A Lie-Poisson bracket formulation of plasticity and the computations based on the Lie-group SO(n)

In this paper we develop a generalized Hamiltonian formulation of a perfectly elastoplastic model, which is a typical dissipative system. On the cotangent bundle of the yield manifold, a Lie-Poisson bracket is used to construct the differential equations system. The stress trajectory is a coadjoint orbit on the Poisson manifold under a coadjoint action by the Lie-group SO(n). The plastic differential equation is an affine non-linear system, of which a finite-dimensional Lie algebra can be constructed, and the superposition principle is available for this system. Accordingly, we can construct numerical schemes to automatically preserve the yield-surface for perfect plasticity, for isotropic hardening material, as well as for an anisotropic elastic-plastic model. Then, we describe an anisotropic elastic-plastic material model without entering the work-hardening range and deforming under a specified dissipation rate, which can be achieved through a stress-dependent feedback control law of strain rate.