PSI - Issue 23

I.S. Nikitin et al. / Procedia Structural Integrity 23 (2019) 125–130 Author name / StructuralIntegrity Pro edi 00 (2019) 000 – 00

128

4





2

τ

s j j  , i j  ,

F

H v v H H

/

1 ( ) (      

( ) ) ( ) ( ) j  

( ) j  

 

s 

( ) j

2

,

i

ij

jj

, i j

, j i

jj

( ) j    τ k j

s j j  , i j  ,

( ) 0 j i   ,

kj kj  

,

( ) j v H H                , ( ) ( ) l ( ) ) / ( 2 ) ( ) ( ) j , k k , j j ( 2 jj v

( ) j j    .

0 x  are used.

l j

0 x  and

In these equations the Heaviside step function and the additional step function ( ) H x : ( ) 0 H x  at

( ) 1 H x  at

3. Numerical method Both formulated systems are semi-linear hyperbolic systems and the numerical solution can be obtained with different explicit schemes. However, the slippage process switches on the non-linear free term with small viscosity parameter in the denominator. The system transforms into the form with small parameter and ordinary explicit schemes will not be stable. To overcome this problem, the usage of explicit-implicit method is proposed (Nikitin et al., 2019). The implicit approximation is used only for equations that contain small term in the denominator. All other equations are approximated with the explicit scheme. The implicit approximation of the first time order for tangential stresses on the slippage plane is:   1 2 1 1 1 1 2 , , ( ) ( ) / 1 n n n n n n ij ij i j j i ij s t v v F                   τ , 1 1 1 n n n kj kj k j         τ The solution of this non-linear algebraic equation can be found with the decomposition on powers of the small parameter  . Limiting to the first term in this decomposition the expression for 1 n ij   was obtained with the fixed precision:   1 1/(1 ) 1 1/(1 ) 1 1 ( / 1) / 2 sign n q n q n ij s ije s ije               while 1 / 1 0 n ije s      , after the “elastic” calculation. The next step, in fact, is the adjustment of “elastic” stresses for the “yield strength” with viscous corrections. With the same approach a set of formulas can be obtained for isotropic and anisotropic media with different slippage conditions. We used here the simplest formula for 1 n ije   only to illustrate the derivation process. In this research we used the grid-characteristic method (Golubev et al. 2015; Khokhlov et al., 2019) on hexahedral meshes with the third order of the spatial approximation. The monotonization was done based on the grid-characteristic monotonicity criterion, based on the internal structure of the linear transport equation. It decreases the approximation of the method to the second order on discontinuities. It allows us to increase significantly the precision and the robustness of the numerical simulation. The seismic response from the cluster of fluid-filled cracks in the homogenous medium was calculated. The geometry sizes of the problem are represented at the Fig. 1. The point source with the 30 Hz Ricker time function was used. P-wave velocity of the medium was 4500 m/s, S-wave velocity – 2250 m/s, density – 2500 kg/m 3 . Spatial discretization was 5x5 m and time discretization was 1 ms. To describe the dynamic behavior of the medium inside 1 ij ije     n n 1  while 1 / n ije s    1 0   where 0 / t     , 0 1/ ( ) t   , 1 t 1 i j j i v v t           – the tangential stress at the next time’s layer 1 1 , , ( ) n n n n ije ij