- Email: [email protected]
A diffusion problem in a two-medium system
A DIFFUSION PROBLEM IN A TWO-MEDIUM SYSTEM* E.M.
KARTASHOV,
G.M.
BARTENEV and
I.V.
RAZUMOVSKAYA
Moscow (Received
November
4
1966)
THE two-dimensional diffusion processes in a system composed of two anisotropic media separated by a moving interface are of considerable Elementary solutions are known Interest in some branches of physics. for this type of problem for the case of constant values of the initial and boundary functions. We shall here consider a more general solution with no restrictions on the functions describing the behaviour of the reoulred concentration on the Interface and at the Initial instant. Consider one of the diffusion theory of the strength of rigid surface-active medium [2 - 41.
[II
problems of this type occurring in the bodies close to destruction in a weakly
The general physical picture of this problem is of diffusion in a system of two media, separated by an interface in uniform motion. The medium to the left of the interface (z < vt, y > 0) is anisotropic, with diffusion coefficients kl and k2 In two perpendicular directions, and with initial concentration distribution fll(z, y); the medium to the right of the Interface (z > vt, y > 0) is isotropic, with diffusion coefficient kl and initial concentration distribution fgg(z, y). There is dfffuslon contact on the interface z = vt, i.e. the concentration of the diffusing substance and the diffusion flux vary continuously In the nelghbourhood
II
of the
Interface.
Given the diffusion flux at the boundary y = 0.
To find the time distribution 1, 2, of the diffusing substance *
Zh.
vychisl.
Mat.
mat.
t)
qlI(z,
Fiz.
in medium I and g~gg(z,
of the concentration in both media. 7,
6, 269
1423
-
1429,
Ci(z,
1967,
t)
y,
in medium
t),
i =
E.M. Xartashov
290
The problem amounts diffusion equations
a2ci
a=ci
at
a22
w
a-c, -=
avz
to the
-kk2-,
initial
to
z >
of two-dimensional
t > 0;
0,
Y >
and boundary
a
vt,
=Cz(Z,Y,t)IL=11I;
Y >
t > 0,
0,
conditions
I =
(6 Y, t)
i 5 1, 2;
wr (z, t),
aY C~(z,y,t)l,=,t
a set
SOlViUg
(1)
w
Cf (z, Y, t) I t-o = fff (2, Y),
al.
z < vl,
a2c2
k iF+k,-,
at
subject
msthematically
acj
-=kk,-
et
u-0
(2:
aci (z, Y, t) 1
az
acz (z, Y, t)
1 zsot=
a2
II r=vt
in diffusion across an interlace with a Such problems arize, e.g. crystalline structure on one side and amorphous structure on the other. Movement of the interface corresponds to movement of the crsstallization process is not directly confront with constant speed V; the diffusion nected with the interface movement. A closely related problem concerns the growth at a constant rate v of a crack at the interface between an Isotropic material of diffusion COefficient kl, and a surface-active medium of surface diffusion coefficient k2 c41. We introduce new function
a new variable x = z - vt ‘and replace t) according to the equation Ni(x, y, V
Cj(z,y,t)= Problem
exp
- (2)
(1)
--x-2k,
a2Nf
aNt
-5
ki
at
w
a2Nz z
+
a2Ns ki -,
x < 0, 2
>
au2 NI(x, 19 t) It-o = fr(x, Y), aY =
1,
t) by a
2.
