ANALYTIC AND NUMERICAL SOLUTION OF MIXED UNSTEADY SPACE, PLANE
AND ANTIPLANE PROBLEMS OF CRACK IN ISOTROP PLANE AND SPACE
A.G. Bagdoev, A.V. Vardanyan, S.V. Vardanyan, A.N. Martirosyan
Abstract
In this paper the analytic solution of mixed unsteady boundary value space, plane and
antiplane problems for semiinfinite crack in composed elastic plane and space is done. There
are applied methods of integral transforms of Laplace by time and Fourier by longitudinal
coordinate along crack. The sixth, fourth and second order Wienner-Hopf systems
correspondingly to mentioned problems are obtained. The singularities of their matrices are
improved and solution of problems is brought to Hilbert problem with sixth, fourth and
second order continuous matrices, and furthermore to solution of sixth, fourth and second
order Fredholm integral equations. The numerical solution of mentioned problems is given by
solution of Fredholm equations and then by calculations of integrals in Smirnov-Sobolev
forms representing stress intensity coefficients near edges of cracks.
Introduction
The solution of problem of horizontal displacements given on boundary of halfspace along
the halfplane, when on remainder of boundary plane stresses are equal to zero, and along of
whole boundary normal stresses are zero, by method of integral transformations and Hilbert
problem solution is obtained analytically and numerically in [1].
The plane problem of smooth punch is solved in [2]. In paper [1] is used method of reducing
of homogeneous Hilbert problem or, which is the same, problem of matrix factorization, to
system of two integral Fredholm equations [3], in [1] factorized matrix is continuous along
whole real axis, which brings to known singularity of solution [2]near edge x  0 .
The dynamic, harmonic in time mixed problem of finite crack bounded by elastic halfplanes
on upper bank of which are equal to zero stresses and on lower one are given displacements
by method of solving of system of singular integral equations solved for originals is
investigated analytically in [6]. In present paper the unsteady space, plane and antiplane
problems of semiinfinite crack in form of halfplane and halfdirect z  0, x  0 , bounded by
elastic halfspaces or halfplanes respectively z  0, z  0 with different elastic constants,
when upper banks of cracks are free form stresses, and on lower banks are given vertical
displacements, by using the method of [1] are considered. In contrasts to [1] the matrices of
Wienner-Hopf equations of mentioned problems of sixth, fourth and second orders
respectively, obtained by method of Laplace and Fourier transforms from boundary
conditions are discontinuous at infinity and is carried out improvement of it by the methods
similar to [3], [6-8]. Since the matrix method of [3] is rather complicated in present paper for
Fourier transforms at infinity is done method of separation of equations on scalar ones and in
them is avoided singularity of Wienner-Hopf systems. Then problem for the all axis is
brought to Wienner-Hopf system with continuous matrix which is led to solution of sixth,
fourth and second order Fredholm integral equations. They are solved numerically for all
mentioned cases. The inverse integral transformations are brought solution for stresses on
lower bank of cracks to Smirnov-Sobolev form. The stress intensity coefficients integrals are
calculated. By other methods problems of dynamics of inclusions in elastic plane are solved
in [6,10,11].
1. Mixed boundary unsteady problem of halfplane crack between two elastic halfspaces.
Let us choose x, y coordinates in plane of separation of halfspaces, z axis is normal to
plane. u x1 , u y1 , u z1 and u x2  , u y2  , u z2  are displacements components in halfspaces z  0
and z  0 , a1 ,b1 and a2 ,b2 are corresponding elastic waves speeds in them.
One must solve Lame equations
1, 2
 2 u x1, 2 
a12, 2  b12, 2
 b12, 2 u x1, 2  
,
x
t 2
2 1, 2 
1, 2   u y
2
2 1, 2
2
a1, 2  b1, 2
 b1, 2 u y 
,
y
t 2
1, 2
 2 u z1, 2 
a12, 2  b12, 2
 b12, 2 u z1, 2  
,
z
t 2
1, 2 
u 1, 2  u y
u 1, 2 
2
2
2
1, 2  x 
 z ,  2  2  2,
x
y
z
x
y
z
1, 2  21, 2
1, 2
a12, 2 
, b12, 2 
1, 2
1, 2






(1.1)
where 1, 2 , 1, 2  are elastic constants and 1, 2  are densities for upper and lower halfspaces

 1, 2  u 1, 2 
correspondingly. Initial conditions vectors for u ,
are zero. We shall consider
t
space problem for crack occupying halfplane x  0, y  , z  0.
Boundary conditions are as follows
u x1  u x2  , u y1  u y2  , u z1  u z2  ,
x  0, z  0, y  
 1
2 
1
2 
1
2 
 xz   xz ,  yz   yz ,  zz   zz ,
(1.2)
u x2   0, u y2   0, u z2   C0 H x H  y  H t ,
x  0, z  0, y  
 1
1
1


0
,


0
,


0
,
 xz
yz
zz
where C0 ,  are constants, H t   is Heviside unit function. Denoting by u x1, ,y2, z Laplace
transforms on t from u x1,,y2,z one can look for solution in form
2  
1, 2 z
u x1, 2  , u y1,2  , u z1, 2      eixiyi n
1  
An1, 2  , Bn1, 2  , Cn1, 2 dd
(1.3)
where after placing of (1.3) in (1.1) one obtains
 n1 
2
2
2 
2
2




,


  2  2 ,
n
2
2
cn
dn
c1  a1, c2  b1, d1  a2 , d2  b2 , s  i Laplace transformant parameter. From (1.3) and
(1.2) one can obtain system of Wienner-Hopf equations
2
 
1 
11  12
1 1   zz
2
2
  2      21  2  2 
 ib1 1
R,  

2
2
2
4 2 11  21  2 2  21    21   2   2   21   2  1 


  zx 

1
 2
ib12 1
 
 

1
  12
1 1   yz
2
2
 1  3 2      41  2  2
 ib1 1
2 
(1.4)
 u
u x 1

x 1


,

i
 i
2
2  11 
1 
1
1 

  zz  1  2  1 1   2  2   12   zx 
1 2
2
 ib121
b121R,  ib121 R,  
 
 

  
1 
zx
ib12 1
(1.5)
 
