JPRS ID: 9371 TRANSLATION METHOD OF DYNAMIC TESTING OF RIGID POLYMER MATERIALS BY SEMEN MIKHAYLOVICH KOKOSHVILI

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JPRS L/9371
28 October 1980
,
~ Transl~tion ~
METHODS OF DYNAMIC TESTING
OF RlGID POLYMER MATERIALS
By -
Semen Mikhaylovich Kokoshvili
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JPRS L/9 371
28 October 1980
METHODS OF DYNAMIC TESTING -
OF RI~ID POLYMER MATERIALS
~ Riga METODY DINAMICHESKIKH ISPYTANIY ZHESTKIKH POLIMERNYKH MA-
TERIP,ZOV in Russian 1978 signed to ~ress 29 Jun 78 pp 4-182
[~ook by Semen Mikhaylovich Rokoshv~ili, Institute of Mechanics ~
of Polymers, Lithuanian Academy of Sciences, Izdatel'stvo
Zinatne, 1300 copies, 182 pagea, UDC 620.178.7]
CONTENTS
Annotation ~ ~ 1 .
- - : .
. - - - - - .
Preface ~ 1
Introduction 2
Part I. Loading and Recordin~ Facilities for Dynamic Tests 6
Chapter 1. Loading Devices 6 -
1.1. Traditional Loading Devices 7
1.2. Magnetic Pulse Facility 10
1.2.1. Loading by Electrodynamic Forces 13
1.2.2. Loading by an Electric Discharge in Liquid 16
Chapter 2. Recording Devices 1~
2.1. Measur~ment of Load and Pressure 17
2.1.1. Piezoelectric Dynamometers 18
2.1.2. Waveguide Dynamometers 21
- 2.1.3. Capacitive and Dielectric Pressure Sensors 24
2.2. Measurement of Displacements and Deformations 25
2.2.1. Photoelectric Methods 26
2.2.2. Strain-Gage Methods 28
2.3. Measurement of Velocities 32
Part II. Dynamic Tests of Rigid Polymer Materials ~6
Chapter 3. The Hopkinson Split Bsr Method 37
3.1. Governing Principles of the HSB Method 39
3.2. Hardware of Variants of the HSB Method 43
3.3. Validation of the HSB Method 47
Chapter 4. Methods of T;esting Polymers With Dynamic Bending 53
4.1. Transverse Impact of a Freely Thrown Mass 56
. 4.2. Experimental Studie~s of Transverse Impact 62
4.3. Wave Phenomena in the Striker Upon Transverse Impact 66
4.4. Pulse Loading by ~ler_tromagnetic Fields ~2
4.5. Standard Methods 76
~ 4.6. Oscillographic tiethods 81
- a - [I - USSR - I FOUO]
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Chapter 5. Methods of Testing Annular and Tubular Specimens 86
5.1. Testin~ Thin Rings 86
5.1.1. Use of Mechanical L~ading Devices, Explosives -
and Magnetic-Pulse Faci~lities 87
' S.1.2. Loading by Electrical Discharge in a Liquid 89
5.2. Test for the Complex-Stressed State 95
5.2.1. Use of Mechanical Loading Devices 96
5.2.2. Use of the Magnetic-Pulse Facility 98
Part III. Generalizing the Results of I?ynamic Tes~s of Rigid
Polymers 101 -
Chapter 6. Influence of Strain Rate on the Mechanical Properties
of Polymer Materials 102
- 6.1. Amorphous Polymers 102
6.2. Crystalline Polymers 111
6.3. Reinforced Materials 116 -
Chapter 7. Models of Mechanical Behavior of Polymers 118
7.1. Calculation of Relaxation Spectra 118 ~
7.2. Validation of the Model of Mechanica? Behavior of Polymers 125~
7.2.1. Model Based on Tests for Mechanical Vibrations 126
7.2.2. Model Based on Quasistatic Tests 128
7.3. Validation of Model From a Study of Propagation of Added Load ~
- Pulses 140 '
References 146 ~
' Subiect Index 159
i
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~
ANNOTATION
[Text] This book deals with dynamic testing techniques _
that gi.ve results used as the basis for designing poly-
mer parts and components that operate under impact _
loading conditions. Particular attention is given to
tests of rings and beams, and also to the use of the
split pressure bar method proposed by Hopkinson. The
theoretical basis of inethods of dynamic testing is
outlined, and a detailed description is given of tech-
niques for loading by electromagnetic fields and an
electric discharge in liquid. The book contains ref-
erence material on the mechanical behavior of polymer
materials over a wide range of temperatures and strain
rates. Methods are given for constructing and ve,~,ifying
models of viscoelastic behavior of polymers. Tables 8,
~ f igures 148, references 170. _
PREFACE
The introduction of modern synthetic materials requires extensive in-
vestigation of their mechanical properties, and above all demands devel- _
opment of sound methods of testing under various loading conditiona.
Practically and theoretically sound techniques and generalized reference
books are available for static tests of reinforced plates, but in the
field of dynamic tests an acute need is felt for ~ust such developments. _
Dynamic tests for pre~ent-day polymer and composition materials are
especially necessary as a consequence of the sharp sensitivity of these
materials to changes in loading rate.
This book generalizes the experience of dynamic testing of polymer and
composite materials that has been accumulatEd in the Laboratory of
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- Dynamic Properties of Polymer Materials at the Institute of Mechanics of
Polymers, Lithuanian SSR Academy of Sciences. Considerable a~~tention is
- given to methods of high strain-rate testing. However, the materials of
- research done by the author are naturally of special interest, in ~ar-
- ticular the development of inethods of loading by electromagnetic fields
and the electrohydraulic effect, improvement of the Hopkinson split
~ pressure bar system, investigation of the properties of ring specimens
and beams made up of reinforced plates. Although the data in the book
are based on the study of polymer behavior, it is hoped that the proposed
methods for dynamic loading and registration of 1_?~gh strain-rate processes
will find application in the investigation of other str~ictural materials.
All comments will be appreciated, and should be addressed to the author
or the science editor at: 226006, Riga, Institute of Mechanics of
Polymers, Lithuanian SSR Academy of Sciences, 23 Ayzlcraukles Street.
V. P. Tamuzh
(science editor)
INTRODUCTION
A typical peculiarity of the behavior of polymer materials is the con-
siderable time dependence of their mechanical properties. Inelastic
, behavior of polymers can be observed even in cases where the loading time
amounts to days or is measured in microseconds. A fairly complete de-
scription of the principles governing mechanical properties of polymers
is attained in complex mechanical tests that include changes in strain
rates and temperatures.
Methods of inechanical tests of materials, regardless of the form of
stressed state of the investigated object, can be differenr.iated into
static, quasistatic and dynamic methods.
Static testing methods for polymer materials are based on using standard
loading devices and recording equipment. The range of strain rates that
are covered by these method~ is fairly wide. In creep studies, strain
= rates are of the order of 10-6-10-1 s'1, and in the case of uniform ten-
sion 10-3-10-1 s-1. Static testing methods are not considered in
this book.
Quasistatic testing methods cover strain rates of the order of 10-1-104
s-1. These methods are realized by identical loading devices and iden-
- tical recording facilities. The loading devices must meet the require-
ment of transmitting high energy to the specimen over a time that varies
over wide limits (from a few milliseconds to *nicroseconds). The record-
_ ing equipment is quick-response sensors, wide-band amplifiers and oscil-
lographic devices.
In describing the loading facilities, particular attention was given to
devices that use electric energy, which have a number of advantages over
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conventional loading devices, one of these advantages being the capability
of loading the surface of a specimen by forces with a given distribution.
Methods of transmitting electric energy to a specimen are based on load-
ing by powerful electromagnetic pulsed fields or by action through a '
transfer medium (free electric diacharge in a liquid). Both loading
methods have found ~pplication fairly recently in mechanical tests, and
have a good outlook, Loading and recording facilities selected with con-
sideration of anticipated strain rates, and also with consideration of -
the shape and dimensions of the specimen, comprise the testing method.
The same test involving short-term loading can be taken as quasistatic or
dynamic, depending on loading duration and the geometric dimensions of
the specimen.
Dz~npmie testing is done with loading of the s~ecimen by an intense short-
- duration pulse load that produces a nonuniform stressed state in the
specimen as a consequence of stress wave propagation. Dynamic methods
i involve loads in which considerable inertial foi`,~es act on the specimen.
Investigation of wave propagation is the most devEloped method of dynamic
testing. In a number of cases dynamic methods do not involve wave propa-
gation, even though they are based on accounting for inertial forces.
One form of the method is testing of a thin r~ng under the action of
internal uniform pulse pressure.
The use of quasistatic testing methods is based on the assumption of
uniformity of the stressed state of the specimen in time at any point,
i, e. without consideration of wave processes. Therefore the results
of quasistatic experiments are processed ~ust as for static tests, al-
though we will show that the loading and recording facilities of the
methods to be compared are quite different. Quasistatic tests are done
over a wide range of strain rates. Close to the upper limit of this
rang~:, the tests differ more an.d more from the static case. For example,
the influence that inertial for"ce~ have on the process of deformation
becomes more pronounced and apparent, as'shown by recordings of the -
transient process. -
~.~r-' ~ 6j .
_ E
9 ~
_ ~o~Co ' ~
_ �~~~r . .
_ . e -V%t ~
o~~ ~ t ~ r
4
Fig. I.1. Displacement-time dependEnce in cross section x= L/2 for an
elastic (a) and a viscoelastic (b) rod
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As an example, Fig. I.1 shows two solutions of the problem of stretching
of a uniform elastic rod of length L with one end fixed ar.d the other
_ moving at constant velocity Vp. The straight line e= Vpt/L is the result
of the quasistatic approach to the solution. The graph showing the step
= function is tl~e solurion of the dynamic problem for croas section x= L/2
of the rod. The dynamic and static solutions coincide at times Ln/cp
(n =C~, 1, 2...), where cp is the rate of propagation of longitudinal _
elast.ic waves in the rod. Experiments with a polymer rod show that vis-
cosity effects are the cause of the observed e-t de^endence shown anal-
ogou~~ly in Fig. I.1, b. Region A-A can be taken as a transition zone.
When this region covers a considerable part of the process of deforma- .
tion, the test should be considered dynamic. Let us note that depending
on the length of the rods, the same testing conditions may lead to dif-
ferent results that must be treated in accordance with quasistatic or -
- dynamic u:ethods. A combination of dynamic and quasistatic processes of
- defarmation in mechanical tests is observed in a system made up of two
long rods separa~ed by a test specimen in the form of a thin gasket.
Loading such a system by a pulse of certain length leads to quasistatic
deformation of the gasket. At the same time, lengthwise nonuniform
st~:essed state of the rods makes it necessary to treat their deformation
as a dynamic process. This testing arrangement is realized by the Hop-
kinson split pressure bar method. Material testing by this method can ;
be: classif ied as intermediate between quasistatic 3nd dynamic studies.
I_
Conclusions about the stLessed state of a specimen at strain rates that '
~ipproach the limit of applicability of these techniques require additional
proofs. Adequately precise definition of the quasistatic nature of de-
formatio*~ is complicated (compared with the above example) by nonlinear
behavior of the test material, the need for accounting for radial move-
ments of the specimen and so on. The final decision on the validity of
the quasistatic approach to the treatment of test results is based on
solution of the problem of propagation and interaction of stress waves
in the speci~en (see Fig. I.1). Testing by the Hopkinson split pressure
_ bar methed is substantiated by solution of the prablem of wave propa- -
gation in the split bar. While effortg on the part of experimenters to
eliminate the influence of wave processes in dynamic tests could hardly
be expected to succeed, a change to new unconventional testing methods
- does make it possible to disregard wave effects in a specimen up to very
high strain rates. One such method, on which devel.opment began quite
recently, is testing of a thin ring with loading by a uniformly dis-
tsibuted impulsive foce. The dynamic response of the ring specimen to
, such loading makes it necessary to consider the ring as a mechanical
oscillatory system, which requires registration of accelerations to com-
pute the stresses in the ring.
New testing methods are now supplemented by quasistatic tests in the
complex-stressed state. Such methods are still in the early develop-
mental stage. In view of the complexity of solving problems on propa-
gation of combined waves and the undsveloped state of the art, it could
be assumed that there is no sound basis for quasistatic testing methods _
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in the complex-stressed state. Nevertheless we feel that it is necessary _
to examine the corresponding techniques. The first attempts to develop
a biaxial version of the Hopkinson split pressure bar are especially
promising in our opinion. It is apparently in this direction that fur-
ther development of inethods of dynamic and quasistatic tests will take
place.
Has there been progress in the traditional method of testing for impact
toughness which is standardized in the Soviet Union and elsewhere? Ab-
solutely. More and more frequent use is now being ma.de of standard
methods that include registration of force and displacement. The impact
toughness method is still the most accessible technique for qualitative
evaluation of materials that are being compared. The breaking energy is -
calculated in accordance with the method of impact toughness testing
without consideration of forces of inertia, i. e, such tests can be .
called quasistatic.
