7 Features of Subsonic Stage of Contact Interaction …
103
and the original of the function W σ is written down as follows:
W σ (α, τ ) = e
−ϑτ F(α, τ )
(7.14)
Green function G 3e (x 1 , α) can be represented in the form (Gorshkov and
Tarlakovski 1995)
G f (x 1 , τ ) =
G f r (x 1 , τ ) + G f s (x 1 , τ )
H (τ − |x|),
(7.15)
where
G f r (x 1 , τ ) = G r 1 (x 1 , τ )H (η|x| − τ ) + G r 3 (x 1 , τ )H (τ − η|x|),
G rk (x 1 , τ ) = G f k (x 1 , τ ) − G f s (x 1 , τ ), k = 1, 3
G f 1 (x 1 , τ ) =
η
4
π
x
2
1
η
2 x
2
1 − 2τ
2
2
P 4 (x
2
1 , τ 2 )
τ 2 − x
2
1 ,
P 4 (x, τ ) =
η
2 x − 2τ
4 − 16τ
2
τ − η
2 x
(τ − x),
G f 3 (x 1 , τ ) =
η
4
π
x
2
1 R 22 (x
2
1 , τ
2
)
P 4 (x
2
1 , τ 2 )
τ 2 − x
2
1 ,
G f s (x 1 , τ ) =
a s τ
x
2
1 − c
2
R τ 2 , a s =
R 22 (c
2
R , 1)
1 − c
2
R
πη 4 P 2 (c
2
R , 1)
,
R 22 (x, τ ) =
η
2 x − 2τ
2 + 4τ
√ τ − x
τ − η 2 x,
P 2 (x, τ ) = x
2
− 2α
2
τ x + β
2
τ
2
,
α
2
= 2(1 − κ) − c
2
R
2,
β
2
= 16
η
2
− 1
η
8 c
2
R
,
c R is Rayleigh wave propagation velocity.
Taking into account (7.10), (7.14) and (7.15), we can write down the expression
for viscoelastic half-plane Green function G 30 (x 1 , τ ) in the form
G 30 (x 1 , τ ) = G 30,r + G 30,s ,
G 30,r = e
−c 1 aτ G 3e,r (x 1 , τ )
+ e
−ϑτ
τ
|x 1 |
G 3e,r (x 1 , α)e
−(c 1 a−ϑ)α
∞
m=0
d m+1 (α)
m!
(τ − α)
m dα,
G 30,s = e
−c 1 aτ G 3e,s (x 1 , τ )
+ e
−ϑτ
τ
|x 1 |
G 3e,s (x 1 , α)e
−(c 1 a−ϑ)α
∞
m=0
d m+1 (α)
m!
(τ − α)
m dα. (7.16)