1

11  1 1 12
u   C  ei u z 1
2
2  yz

 2 1  2   2      2  z 1  0 12
 ib1 1
R,  
i
i
4 s





12    22   u y
b22  2  2  b22 2 12   b22  22  2 u x
 2 


i 12   22    2   2 i
b1 1  b22 12   22    2   2


2
2
(1.6)
 212   22    22    2   2     212   22    22    2   2 
i


C
e
u

 z  
 0
 
 21 ,

2  2 

2
2
2  2 
2
2
2
i 
1  2    
1  2    
4 s b1 1i

   
1 
zz
ib12 1




b22  2   2  2 2 b22  2 2 b22  2b22 12   22  u x


i
b12 1 
b22 12   22    2   2




 2  2 2 b22  2 2 b22  2b22 12   22  u y
 22 2 u z
u z 



i b22 12   22    2   2 i 
b22 12   22    2   2



C0 2  22 e i

2

(1.7)
 3
 b  i,
 2        
   
4 2 sb22 12   22    2   2
11   21   2   2

R, 


1
1
1
2
2
1 1
1 
zz
ib12 1
 

2
1
 1
 1 1  3 21   2   2  411  21  zx2 
 ib1 1
 2 R,  
 1
 2 2  12    12   2   2   12   2   4 2  1  1 
 1 1
 2
 2


2
1 2 




 2 R,  

u y
i

u y
i
 

 yz1
ib12 1
(1.8)

1 ,
3
  
1 
yz
ib12 1




   2    2  u  2  b 2  2  2   b 2  2  2  u y
1
1
x
1
2
1
2


 2  2 
2
2
2 2  2 
2
2
i

i
b1 1  2    
 1  2    
(1.9)
2
2
 212   22    22    2   2     212   22    22    2   2 
i


 uz   
 C0 e
 
 22

2  2 

2
2
2  2 
2
2
2
i 
1  2    
1  2    
4 s b1 1i

where
b 2
 22 2
b1 1


2
  2   , R,    22   2   2  4  1  2  2   2 ,
a12



1
 1
u x, y , z  2  dt  dy  e ix iy  st
u x , y , z  u x2, y , z
dx,
x
4 0   0
z 0



1
1, 2,3  2  dt  dy  e ix iy  st  xz1, yz , zz   xz2 , yz , zz
dx,
z 0
4 0   0


0

1
 zz1, xz , yz  2  dt  dy  e ix iy  st  zz1, xz , yz dx,
z 0
4 0    

2



0
u x , y , z
1
u x, y , z  2  dt  dy 
dx,
4 0   x
   
1
1











(1.10)

z 0
where index “+” denotes analytic in upper halfplane  functions, and “–” denotes analytic in
lower halfplane  functions. In derivation from Wienner-Hopf system of equations (1.4)(1.9) Hilbert problem one can see that in these equations matrix has discontinuity of first
order on infinity, which one must avoid. To do it we shell bounded by homogeneous case i.e.
 
to take 1  2  , a1  a2  a, b1  b2  b. Also we denote 11

 1 . in equations (1.4)-
(1.9), and dropping out the free term with C0 for    one can obtain system of equations
 
 


 xz1
 zz1
b2
a2
 sgn 

 sgn  u x  sgn  u x
2
2
2
2
2
2
2 a b
b 
2 a  b ib 


 


 


 zz1
a2
b 2 sgn   xz1

 sgn  u z  sgn  u z
2
2
2
2
2
2
2 a  b ib  2 a  b
b 


  

 sgn 
1
xz
(1.11)


(1.12)

1
2ia 2 
2b 2

 2
ux  2
sgn u z   2 sgn 
b 2
a  b2
a  b2
b
  

 sgn 
1
zz
b 2


3
2b 2 sgn  
ia 2

u

u


sgn 
x
z
a 2  b2
a 2  b2
b 2
  
1 
yz
2
ib 
 sgn  u y  sgn  u y
  

i sgn 
1
yz
(1.13)
(1.14)
(1.15)

i sgn  2
b 2
b 2
From (1.15), (1.16) one obtains, repeating calculations of §3, for   1
3
1
1  ,  2  ,  is 2  1, 2
4
4
 u y  
(1.16)
(1.17)
4
One can introduce new functions for which there are no singularities in equations (1.15),
(1.16)
  
1 
yz
2
b


 
1  
 1
2



2 1
,
 
2 2
   1
 
 
 
 
 
 
(1.18)



2 2
1   2 1
(1.19)
 1
   1
,
2
2 1
2 2
1
u y  1 1
  2 1
,
(1.20)
2
2 2
2 1   2 1
(1.21)
 1 1
  2 1
,
2
b 2
The same examinations can be made for equations (1.11), (1.12), (1.13), (1.14). Repeating
calculations of plane problem of §2 one can from (1.11)-(1.14) obtain scalar equations for
functions 1, 2,3, 4 , 5,6,7,8 and avoid singularity in them.
Then one obtains
22
24
2 3
1   21
u x 
5  1
 7 1
 6 1
 8 1
,
8
(1.22)
1   21


 2 3

 22

 2 4
ux  
1 1
 3 1
  2 1
  4 1
,
8i
2 3
2 2
2 1
1   2 4
u z 
 6 1
 8 1
 5 1
 7 1
,
8i
(1.23)
1   2 3


 24

 2 1

 2 1
uz 
3 1
  4 1
 1 1
  2 1
,
8
u y 

 
 
 
 
 

1
   
b 4

1
   
b  4i
   1   
b
4i

1
2

3
2
1
  
1 
zz
2
 
 
 
 
 
 5 1

6
 24
1
 5 1

1

 
 2 4
1
 2 1
1
 
22
 
22
 
  2 1
 
2 1
 
 8 1
22
2 3
 

8
 2 3
1
(1.24)

7
 2 1
1
(1.25)
23

4
2 3
  4  1
 3 1
 
22
1   2 1
1  1
  2  1
 3  1
4
b
where singularities numbers are as in plane case


  
  
    ,
    ,
    ,
 
 
 7 1

 
 

6

xz
2
 
 2 4
1
 
24
,
(1.26)
(1.27)
1 1
3 1
a2  b2
(1.28)

ln ,  2, 4  
ln ,  
4 2i
4 2i
a2  b2
Placing (1.23)-(1.26), (1.27) in (1.4)-(1.9) one obtain system of sixth equations WiennerHopf with continuous matrix. It is convenient introduce dimensionless variables
1,3 
1


