A singular problem in incompressible nonlinear elastostatics

This paper represents a contribution to the numerical treatment of problems in incompressible elasticity theory for large deformations. We are especia lly concerned about the solution of plane problems with corners. A review o f the literature on these problems indicates that the behavior of the solut ion in the vicinity of a corner is given little attention. We investigate t he solution of the compressed bonded block problem corresponding to the com pression of an incompressible elastic block of rectangular cross-section an d infinite transverse length between two opposing bonded rigid surfaces, wi th the two remaining lateral faces traction-free. We are especially interes ted in the behavior at a corner where a bonded end is adjacent to a free la teral side. We employ a finite element method based on a reduced and select ive integration technique with penalization to construct a numerical soluti on for this problem. Our computational method converges everywhere except i n a small neighborhood of the corner. We appeal to an elementary a priori i nequality concerning the angle of shear to show that the numerical calculat ions in this neighborhood are inaccurate and need a more refined study. Bas ed on the inequality, we offer a conjecture concerning the local shape of t he deformed free lateral surface at the corner.