PSI - Issue 43

V.I. Golubev et al. / Procedia Structural Integrity 43 (2023) 29–34 V.I. Golubev et al. / Structural Integrity Procedia 00 (2022) 000 – 000

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Araya-Polo et al., 2019; Park et al., 2020; Stankevich et al., 2021) to identify large objects with the mechanical properties that were significantly different from the background media. The current paper contributes to the field by applying the same technique for the relatively small, fractured region location and trying to identify the properties of the region together with its spatial position. 2. Mathematical model and numerical method We consider the elastic medium with a system of periodically repeated slip planes. Their orientation is defined by the normal vector ⃗ . The distance between planes is constant and equal to . The material density is equal to , and elastic moduli and are known constants. If the stress tensor is defined as , then the tangential stress is equal to = ∙ ⃗ − ( ⃗ ∙ ∙ ⃗ ) ⃗ , and the normal stress is equal to = ( ⃗ ∙ ∙ ⃗ ) . We consider the compressed mode with < 0 . We notify the slip velocity as . Its connection with the velocity gap [ ] is set by = [ ⃗ ] . Instead of simulating the interlayer behavior directly, we replace it by the appropriate contact conditions. The Coulomb friction condition with small viscous additive is used. For < 0 we have = | | ( | ⃗ ⃗ ⃗ ⃗ | + ) , (1) or, expressing as a function of = 1 | ⃗ ⃗ | 〈 | | ⃗ | | − 1〉 , (2) where is the viscous coefficient, is the Coulomb friction coefficient, 〈 ( )〉 = ( ) ( ) , ( ) is the Heaviside function. Further, we consider as a discontinuous function of spatial coordinate and time. In this case, the inelastic strain tensor can be defined by = ⃗ ⨂ ⃗ + ⃗ ⨂ ⃗ 2 . (3) The full strain tensor relates to both elastic and inelastic deformations = + , = ∇⃗ ∙ ⃗ +∇⃗ ∙ ⃗ 2 , (4) where is the macroscopic particle velocity, is the elastic strain tensor, connected by the Hooke’s law = ( : ) + 2 . (5) The final equation of the mathematical model is the standard motion equation ⃗ = ⃗∇⃗ ∙ . (6) The govern system of equations is hyperbolic semi-linear one, and the numerical solution may be constructed based on different approaches. However, commonly used explicit finite difference schemes will be unstable. It is the consequence of the non-linearity of the free term with the small parameter in the denominator. The system is become rigid. The authors proposed to use an explicit-implicit scheme to overcome this problem. The implicit approximation is used only for those equations that contain the non-linearity; the others are discretized with the explicit scheme. Let ’ s consider the equation for ̇ 3 in the compressed mode ( 33 < 0 ): ̇ 3 = ( 3, + ,3 ) − 3 |⃗ | 〈 | | ⃗ | 33 | − 1〉 , | | = √ 3 3 , = 1,2 . (7) Earlier, the first-order approximation procedure was successfully used to construct the explicit-implicit scheme, equivalent to the nodewise correction of the isotropic linear elastic system solution (Nikitin et al., 2022): For | +1 | ≥ | 3 3 + 1 | 3 +1 = | 3 3 + 1 | 3 + 1 |⃗ +1 | 1+ | ⃗ +1 | 1+ | 3 3+ 1 | , (8) +1 = 3 + 1 − 3 +1 Δ , 3 + 1 = 3 + Δ ( ,3+1 + 3 , + 1 ), = 1,2 . (9)