Generation and Propagation of SH Waves Due to Shearing Stress Discontinuity in Linear Orthotropic Viscoelastic Layered Structure

Abstract

This present paper deals with the generation and propagation of SH wave due to shearing-stress discontinuity at the common interface of a linear orthotropic viscoelastic layer of finite thickness overlying linear orthotropic viscoelastic half-space. Laplace and Fourier transformations in combination with the modified Cagniard-De Hoop method have been adopted as the solution technique for the present problem. The free surface displacement field has been obtained in integral form for four distinct cases of shear-stress discontinuity (Case 1, 2, 3 and 4) at the common interface of stratum and substrate of the layered structure. Numerical computation and graphical demonstration have been carried out to observe the profound effects of various affecting parameters viz. time of disturbance, distance from the source of disturbance, attenuation parameter and angular frequency on free surface displacement field which serve as major highlights of the present study.

Appendix

$$ \begin{aligned}& \overline{c}_{44} = \left( {c_{44}^{R} + pc_{44}^{I} } \right)\,,\,\,\overline{c}_{55} = \left( {c_{55}^{R} + pc_{55}^{I} } \right),\,P_{1}^{2} = \frac{{p^{2} \rho_{1} + \overline{c}_{44}^{\left( 1 \right)} \eta^{2} }}{{\overline{c}_{55}^{\left( 1 \right)} }},P_{2}^{2} = \frac{{p^{2} \rho_{2} + \overline{c}_{44}^{\left( 2 \right)} \eta^{2} }}{{\overline{c}_{55}^{\left( 2 \right)} }}, \\ & D_{1} = - \left( {\overline{c}_{55}^{\left( 2 \right)} } \right)^{2} \frac{{\beta_{11}^{2} }}{{\beta_{12}^{2} }} + \left( {\overline{c}_{55}^{\left( 1 \right)} } \right)^{2} \,,\,\,\,D_{2} = \left( {\overline{c}_{55}^{\left( 1 \right)} } \right)^{2} \phi_{1}^{2} - \left( {\overline{c}_{55}^{\left( 2 \right)} } \right)^{2} \phi_{2}^{2} ,\,S = \frac{{\sqrt {y_{1}^{2} + h^{2} \phi_{1}^{2} } }}{{\beta_{11}^{2} \phi_{1}^{2} }},\\& E = \frac{{y_{1} }}{{\beta_{11}^{2} \phi_{1}^{2} }},\,\,P = \frac{h}{{\beta_{11} }}, \\ & \alpha = {\text{tan}}^{ - 1} \left( {\frac{{y_{1} }}{{h\phi_{1}^{2} \beta_{11} }}} \right),\,\beta_{11} = \left( {\frac{{\overline{c}_{55}^{1} }}{{\rho_{1} }}} \right)^{\frac{1}{2}} ,\,\,\,\beta_{12} = \left( {\frac{{\overline{c}_{55}^{2} }}{{\rho_{2} }}} \right)^{\frac{1}{2}} \,,\,\,\,\,\phi_{1} = \left( {\frac{{\frac{{\overline{c}_{44}^{1} }}{{\rho_{1} }}}}{{\frac{{\overline{c}_{55}^{1} }}{{\rho_{1} }}}}} \right)^{\frac{1}{2}} ,\,\,\,\phi_{2} = \left( {\frac{{\frac{{\overline{c}_{44}^{1} }}{{\rho_{2} }}}}{{\frac{{\overline{c}_{55}^{1} }}{{\rho_{2} }}}}} \right)^{\frac{1}{2}} , \\ & G^{\prime}_{1} \left( {\xi_{m} \left( t \right)} \right) = {\text{Im}} \left( {F^{\prime}_{2} (\xi_{m} (t))\frac{d}{dt}\xi_{m} (t)} \right)\,H\left( {t - \frac{{\left( {y_{m}^{2} + h^{2} \phi_{1}^{2} } \right)^{\frac{1}{2}} }}{{\beta_{11}^{2} \phi_{1}^{2} }}} \right), \\ & F^{\prime}_{2} (\xi_{m} (t)) = \frac{1}{{\left( {\xi_{m} } \right)\left[ {\overline{c}_{55}^{\left( 1 \right)} \left[ {1 + \xi_{m}^{2} \phi_{1}^{2} } \right]^{\frac{1}{2}} + \overline{c}_{55}^{\left( 2 \right)} \left[ {\frac{{\beta_{11}^{2} }}{{\beta_{12}^{2} }} + \xi_{m}^{2} \phi_{2}^{2} } \right]^{\frac{1}{2}} } \right]}}. \\ \end{aligned} $$