100
L. Igumnov et al.
x 3 + l − u c3 = f (x 1 ),
from which contact area radius is
b 2 = f
−1
(l − u c3 )
(7.5)
Besides, we must require the displacements to be constrained.
Integral equation connecting half-plane boundary normal displacement and
contact stresses has the form
w(x 1 , τ ) = G 30 (x 1 , τ ) ∗ ∗σ 330 (x 1 , τ ),
(7.6)
where the function G 30 (x 1 , τ ) corresponds to the problem (7.1), (7.2) and boundary
conditions
σ 13 | x 3 =0 = 0, σ 33 | x 3 =0 = δ(x 1 )δ(τ )
(7.7)
On the other hand, the following relation connects half-plane boundary normal
displacement with the striker displacement:
w = u c3 + f (x 1 ) − l
(7.8)
So, the resolving equation system has the form (7.4), (7.5), (7.6), (7.8), and for
its solution we have to determine Green function G 30 (x 1 , τ )
7.3 Green Function Construction
Equation system (7.1) solution at initial conditions (7.2) and boundary conditions
(7.7) is sought in accordance with the statement proved in (Ilyasov 2011) in the form
u(x 1 , x 3 , τ ) =
∞
0
u e (x 1 , x 3 , α)W σ (α, τ )dα
(7.9)
Here, u e is constrained displacement vector with coordinates u e1 and u e3 ,
corresponding to two-dimensional elastic problem
β
2 ∂θ
∂ x 1
+ γ
2
u e1
= ¨
u e1 ,
β
2 ∂θ
∂ x 3
+ γ
2
u e3
= ¨
u e3 ,
u e | α=0 =
∂u e
∂α
α=0
= 0,