Convolved action principles for couple stress elastodynamics

Abstract

A novel set of variational principles and integral theorems is developed for consistent couple stress elastodynamics, including the principle of virtual convolved action, the principle of stationary convolved action, the theorem of dynamic reciprocity and the integral representation theorem. Both time-domain and Laplace-domain versions are presented. Most notably, all of these are based on the concept of convolved action, involving temporal convolutions, and are applicable to the most general case of bi-anisotropic couple stress materials. Furthermore, the principle of virtual convolved action extends beyond elastodynamics to viscoelastic and plastic material response. In addition, the two-dimensional infinite space fundamental solutions and kernel functions are derived in the Laplace-domain and used to formulate a new boundary element method (BEM) for transient response, which also originates from a convolved action. The resulting BEM is applied to four prototype examples to demonstrate the validity and robustness of the method, and to explore interesting behavior of consistent couple stress elastodynamics in the isotropic case. Although the focus here is solely on couple stress elastodynamics, the convolved action framework extends to a broad range of time-dependent problems, including those of a multiphysical nature.