106
L. Igumnov et al.
The expression for Green function G 30 (7.16) can be rewritten separating the term
containing integral of elastic half-plane Green function:
G 30 (x 1 , τ ) = G 30,1 (x 1 , τ ) + G 30,2 (x 1 , τ ),
G 30,1 (x 1 , τ ) = e
−c 1 aτ G 3e (x 1 , τ ),
G 30,2 (x 1 , τ ) = e
−ϑτ
τ
|x 1 |
G 3e (x 1 , α)e
−(c 1 a−ϑ)α
∞
m=0
d m+1 (α)
m!
(τ − α)
m dα.
Then, the coefficient a nm is written as follows:
a nm = a nm,1 + a nm,2 ;
a nm,1 =
nδ τ
(n−1)δ τ
e
−c 1 a(nδ τ −τ ) dτ
−τ +(m+n)δ ξ
τ +(m−n)δ ξ
G 3e (mδ ξ − ξ, nδ τ − τ )dξ ;
a nm,2 =
nδ τ
(n−1)δ τ
e
−c 1 a(nδ τ −τ ) dτ
−τ +(m+n)δ ξ
τ +(m−n)δ ξ
nδ τ −τ
|mδξ −ξ |
G 3e (mδ ξ − ξ, nδ τ − τ )e
−(c 1 a−ϑ)α
·
·
∞
m=0
d m+1 (α)
m!
(nδ τ − τ − α)
m dαdξ.
So, finally difference scheme for the system (7.17) at the point τ = τ n , x 1 = ξ m
has the form
σ nm =
n−1
i=1
q i
j= p i
a n−i,2m− j σ ik j −
w nm
a nm
w nm = h n + f m + u c0
h n = h n−1 + V n−1 δ τ
V n = V n−1 + R n−1
δ τ
m
R n = 2δ
⎛
⎝ σ n 0 + 2
l n
j=1
σ nj
⎞
⎠
b n = f
−1
(−h n )
(7.18)
Initial conditions for the system (7.18) have the form
u c3,0 = u c30 , V 0 = V 30 , σ 0m = −V 30 .