Constrained Elastic Bodies and Universal Solutions in the Class of Plane Deformations with Uniform Transverse Stretch

Abstract

The determination of static fields that are producible in every isotropic, homogeneous, incompressible, elastic body under the sole action of surface traction is known as Ericksen's problem. In this paper, the authors consider a family of isotropic constraints including incompressibility, and for each class in the family, they obtain the complete set of plane deformations with uniform transverse stretch that are universal; that is, producible in any elastic body of the particular class under the sole action of surface traction. In fact, in the case where the plane stretches are not constant, the authors distinguish the class they consider only in the last step of their construction of a solution, and this renders possible a general analysis of the problem. The authors stress the singular aspects of nonhomogeneous solutions for which all stretches are constant, showing that for nearly all constraints, they are universal.