a2N*
0,
aNi (3, I/I t)
.Ni(~,y,t)l+-o
i =
y,
now becvacs
at--ki-+k2--,a22 aN2
3 N1(5, Y, q,
4k, 5
Ci(z,
Nz(z,
= ‘pi@, t), I Y-=0
~,t)lx=-o;
0,
t > 0:
(3)
Y > 0,
t > 0:
(4)
Y >
f 5 1,2;
(5)
1-
(6)
aN1(I, ur q az
1, 2;
= Z.-O
W(x, ax
y, t) , g-0
('1
A
diffusion
problem
in
a two-medium
291
system
t) are new (known) functions. We seek Nl(r, y, t) where fit%, y). qi(r, capable of Fourier transformand Nz(z, y, t) in the class of functions ation with respect to x, and t. Assume that
y
flux on the interwhere ~i(y, t) is the unknown function of the diffusion face, which has to be determined later. We apply to (4) the twodimensional infinite Fourier cosine transformation [51 m
m cos Es dz
R(E,rl.f)=;S Using (6)
where
z(f,.q,
and (8),
s
N2 (5, Y, t)cos
we now get
t) = I(T), t) + F2(f, t),the solution
formed initial
(9)
qg dy.
0
0
condition
of which,
with the trans-
%*((5,% 0) = f=2(&u) is
si(E,q, r)exp 1-h
(E2+ q2) (t - ~11dz +
0 -t-
Application
Nz(s,v,t)=-
to this
;,,j,
of the inverse
j “(“;B’(exp[
t- WE2 +
VII.
cosine-transformation
gives
_P7e+p3)2]+
0
0
To find Nl(z,
%$,,q)exp
y,
t) we replace
the variable
x = -LA, and transform
E.M.
292
the
region
x
(8) as ki -
at
Ni(--u,
w
= fit-4
I/,
au Application
of
2nd medium,
4n )I(kikz)
h(~-_B)~+k2(~+a)~
k,(u
-
to
conjugation
(ll),
(13),
OD
(‘12)
t);
as in the
-
case
1 +
(131
0
kt(d-P)2+kz(u-a) 1 II
.___
----
+ exp
4k,kpt
bI>
dS t
ks2
+exp
-
k,(u + a)2 + hP 4k,k,(t
[
working [61 that the and (3) respectively, conditions.
condition Ni t-0,
We have
‘pi(--4
4kikzi
1
The first
I =
ki(y-fi)-‘+kziv.-aJ2
a)Z + kly2
It can be shown by direct (10) satisfy equations (11) sponding boundary and initial
(11;
=
(9)
4kik,t
k*(y + a)‘+
and
I - ‘u.tv, 4. -.u-o
t)
0
0
(61,
u-0
DD
--.T--
(5),
t > oi’
0,
Y >
ai
the transformatior gives oi)
Xi(---u,y,t)=
(3).
write
dNi (-u, Y, t) --
Y);
aNi (-9
of the
now
u > 0,
fkz--*
au2
y, t) It-o
al.
@Ni
PNi
dNi -_=
et
0 < u < + m. We can
0 into
<
Kartashov
(7) Y,
is
used
to
t) = Nz (+O, Y, t).
-z)
functions and also
find
y(y,
da. I>
(13) and the corre-
t):
A diffusion
problem
in
a tso-medium
system
293
where
ki(v -___
-
B)” + kza2 4k,k,t
To solve the integral equation second factor in the inner Integral
c
l! TfxP -
and we apply
(14) in ‘Y(y, t), thus:
a)”
(Y -
we transform
the
=
4ki(t--7)
= exp The equation
1+
-
y2 +
a2
4ki (t -
t)
ch 1
w 2ki (t -
T)
I
(i = a, 2).
now becomes
to
It
the
infinite
Fourier
cosine
trSnSfOrmatiOn
. . . cos qy dy. 0
Changing
the
order
of
integration
and evaluating
the
inner
InteiZralS
294
E.M.
c71,
we
Kartashou
et
get
OD
1
-
3t
1 - --exp[--k&(t-T)Jdt * ,‘(t - T)
za
1 -
we can write
the
last
s 0
Y(a, z)cos aq da = F(q, z),
as
expression
The Carson - Heaviside
to this
equation.
transformation
b--kWt - T)] a$=
1 ,‘(P + kir1’)
where
P s ,. . e-PJdt 0
hl
F(~, t).
can now be
We get
Ikiy (q, P) whence
Y(cz, z)cos c&r) da = P(q, t).
s
3% 8
---e=p
applied
al.
