A SINGULAR PROBLEM IN INCOMPRESSIBLE NONLINEAR ELASTOSTATICS

Abstract

This paper represents a contribution to the numerical treatment of problems in incompressible elasticity theory for large deformations. We are especially concerned about the solution of plane problems with corners. A review of the literature on these problems indicates that the behavior of the solution in the vicinity of a corner is given little attention. We investigate the solution of the compressed bonded block problem corresponding to the compression of an incompressible elastic block of rectangular cross-section and infinite transverse length between two opposing bonded rigid surfaces, with the two remaining lateral faces traction-free. We are especially interested in the behavior at a corner where a bonded end is adjacent to a free lateral side. We employ a finite element method based on a reduced and selective integration technique with penalization to construct a numerical solution for this problem. Our computational method converges everywhere except in a small neighborhood of the corner. We appeal to an elementary a priori inequality concerning the angle of shear to show that the numerical calculations in this neighborhood are inaccurate and need a more refined study. Based on the inequality, we offer a conjecture concerning the local shape of the deformed free lateral surface at the corner.