_ Dynamic and quasistatic studies are aimed at getting fairly exact me-
chanical characteristics. Generalization of the results of these tests -
in the form of models of inechanical behavior is cf first-rank importance
since engineering calculations are based on them. Materials of theo-
~ retical and experimental studies of wave propagation in rods can be used
~ to verify a model obtained with consideration of the results of quasi-
static experiments. The justification procedure is done in the follow-
ing sequence: the problem of wave propagation in the rod is solved on
the basis of the proposed model (which must necessarily conform to an
equation of motion of hyperbolic type), and then an experiment is done
to study wave phenomena in the rod. A decision as tc applicability of
the model is made on the basis of comparison of theory and experiment.
One of the important features~~f the mechanical behavior of polymers is
~ considerable dependence of all mechanical parameters on strain rate.
' The way that strain rate affects the mechanical properties of polymers
of different classes cannot be reduced to a single formulation. For
example, studies of the mechanical properties of amorphous and crystal-
line polymers has shown that their mechanical behavior depends on the
nature of the relaxation process that predominates during deformation.
The problems to be solved in this book were determined on the basis of -
the foregoing presentation:
1) the development of modern facilities for loading and registration;
2) verification of quasistatic methods of inechanical testing on the
basis of theories of wave propagation;
3) construction of a model of inechanical behavior that reflects response
of the material over a wide rartge of times of loading action, and veri-
fication of this model by studying wave propagation;
5
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i vi. v. i i~.icw u..ra: vl~i~~
4) establishment of principles that govern the influence of strain rate
on deformarion and strength of polymers.
Tr~e author takes this opportunity to express sincere gratitude to the
science editor of the boo k, Professor V. P. Tamuzh, and also to Frofessor
G. S. Shapiro, Doctor of Technical Sciences V. V. Kovriga, Candidate of -
Technical Sciences V. V. Viktorov and Candidate of Physical and Mathe-
matical Sciences R. A. Basin for discussing the results. The author
thanks P. V. Tikhomirov for assistance, and also Yu. Ya. Riba, P. P. Kal-
ninya and D. A. Abolinyu.
PART I -
LOADING AND RECORDING FACILITIES
FOR DYNAMIC TESTS
Dynamic tests utilize facilities that differ in the form of cumulative _
energy. Loading facilities that utilize mechanical energy (swinging and
= rotating hammers) and compressed gas energy (pneumatic hammers) have been
fairly well developed. To hurl masses with velocity of the order of
- several kilometers per second, gas guns are used [Ref. 4, 64, 65]. Work
- has been successfully begun on using electrical energy accumulated in
, capacitors for dynamic tests. i
Recording of dynamic quantities is made difficult by the brevity of the
, process, and therefore the use of rapid-action facilities enables reduc-
tion of response lag in accordance with the anticipated strain rates.
At very high strain rates it is advisable to use non-contact measurement
instrumentation distingnished by a comparatively small mass and short
- measurement base.
Without going in~o the question of reproducing equipment, let us note .
that modern cathode-ray oscillographs (S1-29, S1-33, S-37, etc.) that
have taken the place of galvanometer oscillographs only partly meet the
requirements of researchers because of the inadequate screen. The DL-905
instrument made by Datalab, which stores a pulse and reproduces it by -
a two-coordinate chart recorder in the necessary time scale could serve ~
as the ideal reproducing device for dynamic tests.
Chapter 1 _
LOADING DEVICES
Depending on the kind of cumulative energy, facilities for quasistatic
and dynamic tests can be classified as follows: 1) mechanical; 2) pneu-
matic and hydraulic; 3) using explosive energy; 4) using electrical _
- energy. The use of these latter as loading devices has begun fairly
recently, and therefore the most space in this chapter has been devoted
to their description. A detailed descriptian is given of two loading
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_ methods based on the interaction of pulsed electromagnetic fields and
use of the electrohydraulic effect.
1.1. Traditional* Loading Devices
The simplest device in this category is the drop hczmmer. This mechanism
transmits the energy of free fall of a massive body to the specimen.
Because of the restricted height from which the mass descends onto the
specimen, the rate of impact of the free body on the specimen does no t
- exceed 10 m/s. To increase the initial impact velocity, a slingshot
_ device is used in which the energy of a stretched rubber band supplements
the energy of free fall [Ref. 109J. The Ansler ham~ter was one of the
first devices based on using free fall of a system with the specimen that
stops when a mass fastened to the specimen impacts against projections of
a massive base. Stretching of the specimen is accomplished by inertia 1
forces. To increase the capacity of the device, it is necessary to in-
- crease the mass fastened to the specimen.
A more up-to-date design is proposed in Ref. 28, in which the weight is a
hollow cylinder and stretching is accomplished by impact of the weight
against a crosshead fastened to the specimen. The design proposed by
N. N. Davidenkov is improved in Ref. 20.
Another extensive group of facilities using the energy of free fall is
pendulum hammers with principal parameters defined by State Standard
GOST 14703-73 (hammers, pendulum, for determining impact toughness of ~
plastics). In recent years Japan, East Germany and other nations have
been producing hammers for research purposes on which tests for canti-
_ lever bending and impact stretching can be done in addition to two-
support bending tests. Such hammers are equipped with devices for mea- "
- suring elongation of the specimen and t:ie impact force; however, in some
- cases the results do not meet research requirements. The Soviet Union
makes BKM-S-1 and BKM-5-3 hammers. Interchangeable pendulums and supports -
on the model BKM-5-1 allow tests for two-support bending, impact stretch-
ing, cantilever bending and shear.
Amung the disadvantages of hammers with a falling weight are: 1) the
limited impact velocity, which cannot be increased without increasing
the dimensions of the hammer; 2) bending of the specimects due to pendulum
travel on a circular arc; 3) repeated impacts of the loading element with -
the specimen. Nevertheless, the hammers meet requirements for getting
estimates of the characteristics of materials to be compared by standard
techniques. ~
- Rotarz~ hairnrters are based on using the energy of a flywheel accelerated to _
a predetermined speed. The Werkstoffprufmaschinen testing machine plant
*For more detailed information on the loading devices here termed
"traditional," we refer the reader to monographs in the reference list: -
20, 22, 28, 29, 36, 109.
P
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1'Vl~ VL'1'LVlnl~ VJli V1YL1
r
in Leipzig, East Germany, makes the series-produced PCO facility. In
testing, one end of the specimen is held in a fixed clamp, and a light
crosshead is fastened to the other end. The striker of the hammer is
restrained by a pawl. When the rotor reaches the set speed, a signal
energizes an electromagnet that releases the pawl. The striker thus _
freed swivels about a pivot under the action of centrifugal force, and
occupies its working position by the instant when it meets the crosshead.
The disadvantages of the hammer are the same as f~r the drop hammers.
Adequate precision of results can be achieved with r~liable registration _
of forces and deformation, and estimation of bending deformations with
stretching of ~he specimen. Despite the capability for impact tests with
velocit}~ up to 50 m/s, sufficiently reliable results can apparently be -
obtained at velocities that do not exceed 7-8 m/s.
The inadaquacy of the rotary hammer has been eliminated to a great
degree in the design described in Ref. 37. An overall view of the unit -
is shown in Fig. 1.1. Specimen 1 with dynamometer 2 is fastened and
cer.tered in cylindrical gu.ide 3 hinged to lever 4. At the instant
IA 19 J J6 r t 7 6 J P9 TJ
~ ~ ~ ~1
4 � ~ ~
~ ~ I
i 14 1 I
I V ~ _
_ ~15 `
I / ' - i .lO I~~. . y - - � -
u
.i.
~ ~ 9 ~ ' ll !2 ~ 10 1
~ ~ � _
~ ~ P1 /
I
i .
~ t
.~.~--._...'-"''I`.-' . d~~3'--._�J
i
- I
~
I l7
- i ,
_ Fig. .':_.1. Diagram of rotary ha~ner [Ref. 37). See the text for details.
preceding impact, wedge 5 with springs 6 produces slight tension (about
10 kgf) o n the specimen with dynamomster through clamp 7. When rotor 8
has reached the required speed, a controlling signal from the control
panel through arcless breaker 9 releases the lever from pin 11 by means
- of electromagnet 10. Under the action of spring 12, the lever puts con-
nected roller 13 under the thrust of striker 14 and, turning about pivot _
15 through an angle determined by gap ~Z, transforms the rotary motion ~
of the ro tor to translational motion of clamp 16 that has a velocity _
= range of 2-30 m/s. After destruction of the specimen, motor 17 auto-
matically switches to self -braking, and in the case ~f a single strike
lever 4 is held by a special catch (not shown) in a rosition that
8 -
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prevents idle knocking of the striker against roller 13 during deceler-
- ation of the rotor (5-6 s). Lever 4 i~ set in the initial position by
worm gear 18, 19. Gap ~Z can be regulated by motor 20 through worm gear
21, 22 and eccentric shaft 23. By regulat~.:g this gap, rhe force that.
~rises in the specimen upon impact can be varied with fixed rotor speed.
- flydrauZic ar~d pneumatie facilities based on the use of energy trans-
mitted by a liquid or compressed gas enable testing over a fairly wide
_ range of strain rates. ?hese devices are not industrially produced in
the Soviet Union, and therefore a number of laboratories have developed
- original installations [e. g, aee Ref. 16]. In the United States, same
recent developments of the MTS system are used in quasistatic tests.
Pneumatic facilities [Ref. 7, 20, 70] enable tes*_ing to a fairly high
strain rate (10`' s-1). It should be mentioned that A. A. I1'yushin (Ref. ~
86J was the first in the USSR to develop a pneumatic instalZati.on with
technical parameters that satisfy modern requirements right up to the
present. Ref. 94 deals with a hydraulic facility that operates over a ~
wide range of velocities, and can be used for both static and dynamic
testing (E = 50 s-1) .
. The simplest design of a pneumatic facility (hammer) includes the follow-
ing principal elements: a cylinder, a piston, a system for emptying the -
= cylinder and a device for fastening the the specimen. To increase the
_ speed of hammer action, the emptied part of the cylinder must have a
minimum volume. The speed of the piston depends on its mass and the
elements attached to it, and on the speed of operation of the valve that -
= bleeds pressure in the cylinder. A low-pressure line is provided for
regulating the position of the piston before the test (fastening the _
specimen, pretensioning, compression and so on). Systems with replace- -
able diaphragms that rupture for rapid discharge of gas from the cylinder
have given a good account of themselves as emptying devices. Ref. 70
- describes a dev ice in which two diauhragms are ruptured (in two stages).
First a blade slits one of the diaphragms, which leads to a rise in
pressure on the other diaphragm. The abrupt bursting of the second dia- -
phragm empties the cylinder at the appropriate rate. The kinematics of
pneumatic facilities are considered in Ref. 70, 94.
_ The use of energy of powder and explosives for dynamic testing was started '
_ long ago because of a number of advantages of the method. Ordinary
firearma are the simplest means used for impact loading of a specimen.
Davis [Ref, 29~ noted that different force-time relations can be realized
by changing the shape of the tip of the bullet. A disadvantage of Che
method of direct impact of a bullet against the end of the specimen is
that contact phenomena arise that are difficult to account for in pro-
cessing results. This disadvantage has been eliminated in the loading
methods described in Ref. 45, 77. G, M. Kozlov [Ref. 45] has proposed a
facility in which a tubular specimen is subjected to tension upon impact
of a lead bullet traveling at 300-800 m/s through a bullet guide con-
nected to the loaded end of the specimen. Ref. 77 describes a facility -
that can strain a specimen at rates up to 1200 m/s.
9
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rUK ~rrlLttu. UJ~ U[VLY
The use of explosives for loading a specimen does nat require special
devices, excepting those cases where the force is transmitted through a
- transfex� medium (for example a liquid) . Ref. 41 describes methods of
ioading a specimen by forces of inertia of a masa with detonati~n of an
expiogive charge on its surface, Ref. 141 dewonatrates the use of ex-
plosives as a source of energy for loading a Hopkineon bar, and Ref. 169
shows the use of explosives for studying surface waves.
The use of explosives is accompanied by electromagnetic pickups, and as a
result registration of the stressed state is done either at a point
removed from the blast point, or the surface of the specimen is studied ~
by high-speed photography. We should point out the similarity of loading
by explosives and by electrical devices, which is especially apparent
_ when forces are transmitted through a transfer medium. "
1.2. Magnetic-Pulse Facility -
. It is comparatively recently that electric energy has been used for
mechanical tests. The basic principle of dynamic machir.es accumula- _
_ tion of considerable energy and release over a short time has been
most successfully realized in facilities intended for dynamic and quasi- ;
static testing. Energy is accumulated by capacitor banks, and discharge ;
is accompanied by release of the energy. The device used for accumu- ~
lating and relea~ing energy is called a pulse current generator. ltao ~
methods are known for transmitting the discharge energy of capacitors to
_ a test specimen. The first is based on deformation of specimens by
rather high-power pulsed electromagnetic fields [Ref. 9, 21, 34, 44,
136]. The technique is most simply realized by special inductors. Upon
discharge, electrodynamic forces repel a metal body that is freely set
on the inductor and is used as the striker on the test specimen. The
- othar method of transmitting energy consists in using the so-called elec-
trohydraulic effect, resulting in intensive pulse pressure in a liquid as
a consequence of electric discharge.