+ __I-.
I
li(P + km
I
= F(rl, P),
A diffusion
problem
a two-medium
in
1
OL
0
-$s I
To find
the
rl)
fl(--a,
original
ml,
C
p J’(P +
F(q,
-$l@+k2r12)
kzrl”)
exp
t) we write
the
I’(P + W)
P) =
’
1 (P + kiq2) + V(P + kz?)
+
da
i
image
ml,
295
system
T(%P)c
thus
PI+ ’ 15)
I'(P
+ kir12)
P),
-------FS(Tl>
-I’(P + kN)+
I’(P + kzq2)
where 02
E(q,P)=-
F,(q,
v
+
&(a,
1
-$(P
p)ew
fii(-u,
p)=: I/:
+ ktrt2) Ida
i
L
0
t
[-~V(p+kzt12) Ida-
p)exp
Cl
-
al
-G 1 fi (-a,
rl)P e*p
0
The originals
FzCy.
t)
and 1
Pn(& t)=
Fs(y,t)=-
l’kz 2nk,:i
-2
2k,n
s
0
dz ‘_
(t--)2
s s o t
~
&
o (t-t)’
~‘s(,Y,
t)
I
a2 + ii2
m os
+ k2q2)
aq2(a, 7)exp
7)exp
da.
are
-
acpl (-a,
$I(P i
-
-
ha2
da +
z) I
+ du 1-
4kikz (t -
-
4ki (t -
b(y
p_-._
by2 t)
-
p)” + k2a2
4k,k,t
1+
_ _k_i_!y + B)” + ha2 4k,k,t
1
I
J dB.
B.M.
296
Evaluating
the contour 1
et
ab.
Integrals
Y+im
--s 2ni
Knrtashov
1
I@ 4- kzW
y--imT
V(p+
k2q2)+
I@
- exp (pt) 4 +
kiQ"l
and
where v isa lties
of
this
constant,
,:rcater
integrand.
than the real
we find
?‘(P+ JwP)-+-
1
2ni
lip + kN) + ,‘(P + k2n2)
P”+Yos
convolution
to
of all
1
V’(P+ W?
I;----------.------c----_ V_i,~ I’& + kN)+ Y(P + k#)
x2 + ki u2(xf = kz x.‘+x Applying
parts
(1.5).
the
SlIIgUlar-
that
,
t+(x)
=
exp W) dp -
x2 + kz
k, ~.
s2+kt
the original
of y(q,p):
is found to be
m
X
(s
xW(x)q2
7 t~2+kzf(~2+klf2
exp I-- M(r) (t -
t)q2]
dz
& f
applying the Fourier cosine transformstiti3 inversion %I farther changing the order of the integrals, and theorem to this expression, evaluating the inner integral, we finally get
A diffusion
+ (
prob
Zen
a two-medium
(Y-m2
1_
K
26(x)
(t -
= 1o Fs(B’ ‘)-(
) [ exp
.c)
(Y
-’
+ (
46(x)
-
The required expressions fixed
Due to
for
the
the
the
constants, -
such
presence
y,
of
t),
are
i = I,
2) /
t)
(i
2,
(Y+ 4a2(x)
= 1,
21,
now easily after
the
exponentials are
fi(x,
and (13)
since
the
integrals
functions (10)
(7),
Ni (x,
y,
(21,
-
),,, [ -
(t -
’
+
I
(yf!c
1 _
Ci(t,
(1)
PI2
k2)2a(x)
]] dx }dB. (t-r)
satisfying
obtained
again
the
from the
returning
to the
system.
improper
and boundary
to
our problem
coordinate
(13),
(31
concentrations
of
3 (x” + k,) (xr;
(t -T)
28(x)
conditions
297
system
I---
Ikn(ki - kz) * dz -___ .s ____ 2n31s o (t-Z)%
X
in
y)
and ~i(“,
represent
Fourier
in the
convergent
for If
t).
of
class
these
“generalized”
transformation
kernels
a wide
and initial
functions
solutions
obviously
(10) of
of
cannot
are problem
be applied
functions.
