while at the plane of symmetry for x = 0
the.shearing stresses 7 are equal to zero. Consequently, these shearing
Figur.-_? 49.
ningrarr
ling of an extrusion process.
stresses vary in the direction of
incident light and the flow process
is definitely not a plane one. It
must be noted that, as was previously
mentioned, the shearing stresses 7-
x z
and 7- do not reveal tneir presence
zx
through an optical effect in the pre-
setat
howevi,,r;
-
���
causes the distribution of all other
components of the stress tensor to be
different from a plane one, and the
optical
,-)*4-41rork
obtained experimental-
ly does not correspond to the case of a plane problem. In
final analysis
the distortion in the plane flow is due to the frictional forces on tae side
surfaces of the apparatus.
In order that the state of deformation in the three-dimensional process
approximate a plane state of deformation, it is obviously necessary to reduce
tile effect of the side
walls on the distribution of flow: of IrPacc!itiP.q and
stresses. This. can be achieved in two ways. One method ccusic.s of reduction
of the friction on the side walls to a minimum 1)
utilizing a lubricant. How-
ever, this methou is not very useful with models made of resin since trans-
LI
uuv.ivethe rND44
.
In addition to that, it becomes im-
possible to take friction into account in the presence of a lubricant. Another
102
STAT
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method consists of increasing the dimension of the model in the direction of
the incident light making this dimension considerably larger than its other
dimensions. From Figure 49 it can be seen that as we increase the ratio d/b
the effect of the side walls on the process of flow decreases compared with
the edge effect of the periphery, and as a result the process approaches a
state of plane de-formation. It is true, of course, that the distortion of
the plane process near the walls occurs for all thiek.,...e. the model;
however, the optical pattern being observed is a result of the cumulative
difference in the path of rays in the direction of the light across the entire
thickness of the model, and, therefore, it reflects the averaged values of
stresses in the specimen. This averaged value approaches the value of stress
which we obtain in the plane flow process, as the thickness of the model in-
creases. More precisely, observed optical pattern approaches the pattern
11 which corresponds to a plane flow procezs
inreELEink': the thickness of the
11
model relative to its other dimensions enables us, in principle, to achieve
any degree of approximation of a plane flow process; however, experimental
difficulties arise increasingly when this is done. In a number of cases it
proves to be possible to evaluate the degree of alp-proximation of the process
to a plane flow process and by these r.eans to select rationally the thickness
of the model.
Let us consider the conditions of approximation of the process to a plane
flow process for the case of extrusion of resin through a channel of rectangu-
lar cross section.
The experimental setup is analogous to that shown in
First of all, let us establish how .the ratio of the
tangential
4-nv,
-
ces acting on the side and peripheral surfaces of the channel varies with its
cross section and with the ratio d/b. Here d is the dimension of the cross
section of the channel in the direction of incident light and b is the width
of the channel. The flow of the material in the immediate vicinity of the
Pressure washer is analogous both for the peripheral walls and the side walls.
rtlt
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The latter becomes clear if we take into account that the material under con-
sideration adheres both to all the walls of the container and to the surface
of the washer whose velocity naturally is constant with respect to all the
walls. In view of the fact that the shearing stresses are functions of the
I
flow velocities, these stresses on all the walls approach one and the same
value as we approach the pressure washer, and depend only upon the distance
to the pressure washer. This question is treated.in greater detail in sec-
tion 4 of the present chapter.
Let us designate the tangential force acting upon a unit length of wall
by t with a subscript indicating the width of the wall under consideration.
Corresponding to this definition, we have:
O II
d itp. = C T,4y.
w j -
I ..._ C 4: ri t� �
I Since in the vicinity of the pressure washer er and r approach one
yz xz
11
I and the same value which depends only upon the distance from the preislirr.
1
washer, the ratio of the tangential forces acting upon the peripheral surfaces
I i
to the tangential forces acting upon all the walls will approach the following
magnitude in the vicinity of the pressure washer:
11
I .1.
This ratio will decrease with the distance from the pressure washer.' (49)
As was previously shown, there is observed a distribution of velocities
-
m_ =
b + d
11
II satisfying the following equation at distances from the pressure washer which
exceed the width of the channel:
Avs = const . (50) II
II
11 40 Utilizing the solution of this equation for a rectangular region of an experi-
mental case cited in section 2, we can obtain the values x2 for sections at
II
be
some distance from the pressure washer. The computation of x2
iok
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Vr1 CI IT
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!STAT
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out by the following approximate formula with a sufficient degree of pre-
cision for the condition that d/b '4, 2.
x3 =
1 �33
0'6 d
1 � 0,106 7
(51)
The values of x1 and x2 obtained constitute two limiting values of x.
All the observed values of x must be confined beween them.
Experimental confirmation of these deductions based on the optical pat-
I
terns for d/b = 2 proved their correctness. In Figure 50 there is shown a
I
from formulae (49) and (51). As can
graph of values of xi and x2 computed
zi
be seen from the graph, all Possible
II
values of x for arbitrary wilueR of
11
6 7 narrow limits, which ennhipR 13s to
II
d/b are situated m441-.4tn vs.ftics44�r.e.l.tT
vw.t,vAA-11-s& .....,.,
estimate alctroximately that part of
II
11
q8
(12
ar:
t
I
0
2
3
4
Figure. 50
tangential forces which acts upon the side walls.
However, the value of x
of the degree of approximation of the process to a plane flow process. In
order to have a complete understanding we must also know to what alrfont the.
plane process. As was already shown, a plane distribution of velocities* of
observed optical pattern of the entire field of the model corresponds to a
flow, and consequently, of the stresses, will only exist in
the absence of
Figure 49, the velocity of flow in the chlannel .'.'ill detend only upon y. Let
frictional forces on the side walls. in that case, using the designations of
us seek a solution of equation (50) in the form v = v\ for boundary
z(y,
v v
.1.y2
mmilimmimmimmilmmEmmi
still does not give us a complete understanding
conditions
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As can be easily demonstrated, the following expression is such a
solution:
vs = k
we shall components of the tensor of the
1 o ��,
where k is a constant;
b is the width of the channel.
l,ta4n 4.,,,..
x Y
to = 21ov; 1
t :_-.- t =s0
xi ,t, 9
At the same time, the fringe order is determined I
= 2kiidY
n I
.c, �...,�../
la-3:
nt4140,4Ing
St
stresses:
(7),
Consequently, 7-
17' 17!
4/ ALI
by the expression
MM.
Since all the quantities in the right hand side of this expression other
than y do not derend upon yl it follows that for any transverse cross section
of the channel in the case of plane flow process the fringe order n is pro-
portional to y, and the fringes will be uniformly spaced. In Figure 51, are
shown photographs of fringes obtained for channels with various ratios of d/b. II
h I
It can be seen in all the photographs that the fringe order increases in
nnnrol and toward the
pressure washer. As WAR previously established, this is due to the fact that
the coefficient of internal friction increases with hydrostatic pressure.
As
regards the distribution of fringe orders at an arbitrary transverse section,
I
which are situated r'lr' to the end I
of the specimens As the ratio d/b increases,
it proved to be similar for each given ratio of d/b except for those sections
lob
thee distribution of fringes at a transverse section approaches a uniform one.
In Figure 52 are shown gra-1,hs of the ratio 2174 obtained from the data
STAT
n(b,
ONIMMIIIIII1111111111
the direction away from the outer section of tlIgt
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of the optical patterns with indicated ratios of d/b.
Each of these curves
was obtained on the basis of measurements of four photographs.
a
Figure 51. Fringe patterns in
the field of te channels for various ratios
f sides of cross sections: a ��� ��� channel with a section of 5 x 10 mm
b -- 5 x 5 mm (11 = 1); C -- 10 x
5 mm
= 2) �
(s.1
It is clear from the examination of the curves that when we use thin
models (d/b being small) the distribution of fringes is far from that corres-
YN^nAivim to a plane flow pr As the ratio dih increases the distribution
of fringes in the field of the channel approaches the plane case and only
approaches closely the distribution corresponding to a plane process for 17 = 2.
These data certainly do not exhaust the problem of the rational selection
of the model thickness and the evaluation of the errors introduced by the pre-
sence of frictional forces on the side walls. However, in modelling various
technological processes involving deformation they may be utilized
Cesr
pur�
poses of orientation. Thus, in modelling the process of extrusion we can let
the width of the container equal the width of the channel as a first approxi-
mation. Obviously, the selection of the model thickness depends upon the re-
quirements which specify the precision of the results.
107
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STAT
I.
�
�
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In our opinion, in order to obtain
reliable quantitative results, the
thickness of the model must be at least
twice as large as its other dimensions.
For a qualitative study of the dis-
tribution of stresses we may limit the
thickness of the model to approximate-
ly its width.
Legend:
) Diane problem
4. The Simplest Plane Problems of Figure 52. Relative fril..ge order at a
transverse section of c.lannels for va-
Viscous Flow. rious ratios of sides of section (the
straight line corresponds to fringe
One of the ways In which it is order in the case cf a plane
cess.
possible to determine whether the
pliotoplastic method is applicable to the study of processes of viscous flow
is to compare the computed and experimental values of one and
the same prob-
lem. Certain similar comparisons were made in the preceding paragraphs of
this chapter. In the following material there is given the solution of four
plane problems of viscous flow and a comparison of these solutions with ex-
perimental results obtained by the photoplastic method is presented. All the
problems cited are solved for the case in which the coefficient of internal
friction may be presumed to be independent of the value of mean stress.
Problem 1. A Viscous Medium Com ressed Between Flat
This process is showndiagramatically in Figure 53.
YIPPI;7>xtrilmt%111/1/7/2TIAMM1d//T/71777777777777P
Parallel Plates.
Figure 53. Diagram of compression of material between flat plates and the
computed fringe pattern.
Let the plates move towards each other with a velocity vo each, and let
108
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the distance between. them (the thickness of the strip) at the given moment
equal 2a. Because of the adherence of the material, the velocity of the
layers of the medium adjacent to the surfaces of the plates is equal to the
velocity of the plates.
Thus, the boundary conA-;tions for the flow velocities may be written
in the form:
Let us differentiate equation (10a) with respect to yl equation (10b)
with respect to x and let us subtract the latter from the
ceiling the term
11
we shall obtain
o
Av �2 -Liv =0.
dy ox Y
. Let us ex-press v in the form v = v (y).
Then on the basis of (9)
dv
VT x 1(y).
dy
Here f(y) is an arbitrary function of 7.
Since along the axis of symmetry vx = 0 for x = 09 f(y)
v x � x dv, .
dy
� /lb 411. 111.
former. After can-
(53)
(54)
(55)
0
(56)
11
(57)
1
(5)
STAT
Substituting. (51f) in (56) into (53) and cancelling x, we obtain
diet) y
dy4-
A general solution of this equation has the form
Co + C1y C2y2+
The
The process of flow is symmetrical with respect to axis of x and conse-
quently, Co
%ay
2 -
O. On the basis of boundary conditions
109
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3ve
2a
C3
Thus, taking into account (59) and
flow velocities:
V:,
Vx
IOW --�
2a*
fr,e1
l70),
we shall obtain the
3v.
AIM
x, a
(59)
following
Utilizing (7), we obtain tlie components of the stress tensor:
3V Olt �
a k
31/071 i
a
`2401
�
Ve(1.1
"4:1,731=
�a
xy.
a
Applying the known formula of the theory of plane stress
V.
��������001/
be Aft Imi.e..41.0. 4..4.1-tc, pp tY Lk VI 0"..01MAI
WU WW141,10..16.44. 4614. ebiV,rosai
4:2.ry
(60)
(61)
31412 (a2 � y2) 2 + x2y2 .
as (62)
Consequently, the equation of the family of fringes will have the form
3v
n so, 0-r-
ale�
a'�y')' + Je2y1 .
None of the quantities in the expression (63) which are in the term
(63)
preceding the radical depend upon x and y; therefore, using the desirznation
equation (63) may be written in the form
110
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k2 n2 (a2_ y2)2.4. x2y2.
(64)
The family of fringes corresponding to equation (64) is shown in Figure
53. In Figure 54 is shown a photograph of the fringes obtained at the in-
stant when the specimen is compressed 7 mm with the thickness of the layer
in the direction of light being equal to 40 mm. The agreement between the
experimental and computed fringe patterns may be considered satisfactory.
Figure 54. Fringe pattern for compression between flat parallel plates.
Problem 2. Flow of viscous material in an angle formed by two mutual� a
perpendicular walls, one of which is moving in its plane. The diagram of
the process is shown in Figure 55.
Vo
Figure 55. Diagram of the process and the family of fringes for flow through
an angle.
Legend: a) neutral line
Both walls are assumed to be infinite. The flow in dihedral angles formed
by the pressure washer and walls of a container which serves to represent the
process of extrusion, is analogous in character.
ill
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In order to solve this problem it is most convenient to use a polar sys-
tem of coordinates in which the equations analogous to equations (10) for the
case of plane problem have the following form 4777:
as 4. TA a"-v, + a2v, I OD, 2 av, v,
dre r rt) -0;
�
1 03 jdtv, 1 &vs, I dv, � 2 av, �v? Ii.o.
r
____.+1 �momilimmmomM� + . J..... + v.,.
e aft AA 4 7 - ar r
Or rt d? r2 1 (65)
Here vr and v, are respectively the radial and tangential components of
velocity.
The equation of incompressibility in the given case is transformed into
cro.fir I ()pc:, , �Ur
= O.
dr . r
TO
. The components of the stress tensor are defined by the expressions
dv,
+27lOr
;
z":+ 211e vu
- Ovt
r rr�
tav .11 dv7
r
(67)
Let the wall a be fixed and let the wall b move in its plane with the
velocity v0. Taking into s%ccount the adherence of the material to the Walls
and utilizing. the designations indicated in Figure 55, let us set down the
boundary conditions in the following manner:
for
-
for--= O.
V? ---
2
Let us seek solutions for eP vr and v in the form:
112
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(68)
STAT
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tea
Under conditions ( 68) equatio:as (65) will be rewritten in the following
manner:
(313 ( tile � 2 de
IP �
VP
or r2 dee drp
ae I)((Iv v
-F 2d' �ecp) =0.
the r kdc dip
(69)
Equation (66) may be written as follows when we take into account (68):
dv, +
(70)
After differentiating equation ( 69a) with ....F=%.t to er- and equation
(69b) with respect to r, the latter is subtracted from the feyrbytev.. Atter
cancelling 77 and taking into account equation (70), we obtain the following.
nr--Ainsirtwy differential
equation for -;:k
-T
d4v., day
+ 2 �?- + v,
44 de
whose general solution will be given by
v, = A sin? B con+9(Csin, D cos?),
(71)
(72)
where A, B, C, D are arbitrary constants. On the basis of the boundary con-
ditions the constants are
ulCGC11.11ALVaa
vir.a.u44
1.0A1=11Airtf
fn ni ,crn�
A - -1.675 vo; B = 0; C = 1.062 vo D = 0.675 vo.
Thus, the velocity components are determined by the expressions:
=v01-1,675 sin? + ?(1,062 sin? + 0,675 cos?)j;
v,� volcos? � 1,062 sin? � y (1,062 cos? � 0,675 sin?)].
(73)
The components of the stresE, tensor may be expressed 1n the followinr form
on the basis of (67):
�
f 1
�
rc 4-1,0 r-rly V� viv(ya..
