Some theorems in the theory of small deformations superimposed on a finite deformation of a hyperelastic material of grade 2

Abstract

The equations for small deformations superimposed on large deformations of a hyperelastic material of grade 2 are formulated and applied to derive a basic integral relation that is used to establish generalized Betti, Clapeyron, work and energy theorems. Theorems of minimum and complementary energy are deduced essentially from an energy criterion of super-stability, and these are used to prove uniqueness of solutions to the static and dynamic, mixed incremental boundary value problems. These results use a certain generalized kinetic energy functional that is assumed positive definite; and this property and the reciprocal and energy principles are exploited further to establish some theorems in the theory of small free vibrations.