Numerical Solution of the Plane Dynamic Problem of Elasticity in Terms of Stresses

This study addresses the formulation and numerical solution of a plane dynamic elasticity problem in which the stress tensor components are the primary unknowns. The objective is to develop a numerical algorithm for directly determining the dynamic stress field in a homogeneous isotropic elastic medium. The equations of motion, strain–displacement relations, and Hooke’s law are used to derive a system of three differential equations in terms of stresses, supplemented by initial and boundary conditions. Spatial discretization is performed using the Galerkin finite element method. Time integration is carried out using the Newmark method in a C++ program and a finite difference scheme in a FreeFEM++ implementation. A rectangular domain is subjected to a time-dependent normal load on its upper boundary, while the remaining boundaries are traction-free. Spatial stress distributions and the time history of the normal stress at the centre of the domain are obtained. Comparison of the two implementations shows qualitative agreement in the stress distributions and their oscillatory behaviour. The results demonstrate the applicability of the approach to the numerical investigation of plane dynamic elasticity problems formulated in terms of stresses.