� � Z 011t
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(74)
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Since ar - (1,sseri =7;
and consequently, the trajectories of the
T 12);
maximum shearing stresses may be represented by radii and circular arcs, and
the trajectories of the principal stresses form two families of logarithmic
+,
qpirals of the form r = ce
Substituting (74) into (
fringes
7)P.1
^
v...4
an equation of the famil
of
Ii 2,52Y4 vo
= . sin(tp � 57�301.
(75)
to
Thus, the fringes represent circular arcs which are tangent at the
origin of coordinates to the straight line clig = 57�30'. This straight line
is a neutral line, since shearing stresses are absent from it. Figure
shows = family of fringes corresponding to formula (7,.�1/4 Ji.
cc;
in Figure 7t) t,nere
is shown a photograph of fringes obtained experimentally for an angle formed
by the wall of the container and the pressure washer.
Comparison of Figures 55 and 56 shows that the computed fringe pattern
near the vertex of the angle coincides satisfactorily with the experimental
one. These patterns diverge more and more with the distance from the vertex
of the angle since the computations were carried out for the case in which
the extent of the medium in the radial direction is large compared with the
A4A,400Mlin^.Ca
Ul1COsoGktavv....
from the vertex to the points in the model being considered.
Figur* 56. Fringe pattern in the angle Figure 57. A schematic diagram of load
1
formed by the wall of the container and applied to an elastic plate for which the the pressure washer, family of fringes is the same as that for STAT
flow of a viscous medium through an angle
(Figure 55).
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It is interesting to note that the fringe pattern obtained is completely
analogous to that which we observe in loading the vertex of a right angle of
an elastic plate with a concentrated load in the direction of one of the sides
of the angle. A schematic diagram of such a loading is shown in Figure 57.
Although the fringe patterns in both of these cases are the same, still the
states of stress are basically different. In the case of the loaded elastic
plate the trajectories of the principal stresses coincide with the radii and
circular arcs, and the trajectories of the maximum shearing stresses form
logarithmic spirals. Thus, the trajectories of the principal and maximum
shearing
stresses interchange their loci in the transition of one process to
the other.
Problem 3. Flow of a Viscous Medium throu.h a Narrow Slot. A schematic
diagram of the trocess is shown in Figure 58.
The proble= may be stated in the following manner. The volume bounded
by planes z = 0 and z = d (situated at the top in Figure 58), is filled with
a viscous medium which elows through a narrow slot of infinitesimal width,
the slot beinr situated along the axis of z. The flow process is a plane one
and the flow velocity does not depend on z. The boundary conditions are:
for
=0.
It is natural to suppose that the flow velocity in this case is directed
along the radii. _nerefore we shall seek a solution in the form:
= 0;
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(76)
STAT
1
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Substituting values of vr
and vte in the form of
do . c/10
-t-
ar 0 do?
(la 271 d43)
a
(, r2 dp 77)
Let us differentiate equation (77a) with respect to cr and equation
(77b) with respect to r and subtract the latter from.the former. After
O.
(76) into (65),
we obtain:
�
cancelling 7/ we shall obtain the following equation for (1):
dq31 d?
whose solution has the form
(11= Acos 2? Bsin 2? 4: C.
Since the process must be symmetrical with respect to axis
On the basis of the baundary conditions C = A.
Thus, the flow velocity may be written in the form
2A cost?
V,==
The constant A may be expressed in terms of the volume of the
U
flows through the .
opening in m unit -f t4-e
sal J.44L t Ji = -ff . Here U
of the medium flowing through the opening in a unit of time.
and (67) we shall compute the components of the stress tensor:
On the
hnicict
Consequently
Trip
4rUn cos2?
c 1.1
4U-ri cos29
rz
21/1 Sint?
TC e �
a
(78)
of x, B = O.
(79)
medium which
is the volume
Utilizing (79)
of (80) the maximum shearing stress will be equal to
=.�
Tatar
_
COS?
the equation of the fringes has the form
116
(81)
1
MN
1
STAT
mommommomimmil
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ILIIIIIIMIMI Declassified
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n
41174 cosp
o
...����������+��� -�-���� � '�� � �-�� - � - � �
(82)
This equation is an equation of a family of similar ovals which are tan-
gent to the straight line 9? =
rr
2
at the origin of coordinates. The dimen-
sions of these ovals vary inversely as the square roots of the fringe order.
As indicated by the statement of the problem, in comparing the computed
Irstill1^a mi.fh
experimental data we may expect them to coincide only at points
sufficiently far removed from the opening. More precisely, the distance of
the points from the slot must be considerably larger than the width of this
slot. As will be seen from the following exposition, the fringe pattern is
close to that described by equation (82) even for points which are situated
at a distance of twice the width of the it measured
the cont.
from the middle of
Moreover, as was already repeatedly indicated, in order to assure that
the flow process be plane it is necessary that the tangential forces on the
side walls be eliminated.
In practice this is impossible to achieve. How-
ever, at distances from the slot which are small in comparison with the dis-
tance between the side walls (d> r), the relative effect of the side walls
on the flow process is small and the observed optical pattern must be close
to that yielded by equation (82). In Figure 59 there is shown a photograph
of fringes obtained in extruding resin through a slot 0.8 mm wide. di
tance between side walls in this experiment was equal to 28 mm.
In Figure 60 there is shown for purpose of comparison a computed fringe
pattern. With the exception of the region in the Immediate vicinity of the
slot the coincidence of these two patterns must be accepted as satisfactory.
Problem 4. Flow of tedium throuch a slot of finite width. This prob-
lem differs from ii rceding one in that in thi cao e '30 not ne lect the
the immediate vicinity of tne slot as well. As a conseouence, 11
dimensions of the slot and thus obtain a description of the flow process in
we may obtain
11
STA
117
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Figure 59
Figure 60
certain new results which are applicable to actual processes:
Let us formulate the problem in the following manner. The space occupied
by:the viscous medium is divided by a thin plane wall into two parts. The
wall has a slot of a constant width. The pressures in the medium on either
side of the wall are constant at infinitely large distances from the wall but
are different in magnitude. In view of the difference of the pressures the
medium is extruded through the slot and at the same time it adheres to the
wall. In the entire volume filled by this medium a certain field of flow
velocities and stresses is established. The process is assumed to be a plane
one. The computed pattern for the plane xy is shown in Figure 61. The pur-
pose of the problem is to determine the indicated fields of velocities and
stresses.
As will be seen from the following, this problem can be solved most con-
veniently by means of elliptical coordinates. To simplify the calculations
the width of the slot is assumed to equal to 2.
Let us consider the process in the coordinate system given by A and
ii which are related to the coordinates x and y through the equations:
_ 1 1 I 1rkl�
�1 rk r up,
312 a!rt.
(83)
Excluding from equations (83) the term AL , we obtain the following
1.18
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equation of one family of coordinate
lines:
x2 .Y2
1+ 4-
(84)
As A varies from 0 to 00 each
of the lines of this family represents
an ellipse with foci at points (- 1.0)
and (1.0), is equal to the square Figure 61.
indicating equal maximum shearing stresses
of the minor semiaxis. for a plane flow of viscous medium through
a slot.
Computed
t...t 0.1.11
of
-
61.1C
lz
.L.Lnes
analogous manner having
excluded A from (84), we obtain the second family of coordinate lines in
the form
1.
(85)
As 44 varies from -1 to 0 this equ'ation forms a family of hyperbolae
having foci at precisely the same points, (-1.0) and (1.0). In the present
case the absolute value of I-A, is equal to the square of the distance to the
vertex of the corresponding hyperbola from the nearest focus.
It is easy to demonstrate that the families of ellipses and hyperbolae
being considered are mutually orthogonal. Through each point of the plane wy
there passes one ellipse of the family (4) and one hyperbola of the family
(85). Thus, the quantities 1k andll determine with respect to sign any
point within the limits of a single ci,-.Arant of the plane and these quantities
may be considered as the coordinates of the point.
Let us note that for these coordinates the ecuation of the axis of x
between points -1 and 1 be
= 0, the ec.uation of
the axis of x
tiae limits of te inuicatea interval will be --g = &1 th eouatioli of
the axis of y will be = -1.
In utilizing the curvilinear coordinates a very important role is played
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by Lamfits coordinates, which represent arbitrary arcs of a coordinate line
for a given coordinate. For the chosen system of coordinates, Lame's co-
efficients have the form:
hk== I .1 ir -
9
t4
Ca 2 V (I +')x
�
dsx 1
4 2 V �(1+u)
(86)
The components of the stress tensor may be presented in the following
form analogously to the preceding cases:
�
FkIL tip.*
(87)
where in accordance with the theory of orthogonal curvilinear coordinates
(see for example [3-27) the components of the "viscous" tensor of the stresses
410 for coordinates A. and g are given by the following expressions:
�
dvx
ffkOk
.�223 I dvp,
51 i
30 tall
1�V), 1 ov,
Hp, (hi 4-HA OX
1 aff),aff
11), 11�t al., +VP- OkkL)
(88)
Equations of motion of a very viscous incompressible medium in the same
coordinates may be written In the form:
Oa
ji-f '
=_-. :5
k
Hp.
1-4. ay,
�
120 STAT
Ell
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�
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Here
1
,L))14.
The equation of
Let us
to the wall
do
� �
Os
IL
, alimHA
CiL) ()IL
do'
H�, � Cie 44. 9
� aH,
are determined by expressions (8E).
continuity in the given case has the form:
now consider
1 v..
extressed
The conditis at
for
�a ( H, v), � �a (HA va) = 0.
ak dp.
(89)
(90)
t;.e boundary conditions. The condition of adherence
in the following manner:
infinity may be
vx =0 and v
obtained
from
the
(91)
following considerations.
At infinity ( )c--4 00) the distribution of flow velocities into a slot of
a finite width
flow throur7h a
the velocities
must become the same as the distribution
slot of infinitesimal
of flow in the form:
width (79).
VA =-- VA P., 10; V1 =-- 0.
In other- wcrds, let us assume that the
hyperbolae of the
1.1bstitutinfj
whence, taking
family (85).
( 9
\
) into
into account
(9o)
a
�.
we obtain
(1.4
17/.
......_
�
the value of F
ru,==f()
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Let
us
which occurs for a
seek a solution for
lines of flow coincide with the
from (E6),
1� (1 � p.
A �
121
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(93)
STAT
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Here f (g) is an arbitrary function of g and is determined by the condition
411 of transition at infinity of equation (93) into (79).
As can be easily verified, the latter is achieved for
f (10 = A
where A is an arbitrary constant�
form
tis
Thus, if the supposition (92) is correct,
p
the velocity of flow has the
�
(94)
Tae velocity of flow in form (94) satisfies both equation (90) and tile boun-
dary conditions. It remains to be shown that it also satisfies equations (89).
In order to do this, and substituting (94) into (88) we obtain
}IV (I +X) X.
ELY
00. 2A1
419.4y,X (i p.
2
(95)
Further, taking into account (86) and (95), equations (89) may be rewritten
in the form:
(k V(1+ A)A a
(96)
Differentiating the first of these with respect to AL and the second with
respect to
and subtracting the latter from the former on both the right
and left sides we obtain 0 identically. Thus, the flow velocity in the form
(94) is actually the solution being sought. The constant A is related to the
volume of the medium passing through the slot of unit length in a unit of time
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by the relationship
A 2U
411
�
(97)
Let us compute the value of the average stress cf
From the conditions of the problem it is aT:parent that the value of
is determined only with a precision given by an arbitrary increment? eince
the flow of the medium is caused by a difference in the mean stresses (dif-
ference in pressures) on both sides of the wall. From the expression (96b)
it follows that the mean pressure along the slot (for A. = 0) is constant.
Let us assume that it is equal to zero. Then the mean stress may be obtained
by integrating (96a) with respect to A. from 0 to A.
a =
4 Ui ilis.
I i'1-t- A)
(A
(98)
As A 00 the mean stress approaches a constant independently of the value
of g
4U-ri
1:
(98a)
The latter defines the relationship between the volume of the medium passing
through a slot of unit length in a unit of time, and the difference in the
pressures.
Filrther, when we substitute (95) into (87) with (97) taken into account,
the components of the stress tensor finally assume the form:
_ 4UT, k 1/(1 +
�������
�
T4 (-A 20 111:17+ �A,
� ,
4 LIT; o.
(1 - Op.
0- �1-LY
(00)
Here the utter sign corresponds to that portion of the region from which the
medium flows, while the lower sign correspendn to the
123
region
which receives
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1
STK
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the flow.
Let us pause briefly and consider tne bounds of applicability of the
computed data thus obtained. The solution obtained is precise for the case
wnere tile medium occupies the entire volu=e s,n both sides of the wall. Let
us Immediately note that such a character of flow is impossible to achieve
in the vicinity of the slot in that part of the region which receives the
flow. Indeed, it is apparent from (99; taat tae normal stresses attain arbi-
trarily large values on the wall (
slot
g= 0) near the points which define the
However, a given medium can support without failure only certain tensile
stresses. Thus, in that part of semi-infinite volume which receives the
=1 ett1_j
in the vicinity of the points forming the boundary of the slot, there must
'occur a failure of bond between the medium and the wall.
We are usually interested in the case were the medium occupies the
space on one side of the wall and flows freely through a slot. In the initial
stage of the flow process when the medium occupies the space only on one side
of the wall, the solution we obtained is exact. After a certain amount
of
flow of the medium through the slot has taken place, the effect of that por-
tion of the medium on the distribution of velocities and stresses in the semi-
infinite medium. on one side of the w..11
. is quite small and in practice its
effect is the same as in the initial stage
Let U8 evaluate the applicability
of these results to those cases in
which the medium undergoing deformation occupies a finite space. In the pro-
cess just considered the major
n^ri-inn
of the energy of deformation is ex-
pended near the slot. In this region we also find the largest chanc;es in
stresses. In Figure 62 are shown the graphs illustrating the changes in the
ratios
�
ws" -
9
aim
121+
�
;Wax
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along the y axis obtained on the basis of (98) and (99). From these graphs
it is seen that at distances from the slot equal to twice its width th.-
stresses are already close to their limiting value, and at distances of the
order of 3 widths the mean value of stress is 98 percent of
Q4
Ct2
6TT
I
/I
.�-wW
IC,
I
0
2 3 4
,e
Figure 62. Graphs of the ratios of CI 10,e and 7-max to the difference
between stress at infinity and that at the slot for a flow of viscous mediums
through a slot having a width 2a.
Thus, the results obtained on the distribution of stresses for the case
of flow of medium through a slot are applicable to concrete examples of pro-
cesses of flow of a viscous medium in those cases where the dimensions of the
space occupied by the medium undergoing deformation exceed the width of the
slot by a factor of 3-5.
On the basis of the data on the state of stress given by (99) we can
obtain a computed fringe pattern. The order of the fringes n can be given
in the followin manner on the basis of (99), (28) and (98a) as it was done
in the preceding problems:
n
or, adopting Cartesian coordinates, by:
n =
-0
-
(x_0312
I (x2 +y2 I )2 4y2)
'
( 10 0 )
Figure 63 shows an ex-,erimentally obtained fringe pattern for a slot
,0
width of 5 mm and a thckness of layer d of c.0 mm. As can be seen from the
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2
direct comparison with the family of fringes (Figure 61) corresponding to
equation (100), the computed fringe pattern coincides quite satisfactorily
with the experimental one.
In all the problems cited, one can compute the trajectories of principal
stresses and isoclinics.
Alonir with f-4----
c�-%
L.+ -
it.
would _
uc
possible to compare
computed and observed isoclinics; however, isochromatics are always obtained
experimentally with greater precision than isoclinics and this aAslirPrz
reliable conclusions regarding their correspondence to computed values.
