SCIENTIFIC ABSTRACT SHERMAN, D.G. - SHERMAN, E.M.

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• Publication Date: December 31, 1967
• Content Type: SCIENTIFIC ABSTRACT

ACC NRs
AP6033538 SOURCE CODE: UR/0170/66/011/00410516/0520
AUTHOR:- Lyubchenko, A. P.; Tsarina, I. V.; Sherman, D. G.; Shukhov, A. S.
ORG: Transportation Machinery Plant, Eharlkov (zavod transportnogo mashino-
stroyeniya)
TITLE: Method of determining temperature fields of machinery-part surfaces
inaccessible during operation
SOURCE: Inzhenerno-fizicheskiy zhurnal, v. 11, no. 4, 1966, 516-520
TOPIC TAGS: temperature, temperature dependence, temperature measurement,
diffusion method, diffusion parameter, temperature field
ABSTRACT: A method is proposed for determining the temperatures and
Lopologies of the temperature fields of objects which are inaccessible during
operation. It is based on the use of the critical dependence of the diffusion
parameters of materials on temperature. The method was tested on simple and
complex multicomponent heterophase alloys over a wide temperature range (the
lowest temperature was 200C). The alloys tested were 65G, Hhl2M, Ehl8N9T,
E1283, and AK-4 grades, with partial reference made to parts of internal
r-.A 1 / 2 UDC: 536.5

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- . I . L:-. 111 . J,
-;t, 'i.- - ', - , . .. .-. . .- --- . . ) ) I I I-

Verman, D. I. On the pro0agation of wjavpq In
a fluid
Sci. TjRojS.-.
yer covering an elastic balf-snace--: Acad.
-
Publ. [Trud'511 Jnst.1S6ismoI4j.-no-. 115, 43 pp- (1945).~,,
(Russian)~
Dans ce, rnkmoir6 I-auteur kti& le, pro.Wnie de dker~
'
mination des ondes superficielles
& Rayleigh sur les limites -
st m
d mftegx~ Le pr6b1ilme e effective

it-160it.-A
r jqu
JA
e-
ara
VP Ad -ifir%f
- -
he"
m e- a cou
fortnules obtenms sont discdt6es, et les oscillations Prin.
'-
'
~
ffisammek m-andes
cipaie et r6i6ellc, A des disitah
su
ces
des points d'a-pplication de-i .1a force Oriodique superficielle,
sont calcul&s. V, A. kostit. (Paris).

"Core -~r- ~ n,6 er ta-I -) 7-T-Cbl-rs of "he Statl c T 1, ~- cry r,-T' 7la-= tic I:'Cr the 1--'a1 f-Spac e Zz e/
;'cr --..ic 7--,crccn-lec~e-i Falf-Spaces Zz- e 'th Different Elas`c ---rcnerties.11 Iz. 1k.
ancl I - w-,
-7
SiSh, Ct-del. Tekl--.
1945- Su*u7-tte6 7 jun 1945.
Report U-1532, 6 Dec 19151.

VOW
he i D On a- ka jWb1-Gm--'d
S
of potendil togn Integ"Witt -don. 0 If
U
Acad. Sci.:URSS. &r.~ Math.
-[1tva& AW. Nauk
SSSRJ9,357-362 (1945). . (RuWap. English-Summary)
Let S be a finite Plane,domain'.whose boundary L consists
of a finite number of closed curves of continuous curvatum
A lie autlsor considers the proble-in of determining a function
u(x y)~ harmonic- in S:and-satisfying on L- a differ6ptial
cQitio-n of the'form awau/6-408646 &)' AS),
+
or More generally of the form,
E Ea
k axk-jayi
k-0
where aq(s) and f(s) arc suitably restricted functions of the
arr length; lie redlices the problem to the solution of-a
-V~--Ihofm equation. E. F. Baken&aCll'
z;. D,
2

Sherman D On some problems, of the theory of station-
-AwRrrahlons. Bull., Acad. Sci. 4 URSS. Sdr Math.
[Izvestia Akad. Nauk SSSR] 9, 363-.!370 (194S), (Rus-
Sian. English summary)
The method of the paper reviewed above is extended to
treat similarly the problem of determining solutions. of the,
differential equationdu+Vu-0, with boundary condition.
the same as in the formee problem. E. F. Beckenback.'
Vol No. 2
Sourc al Revi-~'Rs,
e S FztheTmUc.
7 -A
H;
RE aK-

