On the Unification of Normal-Mode Techniques for Solving Dynamic Elasticity Problems

Abstract

Normal-mode techniques for solving dynamic elasticity problems with inhomogeneous boundary conditions are divided into two categories—namely, displacement (separation of variables, integral transforms) and acceleration (mode acceleration, Mindlin-Goodman, Williams) methods. In general, acceleration methods exhibit superior convergence properties relative to displacement methods. In this study, the solution of the complete linear dynamic elasticity equations of motion with inhomogeneous boundary conditions in an arbitrarily shaped regions is obtained by two methods: (1) a general integral transform technique (based upon Green's extended identity), and (2) a general Mindlin-Goodman process. It is then shown that both of these solutions are actually just different ways of writing the convolution integral in the complex (Laplace transform) domain thereby establishing the unity of all normal mode techniques. Finally, a means for obtaining the quasistatic solution from the displacement (integral transform) solution without the necessity for solving the quasistatic problem, as required in acceleration methods, is then given. [Work supported by the U. S. Atomic Energy Commission.]