Discontinuous bifurcation states for associated smooth plasticity and damage with isotropic elasticity

For many constitutive equations the tangent tensor consists of a rank one modification to the isotropic elasticity tensor with a total of two elasticity parameters and one parameter describing the current state of inelasticity. For small deformations, general expressions are derived for the loss of ellipticity, the corresponding normal to the bifurcation plane and the mode of discontinuous bifurcation for the velocity gradient. If the principal basis of an evolution tensor is used, the current stress or strain state is characterized by two additional parameters. The small number of material and state parameters makes it feasibk to use contour plots to illustrate all possible combinations that can provide a discontinuous bifurcation. These bifurcation maps can be used to illustrate the bifurcation properties of a particular plasticity or continuum damage constitutive model. Conversely, the bifurcation maps can be used in conjunction with experimental data on bifurcation features to assist in the development of constitutive equations that provide the correct failure criterion for a given material under all possible stress paths. Published by Elsevier Science Ltd.