Sherman D. I Certain problems of the theory of poten-
~Tath. Met-h. [Akaff. N.Ltfk SSSR. Priki. Nlat.
Nlech.] 9, 479-189 (1945Y. (Russian. English Win-
in.try) [AlF 15441]
The authior shows flow the p2tential fitlicting II(x,y).
which. is regular in the finite simply-eanuected reizi6n S and
which satisfies tile houndary condition a(s)n~+b(s)uY=f(s)
oil the Coll tilmously curved boundary L of S call br.1found
a-, the soltition of an equivdent. Fredholin equation. Assum-
ing aT') +h'(S) = I aII along. L, two cases arisr, according.to
whether in th ation a =cos w(s), -b-sin W(s) the
C represent.,
function w(s) inci-eases by a - nonpositivt or by a positive
multiple of2jr. when. the colrtour L is described once in the
positive sense. In the first case the boundary-valoc problem
.11%hyw has a solution, in the second case there ire'non-
trivialsolutions of tile homogeneous problem.
By tile 6wnc ntethod Poinearla's i3roblern where the
boundar condition, is du7a +7p T)dtt1ds+qt').=f(.')' i.
redu cet I to a F red h of in eq tia t ion, and so are certtin singular
ations.
integral equ, AL Gelo4nb (Lafayette, Ind.).
Source. 1948' Vol No.
MathematiQal Reviews, 3

-cncern':-_ Cn--- cafe cf '.,ar-_'at_,-cn cf, an
.-anic-, Acadiemv ef Sciences,
tute of :-,ec,
Ctdel. "au', ':c- 10-11, 194-5. C:Utip-'tted 16
-,eport, "l-1532, () Dec 1,"51.

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. : i 1 1L '. . -1 , - - I , ~): I _: ~ fl~: j ~*,
.1 " : : ~- . .. , ,,:i ~ I __, -' _,,I '' I ..-" :; ~) .. ; L:" (: 1, ~* .: -)
- . .,: . . ., II I- ; I .. " ., ] -;. . i : - !.' - : . 117:..

* - i: . - 1 11 1 - I L0
I . i ! , - i : , 'i -.- I
- I - r I . ., . , . . ~ I

*Schermann, D. 1. Oscillation du 8 ildstique
aux diplacements ou aux forces A
la frontl6re. A-cad. Sci, UIL%. Nbf.- [Trudy] Inst-
no. 118, 47 pji, (1046). :(Russian)
This paperi.ernitaing 41 11M niethiXI of solution of the -
threv-dimemimial problvin of propngation, of iiibrations in
an Qlastic: half-spacc~when either.the displacements orexter-
nal forces are six-vifiLd on the boundary. For.simplicity,
initial displaMountsand velocities are assumed to vanish~
The 111COIM (if solutioll Is based on Ia Wrect application. of
Follrier iWegrals. After a acqu0ce,lof transformations on
seiI rIand vvetor jvwnti als both boundky, -valtie problem s
art, educt4itoat, jiF;itl(!r.,itiniit)fccitai~qtiadrupI integrals,
a ~04
wl-.icliyiet(l;iqu;tlititive;tiialysisof thephenomenonof wave
propagation. 1. S. Sokolnik6l (Los Angeles,. Calif.).
ISOUrce: tathematical Revie,,is,
Vol No.6

USSR/Mathematics Feb 1946
Potential theory
ft
"The General Problem of the Potential Theorem,
D. I. Sherman, 14 pp
"Izv Ak Nauk Ser Mat" Vol X, No 2
Determination of a function u(X,y) harmonic in a
finite (simply or multiply connected) domain S in
the plane z = x + iy, satisfying certain conditions
on the boundary L of S.
13T94

