Dynamic analysis of nano-heterogeneities in a finite-sized solid by boundary and finite element methods

Abstract

This work addresses the elastodynamic problem for a finite-sized, elastic solid matrix containing multiple nano-heterogeneities of arbitrary shape, number and geometric configuration. The problem is formulated under plane strain conditions, and time-harmonic motions are assumed to hold. The aim is to evaluate the non-uniform stress and strain fields that develop in the solid matrix and to identify zones of dynamic stress concentration for the case of dynamic loads applied along the matrix boundary. The mechanical model used here is based on a combination of classical elastodynamic theory for the bulk solid under non-classical boundary conditions, supplemented with a localized constitutive equation for the solid-inclusion interface in the framework of the Gurtin–Murdoch theory of surface elasticity. As computational tools we use (a) the 2D boundary element method (BEM) with frequency-dependent fundamental solutions for the bulk solid and (b) the finite element method (FEM) software package ANSYS augmented by a macro-finite element for representing surface effects on the contour of the nano-inclusions. At first, accuracy of the numerical solutions obtained for the dynamic stress concentration factor (DSCF) and for the diffracted displacement wave field is satisfactorily established. Next, comparison studies are conducted to gauge the BEM and FEM separately. These are followed by extensive numerical simulations that show that both BEM and FEM are able to capture the dependence of the diffracted wave field and of the DSCF on the type of the inclusions, their overall configuration and the nature of applied dynamic loads. It is concluded that the interaction effect between the nano-heterogeneities and the external perimeter of the bounded solid matrix is of paramount importance.