- The use of facilities based on electric energy stored in capacitors is
dictated mainly by high efficiency and the capability for producing short
and fairly intensive loads. An electric diagram of a pulse generator is
R Fig. 1.2. Electric diagram of a pulse
_ ~~p generator: T--high-voltage step-up trans-
~ 3 H== former; B--high-volCage rectifier; C--
b capacitors; P--spark gap
shown in Fig. 1.2. The element for electrical-to-mechanical energy con-
- version (denoted by H on the diagram) in this case is based on free dis-
_ charge in a liquid. The most important characteristics of the generator
are the installed power and energy of the discharge, and also the period
of the discharge. The discharge energy is
G
oz
E = z C,
10
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FOFt OFFTCIAL USE ONLY -
- where Up is the working voltage. The frequency of capacitor discharRe
~ determines the shape of the pulse acting on the test specimen.
MIU-40/5 faciZity. The schematic electric diagram of a facility used
_ at the Institute of Mechanics of Polymers, Lithuanian SSR Academy of
Sciences, is shown in Fig. 1.3. The circuit operates as follows.
I i
i2 p! R ~ , -
~ 4-4 6 ~ ~ i
~ ,
i i
i i
zo ; ; r
,+n-so R R ; ' ~at,
K~ Qv Tp/ Rr ~ rpt?
g K6 ~
Ipy I p~ u
K
R R
. /yr1 ~7}
YO ID ~0 X, R qw ~i
X i
; JO K ~
r
o~ i a
, 61l~ GK,
' ~ D~-D,. IP [rar Ru
. 10 1
~21.~[l~/Nit AE~9~A~9 ?p ~a pisR 7P
~
; w -
� i ~ ~ PQ.~9 JP
_ I1--set awitch ~ v
~2 -iii~r;h voltage ~p 2p 3P
;3--~afety p
Fig. 1.3. El~ctric schematic of the MIU-40/5 facility at the Institute
of Mechanics of Polymers [see explanation in text] -
_ Pressing the "charge" button jaapttg] engages contactor K1 that sends
voltage to step-up transformer Tpl. The secondary voltage of the trans-
former is rectified by high-voltage rectifier BB and is fed through
current-limiting resistors R1-R4 to sections of the capacitor bank
1~1 and KG2. The charging voltage level is monitored by microammeters
connected in the circuits of each of the sections of the capacitor bank.
Charging of the capacitor banks is stopped by pressing the "STOP" button
jC7n17]; when this is done, capacitor K1 is disconnected. The "discharge"
button [pa3pRyJ is pressed to discharge the sections of the capacitor _
bank to the inductor. Doing this energizes the ignition system consist- -
ing of elements Tp2, ,Lj1, ...,,L16, C1, C6, R~, C7, PI and P6. The igni-
tion system produces a high-voltage pulse of 25 kV that is fed to the ig-
nition electrodes of transformers Tpl, Tp2. The pulse ionizes the gap -
between the electrodes, resulting in discharge of the capacitor bank to
the inductor. A more detailed description of the principal components
of the circuit is given below.
11
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- , ~
- Car�acitor bank. The MIU-40/5 facility uses IMU-140-5 capacitors dis-
tingui~hed by low self-inductance. The capacitor bank is made in the
- fr~rm of two sections of ten capacitors each. The choice of two parallel -
_ operating sections of the capacitor bank facilitates the work of the dis-
chargers. The possibility of non-simultaneous firing of both dischargers
- is prevented ~y a special ignition circuit. In each section, the capaci-
tors are interconnected by three-layer flat busbars made of copper plates
6 mm thick.
Dischargers (Tpa). The facility uses two dischargers operating in
parallel. Dischargers of coaxial type are used to reduce self-inductance.
- Teflon is the insulating material. Practical tests have shown that dis-
charge tracks can be observed on the electrodes after 150-2J0 operations.
The discharger leads are interconnected by a three-layer system of flat
busbars to wnich leads are attached for connecting the load.
_ Step-v.p transformer, high-voZtage rectifier. The primary of transformer
Tp3 (see Fig. ]:~.3) is connected to a 220 V line. The MIU-40/5 uses an
NOM-10 voltage transformer with seconnary voltage of 10 kV and power of
' 750 W. The time for charging the capacitor bank to the full working j
voltage is about two minutes. The high-voltage rectifier rectifies the I
secondary voltage of the step-up transformer. The rectifier is based on
silicon semiconductor diodes. The working voltage of the rectifier is ~
28 kV, which is much higher than the voltage rating of the transformer. ;
The rectifier operates in a half-wave circuit.
- Ignition deviee. The capacitor bank is discharged when a high-voltage
pulse produced by a special system utilizing low-power step-up trans-
former Tp2 (see Fig. 1.3) with secondary voltage of 4 kV is sent to the
controlling electrodes of a trigatron. For reliable firing of trigatrons
the voltage pulse must have a steep leading edge, therefore the ignitior.
voltage Uign is chosen 2-�3 times as high as the charging voltage of the
capacitor bank. The voltage of the secondary winding of the transformer
is stepped up by multiplying the voltage of,L(1,...,,Lj6 and C1,..., C6
_ by a factor of six. Ignition capacitor C7 is charged to a certain level
of voltag~ (24 kV), the capacitance of 0.8 uF ensuring a sufficient dis-
- charge frequency of C6, and thus providing a steep front for the initiat-
ing pulse. When the voltage across capacitor C7 reaches 20 kV, breakdown
of the spark gap takes place in discharger P1, whose second electrode is
- connected through resistor R to the zero wire. When this happens, a
high-voltage pulse is also sent to the controlling electrodes of triga-
trons Tp21 and Tpz2. The selected discharge frequency of tha ignition
capacitor, which is 5-10 times as high as the frequency of the main cur-
rent of the capacitor bank, is conducive to simultaneity of trigatron
- firing and discharge of both�section of the capacitor bank. ,
~2e eontroZ eireuit of the facility is based on DC electromagnetic relays
supplied by step-down transformer Tp3 with rectifier ~(7-I~10 connected in -
_ a bridge circuit. Also connected to the transformer secondary is the
supply to the signal circuits (.lIC-7. ...IIC-3) . When AT!-50 is switched on,
12
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voltage is fed to the facility that goes through normally clossd con-
tacts of relay 3P to the windi.ngs of safety discharge cont3ctors K2 and _
K3. Pressing pushbutton sapRg sends voltage to intermediate relay 1P
through normally clcased contacts of relays 4P and 5P. Intermediate relay
1P energizes contactor K1, which sends line voltage to the primary
winding of step-up transformer Tpl. This results in charging of sections
1{61 and 1~2 of the capacitor bank. The resistance of R4 and R2 is ex-
perimentally determined for simultaneously charging these sections to a _
given voltage.
The voltage level across the capacitors is monitored by two microammeters
connected to high-potential voltage dividers R7, R8 and R11, R12. Resist-
- ances of R19 and R25 are used for calibrating the microammeters with re-
spect to a laborator}~ voltmeter. Capacitors C8 and C9 that shunt the
microammeters protect them from pulse interference during discharge of
the capacitor bank. Charging of the capcitor bank can be terminated by
pressing pushbutton C7n17. This interrupts the supply to relay 1P and K1,
and disconnects transformer Tpl from the line. Charging is also termi-
� nated automatically when a certain voltage is reached, that is set by
_ potentiometers R21 and R23 through which voltage relays 4P and 5P are
supplied from voltage dividers R9,..., R14. Patentiometers R21 and R23
~ are selected so that at thei.r maximum resistance the operation of rela;;,~ _
4P and 5P occurs at a voltage of 5 kV on the capacitor bank, i. e. the
maximum permissible voltage. The next major operation on the facility
discharge takes place after pressing the pushbutton pa3pstg. This
energizes intermediate relay 2P that supplies the ignition unit. _
1.2.1. Loading by Electrodynamic Forces
- The ma.gnetic pulse facility can be a generator of a powerful pulsed mag- _
_ netic field if an inductor is connscted to its output terminals, and the
magnetic field induces eddy currents in metal comvonents in the near
vicinity. Interaction between the magnetic field of the inductor and the
induced field leads to mechanical body forces in the inductor and the
components directed in accordance with the well known left hand rule.
As a result of this interaction, the component is pushed away from the
securely fastened inductor. Ano~her method that is used for transmitting
energy to a specimen from capacitors is based on repulsion of 2 conductors
with currents in opposed directions. At present, both of these methods
of transferring the energy of strong magnetic fields to a test specimen
- can be considered fairly well developed. The former is realized by
single-turn inductors (Fig. 1.4); the latter is based on the use of
Fig. 1.4. Diagram of specimen loading by ~
a single-loop inductor: x NI~GY= to magnetic- x NSI
pulse facility =
many-turn inductors paired with massive bodies. The accelerated masses
may be plates [Ref. 9, 21, 34, 44], cylinders, etc. The working principle
of the single-turn inductor is as follows. A one-piece or composite band
13
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is shaped into a loop. One half of this loop is secured to a massive
base, and the other half rests freely on the surface of the specimen.
The ends of the inductor are connected to the leads of the magnetic-pulse
facility. Discharge of the capacitors of the magnetic-pulse facility to
the single-turn inductor leads to electrodynamic forces of repulsion
between the two halves of the inducror. The use of the single-turn
inductor for testing beams and rings is described in more detail in the
next section. Discharging the capacitors of the magnetic-pulse facility
produces electromagnetic pickups that complicate the recording process in .
tests with single-turn inductors. For thisreason, recording must be done
on the surface of the specimen at a point rather far removed from the .
inductor, or else t?~e stressed and strained state that arises in the
specimen under the action of inertial forces must be recorded (i. e. at
- the instant of termination of' the discharge and hence when the influence
of electrodynamic forces is.absent). Complete independence of recordings
from pickups due to dischar_ge is attained when the force is transmitted '
to the specimen by using many-turn inductors to hurl massive bodies.
To eliminate interference in this case, it is only necessary to choose
the distance covered by the thrown body L before it meets the specimen
such that the relation
y > T
0
is satisfied, where Vp is the mean velocity of the hurled body, T is the
- time of discharge. The impulse of the force acti:lg on the thrown mass is
found from the formula [Ref. 43]
- p_ 2nE~on2 (h'i+Rz) z~~y s s 1'h
- y~ (Rz-R,) ~ I sh 2 ; (1.1)
f~-f-S~R2; G+~' Eia is the coefficient of induction; R2, Rl are tY:e
1
outside and inside radius of the inductor respectively; h= H/R2; H is the
thickness of a turn of the inductor; d is the distance from the inductor
to the mass to be thrown; I is the current in the inductor; n is the
_ number of turns; c is the width of the insulation; b is the width of a
- turn. The velocity with which the mass is thrown is V= F/m. ~
At the Institute of Mechanics of Polymers, Lithuanian SSR Academy of
Sciences, re~earch has been done on throwing a mass by an inductor that
is a copper plate in which Micarta is cemented into a milled spiral gap.
_ Copper bars 8 mm in diameter were welded to the plate as leads of the
inductor. The design of the projectil.e-mass is shown in Fig. 1.5. The
_ way that the velocity of this pro~ectile depends on the parameters of -
the ind.uctor (Table 1.1) and the charging voltage of the capacitors is
shot~n in Fig. 1.6 and 1.7. These figures also show the results of calcu-
lation by formula (1.1).
14
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TABLE 1.1
Variants
Inductor parameters
I II III
Number of turns 3 6 6
Inside diameter 2R1 0 0 0
Outside diameter 2R2 70 70 70
Turn thickness H, mm 2.0 2.5 2.0
Turn width h, ~mm 9.0 5.0 9.0
Insulation width c, mm 1.~ 1.0 1.0
Inductance, n?.~i 0.244 0.833 1.640
Resistance, S2 8.6 27.5 34.4
I Ie
I ~ b
I '
Fig. 1.5. Shape and dimensions of ~
- projectile-mass , .
~eo----- ,
,
~~c 3 V, n/c
V~ ID S ~
e ID~8
III3SS g 1
30 ~ a,~ s0 47,1 666
~ 12 SB
V ~
~ Cl1TV l ~ ~ (
~ .
/ / I ---1-- . , s
- - ---f- 2~ ~ ~
. / ~ ~ 6
( , .
~ ~o - - 20 - - ~
~ i 'k~ I
u, kv ~ ~ u, V
~ ~ U,KB ~ I u, a !
_
- 0 0,4 ~o,e ~2 v 40o r~oo
Fig. 1.6. Curves for projectile velocity as a function of voltage across
the capacitors: 1 and 4--calculation; 2 and 3--experiments using in-
- ductors I and III (see Table 1.1). Projectile mass.30 g.
Fig. 1.7. Curves for projective velocity as a function of voltage across
the capacitors and the value of the mass: 2, 4, 6--calculation; 1,, 3, 5--
experiment
15
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rvn vrrt~lru, u~~, VLVLY
w /
J~ a + e~ c
~ . / s
T ~~2 . eo .
/
--l60--- /
' b~b _ C =_300 uF ~ ~
Y' M~ .
c�aoo ~ na? ao
8~ ~ s ~
20
ao - ' ~ uF
_ '~i ~ �
~ o ~s rso rzs ~oo
40 -
j 75 Fig. 1.8. Curves for velocity of
po _ throwing by an inductor (a) as a func-
~ p,kV tion of the voltage across capacitor~
o' uo~8 (b) and their capacitance (c). Projec- .
� . 2 3 k S tile masses 92 g( ) and 41 g(---)
[Ref . 91] .