 2
(1.29)
  ,   , 1     
a
a
a


 
u y ,  x , y , z take the same form in dimensionless variables,
then all functions u z , u x ,
a
a
a
2
 
only have multiplier   and system can be written as
a
5
 ,   G3 ,  ,   g 2 , 
where
(1.30)
G31 ,   cij , i, j  1,..6, G3  A1 B,
g2 ,   G3 , g1, g1  B1g
 
 d1 

 

 d2 
 0
d 

3
g   , B   0
 d4 
 0
d 

 5
 0
d 
 0
 6

 
d1,2 
3
2
0
 
1
 2
0
0
0
0
0
0
0
0
0
0
0
0
0
 
 2 3
 
 24
0
0
0
0
 
0
0
 22



0 
0 ,

0 

0 
2 1 


0
 
 1 
 1 
 


 2 
  2 
 
  


   3 ,    8  
 4 
  6 
 
  
 5 
 5 

 
   
 6
 7
p1C0 C0i
2C0  a 2 2i p1  C0i
,
 2 , d3,4 



p4
4
42  b2 p4
p4  42
 p  a 2  i
 Ci
d5,6    1  2 2 2i  02 ,
 p4 b p4
 4 
p1  2 2  2  2  22 , p2  32 2  2  2  42 ,
p3  2 2 2 2   2 2  2   2  2 2  2   42  2 , p4   2  2   2 ,



p5  22  2 2   2 2  2  2  2 2  2  42  2 ,
a2
 2  2 ,
2
b
a
a j6
1
1
j5
b j 1  a j1  a j 2 

,
4i
4
8i 8
a j5 a j6
1
1
b j 2  a j1  a j 2 

,
4i
4
8i 8
a j5 a j6
1
1
b j 3  a j1  a j 2 

,
4i
4
8i 8
a j5 a j6
1
1
b j 4  a j1  a j 2 

,
4i
4
8i 8
a j3  a j 4
a j3  a j 4
b j5 
, b j6 
,
2
2
  1  2  2 , 2 
6
 
 
 
 
 
 
c1 j  b1 j 
 2 j
 
 
 
 
 
 
c2 j  b2 j
 2 j
c3 j  b3 j
 2 j
c4 j  b4 j
 2 j
c5 j  b5 j
 2 j
c6 j  b6 j
 2 j
3
 2
    , c  b     ,
, j  1,2,3,4, c  b     , c  b     ,
, j  1,2,3,4, c  b     , c  b    
, j  1,2,3,4, c  b     , c  b    
, j  1,2,3,4, c  b     , c  b    
, j  1,2,3,4, c  b     , c  b    
, j  1,2,3,4, c15  b15 
1
 2
25
35
3
 2
3
 2
25
  2 3
35
 2 4
45
45
55
55
65
65
 2 2
  2 1
 
a11 
3
 2
16
1
 2
16
26
3
 2
  2 3
3
 2
 24
3
 2
 22
3
 2
  2 1
1
 2
26
36
1  2  

3
 2
1
 2
36
46
46
56
56
66
66
1
 2
1
 2
  2 3
,
1
 2
 2 4
1
 2
 22
1
 2
  2 1
(1.31)
,
,
,

p3
p1
2 p2
a 2  b 2 2   b 2 2  2
, a12 
, a13 
 1, a14 
 1,
R1
 2 R1
 2 R1
b 2 p4
    2 
p3
p1
p1
2p2
a15 
, a16 
, a21 
, a22 
, a23 
 1,
p4
p4
R1
2 R1
2 R1
a24
a

2

 b22   b22 2
  2 
p

1
,
a


, a26   1 ,
25
2
p4
p4
b p4
 i 2 p1 a 2  
 p
p 
a31  2
 2   1, a32  2i 5  1   i,
b R1 
  2 R1 R1 
 R1
 p2 ip1 
p
   2
, a34   1 
a33  2i

,
R1 
ip4
p4
  2 R1
a35  2 
ip1
a 2  b22 22
i2a 2
p


,
a


2

i

 1,
36
2
2
p4
p4
b p4
b p4 p4
 i2 p1 a 2  
 p
p 
a41  2
 2   1, a42  2i 5  1   i,
b R1 
  2 R1 R1 
 R1
 p2 ip1 
p1   2
a43  2i 


,
 , a44  
R1 
ip4
p4
 2 R1


ip1 a 2  b22   b2 22
2a 2
p
a45  2 

, a46  2i 
 1,
2
2
p4
b p4
ib p4 p4
 i2 p1 a 2  
 p
ip1 
a51  2 
 2   1, a52  2i  5 
  i,
b R1 
R1 
 R1
 2 R1
 i 2 p2 p1 
ip1    2
, a54 
a53  2


,



R
R

p
p
1 
4
4
 2 1
a55  2 


ip1 a 2  b22   b2 22
i2a 2
p

,
a

2

i

 1,
56
2
2
p4
b p4
b p4 p4

R1  2  2  2
2
  4 
2
2
2

 2 ,
7
 i2 p1 a 2 
 p
ip1 
a61  2 
 2   1, a62  2i  5 
  i,
b R1 
 R1
 2 R1 R1 
 i2 p2 p1 
  2
ip
, a64 
a63  2

 1,

R1 
p4
p4
 2 R1
a65  2 
a
2

 b 2 2    2 2 ip1

,
p4
b 2 p4
a66  2i 
Solution of (1.30) is [3]
a 2 2i p1
 ,
b 2p4 p4


1
X    X    g 2  d
  
2i 


where argument  is not noted. X   are solutions of corresponding homogeneous
problem
X    G3 X  
As it is shown in [3] X   must satisfy to Fredholm system
X   
1  G3 G3    E 
X  d   
2i 

(1.32)
where one can believe that   E is unit matrix. Placing of    in solution (1.27),
(1.25), (1.26), writing in matrix form
 u x 
 
 u z 
  
 21 
b 
  3      ,
 2 
 b  
 u y 
  
 2 2 
b 








   