Sherman, 1). 1. . At istic osciDatidns in the case oi
---VVWTrpl tits on the boundary of the iliediufn.
Appl. Malli. Met-li. [Akmd. Nai~k ',176SR. I'viH. Alal.
N14-ch.] 10-617-622 (11946). (Rus-sian. E'nglish Sulli-
Till. dastic medium filk a fillite, multiply-mimected do-
main S ill (lie complex plativ houndeti by lion ill tersect ing
alld liolloscillating Closed-Curves. Consideration of (he o5vil--
latious arising from given tli~,ila.cemcnts on the boundary.
gives rise toa pair of second -:)f lei pilrdal differential equm-
tious ill the longitudinal and -'irarisversc patentials Subject.
to, prt,-scribed boundary conditions hivolving first partial
Ill-rivalivus of lilt, poluillials. Potentials of a form leading
fit lilt- Sollition of a s\-stem of Fredliolm integral equations
%%illl t-imulillilig Ole frequency X of ust-illatiolls .3s a
paramulcr are sought. The exisivill-v ~if soltitiotm for X=,.O
i,. (-~1;111611vd, and Ill-lit-Y, by .1 theorcinof I'amarkin [Aim.
of Math. t2) 28, 127 -152 (1927)]. twAtitions for ahnivstall X
7. C. MPYIC (I Iallovvr, N. 11.).
140.
"~d

I'T'
USSR/OscIllationB - Theory Feb 1947
Mathematics., Applied
"The Dirichlet and Neuman Problems in the Theory of
Steady Oscillations," D. I. Sherman, 8 pp
"Prik Mate i Mekh" Vol XI, No 2
Reduction of the problems of the multi-connected
domain to the Fredholm equations, which differ
somewhat from the hitherto known integral equations
for the same problems and make possible direct
establishnent of the existence of the solution.
15TT

)AIV, ZD
Scherman. D. 1. Sur une mithode do. r6soudre certalne
proMmes do le. thforle de 1161asticN pour lee doinalnes
doublement coAixes. C. =o- ady) Acad. Sci.
URSS (N.S.) 55, 697-700 (1947).
Let the elastic medium occupy a doubly connected
domain 5 in the plane z=x+iy. The boundary 1, of S
consists of two simple Closed curies, L, And L2 having no
points if) COMM-0t), suppose Ll is In the interior of L~ and
denote by S, and S, two, simply conn ected domains bounded
by Ll and L2, respectively, The domain S, is then infinite.
The determination of stresses throughout the domain S,
when forces are prescribed on Li can be reduced to the
search for two functions j~(;) regular in S, and
riti.,ifyingotiLtliecoii(litioti (1)+7v )+J(I)=j(I),wherc
t is a variable point on L and.f(s) is a known function. The
author indicates a: method of reducing the solution of this
problent to an analogous problent for a simply co-nnected
dornainand is fell to one integral c(Inalion of Fredliolm tvlw
for a certain anxilinry function defined over om. of the
Curves I'l or ['2. 1. S. Sokolnik-off (Los Angvles, Calif.).
Source: Natheriaticil Reviots. ~1948, VOI~ 9 NO

?A 5FT~;2
um/physics M&Y 1947
Vibration
Wthematies, Applied
"Several Particular Cases of a General Problem in the
Theory of Vibrations," D. 1. Sherman, Inst Mechanics,
Acad Sci USSR, 4 pp
"Dok Akad Nauk SSSR, Nova Ser" Vol LVI, No 6
Containg number of mathematical formulas designed to
prove that for any function t4 (X, satisfying
the equation. A %, - X~A
eF
Isom

I Thefunctionsviand le--'W j-_-, no
*i are FeiWrWnQ Eby the qiwin

nda
ry conditions:
u

on Li
;I Berman,
D. n t e state o( elzmetsa hrink-fitted

'(1) +iwo) +T(1)
Wembers. Izvestiya Akad. Nauk SSSR. Otd. Tchn.

'
It010) _100) -*10) -,VVJY)- IVAO
Nauk 1948, 1371-1388 (1948). ~ (Russian
)
i The paper contains a solution of the following twoAimen on, -y, where x and A are constants-determined by the elastic
sional elastic problem. A long hallow prismatic body whose properties of the medium and by the amount of shrink

section UP ~ plane not7mal to the axis of the prism is a square -f. The author reduces the problem, (by means of
along
(with rounded corner3) with a circular hole at the center of analytic condnuation) to the determination of only two
the square, is shrink-fitted on a solid circular shaft. The ji functions.,p(s) and ~(z), analytic in the region S~+& of

