State update algorithm for isotropic elastoplasticity by incremental energy minimization

Abstract

Summary An original state update algorithm for the numerical integration of rate independent small strain elastoplastic constitutive models, treating in a unified manner a wide class of yield functions depending on all three stress invariants, is proposed. The algorithm is based on an incremental energy minimization approach, in the framework of generalized standard materials with convex free‐energy and dissipation potential. Under the assumption of isotropic material behavior, implying coaxiality of trial stress, increment of plastic strain, and updated stress, the problem is reduced from dimension six to three. Then, exploiting the cylindrical tensor basis associated with Haigh–Westergaard coordinates, the problem is recast in terms of two nested scalar equations. The proposed algorithm (i) exhibits global convergence even for yield functions with difficult features, such as not being defined on the whole stress space, or implying high‐curvature points of the yield domain, and (ii) requires no matrix inversion. After the tensor reconstruction of the unknowns, a simple expression for the algorithmic consistent material tangent is derived. The algorithm is validated by comparison with benchmark semi‐analytic solutions. Numerical results on single material points and finite element simulations are reported for assessing its accuracy, robustness, and efficiency. A Matlab implementation is provided as supplementary material. Copyright © 2015 John Wiley & Sons, Ltd.

An Overview of Mixed Finite Elements for the Analysis of Inelastic Bidimensional Structures

“…Initially proposed as a generalization of classical plasticity theory (in that context the normal dissipation hypothesis leads to associative flow law, expressed by the principle of maximum dissipation [32,81,37]), the model of generalized standard material has been then successfully applied and extended to complex material behaviors (e.g., see [58] and references therein). In particular, it has been shown that, in a time-discrete setting, the evolution of a generalized standard material follows the minimizing path of an incremental energy, given by the sum of free energy and of dissipation potential, with respect to the internal variables [59,70,60,62,68,65]. The introduction of the material incremental energy allows a unified treatment of the variational formulations to be discussed in Section 3.…”

“…To this end, the material nonlinearity is restricted to the class of generalized standard materials in the sense of Halphen and Nguyen [36], with special consideration for hardening associative plastic behavior. In particular, the material results to be equipped with an (incremental) energetic structure [59,70,60,62,68,65], which is exploited for a synthetical exposition. Though many of the results here presented are also valid in the threedimensional case and might be extended to the finite strain framework, the discussion is referred to bidimensional structures under small strain assumption.…”

A mixed tetrahedral element with nodal rotations for large-displacement analysis of inelastic structures

“…Symmetric and skew‐symmetric parts of stress tensor are denoted by σ and τ , respectively. The material constituting the body is endowed with a stress potential ψ , such as Helmholtz free energy for hyperelastic constitutive behavior or a suitable incremental energy for standard dissipative constitutive behavior . As usual, the boundary is partitioned into two disjoint sets and , where Dirichlet and Neumann conditions are imposed, respectively.…”

A mixed finite element for the nonlinear analysis of in‐plane loaded masonry walls

“…Following such an observation, a generalization of that approach to 2D problems has been proposed . Specifically, in the framework of membrane structures comprising a hardening generalized standard material, a discontinuous and piecewise constant strain field interpolation has been considered over suitable element subdomains, proving the capability of the formulation to accurately capture structural inelastic effects even using coarse discretizations. However, a further generalization to membrane problems involving a damaging material model, as needed for simulating the behavior of masonry structures, is still missing.…”