Computing FEM solutions of plasticity problems via nonlinear mixed variational inequalities

Moving from the seminal papers of Han and Reddy [3][4][7], we propose a fixed–point algorithm for the solution of hardening plasticity plane–strain problems. The continuous problem may be classified as a mixed nonlinear non–differentiable variational inequality of the second type [3] and is discretized by means of a truly mixed finite–element approach [1]. One of the peculiarities is the use of the composite triangular element of Johnson and Mercier [2] for the approximation of the stress field. The non–differentiability is coped with via regularization [5] whereas the nonlinearity is approached with a fixed–point iterative strategy. Numerical results are proposed that investigate the sensitivity of the approach with respect to the mesh size and the regularization parameter ε. Hardening plasticity may be considered as a mature topic from the theoretical and algorithmic viewpoint. The comprehensive book by Simo and Hughes [8], for example, accounts for various return mapping algorithms (also at large strains) of which stability, consistency and convergence are assessed. There actually exist several variants of strain–driven elastic–predictor plastic–corrector algorithms for all of which the notion of consistent tangent operators is crucial in order to preserve the quadratic convergence of Newton’s method [9]. More limited attention has been paid to complementary mixed formulations making use of the maximum plastic dissipation principle. Along this path, one of the first investigations seems to be [10] where stresses are simply in L, i.e. globally discontinuous, whereas displacements are in H and are therefore continuous. Following [3], we conversely consider the truly mixed approach, as defined in [1], according to which displacements are the discontinuous Lagrange multipliers and the stresses are “more regular”, i.e. belong to the anisotropic space H(div). Convex analysis and the theory of variational inequalities offer the framework for the theory of the continuous problem as proposed in [7]. We will closely follow the contribution [3] in which the FEM approximation is proposed and computable a posteriori error estimates provided. The discretization error as well as the regularization error are therein considered following a technique that was used in [5] in the “non–mixed” case. The paper outline is as follows: we present the mathematical formulation of the continuous problem paying attention to the mixed variational formulation leading to the FEM discretization; the discrete FEM spaces for stresses, plastic strains, displacements and plastic multipliers are introduced. Then we introduce the numerical algorithm for the solution of the elastoplastic problem. Rather than relying on classical predictor–corrector strategies, our approach takes full advantage of the variational structure of the problem. The nonlinear non–differentiable variational inequality is regularized to form a differentiable nonlinear problem that is solved by means of a recursive fixed–point algorithm or by a more classical Newton approach. As a numerical example we consider the Cook membrane problem under the action of a uniform load (rather than the more classical free–end boundary load). From a theoretical standpoint, there exist optimal error estimates concerning the convergence of the discrete non–smooth problem to the continuous one. Figures 1 shows the convergence patterns of stresses and 2 shows the plastic strain relative error versus the iteration number and the plastic strain components pxx and pxy whereas pyy is omitted since in 2D Tr(p) ≡ 0 implies pyy = −pxx that has been applied weakly using our formulation. We have presented a novel numerical approach for the numerical solution of elastoplastic problems that extends the approach presented in [6]. The core of the method is the numerical solution of a regularized mixed variational inequality that is tackled by a two–step algorithm consisting of a fixed–point method within which a standard Uzawa–type algorithm is nested. Convergence rates are numerically assessed along with the sensitivity of the approach to the choice of the regularization parameter ε. Comparisons with more standard approaches are also given to validate the proposed framework.