1
2
1

2
0
 
0
0
 
0
0
0
0
 2 3
0
8
 2 1
 2 1
 
 
1  2 2

2
1  2 2

2

 24
 22
8i
24


4
 24


4i
0
0
0
0
0
0
8i
2 3


4
 2 3


4i
 
 
8
 
8
22


8i
 22

4
22


4i
 2 3
 
 
 


 
 24
 
 
 
 
 
 



8 
2 1 


8i 
2 1 



4
 2 1 



4i 

0


0


 2 1
 
 
 
8
and inverting of transformations of Laplace and Fourier one can write solution on halfplane
z  0, x  0 in Smirnov-Sobolev form






  u x1  u x 2 




x

1
 2 
  uz  uz 


1
x





X

g1    d 




a
1
1
2
J   xz  xz   2 Re  X   0    0  d  
,
x
2

i





0


1
 2
(1.33)
  zz   zz


1
 2 
  uy  uy 


x


 1   2 
yz
yz


at   y    
0 
, g1     G3    B 1    g .
x
where g1 depends also from  . Due to convergence of integral furthermore are taken limits
on  and  (-5,5). Near edge x  0, x  0, 0  , X 0   E , 
obtain
~  1  y   

at
5
5
1
Jat
1
 Re   d   d   X    g 2  

i 5
5
1
1
1


 1
   4
  2
  1
0 0  3

8
8
8
8




3 1
 4 1
 2 1
1 1
 0 0  i 8 i 8 i 8  i 8



1  2
1  4
 0 0  1  3  1   4



1
4
4
4
4

~


1
0
  3
1  4
1  2
1 1 
 0
 
 
  
4i

4i
4i
4i
1
1




  1   2

0
0
0
0
2

2
0
0
0
 1  1

1  2

0
 

2
2

x
where    ~ ,
at



1
0  i02
 
and one can

(1.34)
1 1
3 1
a2  b2

ln ,  2, 4  
ln ,

,
4 2i
4 2i
a2  b2
correspond to singularities of plane problem. One can calculate following integrals
1,3 
9
 y   
  1 

at

5
5
I1, 2,3, 4
 y   
  1 

at

5
5
I 5, 6
1 1, 2 , 3, 4
5
1


d  X    B 1  
5
1 1, 2
5

1

d  X   B 1  
5

1
g  d ,
C0
1
g  d ,
C0
The all functions in integrals on  are also functions of  . There are mode numerical
solutions of Fredholm system (1.32) on determination of X  ,   , then are calculated
b2 1 y

I1, 2,3, 4,5,6,7,8 by (1.34) for values 2  ,
 0,3,
 0,1.
a
3 at
at
2. Formulation and analytical solution of plane unsteady problem.
First is carried out analytical and numerical solution by methods [1-5] of unsteady mixed
boundary value plane problem of elasticity. The plane problem of semiinfinite crack,
bounded from upper and from lower by elastic halfplanes with different elastic constants is
considered. There are denoted by index 1 stresses and displacements in upper halfplane, and
by index 2 in lower halfplane. Then boundary conditions, similarly to [6], for y  0 have
form
 y1 x,0, t    y2  x,0, t ,  xy1 x,0, t    xy2  x,0, t ,
u1 x,0, t   u 2 x,0, t , v1 x,0, t   v 2 x,0, t ,    x  0,
 y1 x,0, t   0,  xy1 x,0, t   0, u 2 x,0, t   0,
(2.1)
v 2 x,0, t   C 0 H t H x  , 0  x  ,
where C0 ,   are constants, H t   is unit Heviside function. Let us introduce jumps of
stresses and displacements
 y1 x,0, t    y2  x,0, t   x, t ,
 xy1 x,0, t    xy2  x,0, t   x, t ,
u1 x,0, t   u 2 x,0, t   u x, t ,
(2.2)
v1 x,0, t   v 2 x,0, t   vx, t ,
Solutions of Lame equations of elasticity are looked for by method of Laplace transformants
on time t with parameter s and Fourier on x .
1 i st
1  j 1 j k   y ix
(2.3)
 j x, y, t  
e
ds
k e
e d, j  1,2
2i i
2 
where on k is carried out summation from 1 to 2,
s  i, k   
 
2
2
 
2
Ckj
, C11  a1 , C21  b1 , C12  a2 , C22  b2
(2.4)
a1, 2 and 1, 2 are speeds of longitudinal and shear waves in the halfplanes y  0 or y  0 .
Then, passing in (2.1), (2.2) to transforms due to (2.3) one can express for
1
 , u2  , v2   by means of  ,  , u  , v   and obtain the
y  0,  y1  ,  xy
relations, which in case of same elastic properties of halfplanes are derived in [6]. In further
calculations are considered for this particular case and indexes of a1, 2 and b1, 2 are omitted.
10
In distinction from [6], where in mentioned relations is done passing to originals and for
considered in it solution for case of finite crack are obtained four singular integral equations,
in present paper is applied more available for semiinfinite crack Wienner-Hopf method [1-5].
There are introduced the notations
u
v
     ,      , u   
, v   
,
i 
i 
(2.5)
i


C
e
u
v
1
     , u 2   
 y1      ,  xy
, v2   
 0
,
i 
i 
is
and are taken into account conditions (2.1) in which are carried out integral transformations,
by index (+) is denoted functions, analytical in upper halfplane  , by index (-) in lower
halfplane. From mentioned relations one can obtain system of four Wienner-Hopf equations
in matrix form, introducing also dimensionless variables by formulae


a2
2
2
  , 1, 2  1,2 , 1    1,  2    2
a
a
b
(2.6)
A   B   C  0
(2.7)
where
a1
0
a 2
 a0



a0
a3
0
 a1 f


2 
a4
 a0
 a1 f
A 0
,

1


 a 1 0  a  1
 a0 
1
 4

2
2


2
2

1
i 
b 
2b
1  2 1 
,


a0  , a1 




1
1
2

2
21 
a 2  a 2  2b 2


a2  
1 
b 2 
2b 2
1  2   22  2 
   2  21  22 2
2 
2
2 1 1 2

21i 
a 
a  2b
a1 f 
  1 2
b i 2
b
 2  2  21 2 , a4   2 i
,
2
21
a 2 2
a
a3  
1 
b 2  4b 2
1  2 
21 2   22  2
2  2
2