The authors
thank
V.I.
Levin
for
his
interest
and for
discussing
the
results. Translated
by D.E.
Brown
REFERENCES 1.
LYKOV, A. V. Theory
of
Moscow, Gostekhizdat,
2.
LIKHTMAN, V.I., mechanics
Moscow,
of
Akad.
heat conduction 1952.
SHCHUKIN, E.D. metals
Nauk
(Teoriya
and REBINDER, P.A.
(Fiziko-khimicheskaya SSSR,
teploprovodnosti),
1362.
Physico-chemica
mekhanika
metallov).
I
E.M.
298
Kartashov
et
al.
3.
BARTENEV, G.M. and RAZUMOVSKAYA,I.V. Time dependence strength of brittle bodies in surface-active media, Nauk SSSR 150, 4, 784 - ‘787, 1968.
4.
BARTENEV, G.M. and ZUEV, Yu.S. Strength and destruction of highly i razrushenie vysoko-elasticheskikh elastic materials (Prochnost’ materialov), Moscow, Khimiya. 1964.
5.
SNEDDON, I.N.
6.
TIKHONOV, A.A. and SAMARSKII, A.L. Equations of mathematical (Uravneniya matematicheskoi fiziki), MOSCOW, Gostekhizdat.
Fourier
transforms,
7.
GRADSHTEIN, I.S. and RYZHIK, I.M. and products (Tablitsy integralov, Moscow, Fizmatgiz, 1963.
8.
DITKIN, V.A. and PRUDNIKOV, A.P. (Spravochnik po operatsioanomu shkola. 1965.
McGraw-Hill,
of the DokI. Akad.
New York.
lg51, physics
1958.
Tables of integrals, sums, series summ, ryadov i proizvedenii),
Guide
to
ischisleniyu),
the
operational
Moscow,
calculus
Vysshara
KARTASHOV,
G.M.
BARTENEV and
I.V.
RAZUMOVSKAYA
Moscow (Received
November
4
1966)
THE two-dimensional diffusion processes in a system composed of two anisotropic media separated by a moving interface are of considerable Elementary solutions are known Interest in some branches of physics. for this type of problem for the case of constant values of the initial and boundary functions. We shall here consider a more general solution with no restrictions on the functions describing the behaviour of the reoulred concentration on the Interface and at the Initial instant. Consider one of the diffusion theory of the strength of rigid surface-active medium [2 - 41.
[II
problems of this type occurring in the bodies close to destruction in a weakly
The general physical picture of this problem is of diffusion in a system of two media, separated by an interface in uniform motion. The medium to the left of the interface (z < vt, y > 0) is anisotropic, with diffusion coefficients kl and k2 In two perpendicular directions, and with initial concentration distribution fll(z, y); the medium to the right of the Interface (z > vt, y > 0) is isotropic, with diffusion coefficient kl and initial concentration distribution fgg(z, y). There is dfffuslon contact on the interface z = vt, i.e. the concentration of the diffusing substance and the diffusion flux vary continuously In the nelghbourhood
II
of the
Interface.
Given the diffusion flux at the boundary y = 0.
To find the time distribution 1, 2, of the diffusing substance *
Zh.
vychisl.