elastic properties of fl-ie shaft are identical with those of the ~1 the fonn
prism, and the lateral surface of the prism is free of stress.
What is the state of stress in the member so formed? If
qw- 00-
f
f
the boundary of the square in the (x, y)-plane is L, and that i 2xi
S
z,
of the circular bole is V, the solution of the problem, follow-
where w(t) satisfies a certain intigral-differential
equation
ing Muscheli9vili, reduces to the search for four functions .
and bars denote conjugate values. The solution of the latter
pj(z) and Pj(a) (j= 1, 2) of a complex variable z=.-C+iY,
i
h
l
i
i equation is obtained in t Ihe form of an infinite series. Xcg~
ana
c
n t
ons Sj, where S, is a doubly-connected
e reg
yt i culation. of the distribution. of normal stress aloij
region bounded by L and 7 and S2 is the circular region
contained in the paper, illustrates the practical value of ihe~
bounded byY.
function-theoretic'methods of solution of elastic problenia~
L S. Sokolsika
(Los Angeles,
Sourcer Mathematical Revi We Vol 10- No. a

U /Zagineering May 1948
Wing Theory
Mathematics, Applied
"The Prandtl Equation In the Theory of a Wing of
Mite Span," D. 1. Sherman, Inst of Mech, Aced Sci
USM?-, 6 pp
"Iz Ak Hauk SSSR, Otdel Tekh Nauk" No 5
Presento solutions for Prandtl's integral differential
singular equation for case when function p(x) is
rational. Also presents approximate method for
solving the equation which will hold true for any
value of the function b(x). Submitted 2 Feb 1948.

USM/Fingineerine Sep 48
Stresses
"The Tension States in Some Pressed Parts," D. I.
Sherman, Mech Inst, Acad Sci USSR, IS pp
"Iz Ak Nauk SSSR, Otdel Tekh Nauk" No 9
A hollow prismatic body whose section is a plane
perpendicular to its axis, Is of quadrate shape.
It is weakened by a symetrioally located circular
hole and by mounting an a solid circular shaft.
72ghtness of fit is given. Elastic properties of
both bodies are considered identical. lateral sur-
face of the prismatic body is assumed to be free
14
USSR/Engineering (Contd) Sep 48
from external forces. Calculates stress dietri-
bution I- body. Coneludes by considering c~,.se,
when elastic properties oA parts are not identical.
Submitted 8 Mar 48.
t~.q lk /4?T23

. - I : . - -,~ ~-*. -,. i ---- ~ -.:- y I . : .- . . , L . i . ,
--'- . . - -' _,I - -.

qr
k,german, D. L On, pethods of solving certain AW&WA6413pecifically, it is .assumed that Aj(l) has a simple zero at,
I Integral eaWLWLs 'Alcad. Nauk SSSR. Prikl. NNW. Aleh. I=a (an L) and the.coc(ficients are analytic at I-& (the
12, 423-452 (1948). (Russian) latter condition can be lightened). It is shown thata reduc.-
ss
This is a study of systems of the form tion to regular Fret holm equations is po ible anti that (1)
his a solution wj 1, 2) continuous (in L. Such results
(1) f1#0) are "tended to systerns
P,
h C'
(k - 1. 2), where L M a simple, closed, "smoot urve (in (2)
the complex plane of x-x+iy), bounding a finite simply-
fa (to)
connected region S; the wj(t) are the unknowns and the
aij, bkj, f& are assigned, suitably differentiable on L; the (k 1, 2), where Tt-fLwy(1)Kkj(1�1o)d1: here the XqC1, to)
-hi In I# and are
integrals are in the sense of principai. values. On letting (and the coefficienW--.tre Holder-d-n-L, t,
;-C1j-akj-bApdAj--a&-j+b - o -me forms Ideterminants A, (cj) 1, analytic at the point to = ve, at which 41 has a zero of multi.
As The extensive literature relating to e4uations of plicity m. The system (1) is also studied when the aki, bkj are
type (1). and of other similar types, is larg,-,Iy concerned constants and L is an open art. IV. J. Trfilzinsk~.
i with transformations into regular Fredholm equations -of.
the second kind, when A,, As (or other analogous functions)
are distinct from zero on L. One of the novel features of this
work is that one of the determinants is allowed to vanish
at some points of L (the other one is assumed not 0 on L).
Sourcet Mathematical Reviews$ Vol No.