The material considered in this discussion confirms the previously ex-
rrinrp
pressed opinion that
111"
.1.. V
possible to solve plane �problems in viscous flow
by experimental means. As regards the solution of three-dimensional problems
this question belongs to the
i 4-
Th.tsmft nrcx
NA is" AN ���
reasons for believing that
the method utilizing scattered light is quite applicable in this case. The
photoplastic method even now may be applied in a number of cases for checking
the theoretical solutions of three-dimensional problems (for instance, as it
was done in considering the flow through a channel).
Figure 63.
5. Sinpularities of the State of Stress at the Periphery of the Model and
Some of the Methods of Reduction of Ex erimental Data.
As was previously shown, we may obtain by the method of photoplasticity
the fringe order and the inclination of the trajectories of the principal
stresses in the entire model field. Knowing the fringe value for a given
material, it is possible to determine the state of stress at any point of the
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9
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model on the basis of these data by one of the methods known in photoelasticity.
In this instance the following formulae of the theory of
but ion are generally utilized:
Txy max =T -sin20;
0 = a � 2rmax � cos 20;
Y
r11 ?I a
stress distri-
(101)
(102)
The shearing stress 1 is determined on the basis of formulae (28)
� xY
and (101) according to the data on the fringe order and parameter of the iso-
--0i- Ox (104) I
6: + atyx = 0.
a..$1 . (103)
1
As regards the normal stresses, they are determined from the optical
data only With a precision given by a constant term in view of the fact that
a
(105)
clinics from the expression txy = n sin 20.
the anisotropy is independent of the mean stress. For a com7tdete determina-
model.
at the point (x0, Y0)4 then on the basis
Let us assume that we know Cr 1
tion of the normal stresses we must have in addition to the fringe and iso-
clinic patterns the value of one of tne normal stresses at some point of the
of equation (103) Cfx is determined by the following expression at any point
on the straight line Y = I
Here (x,y)is d at the
(x, Y cx (xoY starting point having coordinates (x, y,). I
zro)=--- ) dx.
ay
xo
(106)
127 STAT
@ 50-Yr 2014/06� 8 OMMEMEMEMEMMEMMEMMill
Yo:
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The integration in this formula is performed along the straight line y = yo.
In order to compute the integral in the formula (106) we may proceed in
the following manner. Let us divide an interval of the straight line y = yo
A
by the points x1, x2 ... into small increments LiX where 1---1 xi = xi - x
Through the mid points of the increments of the straight line let us pass
straight lines parallel to the y axis. Let us select on each of these straight
lines two points close to each other and situated on both sides of the straight
line y = yo. Utilizing the experimental values of n and 69, let us compute
the values of I at these points. Dividing the difference between the
xy
values of 7- at the indicated points by the distance between them, we shall
xy
W
obtain the approximate value of the derivative in the interval
Further, after we substitute a summation sign for the integral sign in (106)
this formula may be approximated by the following- one:
Ii
Yo) ---- 3x (x., Y.) � E LiTxyl
AY: �
Axi
(107)
Knowing Q, Gr is computed at the same point by means of formula (102).
xY
In order to compute C) along any straight line parallel to the y axis,
Y
an analogous formula will be giver by
at
ay (x0, yn (xo, Yo) 221
(108)
In the calculations we may take as the starting point any point where
the state of stress is known. By keeping track of the movements along inter.
vals of straight lines parallel to the coordinate axes, it is possible to
reach any point in the model and thus to determine the magnitude of the com-
ponents of the stress tensor in the entire model field. Any point along the
free boundary of the specimen undergoing deformation may serve as the point
cf the model where the stresses are known. In view of the fact that external
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loads are absent along the free boundary, the boundary itself becomes a tra-
jectory of one of the principal stresses. The second principal stress whose
trajectory is perpendicular to the free boundary is equal to zero at each
point of that boundary. Thus, the parameter of isoclinics which enters a
free boundary is equal to the slope of the tangent
point of entry. Since the maximum shearing stress
sum of the principal stresses, it follows that the
in the direction
of the boundary at the
is equal to half of the
principal stress acting
of the tangent to the boundary is determined by the following
expression and the known order of the fringe:
02 21ton.
(109)
The method of commutation which was briefly described here is called
the method of differences of tangential stresses. It is described in greater
detail in a monograph
"Photoe.A.a.k..1.ci6N
by N.
FrncJir
When we study flow processes by means of photo-olastic models, we must
keep in mind certain singular attributes of a state of stress at the bounda-
ries of the
model in contact with the apparatus. We have already indicated
in this chapter that
the trajectories
this assertion,
fcr a viscous flow the boundaries of the apparatus are
of maximum shearing stresses. In view of the importance of
we
shall show that it follows fro':
equations (7), (9) and the
condition of adherence. -;iithout limiting the general nature of this proof,
let us select for simplicity the coordinate axes in such a manner that the
tangent to the boundary at te point under considerticn
x axis. In
tus, )1fx
3X
at the same
Applying the
this case,
view of the adherence n4s tnA di
0 at the moInt on
point.
formula of
te boundary
On the
r,arallel to the
iN
()dm
fc=ulae (7a) and we obtain(J,
theory of plane state of Etress
t 20
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=0
a
=0-
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we obtain the value of 0
tg20=---- 00, 9-- -4- 5'.
Thus, at all toints where the deformed edi is in contact with the
apparfic the tr.-
-4'n-- form an
angle of 45�
with the periphery cf the ap;.,aratus, and the periphery itself constitutes a
trajectory of the zaximum shearing stresses. 2te utilization of this proper-
ty of the state of stress along the boundary facilitates considerably in.many
cases the reduction of experimental data and increases the precision of the
results. In planning the reduction of exterimental data, it often proves to
be expedient to compute first of all the state of stress along the boundary.
For the points on the boundary the state of stress can be conveniently ex-
pressed by the stress normal to the boundary and the shearing stress acting
in f!ho ToPne tavIzent to the boundary= This c!..ara?�ing stress being a mp-eimum
shearing stress is computed by formula (2&').
Let us show how the normal stress along tne boundary may be calculated
integrating alonz the boundary. In order to do this, let us express all
11,1.111.
the components of the stress tensors in terms of the average stress, maximum
shearing stress and the angle&
max ' COS 20;
ay a --- T. m a x " iLigC7;
Substituting- expressions (
Txy max � stn 29.
1 in
) and (112) into (103), we obtain
do of) at max
2t,,, � sin 29 � � +cos 2A.
ox
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a
�
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Ot3 0-c
+ 2max� Cos 20 + � sin 20 .13.
ty
(113)
If the x axis is chosen as a line parallel to the tangent to the boundary
0
= - cos = 0,
at the point under zonsideration, then 45, sin 2[7 in -
the derivative with resect to x coincides with the derivative along the arc
of the boundary, the derivative with respect to y will be equal to the de.
rivative along"- normal to the boundary, and
1,11c will be the curvature tr%
dx
the boundary (Figure 64).
4.16Lkc.,ch v�e, into account and expressing the curvature in
terms of the radius of curvature, we shall obtain
, crzmax Cl Tmax
= 0.
as aNr (114)
N
\\ASN Periphery of apparatus
11
X
ll Figure 64
..ere
Noliz?�
lki
ad is a derivatf-;e cf the :_ean stress along the arc of the periphery,
ds
is a deriva:ive of te maximum shearing stress in tne direction of
rs
.
aiN
Thus, on the cf (114) mean stres3 at -- bc-,;nary points may
I
( 2ma
r
T x
sn atm.x
sT,
t1-.e
the normal to the Ierihery; r is tne radius of curvature of tne periphery. 11
._-.
131
.1/.6
Direction of normal
/f1
02
be computed by the fcrimula
c(s) =.s(s0
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1111111111111111111111J
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Taking into account (28), formula (115) may be rewritten in the form
3
2n an
= a(.50) to.r(�aN
� �)
r
it)
Keeping in mind the fact that the mean
the boundary and utilizing (116), we obtain
boundary. As a starting
take any point where the
Point for purposes
stress is equal to that normal to
at once the stress normal to the
of calculations (s - so) we
periphery of the model in
contact with the apLaratus
becomes a free boundary. Often we can assume without
that at such points Cr- 0.
The calculation of stresses along the boundary utilizing
orly mnr.r.gar.inhlA orrnr
the relation-
ship (116) offers a series of important advantages. The most important of
these is the fact that we do not utilize the isoclinics and in doing so we
remove the basic source of errors, while the calculations are considerably
simplified. In those cases where the boundary is a straight line, the calcu-
lations become particularly simple. As a rule, the fringe order is higher
near the boundary than in the center portion of the model in the study of
various processes. Because of this the error in the determination of the
fringe order near the boundary will be less, and this in the final analysis
will improve the precision of the results. Besides that, the state of ntress
along the boundary often is a matter of basic interest since it determines
the loads exerted against the deforming apparatus.
Among the disadvantages of the calculations by the method of integration
along the boundary we must consider the fact that the derivative must
aN
be computed according to the values of n at points lying only to one side of
the line along which the integration is carried out. It is useful to note
from the derivation of formula (116), that it permits us to
stress by the method of integration along any trajectory of
ing stress and thus that it may be utilized in all cases in
132
compute the
tne maximum
which these
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mean
shear-
1
STAT
�
m
�
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trajectories are known. When it is necessary to compute the
tensor
4.0 4ft ,w
over the entire field of the specimen the calculations may be carried out by
the method of the difference of tangential stresses utilizing as starting
points the nearest points on the boundary.
More precise results may be obtained in the following manner. Since the
mean stress satisfies Laplace's equation, it iay be calculated over the entire
field on the basis of the values of stresses at the boundary. In order to do
this we may utilize any approximate method of calculation, for instance the
grid method.
The mean stress, the maximum shearing stress and the value of the angle
eobtained on the basis of isoclinics, determine completely the state of
stress.
133
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STAT
�
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nrratrw7171.n v
MODELS FOR ST= CF 1-RCCESSES CF 2kBRICATION BY PRESSURE
II222PAnz-
At the present time the stamping process is used widely in the national
economy because of its higa productivity, reduction of the waste
cka.a.0
high quality of .the product. This process is successfully employed both in
the fabrication of metal parts by pressure and in the fabrication of plastics.
The flow of metal under the conditions of the stamping process presents
a complex process of plastic deformation. In a number of cases there occurs
during stamping a combination of compression, indentation, and flow. More-
over, under conditions of stamping the process of filling the die is not a
stationary one and tne
product.
The study of
the
rate of deformation varies for various
parts of. the
relationships governing the flow of metal during stamp-
ing is intimately related to the solution of the general problem of stress
distributions and deformations under sharply defined conditions of three-
dimensional compression.
In order to solve the indicated problem the application of the method
of mathematical analysis based on modern mechanical-mathematical concepts of
the theory of plasticity proves to be inadequate. The incomplete knowledge
of the boundary conditions and an entire series of distinctly specific periods
of transition which accompany the stamping process make it difficult to obtain
reliable quantitative results. Therefore, it is expedient to employ an
134
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experimental method of studying stresses In order to solve the indicated
problem the method of photoplasticity.
Let us consider the character of the state of stress in the models un-
dergoing deformation which
are subjected
le�LI
GA&
prepared from
optically sensitive resin and
to deformation in dies of simple and complex forp. In Figure
65 there are shown various stages of the stamping of the model in a die of a
simple form.
Figure 65. Fringe pattern observed
in a model 40 mm thick being deformed
in a die of simple form: a -- initial
stage; b -- stage corresponding to
contact with the side walls of the
die; c stage corresponding to flow
into a flash vent; d stage corres-
ponding to filling of the flash vent.
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In the 4"4t4061 stage of compression when the contact surfaces are lubri-
cated, we observe a uniform state of stress in the model;
SIV1
effect of producing a uniform coloring of the entire field of view of the
model. in silbsfacrAnt loading the uniform coloring changes 2-3 tires after
which we begin to observe individual fringes which indicate the origin of in
homogeneities in the state of stress of the deformed model. This inhomogeneity
in the state of stress appears firSt at the surface of contact. It is caused
by tne friction between the material and the model and the surface of the
apparatus.
Since the lubricant is forced out of the space between the contact sur-
faces during the process of compression of the model, the material bens to
adhere to the surface of the apparatus.
The displacement of the model material
along the contact surface o the
presence of large surface frictional forces. The layer of the material which
adheres to the surface of the apparatus retards the adjacent layers and in
this manner forms zones of low mobility. This factor (retardation of layers
I
I
situated near the contact surface of the apparatus) aids in intensifying the
spread (broadening) of the middle layers of the model in the direction of
least resistance. As a result the model acquires a barrel-like shape.
It can be seen from the examination of the given illustrations that in
the initial stage of the stamping process the lines of equal maximum shearing
stresses are arranged the same as in the case of a simple compression. of a I
deformation increases the region of uniform state of stress in the middle
portion of the model gradually decreases. The number of fringes pzradually
increases. The reduction in the height of the model causes an increase of the I
model whose height is considerably greater than its width. As the degree of
Iend surface at the expense of the material extruded at the side surfaces of
comes in contact with the interior
of the apparatus is retarded considerably due
the model.
The model material flowing sidewise
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side walls of tae die. As the contact is made with the side surfaces of the
die, there .occurs a redistribution of stresses in the interior of the model
which reveals itself by the change of the fringe pattern (Figure 65b). The
new fringes which originate at the places of contact with the side walls
along with fringes previously produced form a new fringe pattern. In this
instance the highest fringe order is observed at the point of entry into the
flash vent.
In these places the maximum shearing stresses are found. It is
precisely in these places where the flow of the excess material of 611C' muuel.
into the opening of the flash vent takes place.
Owing to the adherence of the model material to the contact surfaces of
the apparatus and the resistance of the side walls of the die to the flow of
the material there is created a more sharply defined three-dimensional state
of stress. As a result, at this stage of the stamping process the material
begins to flow Into the portions of the die which are difficult to fill (cor-
ners) and simultaneously into the opening of the flash vent. This is pri-
marily aided by the resistance to the flow of the material into the opening
of the flash vent. If the -,'ter4-1 enccunterz a greater resistance to flow
at the opening into the flash vent than into other portions of the die, then
as a result we shall have a more intensive filling of the still incompletely
filled portions of the die. As the unoccupied portions of the Ai4' are filled,
the fringes also disappear. As the entire interior of the die is filled the
state of stress in the upper and lower Portions of the model becomes more
uniform and a uniform coloring is
.1
vinich
indicates the rresence of
a uniform state of stress (Figure 65d).
The dimensions of the flash veLt must be such that it offers .;.."ficient
resistarce to the flow of the material into the vent aLa tnat it assures by
this the c^ri-lete filling of
t1'141.
4116
-rs
of effort. Excessive resistance tr
4.77,
r...nditure
flow cf material into the vent above
the necessary value will cause an excessive expenditure of energy ars and an
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unnecessary increase in the deforming force. Therefore, the regulation of
the flow of the material in the die during stamping may be achieved by varying
the width of the flash vent and the thickness of the flash ribbon. The values
of these trinnfitian (parAmptprR) have a substantial effect on the magnitude of
the pressure causing the flow of the material into the interior of the die.
We shall observe either complete or incomplete filling of the interior of the
die depending upon tne relationships of these quantities.
Figure 66.
In the final stage of the stamping process we observe a uniform darkening
of the field of view in the upper and lower portions of the model (Figure 66).
This points to the presence of a uniform state of stress in these portions of
the volume of the model. The middle portion of the model has a uniformly
light coloring which indicates that in this portion of the model we have equal
maximum shearing stresses of small magnitude. The portions of the model situ-
ated near the entry to the flash vent are subjected to the most nonuniform
state of stress. In these places we observe concentration of stresses and
deformations..