Mat.
mat.
t)
qlI(z,
Fiz.
in medium I and g~gg(z,
of the concentration in both media. 7,
6, 269
1423
-
1429,
Ci(z,
1967,
t)
y,
in medium
t),
i =
E.M. Xartashov
290
The problem amounts diffusion equations
a2ci
a=ci
at
a22
w
a-c, -=
avz
to the
-kk2-,
initial
to
z >
of two-dimensional
t > 0;
0,
Y >
and boundary
a
vt,
=Cz(Z,Y,t)IL=11I;
Y >
t > 0,
0,
conditions
I =
(6 Y, t)
i 5 1, 2;
wr (z, t),
aY C~(z,y,t)l,=,t
a set
SOlViUg
(1)
w
Cf (z, Y, t) I t-o = fff (2, Y),
al.
z < vl,
a2c2
k iF+k,-,
at
subject
msthematically
acj
-=kk,-
et
u-0
(2:
aci (z, Y, t) 1
az
acz (z, Y, t)
1 zsot=
a2
II r=vt
in diffusion across an interlace with a Such problems arize, e.g. crystalline structure on one side and amorphous structure on the other. Movement of the interface corresponds to movement of the crsstallization process is not directly confront with constant speed V; the diffusion nected with the interface movement. A closely related problem concerns the growth at a constant rate v of a crack at the interface between an Isotropic material of diffusion COefficient kl, and a surface-active medium of surface diffusion coefficient k2 c41. We introduce new function
a new variable x = z - vt ‘and replace t) according to the equation Ni(x, y, V
Cj(z,y,t)= Problem
exp
- (2)
(1)
--x-2k,
a2Nf
aNt
-5
ki
at
w
a2Nz z
+
a2Ns ki -,
x < 0, 2
>
au2 NI(x, 19 t) It-o = fr(x, Y), aY =
1,
t) by a
2.
a2N*
0,
aNi (3, I/I t)
.Ni(~,y,t)l+-o
i =
y,
now becvacs
at--ki-+k2--,a22 aN2
3 N1(5, Y, q,
4k, 5
Ci(z,
Nz(z,
= ‘pi@, t), I Y-=0
~,t)lx=-o;
0,
t > 0:
(3)
Y > 0,
t > 0:
(4)
Y >
f 5 1,2;
(5)
1-
(6)
aN1(I, ur q az
1, 2;
= Z.-O
W(x, ax
y, t) , g-0
('1
A
diffusion
problem
in
a two-medium
291
system
t) are new (known) functions. We seek Nl(r, y, t) where fit%, y). qi(r, capable of Fourier transformand Nz(z, y, t) in the class of functions ation with respect to x, and t. Assume that
y
flux on the interwhere ~i(y, t) is the unknown function of the diffusion face, which has to be determined later. We apply to (4) the twodimensional infinite Fourier cosine transformation [51 m
m cos Es dz
R(E,rl.f)=;S Using (6)
where
z(f,.q,
and (8),
s
N2 (5, Y, t)cos
we now get
t) = I(T), t) + F2(f, t),the solution
formed initial
(9)
qg dy.
0
0
condition
of which,
with the trans-
%*((5,% 0) = f=2(&u) is
si(E,q, r)exp 1-h
(E2+ q2) (t - ~11dz +
0 -t-
Application
Nz(s,v,t)=-
to this
;,,j,
of the inverse
j “(“;B’(exp[
t- WE2 +
VII.
cosine-transformation
gives
_P7e+p3)2]+
0
0
To find Nl(z,
%$,,q)exp
y,
t) we replace
the variable
x = -LA, and transform
E.M.
292
the
region
x
(8) as ki -
at
Ni(--u,
w
= fit-4
I/,
au Application
of
2nd medium,
4n )I(kikz)
h(~-_B)~+k2(~+a)~
k,(u
-
to
conjugation
(ll),
(13),
OD
(‘12)
t);
as in the
-
case
1 +
(131
0
kt(d-P)2+kz(u-a) 1 II
.___
----
+ exp
4k,kpt
bI>
dS t
ks2
+exp
-
k,(u + a)2 + hP 4k,k,(t
[
working [61 that the and (3) respectively, conditions.
condition Ni t-0,
We have
‘pi(--4
4kikzi
1
The first
I =
ki(y-fi)-‘+kziv.-aJ2
a)Z + kly2
It can be shown by direct (10) satisfy equations (11) sponding boundary and initial
(11;
=
(9)
4kik,t
k*(y + a)‘+
and
I - ‘u.tv, 4. -.u-o
t)
0
0
(61,
u-0
DD
--.T--
(5),
t > oi’
0,
Y >
ai
the transformatior gives oi)
Xi(---u,y,t)=
(3).
write
dNi (-u, Y, t) --
Y);
aNi (-9
of the
now
u > 0,
fkz--*
au2
y, t) It-o
al.
