Finite element stress formulation for wave propagation

Based on a variational principle due to Gurtin, for linear elastodynamics, a finite element method in terms of stresses is developed for wave propagation problems. The finite element equations are simultaneous integral equations in time, with the peculiarity that they are equivalent to simultaneous linear differential equations with zero initial conditions. Written as differential equations, the finite element equations are of the form where [H] is a symmetric positive-semidefinite matrix, [Q] is a symmetric positive-definite matrix and the stresses are represented by {s}. This is, of course, the same form as the equations for the displacement formulation. As a demonstration of the validity of the formulation numerical results are compared with a solution for a triangular-shaped strip load applied to an elastic half-space as a ramp function of time. This solution is obtained by numerical integration of the exact solution for Lamb's problem of a line load suddenly applied to a half-space. The agreement is found to be generally very good. As a further example, the case of a plate subject to uniform tension on two ends and containing a hole in the centre is presented. The results are found to be reasonable in that the characteristic stress concentration occurs near the hole, and away from the hole the results are similar to the solution for an infinitely wide plate.