Figure 670
138
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We may draw the very same conclusion from examination of Figure 67, in
which there is shown the distribution of trajectories of tangential stresses
(characteristics) throughout the volume of the model obtained by illuminating
it in nonpolarized light.
The qritiT�A volume of the model in one of the last stages of a stamping
process is divided as it were into three zones. The first zone corresponding
to the concentrated nonuniform state of stress is situated near the openings
of the flash vent. In this zone are situated the most closely spaced trajec-
tories of tangential stresses. The second zone occupies the middle (central)
portion of the model. In its external appearance it has a lens-like form. In
this zone we observe a more uniform state of stress (Figure 66). The third
zone presents itself as an envelope in which is included the lens-like zone
of the more uniform state of struso. Plastic deformation is absent in the
third zone and there exists In it a uniform and homogeneous state of stress
(hydrostatic pressure). In this zone the deviation of the state of stress may
be set equal to zero.
We can verify the above by considering systems of isoclinics observed in
linearly polarized light. As we rotate the plane of polarization from 0 to
900 we find that the completely darkened regions (isotropic) are the upper
and lower portions of the model in which we observe a uniform state of stress
(Figure 68).
Figure 68. Isoclinics observed " one of the fina.l. stages of
mu
00 isoclinic; b 450.
We find still another confirmation of the results obtained with resin
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models from the examination of deformed models of lead having an engraved co-
ordinate grid 1227 (Figure 69).
�
Figure 69. A lead model with a coordinate grid deformed in stamping (from
E. S. Bogdanov).
Let us consider the fringe pattern which accompanies the process of
stamping for a model in the presence of a slight excess of the amount of
material needed to fill the mold, that is, when the model fills the interior
of the die almost completely. In this case the fringes are observed only at
the entry into the flash vent (first zone). In the second zone the stresses
have a small value. This is evident from examination of Figure 70 in which
there are shown two stages of the flow of the excess part of the material into
the flash vent.
The volume of the model was only sufficient to fill the interior of the
die and the flash vent (stamping with a small flash). The field of the model
and the fringes are observed to grow lighter only in a limited zone near
the entry to the flash vent. In the remaining portion of the model
1. U.
UAL=
ing remains uniform which indicates a more homogeneous state of stress, This
shows that only those portions of the model which are situated near the entry
to the flash vent and the center portion of the model are deformed. The flow
into the flash vent is due only to these regions, that is, only the first and
the second zones feed the flash.
The above is also confirmed by the system of isoclinics. In Figure 71
there is shown the field of isoclinics at the stage corresponding to filling
of the flash vent. The isotropic regions are observed in the upper and lower
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Figure 70. Fringe pattern observed in stamping of a model having a small
excess of required material: a -- contact of model material with the edges
of the opening of the flash vent; b -- the stage corresponding to complete
filling of the flash vent.
portions of the model (third zone). In the middle portion of the model (in
the plane of separation of the die) we observe an increase in the light inten-
sity as the angle of the plane of polarization rotates from 30 to 600, which
indicates the presence of maximum shearing stresses of small magnitude.
Figure 71. Isoclinics at the stage corresponding to the filling of the flash
vent: a -- isoclinic of 00 b 150; c 300; d
The very same results are obtained when we observe models of lead having
an engraved coordinate grid. The distortion of the lines of the coordinate
grid (for a small excess of required material) is observed only at the entry
into the flash vent (Figure 72). In the remaining portions of the volume of
the forging a coordinate grid remains practically unchanged (from
danov and M. I. Kalachev).
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E.
Se Bog-
STA
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Figure 72.
In the final stage of the stamping process the entire field of the model
presents an isotropic region (Figure 73). The rotation of the plane of polari-
zation changes the character of the disposition of the isoclinics only near
the entry to the flash vent and within the vent itself. In these places there
is concentrated
�
the final stage of the stamping process a plastic flow and
there is observed an inhomogeneous state of stress. In the isotropic region,
however, which now covers the entire field of the model, uniform normal stres-
ses are acting while the shearing stresses are equal to zero.
Figure 73: Isoclinics observed at the final stage of the stamping process:
a -- 0 isoclinic; b 4,
In Figure 74 there is a
-h-r-
i"of isoclinic& obtained for the stage
field
corresponding to the filling of the interior of the die as indicated in Figure
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65d. The isoclinics are given for increments of 15� both for the interior of
the die and the flash vent.
Figure 74e 1
Since we have the pattern of isoclinics, it is possible to construct a
sample grid of trajectories of the principal normal stre esboth for the in
tenor of the die and the flash vents (Figure 75). In Figure 76 there are
. I
trailik*AmIrl,
shown magnified views of the iso.
clinics observed at the flash vent
and the trajectories of thd principal
41162074111
Valf..77111/11MMIPP.�---
normal stresses constructed on the 1
basis of these lines. eang6
0/
As can be seen from examination 1
I
their direction is observed at he entry '-
region of the first zone where concentration of stresses exists.
Thus, the exa ination of the state of stress in the final stages of the I
1. " flash vent, that is in the
of isoclinics and also the trajec-
tories of the principal normal
stresses, the most abrupt change In
Figure 75. Grid of trajectories of
principal normal stresses in die of
simple shape during stamping process.
stamping process enables us to draw the following conclusions:
(a) There is observed throughout the volume of ti
geneous btate of stress during the stamping process,
forging an inhomo-
Eij
(b) The intensity of pressure is uniformly distributed over the contact
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surface of the main body of the forging,
� (c) The pressure at the last ctage of
trAly
stamping process does not de-
pend upon the form of the forging, but upon the geometry of the flash vent and
primarily upon its height.
Let us consider the state of stress of tat portion of the model which
flows into the vent.
Figure 76.
As the excess material of the model flows into the flash vent there is
observed in it a very definite fringe pattern, and therefore a very definite
state of stress. The character of the state of stress in the flash vent de-
pends on how completely the interior of the die is filled. If the material
enters only the flash vent we observe the fringe pattern shown in Figure 77,
while in the presence of other openings in the interior of the die into which
the model material may flow, we observe at the flash vent the fringe pattern
shown in Figure 78.
As can be seen from Figure 77, as the material flows into the opening of
the flash vent, the fringes originate at the corners of the opening. As they
are developed further, they go round the corners and show a tendency to line
up along the boundaries of the flash vent. In the front part of the flow vent
the fringes turn toward the boundary of the vent. In its middle portion there
is observed a dark neutral fringe which divides the general fringe pattern into
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two symmetrical parts. At the outer end of the vent the neutral fringe be-
comes wider. Along this fringe, the magnitude of the MaXiMUM shearing
stresses in the plane of the ray of light is equal to zero. These stresses
increase as we proceed from the neutral fringe to the bourdary of the flash
vent and attain their maximum value at the boundary. The value of the maximum
shearing stresses is determined by the order of the fringes, and the order is
reckoned from the neutral fringe. As the load is removed, the change in the
fringe pattern proceeds in the opposite direction.
Figure 77.
In the case of Figure 78 the fringes are arranged in the same manner
for a wide model being compressed between two moving plates. At the same time
the fringe pattern is symmetrical with respect to a certain vertical plane
which we may call a neutral plane. This shows that as the boundaries of the
flash vent approach each other, there is a period when the resistance to flow
into the flash vent is blocked, and this curtails the flow of the material
from the interior of the die into the flash vent.
Figure 78.
The Just-atilt during the stamping at the material ceases to
Ifts.^^..mm
flow from the interior of the die into the flash vent is called by the authors
the instant of locking of the material within the die. The conditions of the
state of deformation which determine this instant are called the conditions
of locking. From this moment the material flows only into the vacant spaces
in the interior of the die and thus assures that it becomes completely filled.
The observed neutral plane may be situated at various portions of the
flash vent. Its location depends upon the dimensions of the flash vent and
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the amount of excess material being fed into it. For greater heights of the
flash vent (earlier stage) the neutral plane is nearer its opening. As the
height is reduced (other things being equal), the neutral plane is shifted
toward the interiors but b-.4 a distance not exceeding half of the width of
the vent.
The neutral plane in the flash vent is observed only 1110 to the stage
when the interior of the die is completely filled. As soon as all its re-
cesses are filled, the fringe pattern at the flash vent changes. The fringes
begin to arrange themselves along the boundaries of the vent. This shows that
the interior of the die is filled and that the excess of .the material flows
only through the opening of the flash vent. One fringe pattern is trans-
formed gradually into the other.
The problem of determination of the moment of locking or locating the
neutral surface (critical surface in the terminology of I. A. Tarnovsky and
0. A. Ganazo) was studied by a number of investigators g9 and 307. They
determined the moment of locking by comparing the amounts of material flowing
into a flash vent and into auxiliary recesses. The moment at which the amount
of material flowing into the flash vent decreases while the amount flowing
into an auxiliary recess increases, was considered the moment of locking.
This determination required the performance of a large number of experiments.
The location of the neutral plane could not be determined at the same time.
The application of the method of photoplasticity furnishes visual data for
determining the moment of locking and the location of the neutral surface.
The state of stress in the billet undergoing stamping depends exclusively
upon the geometry of the branches of the die. We shall show this in the fol-
lowing examples.
In Figure 79 one can see the character of the origin and distribution of
fringes when a model of rectangular form is being deformed In a die having
openings in the upper and lower parts. The width of the entry into the
146
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openings is equal to 8 mm. The height of the flash vent at the surface of
contact of the two halves of the die along the plane of separation is equal
to 2.6 mm.
Even in the initial stage of the plastic deformation one can determine
the sources of the fringes situated near the corners of the openings. As the
process develops the fringes join in the middle portion of the model forming
a system of lines which is determined by the form of the given deforming ap-
paratus (Figure 79a). In the side portions of the model we observe a uniform
coloring which indicates a state of uniform stress caused by the compression
of these portions of the model. As the load increases the fringes become
more concentrated. A change in the height of the model causes a contact of
its sides with the walls of the interior of the die as the process develops.
As a result there occurs a redistribution of stresses within the model which
fact leads to the change of the fringe pattern (Figure 79b, c). The non-
uniformity of stress distribution becomes more pronounced. Due to the greater
resistance to flow 'into the flash vent the flow of the material of the model
is channeled into the supplementary openings.
The fringe pattern observed at all stages of the stamping process near
the upper opening is a mirror image of the fringe pattern observed at the
lower opening. The flow of the model substarce into the indicated openings
proceeds simultaneously. The amount of the subRtanetb_ which flows
ir.t^ the
upper and lower.openings.of the die is identical. This is valid only in the
case where the die is symmetrical with respect to the plane of ser-aration and
the openings have the sane form and dimensions and the surfaces have the same
degree of finish. In the oplosite case the observed optical patterns will
not coincide completely.
The material flows into the openings from the center portion of the
model. At the openings of the flash vents the flow is slight and the material
undergoes almost no deformation. This may be seen from examination of
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Figure 790 Fringes observed in a model
being deformed in a die with two supple-
mentary openings; a -- initial stage;
b -- stage corresponding to contact with
side walla of the interior of the die;
c -- stage corresponding to flow into
the supplementary openings and the flash
vent.
isoclinice observed for increments of 150 and shown in Figure 80. As the plane
of polarization is rotated from 0 to 90� completely darkened regions are ob-
served to the left and right of the zone situated along the
openings.
MUs eporkma+mai, Able+Ani.
J.-Lava
mw4m of the
of illumination is observed at the middle
portion and the least at the openings of the flash vent. It follows then that
the flow of material into the upper and lower openings proceeds from the middle
portion of the model.
tli
Declassified
in Part - Sanitized Copy Approved for Release
Figure 80. Isoclinics observed in a
model being deformed in a die with two
supplementary openings: a -- Ou iso-
clinic; b 150; c -- 45�.
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This is also confirmed by experiments gg carried out with metal models
having a grid of coordinates (Figure 81). The specimen had an initial form
shown in Figure 79c. After the test the grid of coordinates was deformed
only at the upper and lower openings and at the flash vent.
In the final stage of the stamping process when the openings are complete-
ly filled there is developed throughout
the volume of the billet a state of
uniform stress. The fringes disappear
over the entire field of view of the
model and are retained only at the
Figure 81. A lead model with a grid flash vent.
of coordinates undergoing deformation
in a die having two supplementary In Figure 82 there in chown the
openings (from E. S. Bogdanov).
character of the origin and distribu-
tion of fringes observed in the stamping of a gear-like model (stamping ac-
companied by indentation).
Even in the initial stage of the plastic deformation fringes develop
along the points of the projecting portions of the die (Figure 82a), while
the number of fringes increases with an increase in load. We observe two
broad, indistinct, dark fringes to the left and right of the middle portion
of the model. These are neutral fringes in which the value of the maximum
shearing stresses is zero. These fringes form the boundary between the region
of compression and the region of tension. The peripheral portions of the model
are light. They are subjected to tensile stresses of small magnitude. With
further increase in load the tensile stresses attain considerable magnitudes.
In Figure 82h can be seen dark fringes of the first order which are situated
along the unloaded periphery of the model.
As the load increases the region of uniform coloring in the middle por-
tion of the model decreases. The concentration of fringes in this instance
occurs until the sides of the model come in contact with the walls of the
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interior of the die. Then a redistribution of stresses within the model takes
place and the fringe pattern covering the field of the model changes.
4.1%a
ULAV
given case this leads to still greater nonuniformity in the distribution of
stresses which becomes intensified as the process develops (Figure 82c). As
the interior of the die is filled the state of stress becomes more uniform
and the fringes disappear.
The field of the model acquires
gs.ritnifnr.m enlnrinz
and the stresses become more uniform. The fringes are preserved only near the
openings of the flash vent where the flow of the material still continues.
Figure 82. Fringe pattern for a resin model during a stamping process accom-
panied by indentation: a -- initial stage; b -- appearance of fringes of the
first order along the free boundary of the model corresponding to tensile
stresses; c -- disposition of fringes at the stage corresponding to flow into
a flash vent (nonuniformity of the stress distribution became intensified);
d -- final stage of the stamping process.
For a given geometry of the branches of the die certain
lifteher4^1.1=
in which the shearing stresses are zero. These are isotropic points which are
subjected to either a triaxial compression or tension. Formation of such
points or distinct regions is observed in dies of a complex form. Such points
may be seen in Figure 79 (a, bl c) and 82c (in the given ease these are points
of triavisa compression).
In all the cases considered for models undergoing stamping in dies of
simple and complex form, each configuration of the die gives rise to its own
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characteristic state of stress. A complete picture of the stress distribution
may be studied at various stages of the plastic process of change of form. In
each individual case we may visually observe points of stress concentration at
which the wear of the dies is accelerated. This gives to both the technolo-
gist and the designer of the dies the needed
material for
u:ac
proper rational form of the billet and the development of the most rational
geometry of the Interior of the die.
From the above it is apparent that a state of uniform stress develops in
the entire villums. of
the billet in the last stages of its processing indepen-
dently of the geometry of the billet. The smaller the volume of the forging
billets the more uniform is the deformation of the billet in various stages
of its fabrication.
The examination of the isoclinics in the final stage of the stamping pro-
cess shows that the less uniform state of stress is replaced by. a more uniform
and homogeneous state of stress (hydrostatic
in
the greater palt of
the volume of the model. The only exceptions are the regions in the vicinity
of the entries to the flash vents.
The investigations conducted by the authors show that the primary atten-
tion in designing dies should be. paid to the geometry of the trough of the
flash vent. In the -region of the flash vent there is concentrated a plastic
flow from the beginning to the end of the stamping process. In these places
there occurs a high concentration of stresses and deformations which may cause
an Accelerated wear of the apparatus.
