Discrete Variational Formulation and Finite-Element Implementation

Abstract

In this chapter we address in detail the variational formulation and numerical implementation of classical plasticity and viscoplasticity in the context of the finite-element method. As noted in Chapter 2, the variational setting of classical plasticity leads naturally to a variational inequality typically formulated in stress space. This is the framework adopted by several authors, notably Johnson [1976a,b, 1978]. On the other hand, our formulation transforms this inequality into a variational equal-ity by introducing a Lagrange multiplier at the outset which is interpreted as the consistency parameter. Furthermore, the yield condition is formulated in strain space. We show that these steps are in fact crucial to obtain a variational frame-work suitable for the implementing the strain-driven, return-mapping algorithms examined in detail in Chapter 3. By now it is well established that displacement-based, finite-element methods may lead to grossly inaccurate numerical solutions in the presence of constraints, such as incompressibility or nearly incompressible response; see e.g., Hughes [1987, Chapter 4] for a review and an illustration of the difficulties involved in the context of linear incompressible elasticity. As first noted in Nagtegaal, Parks, and Rice [1974], the classical assumption of incompressible plastic flow in metal plasticity is the source of similar numerical difficulties. Finite-element approxima-tions based on mixed variational formulations have provided a useful framework in the context of which constrained problems can be successfully tackled.