2 2i 
a  a  2b
2

1

0
B   E, E  
0

0


,

2
2

1  a 2b2b
2

0


0 0 0



0


1 0 0

,   b 2.
,
C

0

0 1 0




C0 i a  


e
0 0 1


 s

2
2

12  ,


(2.8)
 
 
 
 
  
  


    ,     
u 
u 
 v 
 v 
 
 
One can for determinant of matrix A obtain
11




A  a02 a02  a  2 f  a3 a 4  a1 2 f a02  a3 a4  a12 f  a 2

1 2
a1 fa4  a02 a4  a42 a3
2

All summands contain only terms with 12, 2 , 2 which on passing on contour in  plane
 , , come to their initial values, and since argument of determinant
A for    is
same, also as in [1,2] index of A is equal to zero. For    ,
1
b 2 i sgn 
a2  b2
a0  , a1  
, a3  a 2  
sgn ,
2
2a 2
ia 2
(2.9)
b 2 i sgn 
a2  b2
a1 f 
, a4  
i sgn 
2a 2
4a 2
i.e. A  const
(2.10)
The solution of system (2.7) can be replaced by Hilbert problem
0




0



(2.11)
   G   g , G   A 1 B, g  A 1 
0

i 
 C0 e 





One must note that in Laplace transformation  has small imagine part [2], therefore roots of


Releigh equation   
, entering in form of brackets  
in denominator of
CR
CR
G     A 1  B  , are situated in complex plane out of real axis and matrix G  for
finite  has no singularities on real axis, also abovementioned is related to branch points


   ,    . The same conclusions are related to matrix G , where integrations on
a
b
a
a
, 1,  must be carried out near them
 in solution (11) [3] near mentioned points   
CR
b
in complex plane.
Due to (2.9) matrix A , as well as matrix G , is discontinuous for    , and one must bring
(2.11) to Hilbert problem with continuous coefficients [3], by analogy with calculations of
[6,8], only by other method, developed there for originals. System (2.7) on account (2.9) one
can write for    in form


   i  a 2  b 2 sgn  a 2  b 2

sgn  u   iv      i  ,
2
2
2
a
a
2
2


a b
  i
u   iv  a 2  b 2 sgn 

sgn


 u   iv  ,
2
2
4
2
a
a
One can introduce the functions



 2iu




1, 2    i  2i u  iv ,  
 3, 4     i 

 iv 
a2  b2
,
a2  b2
(2.12)
(2.13)
Placing    i  , u   iv  from (2.13) to (2.12) one obtains
12
1, 2  1 
b2
b2
b2
sgn


i
1

1

sgn ,
a2
a2
a2
b2
b2
b2
sgn


i
1

1

sgn ,
a2
a2
a2
1

1
1
1  2  2     i  , 
11 
 2  2  u   iv  ,
4
4
8i
8i
3

1
1
 3  4  2     i  , 
 3 3 
 4  4  u   iv 
4
4
8i
8i
Similarly to (2.13) one can assume
  6
  8
   i   5
, u   iv   7
,
2
4i
  8
  6
   i   7
, u   iv   5
,
2
4i
From (2.14), (2.15) one obtains equations
  1
 
1
1, 2 1, 2   6 ,  3, 4  3, 4   8 
2
 5  2
 7 
 3, 4  1 
(2.14)
(2.15)
(2.16)
One must multiply equations (2.16) on such multiplies, that they, multiplied by 1, 2 ,  3, 4
1
 0 functions.

Let us examine the function
will give continuous for
2is


1
1 
2is2 ln  i arg 
 1  2
 
1 
1

 
(2.17)

e
, 1    1
  
 1 
Though these examinations may be carried out also for dimensionless variable  , it is more
convenient consider initial variable  , and 1 
from   to  arg
1
1
varies from 
 

1 , 1    . In variation of 
a
a


to
and rate of expression (2.17) for    to
2
2
those for    will take value e 2s2 .
Then one can show, that after multiplication of equations (2.16) on (2.17), where respectively
for first and second equations or for  1, 2 and  3, 4 , in these multipliers
1 1
3 1

i ln , is 2   2   
i ln ,
4 4
4 4
3 1
1 1
is 2   3   
i ln , is 2   4   
i ln ,
4 4
4 4
is 2  1  
(2.18)
2is
 1  2
the rate of 1, 2,3, 4   
taken for    to those taken for    will be equal to 1, i.e.
 1 
coefficients of equation (2.16) after mentioned multiplication become continuous for
functions
13
 
 
1, 2,3, 4 1
2 1, 2 , 3, 4
 2 1, 2 , 3, 4
1
 1, 2,3, 4 ,
(2.19)
 2,1, 4,3
 6.5.8.7

System of (2.7), (2.8) in open form yields
a0    a1   a2 v    
(2.20)
a1 f   a0    a3u    

a4    a0 u   a1 f 2 v   u 
1
(2.21)
(2.22)
s
C0 a 


a 4 1    a1 1 u   a0 v  
e
 v
2
2
s
(2.23)
u
v
U ,
 V  , where U  , V  are Laplace and Fourier
i
i
transforms from displacements, then in equations (2.20), (2.21) at point   0 for U  , V 
there are continuous coefficients and for   0 it is natural believe, that U  , V  are finite,
Let as assume, that
u   0, v   0 , then from (2.13) it is seen that at   0 , 1   2 ,  3   4 and in equations
(2.24) for   0 , on account (2.8), one can see that singularity is avoided, then in all further
analytical calculations, one can ignore fictive pole in coefficients of equations in   0 , in
numerical calculations exclude point   0 and near it carry out calculations of integrals out
of real axis.
Placing (2.13), (2.15), (2.19) in (2.20)-(2.23), and in accordance with calculations carried out
for    , which allowed from (2.12) obtain (2.16), for arbitrary  , after multiplication of
(2.21) and (2.23) by i and on addition of them to (2.20) and (2.22), one obtains four
equations, in right-hand side of which will be    i  , u   iv  , and furthermore, after
multiplication of obtained fourth and third equations by 2i and on additions of them
respectively to first and second, one can obtain resulting system of equations
s