@Ni
PNi
dNi -_=
et
0 < u < + m. We can
0 into
<
Kartashov
(7) Y,
is
used
to
t) = Nz (+O, Y, t).
-z)
functions and also
find
y(y,
da. I>
(13) and the corre-
t):
A diffusion
problem
in
a tso-medium
system
293
where
ki(v -___
-
B)” + kza2 4k,k,t
To solve the integral equation second factor in the inner Integral
c
l! TfxP -
and we apply
(14) in ‘Y(y, t), thus:
a)”
(Y -
we transform
the
=
4ki(t--7)
= exp The equation
1+
-
y2 +
a2
4ki (t -
t)
ch 1
w 2ki (t -
T)
I
(i = a, 2).
now becomes
to
It
the
infinite
Fourier
cosine
trSnSfOrmatiOn
. . . cos qy dy. 0
Changing
the
order
of
integration
and evaluating
the
inner
InteiZralS
294
E.M.
c71,
we
Kartashou
et
get
OD
1
-
3t
1 - --exp[--k&(t-T)Jdt * ,‘(t - T)
za
1 -
we can write
the
last
s 0
Y(a, z)cos aq da = F(q, z),
as
expression
The Carson - Heaviside
to this
equation.
transformation
b--kWt - T)] a$=
1 ,‘(P + kir1’)
where
P s ,. . e-PJdt 0
hl
F(~, t).
can now be
We get
Ikiy (q, P) whence
Y(cz, z)cos c&r) da = P(q, t).
s
3% 8
---e=p
applied
al.
+ __I-.
I
li(P + km
I
= F(rl, P),
A diffusion
problem
a two-medium
in
1
OL
0
-$s I
To find
the
rl)
fl(--a,
original
ml,
C
p J’(P +
F(q,
-$l@+k2r12)
kzrl”)
exp
t) we write
the
I’(P + W)
P) =
’
1 (P + kiq2) + V(P + kz?)
+
da
i
image
ml,
295
system
T(%P)c
thus
PI+ ’ 15)
I'(P
+ kir12)
P),
-------FS(Tl>
-I’(P + kN)+
I’(P + kzq2)
where 02
E(q,P)=-
F,(q,
v
+
&(a,
1
-$(P
p)ew
fii(-u,
p)=: I/:
+ ktrt2) Ida
i
L
0
t
[-~V(p+kzt12) Ida-
p)exp
Cl
-
al
-G 1 fi (-a,
rl)P e*p
0
The originals
FzCy.
t)
and 1
Pn(& t)=
Fs(y,t)=-
l’kz 2nk,:i
-2
2k,n
s
0
dz ‘_
(t--)2
s s o t
~
&
o (t-t)’
~‘s(,Y,
t)
I
a2 + ii2
m os
+ k2q2)
aq2(a, 7)exp
7)exp
da.
are
-
acpl (-a,
$I(P i
-
-
ha2
da +
z) I
+ du 1-
4kikz (t -
-
4ki (t -
b(y
p_-._
by2 t)
-
p)” + k2a2
4k,k,t
1+
_ _k_i_!y + B)” + ha2 4k,k,t
1
I
J dB.
B.M.
296
Evaluating
the contour 1
et
ab.
Integrals
Y+im
--s 2ni
Knrtashov
1
I@ 4- kzW
y--imT
V(p+
k2q2)+
I@
- exp (pt) 4 +
kiQ"l
and
where v isa lties
of
this
constant,
,:rcater
integrand.
than the real
we find
?‘(P+ JwP)-+-
1
2ni
lip + kN) + ,‘(P + k2n2)
P”+Yos
convolution
to
of all
1
V’(P+ W?