Curves for the velocity of the projectile mass as a function of capaci-
~ tance of the capacitors as plotted by G. V. Stepanov [Ref. 91] using an
- inductor consisting of 28 turns o:f PE 2.0 wire are shown in Fig. 1.8. i
1.2.2. Loading by an Electric Discharge in Liquid
A method of producing high pulse pressures in a liquid by an electric
discharge was discovered by L. A. Yutkin [Ref. 111] and called the elec-
trohydraulic effect. The use of the electrohydraulic effect for testing
beganfairly recently [Ref. 3]. At the present time, electrical energy
is converted to mechanical energy by two equivalent techniques: by ex-
= ploding a wire in the liquid, or by an electr~c discharge in the liquid
[Ref. 14, 40J.
EZectric discharge in ZZ(~'7,i2CZ. The process of electric discharge in a I
liquid is characterized by a number of sequentially occurring effects.
First electric breakdown takes place between electrodes with the forma.- -
tion of a spark channel in which high temperature leads to the formation
of a gas-vapor cavity with high internal pressure that causes abrupt
expansion until the pressure inside the cavity is equal to the hydro-
static pressure of the ambient liquid. Then the radius of the cavity ~
- fluctuates, attenuating in time. The abrupt enlargement of the gas-vapor
cavity produces a shock wave in the ambient liquid.
In addition to formation of a shock wave and external fluid flow around
the gas-vapor cavity, the discharge is accompanied by acoustic, ultra-
sonic, x-ray [Ref. 18] and other phenomena that have no significant
influence on the process of deformation of the test speicmen, but do
affect the recordings made during testing. The numerical values of the _
pressures on the shock wave front during free electric discharge in a
liquid depend on many factors. Among these, we note the size of the
16
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~
FOR OFFICIAL USE ONLY
P,an,,, ,
P, atm ~
au~
eoo -
O
Q00
o s
i
400
~
r
200
U0, V
_ ~ U� e
sao +soo tsoo
Fig. 1.9. Pressure in liquid accompanying electrohydraulic effect as a
function of the discharge gap 8 and initial vol~age acorss the capacitors.
The pressure was measured at a distance of 200 mm from the initiator. -
discharge gap, the voltage across the capacitors, the inductance��and
capacitance of the capacitors.
Fig. 1.9 shows the way that pressure depends on the discharge gap and
voltage across the capacitors of the MIU-40/5 facility. The pressure
- was determined at a distance of 200 mm from the discharge source by
measurement of deformation on the surface of an aluminum rod with its
end immersed in the liquid.
Chapter 2 ~
RECORDING DEVICES
Recording during dynamic testing is faced with a number of difficulties
due to the brevity of the processes to be recorded. The measurements are
influenced by the time lag of the recording sensors, wave phenomena in
the specimen, and the frequency responses of the recording and amplifying
equipment Development of correct and sufficiently reliable recording
methods is aimed primarily at eliminating the influence of these effects
on the parameters to be recorded.
2.1. Measurement of Load and Pressure
The load must be recorded in addition to strain measurements for plotting
stress-strain diagrams in quasistatic and dynamic tests. When forces
are transmitted to the specimen through an intermediate medium, the
pressure is measured. .Linear forces are determined by sensors (strain
gages, piezoelectric sensors, capacitive, inductive, dielectric). The
17
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- ? vi. v. L~ 1~.~.na, u.)l:. varL,1
main requirement to be met by the dynamometer is that readings must be
independent of strain rate, i, e. the static and dynamic calibration
should coincide. Force registration by dynamometers reduces to measuring
deformation of an elastic element by sensors of a type determined by the
anticipated magnitude of the force as well as the time of its action.
- The elastic element of the dynamometer may be the thickened portion of
the specimen [Ref. 7, 20] or a sgecial structural component that goes
beyond the limits of the specimen. The simplest dynamometer of the
latter type is a long waveguide rod. The main requirements to be met
_ by dynamometers that are a thicker~ed portion of the specimen are: the
stresses in the thickened part must not exceed the yield point of the
material, which should show pure elasticity. Therefore the given method
of force measurement is used only in testing metals. The values of the
force measured on the thickened part of a specimen in quasistatic tests
- of pclym~rs cannot be considered reliable because of the dependence of
the mechanical properties of the specimen material on strain rate.
2.1.1. Piezoelectric Dynamometers
- Dynamometers may be placed on the striker or in direct proximity to the
stationary clamp. Combining the dynamometer with a massive striking body ,
is most ~ustified in recordings that accompany tests for dynamic bending.
I
: Fig. 2.1. Diagram of the simples~ i
dynamometer
,
, I .
. I - _ .
A ~ A.~ . _ _
~ ~ ~C""~ R3 �1!
~oR Jl~
rs~n
~ nro: nap
I ~ R6
i L R2 ~ ~ 1.3K
~ i Q� -lt
~ Fig. 2.2. Circuit of the simplest
piezoelectric amplifier [Ref. 66]:
_ exog = input
The simplest dynamometer located beyond the limits of the specimen is
shown in Fig. 2.1. The construction of the elastic element depends on
the method used for measuring deformation. The dynamometer has the -
18
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i
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simplest construction when deformations of the elastic element are mea-
sured by strain gages placed as cZose as possible to the clamp.
Regardless of the measurement method, the dynamometers must meet the
following requirements: 1) the frequency of thei�r norsal modes must be
suff iciently high; 2,1 the readings of the dynamometer must depend linearly
on the applied load. These requirements are interrelated since they
- reduce to the necessity for high rigi~ity of the dynamometer, which in !
turn leads to a reduction in sensitivity of strain gage registrations.
As a result, a trend was noted long ago to force measurement by piezo- .
electric pickups with sensitivity 2-3 orders of magnitude greater than
that of recording by strain gages under conditione of identical dyna-
mometer sensitivity. A definite advantage of using piezoelectric sensors
is the capability for measuring weak forces, which is very important in
tests of low-modulus materials, filaments and so on. Another advantage
of piezoelec~ric dynamometers that is no less important is practically
total shielding from pickups due to ionization of the air when explosives
are used, and also with the use of electric energy stored in capacitors.
Piezoelectric measurements are based on the piezoelectric effect, and on
the charge that arises when a load acts on a piezoelectric element. -
Quantitatively, the piezoelectric effect is determined by the relaticin
q = aP~ ~1.2)
- where a= const is the piezoelectric constant. The piezoelectric effect is
displayed by materials among which quartz occupies a special place due to
high mechanical strength. A number of ceramics that have lower mechanical
strength have a piezoelectric constant an order of magnitude greater than
that of quartz (Table 2.1).
The emf that arises upon impact loading in a piezoelectric element can be
recorded by a reproducing device without intermediate amplification.
However, for purposes of static graduation of the piezoelectric element
an amplifier with high-impedance input is needed to reduce the rate o� ~
bleeding of the charge from the element.
Piezoelectric amplifiers have practically zero lag. Fig. 2.2 shows a
schematic diagram of the simplest amplifier [Ref. 66]. The standard PM-1
- set (East Germany) includes a piezoelectric amplifier with high�-impedance
input. The device can be used for static graduation of piezoelectric
elements, and has a wide frequency passband. Standard dynamometers
called force sensors or pressure sensors are also produced in the United
States and Denmark. Practically aZl standard force sensors have a low
level of ineasurable force, and therefore are used mainly in testing
materials for harmonic oscil~.ations (from 0.5 g to 5-10 kg). -
D~nczmometers for tensiZe tests. In contrast to designs with annular
piezoelectric elements in standard non-Soviet dynamometers, the devices
19 -
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rvn vrrl~irw u~~ VLVLs
m ~c~ I~~ ~ i ~I
I o
~ tl $ r
N
_ ~ .
m
V
F .
A Q
~ ~ T
O ~ '
s V ~ ~ Ca C~
II II ~
~r1 C m $ �
v CV
_ n O~$ p .
~ n _ I m ~ ~
tl.a0. ~
- .'iCV . .
9 � . '
7
_ C
u � . '.~.r
~ ~ \
� - t~ p U~
M U OcVt~ ~ O I N O
~ ~ II II - ~
~ n ~ ~ ~ ~ r, ,-{c~
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~ - i--I tA U
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- v . ~
~ � N 4.~ N N
a ~ ~ ~ ~ H~
~ ' 1J \ ~ \ i
-0, p
' b cd ~ GfI b0
O N .
~ h C aQ. ~ g~~ I vI tn 00 ~G aG
~-I N m O N N.L" .C
Q U ~ ~..i ~
cv . , , ~ , ~ u w v~i ~ ~ (
W _ _
~ t) c~v u c~v o ~ t~.i y a1 1~-+ N
~ ~ N ~ 41 J-1
H ~ u~ II ea O M ~ ~ ~ U N m cn
~ M ~ m v o0
cv - u ~ ~ ~ r-1 ~
- �rl O tA rl rl
q N~ a tA TJ '
_
a u~ u~ ccOV ~~ri U O d N
, ~ e~
1 A W~ U E+ o~t
`~j~W O n I I~ i i i i i i
~ F n�a N t~ ao rn O'-1
F ~ ~
M
~
0 p p
M~I~ N O lT! pQ
F CQ N M CV tD ~ i
N ~ I
L"+
cd _
~ ,
~
~r~l
~ ~ CV Q G y ~1 ~
~ t0 N 4 ~ ~ 1~ QJ I
~ N ~ M ^ ~ Ri 41 ' -
cd q ~u
ci u1 O cb
Q~1 t~.~ 3a O -
� ~J ~r~l ~ri
~ 0 ~ Q N N~ N G"+
F- ~ 1J ~
vi ~ ~ 1- q (d 3-1 rl 'U 'U
C ` ~ ~ a ~i ~ ~ ~ ~ ~
~b ' ~ ~ V ~ ^ QI & sa a a
~ a ~ c�CV~~ $ i I I I I I
~a .a i, v u,p . ,p rl N rl ~7' ~1
. ~ V p ~N v~:.. va V^ -
O i �-Spu pc~jp ~
4 L
.C X q Y O, X p, ~L
o 1-~~ ~`hO.` !~o+`FO-~u ~
- .1 . 'Qi .1 i .1 .1
. /1. /ti r\ ~ , /1
~D~f~ CO Q1 ~ ~
u v ..i u *-I rl
v v
2~
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~ ,
, a
~ ~
~ ' ~o-.w~ ~ _
- ~ ~
I ?
" J . ~ ,
- ~
i i
e ~ ~ _10 1.
~ -
~ m i ~ ` ~
'~'%~4~:: .~cmc~
~
Fig. 2.3. Con.struction of piezoelectric dynamometer for tensile tests:
1--piezoelectric sensor; 2--specimen; 3--strain gage
Fig, 2.4. Construction of piezoelectric dynamometer for compressive
- tests: 1--piezoelectric sensor; 2--specimen
desr.ribed below use piezoelectric elements in the form of disks that can
take considerable forces. The time lag of the force sensor is evaluated
on the basis of dynamometer design data, including the clamp for holding
the specimen if the dynamometer is intended for force registration under
tension. Fig. 2.3 and 2.4 show dynamometer designs in which the piez~-
electric sensor is located as close as possible to the specimen in the _
clamp, which leads to an increase in the natural frequency of the dyna-
mometer [Ref. 50, 51]. The piezoelectric element is installed in a
window in the elastic element, which is combined with the clamp and
precompressed by a screw. As the specimen is stretched, deformation of
the elastic element relieves the pressure on the precompressed piezo-
electric sensory and an emf arises. Piezoceramic pellets 8 mm in diam-
eter and 5~n thick are used as the piezoelectric element. The natural
frequency of the dynamometer depends on its geometric dimensions.
Dp~namometers for compressive tests have a design analogous to that de-
~;cribed above. The only difference is that there are no clamps, which
increases the natural frequency of the dynamometer. Fig. 2.5 shows a -
,typical oscillogram produced by a dynamometer with natural frequency of
_ SO kHz [see Fig. 2.4].
2.1.2. Waveguide-Dynamometers
The lag time of dynamometers described in the preceding section limits
- their applicability in tests with strain rates of about 20-50 s-1. The
' range of applicability is expanded by using waveguides, the main purpose
being to prevent the natural vibrations of the dynamometer from being
21
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_ .
~ iu~~l
. .;d' p:' . ,yy:.
I~1'.
, {
~ :~~1.. ',~I
':.f.:.l~' .
.jy,
5 y_� .
- ~ q~ ;w
~ , . ?
~f '~1 ~.;.r .i~, ~ ~�.,~i
I : ; 4,...
