Bifurcation in elastic-plastic solids in plane stress
A sufficient condition for bifurcation of equilibrium for an elasticplastic solid under a state of plane stress is established. The bifurcation is found to occur in any of the following modes: (i) symmetric mode corresponding to necking or bulging, (ii) antisymmetric mode corresponding to buckling and (iii) mode of deformation localized at the surface. Introduction. The stability and bifurcation of rigid-plastic solids in plane-plastic flow has been studied extensively [1-3]. Similar work for elastic-plastic solids in plane flow is also available [4], However, very little work seems to have been done on the problems of stability and bifurcation of an elastic-plastic solid in three-dimensional flow. A similar investigation for rigid-plastic solids in three-dimensional flow is limited in scope due to the restricted choice of the velocity field. Prager [5] found that for a von Mises solid under a state of uniform stress, the only admissible velocity field in three-dimensional flow is of quadratic form. Hence some form of instability may be excluded when the behaviour of the solid is postulated to be rigid plastic (example—surface instabilities, buckling). In the degenerate case for a von Mises solid, when one principal component of the stress deviation is zero as in plane strain, or for a Tresca solid, a transcendental form of the velocity field is admissible. Even then it would appear that a rigid-plastic solid does not exhibit instability localized at the surface. In this analysis, an elastic-plastic plate in a state of plane stress is considered. In the constitutive equation, which is due to Hill [6], the plastic component of deformation is expressed in terms of the components of the unit normal to the yield surface at the local stress point. Hence the analysis is applicable to a wide class of inelastic solids. As an illustration, we consider a plate that is under equal biaxial stress and obeys von Mises' yield criterion. This assumption results in considerable mathematical simplification. It is found that the plate may become unstable in any of the following modes: (i) symmetric mode, (ii) antisymmetric mode or buckling mode, and (iii) instability localized at the surface. Formulation of the problem. Consider an incompressible isotropic elastic-plastic disk in a state of plane stress at a certain time t = 0 during a process of continued deformation. In the current configuration, also taken as reference configuration for incremental deformation, the plate is of in-plane dimension 2a X 26 and of thickness 2c. A fixed coordinate frame coinciding with the axes of symmetry of the plate is taken as the reference frame. Wherever convenient the coordinates xx , x2 , x3 will