# Elastic and Pseudo-Elastic Instability and Bifurcation

## Abstract

This article is concerned with the analysis of stability, loss of stability and the associated bifurcation phenomena in elastic and pseudo-elastic solids subject to large deformations. The basic equations of nonlinear elasticity are summarized prior to a discussion of the linearized incremental equations, which are used as a basis for the analysis of (linearized) stability with particular reference to the dead-load boundary-value problem as a key exemplar. Both compressible and incompressible materials are discussed. Global aspects of the general problem are then considered from the viewpoint of inversion of stress-deformation relationships and the theory is applied to two representative problems involving homogeneous deformation. An illustrative example in which bifurcation from a homogeneous state of deformation (namely, simple shear of a plate) is associated with inhomogeneous incremental modes of deformation is then examined in detail. Next, a new theory of *pseudo-elasticity* is discussed. This theory enables changes in the (elastic) constitutive law associated with damage to be invoked. A general form of the theory is described, and aspects of local and global stability associated with the theory are discussed. The theory is then exemplified using a specific model in the context of isotropic material response. The model, which is associated with a discontinuous change in material properties, is applied to the problem of inflation and deflation of a thick-walled spherical shell of incompressible material. Specific results are obtained for the residual deformation and stress distributions remaining after removal of the inflating pressure.

## Keywords

Deformation Gradient Nonlinear Elasticity Polar Decomposition Incompressible Material Isotropic Elastic Material## Preview

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