7' ,i,-
/ I� �
:
~ i
/ !1 , '
~
~ ~
~ J :
J
yl . i ~.~~r. ,i ~ I
..1. . . . ( ~ . . ~
I
Fig. 2.5. Typical force-time oscillogram produced by a dynamometer [see
Fig. 2.3]. Testing of glass-textolite at e= 5 s-1 (AST fabric 50%,
EDT resin 50%).
passed on to the specimen. A diagram of force measurements by waveguides
_ is shown in Fig. 2.6. The length of the waveguide is selected so that
Fig. 2.6. Diagram of force regis- �
tration by a waveguide in compres- ' ? ~ y
sion tests: 1--waveguide rod; 2-- rooo~-~--~-- ;
support; 3--strain gage; 4--specimen. ~-8 ` '
_ . _ _
Cj 6s i
the reflected waves from the left n ~ ,
end do not reach the specimen during =D
testing. If the lengt:~ of t~?e p
specimen is more than an ord~er of ~y,~y
magnitude less than r..he lengr:;~ of
" the waveguide, a qu~isistatic state
may be rapidly reached in the specimen, whereas there is no such state in
the waveguide thr%~ughout the course of the experiment. The simplest ver-
sion of a waveguide is a specimen with an elongated dynamometric section -
[Ref. 76, 9 2]. The advantage of a force sensor with a long waveguide rod
22
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is obvious, and shows up in the absence of distoztions caused by natural
vibrations of the force sensor and elements of the facility [Ref. 23]. -
Methods have been developed at the Institute of Mechanics of Polymers,
- Lithuanian SSR Academy of Sciences for recording quasistatic compression
with a strain rate of up to 100 s'"i by using a waveguide [Ref. S:i].
Given below is the experimental technique and a comparison of the results
- of ineasurements by the dynamometers described in 2.1.1 and by waveguides .
Stress-atrain diagrams obtained by these devicea are shown in Fig. 2.7,
showing the feasibility of plotting a reliable diagram right up to
e = 100 s" 1.
G~~` , Fig. 2. 7. Diagrams of Q/ob vs e in which _
oq _ _ the values of Q were found by a waveguide
~ and by a piezoelectric dynamometer
,p~ in compression tests: 1--strain rate
2~ 26 s~l; 2--5 s-1; 3--10-3 s-1; Qb is the
- o? _ _ static breaking point.
~ 3
J.~ -
' A steel rod (waveguide) 10 ~ in diameter and
-1- ~~y' 1 m long [see Fig. 2.6] was placed in Teflon
0 0,~ n,2 0,3 0,4 o,s bearings and put into contact with a test
specimen resting against a rather massive
plate. The specimens were bars with a cross section of 1Ox10 mm and
length of 30-40 mm. The loading device was an MK-30 pendulum haBUner.
The signals from strain gages A and B cemented close to the contacting
ends of the rods were amplified by a T11-M device and fed to the inputs _
of two 51-29 memorizing oscilloscopes. Considering that the steel rod
was elastically deformed, the experimental relation between strain and
= time (eI-t) on the end of the steel rod was transformed to a relation -
- between force and time (P-t). Dynamic stress-strain diagrams were -
plotted from relations eIi-t for a polymer rod, and from P-t curves.
Pressure sen3or~. The necessity for pressure measurement arises in cases
where loading is done by an explosive charge or by underwater explosion
of a wire. In either case the dynamic pressure is transmitted to the
specimen through an intermediate medium (liquid) in which the pressure _
_ is recorded. Pressure sensors produced by a number of non-Soviet enter-
prises and companies have a fairly wide range of ineasurable pressure. -
_ The natural frequency of the sensors is no more than 45 kHz. Quick-
response instruments based on waveguides have been used for recording
pulse pressures over the last few years. A version of pressure recording ~
using a waveguide is shown in Fig. 2.8. The waveguide material is chosen
so that the acoustic stiffness pc (Table 2.2) of the piezoelectric element -
- is not appreciably different from that of the waveguide. The sensor for
_ measuring pulse pressure [Ref. 66J (see Fig. 2.8) consists of a waveguide
and piezoelectric element that have the same acoustic resistance.
23
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. v.. va a a.~.itu~ uJ1i v1ll.il
TABLE 2.2 -
'~Javeguide pc�10'4 Piezoelectric pc�10'`'
material g/s�cm2 material g/s�cm2
Bismuth 214 H6C-1 200-240 -
Cadmium 240 H6C-3 200-230
Zinc 296 uTC-19 290
Aluminum 169 quartz 152
Steel-3 454 - -
.
. . ,
j �
P 9 I 2'
Fig. 2.8. Piezoelectric pressure sensor: 1--elastic rod (waveguide); I
2, 2'--versions of placement of piezoelectric elements; 3--casing; 4--
insulating tube
The sensor for measuring pulse pressure, including the piezoelectric
ti~ element, can be statically calibrated.
The simplest version of the pressure sensor is a metal rod waveguide
with cemented strain gages (results of application of the sensor will be
considered later on).
2.1.3. Capacitive and Dielectric Pressure Sensors -
_ Below we describe methods of recording dynamic forces and pressures that
have not been extensively used because of the complexity of realization
and limits of applicability. Nevertheless, we will consider them in this
section since experimental practice does not exclude cases where just ~
such methods are the only ones that can be realized. ~
A capacitive sensor of diaphragm type is described in Ref. 36. The pulse
pressure to be measured by the sensor compresses its elastic element
(e. g. a mica ring), whose thickness determines the gap in a capacitor.
The sensor is connected in a tank circuit to which high-frequency voltage
is sent from a stabilized quartz-crystal oscillator. The tank is tuned
_ slightly off the frequency of the oscillator. Change in the capacitance
of the sensor under the action of a load increases or decreases the mis-
match of the tank frequency. The amplitude of the rf voltage on the tank
varies as a f unction of the magnitude of the force acting on the dia-
phragm, and as a result the carrier frequency is amplitude-modulated, and
- 24
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i~
_ ~ p ~ j eooe earrn 6 I
+ ~ ~ ?~o ~
~u' _ ~ I
~ ~ ~ ~ :
, Y ~ 43
_ P to I
Fig. 2.9. Diagrams of a dielectric ~ 73 I
- pressure sensor (a) and a cathode ~
follower (b) [Ref. 93]. ~-----------___..i
then goes to an oscilloscope input through a bandpass amplifier and
detector.
DieZectric pressure sensor [Ref. 79, 93]. This is a capacitor formed by
aluminum foil (0.01 mm thick, 12 mm in diameter) and ad~oining metal
- surtaces with a dielectric film between them (see Fig. 2.9,a). Preliminary
electrical polarization of the dielectric compressed between plates under
a potential of 800 V through a resistor of 4.3 MSt ensures constancy of
the charge accumulated on the capacitor of the sensor and the capacitive
divider. The dynamic component of the voltage across the sensor due to
the change in capacitance with compression of the dielectric is sent _
- thro~}gh a cathode follower (Fig. 2.9, b) to the input of the oscilloscope.
2.2. Measurement of llisplacements and Deformations
The basis of photoelectric methods of ineasurement in quasistatic and _
dynamic tests is the use of photoelectric devices photocells and
photomultipliers that convert radiation in the visible, ultraviolet _
and infrared regions of the spectrum to an electric signal. This con-
version is based on the photoemissive effect in which a flux of radiation
incident on the surface of certain semi.conductors and metals produces
emission of electrons (the physical nature of this effect is described
in Ref. 71). Reasons for using photocells and photomultipliers in dynamic
and quasistatic tests are: 1) practically zero response time; 2) capa-
_ bility of total shielding of the measurement system from pickups caused
- by the discharge of capacitors and ionization of air when explosives are
used; 3) high sensitivity; 4) capability of direct static calibration.
The simplest circuit for photocell connection is shown in Fig. 2.10.
Displacement is registered by a display component (flag) fastened to a
moving element that varies the light flux from the source. Between the
flag and the photocell is a diaphragm (Fig. 2.11), which is a metal~plate -
with a specially shaped opening. Because of the difference in sensitivity
_ of cathode sections relati~�e to the center, the opening has a shape like
that shown in Fig. 2.11 to make the sensor readings linear. Sensors for
registration of 3isplacements in quasistatic tests differ with respect to
the distances between the photocell and the light source. In the diagram _
~
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a) , r
Q1,4 aw ,
. *
2 4 3 .
s)
~ ~ -
pM ~4
1 ! ~ +
- Fig. 2.10. Systems for recording displacements by photocells with an
optical lens system (a), and without one (b): 1--display element; 2-- `
light source; 3--photo~ell; 4--diaphragm; ~A--photodiode
t ~
~
i of Fig. 2.10, b, this distance is set at a
minimum, and the light source and photocell
I are accommodated in two housings fastened by
~ bolts. The clearance between the housings !
permits passage of the flag. A diagram of a
~ displacement sensor in which the light source
~ is more distant from the photocell and the
' photomultiplier is ahown in Fig. 2.10, a.
Fig. 2.11. Construction In contrast to the previously described
of diaphrs~,gm (1) with sensor, thi.s one is equipped with a lens
frosted ~;iass (2) system. Registration of displacements by
photocells and photomultipliers with the use
- of flags is applied in the case of gnasistatic testing of materials.
The indicator flag may be a moving part of the loading device.
2.2.1. Photoelectric Methods i
i
Use of tha method described above is limited to impact velocities at _
which vibration of the indicator component begins to distort the results, ,
and moreover it cannot be used to register small displacements (of the
- order of 0.01 mm). ~
7`he interferometric method of ineasuring longitudinal extension of a '
specimen under tension enables determination of quantities with very high
precision. Ref. 155 describes the use of an interferometric method for
measuring elongation and the angle of turn of a specimen sub~ected to ~
torsion and tension. Fig. 2.12, a shows a system for recording elongation
of a specimen in quasistatic tests. The measurement links were
26
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Fig. 2.12. System for interfero-
metric registration of elongation
of a specimen (a) and stages of
obtaining the dependence in time
of elongation of a specimen under
~1~~ tension (b) [Ref. 154]: 1--specimen;
2--photodiode; 3�--counter; 4--pulse
� generator; 5--integrating circuit
(2) (3) (4).. ~5>
y~,~ a,~�~,~� ~ ~ c, -
A A' B
- 6) Cu2xan sZgnaz
~ interferometers. Fig. 2.12, b shows ~`"A
the corresponding signal diagrams ob-
tained in the circuit elements of c~.~,a' 'll
Fig. 2.12, a. The signals from the ~
photodiodes are sent to a binary- f,i
decimal counter, which is a device c~~a ~ I , ,
~
based on discrete measurement by r
coding of the quantity to be measured c~+~e I IIII l+ I~
(designs of counters are described in
Ref. 71). The pulses have the same
amplitude and rise time and are used
to trigger an oscillator i_n which.each trigger pulse produces a pulse of
constant amplitude and duration at the output. Thus the passage of each
interference band produces a pulse of. predetermined amplitude and dura-
tion. The pulses are �ed to an integrator, and from there to an oscillo-
scope. The interference photoelectric method used for recording defor-
mations in tests of materials [Ref. 5, 115, 116] is now under intensive
development, and certainly deserves attention.
Registration of deformations is done as follows. A diffraction grating
is formed on the surface of the specimen, at which a laser beam is aimed. _
- Upon reflection from the grating, the beam is diffracted and split into
several beams that satisfy the Bragg condition (see Ref. 115):
_ (2.2)
where a is the wavelangth of laser emission, On is the angle of deflec-
tion of the n-th order beam, dp is the lattice constant for the grating
on the undeformed specimen. As the specimen is strained, the lattice
constant changes, causing a change in angle On. Thus the registration of
� 27
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~ r Vc~ vr r tl.trw u~~ U1vLY
deformations during testing reduces to measurement of the angle On. The
theory of the diffraction grating is considered in Ref. 115. The system
for converting the input quantity to recorded strain is not different
from that given above for all practical purposes.
2.2.2. Strain-Gage Methods
_ The change in resistivity of electrically conductive materials uilder the
action of inechanical forces (strain-gage effect) is the basis for record-
ing deformation by metallic strain wires. The most pronounced strain
effect is displayed by semiconductor materials (germanium, silicon, indium _
antimonide, etc.). Both metallic and semiconductor strain wires are _
currently in use as strain gages. The principal equation of the strain
effect for a metallic strain gage is
~R =Se, (2~3)
R
where e is relative deformation of the wire, R and ~R are deformations
[sic] of the unstrained wire and the increment in its resistance under '
the action of a force; S is a coefficient of strain sensitivity (ranging -
from 1.9 to 2.9 for high-resistance wirF~s). -
Tfao kinds of inetallic strain gages wire and foil are produced by
Soviet industry. Wire strain gages are made from annealed constantan i
(GOST 492-52). The strain gages have a comparatively low coefficient.of ~
i_
strain sensitivity (about 2.1). Other disadvantages of wire strain gages
are an inadequate range of ineasured deformations (less than 0.3-0.5~),
sensitivity to transverse deformation and low permissible current.
Foil st.rain gages are made from constantan foil 2-10 um thick. Their
main advantage is lower sensitivity to transverse deformations than wire
strain gages. The use of annealed iron for foil strain gages increases
the maximum measurable relative deformation to 10% [Ref. 35].
- Uf the semiconductor materials tnat have been stuuied at ~he present ~ime
the most suitable for making strain gages are germanium and silicon. The
strain gages made from these materials are called gedistors and krem- ~
nistors respectively. Gedistors are not nearly as good as kremnistors in
_ their operational parameters. The coefficient of strain sensitivity of
semiconductor strain gages reaches 200, which gives a signal at the out-
put of a bridge circuit of the order of a few volts with power of -
hundreds of milliwatts, obviating the need for amplifiers. ,
Measurement and amplification cireuits for registr,ation of deformation by -
strain gages. The strain gage senses the deformations to be measured ~
and transforms them. In order to register changes of resistance, they
must be converted to the corresponding current or voltage. To do this,
the strain gage is connected in an electric measurement circuit that is
capable of performing these transformations. ~ao strain-gage conversion
= 28
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circuits have been developed: the potentiometric circuit and the bridge -
circuit. These produce electric signals with a change in resistance of
the sensor. When metallic strain gages are used, these signals may be so
weak that they are not felt by the recording device (oacilloscope). In
such cases, resort must be had to special amplifying devices that raise
the signals to the necessary levels.
