Five Different Formulations of the Finite Strain Perfectly Plastic Equations
Chein-Shan Liu;

Source CMES: Computer Modeling in Engineering & Sciences, Vol. 17, No. 2, pp. 73-94, 2007
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Keywords Elastoplasticity, Corotational stress rate, Lie algebra, Lie group, Poisson manifold, Generalized Hamiltonian system, Lie-Poisson system, Coadjoint orbit, Two-generator
Abstract The primary objectives of the present exposition focus on five different types of representations of the plastic equations obtained from an elastic-perfectly plastic model by employing different corotational stress rates. They are (a) an affine nonlinear system with a finite-dimensional Lie algebra, (b) a canonical linear system in the Minkowski space, (c) a non-canonical linear system in the Minkowski space, (d) the Lie-Poisson bracket formulation, and (e) a two-generator and two-bracket formulation. For the affine nonlinear system we prove that the Lie algebra of the vector fields is so(5,1), which has dimensions fifteen, and by the Lie theory the superposition principle is available for this system. Although the plastic equations are nonlinear in stress space, we can develop some methods to transform them into the linear systems in the augmented stress spaces with a canonical form and a non-canonical form in the Minkowski space. On the cotangent bundle of yielding manifold, we can introduce the Lie-Poisson bracket to construct an evolutional differential system of plastic equations. The stress trajectory traces a coadjoint orbit in the Poisson manifold under a coadjoint action of the Lie group SO(5). Then, we prove that the plastic equations admit two generators: one conservative and one dissipative, as well as two brackets: the Poisson bracket and dissipative bracket. From a dissipation point of view the yield function is a Casimir function of the dissipative bracket system.
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