Effect of SH-type waves and shear stress discontinuity on a moving loaded composite structure

Abstract

This paper investigates the displacement of SH-type waves at the free surface of a layered media composed of transversely isotropic fiber-reinforced layer overlying orthotropic homogeneous half-space. The inverse Fourier transform is obtained by deforming the path of integration. The mathematical expression of the displacement is obtained in the form of inverse Laplace transform by using the convolution theorem. The impulsive normal line moving loads in terms of the delta function is assumed at the free surface. Two dynamic stress discontinuities at the interface of layered media are assumed in such a way that: (i) the uniform motion of stress discontinuity is in the direction of propagation of SH-type waves; (ii) the stress discontinuities are created and then expanded uniformly along the path of propagation of SH-type waves. The analytical form of displacement is evaluated using numerical integration technique (Simpson’s 3/8 method) and further plotted using MATLAB software. The variation of displacements with respect to time and fiber orientation has been studied. Some important results have been depicted using graphs. Furthermore, some special cases are derived to validate the model with the revealed literature.

Appendix

The value of dimensionless parameters are defined as,

$$\begin{aligned} t&=\frac{H \tau }{\beta _{T}},\quad X_{1}=\frac{x_{1}}{H},\quad X_{3}=\frac{x_{3}}{H},\quad U_{2}=\frac{u_{2}}{H},\quad V_{2}=\frac{v_{2}}{H},\quad \phi =\frac{Q_{1}}{Q_{3}},\quad \phi _{2}=\frac{\phi _{1}}{\phi },\quad \phi _{1}=\frac{\beta _{t}^2}{\beta _{s^2}},\end{aligned}$$

(83)

$$\begin{aligned} \beta _{s}^2&=\frac{Q_{3}}{\rho _{1}},\quad \beta _{1}=\frac{\beta _{t}}{\gamma _{1}},\quad F_{1}=\frac{F}{H \mu _{t}},\quad Q_{4}=\frac{Q_{1}}{\mu _{t}},\quad P_{2}=\frac{P_{1}}{\mu _{t}},\quad a=\frac{a_{0}}{H},\quad \eta =\left( \frac{P}{R}-\frac{Q^2}{R^2}\right) ,\end{aligned}$$

(84)

$$\begin{aligned} q_{1}&=\frac{\sqrt{X_{1}^2 \eta +(2n)^2\eta ^2}}{\eta \sqrt{R}},\quad Y=X_{1}+ \frac{Q}{R},\quad Y'=Y-b,\quad Y''=Y-a,\quad q_{2}=q_{3}=\frac{\sqrt{Y^2 \eta +(n)^2\eta ^2}}{\eta \sqrt{R}}. \end{aligned}$$

(85)

The expressions for some functions are given as follows,

$$\begin{aligned} G_{2,n}(g_{2,n}(\lambda _{2}))= & {} {\textit{Re}}\left( F_{2,n}(g_{2,n}(\lambda _{2}))~\frac{d g_{2,n}(\lambda _{2})}{d \lambda _{2}}\right) ~ K_{2}^n~H_{1}(\lambda _{2}-q_{1}), \\ F_{2,n}(g_{2,n}(\lambda _{2}))= & {} \frac{1}{\left( \sqrt{\eta g_{2,n}^2 +\frac{1}{R}}\right) \left( \beta _{1} +i g_{2,n}\right) },\\ g_{2,n}(\lambda _{2})= & {} \frac{1}{X_{1}^2 + (2n)^2 \eta }\left( iX_{1} \lambda _{2} + 2n \sqrt{\eta \lambda _{2}^2 - \left( \frac{X_{1}^2 +(2n)^2 \eta }{R}\right) }\right) ,\\ G_{3,n}(g_{3,n}(\lambda _{3}+\frac{\beta _{t}a}{\gamma _{2}}))= & {} {\textit{Re}}\left( F_{3,n}(g_{3,n}(\lambda _{3}))~\frac{d g_{3,n}(\lambda _{3})}{d \lambda _{3}} \right) ~ K_{3}^\frac{n-1}{2}~H_{1}(\lambda _{3}-q_{2}),\\ F_{3,n}(g_{3,n}(\lambda _{3}))= & {} \frac{1}{\left( \frac{\beta _{t}}{\gamma _{2}}+ig_{3,n}\right) \left( R \sqrt{\frac{1}{R}+g_{3,n}^2 \eta }+ Q_{4} \sqrt{\frac{g_{3,n}^2}{\phi }+\phi _{2}^2}\right) },\\ g_{3,n}(\lambda _{3} + \frac{2\beta _{t}}{\gamma _{2}}a)= & {} \frac{1}{ Y^2 +n^2 \eta }\left( iY (\lambda _{3}+\frac{\beta _{t}}{\gamma _{2}}a) + n\sqrt{\eta (\lambda _{3}+ \frac{\beta _{t}}{\gamma _{2}}a)^2 - \left( \frac{Y^2 + \eta n^2}{R}\right) }\right) ,\\ G_{4,n}(g_{4,n}(\lambda _{4}))= & {} {\textit{Re}}\left( F_{4,n}(g_{4,n}(\lambda _{4}))~\frac{d g_{4,n}(\lambda _{4})}{d \lambda _{4}}\right) ~ K_{4}^\frac{n-1}{2}~H_{1}(\lambda _{4}-q_{3}),\\ F_{4,n}(g_{4,n}(\lambda _{4}))= & {} \frac{1}{\left( \frac{\beta _{t}}{\gamma _{2}}+ig_{4,n}\right) \left( R \sqrt{\frac{1}{R}+g_{4,n}^2 \eta }+ Q_{4} \sqrt{\frac{g_{4,n}^2}{\phi }+\phi _{2}^2}\right) },\\ g_{4,n}(\lambda _{4} )= & {} \frac{1}{ Y^2 +n^2 \eta }\left( iY(\lambda _{4}) + n\sqrt{\eta \lambda _{4}^2 - \left( \frac{Y^2 + \eta n^2}{R}\right) }\right) ,\\ G_{5,n}(g_{5,n}(\lambda _{5}))= & {} {\textit{Im}}\left( f_{5,n}(g_{5,n}(\lambda _{5}))\frac{dg_{5,n}(\lambda _{5})}{d\lambda _{5}}\right) ~ K_{4}^\frac{n-1}{2}~H_{1}(\lambda _{5}-q_{5}),\\ f_{5,n}(g_{5,n}(\lambda _{5}))= & {} \frac{1}{g_{5,n}\left( R \sqrt{\eta g_{5,n}^2 +\frac{1}{R}}+Q_{4} \sqrt{\frac{g_{5,n}^2}{\phi }+\phi _{2}}\right) },\\ g_{5,n}(\lambda _{5})= & {} \frac{1}{Y''^2 + \eta }\left( iY'' \lambda _{5} + \sqrt{\eta \lambda _{5}^2 - \left( \frac{Y''^2 + \eta }{R}\right) }\right) . \end{aligned}$$