~r e ~
,b q6 ~ J~ a -
6 D 9 ~
~ ~ f~- Q~ r
.
u~'I ' ~
' � u a a 6
~r'~,
Fig. 2.13. Potentiometric strain , P ~ Q
i
gage amplifier circuit [Ref. 110] ~ J~
w
~
V~
Fi 2.14 . Strain ~~'~f ~ ~~`~~'~t~l
g. gage amplifier
bridge circuit [Ref. 110] ~
The potentiometric conversion circuit is simplest (Fig. 2.13). The two-
armed voltage amplifier consists of strain gage RT and ballast resistor
R~ connected in series. The circuit is connected to a DC source with~
voltage U~. Thanks to a capacitor that blocks the constant singal com-
ponent, the potentiometric circuit becomes suitable for measuring dynamic -
processes. The v~ltage ~UT at the output of the amplifier depends on the
voltage of the supply circuit U~ and the resistance of strain gage RT.
The relation between ~UT and ~RT takes the form [Ref. 110]: -
euT=aRT ~R6 ~RT)~. (Z.a>
and since U6 = I(R6 +RT), ~RT = SeRT, it reduces to the form
e uT - r RT P,S
Rr ' (2.5)
1-}- R~
- In practice we take RT = R6, whence ~UT = 1/2IRTeS. We can see from (2.5)
that increasing the supply current of the sensor causes a proportional
increase in amplitude of the signal being recorded. Brief high-current
- loading of the strain gage (impermissible with prolonged duty) during
dynamic tests is extensively used [Ref. 20, 35]. The sensor does not
have time to change temperature in this period.
The bridge circuit for resistance measurement (Fig. 2.14) is based on a
very important property of the bridge: at a certain resistance ratio in
the arms (R1R3= R2R4) the voltage at the output vanishes even when there
is input voltage. The state of electrical equilibrium of the bridge is
29
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run Vrrtl:l[~L USr; UNLY
very easily upset with the slightest change in the given position, so
that measurements with the circuit are very aeneitive.
Strain ampZifiers. There are three types of standard strain amplifiers
used in strain measurement: DC amplifiers, alternating voltage ampli-
fiers and carrier frequency voltage amplifiera. The DC amplifier has
considerable advantages over the other types, the main advantages being
absence of frequency and phase distortions due to the fact that there
are no reactive elements in the circuit. However, the DC type has the
greatest zero drift, which is not very detrimental for recording rapid
processes. Three-channel and ten-channel transistorized DC amplifiers
have a passband of from 0 to 20 kHz.
Alternating voltage amplifiers,
_ K""�~�"""p�d�'"'�"s' while highly sensitive over a wide
~ 1 ~ uc.~nowwry nuRrorKM
. frequency range, are subject to
p~ '0~ strong influence from various kinds _
(2r nn�2ee HQ~Ov 3 of interference. The use of carrier
~ cr�rc ~ ~
frequency voltage amplifiers for
recording fast processes is limited
~ Q~ p~ since the limiting frequency of the ~
carrier signal (close to 50 kHz) as ;
a consequence of elevation of the
- Fig. 2.15. Simplest transistorized carrier frequency causes an increase
strain amplifier circuit [Ref. 35]: in phase shifts that impede balan-
1--to stabilized power supply cing of the bridge with respect to _
2--switch the reactive component and reduce
3--to S1-29 oscilloscope input its sensitivity. The described
- strain measuring equipment [Ref.
20] is based on series produced oscilloscope EO-7 supplemented by a block
of input amplifiers with carrier frequency oscillator and commutating
device, and a series produced rectifier with electronic stabilization of
the anode voltage.
The working principle of the amplifier shown in Fig. 2.15 is as follows
[Ref. 35]. Strain gage RT is connected in the measurement circuit in a -
potentiometric arrangement. The nonlinearity of the input circuit is
less than 0.5% at a deformation of about 10%. Calibration is by vertical
devlection of the oscilloscope Leam when the strain gage is shunted by a
, calibration resistor. No phase or amplitude-frequency distortions are
- observed at a pulse rate of up to 500 kHz.
Influence that the base of the strain gage has on recordings in quasi-
sl:atic tests. The mechanical distortions that accompny rapid processes
ai~e due to various effects, the most significant being wave propagation.
Let us consider the influence that these distortions have on registration
of the deformation of a strain gage resistor of length Z cemented on the
surface of a specimen subjected to impact. A mechanical quantity _
- deformation is converted to an electrical quantity change in
30 ~
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rr>~c �~~`x(:.~~t. l~5~~.� u~~.Y
resistance of wire. If e is deformation at the point with coord.inate x,
then the complete equation of strain gage resistance is
- i--
~R = S f edx ( 2. 6)
R 1 ~
0
- ~ where S is the coefficient of atrain sensitivity, and R is the initial
resistance of the strain gage.
For a plane wave propagating at velocity cp, deformation is e= f(x - cpt).
The function f(~) depends on boundary conditions. For the case where a
load is applied to the end of the rod that causes a constant strain rate,
E_ (cot'-x). (2.7)
Limits of integration in (2.6) equal to 0 and cpt correspond to the time _
interval during which the pulse front passes through the sensor, which
gives on the first stage at t< L/cp:
_ - _
AR eo r~~~_x) dx= - 21 c~t'. (2 .8)
RS ~ col J
o _
Denoting T= Z/c~, we rewrite (2.8) as
aR t~ (~.9)
_ RS 2z ~ 1~.
The stage is terminated at t= T:
- eR eoY ~2.10)
RS I~_~ 2 .
The readings of the strain gage on the second stage, i. e. t> T:
. ~ _
e~R ~r ~cdt+-x) dx= - 2 (2t-~c),
RS - Ico 0
i, e. the solution is the piecewise function
eR . 12 -
I~S _ _ g~ 2T ~ r~c.
Both stages are matched both with respect to amplitude and with respect _
to the first derivative: ~
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1'Vi\ VL'1'LtJ1t1L UJL~ V1VL1
eR _ ~z d /oR~ � ~
RS - 2~ dt` RS ~~s=eo. (2.12)
It can be concluded from (2.11) and (2.12) that: 1) distortion of strain
registxations as deformation uniformly increases is expressed in the fact
that on the first stage at t< T the signal e= ept is converted to the
function ept2/2T as a consequence of the fact that the leading edge of
the wave passes through the sensor in finite time T; 2) for times longer
than T, the recorded strain rate coincides with the actual value. Elec- -
trical distortions on the second stage (at t> T) are expressed in the
fact that the recorded signal is less than *_he actual strain by an amount
epT/2; 3) in contrast to other loading conditions (for ~xample e= const,
e= Ae-t~T), under uniform deformation the function e(t) does not have a
discontinuity of the first derivative.
The amplitude error of ineasurements as a function of strain rate ep, base
of the strain gage Z and velocity of wave propagation cp in the investi-
gated material is computed from the formula i
i
EOT I OO ~~p - E0~'-----I OU%o ~ 2..13 ~
~1~- 2 � _
- EMn~rc 2COeMauc ~
i
where eMaKC is the maximum value for the strain gage at which it retains
its linear readings. It can be concluded on the basis of (2.13) that the
strain gage base Z must be selected in accordance with the anticipated
strain rates.
2.3. Measurement of Velocities
Methods of testing at low (less than 10 m/s) and moderately high veloci-
ties (up to 100 m/s) differ not only in the necessity of using wide-band -
equipment for the high-speed tests (with passband of more than 4 MHz),
but also in construction.
Discrete registration of veZoeities is based on using oscilloscope pips I
that are generated by two or more electric contacts with known distance
between them. Discrete velocity sensors (which are called electrocontact -
sensors) are most justified in measurement of the velocity of the body
striking against the specimen. Ref. 36 describes the electric circuit
for measuring the velocity of a striker up to 5 m/s used in the construc- -
tion of a horizontal hammer with rubber striker accelerator. The system
includes two independent electric circuits, one of which triggers an
- oscilloscope, while the other sends pulses from an electromagnetic sensor
to the oscilloscope during motion of the striker preceding impact.
Ref. 79 describes a method of ineasuring velocities exceeding 300 m/s by -
electrocontact sensors (Fig. 2.16) made in the form of steel wires 0.5 mm
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O
_ c~�z~ rJ-2a
. ' . (Wt-1Jl -
~i Q, ~1 .
M~ Mi IM~ ~
N
S '
R~ 11~' Rt' A~ . ~
H------~ -i- ~~i . N
. .~_.-y _
L-----~ o
Fig. 2.16. Diagram of discrete registration of the velocity of a pro- -
jectile [Ref. 79]
Fig. 2.17. Induction sensor [Ref. 69]: a--construction; b--sensor in a
magnetic field
in diameter stretched perpendicular to the direction of motion of the
, striker. The measurement circuit includes multivibrators that operate
in the slave mode. The multivibrators receive trigger pulses from a -
three-armed voltage divider when the wires are broken by the striker
immediately before collision and after breaking the specimen. The multi-
vibrators generated pulses with duration of 9.1 us sent through cathode
followers to the oscilloscope input. The slave sweep of the oscilloscope
was triggered from a potentiometric circuit when the sensor was opened. -
The use of three sensors enables measurement of two time intervals and
computation of two velocities, which improves the reliability of the
measurements.
Discrete registration of velocities is satisfactory only in the case of
_ measurement of kinematic quantities, in particular the velocity of a
large mass striking against a specimen, since the velocity of this mass -
during the experiment is taken as constant. In case of necessity of -
establishing the velocity of a cross sectional surface or point of the
specimen, continuous registration is advisable, a particular advantage of
which is that where necessary the velocity registration data can be sup- -
plemented or replaced by displacement data by means of an integrating
_ circuit. Two methods are known for continuous velocity registration,
~ based on using inductive and capacitive sensors.
The inductive sensor (Fig. 2.17) is a rectangular mica frame on which
= 20-30 turns of fine wire are,wound. The frame is fastened to a rod
. (wire), and one end is placed in the uniform magnetic field of a permanent
magnet. With displacement along the long axis, an emf proportional to
the velocity of displacement is induced in the frame. The signal from
the sensor is sent without intermediate amplification to the input of a
33
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1'VL~ vi'l1V1t11. UJ~ V1VL1 -
cathode-ray oscilloscope. The small dimensions of the sensor, and con-
sequently its low mass, enable investigation of wave propagation in rods
[Ref. 48, 49, 69, 155]. In case of necessity, the signal from the sensor
_ goes to the input of an RC integrator. This gives a recording of dis-
placement as a function of time [Ref. 38). The range of displacement
measurements is limited to a quantity somewhat shorter than the sensor.
The sensitivity of the sensor depends on magnetic field intensity, the
- distance between poles of the magnet, the number of turns and the width
of the sensor. The use of the sensors is limited by the finiteness of
their mass. High accelerations cause development of considerable inertial -
forces capable o.f deforming and destroying the sensor. The simplest
method of calibrating inductive sensors is to compare recordings of the
velocity V of some cross section of an elastic rod with data of strain
measurements in the same cross section of the rod by a strain gage. In
doing this, a relation is used from the one-dimensional theory of wave
propagation:
- -V = cpe,
where cp is the velocity of elastic wave propagation.
Capaeitive sensor. Recordings by capacitive sensors are zero-lag as a I
consequence of the fact that the sensor is non-contact. The convenience
of ineasuring velocity or displacement of a free surface has led to use i
ot- a capacitive sensor in a system called the Hopkinson bar. A sensor
was first used for this system by Davis [Ref. 29] who applied it for de-
termining the displacement of the free end of a rod used as the grounded
plate in a flat capacitor (the other plate was stationary while the r.ad
was dynamically loaded). The change in capacitance of the capacitor thus
formed in proportional to displacement of the end of the rod.
S 4 3 ' _ ?
. /
-
~
' < <
Fig. 2.18. Kolsky bar [Ref. 46]: 1--specimen; 2--flat sensor; 3--
cylindrical capacitor sensor; 4--rod (waveguide); 5--explosive charge
with cap
Further improvements in the method of determining stresses by recording
the rate of displacement by capacitive sensors are given by H. Kolsky
[Ref. 46]. Fig. 2,18 shows a diagram of what is sometimes called the
Kolsky bar. This design includes two rods between which a pellet speci-
men is placed. Pulse pressure is applied to the free end of one of the
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~ a~. ~ _
rods. The pulse.,propagating through the rod compresses the specimen.
The incident pulse is recorded by a cylindrical capacitive sensor, which
enables determination of the relation between velocity and time. A flat
sensor measures the displacement of the free end of the anvil rod.
