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An efficient return mapping algorithm for elastoplasticity with exact closed form solution of the local constitutive problem

Fabio De Angelis (Department of Structures for Engineering and Architecture, University of Naples Federico II, Naples, Italy)
Robert L. Taylor (Department of Civil and Environmental Engineering, University of California, Berkeley, California, USA)

Engineering Computations

ISSN: 0264-4401

Article publication date: 2 November 2015

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Abstract

Purpose

The purpose of this paper is to present an efficient return mapping algorithm for elastoplastic constitutive problems of ductile metals with an exact closed form solution of the local constitutive problem in the small strain regime. A Newton Raphson iterative method is adopted for the solution of the boundary value problem.

Design/methodology/approach

An efficient return mapping algorithm is illustrated which is based on an elastic predictor and a plastic corrector scheme resulting in an implicit and accurate numerical integration method. Nonlinear kinematic hardening rules and linear isotropic hardening rules are used to describe the components of the hardening variables. In the adopted algorithmic approach the solution of the local constitutive equations reduces to only one straightforward nonlinear scalar equation.

Findings

The presented algorithmic scheme naturally leads to a particularly simple form of the nonlinear scalar equation which ultimately scales down to an algebraic (polynomial) equation with a single variable. The straightforwardness of the present approach allows to find the analytical solution of the algebraic equation in a closed form. Further, the consistent tangent operator is derived as associated with the proposed algorithmic scheme and it is shown that the proposed computational procedure ensures a quadratic rate of asymptotic convergence when used with a Newton Raphson iterative method for the global solution procedure.

Originality/value

In the present approach the solution of the algebraic nonlinear equation is found in a closed form and accordingly no iterative method is required to solve the problem of the local constitutive equations. The computational procedure ensures a quadratic rate of asymptotic convergence for the global solution procedure typical of computationally efficient solution schemes. In the paper it is shown that the proposed algorithmic scheme provides an efficient and robust computational solution procedure for elastoplasticity boundary value problems. Numerical examples and computational results are reported which illustrate the effectiveness and robustness of the adopted integration algorithm for the finite element analysis of elastoplastic structures also under elaborate loading conditions.

Keywords

Acknowledgements

The authors wish to thank the reviewers of the paper for the stimulating comments and valuable suggestions.

Citation

De Angelis, F. and Taylor, R.L. (2015), "An efficient return mapping algorithm for elastoplasticity with exact closed form solution of the local constitutive problem", Engineering Computations, Vol. 32 No. 8, pp. 2259-2291. https://doi.org/10.1108/EC-06-2014-0138

Publisher

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Emerald Group Publishing Limited

Copyright © 2015, Emerald Group Publishing Limited

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