Effect of Loosely Boundaryon Reflection and Transmission of Planes waves between two Couple Stress Elastic Half-Spaces

Page: 09-11

Suman Sindhwani and Rachana Gupta (Department of Mathematics, Hindu Girls College, Sonepat, Haryana)

In the classical theory of linear elasticity, we are mainly concerned with the behaviour of continuous elastic bodies under the influence of loads applied on it within certain limits.This theory is based on assumption that the loads are transmitted through an area element dA in body by means of stress vector p→dA only. But the interaction between two parts of body through an element d A is transmitted not only by the action of stress vector but also by a couple stress vector. Voigt (1887) was the first who tried to remove the discrepancies of the classical theory of elasticity assuming that loads are transmitted through an area element dA by means of not only the force vector but also by the movement vector, such an assumption would mean that not only the force stresses but also the couple stresses acts on the faces of an elementary parallelopiped. From the equilibrium conditions for the parallelopiped it results that both type of stresses are symmetric. However the complete theory of asymmetric elasticity was developed by Brothers Cosserat and Cosserat (1909). They assumed that bodies composed of interconnected particles in form of rigid trihedron. During the process of deformation , each particle is displaced by u (x,t) and rotated by φ (x,t), the functions of position x and time t. The elastic continuum is thus described in such a way that it’s points possess the orientation and for which we speak of as rotation of a point. The novelty of the idea and of the work of Cosserat and Cosserat remained un noticed for a long time perhaps due to 1. Theory was non linear from the beginning. 2. Notations were unclear. 3.Complicated theory in comparison to theory of elasticity. The theory presented by Cosserats involves mechanics, optics and electrodynamics and combined with a general principle of least action and this is the reason the attempt to create the unified theory was done. Murty proposed a theoretical model for propagation of waves at the loosely bonded interface of elastic half spaces . It was assumed that there is a thin interface layer of viscous fluid between two half spaces such that a finite amount of slip can take place at the interface . In this paper the effect of loosely boundary on reflection and transmission of planes waves between couple stress elastic media has been studied and the amplitude ratios for reflected and refracted waves to that of incident waves are obtained. Some particular cases have been deduced from the present case of effect of loosely boundary between two couple stress elastic half space.