104
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7.4 Contact Problem Solution Algorithm
So, finally contact problem for the subsonic stage of interaction is determined by
closed equation system
m ¨
h = R e + R 3 ,
˙
h = v c3 , h| τ =0 = 0, ˙
h
τ =0 = v c30 ,
R 3 =
b(τ )
−b(τ )
σ 330 (x 1 , τ )dx 1 ,
b 2 = f
−1
(h),
w(x 1 , τ ) = G 30 (x 1 , τ ) ∗ ∗σ 330 (x 1 , τ ),
w = u c30 + h + f (x 1 ).
(7.17)
Here, h = u c3 − u c30 is striker indentation depth, v c3 is striker velocity, and v c30
is initial striker velocity.
The solution of the problem is constructed using time and coordinate mesh representation of the resolving equations integration area and the following construction
of difference schemes for these equations and quadrature formulae for the integrals.
We shall mesh the plane R
2
τ x 1
using mesh with time step δ τ and coordinate step
δ ξ :
τ i = iδ τ , ξ j = jδ ξ (i = 0, 1, 2, . . . ; j ∈ Z ).
In the general case, δ τ = δ ξ .
We shall assign the functions of one and two variables in the relations (7.17) to
mesh functions
h i = h(τ i ), b i = b(τ i ), v c3,i = v c3 (τ i ), f j = f (x 1, j ),
σ i j = σ 330 (x 1, j , τ i ), w i j = w
x 1, j , τ i
.
Then, the integral relation for the normal displacement of the half space at the
point τ = τ n , x 1 = ξ m has the form
w nm =
¨
D nm
G 30 (ξ m − ξ, τ n − τ ) · σ (ξ, τ )dτ dξ.
Representing integration area D nm by a polygon analogous to (Gorshkov and
Tarlakovski 1995), we approximately replace the integral by the following sum:
w nm ≈ I
r
nm + I
s
nm + ε nm I
0
nm ,