Deformations of Neo-Hookean Elastic Wedge Revisited

Abstract

We reconsider a semi-inverse problem for nonlinearly elastic wedges, versions of which have been studied by several authors over the past 40 years. An intriguing feature of this problem is the existence of "non-unidirectional" deformations of the wedge, i.e. deformations for which material draws radially inward toward the apex in certain sectors of the wedge while material bulges radially outward in complementary sectors. We observe that such deformations are possible for a much larger class of materials than has been previously noted. We explore more carefully the relation between the displacement boundamy conditions and the "directional" properties of defonnations. Also, we consider the problem under alternative stress boundary conditions that must be satisfied pointwise; most previous treatments of the problem considered either pure displacement boundary conditions or stress boundary conditions satisfied in some average sense. Finally, we investigate a much broader class of generalized plane-strain deformations, consistent with these alternative boundary conditions and with a certain natural physical restriction on the pressure, and we show, surprisingly, that no additional solutions exist within this broader class.