Normal configurations and the nonlinear elastostatic problems of bending, torsion, expansion, and eversion for compressible bodies (Q805343)

The best known universal solutions are those for the class of incompressible isotropic elastic materials with a homogeneous structure. Clearly, relative to a homogeneous reference configuration, all homogeneous deformations are universal solutions. Other than this trivial family, five families of inhomogeneous universal solutions have been found and thoroughly analyzed. In the paper there are no additional restrictions on the class of materials. The results are applicable not only to all compressible isotropic elastic materials with a homogeneous structure but also to all isotropic or transversally isotropic elastic materials with certain special inhomogeneous or even materially nonuniform structures. The solutions are obtained by decomposition of an equilibrium configuration into a disjoint union of a one-parameter family of surfaces, called lamina surfaces. All of these lamina surfaces be two-dimensional universal solutions for homogeneous isotropic elastic membranes. In the paper there are given some specific examples of the five families of deformations, including solutions to bending, torsion and eversion problems for compressible bodies.