As can be seen from the exteriments, the extent of filling of the interior
of the die depends exclusively on the conditions of locking of material in the
flash vent. It is determined by the dimensions (parameters) of the flash vent
and the conditions of external friction. The conditions which determine the
locking of the material in the flash vent determine the required stamping force
the power parameters of the press, and the stability of the die itself.
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The picture of the states of stress and deformation in the stamping pro-
cess obtained by the method of photoplasticity coincides completely with the
picture which we obtain on the basis of the study of the state of deformation
by the method of a coordinate grid engraved upon metal models.
the coordinate grid
of
deformations
4.%
in n iCt�
,. Jenthe
4� z5,cul�Lma
The method of
character of the distribution
model undergoing stamping while the method of pho-
torlasticity confirrLs these results, substantially supplements them, and in-
troduces a certain degree of visual clarity.
ROW let us pass to the consideration of the numerical values of normal
and shearing stresses during the stamping process.
Since in the present case we are solving a problem of a ;lane state
of
stress, we may, therefore, utilize the concept of forces distributed along a
line instead of forces distributed over a surface. Therefore, the magnitudes
Cfx'
stress tensor
Cr , and T will represent those values of the components of the
xy
when acting upon a model 1 cm thick create an optical
effect corresponding to reality.
. In the experiment we
of T and the angle e
max
stress with the axis of x.
to determine the components
out by the method described
shall obtain at each point of the model the value
formed by the trajectory of the principal normal
Having the values of these quantities enables us
of the- stress tensor. Their calculation is carried
in Chapter IV 5ee formulae (102), (105), (10617.
The experimental values of the fringe order n and
clinics (3 which define' x , 1 and I at various horizontal sections, are
y xy
taken froangures 65d and 74 which corresrond to one and the same stage of
stamping process and loading.
foil=
rarameters of the
In Figure 83 are shown distributions of Cf
horizontal sections of Figure 65d.
and 1". for one of the
xy
From examination of the graph it is clear
that the stressesx and acting within the model are compressive stresses,
2'OP in absolute value. The maximum dispersion between the values
with Cr
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Figure 83. Distribution of C,
r 01' and
x rxy. for the section X-X.
Crx and 01y occurs at the entry to the flash vent that is, In the
zone.
In a certain portion of the flash vent
greater part of the horizontal section (second zone) the
first
r 71 0. For the
xy
values of qx and
cr are different, but the difference has almost a constant value. The dif-
ference
-
.11.1
the numerical values of Cr
e<
and Li varies approximately from 5.3
kg/cm in the region of transition from the first zone to the
kg/cm along the vertical axis of symmetry. It is this T11.111
determines the almost constant coloring in a greater part
the second zone.
Thus, it follows from the discussion that the model
interior of the die .is in a state of nonuniform triaxial
Let us determine the summation of the forces along
83). In order to do this let us sum tel.) all the values o
the given section and multiply the value obtained by the
second to 2.8
difference which
of the volume of
material within the
compression.
section XX (Figure
f at 30 points of
length of the inter-
val Ax between the points under consideration.- In our case
= 0.1 cm.
Summing up the values of cr at the sect iGn X-X, we obtain a magnitude equal
to 895.7 kg. Then the force P = 895.7 x 0.1 = 89.57 Z9.6 kg. The applied
force for the stage of the stamping process indicated
kg. Thus, the value of t.-,e force r obtained frc!:.
in &he drawing
calculat.Lol.s was
the value of aptlied force by 15.4 kg, a difference of about
The observed discrepancy may be explained by the effect
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153
was 105
less than
1
11
1
percent.
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stresses
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actinic in the plane of the glass plates which do notreveal themselves opti-
cally. Therefore, the optical pattern of the lines of equal maximum shearing
stresses observed in Figure 65d corresponds to the fringe pattern for a load
of 89.6 kg for a plane state of stress.
Let us determine what factor influences the precision of the determina-
tion of 6, and le to the greatest extent. Since we utilize fringes
xy
and the parameter of the isoclinics for their determination it is natural
that the precision of determination of 01x and 7- will depend upon how 11
7cY
accurately we determine at each point of a given section the fringe orders n
and the parameter of the isoclinic 9. However the effect of these quanti-
ties may vary.
Let the true value of the difference between the principal normal stres-
ses at a given point of the model be q and the error in its determination be
q Then
.
In an analogous manner,
isoclinic and 44
tie is the value
if eis the true value of the parameter of an
obtained from an experiment, then
48.
eg.==
The shearing stress determined from experimental data will be
and the true value
(q ad')
71 sin (20
q
txy sin 20.
IN*
PO
how'
Let us find an expression for the error in the determination of the
shearing stresses:
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: CIA-RDP81-01043R003900240001-8
STAT
���
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q -
TAW - q
- sin944 --
9 sin (20 -- 2.4)
9
2
q sin28 q -2 Aq I sin9vi c(WIA*41- cos9A0 s1n210).
�
A Jel A
T1cng into ac count the fact that tne quantities LI OD' and L-1 q are small,
the latter exl:ression may be simplified and then
11.7
yry Siri.90 +.1/.114
..
The relative error in the determination of the tangential stresses will
have the following value:
2Aq � sin2R 2q cos244
L.
2q sin2e q sin20
Simplifying the latter expression, we finally obtain
2.9 ctg20. �
The first term gives the value of the relative error in determination of
7e whicn depends upon the precision with which we determine the fringe or-
xy
der at a given point:
The second term defines the relationship of the relative error in the
determination of r to the parameter of isoclinic eand, therefore, to the
xy
angle of inclination of the principal normal stresses. From expression (117)
it is clear that the second term approaches zero as the
parameter
f
the iso-
clinic ar,rmacnes 450. In this case tne relative error in the determination
of 7" will depend exclusively upon tie T.recision of determination of the
xy
parameter yl of the isoclinic. This cf.rcuEstance demonstrates the necessity
of deter=ining quite accurately the -r.arar_eter of isoclinics near the values
of angles of 0 a 900 and Unf. :hotomranhinp- them (or sketching
them) every 3L50. In addition to that, in order to obtain more accurate
quantitative results it is necessary to select sections in such a manner as
155
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to reduce significantly the number of points situated near 0 and 90� iso
clinics and to exclude the possibility of having either isotropic points or
regions upon the selected section.
In Figure 84 there is shown a curve cf distribution of narmal stresses
for one or the vertical sections in the vicinity of the flash vent computed
in accordance with equation (108). As can be seen from the graph the values
of 0# along the upper part of the section have a constant value (third zone).
As we approach the opening of the r1A=;, vent
GJW.61.11G/
Ile 1.11G
valua
of ay increases and atains its maximum value at the level of the boundary
of the flash vent. The least value of this stress exists along the horizon-
tal axis of symmetry.
13
f2
if
10
9
8
7
6
5
4
3
21
25 26 2728 29
Figure 84. Character of distribution
of normal stresses along a vertical
section in the vicinity of the flash A typical
vent (Figure 65d).
In the combined drawing and Table
85 there are given the numerical values
of shearing stresses for one portion of
the volume of the die. The values of
er are different at all points of
xy
rection of
then they
the second and third zones.
the selected sections and the greatest
values are observed in the first zone;
gradually decline in the di-
curve of distribution of
maximum values of 7r in the volume of
xy
the entire model of resin subjected to deformation is shown in Figure 86.
Thus, it follows from the cited example that it is possible in principle
to determine at each point of the billet the values of the components of the
stress tensor. The precision with which these values can be deter=ined will
depend upon the accuracy of determination of n and t)
:Ipon the experimental technique.
the values n and e at the initial points along the free
which in turn depend
It is important to determine with precision
11
boundary. This is
156 STAT
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POOR ORIGINAL
9(4
d Jr
VA/7/1/ft
0 01.11 AIM _13% 1s30
a.r4
211 213 IVY
fri Of )0u ',OAS
43 125 '1:11,4 0.S0
n18
CIO
4. flirST"rerg"1346 237 149 122 ouoc 02ft
la 54. Sis -3 /2 453 44 OS) 0 70 loso {an
S35 702 $U .3M t71.4 1107 0$2
3ic In 71 fits%
3se pn 7.4 I S4 MS 10 SS 103 VI MI
O 0 W...0 010 ;0 10 10 *
I 2 3 4 5 6 ? 6 9 M
. section
7
6
Figure 85. Numerical values of Tx at points of one portion of the model
corresponding to the stage of the gtamping Process shown in_Figure 65d.
rendered difficult because of the appearance of a shadow along the periphery
of the curved surface of the flowing portion of the model material. The er-
ror in the determination of n and e along the free boundary will reveal
itself in all the calculations of the components of the stress ter-ors at all
interior points of the model. However, this circumstance may be avoided in
principle if we utilize an immersion fluid with an index of refraction equal
to the index of refraction of the resin from which the model was premared.
Then the boundary of the model will not be darkened and the values n ande
may be determined with precision.
2. Extrusion.
Basically the purpose of model- S TAT
ling the extrusion process is to study
the possibility of utilizing the
mE,tnod of photoplasticity for a quan-
titative determination of stresses.
In modelling the extrusion
Figure 86. Typical curve showing distri-
bution of maximum values of in a
model undergoing deformation (PiFure 65d).
157
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4. �
process the authors utilized a fiat container with a rectangular die. The
schematic diagram of the process and the basic dimensions of the container
are shown in Figure 87. The thickness of the layer in.the direction of the
incident light exceeded the w4f4t.-- of the containcr
LV A
-a �
_t
fact3r of threc, and
this assured sufficiently close approximation of the process
rmA,
The experiments were conducted with specimens of resin. The force was applied
to the pressure washer by the loading mechanism of the polariscope installa-
tion and the load was maintained at a constant value during each individual
experiment. The patterns of the f'-inges and isoclinics were photographed
with the light of the yellow line of mercury and the photograph was taken after
the fringes had become stable. The Process of stabilization of the optical
pattern usually took 10-15 seconds.
Figure 88 shows fringe patterns obtained for different forces applied to
the pressure washer. As can be seen from the given photographs, as the load
is doubled, each previously obtained fringe is replaced by a new one whose
�
order is twice as high, and in the space
between each pair.of fringes a new one
appears. S Consequently, the distribution
of stresses remains similar with a
change In load, and the stress at each
point changes in proportion to the
applied
le.ketA
These data confirm the
previous assertion that in the case
Figure 87. Schematic diagram of the where the viscosity of the entir-AT model
ST
is constant the distribution of stresses
in it is determined only by the geometric characteristics of the process.
Figure 89 shows fringe patterns obtained at various stages of extrusion
with a force of 180 kg applied to the pressure washer. As we examine the
photographs we first of all note the fact that the position of the fringes in
modelling of an extrusion process.
158
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043R003900240001-8
401
C.
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POOR ORIGINAL
It�
11111111maskakimmonanammimismhe
#0=7.701-montn,
N)(C-
.e
�-�
'.."4 �
Figure 88.
trusion:
90 kg; c
Wirre
Fringe' patterns during ex
for a load of 45 kg; b
180 kg.
aON OM
�
STAT
Figure 89. Fringe patterns at various stages of extrusion modelling.
159
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the field of the container exclusive of the region in the vicinity of the die
is similar to the distribution of frin-
ges in the field of the channel. V44=
is particularly noticeable in those
stages of the process where the height
of the column of the material in the
container exceeds its width by a factor
�.r
Figure 90. View of a model after its
height in the container has been re-
duced in half by extrusion.
m^ima_
InV di VI MIO AI W. WOW WO W
Tr, +hia ^Jana +ha frin�
gee in the middle portion of the con-
tainer are disposed parallel to its
axis and are uniformly spaced, that is,
they are arranged precisoly in the same manner as for the case of a plane flow
in the channel.
This can be seen from the comparison of Figures 51 and 47. The fringe
pattern at the die opening and in its vicinity and also near the pressure
washer depends but slightly upon the height of the material in the container.
For practical purposes, the pattern only begins to change when the distance
between the pressure washer and the die becomes less than the width of the
container. The points of concentration of shearing stresses and, consequently,
of deformations are the corners formed by the pressure washer and the walls of
the container and also the corners at the entry to the die opening. The dis-
tribution of stresses near the corners of the pressure washer corresponds to
that obtained in the solution of the problem 2 of section 4, Chapter IV. The
photographs reveal the presence of several isotropic lines and points. Tiirr
following points prove to be isotropic: the point at the center portion of
the die opening, the corners formed by the die and the container, and the
point upon the axis of symmetry passing through the pressure washer. The
4IM fringes which radiate from the corners of the pressure washer and which ter-
minate at some distance from these corners also represent isotropic regions
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POOR ORIGINAL
or fringes of the order zero. In those cases where the height of .the column
41 of material in the container exceeds its width by a factor of more than one
and one-half; an isotropic line is also observed along the axis of symmetry
in the center portion of the container. This line disappears in subsequent
^4% -4--1*-1Aa
%a *of -a. 4 � NO Nor .0% d %oft ���� de. mkp, � �
ni'OCeSAe
We may obtain a picture of the lines of flow by examining Figure 90.
This photograph was obtained with ordinary light after the height of the spe-
cimen
4 v.
A.S.&
the container was reduced in half by a process of extrusion.
Now let us pass to consideration of quantitative data on the state of
stress. We shall cite certain calculations for the stage of pressure which
corresponds to the fringe pattern shown in Figure 91.
The isoclinics corresponding to this case are shown in Figure
C)
./1.. �
At first let us compute the state of stress in the horizontal section
bb' which passes through the middle of the die. It this cross section there
is present an isotropic point. According to the data obtained from the field
of isoclinics, at this sectionA= 450 at every point.
- The value 7;cy computed
from formula (101) is given in Figure 93.
1111111111111111V"Matmemomme
STAT
Figure 91. Fringe pattern utilized in Figure 92. Field of isoclinics obtained
the calculations given above, experimentally in modelling an extrusion
411 process.
In order to compute normal stresses let us utilize formula (106) which
161
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POOR ORIGINAL
we shall rewrite in the form
At the points of the section under consideration the tangents to the fringes
are parallel to the y axis and, consequently,2h. = 0. In view of the fart
-
that at this section cos269 -4 0 and Cjx Y.
Thus, along the section bb' the integral in the formula (118) is equal
to zero and C) = CI; constantly. The value of may be obtained from the
condition of equilibrium of the lower part of the model bounded by the sec-
tion bb'. This portion of the model is subjected to the action of shearing
forces in the direction of the y axis along the side walls of the channel and
to the action of normal forces at the section bb' from the direction of the
upper portion of the model. The former of the indicated forces are defined
by the expression
Here the integration is carried out
along the walls of the die opening from
its outer end to the section under con-
sideration. The value of the integral
may be computed on the basis of experi-
/
W
44
A
lir
mental data by means of numerica'sTmate-
gration. The normal force at the section
bb' will be given by the following ex-
Figure. 93. Shearing stresses at thepression in view of the fact that Cr
section bb' (Figure 91).
is constant:
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Ft= 203y,
where c is half of the width of the die opening.
Since F + F2 = 0 it follows that
1
Fl
er
The value N, computed in this fashion at the section bb' is equal to
20.6 kg/cm.
Further, it is possible to compute the stresses along the axis of sym-
metry (y axis). In order to do this, it is expedient to utilize a formula
analogous to (118). Taking into account the fact that along the axis' of sym-
metry t) = 0, such a formula has the form
a
= cy(xosy 0) � 2�co S n dx dy.