C
a111  a12 2  a13 3  a14 4  4 5  2 0 e a ,
s
s
a 211  a 22 2  a 23 3  a 24 4  4 6  2
a311  a32 2  a33 3  a34 4  4 7 
C 0 a 
e
,
s
s
C 0 a 
2
e
,
(2.24)
s
s

C
a 411  a 42 2  a 43 3  a 44 4  4 8  2 0 e a ,
s
where
a1
a a



(2.25)
 ia1 f  2  3  2a4   2a4 1   ia1 2  ia1 1
i
2 2
2
1
2
and other coefficients in (2.24) aij can be obtained from a11 by changing of signum + on
signum – for following summands:
in a12  2 for 3,4,7,8; in a13 for 1,3,5,7;
a11 
14
in a14 for 1,4,5,8; in a21  2 for 5,6,7,8;
in a22 for 3,4,5,6; in a 23 for 1,3,6,8;
in a24 for 1,4,6,7; in a 31 for 2,4,6,8;
in a32 for 2,3,6,7; in a33 for 1-8;
in a34  2 for 1,2,5,6; in a41 for 2,4,5,7;
in a42 for 2,3,5,8; in a 43 for 1,2,3,4;
in a44 for 1,2,7,8.
System (2.24) can be written for    , and due to (2.9) one can show that in first equation
remain only terms with  2 and  5 , in second 1 and  6 , in third  4 and  7 , in fourth  3
and  8 , and they coincide with (2.16) without free terms. Thus as it is shown, matrix of
coefficients in (2.24) written in new functions (2.19) must be continuous and system of
Wienner-Hopf (2.24) become
A1 1  B1 1  C1  0
(2.26)

 
 4 
A1  aij 1
B1
2  j
 2 j
1
, i, j  1,2,3,4,
 ij , i, j  1,2,3,4,
(2.27)
1
C1 
where  ij  0, i  j ,  ij  1, i  j ,
 
s
C 0     1

e
,
  1
 
1
 
s



1  1, 2,3, 4 , 1  1, 2,3, 4
From (2.26) one can, as for (2.11), obtain Hilbert problem
 1   G1  1   g1 ,

G1    A11 B11 , g1    A11C1
Solution, bounded on infinity [3], is as follows
1 g 1  
X  
 1   1
X 1  
d

2i 



(2.28)
(2.29)
(2.30)
where X 1   X 1  is solution of problem
X 1   G1 X 1 
Solution of this problem [3] can be reduced to
1  G11 G1    E 
X 1  
X 1  d   
2i 

(2.31)
where   represents asymptotics of X 1  for large  and since as it is shown for
  , G1   const , one can assume that X 1  const and   E [3].
One can show that from (2.29), (2.27) one obtains
15
1
 
  1
1
(2.32)
g1   g 2 
g 2   C 0 A1  
1
s
 
1
 
To pass on initial functions one must do reverse transformations of Laplace and Fourier and
for y  0, x  0 for
s
1  
e
,
  y1 x,0, t  


  
 
  xy1 x,0, t  
  i
 i  x
  


1

(2.33)
J x, t      u 2 x,0, t    2  e st ds  e a4
  d
a
4

i
u


i



x
 


 v 
v2 x,0, t  

 


x


s  i
where the last vector is given by (2.15), (2.28), (2.30), (2.19), after calculating first integral,
 
 
and then integral from  t  x    [1], one can write down closed solution in Smirnova 
 a
Sobolev form.
For x  0 one can obtain solution for y  0, x  0 in form
  x  2
 
~ 
  at


2
  x 
~
5
at
1
1   at

J x, t   2 Re  ~  
2
C 0 
8
5  
i  x 



~
 2  at



2
 1  x 
 2  at
~ 



1
 
1 1
  1

 X   B1  d  ,
1
 
1
 

1
4
 x 
 ~ 
 at 
1
 x 
 ~ 
 at 
i  x 
 ~ 
2  at

 x 
 ~ 
 at 
4
 x 
  ~ 
 at 
1
1  x 
 ~ 

2  at


1
i  x 
 ~ 
2  at

1  x 
 ~ 

2  at







3  
i  x  
 ~  
2  at
 
3
1  x  
 ~  
2  at
 
3
4
4
 x 
 ~ 
 at 
3
 x 
  ~ 
 at 
(2.34)
~  1   

at

 1, 2 ,3, 4 are given by (2,18).
The problem is reduced to calculation of I1, 2,3, 4 

  
 1  at  

12 ,1, 4 , 3
X 

1
1
B11 d

  0 . Singularities
a
for solution for x  0 in (2.34) are in accordance with solution of static problem. The results
of the calculations of last integrals, carried out, after solution of system (2.31), for [7-9]
b 2 1 
 ,
 0,2, t  0,1 , where are taken limits of integration (-5,5), are done in table 1.
a 2 3 at
From properties of Laplace transformations one must assume that t 
Table 1
I1
8.256+8.63 i
-8.15+10.65 i
-0.843+0.98 i
-4.873+1.11 i
I2
-0.566+0.36 i
-1.434+2.17 i
4.388-3.157 i
5.676+4.464 i
10.836+6.3 i
-6.26+9.94 i
0.855+6.62 i
-1.405+4.39 i
7.291+9.69 i
-1.064+7.7 i
3.82-11.81 i
7.584+0.606 i
16
-8.296-0.53 i
-0.374+6.12 i
0.756-2.685 i
0.495-1.1 i
11.87+7.01 i
-6.16+10.4 i
-9.88-2.05 i
-4.99+9.7 i
-0.784+7.92 i
-2.003+5.87 i
-0.773-3.077 i
-4.245-5.84 i
-1.737-0.35 i
-0.412-1.91 i
-0.205+4.47 i
2.795+1.36 i
-9.87-1.277 i
-3.28+10.04 i
-0.601-2.69 i
-4.485-3.88 i
5.31+13.41 i
-2.815+8.51 i
-3.235-3.5 i
-18.56-3.25 i
6.611-13.08 i
9.023+0.871 i
-1.277+10.46i
13.83+10.47 i
8.226+8.65 i
-8.09+10.74 i
-8.236-0.44 i
-0.263+5.94 i
-0.919+0.8 i
-5.016+1.05 i
0.803-2.717 i
0.688-1.039 i
I3
-3.697-2.616 i
-15.66+1.14 i
0.35+9.59 i
13.93+6.296 i
-0.899+0.11 i
-1.4799+2. i
-1.644-0.3 i
0.0672-2.17 i
4.438-2.894 i
5.678 +4.546 i
-0.23+4.301 i
2.363+1.289 i
I4
3. The unsteady mixed antiplane boundary problem for semiinfinte crack. The mixed
unsteady antiplane problem for elastic space made from two half spaces with different
modulus, when on their junction plane there is crack, on upper bank of which stress
component is zero, and on lower one is given displacement is considered. There are carried
out, after application to motion equations and boundary conditions of Laplace on time and
Fourier on coordinate transformations, the two Wienner-Hopf coupled equations, which are
brought to Hilbert problem with matrix of second order. The singularity of matrix on infinity
is avoided and the solution of mentioned problem is reduced to system of two Fredholm
integral equations. The inverse integral transformations are carried out and solutions for
stress and displacements are brought to Smirnov-Sobolev form. There are carried out
calculations of solution of Fredholm system and of obtained integrals. The corresponding
solution for finite crack and harmonic on time mixed boundary condition for given stress on
its bank by other method is solved in paper [12].
It is considered the elastic space in x, y, z Cartesian coordinate system consisting from two
elastic halfspaces made from different materials, with shear modules 1 and  2 respectively
for y  0 and y  0 halfspaces. On their junction plane y  0 there is crack upper bank of
which is free from stress, and on lower bank acts semiinfinite punch. Only there are z
components of displacement u1, 2  x, y, t  and are used  yz1, 2  x, y, t  components of stress.
Indexes 1,2 are used for upper and lower halfspaces.
The equations of motion for antiplane problem are
2
 2u j  2u j
1  uj