I~ur.~hc~r dcvel.o~~m~~cte al' e.Lectrie c:ireuitR, including capa~itive HE!1180r8~
has shown that the most convenient for dynamic measurements is an ar-
rangement in which the potential difference across the sensor electrodes
Up is proportional to the velocity of displacement. The signal amplitude
from the capacitive sensor is [Ref. 93]
z
eu=udR dE =kuoRV oz,
where R is the load resistance of the measurement circuit, V is the
velocity of the free surface, C= kd2/x~ is the capacitance of the sensor,
k is a constant, d/xp is the ratio of the diameter of the free electrode
to the gap between electrodes. An increase in Up, R and d/x~ increases
the sensitivity of the sensor. The maximum voltage Up is limited by the
electric strength of the air gap. The permissible value of Up can be
increased to several kilovolts by using a thin layer of dielectric to
_ insulate the electrode from the surface of the specimen. Circuits for
veloc ity measurement at Up = 1500 V were developed in Ref. 93. Electrodes
with dimensions d= 25 mm, xp = 2 mm were used for measurements. The same
voltage was applied to a guard ring with inside diameter of 28 mm and
_ outside diameter of 96 mm. To prevent electric breakdown, a triacetate
- - . .
' b~ Oscillosco e
a) 'JO-3 CN-17;d
~
' 40M I,OK C~
~ J"8 RJ R4
- 13008 4,0
,q lOYY !b0
zo ,q 20~ �
, ,
Fig. 2.19. Electric diagram for measurement of velocity by a capacitive _
sensor with initial voltage of 1500 V across the plates [Ref. 93] (a), and
with initial voltage of 5000 V[Ref. 167] (b) -
f ilm 0.2 mm thick protected the electrodes of the sensor, the guard ring
and the surface of the specimen (see Fig. 2,19, a). From the sensor, the
signal went to the input of an oscilloscope through an amplifier with
input impedance of 75 ohms. Ref. 124 and 167 give circuits for measuring
veloc ity by a capacitive sensor with voltage of 5 kV applied to the
plates (Fig. 2.19, b).
In concluding our survey of inethods of velocity registration in dynamic
tests, let us note that new non-contact facilities have been daveloped in
recent years that are based on the use of laser interferometry [Ref.
11, 32, 33, I13, 163].
* * ~
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rUlt Vrrll:lA1. U5~ UNLY
Since no universal facility has as yet been developed ~ar testing materi-
als over a wide range of strain rates, studies are done on two or three
- testing machines. Quasistatic tests are done by well developed facili-
' ties that use energy of various kinds (compressed gas, accelerated mass).
- The main disadvan:tage of devices based on accumulation of inechanical
energy (hammers with a falling weight, pendulum and rotary hammers) is -
bending that cannot be eliminated. Pendulum and rotary hammers are in-
tended for testing specimens with geometric dimensions that do not always
satisfy the researcher. Hydraulic and pneumatic devices at present are
the only facilities for quasistatic tests over a fairly wide range of
strain rates. -
_ Devices that use electrical energy have a number of advantages over the
conventional loading devices: the capab ility of transmitting loads with
distribution over the surface of the specimen that can be predetermined,
and universality that ensures wide variation of the kinds of tests. An
important advantage of the devices is also the capacity for transmitting
considerable energy to a specimen over a quite brief time interval. The
studies given in Part I of different modes of operation of the magnetic-
- pulse facility show the feasibility of throwing considerable masses at
high velocities, and transmitting intense forces by electromagnetic fields ;
as well as by free discharge in a liquid.
- Dynamic tests are done for fairly precise determination of functions that ~
characterize the stressed and strained state of a test specimen under
short-term intense loads. This purpose may be attained when the tests are
accompanied by correct and exact measurements, and therefore the record-
ing facilities must meet requirements that minimize response time. Dyna- _
mometric devices combined with the clamp of a loading device with maximum
frequency of up to SO kHz may find application in quasistatic tests with
strain rates of no more than 50 s-1. Dynamometers in the form of wave-
guides permit testing of materials at rates up to e= 103 s-1. Noncontact
methods of ineasurements by capacitors and interferometry have practically
- no limits of applicability with respect to e.
- PART II
DYNAMIC TESTS OF RIGID POLYMER MATERIALS
- Testing of rod specimens is complicated by the presence of wave phenomena,
stress concentration, local strains and so on. Therefore attempts have _
b een made recently to develop reliable methods for taking consideration _
_ of rhese effects, if not eliminating them entirely.
The most d~veloped method is the Hopkinson split-bar system based on the
use of specimens of the simplest shape (pellets), enabling consideration -
of wave phenomena in the specimen. In the case of axisymmetric loading
of a thin ring by a distributed dynamic load, wave grocesses in the
specimen can be disregarded, which makes the method especially valuable.
The method of dynamic bending of beams gives important information on the
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- resistance of polymer materials to shearing loads when studying trans- -
verse impact. The investigation of inechanical properties cannot ~.n any
way be considered complete without studying the behavior of polymer
materials in the complex-stressed state.
In dving Cests and developing methods, particular attention was given ~
to substantiating the quasistatic treatment of the results of experiments,
which is based on solving problems of wave propagation.
Chapter 3
THE HOPKINSON SPLIT BAR METHOD
A method of calculating dynamic stresses in an elastic rod by :neasuring
the velocity of the free end was proposed by Hopkinson [see Ref. 46;, and -
later improved by Davis [Ref. 29] who provided a velocity sensor for the
method, and made a careful analysis of wave pr~pagation in the rod. In
1949 H. Kolsky suggested using the Hopkinson bar to measure stresses and
_ strains of a specimen in the form of a thin gask.et. This marks the be-
ginning of development of the Hopkinson split bar method for testing
materials at co~siderable strain rates (up to e=~ 10`' s' 1) . By the ef-
forts of experimenters [Ref. 5, 116, 126, 132, 145, 147, 168] a technique
was developed based on using modern loading devices and recording fa-
ciliCies. The Hopkinson split bar [HSB] method occupies an inr_ermediate
position between quasistatic and dynamic tests, since the stressed state
of the specimen is uniform within its range of applicability, and the
recordings that accompany an experiment are mad e ~;ith consideration of
- wave processes in long metal rods, Fairly well developed variants of the
HSB method are ~ompressive, torsional and rensile. Reszarch has started
on development of a biaxial variant of the technique.
- All variants of the HSB method are based on using a system of split bara. _
These systems comprise two eiastic rods (a pres sure transmitter and an
anvil) between which the specimen i~ placed. Recorded data on strains
ei and eil enable determination of the stress and strain in the specimen. _
The rods are chosen of such a length that stres s~~waves reflected from the
free ends do not introduce distortions. An indisputable advantage of the
method (as compared with conventional techniques) is the capability of:
- 1) tuning out interference due to the resp.nse time of the force sensor,
- vibrations of the installation, the foundation, etc.; 2) determining con-
siderable deformations of a specimen measured iri tens of percent; 3) pre-
venting bending of a specimen in tensile and compre~sive tests; 4) cor-
- rect validation of an experiment based on a simple scheme of force trans-
mission to a specimen through rods.
Stresses and strains in the specimen in tests by the HSB method are cal- -
culated on the basis of a one-dimensional theory of wave propagation.
According to the one-dimensional theory of elastic waves in semi-inf.inite -
rods, strain and stress are related to velocity by the expressions
37
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rux ur'r~lc;ltw U5~ UNLY
~=-P~ov; (3.1)
v -
Q~- ~o~ (3.2)
where p is the densi~y of th~ rod material.
Let a rod consist of. two sections with area F1 and F2, i. e. at some
coordinate the rod is broken by a discontinuity of cross section. If _
the rod is loaded by stress ~J, then a reflected wave will propagate
_ from the coordinate of sudden change in F: -
Q,~= _ F~-Fi Q~~ (3.3)
Fs-{-F~
- and a transmission wave will enter the rod with area F2 wi.th stress
T 2F2 f (3.4)
Q = Fi+F2a .
~Lf the sections of `::ods F1 and F2 are made of different mater.ials with ~
- acoustic stiffnesses plcl and p2c2, then the functions relating QR ancl QT i
to aJ, according to the one-d~mensional theory, take the form ~
'
- ~2~2F2_~~~~~F~~f1 I
- R
v (~i~~F~+ f~z~2F2 _
2p2c2F2_Q, ( 3 . 5 )
~T _
pic~F~�+'p2c~FZ
- The correspondence between (3.5) and experimental results is demonstrated
in Ref. 78, 132. The simplest refinement of the one-dimensional theory
is a Love equation [see Ref. 22] derived by the Hamilton principle with -
consideration of the kinetic energy of radial motion. De Vault [Ref.
127] gives asymptotic forms of the solution of the Love equation.
At present no rigorous salutions have been found for the three-dimensional _
problem of wave propagation due to the complexity of simultaneously
- accountin~ for boundary conditions on the ends of the rod and ies lateral
- surfaces. The solutions of Po ghammer and Chree for a one-dimPnsional -
circular rod [see Ref. 22] are considered the best approximation. The
complicated form of the frequency equation brings about c~rtain diffi-
- culties for getting numerical results. Besides, since the solutions
are represented as infinite waves, they cannot describe the propagation
af discontinuities. Less rigorous theories are an attempt to simplify =
the mathematical description while retaining the m~st important con-
clusions of the three-dimensional theory. The precision of this theory
is usually evaluated by comparing the corresponding spectra with those
- derived on the basis of the Poghammer-Chree theory. At present, the
38 -
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- exact equations are being successfully solved by numerical methods. In
spite of the fact that in many cases they do not permit direct comparison -
of results with the exact Pogha~ner-Chree solution, the reliability of
numerical resulte can be valid~ted by existing methods of evaluating the
accuracy of numerical methods. �
Data found on the basis of the elementary theory do nut agree with ex- _
perimental data where the time of action of the load is commensurate
with or less than the time taken by a wave to travel a distance comparable
to the diameter of the rod. To describe the dispersion nature of wave
- propagation in rods, methods have been developed that give asymptotic
solutions of exact equations of motion with mixed boundary conditions
(displacements and stresses are given on the ends). It was found that
- the exact theory and all approxima.te theories (with the exception of
the one-dimensional theories) lead to the same asymptotic solution for
the head part of the pulse corresponding to the first mode of oscilla- ~
tions. These solutions take the form of an Airy integral, and are found _
on the assumption that flat cross sections are not curved during wave
propagation, and therefore radial, circular and shear stresses are equal
to zero. The curvatures of cross sections were first considered in an
- examination of waves in rods of rectangular cross section under the action _
of a suddenly applied load [Ref. 30]. An a.symptotic solution of the
problem for circular rods based on exact equations was found in Ref. 31 _
for the case of loading by stepwise pressure or by a velocity pulse.
Numerical solutions have been found for the problem of wave propagation
in an elastic rod when the end is loaded by pressure that depends on
coordinate [Ref. 39].
Exact equations were used in Ref. 107, 108 for calculating dynamic
stresses in a split bar. The results of this work can be expressed in
the form of recommendations on loading a Hopkinson split bar and using
it for recordings: 1) the head part of the pulse in a cross section not
far from the end of the rod that is struck contains higher modes of vi- ~
brations that die out with distance from the end. Deformations on rods -
- are measured in cross sections not more than twenty diameters from the
ends; 2) the end face of the bar in the HSB system should be loaded by
uniformly distributed pressure; excitation of higher modes incr~ases
when the rod is loaded by a concentrated load; 3) specimens for tests
by the HSB meth~d must be of such a shape and size t~-iat the free surfaces
of the end faces of rods in contact with a specimen have minimum areas;
4) the HSB must not be loaded by an abrupt intense pulse of a length
commensurate with the diameter of the specimen.
3.1. Governing Principles~of'_the HSB Method
Let us consider loading of a system of three rods (Fig. 3.1). The load
is applied to the end of a rod. With regard to the one-dimensional
theory of propagation of elastic waves, deformation in some section of
the rod on the wave is related to mass velocity by the expression
39
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j~~` Fig. 3.1. Diagram of the Hopkin-
~o`~' son split bar method: 1--pres-
c~ c~ e* -
. -�p, sure transmitting rod; 2--anvil
, ~ ~I~ rod; 3--specimen; 4--damper; 5--
_ , striker; 6--supports; 7--piezo-
~ y e~ t y electric crystal; 8--system for
measuring the striker velocity; `
~-~-,-b I~a ~ 9, 10--amplifier and transducer ~
~ B ~ of the signal from the piezo- -
L~~__J v v electric crystal for triggering
9 the oscillosco e� e e strain
P ~ I~ II--
~o U gages
e~r- ~o . ~U~
(3.6)
r
whence U= -co f edl', where cp is the velocity of propagation of elastic
d
0
waves in the rod. The displacements of the ends of the rods U1 and U2
are found from the formulas
r r e
-Ui=co re~dl'-{-co reRdt'=co f (e~-{-eR) dC;
.l J J
0 0 0
~ (3.7)
U2= -co f ETd~'~
0
- where eJ, eR, eT are the deformations in the incident, reflected and
transmission waves. Deformation of the specimen es, considering its ,
stressed state to be uniform, is defined as follows:
- ~ i_
Uz-U~ _ _ ~o ~er+eR_e'') dt', I
es= L - (,J ~3.8)
0
where L is the initial length of the specimen.~ Since eJ, eR, eT, we
rewrite (3.8) in the form
r
Ea=- 2L f endt'. (3.9) -
0
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Let us go on to determine the stress in the specimen. For forces Pi and
P2 (see Fig. 3.1)
P,=EF (e'-cR); P~=EFe'',
(3.10)
'where E and F are respectively the modulus of elasticity and the cross
section of the rods. Since P1= P2, we get QS =(P1 + P2)/ZFS. The stress
in the specimen as a consequence of continuity of forces on the ends of
the specimen is -
cta=2
F(eT~-~e'-eR)= FF$T, (3.11)
~
- where FS is the cross sectional area of the specimen. Expressions (3.9)
and (3.11) are the starting relations in the case of utilization of the '
HSB method. The purpose of the method is to calculate QS and os from
_ data of registration of eR and eT. Equality of the forces on the ends
of the specimen occurs in cases where the accelerations and the inertial
forces that they produce are vanishingly small. Otherwise they must be
taken into consideration,
I
Computing stress css according to a two-dimensionaZ theory. The formula
for computing stress with consideration of axial and radial inertia of
the specimen from the HSB method is given in Ref. 126. However, this
relation can be derived in a,impler way. Suppose that the axial and
radial velocities of a specimen in the form of a cylinder of height L and
radius R= D/2 are related to axial strain rate es by the expressions -
; ~x = ~1- EiX;
(s.ia> _
Ut~ -VefCii
where vS is the Poisson ratio, U1 is the velocity of cross section with
ordinate x= 0. The x-axis is directed along che axis of the split bar.
