Flexural vibrations of a thin circular elastic inclusion in an unbounded body under the action of a plane harmonic wave

Abstract

The interaction of a plane harmonic longitudinal wave with a thin circular elastic inclusion is considered. The wave front is assumed to be parallel to the inclusion plane. Since the inclusion is thin, the matrix-inclusion interface conditions (perfect bonding) are formulated on the mid-plane of the inclusion. The bending displacements of the inclusion are determined from the bending equation for a thin plate. The problem is solved using discontinuous Lamé solutions for harmonic vibrations. Therefore, the problem can be reduced to the Fredholm equation of the second kind for a function associated with the discontinuity of normal stresses on the inclusion. The equation obtained is solved by the method of mechanical quadratures using Gaussian quadrature formulas. Approximate formulas for the stress intensity factors are derived. Results from a numerical analysis of the dependence of the SIFs on the dimensionless wave number and the stiffness of the inclusion are presented Keywords harmonic wave, elastic inclusion, discontinuous solution, stress intensity factor Introduction. In studying the vibrations of elastic bodies containing a thin inclusion, it is often assumed perfectly rigid. Though such an assumption greatly facilitates the mathematical solution, the effect of the elastic properties of the inclusion on the stress concentration near it cannot be examined. That this effect can be significant was shown in [3] where the vibrations of an unbounded body with strip inclusions were studied. In the present paper, we will use an approach based on discontinuous solutions to study the effect of plane elastic waves on an inclusion in the form of an elastic circular plate. Similar problems for inclusions of low stiffness were addressed in [2,10]. Note also that static magnetoelastic problems for bodies with inclusions and an elastic orthotropic medium with an ellipsoidal inclusion were solved in [7,9]; the stress state of viscoelastic plates with elliptic inclusions and plates with a multilayer inclusion was studied in [8,11].1. Problem Formulation. Consider an unbounded body (matrix) with an elastic inclusion in the form of a thin circular plate of thickness h. The inclusion is represented by a circle of radius a in the plane z = 0 in a cylindrical coordinate system with the origin at the center of the inclusion. The inclusion is acted upon by a plane expansion-compression wave with front parallel to its plane. The potential of and the displacements in this wave are given by