Yo
We may take as an initial point (x0, y0) the point of intersection of
the axis of symmetry with the straight line bb', at which point the state of
stress is known from the preceding discussion. The value of 0" at the axis
of symmetry commuted by means of numarical integration is shown in the graph
of Figure 94.
The same figure also shows the value ofrf which in the given cane,
being the second principal stress, is computed by the formula
� n
ax 0--
STAT
Applyinr: the nethods described in Chapter TV and utilizing as initial
points the Toints of the axis of symmetry, it is possible to calculate the
stresses at various sections parallel to te x axis. The stresses in the
plane of the pressure washer (section aa') are shown in Figure 95. In the
middle portion of the container the state of stress is
163
to one defined
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POOR ORIGIN AL
by the expression
CI, 'ix -AY).
This can be easily demOrtni-rAtd by bRing the axnarimaintril data into
formula (118). Of particular interest is the distribution of pressures on
the walls of the container (Figure 96).
-
Ng.
20
T""m".4.44441.-
i
i
t
1
0 0.5 SC5
Ptrammum, 079 MUMVUUbi . CM
Legend: .a) kg/cm
b) distance from die,
in cm
c) pressure washer
Figure 94. Stresses along the axis of symmetry.
nonmexua
Ovora.
(42
44 48
C. pace/move ofn i CL CM.
Figure 95.
Legend: a) *kg/cm
b) half-width of
opening
c) Distance from axis
of symmetry, in cm
STAT
In Figure 97 there is shown the quantity 6; at the boundary between the
medium undergoing deformation and the pressure washer.
The calculations were
carried out by means of numerical integration from the axis of symmetry with
formula (118). Since at these points e = 45it follows that
.164
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POOR ORIGINAL
0 1.5 � IP 45
POOL1710.1444u9 esconikutp- c?4
and
Legend: a) kg/ cm
b) Distance from die,
c) Pressure washer
Figure 96. Normal stresses acting on walls of container.
zx(0) dn
cA
T
0 ay
0
This method of calculation cannot be used near the corners formed by the
pressure washer and the container since
:
:
1
!
4
egend:
) kg
) wall of
ontainer
Figure 97. Distribution of normal
stresses acting upon the pressure
washer.
the fringes at these regions cannot be .
obtained by theory.
However, as was already shown, the
distribution of fringes in these regions
is close to the one which follows from
the solution of problem 2 dealing with
the flow of a viscous medium in a corner.
Thus, further calculations ad analysis of the state of stress near the corners
STAT
between the container and th4. 7ressure washer may be carried out by utilizing
the solutions of this proble=
It must be noted that it follows from the solution considered that as we
approach the vertex of the corner all components of the stresses increase in
inverse proportion to the distance from the vertex, that is, they approach
165
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.41
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POOR ORIGINAL
�
infinity. Moreover, this law of distribution of stresses brings us to the
observation that the summation of the forces acting between the model and the
Pressure washer also approaches infinity. In other words, it follows that
the movement of the pressure washer with respect to the container cannot pro-
1 M1
%MY
finite ValOCitY This L..1111%.%,&4
�40
refuted by experiments.
In
actual practice this solution describes the process correctly only within
those limits where the coefficient of internal friction may be considered to
be constant. As was shown in Chapter IV, the relationship between the shear-
ing stresses and the rates of deformation was not a linear one. As the rate
of shearing strain increased the coefficient of internal friction decreased.
In addition td that, as the vertex of the corner is approached the temperature
of the material increases due to the energy of deformation which increases as
a result of the increase of the rate of deformation. (As we know, the energy
of deformation per unit of time is proportional to the square of the rate of
deformation for the case of viscous flow.) In its turn, the heating of the
material lowers its viscosity and this leads to the reduction of the stresses
compared with the values given by solution (74).
The presence of heating at the points of concentration of deformation
reveals itseif in another way. In all the photographs small zones near the
corners where the concentration of deformation is particularly large, are
4
dark.. The magnitude of these zones, other things being constant, varies de-
pending upon the diaphragm opening of the lens during the photographing. As
the lens diaphragm is opened the dimensions of these zones decrease as though
they contracted toward the vertex of the corner. This phenomenon can be ex-
plained in a logical manner if we take into account the fact that theTeMpera-
ture of the medium in these zones increases toward the vertex of the angle.
The medium becomes optically inhomogeneous. Its optical density decreases
as we approach the corner. The Incident rays of light passing through these
zones are deflected in the direction of the gradient of the optical density
166
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and in passing through may miss the objective lens. As the diaphragm openin
of the objective is increased the rays having lesser deflection pass through
it and the observed magnitude of the dark zone decreases. A typical path of
the rays passing through the model at the points of concentration of deforms
tion is shown in Figure 98. -
As applied to the case under consideration, the distribution stresses
near the corners of the pressure washer is satisfactorily described by for-
mulae (74) with the radii varying within the limits of 0.6 to 3 mm. The ab-
sence of data on the state of stress in the immediate vicinity of the corners
formed by the pressure washer and the container and on the resistance en-
countered by the pressure washer in its movement by virtue of the formation
of an envelope deprives us of the possibility of comparing the computed force
upon the-pressure washer with the force applied by the loading mechanism of
the machine.
toire
1
I'NUMMI",
06beiffnu60
1� fie;71..-72 p� �
1.2"fiCre --..�
.N..................
Figure 98.
Legend: a) lens diaphragm
b) dark zone
c) instrument
d) .model
e) light
3. Indentation of a Punch.
An analysis of the distribution of stresses caused by indentation of a
punch
was
carried out by means of models of rectangular form prepared from
resin and chloric silver. The models were tested in a flat container (Plgre
31) and the Mad was applied in increments of 20, 60 and 100 kg.
Ar
the rigid punch is pressed into the model Prepared from an elastic
material, we observe a system of fringes in the form of cirr-1...: which
through the
is known
boundary points of the punch. From the theory of elasticity it
that the pressure at the surface of contact 2or the boundary points
167
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'
�
must have theoretically infinite
le,0112Aft
From examination of Figure 23, it
is seen that considerable concentration of stresses is observed precisely at
the boundary points. Even after application of a small load to an elastic
body, plastic deformation near the corners of the
"" � LI fro 4a.
W164.4.7110,
This is confirmed by experiments carried out with specimens of mild steel
subjected to a concentrated
slip lines radiating
46.16 %Jilt
lociel /8/.
In this (tang' we observe
a system of
corners of the punch. In more brittle bodies,
destruction of the material also is initiated at these corners. This pheno-
menon is observed also for plastic material of the resin type.
First let us consider the fringe pattern observed when the punch is
pressed into a model made from resin for various ratios of width to height
of specimen, and also for various dimensions of the punch.
The character of the state of stress observed when a punch is pressed
into a body depends both upon the dimensions of the models and the dimensions
of the indenting instrument. This is revealed by the number and arrangement
of the fringes caused by the.
Aafeviomo4-4,16,% ^f 0010.�Ao�11
Nog. IraiL�G 444W41.41Va.40
Figure 99. Fringes observed in a model
having a ratio of height to width = 1
for loads: a -- 20 kg; b 60 kg;
100 kg.
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P60 R ORIGINAL
Figures 99. 100, 101 and 102 show fringe patterns observed during the
indentation by a punch 10 mm wide into models having ratios of height to
width of 1.0, 0.75, 0.5 and 0.25. The width of the model in all these cases
was equal to 40 mm.
At the initial moment of application of load there occur at the boundary
points of the die fringes having the form of closed curves of oval form.
major axis of these curves ("peacock spots") is directed at
proximately 500600 to the contact surface of the punch. As
The
an angle of ap-
those fringes are
displaced, those radiating from the corners of the punch merge in the center
portion of the
model and then diverge. One part of the fkiages is
shifted to-
ward the punch in the form of arcs convex away from the punch, and a second
�
part is shifted from the locus of merging fringes both to the left
and right
of the axis of symmetry. The left and right portions of the model which are
not loaded, are not stressed in the initial stage of the loading. These
regions are separated in the following stages from the remaining portion of
410 the model by a neutral fringe. After this even these portions of the model
0
are subjected to deformation. The sign of the stresses acting
gions is different from the sign of the stresses acting in the
Figure 100.
for a ratio
for a load:
169
in these re-
major part of
STAT
Fringes observed in a model
of height to width ap 0.75
20 kg; b 60 kg;
100 kg.
a --
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�
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POOR OgidiNAL
Figure 101. Fringes observed in the model for a vb=7-.^
.6
= 0.5 and a load: a-- 20 kg; b-.. 60 kg.
ato a
1LAL
IS .Fa sa7.�
=P.M.' 1 A 1110
_
to width
Figure 102. Fringes observed in
a model for ratio of height to
width = 0.25 for a load: a ...20
kg; b 60 kg; c -- 100 kg.
the model. An investigation by means of a compensator shows that tensile
stresses exist along the free boundary of the model. Thus there must exist
an intermediate region with both tenstle and compressive stresses between the
free boundary and the central portion of the model where we have only com-
pressive stresses.
As the model is loaded the regions in which tensile stresses are present
grow lighter.
This indicates that the tensile stresses increase.
In Figure
102c the tensile stresses attained considerable magnitude..�AP
eg1ANSWICI 14.1. 61117
first order corresponding to these stresses, which only became apparent �-17a
Figure 102b, are shown to have spread over the entire height of the model in
11110%... at-
a. A....Li.
r4zure avcg au bum aura' ua eca-Law .16.1v.16.1.4sual
the region of nnnlinifnrm
compression from both sides. From consideration of Figure 102 it is apparent
that the number of fringes under the punch differs from the number of fringes
4111 in the other portions of the model. The larger the load on the model, the
larger is this difference. The order (n) of the fringes under the punch is
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POOR ORIGINAL
always lower than in the remaining portion of the model, with the greater
portion of the model having all the fringes from the first to the highest
order; however, in the region below the punch we observe fringes ranging from
some intermediate order, depending upon the load, to the highest order.
Fringes 1-ergiv,g from the first to the highest order
es..Mes 1-W.ICACCITITrelrl
.td 4. � ���� a. � �...� A4-1.
in that portion of the model which is bounded by neutral fringe:s. The mate-
rial underneath the punch is in a state of triaxial nonuniform compression
and the neutral fringe is absent in this region, since regions of tenSile
stress are absent in it. Therefore, in this region we do not observe any
concentration of fringes. As the shearing stress increases here, the diffe-
rence in the paths increases and we observe corresponding fringes of a certain
A
farance
the in the paths increases, the fringe of the first or-
der
disappears
first, then the second one, etc. As the loading is discon-
tinued or as it is removed, fringes begin to disappear starting with the high-
est order down to the first
order.
The disappearance of the
responds to complete unloading of the model (that is, to the complete removal
of the state of stress). This phenomenon is repeated under repeated applica-
tions of load: Table, 4 gives the values of the fringe orders at various
portions of the indicated models for various loads.
Thus, if in Figure 100a the ratio of the fringes at various portions of
the model is equal -4 4..1
--
1.1 then
1
in
.A_.J%J�J 4-
164 is equal to 4:5, and in
Figure 100c it is equal to 9:12. For a load of 60 kg we observe a difference
of one fringe, and for a load of 100kg-
the difference is already equal to
STAT
three fringes. Still greater difference is observed in models whose height
is considerably less than the width (40x10x5 model). From the given thoto-
gral:hs and Table 4 it follows that saller the ratio cf height of the
mcd,0 to its width, the Is the defor'r.ed r ion of the rfdel. -In view
of this, the major portion of the nodel is not deformed (free of any stresses)
last -Pvennms3 e.nr.
and has a dark appearance.
This can be clearly seen in Figure 102. Since
171
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for one and the same load the deformed region of the model decreases, it
follows that the concentration of stresses in the deformed part increases
and this is revealed in a noticeable increase in the number of fringes. Ex-
periment shows that the free portions of
1Lo. 1,n11AA1
bigwuallor
do not uLicuAme
the
form of
the deformed portion. The magnitude of the latter depends upon the ratio of
the dimensions of the model and the instrument.
While in the initial stage of loading, when we observe only elastic de-
formations, the fringes are almost of circular form, in the second stage of
20....,
100
loading the similarity of the fringes to circles is destroyed. In this stare
and the following stages of loading the fringes become ovals elongated in the
direction of the applied load.
Table 4 1
Dimensions Width of Load in
f ~male Punch
n
in mm
in min
a.
bs�
alb
a
40x40x5
40x30x5
40x20x5
40x10x5
� �
10
10
in
10
4.0
4.0
4.0
3.0
1
1 54 1
6.0 1
1
8.0
a -- number of fringes under the punch
number of fringes in the other portions of the model
9.0 1 12.0
9.01 12.0
In Figure 103 there is shown a fringe pattern for an indentation by a
Figure 103. A fringe pattern observed
for indentation of a punch 5 mm wide
into a resin model.
punch 5 mm wide into a model having
dimensions of 40x40x5 mm.
L
111 t,LIC
given
case the entire discussion presented
above for a punch 10 mm wide is corn-
STAT
pletely applicable to this situation
as well.
Figure 104 shows a fringe pattern
^1.=erviati for An indentation of a punch
20 mm wide into a 40x20x5 mm nodal.
The colored isochromatic pattern of the stress distribution for indentation
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PoOR- ORIGIN AL
�
of a punch
4 C!
-A.
shown in Figure 105
(see insert between pages 178-179 in
text). The presence of a colored
isochromatic pattern enables us to
determine easily the neutral fringe
(fringe
f
the zero order) which
always has a dark color, the order
Vd6a46,1
111.a. 141.41. VI V
place of its origin.
If the model
V,
mi. la'
line (each order
,
IllnletnActA
anri i
Figure 104. Fringes observed in a model
for an indentation by a punch 20 mm wide.
is bounded by a green line), and the
loaded again after removal of the state
of stress, the character of the origin, development and distribution of frin-
ges will be the same as during the first loading of the previously untested
model. This may be seen by comparing
Figures 106 and 99c obtained from one
and the same model. Figure 106 shows
the fringe pattern after the model
corresponding to the stage of Figure
99c was unloaded, and subsequently
loaded again. The fringe patterns in
these figures, corresponding to one
and the same stage of loading (100 kg), are quite similar. The order of the
fringes both in the region under the punch as well as in the remaining por-
tions of the deformed models is one and the same, and the orders are reg_mc-
Figure 106. Fringe pattern obtained
in the model of Figure 99c after it
was unloaded and reloaded at 100 kg.
tively 9.5 and 12.5.
Then we test models of resin we observe
na�o�sacsa.vs.a.I.
thc.4r
failure �nriar,
load. It is characteristic of these models that the cracks which originate
at the points of stress concentration develop along lines of equal maximum
shearing stresses.
As was shown by L. A. Rapaport J17.1 under the indentation (dynamic) of
3:73
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�
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m punch of rectangular form having dimensions of 6.520+ mm into a specimen of
steel of grade 45 in a form of a parallelepiped having dimensions 8x8x6e5t
cracks develop in the specimen which coincide with the loci of the maximum
localized deformations (Figure 107). The line along which failure occurred
is similar in form to the lines obtained
material. This may be
e=actn b
in pressing the rianf�h into a plastic
comparison with Figure 108, which gives a
schematic diagram of the arrangement of fringes for a similar model. The
line designated by number 5
stress under a given load.
of comparatively fine grain
corresponds to the maximum value of the shearing
Along this line there occurs in crystalline
structure, localization of stresses and deforma-
tions and failure when stresses of a certain magnitude are attained.