We take the cross sectional areas of the specimens and rods to be the
same (F = FS =~rR2). The coordinate origin is at the center of the end
face of the pressure-transmitting rod in contact with the specimen.
Let us use the Hamilton principle
- %1-F-1 ' _ _ jl
. i i% +
i Lv~ '
_ ~ . ~~1.f�q~Ol,c
_ a~_~
~ ~ t�10-4/2.07 s
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m/M increases, there is a corresponding increase in the number of
collisions; for m/M-~0 the time intervals beLween collisions decrease
rapidly; after 4-5 collisions the striker moves into uninterrupted con-
tact with the beam. The contact force is compriaed of polyharmonic os-
cillations (the sum of the odd harmonics of the beam) around the straight
line P=Apt, where Ap = 48EIVp/L3. The influence of shears and rotational
inertia shows up as a small change in amplitude and in delay of the onset
of the repeat collision (see Fig. 4.4).
s
4.2. Experimental Studies of Transverse Impact
Experiments [Ref. 61] on recording reactive forces, deformatians of the
outside line and contact force in dynamic bending of a beam were done
with impact by a mass of 590 g in the form of a rod freely dropped from
a height of 5, 10 and 15 cm.
_ A method of determining the static modulus of interlaminar shear from
tests of bending of a beam by a concentrated load P is proposed in Ref.
96, 98. G is calculated from the formula
3 �
y 98L:1~1+ 2 (L~ GJ' (4.18)
where E/G is the ratio of the elastic modulus to the shear modulus;
H/L is the ratio of the cross section of the beam to the length of the
span. Numerous experi~ents done under static conditions as well as with _
transverse impact have shown that using (4.18) for calculating the shear
~ modulus leads to considerable errors and scatter of results as a con-
2
sequence of the fact that the order of magnitude of quantities Z~L~ G
- is commensurate with the accuracy of the experiment . We consider below
a method of calculating the dynamic shear modulus with high accuracy from
data of registration of shear wave propagation.
Reactive foz~ces and bending rrbments in the case o f dynamic bending.
The propagation of bending moments was studied in Ref. 61 from data of
registration of the deformation of the outside line of a beam by strain
gages with a base of 10 mm. The results of numerical calculations ac-
cording to (4 .1) and also experimental data are given in Fig. 4.5, which
shows in particular that the main portion of the signal M vs. t propagates
at the velocity of the shear wave; the part of the pulse that propagates
at velocity cp is recorded in the form of a forerunner.
. The reactive force was determined from~~ a dynamometer combined with a
support (Fig. 4.6). It was established by preliminary experiments that
the form of the support fasteners has a noticeable effect on the study
results in the case of dynamic bending. It was ~ound that th e reaction
of the support is most sensitive to the f orm of the support fastener.
62
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Development of a support design (see
Fig. 4.6) that produces supporCing E'10~ ~I
_ conditions fairly close to hinge ~.,i
~ ~ 2 I
support is dictated by the need for i
comparing data of experiments with ~
the results of numerical calculations 2--- -
7V i
for the case of bending of a hinged I ~
beam. Two piezoceramic pellets are
_ ---I__2
placed in the opening of the support ~ v
and compressed by a screw. Signals ~ i t�10~MKC I
from the piezoelectric elements are o 1 4 s us ~
_ sent through an amplifier to the ~
input of an oscilloscope. The ar- ~__~_y~ ~ A l_J
rangement permits static calibration.
Triggering of the oscilloscope was
synchronized with the beginning of Fig. 4.5. Theoretical (1) and
contact of the striker and the beam. experimental (2) curves for _
Fig. 4.7 shows typical oscillograms e vs. t for cross section -
- of R vs. t recorded during impact. x= 18.8 cm. Length of the
We can see from Fig. 4.7, a, that beam 60 cm ~
preceding the registered force pulse
is an oscillating forerunner that
propagates at velocity cp. The os- r
cillograms can be used in particular ~I~ _
to determine the rate of shear wave .
propagation cG. The results of nu-
merical calculation according to
(4.1) and experimental data are , ~
given in Fig. 4.8. Steel and fiber- ~ I
glass beams were tested to check out �
the method. The problem was posed ~
of calculating the values of cG and L
G from propagation of the transverse
force, and also experimentally con-
firming the influence that the shape Fig. 4.6. Design of the support
_ of the beam cross section has on the for studying dynamic bending of
velocity of shear wave propagation beams
stemming from the relation cG
, where k' is a coefficient that accounts for the shape of the cross sec-
- rion [Ref. 125]. Data on the shape and dimensions of specimens are
summarized in Table 4.1. The values of G3, cGT given in the Table were
calculated from the expressions
s ,
_ Ga = p ~ k
~ ~ ~G,T = k ~ ~ ~ 4 .19 )
U P
~~v)
where v is the Poisson ratio equal to 0.3. It can be concluded from a
comparison of the data given in the Table that the proposed method of
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Q 1~'~v,,a ~ Y ~ tii 4. irfl`'` . yi..'~Y ti^t�ri ,~'P.M . _
~~r C. h' ~L ' y.~:. . .~.y.. ~ `~ir: ~ ~
f ' ~ ~ 3.~ �fF ~ ~.-I ~ � ~ 4
. ' . . r ~
Y:_~ ~ .
; � :
- . d-� � ,
. . .
. ~ ' ~'s
R
� y ~ 50 h1KCr ~t
.~.�+r.~ ~+~~"rr.~pl ~
_ 50 us
Fig. 4.7. Typical oscillograms of R vs. t with dynamic bending of steel
(a) and fiberglass (b) beams 88.8 cm long
determining the velocity of shear waves
is reliable. The theoretical position on -
fd,~rc (kgf) i the influence that ~he shape of the beam
-0__ has on the velocity of shear waves can be
'I 2 taken as theoretically proved. Registra- _
tion of the velocity of shear waves
? enables determination of the shear modulus -
'c6 2 for glass-textolite where the reinforcing
co t�~D,N~~. layers are perpendicular or parallel to
� 2 ~ y s the acting force. In the former case the
given characteristic is termed the inter-
? laminar shear modulus. For glass-textolite
the computed value of the ratio E/~G was -
5.80. The static rs:Lio of these quantities
_ ~ is ab4ut 10.
Registra~ion o f contact forces . One paper
on studying contact force [see Ref. 22]
describes a method of recording it. A
Fig. 4.8. Theoretical (1) steel ball fell on top of a steel hemis-
- and experimental (2) curves Phere fastened on the upper surface of a
- for R vs. t for a beam 60 cm beam. A barium titanate pellet secured to
- long. The arrows indicate the beam measured the contact force. How-
times of arrival of waves ever, such a beam canna t be used to record -
cp and cG repeated collisions.
At the Institute of Mechanics of Polymers, Lithuanian SSR Academy of -
Sciences, a special dynamometer (Fig. 4.9) has been made far studying
contact force. The device consists of a long (750 mm) cylinder l0 mm in
diame ter terminating in a short thicker cylinder in which an opening has
been c ut to accommodate two piezoelecttric ceramic pellets. The thick _
part of the dynamometer is tipped with a hemisphere 3 cm in radius.
64
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~ ~n ~ o _
~ N '-1 a0 u1 O
_ W N N tt1 ~t N N '
~ ~N
I ~ O O crt ~p M ~p
O M N ~ tf'~ ~y ~p
y,~ ~ O O O
O O O O O O -
~
F v1 -
\g O~ M M
C7,SG N N ~ I
- U ~ '
j ~N
~I U O O N G N~ N G aJ p~p
O~ � � .
W~ N N O~ t~.i ~ O 1.~
c0 'n O
~ O O c'~1 ~ 4J ~ c'n C Gl ~ ~ 00
o N . ~ .
U~ v1 U1 ~t r~i 1~.~ ~~,7 +-1 ~ N O
R1 'n cti _
~i ~
~ m\ ~ ~ o 0 0
~7 ~O M O
W C7 ~ C~'1 N r-I r~ r-I N
a u
6 n o0 c"1 M M M =
H o0 ~7 r1 c*1 M M _
,y, ~ ~1 00 00 00 00
O O O O O O
N r-{
N O r-1 rUi
N N ~ 3~-+ N
~ aa ~ a~ ~ ~ a ~
N fn 1J 00 (n U N r-1 u1
i-~ U t~+ 1~+ rl D, 'b ~ II
ro ~ q~ cd N ~ c7,U w .C cVU ~ H q ~
ro
o ~n a~ ~.c q.. oo ~c o 0 0 ~
~ ~ ~ ~ oo v p oo ~ a~ a~i ~ ~ ~ ' v ~
~ n II ~ q o ~ G u ~ ~ ~ ~ u ~
N A O�~ ~ ~0 N�ri U d0 O~~ b00p
- N r'+ N~..I U r-1 U
Equation (7.37a) corresponds to the behavior of the model shown in Fig.
7.1. To write the equation of motion for this model, we assume that each
element is acted on byninertial force ~o p~,t, where p is the density
of the material, Eo= ~ E~; Eo is the instantaneous modulus of elasticity;
ia,
Ei is the modulus of elasticity of the i-th element. Summation over the
individual elements of the model yields
~--n--E{ ---av av
~ Eo P ac -P ar �
-
The equation of motion of the i-th element is
dz~ __~PEt dV ~ _
o dt
= with considerstion of strain law (7.1) this is rewritten as
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- a-Q, F; a2E i a~Q{ a~ i
~ ~3.rz - p Eo dtz _~P Eo ~ Ei ~Iz + d
t EiYt ' ( 7. 38) -
Finally, (7.38) can be represented as follows:
,azQi `aZat ~ a~t ~~o= Eo
aX2 acZ + ar ~ ~ ' c ~ . 39 ) -
- Corresponding to each equation (7.39) is a pair of characteristics
dx=�codt (7.40) ,
and the conditions thereon
' d crt Eo pcod V- art T
t. ( 7. 41)
Considering that ,
t ~~E _ .
Qi=E~E- f E{e ~i E~~e~d~~ (7.42)
d
0
by summing the stresses over the elements and passing to the limit, we get
. ,
dQ=~pcadY-e f H(~z didt+
0
~_E
-~-rtt f ~H,~~tlTe~ t e~~)d~. (7.43)
J
0 0
Using the notation
_ ~
(0) = f H~ ~ d~r;
0
t -
~t~ - dl,f H~rT, e~ T dt= - f�H~TS) e` d?~
0 0
.q
~ (7.44)
_ = H~ ) e s d~r; . . .
u �
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we rewrite (7.43) as
.r
z=r~--- .
~'o
(7.45)
On the characteristic dx = 0; x= x*
_
da=Eod~-dl[etY(0) - f '~'(z*-~)e(g)dEJ~ (7.46)
0
~
where z* = t- ~ . Let us find the change in stress on the leading edge
~
of the wave. Since on the leading edge
~(a~ _ -pcd[V]; =~Ed[e~~
we get from (7.46)
2d [v] _ -dt~~~ [a],
, ~r~p~ (7.47)
where Ep = pcp =4'(0) is the instantaneous modulus of elasticity (the sym-
bols in brackets denote discontinuities of the corresponding quantities).
Integration of (7.47) gives*:
_ ~ .
[c~J =voexpr - 2~~~0~ � Col � �(7.48)
L J
~ Let us note that in order to account for equilibrium pliability it is
necessary to substitute the constant 'Y(0):
.f.~, -
l~~(0) = f H(t)dinT=Eo-E~.
Relations (7.45), (7.46) and (7.48) are sufficient for numerical solution
of the problem with the given initial and boundary conditions.
It should be emphasized that the choice of functions 'Y(t) at which
`Y(0)-~~ leads to infinite attenuation of discontinuities on the leading
edge. Infinite attenuation of discontinuities of stresses on the leading
*The same result, obtained by the Laplace transform method, is given _
in Ref. 63. -
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, edge of a wave takes place in the case of models with wealcly singular
waves. Let us assume Chat in the strain law
r. .
_ Q=E~e- (Eo-E�) f ~'(t-T)e(T)dt
J
0
the function `Y'(t) is selected according to Ref. 88:
a~e-nt ~~.49)
~f~~~~ 1'~k~~tl--IS
(a~ 0, 0< S