SCIENTIFIC ABSTRACT NAYMARK, B. M. - NAYMARK, M. A.

• Document Type: CREST
• Collection: Scientific Abstracts
• Document Number (FOIA) /ESDN (CREST): CIA-RDP86-00513R001136220013-2
• Release Decision: RIF
• Original Classification: S
• Document Page Count: 100
• Document Creation Date: January 3, 2017
• Document Release Date: July 31, 2000
• Sequence Number: 13
• Publication Date: December 31, 1967
• Content Type: SCIENTIFIC ABSTRACT

25475
3/02OJ61/139/001/007/018
B1041B231
Naymark, B. M.
TITLEs Some nonlinear boundary value probiems In the theory of the
Maxwellian bo,4y
PERIODICALe Akademiya nauk SSR, Dokladyq v. 1390 no. 1p iq6ij 63'- 66
TEXTs Some physical problems dealing with the motion of an elastic body
exhiblting relaxing stresses involve finding the displacement vector
T(x,l x2, x39 t) with the components u 1, u2P and uY and the stream '
(x11 x2, x3s t), I, J, 1, 2, 3. These satisfy equ.atlonle
tensors G~i
i
I
j
0'a Nis No exp ds 2 a dv,
~Au + (X + Ii) grad d1v u + pF 01 (u),
Card 1/8

25475 -7~4 -::-i
3/02Y61/139/001/007/018
Some nonlinear boundary value.: B104 B231
Here,, the ?form the columns with the components Ou du dX30
ildxi 3/
du~/d 4-9u 1dxj, du + 'Du ilmn
X2'+ ()u2/)x,' Dul/~X3 3 2/"?x3 3/ax 2; ris the col
with the components 6~~ X P 6- (51
ISIIX X X X (5'X X 3 9 IX X -, Xand
1 1 2 2 3 31 1 1.2 2 3
stand for the Lam6 constants. ?(Z ie ) is the positive density
10 x2t 3 45
funotionj -;(x I? x29 x3#' t) is the vector of the volume forceal
T(x,t x 2P 'r3t tt 5") is the relaxation time. Furthermore# the foll4ug
.formulas apply to I it H2 9 and the vector I)t(q)g
X+2P
X
X
.,.o
0
0 1
X + 214
%
1 0
1.0
0 X 0 0 0
)6 0 0 0
+ 2p 00 0
0 0 0.0
0 0 0 0
0 0 0 14 t -t7* .1 0 0 0
-t 2-t 0 0 0
-t '-f. 2 0 0 0
~O. 0 0%0 0. (2)
0 0j 0 Ott 0
0 0 Ott$
Card 2/8

S
/001/00
/018
7
,
OM

Some.nonlinear boundari ;alue... B
B104 B231

0a (U) ak exp (2 dr +
T 3T 7x, T'rk (3)
ds 14 au, am/ P[ di
-' + a"A-) dr,
d-r + ex 4
~] ~I
+ a ~exp[4~1 + N
X, r r

The author assumes, moreover, that the point X11 x2t jr.3-is positioned
in a bounded region D of e* thresdimensional speoe and that the boundary-
th
of this region 1 11 a lane whose curvature is continuous. Moreover, it
F f
t
i
h
d
ti
rom
s assumed that + and t
a
the boun
ary separa
ng
1 2 1 2

constitutes a smooth curve. The following three boundar value problems~
are investigatedt 1) Finding the vector ~X(xit.%~t X30 t~ and the stress
.tbneor -(x t), which satisfy equation (1) and boundqry
Xi x It X20 X39

Card 3/ 8
A

25475
S/020/61/139/001/007/018
Some nonlinear boundary value... 3104/3231
cand
1tions a) at t), SSF, a, t) is a given vector* 2) Finding
& tr .
a %~qotor Uand a a see tensor 6- whic4 satisfy equation (1) and
x X4
i
.boundary'oondition 6. coonx + i~l coenie + 6' oosnx - Yst.-Ot
x x I x , x 2 x x 3
1 1 2 1 x
SEF i i - 1, 2, 3'~ Her's coo nx are-the direction cosines of the outer
i
perpendicular to( I Xi is,a, given vector (vector of outer forces).
3) Finding a vector ~t and a stress tensor ly which satisfy (I ) "d
x x
z`-
boundary conditions of 1) with respeot to.F and those of 2) with respatol1
to For the purpbse of solving thsee boundeLry'value problems a Hilbert.,
space 7tis introduced as well as a linear norialized space *a(t Pf
11 t2
the vector functions with values in In 'these spaces
Card 4/8

25475
Is/02OJ61/139/001/007/018
some nonlinear boundary value*** B104/B231
ra, -C1,
W(U. V) X
++ + +
la 0% acts Oug (?US ave +a &I du
+ 2t, + + a
+
+ 21,? OF + 214
+ dri digs
+ + dx
+ + h,,G) (y,-
is deslgn~ted as Solution of equations,
'is validg and a pair u,
0.) with one of the boundary conditions from 1) to-3), so that the pair
t?16'j will satisfy equation
-et dxs
% V) IlVi Po i0t, P~. V) dxi d,
tiv. Pi.,#] dxt dkidx. +
Card 5/
30

25475----
S/02OJ61/139/001/007/018
Some nonlineV boundary value... B1041B231
40
+ exp N dv, dxs dxt dxs
[ I Ve x p ; rer d v. Pd x i d xt d, ri, +
.45
+ 1i INS ~r r ;Zdv. poiPCA
50
for any pair Tell The solution 6NO, t i7ound in this way Ia.
Y n 0
referred to go War's open polygon or problem (1) with the boundary
conditions from 1) to 3). To oong'ludep the following Is=& and the
following theorems are established and provedt Lemmas ftler's open polygott 5!3
system is,equally limitod and continuous to an'equal degree in 7Lon
section 0 r, t C t if conditions
0

25475
i5/020 61/139/001/007/018
Some nonlinear boundary value*** -B.04/423
sup W (Ut. U') < Co. Inf r (xi, xt. xs. 0) > 0. (7) 60
V
are met. Theorem It if (7) to mot and
'U1% I I