On the diffraction of nonstationary SH-wave on semi-infinite crack in porous elastic medium
type: Item
instance of: scholarly article
zbMATH Open document ID: 1050.86518
author: Kholmatzhon Imomnazarov
author: A. Kh. Matevosyan
publication date: 2001-07-22T00:00:00Z
Mathematics Subject Classification ID: 86A15
Mathematics Subject Classification ID: 86A17
Mathematics Subject Classification ID: 74B10
Mathematics Subject Classification ID: 74J20
zbMATH DE Number: 1622425
zbMATH Keywords: porous medium
zbMATH Keywords: plane SH shock wave
zbMATH Keywords: semi-infinite crack
zbMATH Keywords: diffraction problem
zbMATH Keywords: formula for wave scattering
MaRDI profile type: Publication
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title: On the diffraction of nonstationary SH-wave on semi-infinite crack in porous elastic medium
published in: Bulletin of the Novosibirsk Computing Center. Series: Mathematical Modeling in Geophysics
review text: The authors study the diffraction problem for a plane SH shock wave on a semi-infinite crack in a porous medium for the case of energy loss due to intercomponent friction.NEWLINENEWLINE An unbounded porous isotropic medium in the space \(\mathbb R^3\) is considered which is characterized by the elastic constants \(\lambda\), \(\mu\), \(\alpha\), friction coefficient \(\kappa\), partial density \(\rho_s\) of the elastic body, and partial density \(\rho_l\) of the liquid. It is assumed that the medium under consideration has a crack (defect) of the form NEWLINE\[NEWLINE z = 0,\qquad 0\leq x <\infty, \qquad -\infty < y < \infty. NEWLINE\]NEWLINE An SH-wave is incident on this crack where the wave is defined by the formula NEWLINE\[NEWLINE u^0(t,x,z) = \tilde{\varphi}\left(t - \frac{x\sin\alpha - z\cos\alpha}{c_t} \right),\qquad c_t^2 = \mu\rho_s^{-1}. NEWLINE\]NEWLINE Here \(\tilde{\varphi}(\xi)_{\xi < 0} \equiv 0\) (the incident wave \(\tilde{\varphi}\) has a leading edge), \(\alpha\) is the angle of incidence of the wave. In the absence of a crack, this wave causes the following tangential stresses in the porous isotropic medium: NEWLINE\[NEWLINE \tau^{0}_{yz}(t,x,z) = \mu\frac{\partial u^0}{\partial z} = \mu\cos\alpha \left|\frac{d\tilde{\varphi}(\xi)}{d\xi}\right|_{\xi = t - (x\sin\alpha -z\cos \alpha)/c_t}. NEWLINE\]NEWLINE The following problem can be posed: Find the distributions of stresses and displacements of the elastic body under consideration, and the quantities \(\tau_{yz}(t,x,z)\), \(u(t,x,z)\), \(v(t,x,z)\) after the incidence of the SH-wave on the crack where the functions \(u\), \(v\) satisfy the system NEWLINE\[NEWLINE \begin{alignedat}{2} \frac{1}{c_t^2}\frac{\partial^2u}{\partial t^2} - \frac{\partial^2u}{\partial x^2} - \frac{\partial^2u}{\partial z^2} + \frac{\chi}{c_t^2}\frac{\rho_l^2}{\rho_s}\left (\frac{\partial u}{\partial t} - \frac{\partial v }{\partial t}\right) &= 0, &&\qquad | x| < \infty,\;z\neq 0, \;t > 0, \\ \frac{\partial^2v}{\partia…
reviewed by: Vladimir N. Grebenev
description: scientific article; zbMATH DE number 1622425