The fringe patterns enable us to construct curves showing distribution
of equal maximum shearing stresses As can be seen from Figure 109, we
obtain a
sharp increase in the curve at the point of concentration of stres-
ses (a curve with two uaxima) at the section which is close to the contact
plane of the punch, that is, for the boundary points. The further from the
contact plane of
vwc.- punch is the section under con3ideration, the less sharp-
ly does the character of the distribution change. This curve becomes smoother
and finally becomes a curve having- one maximum value which is now situated at
the center portion of the model.
Figure 107. Formation of cracks
at an indentation of a punch into
a specimen of steel of grade 45
(from L. A. Rapaport).
STAT
Figure 108. A schematic diagram of
distribution of fringes for a punch
indenting a model of resin with a
ratio of height to width equal to
0 5
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POOR- ORIGINAL
4-Mtiucne
d\
2
nonot
00
I11
/\
fo,-)to
19111
Ii.
-4
IIIKAlk
I
'
MIL
Z 0
6 8 10
Legend: a) ainonff fringes
As was already showily one can ob-
tain by the method of photoplasticity
a system of isoclinics which permits
us to construct a system of trajecto-
ries of normal stresses (isostatics)
and a system of trajectories of shear-
in stresses (characteristics) for
Figure 109. Distributions of maximum models subjected to finite plastic
shearing stresses at sections A-A and
R-R for the mnr1411 shown in Figure 108. deformations.
The
shown
in
field
indentation of the punch into a model of resin gives isoclinic lines
Figure 110 (see insert between pages 178-179 of text). The entire
of isoclinics obtained with the plane of polarization rotated from
0 to 900 is shown in Figure 111.
_
t)
Figure 111.
STAT
Fir-ure 112. Arrang-ement of traec-
torie of Drincital norr-1 stresses
for a resin model indented with a
punch.
ries of of shearing stresses (character-
istics) for a resin nna4,1 inriAn+pA
with a punch.
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The grid of the trajectories of principal normal and shearing stresses
AP for the given case is shown in Figures 112 and 113. In Figure 112 we observe
an appreciable curvature in the trajectories ^I the principal normal stresses
in the regions situated in the vicinity of the extreme points of the punch.
The.
effect of friction is also shown in the character of the disposition
of trajectories of the shearing stresses.
In the preceding part of the chapter we considered the character of the
distribution of stresses for a resin model indented by a punch. Now let us
consider the character of the state of stress for a loading of the indicated
type in models of a plastic material of crystalline structure -- a model of
chloric silver.
In models of chloric silver having a coarse grain structure we observe a
disordered motley isochromatic pattern for indentation by a punch (Figure 10b)
Each grain is deformed in a different fashion depending upon its strength,
location and magnitude, and therefore, the entire field of the model presents
a disordered colored mosaic. In this case
do not observe a system of con-
tinuous fringes and isoclinics as in the case of an amorphous material.
Figura 114 shows a pattern obtained with a model with a finer grain
structure. In this case we also do not observe discrete continuous fringes,
but we do observe neutral fringes which bound the deformed region.
If we reduce the dimensions of the
grains, we can obtain sufficiently
sharply defined continuous fringes
having completely defined form arlrAio-
cation. This may be confirmed by the
Figure 114.
photographs in Figure 115. In Figure 115a we observe In addition to neutral
fringes two continuous fringes of regular form. Figures 115b, c, d show an
increase in the number of fringes with an increase in load. In Figure 115d
we can count up to 11 fringes. Under the die there occurs bulging of the
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�
�
Figure 115. Fringes observed in deforming a model of polycrystalline chloric
silver.
material in view of the plastic deformation taking place. As the load is
removed not all the fringes disappear.
The form et the
remaining fringes is
-
the same as under the load.
Still more convincing is the fringe pattern shown in Figure 116. Since
the punch did not come in contact with the top surface of the model over its
STAT
Figure 116. Fringes observed in deforming a model of polycrystalline chloric
silver with a finer grain structure.
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Polo R-r ORIGINAL
11,
entire surface, the fringes shifted somewhat to the right. In Figure 116a
4.1.4s7AVOCI MV6g1
V A
clearly visible two continuous fringes and a third one located in
the general background of the portion of the model being deformed. In Figure
116b, three fringes are already observed and a fourth one is being initiated.
four fringes. In Figure 116d the fringe
In Figure 116c there are already
pattern directly beneath the punch is distorted due to bulging of the material.
�� sr% _
vat iri) ca
V V a.�
model was unloaded a part of the fringes disappeared due to
removal of the elastic component of the deformation (Figure 117), but two
continuous fringes which determine the character of distribution of the re-
sidual stresses of the first kind remained on the right hand side after the
plastic deformation had taken place. These fringes had the same form and
location as were observed in the loaded model.
Figure 118 shows the fringe pattern obtained for a model of considerable
dimensions with a ratio of height to width equal to 0.5, and for a punch 10
mm wide. In this case we succeeded in obtaining a system of continuous frin-
ges covering the entire field of the deformed part
the model. In
cases the fringes had an oval form (ellipses) elongated in the direction of
applied load.
In the cases considered we observed a pattern of continuous fringes
even for a grain size of the order of 0.06-0.03 mm. For still finer grain
structure, the pattern must be even more regular.
Figure 117. Residual stresses re-
maining in deformed model of poly-
crystalline chloric silver after
unloading.
Figure 118. Fringe pattern obtained
for a model of polycrystalline chlo-
ric silver with a ratio of height to
width equal to 0.5.
178
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�
...,��������
Figure 105. An isochromatic pattern observed in a resin model indented by a
punch.
STAT
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,
~VW%
�
,
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-
0
11%
0
STAT
Figure 110. Isoclinics observed in a
rgsin model indented gy a punch: a
0 isoclinic: b MI MM. 10 : c -- 20�: d A= NM
v; e -- 45�
30 .
IMP ION
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POOR ORIaIN AL
�
Figure 119. Isochromatic lines observed
in a gelatin-glycerine model indented by
a punch.
VAT I \VI
1.1 L_
low Ar At t: AL a
EL kr16,1
71.'117'v
11101111011111NralINIROMNIMMIINMINNIM
Figure 120. Isoclinics observed in a gelatel-
glycerine model indented by a punch: a 0
Gn�
isoclinic; b uw e
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STAT
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POOR oRtaiNAL,
The observed pattern of continuous fringes of regular form may be inter-
preted in the following manner.
When a model consisting of uniform grains of constant length of :axis is
deformed, a sufficiently large number of the grains located in the path of
averaged optical (affect consisting
thepolarizedlightimustpro"ce. caL certain
of the optical effects introduced by the individual grains. TIm
a-a&
=nif-A of the
fact that each grain is deformed in an Individual fashion, depending upon its
individual properties and arrangement (orientation), at each point of the
model there will be observed a certain mean optical effect which is indepen-
dent of the orientation of the individual grains and their properties. The
optical effects introduced by individual grains will be summarized by the
passing ray and averaged, while their Individual properties will be minimized.
The optical effect will then depend upon a certain mean value of all the grains
in the field of state of stress. In an optical and mechanical sense the model
will behave as a solid and homogeneous body. TL-m4-t11%=1,1r
this tate will be
achieved only when a sufficiently large number of grains is present in the
path of the ray of light.
As can be seen from the examples considered, a material of crystalline
structure consisting of fine grains may yield a macroscopic pattern of stress
distribution which coincides sufficiently close with the pattern obtained
with models of plastic noncrystalline material.
The authors subjected models of gelatin-glycerine material to indenta-
tion by a punch which produced considerable plastic deformations. The arrange-
ment of the fringes and isoclinics for this case is shown in Figures 4.4.7 and
120.
However, it is premature to draw any conclusions regarding the state of
stress in the given case since the relationships between the observed optical
effects for considerable plastic deformations in gelatin-crystalline material
are not yet adequately known.
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MN
CULAPTIM VI
PRACTICAL SIGNIFICANCE AND PROSPECTS OF THE KETHOD OF IPHOTOPLASTICITT
1 Practical_Alwallicance of the Method 2.1_EARIsytasticiIi.
The method of photoplasticity even at the present time may be called
upon to solve the number of practical problems.
The most important of these problems are:
1. Modelling of geological and geophysical phenomena involving 1)114=t4eb
WIGAWilliGalk0i%0440
In this case, as a rule, the
problem of
modelling rmalmao=
0291:01M+45aM
.1-0 IR:: 7.0 dii6a
of strata which are inhomogeneous in mechanical properties and rheological
behavior. vie may construct suitable models by combining optically sensitive
resins of various viscosities which assure similarity to the strata for se-
lected dimensions of the model.
Similar models may be utilized in certain cases for the study of pheno-
mena observed in mining operations. At the present time the study of prob.
lems dealing with pressures encountered in geological formations is carried
out by the method of Dhotoelasticity. It is quite expedient also to call
upon the method of photoplasticity in solving such problems. STAT
2. Modelling of the technological processes of deforming of plastics.
The application of parts of considerable dil_ensions made of plastics has
recently become quite widespread. Plastics are used for fabrication of boats,
. parts of small airplanes, and other parts hPinc- used in the most varied bran-
ches of technology.
180
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POoR ORIGIN AL
However, the technology of production of these plastic parts is still
limited by a number of substantial shortcomings. These shortcomings must be
eliminated both by perfecting the existing technological processes and also
by assuring further progress of this extremely important branch of technology.
Th4a tec1es
,11^1
et GPM'
sometires isfili7,pfg transparent optically sensitive plastics,
or plastics whose rheological behavior may be simulated by homogeneous amor-
phous media which show double refractivity.
Therefore, the data which we obtained in the study of processes of de-
formation of the homogeneous plastic under conditions of viscous flow may be
utilized by the industry for perfecting the existing processes and development
of new technological processes for manufacture of parts made of plastic ma-
terials.
3. Modelling of technological processes involving deformation of metals
(primarily those involving pressure and cutting).
whir_
-C the plastic
of
plastics
takes place primarily under
conditions of viscous flow, the plastic deformation of metals occurs primarily
under conditions of plastic flow and In certain cases under conditions Gf
viscous-plastic flow. In addition to that, in dealing with plastic deforma-
tion of plastics we have as a rule a homozeneous medium. However, in.produc-
ing deformation.in crystalline bodies the medium is inhomogeneous and at best
may be regarded as a conditionally homogeneous medium only for a sufficiently
fine grain structure and
n 1..nriA1-%m
crystallographic orientation.
All this affects the distribution of stresses in viscous and plastic
STAT
bodies. Therefore, for one and the same method of external loading the dis-
tribution of stresses in a viscous body may be substantially different from
distribution of stresses in a plastic body. aowever, in many cases we need
not be concerned with the properties of the medium. Vie can do this In cases
when we are interested not in the details of the distribution of stresses in
the deformed body and the quantitative aspect of the problem, but in a
161
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POOR OglaiNAL
qualitative picture of general character. In order to obtain such a picture,
if is possible to utilize amorphous homogeaeous media instead of crystalline
bodies, as was shown by our experiments. Moreover, the clearest qualitative
picture giving a general view of distribution of stresses in a deformed body
yielded by models of amorphous materials which are deformed under con-
ditions of viscous flow. Therefore, in solving by means of models such prob-
lems in which the basic task is to obtain a most clearly 'visible qualitative
macro-pattern of distribution of stresses, we must utilize models made of
� �
optically sensitive homogneous amorphous materials undergoing deformation
under conditions of viscous flow. As an example we shall cite the following
problem.
During a stamping process we observe at times separation of the metal
into layers at the points of maximum concentration of deformations. This
thenomenon is called �At7-nfifiri-inli,�
mh. .�*.r...1- 4 4" 4- 4 - _
vva.c.4.%.4%..LLJLI
op.LiLti -Lae forging
into two parts and constitutes a defect which cannot be corrected. In order
tc identify the causes of this phenomenon the authors studied the process of
stamping by means of models of optically sensitive resin. The distribution
of stresses and deforrations in.this case are described in a preceding chap-
ter (see Figures 65, 67, 69).
Comparing the photographs show ng forgings of the saiLe form, we Llay draw
the conclusion that the concentration of deformations increases with the
amount of meal passing into the flash. When the amount of the excess mate-
rial of the billet reaches a certain value, there occur in the vicinity of
STAT
the flash vent deformations of intensity which are sufficient to cause failure
of the metal. In the vicinity Cf the imminent failure there occurs a concen-
tration of stresses and deformations and this causes the splitting of the
billet.
In Figure 121 there are shown photographs of macroscopic polished sections
of forgings which had a considerable amount of metal in the flash vent. The
182
�
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Poo g Rid IN AL
macro-polished sections show clearly a dark
1 4
which was
obtained in the
loci of maximum concentration of deformations and which indicates the place
of stratification.
Figure 121 Formation of stratification of metal within billet during stamp-
ing process: a -- by means of hammer; b -- by means of pressuree
The investi gAtion carried out permits
sions.
/UV
UM0.16y
it
1,01ashamov
"f
ii
164160
t^
draw the following c*'nclu-
stratification is the presence of excessive
in the metal billet. The stratification may be eliminated by selecting the
proper billet volume.
The distribution of deformations obtained by the method of a coordinate
grid engraved on metal models corresponds completely to the distribution of
stresses obtained with resin deformed under conditions of viscous flow.
We can solve the following problems related to the processes of forming
of metals by using models of resin under pressure:
1. Find the most suitable geometry of the instrument producing deM[-
mation.
Figure 122 shows fringes obtained during extrusion of specimens of resin
through a die opening with different angles of inclination. The fringe pat-
terns were photographed for a constant value of external load.
Figure 123 shows trajectories of maximum normal stresses obtained by
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POOR OR;GittJAL
�
Figure 123. Trajectories of principal normal strssses for n angle of incli-
nation of die opening of: a -- 45 ; b 900.
1.
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POOR oie,GIgAL.
reducing the data in the corresponding photographs of isoclinics.
The examination of this material permits us to conclude that the angle
of inclination of the die opening affects materially the distribution of
stresses in the material being deformed. (Since quantitative data were not
Oa
required, these experiments were carried out with thin models.)
2. Determine the zones in which the deformation process is difficult
and define tae regions of hydrostatic pressure.
3. Determine the places of localized plastic deformation.
� 4. Determine the places of stress concentration.
5. Formulate a concept of the form and internal boundaries of the
regions of origin of the plastic deformation.
6. Obtain a general qualitative picture of the distribution of stresses.
2. Prospects for L.Lle mcl.uou of Photoplasticity.
Tr....11.1..
The method of photoplasticity may. be utilized for
k,Liu cit,IALij of the nature
of plastic deformation of a substance and the
c+ 11A17
of the stress distribution
under conditions of plastic and visco-plastic flow.
Utilization of the method of photoplasticity for the study of the nature
of plastic deformations may aid us in gaining a broader concept of this prob-
ft
.11.41.61141.
�
We know that it is possible to obtain a well defined pattern of the dis-
tribution of stresses in a homogeneous amorphous medium whose elementary par-
ticles have a molecular weight of a magnitude measured in hundreds of units,
And this will enable us not only to study the mechanism of flow in an amor-
S TAT
phous body but to describe it in mathematical terms. This circumstance has
a great significance, since studies in recent years have indicated that the
amorphous mechanism is applicable to all solid bodies regardless of their
nature and constitutes one of the basic mechanisms of plasticity.
Even with substances of mycelian structure we observe a fairly well
defined pattern of stress distribution (Figure 19). Among such substances
185
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are, for example, gelatin-glycerine materials. For large rates of loading
these substances behave as quite elastic bodies which develop large elastic
deformations measured in tens of percent. When large deformations are de-
veloped under comparatively high rates of loading, gel fails without showing
any noticeable plastic deformations. However, such a failure cannot be called
a brittle failure
44-
nl�nn^PAs quite "sluggishly' (thin is a distinguish-
able type of failure). The large elastic compliance of these bodies can be
explained by the elastic flexibility of its mycelia. In the case of slow
loading there is, apparently, sufficient time for an irreversible mutual
displacement of the mycelia.