 2
, j  1,2,
x 2
y 2
C j t 2
(3.1)

u
1, 2
 yz1, 2   1, 2
,
y
1, 2
, 1, 2 are densities. Boundary conditions
where C12, 2 
1, 2
u1 x,0, t   u 2 x,0, t ,  yz1 x,0, t    yz2  x,0, t ,    x  0,
(3.2)
 yz1 x,0, t   0, u 2 x,0, t   C0 H x  H t , 0  x  
(3.3)
where H t  is Heviside unit function, C0 ,  are constants.
The solution in halfspaces is looked for as follows
1 i st
1 
1 j  j  , s  y
u j x, y, t  
e ds
u j , s e ix e
d


2i i
2  
where


0

(3.4)
u j   e  st dt  e ix u j x,0, t dx,
 yzj
2
  j  1  j u j ,  j    2 ,   is.
Cj
j
(3.5)
2
17
Let us denote the jumps of functions by
u1 x,0, t   u 2  x,0, t   u x, t ,
 yz1  x,0, t    yz2   x,0, t   x, t ,
(3.6)
Since on boundary part x  0, u  0,   0 from (3.2), one can write using (3.5) transforms of
u, 
u
1
2 
,     , u  u1  u 2 ,    yz
  yz
(3.7)
i 2
u  x, t 
where   , u  respectively represent transforms of x, t  and  2
, indexes  are used
x
for functions of  analytic in upper and lower halfplanes.
From (3.3) and (3.5) one obtains
C
u
1
 yz
 1 , u 2  , s    2  0 e i
(3.8)
i 2 is
u 
where anew u 2 is function analytic in lower halfplane. From (3.6), (3.5) one obtains
 2 2 u  
 u 
.
, u2   1 1
1 1   2 2
1 1   2 2
Using also (3.7) one obtains from (3.5), (3.8) Wienner-Hopf system
1 u 
u

1
 
 2

i  

C e
u

i
i
1  1 1
,  2  0
 2
1 1   2 2
i 2
is
1 1   2 2
In matrix form (3.9) yields
 u 2 
u  




    ,     ,
 
 1 
A   B   C  0,
u  u1  u 2 ,   1 1u1   2 2 u 2 or u1 
(3.9)
(3.10)
 1 1 2

1
 i
A
1 1   2 2  1 1
  i
 2
On infinity   , 1,2
on infinity free term

1 1 
0


 0  1
(3.11)
, B   1
, C   C 0 i 
0
e 



1 
 i

 is


  and (3.9) gives, after dropping out unnecessary in investigation
1  i
1 sgn  
1
u 
 ,
1   2
1   2
(3.12)
 2 sgn  
1

i
 
u
1   2
1   2
and it is seen, that for    there is jump of coefficients. Introducing like in [12] new
functions

(3.13)
1, 2     u  ,  3, 4  1  u 2 ,   1
2
u 2
18
or   
  4
  3
1   2
  2
, u  1
, 1  3
, u 2  4
, one obtains from (3.12)
2
2
2 
2 
after multiplication of second equation on   and addition to first
 3, 4   1, 2

 
1
1 

i
sgn


1   


(3.14)
2is
 1  2

One can examine variation of rate    , where 1   
on real  axis, s2 is
a1
 1 
unknown constant. Due to properties of Laplace transformation  has negative imagine part.
Since
2is2
 1
1 

(3.15)
 exp 2is 2 ln   i arg 
 
1 
1



and since arg 1 for    is  and for    is 0, and arg 1 for    is
and for
2
2

   is 0, one can see that arg 1 for    is   and    is 0, and difference of
1
 1

 
 1
values of arg
1
1




taken for    and those taken for    is   . Than rate of values of
2is
 1  2
2s2


    taken for    to those taken for    is e , which follows from (3.15). We
 1 
demand that
  i 2s2
(3.16)
e
1
 i
which is satisfied by solutions
1 1
1, 2  is 2    arctg 
(3.17)
2 
or
2 
1 1
1, 2  is 2   
arctg
(3.18)
2 2
1 
This singularity is known from statics and [12]. One can see using (3.16) that after
multiplication of (3.14) by (3.15) and introducing of new functions
 
1, 2 1
 
21, 2
21, 2
 1, 2 ,  3, 4 1
 1, 2
one can obtain for them system with continuous for    matrix. Introducing for arbitrary
values of  dimensionless variables and corresponding functions


 
C1  a1 ,   , 1, 2  1, 2 , 1, 2 
1, 2 
a1
a1
a1
(3.19)
 
21, 2
 
1, 2 1
 1, 2 ,  3, 4 1
one can from (3.9) obtain following system
21, 2
 1, 2
19
1
 