ti C-- I
-Z-
german, D I.- On a caue In the theory of singular equa-
Ong Mlady Akad.- Nauk SSSR (N.S.) 59, 647,::650,
1948). ',Russian)
The auth)r studies the equation
(1) AQo)u1,1o)+(ri)-1B(1a) (1-Q-1w(t)d1
f
+fiv(t)K(1, Qdt=f(Q,
where inteplration isalong' a simple "smooth" contour L;
1, to are on : the co0ficients are essentially of a 1-161der class
on L. In pievious literature (1) has been transformW into.
.a regular: F redholm equation of the second kind? predomi-
nantly uncler the'sujiposition that (2) A2-BIP-10 on L.
The author gives a new. method for effecting such a trtins-
formation when (2) does not hold. The case actually carried
out is die i ~ne when A -B has just one simple zero on L.
The questic n of eqtiivalence of the resulting Fredholm equa-
tion and ol (1) is ekamined. Tito results obtained can be
Cxtended tt systems of equations analogous to (t),
W. J. Tridzinsky, (Urbana, 111.).
8ource; IL-ithematical Reviews, 'Vol v No

DM/PhYi1cs ' ' ' Doe 48
BeaW - StreeB-Analysis
Theory of Masticity
One Torsion Woblem," D. 1. Sherman, Inat of
0,
:-I Mach, Acad Sci USSR, 4 pp
"Dok Ak Nank S M * Vol LXIII, No 5
Gives a method to solve special problems in the
theory of elasticity and hydrodyrgutdca relating
to toralon and curvature In hollow prism-shaped
beams, the cross sections of irkich are areets with
double connections, problems in the theory of
elasticity for Almila areas, and other probleym
55/49T83
USSR/Physics (Contd) Dec 48
including the more general case where an ellipse!
is placed asymmetrically with regard to a cirale.
Submitted by Acad L. S. 3:*ybenzon 14 Oct 48.
55A9T83

D. 1.
USSR/Engineering - mechanics Sep/Oct 49
Elasticity
"Theory of Steady Vibrations of a Mediun for
Given External Forces on Its Boundary," D. I.
Sherman, Moscov Inst of Mech, Aced Sci USSR, 4 pp
"Pr1k Mat i Mekh" Vol XIIII No 5
Discusses steady vibrations of an elastic medium
filling a finite simply conn cted region lying
in the complex plane vhen effective external
forces are acting upon the curve bounding the
region. Submitted 11 May 46.
sw
-I )I cWto
PA 149T42

'V
'termini D. I, and, $af6dkk0, M. Z. Owthetorbion'of.
some prismatic hoirow b"es. Ak-ad. Nauk SMK. a-
lenemyl Sbornik 6,17-46 (1950). (Russian)
The authors consider two prismatic hollow bars with the
following cross-sections: I)- an _ellipse with a concentric
circular hole, and 2) a square with rounded edges with a
concentric circular hole. In both cases the barsare twisted
by a moment Macting on the outside contour. The problems
are solved by conformal mapping, ,%rhich, due to the fact:
that the cross-wtion is a d oubly connected regioni presents
considerable difficlilty. The method employed wascleveloped
bv Serniap lady: Akad. Nauk SSSR 63,499-502 (1948);
CDok
tl~m Rev. 10, 6511. The obtained stress function, and hence
the expressions for the stresses am in a form of compiicated
ries. To make. them useful for -practical applicationsthi
authors- tabulate for both cases up -to twelve cdefficients of
the series for several dimension ratios. T. Leser.
Vol 13 No.
Source- Mathematical Revie'43, 9

I ~). I.
USSR/Physics - Mechanics
Elasticity
158T97
Mar/Apr 50
"Problem of Conformal ReClection," M. Z. Narodetskiy, 1). 1. Sherman, Mosco-.T, 6 pp
"Priklad Matemat i MAW' Vol XIV, No 2
Gives approximate, but sufficiently effective, solution of problem of conformal
reflection, in a doubly connented regi.)n 3 in the complex z-plane aga;.nst a circular
ring. Submitted 31 Dec- 49. -
I 158T97

t A~

Shown to sati&y a Fredholm in'teg-ral equation whose kernel
1, wiCh a known degree of approximation, by
can be vcplace(
-try
a degent-r- us I he f etermination ol theauxili.
mw kernel. Th , .1
terman D I On the %fregges in- a- twisted 4rcular beiam:

:~~
i7l
~ flifictiotIi is reduced to the solution of, a system nf linea, r
f unct,
-ki!6wn
oke thi! 'WARMY lon la
tion
e
l
id
b
we, enw
y a prismatic cavity. Izv6tiva Akid.:
iuk .
s,
,
ra
,
qua
ge
X"
a
S,SSROL(1.1'i,.Iiii.Nattkl95l,969-995(19~1). (Russian) thet6rsion. unctioncan
I.
_
Saint Wnwit's tor.Mon problem for a long &Cular beam tting process is not fully eslib-
of the approxim
~vergen~e

weakened by a rectangular prisaintic cavity with romided lishFd in I he pa wr, extensive numerical computations testify
cornl.,rs is solved I)v the introduction of an ati ~`illllr- to the rernarkable effeeLil. elless Of the proposed method, even
which assumes wi (fie circular innindary the when it is applicd toa thin-walled sectior.
thu complox iorsion fmiction. The 4uxiliary function is S. Sokolnikd (Los Angeles, Calif.).
1-~-Ithefnatical Rev Vol
No.3
'k~

Him
Aerman. D. L Torsion o an tic cYjWde stiffened with
a circular roa. roxf
10, 81-108 (1951). (Russian)
A detailed solution of Saint-Venant's torsion problem is
given for a homogeneous and isotropic elliptical beam -rein-
Mathematical Reviews forced by ~ circulai rod. The rod is welded onto the cylinder
along the lateral surface, and the axes of the rod and bearh
Vol. 15 NO- 4 coincide. This is a generalization of the corresponding tor-
Apr. 1954 sion problem for an elliptical beam weakened by a cylindrical
Mechanics cavity, solved by D. I. Serman and M. Z. Narodeckil
[same Sbornik 6, 17-46 (1950); these ROTH, 886].
L S. Sokolnikoff (Los Angeles, CaliQ.,?.'PI

erman. D, 1. On a case of regularization of singular
equations. Akad. Nauk SSSR. Priki. Mat Meh. 15,
75-82 (1951). (Russian)
A I nan earlier Ante [Doldady Akad. Nauk SSSR
(N.S.) 59,
.
-(1948)
647-650 these Rev. 9. 442J the author indicated a

I new method of t 'nsforming the singular equation (in the
ra
sense of principal valu
CS)
t
1
G
d
~
,
,)
(

into a Fredholm integral,equation. Here L is a closed, suit-
ably smooth curve, bounding a bounded, simply connected
domain; 1, to are points on L; A. B, G, f are givep on L and
are of 1-161der classes. This transformation was possible
when the functions A+B nmish at some points of L. In
this ezirlier work it was assumed that A(lo), B(to), G(1, to),
f(to) are analytic in to at those points. In the present work
the author studies (1), adapting the method of his preceding
.
note to the regularisation of (1) under.less restrictive condt-
tions on the toefficients involved. IV. J. TrjWzinsky.
No.
Sourco: Yatheirotical Reviews, Vo,

USSR/Physics - Stresses in Plates may/Jun 5.1
"Stresses in a Ponderable Half-Plane Weakened by Two
Circular Apertures," D. I, Sherman, Moscow, Inst Mech,
Acad Sci USSR
"Prik Matemat i Mekh" Vol XV, No -s, pp 297-316
Considers elastic isotropic and homogeneous half-
plane possessing 2 openings circular in form which
are sufficiently far removed from the margin. Cf.
G. V. Kolosov's "Application of Complex Variables to
the Theory of Elasticity," 1935, Moscow, and N. I.
Muskheli5hvill's "Certain Basic Problems in the Math-
ematical Theory of Elasticity," 1949, Moscow.%Espe-
cial interest is in the stress near boundary of aper-
tures. Submitted 16 mar 51. i85TIo4

On stresses in a plane heavy medium with
two Identical symmetrically placed circular openings.
Akad. Nauk SSSR. Priki. Mat. Meh. 15, 751-761 (195 1).
(Russian)
A homogeneous and isotropic elastic material fills the
Mathematical Reviews semi-infinite triply-connected domain S, bounded by the
Vol. 15 No. straight line La parallel to the X-axis, and by two non-
Apr. 1954 intersecting circles L, and L, with equal radii R. The centers
Mechanics of the circles lie on the X-axis at a distance f from Lo.'The
material filling S is acted on by a uniform gravitational force
in the direction of the Y-axis, and the boundaries Lo, L1, L,
are free of external loads. A solution of this two-dimensional
elastostatic problem, in the neighborhood of L, and L2, is
obtained under the hypothesis that R