Elastic and Pseudo-Elastic Instability and Bifurcation
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.
KeywordsDeformation Gradient Nonlinear Elasticity Polar Decomposition Incompressible Material Isotropic Elastic Material
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