I;----------.------c----_ V_i,~ I’& + kN)+ Y(P + k#)
x2 + ki u2(xf = kz x.‘+x Applying
parts
(1.5).
the
SlIIgUlar-
that
,
t+(x)
=
exp W) dp -
x2 + kz
k, ~.
s2+kt
the original
of y(q,p):
is found to be
m
X
(s
xW(x)q2
7 t~2+kzf(~2+klf2
exp I-- M(r) (t -
t)q2]
dz
& f
applying the Fourier cosine transformstiti3 inversion %I farther changing the order of the integrals, and theorem to this expression, evaluating the inner integral, we finally get
A diffusion
+ (
prob
Zen
a two-medium
(Y-m2
1_
K
26(x)
(t -
= 1o Fs(B’ ‘)-(
) [ exp
.c)
(Y
-’
+ (
46(x)
-
The required expressions fixed
Due to
for
the
the
the
constants, -
such
presence
y,
of
t),
are
i = I,
2) /
t)
(i
2,
(Y+ 4a2(x)
= 1,
21,
now easily after
the
exponentials are
fi(x,
and (13)
since
the
integrals
functions (10)
(7),
Ni (x,
y,
(21,
-
),,, [ -
(t -
’
+
I
(yf!c
1 _
Ci(t,
(1)
PI2
k2)2a(x)
]] dx }dB. (t-r)
satisfying
obtained
again
the
from the
returning
to the
system.
improper
and boundary
to
our problem
coordinate
(13),
(31
concentrations
of
3 (x” + k,) (xr;
(t -T)
28(x)
conditions
297
system
I---
Ikn(ki - kz) * dz -___ .s ____ 2n31s o (t-Z)%
X
in
y)
and ~i(“,
represent
Fourier
in the
convergent
for If
t).
of
class
these
“generalized”
transformation
kernels
a wide
and initial
functions
solutions
obviously
(10) of
of
cannot
are problem
be applied
functions.
The authors
thank
V.I.
Levin
for
his
interest
and for
discussing
the
results. Translated
by D.E.
Brown
REFERENCES 1.
LYKOV, A. V. Theory
of
Moscow, Gostekhizdat,
2.
LIKHTMAN, V.I., mechanics
Moscow,
of
Akad.
heat conduction 1952.
SHCHUKIN, E.D. metals
Nauk
(Teoriya
and REBINDER, P.A.
(Fiziko-khimicheskaya SSSR,
teploprovodnosti),
1362.
Physico-chemica
mekhanika
metallov).
I
E.M.
298
Kartashov
et
al.
3.
BARTENEV, G.M. and RAZUMOVSKAYA,I.V. Time dependence strength of brittle bodies in surface-active media, Nauk SSSR 150, 4, 784 - ‘787, 1968.
4.
BARTENEV, G.M. and ZUEV, Yu.S. Strength and destruction of highly i razrushenie vysoko-elasticheskikh elastic materials (Prochnost’ materialov), Moscow, Khimiya. 1964.
5.
SNEDDON, I.N.
6.
TIKHONOV, A.A. and SAMARSKII, A.L. Equations of mathematical (Uravneniya matematicheskoi fiziki), MOSCOW, Gostekhizdat.
Fourier
transforms,
7.
GRADSHTEIN, I.S. and RYZHIK, I.M. and products (Tablitsy integralov, Moscow, Fizmatgiz, 1963.
8.
DITKIN, V.A. and PRUDNIKOV, A.P. (Spravochnik po operatsioanomu shkola. 1965.
McGraw-Hill,
of the DokI. Akad.
New York.
lg51, physics
1958.
Tables of integrals, sums, series summ, ryadov i proizvedenii),
Guide
to
ischisleniyu),
the
operational
Moscow,
calculus
Vysshara