This displacement in the present case is
pre-
cisely the basic mechanism of plastic deformation. Moreover, there is a
possibility that for slowly applied forces some permanent changes take place
in the form of the mycelia themselves. This supposition appears to be credible
but it still requires verification.
The study of delurmation of transparent i'rystals
VTIMIP�4A%.ir. 4..._s.. eft.4..eammarw
irw...s.a.Nota=4 U,GWVGARUUlliVID
will enable us to understand more fully the characteristics of the amorphous
mechanism and to establish its specific features both for crystalline materials
and amorphous homogeneous Media. ks expected, the study of deformations of
crystals will enable us to understand more fully the mechanism of. slip and
other mechanisms characteristic of n.,..7f=114ne
uwuJi-CO�
The prospects are
particularly favorable for the study of the behavior of the material between
crrairA
and the phenomena occurring at the grain boundaries.
The study of the nature of plastic deformations by means of the photo-
plastic method has in fact been initiated. This fact is convincinzly con-
firmed by the work of S. C. Tzobkallo and B. A. Kuznetzov /197 wh' was
con-
cerned with the study of the nature of plastic deformation of a polycrystalline
material by an optical method� rr'IP material under study was a yolycrystalline
.chloric silver. The work demonstrated that in the grains the residual stresses
of the second kind are nonuniforway distributed and that stresses are localized
186
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_
POOR OgWIN AL
near the grain boundaries. Figure 124 shows characteristics curves of the
distribution of stresses of the second
, 4.
,n,d
�
However, the distribution of residual stresses has such a character only
under a static load. In the case of cyclic loading, the Concentration of
residual stVesses of the second kind near the graia boundaries is reduced.
After a large number of loading cycles, the traces of slip indicate large
local distortions. The authors have also shown that the distribution of
stresses is quite nonuniform in Individual grains of the polycrystalline ma-
terial. Figure 125 shows the .distribution of stresses in a grain 1.2 mm long.
V. M. Krasnov and A. V. Ste.nanov carried out an interesting study by an
protical method gg of the initiation of failure under the action of a concen-
trated force applied to the surface of a crystal of fluoric casting. Inasmuch
as this work was concerned with residual stresses, and. also with the origin of
cracks, slip and
�
g LIAL1lo VC
classified as an investigation
%ft
.1.11.11
%41-1.G
field of photoplasticity both with respect to its methoa of attack and for-
=Illation of the rrnhi.m.
�
itnr
-
o -2
x-3
�If X
020 42 04 ad a
tO t2
YM?,
560
540
520
500
480
460[ \ftaciori,1.11"1.66.3
44�a cal die 41 6 aaa to' i.2.1
14,4
Figure 124. Distribution of residual Figure 125. Distribution of stresses
stresses in the grains of chloric in a grain under the action of an
extgwv�nsal'as 1r1At4
a4111:-.12.-rift � "rt.
-1
ielA4v4rUnftl
STAT
The facts cited here indicate that the method of optical investigation
of stresses in polarized light is already being utilized in the study of the
nature of plastic deformation. The possibilities of exploiting the method of
ri'ilettnrOm+inif.y will
1.4.0
�
becorLe
quite favorable after we solve the problem of dis-
tribution of stresses under conditions of plastic flow.
187
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Et
For the case of viscous flow we have the solution of the simplest prob-
Its simplicity is due to the fact that the properties of the body do
not change during the process of plastic deformation. In the case of plastic
flow, however, we encounter a considerably more complex problem. Its com-
piexl-Gy LUUC
mr-
VW
A."�
t.
4010.m4.
the pr^pav4.i.01 of the body change during the
process of plastic deformation. In the case of viscous flow we are dealing
with a homogeneous medium, while in the case of plastic flow the medium is
either heterogeneous and anisotropic, or quasi-homogeneous (quasi-isotropic).
V. M. Krasuov and A. V. Stepanov made a study of the state of stress in
a transparent monocrystal of an alloy of 60 mol.% of bromic and 40 mol.% of
iodic thallium under a concentrated load 147. Crystals of this alloy belong
to the cubic system, are optically isotropic, have a relatively high yield
point (2 kg/mm2), and have high photoelastic constants. The pattern of iso-
chromatics obtained for this material under load differs substantially from
that usually observed under a load of this type (Figure 126). The isochro-
matics for the anisotropic medium do
not coincide with the curves of maxi-
mum shearing stresses as is the case
for isotropic bodies, and the aniso-
tropy of the mechanical properties
has a character different from that
of the anisotropy of the.
properties.
V. M. Krasnov and A. V. Stepanov
photoelastic Figure 126. A photograph of a specimen
under a concentrated load illuminated
by circularly polarized light. The dark
liaes in the photograph are the isochro-
matics. STAT
showed that in order to determine the state of stress in an anisotropic medium
one muct have in additIon to the isochromatic pattern, the orientation of the
axes of the optical ellipsoid in the legion under investigation. These inves-
tigators developed a corresponding theory which enables us to obtain on the
basis of experimental data a complete picture of the state of stress in an
188
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anisotropic medium, and in particular enables us to obtain the curve of maxi-
mum shearing stress (see Figure 127).
The problem now arises how to propagate the theoretical concepts of
V. M. Krasnov and A. V. Stepanov as they apply to molycrystalline material
and how to find reliable methods for quantitative deterrination of stresses
in a quasi-isotropic polycrystalline material on the basis of the macro-
pattern of isochromatics obtained during the plastic deformation of such a
polycrystalline material. The solution of this nroble= along with the prob-
lem of determination of the fringe value under the conditions of plastic flow
will be the basis for the study of stresses for plastic deformation in bodies
which change their properties during the process of defomation (plastic
h^Aie.).
otipasio
JP*1
ir
f anell
Nmor
_
37'mm
NW% -
Legend:
edge of specimen
direction of action
of force
direction 5007
Figure 127. Curve 1 -- isochromatic based on excerimental data; 2 theore-
tical curve of,equal silearing stresses. The points on curve 2 are experimen-
tal data.
Solution, of the problem of stress distribution under conditions of plas-
tic flow will enable us to examine a number of most important problems of
the theory of fabrication of metal parts by pressure, which problems are
concerned with a quantitative determination of stresses for those deforma-
tion processes which take place at temperatures corresponding to regain.sTAT
of strength, and even to the state of incomplete loss of strength. The quan-
titative determination of stresses for the deforation processes in metallic
alloys taking place in the presence of incomplete or complete loss of strength
is already concerned with the solution of the -roblem of photoplasticity under
conditions of viscous-plastic flow.
189
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_
POOR ORIGINAL
The solution of the viscous-p16.stic problem presents a maximum of diffi-
culties both with respect to experiment and theory. In this case, just as
in the case of the viscous problem, the resistance to deformation is affected
by the rate of deformation and the level of the mean stress, and at the same
time during the deformation process the structure, and sometilLes even the
properties of the body undergoing deformation, may vary. The plastic prob-
lem is simpler, since in this problem neither the rate of deformation nor
the mean stress level have any effect on the resistance to deformation. The
solution of viscus.-plastic problem heads the list of basic problems which
must be considered for the complete development of the method of photoplas-
ticity.
STAT
190
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Nog ORIGINAL
BIBLIOGRAPHY
1. Chernov, D. K. Obobshcheniye po povodu nekotorykh novykh nablyudenly pri obra.
botke stall. LGenera1izatis with respect to some new observations in the work-
ing of steel f, 1834.
2. Trudy konferontsii po o.oticheskomu metodu izucheniya napryazheniv /Proceedings
of the conference on the optical method of study of StresseJ, NII MM LGU, 1937.
3. Stepanov, A. V., "Zhurnal Tekhnicheskoi Fiziki," 5ournal of Technical Physics7
XX, 1950.
4. Stepanov, A. V., Physikalische Zeitschrift der 5owjetunion ficurnal of Physics
of the Tviet Union _7 8, 25, 1935.
5. Zaytsev, A. K. Opticheskiy metod izucheniya napryazheniy, 175ptical method of
study of stresses!, ProMburo, 1927.
6. Gubkin, S. I:, Teorlya obrabotki metallov davleniyem /Theory of metal working
by-Means of pressurfj , Metallurgizdat, M., 1947.
7. Kobeko,.P. P., Amorfnye vezhchestva amorphous substanceil, Publisher, AN, SSSR
ma T.., 1959.
A
aaucs-L, no, .L. .1.. razrusheniye i. LU tel. TzA. inostrannoy liter-
atury. Plasticity and failure in solid bodies. Publishers of Foreign Liter
7 H., 1954.
ancrohAv.irize, D. 3. and Kir7a1idze. I. n "Zhurnal TeknicHeRkmi Fiziki,"
/Journal of Technical PhysicS7, 1951.
10. Gubkin, S. I. and DobrovolskY,
of Sciences7, XXIII, 1, 1950.
11. Gubkin, S. I. and DobrovolskY, S. 1., Ibid., XXXVIII, 5, 193.
12. Frocht, M., Fotouprogost'. V. I. Gosudarsvennoye izdatel"stvo tekhniko-teoretich-
noy literatury /Photoelasticity, V. I., State Publishers of Technical-Theoretical
Literaturt7M., 1948.
I., "DAN SSSR" /Reports of the USSR Academy
13. Feldman, G. I., "Zavodskaya Laboratoria," LThe Plant LaboratorV7, XVII, 2, 1952
14. Prigorovsky, N. I.., Preise, A. K., and Slutzker, 0. D., "Zavodskyaya Laboratoria,"
',The Plant Laboratori7, XV, 3, 1949..
15. Entsiklopedicheskiy spravochnik "Mashinostroyeniye" /7ncyc1opedia "Machine
Construction" V. I and III, Mashgiz, 1947. .
14. Ibid. , V. IV, M' 104'
1947.
17. Stepanov, A. V., "Zhurnal Tekhnicheskoi Fiziki", I:Journal of Technical Physics7,
XIX, 2, 1948.
STAT
18. Zhitnikov, R. A., Dissertatsiya "Razrabotka optichesk000 netoda issledovaniya
usrednennykh napryazhennykh sostoyaniy v melkozernistykh polikristallakh"
ipissertation, *Development of an optical method of investization of mean stress
levels in fine grained polycrystals,471953.
191
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POOR ORIGIN AL
19. Tzobkallo, S. O. and Kuzetwv, B. A., "Zhurnal Tekhnicheskoi Fiziki" gournal
of Technical PhysiciT, XXIII, 1, 1953.
20. Tzobkallo, S. O., "Zhurnal Tekhnicheskoi Fiziki," gournal of Technical Physici7
XIX, 4, 1949,
21. Krasnov, V. M. and Stepanov, A. V., "Zhurnal Eksperimental'noy i Teoreteicheskoi
Fiziki,"LJournal of Experimental and Theoretical Physicg, 25, 1(7), 1953.
22. Krasnov, V. M. and Stepanov, A. V., "Zhurnal Eksperimentallnoy i Teoreticheskoi
Fiziki," fjournal of Experimental and Theoretical Physic, 23, 2(8), 1952.
23. Sidorenko, Yu. A. "Zavodskaya Laboratoria," frhe Plant Laboratori7, 6, 1954.
24. Eiring, G. Glasstone, S., and Leidler, K., Teoriya absolyutnykh skorostey
reaktsiy trheory of absolute rates of reactiong, M., 1948.
25. Kirpichev, M. V., and Gukhman, A. I., Trudy LOTI, 1Troceedings of the LOT17,
No. 1, 1931.
26. Volkenstein, M. V., Molekulyarnaya oDtika. Gosudarstvennoye izdatellstvo
tekhniko-teoretirheskoy literatury /Molecular optics. State publishers of
technical theoretical literaturej M., 1951.
27. Landau, L. and Lifshits. E., MekhAnika sploshnykh sred. Gosudarstvennoye
izdatel'stvo tekhniko-teoreticheskoy literatury. 'Mechanics of solid media.
State Publishers of technical-theoretical 1iteratilre2 M., 19144.
28, Kantorovich, L. V. and Krilov, V. I., Priblizhennye metody Trysshego nnaliva.
Gosudarstvennoye izdatel'stvo tekhnikoteoreticheslm:r lit eratury /Approximate
methods of advanced analysis. State Publishers of technical- theoretical
terature 7 K.. 1952.
29. Otchet laboratorii obrabotki metallov davleniyem eport of the laboratory on
fabrication of metals by means of pressure 7 FTI AN BSSR, 1953.
30, Tarnovskiy, I. A. and Ganac:o, 0, A. SborniJr .',aschet I konstruirovaniye tavod-
ski� o oborudovaniya," 1,"Design and Construction of Plant Equipment," a Sympos-
ium 48, 1953.
31. Rapaport, L. A., Eikroskopicheskoye issledovaniye sdvigovoy deformatsii polikri-
stallov pri udarnom deformirovanii LMicroscopic study of the shearing strain
of polycrystalline bodies under impact7, 1952.
32. ochin, N. E.,
LVector analysis
U2SR Acacler-c sf
Vektornoye ischisleniye i nachala
and elements of tensor anal7sis.7
Sciences, M., 1951.
tenzornoo
ischisleniya.
House of the
STAT
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Introduction
TABLE OF CONTENTS
- CHAPTER I. METHOD CF 7.;HCTOPLASrICITY
1. Photoelasti-city
2. The Need for Development of an Experimental Method of
Studying the State of Stress with "PlAtie Deformations 10
3. Classification of the Rheolcp-ical Behavior of Solid
Page
3
5 .
5
4. Basic Pro-bier:is of Photoolasticity
Pk
5. Basic Characteristics of the
ViGieLLVu of PlintoplasticitY 30
CHAPTER Ii. MATERIALS UTILIZED IN .TUE 1.:ETHOD CF PHOTOPLASTICITY . 34
l� Specifications for Materials Used in the PhntnAlnA4-.ic
Method . 34
2. Specifications for Materials Used 4,�
the :,:etlIod of
PhotoplasticitY
3. Classification of Materials 37
35
-r� Effect of the Nature of the Material Being Deformed on the
Character of Stress Distribution 52
nueprirzR TTT. SPnT411 FEATURES CP EXEERIMErTAL TECEINIQUE 57
1. Optical Installation and Apparatus 57
P-1
C. �
Fabrication and Machining of Models
STAT
60
3. Experimental Technique 69
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Photographing of Isochromatics and Isoclinic
CHAPTER IV. VISCOUS FHOTOPLASTICITY 74
1, Viscous Flow 74
2. Optical Anisotropy in Conditions of Viscous Flow 83
3. Certain Special Features of the Problem of Plane Models
Undergoing Viscouill Flow
4. The Simplest Plane Problems of Viscous Flow
5. Singularities of the State of Stress at 4..A% Periphery of
the Model and Some of the Methods of Reduction of Experi-
100
io8
mental Data
126
CHAPTER V. morigu FOR STUDY OF PROCESSES OF FABRICATION BY
-
PRESSUPT:�
114
3.� Stamping
134
2. Extrusion
157
'Indentation of a Punch
167
CdAPTER VI. PRACTICAL SIGICIFICAINCE AD PROSPEC;S CF mn-n METHOD
OF IHOTOPLASTICITY
180
1. Practical Significance of the Method of T:hotnplasticity 180
2. Prospects for the Method of Photoplasticity 185
Bibliography
1 rY1
STAT
194
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