Time-Differencing Fundamental Solutions for Plane Elastodynamics

Abstract

In this paper, new fundamental solutions for dynamic analysis of plane elastodynamics are developed. The governing dynamic differential equations are rewritten by replacing the accelerations with the corresponding displacements at different time steps via a suitable finite difference scheme. The known displacements in initial conditions or from previous time steps are treated as generalized new inertia terms. The unknown displacements at the current time step are added to the governing differential operator to form a new operator. The new time-dependent fundamental solutions are then derived with respect to the new differential operator. The required mathematical derivations are presented in detail. The corresponding integral equations and domain load treatments are also presented. Singular integrals of the derived fundamental solutions are also treated. Several numerical examples are solved to demonstrate the validity and the accuracy of the developed new solutions. The results demonstrated the accuracy of the developed new solutions compared to traditional time domain fundamental solutions.