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A general solution method for transient multidimensional problems in solid mechanics

DEPARTMENT OF AERONAUTICAL ENGINEERING TECHNION-ISRAEL INSTITUTE OF TECHNOLOGY, HAIFA, Israel

Abstract

A new approach is presented for the difficult task of evaluating the multidimensional transient deformation of solids, in which waves emanate from impulsive sources and their reflections from boundaries map a strong discontinuity region into the solid. The affected region is traversed by longitudinal, shear, head, and surface waves, all of which are revealed in the solution by the present method.

The method comprises two stages: an analytical formulation and a numerical integration technique. The analytical stage is a kinematical approach, and is therefore valid for any field equations. It consists essentially in superimposing compatible kinematic relations, existing across strong discontinuity wave surfaces, on the multidimensional theory of characteristics. The dynamical part is based on a wave grid, and integration along these fronts is carried out numerically using a finite-difference technique.

The proposed method is illustrated in a representative boundary-value problem where a step-function load is applied to the boundary of a cylindrical cavity embedded in an elastic half-space. For this problem the entire domain of deformation is solved to yield stress distribution and velocity of the particles in the half-space.