Theory of Bifurcation and Instability in Time-Independent Plasticity

The theory is developed for constitutive rate equations which are arbitrarily nonlinear and can thus describe the important effect of formation of a yield-surface vertex. Basic elements of Hill’s theory of bifurcation and stability in time-independent plastic solids are presented. In particular, the condition sufficient for uniqueness of a solution to the first-order rate boundary value problem, the stationary and minimum principles for velocities, the concept of a comparison solid and the primary bifurcation point are discussed. Distinction is emphasized between the conditions for uniqueness and for stability of equilibrium, and between the bifurcation point and an eigenstate. Several recent extensions of Hill’s theory are next presented. It is shown how the property of the comparison solid required in the uniqueness criterion can be weakened and justified then on micromechanical grounds, without the need of complete specification of the macroscopic constitutive law. The question of non-uniqueness and instability in the post-critical range is examined, and respective theorems are formulated for discretized systems and for a certain class of continuous systems. The energy interpretation of the basic functionals and conditions in the bifurcation theory is given. Finally, a unified approach to various bifurcation and instability problems is presented which is based on the concept of instability of a deformation process and on the relevant energy criterion.

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