Wave propagation in a transversely isotropic microstretch elastic solid

Abstract

BackgroundThe theory of microstretch elastic bodies was first developed by Eringen (1971, 1990, 1999, 2004). This theory was developed by extending the theory of micropolar elastcity. Each material point in this theory has three deformable directors.MethodsThe governing equations of a transversely isotropic microstretch material are specialized in x-z plane. Plane wave solutions of these governing equations results into a bi-quadratic velocity equation. The four roots of the velocity equation correspond to four coupled plane waves which are named as Coupled Longitudinal Displacement (CLD) wave, Coupled Longitudinal Microstretch (CLM) wave, Coupled Transverse Displacement (CTD) wave and Coupled Transverse Microrotational (CTM) wave. The reflection of Coupled Longitudinal Displacement (CLD) wave is considered at a stress-free surface of half-space of material. The appropriate displacement components, microrotation component and microstretch potential for incident and four reflected waves in half-space are formulated. These solutions for incident and reflected waves satisfy the boundary conditions at a stress free surface of half-space and we obtain a non-homogeneous system of four equations in four reflection coefficients (or amplitude ratios) along with Snell’s law for the present model.ResultsThe speeds of plane waves are computed by Fortran program of bi-quadratic velocity equation for relevant physical constants of the material. The reflection coefficients of various reflected waves are also computed by Fortran program of Gauss elimination method. The speeds of plane waves are plotted against angle of propagation direction with vertical axis. The reflection coefficients of various reflected waves are plotted against the angle of incidence. These variations of speeds and reflection coefficients are also compared with those in absence of microstretch parameters.ConclusionsFor a specific material, numerical simulation in presence as well as in absence of microstretch shows that the coupled longitudinal displacement (CLD) wave is fastest wave and the coupled transverse microrotational (CTM) is observed slowest wave. The coupled longitudinal microstretch (CLM) wave is an additional wave due to the presence of microstretch in the medium. The presence of microstretch in transversely isotropic micropolar elastic solid affects the speeds of plane waves and the amplitude ratios of various reflected waves.Mathematics Subject Classification74J