1  2  1 1

1   2  i 

 
 
 1 1
1
 2 1

1   2

 

1 1  2  1
1   2  i  
 
  2
 
  2 1
 
 
 2 1
2 2

2 2
 1


 1 1
 i 



    

1
1
i  
1
 2 1
  2
 2 1
  2

1   2
 1


 1 1
 i 



 
2 2

(3.20)

 
i 
  C 0  2 e a1

si
2 2
1
One must note that branch points of 1,2   , since  has small imagine part, are situated in
complex plane out of real axis and in the further are out of integration part in integrals giving
solution of Hilbert problem, taken along real axis  , therefore they do not represent for them
singularities. Passing to variable  on must in mentioned integrals carry out, near mentioned
points integration slow inclining from them in complex plane. Also it is necessary to point
1
out that multipliers
in u  , u 2 are arisen since we demanded that mentioned functions will

be of same dimension as 1 ,   , and they brought to fictive singularity in point   0 ,
therefore in integration on  or  along real axis one must also slowly make deviation of
contour from this point in complex plane.
u
u
 U  ,  2  V  , where U  , V  are finite
Indeed let us assume, that 
i 2
i 2
u   0, u 2  0 , then from (3.13) one obtains that for   0, 1   2 ,  3   4 and in
equations (3.20) for   0 , on account of above written, singularity is avoided, then in all
further analytical calculations, one can ignore fictive pole in coefficients of equations in
  0 , in numerical calculations exclude point   0 and near it carry out calculations of
integrals out of real axis.
System (3.20) can be written in matrix form [8]
 
 
    1 ,     1 
(3.21)
2 
2 
A1   B1   C1  0
  

   2  1   21
1
 1
  i 


A1 

  21
1   2   1

 1 1
 

i



 
 
 
 
 
 2 1

1

 2 1
B1   1
1

 i 
1   2  1,  2   2 

 
1 1  2  1
 i  


 1
 

 1 1
 i 



 
2 2
 
2 2



,



0






i



C 0  2 a1 

 2  2 , C1 
,
e
 


 si 


 a1

 
1
 
 2 2
1

(3.22)
i 
C12
C 22

1
, 1    1
20
Solution of (3.22) can be brought to nonhomogeneous Hilbert problem
   G1   g1 , G1   A11 B1 , g   A11C1
The solution of (3.23) is as follows [2,3]

(3.23)

1
X   X    g1  
 
d
2i 

(3.24)
where X   are solutions of homogeneous problem
X   X  , X    G1 X  
whose solution can be brought to solution [3-5] of corresponding system of Fredholm integral
equations
1  G11 G1    E 
X   
X  d   
(3.25)
2i 

where [3]
1 0

(3.26)
   E  
0
1


matrix   represents behavior of X   for    . Using (3.22), (3.23) one obtains
 

  1  1  
  i   1
1   2
1
G1  

  2   

1
2 1
 1  1 1

i    i  


 
2 1
 
2 2
 
 

22 
 

1 1  2  1
 i  



x
 2
  21 
 1 
 1 

 i   1



 
 
(3.27)
 
 
 2 1
2 2


1
1


x 1
1
  2 1
 2 2 

1
 i  1

i 


Previous investigation for    shows that matrix G1  must be continuous for    ,
and one can justify that it is constant at infinity. hence it can see that the choose of   E
is true. After solution of (3.25), by (3.22), (3.24) one obtains


1
 1  X    X    g1  
 
d,
 
2i 

 2
1, 2 
(3.28)
 
2 1, 2
 3, 4 1
where
 

  1  1  
  i   1
1   2
i
g1  

 
1 21   2    1


  i   1 1



2 1 
 

1 1  2  1
0

 i  





 C0  2 i a1   (3.29)
e
 

2 2
2  2 

si
 1  2  1 1



 i  



  4
Then, after substitution of (3.28) in 1  3
and inversion of integral transforms one
2
obtains stress distribution out of crack
 
 
 
 
2 1
21
  i


i x
1
 
 yz  2  e st ds  e c1
1 d,
c
4 i i
1


  2 1

 2 2




1  

1
1
2 1
1 
or 1   1  1
2
2 2 
1
 
 


 2 1
 
 2 2
1
, 

(3.30)

x

After calculations of first integral and furthermore integral of  t      . one obtains
c1
c1 

the  yz1 for y  0, x  0 .


Near the crack edge x  0, 0  , x  0, X  0   E , X   
1
 yz
I1 
1
x
  11
 x21

 x  1
 x  2 
 2 Re    I1     I 2  , a1  c1t
8 a1  a1 
 a1 



 2 C0
1     2  


  
1
t  
1     2    a1 
I2 




1     2  


(3.31)
 1 1
x

    
    
2 1
2 2
x  1   X 111  2   1  
X 211  2 d,
 i  
 i   






x12 
,
x22 

  
1
t  
1     2    a1 
 2 1
(3.32)
x

    
    
2 1
2 2
x  1   X 12 1  2   1  
X 22 1  2 d
 i  
 i   






 0.1
The calculated values are presented in table 2, where is taken   1,
a1
The jump of displacement in conditions (2.1) and furthermore in (3.3) at point x    can
give  function behavior of stresses. But it can be smoothed replacing H x   by




 x   , x  
generalized function  x     
, where   0 , and after abovementioned
0, x   
calculations of solution one can tend   0 and on x  0 semiaxes one can anew obtain
solutions (2.35), (2.31) and (3.32) on neighburhood at crack tip x  0 .
Table 2
I1
I2
229520. +117443. i
-277956.-100472. i
One must note that known singularity (3.18) does not coincide with obtained by circulant
method [10] singularity.

t
Conclusions. In present paper by method of integral transformations are solved space, plane
and antiplane unsteady mixed problems for semiinfinite crack. The system of sixth, fourth
and second orders correspondingly Wienner-Hopf equations with discontinuous on infinite
matrices are derived. The singularity of the last ones are avoided and solution of
corresponding Hilbert problem is brought to system of sixth, fourth and second orders of
Fredholm integral equation. Their solutions are obtained numerically, as well as are
22
calculated stress intensities coefficients near edges of crack, written in effective SmirnovSobolev form.
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