Keywords: elastoplasticity, mixed formulation, path-following analysis.

A preeminent concern of the finite element literature is the formulation of high-performance finite elements. In the elastic field the number of the proposed alternative finite element formulations is large and different techniques have been evaluated. In the plastic field the formulation of high-performance elements has been focused on the use of mixed or generalized variational frameworks. The researchers' attention was principally directed to the use of non-conventional formulations while the modeling of the plastic part of the constitutive law has not received so much attention and often a pointwise enforcement of the material constitutive laws at the element Gauss points is used. As shown in [1], this choice affects the plastic response of the element unfavorably and, as a consequence, the element performance.

The aim of this paper is to contribute to the deepening of the insight in the formulation of enhanced finite elements for elastoplastic applications. The variational framework used includes the weak statement of the equilibrium equations, of the compatibility conditions and of the plastic loading-unloading conditions. In this context the finite element requires the interpolation of three fields: displacement, stress and plastic multiplier. The first two interpolations can be the same used in the elastic context, than the need to start always with a good elastic implementation of the element, [2]. The interpolation of the plastic multiplier instead can be suitably enriched in order to significantly improve the quality of the element response in the plastic phase.

In the present research some enriched patterns of the plastic multiplier have been tested showing that a better accuracy can be achieved by also containing the computational costs. At the element level in fact the closest point projection problem used in the integration of the elastoplastic relationships, becomes a convex optimization problem subjected to a number of constrains equals to the number of the discrete parameters used for the plastic multiplier interpolation. The problems is computationally very small and its solution could actually be performed by using standard, but also effective and robust, optimization algorithms.

A four node mixed element with two different interpolations of the plastic multiplier will be proposed for the analysis of two-dimensional elastoplastic solids. Both demonstrate a better behavior than the usual compatible elements but also than a previously proposed mixed element of the same kind using a simple constant interpolation for the plastic multiplier.

1

A. Bilotta, R. Casciaro, "A high performance element for the analysis of 2D elastoplastic continua", Comput. Methods Appl. Mech. Engrg., 196, 818-828, 2007. doi:10.1016/j.cma.2006.06.009

2

T.T.H. Pian, K. Sumihara, "Rational approach for assumed stress finite elements", Int. J. Numer. Methods Engrg., 20, 1685-1695, 1984. doi:10.1002/nme.1620200911

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