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Modelling, Analysis and Computation in Plasticity

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Abstract

The typical problem in the mechanics of deformable solids comprises a mathematical model in the form of systems of partial differential equations or inequalities. Subsequent investigations are then concerned with analysis of the model to determine its well-posedness, followed by the development and implementation of algorithms to obtain approximate solutions to problems that are generally intractable in closed form. These processes of modelling, analysis, and computation are discussed with a focus on the behaviour of elastic-plastic bodies; these are materials which exhibit path-dependence and irreversibility in their behaviour. The resulting variational problem is an inequality that is not of standard elliptic or parabolic type. Properties of this formulation are reviewed, as are the conditions under which fully discrete approximations converge. A solution algorithm, motivated by the predictor-corrector algorithms that are common in elastoplastic problems, is presented and its convergence properties summarized. An important extension of the conventional theory is that of straingradient plasticity, in which gradients of the plastic strain appear in the formulation, and which includes a length scale not present in the conventional theory. Some recent results for strain-gradient plasticity are presented, and the work concludes with a brief description of current investigations.

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Acknowledgements

The author thanks the referees for their time and useful comments.

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Authors and Affiliations

  1. Department of Mathematics and Applied Mathematics, University of Cape Town, 7701, Rondebosch, South Africa

    B. Daya Reddy

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  1. B. Daya Reddy
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Correspondence to B. Daya Reddy.

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Supported by National Research Foundation through the South African Chair in Computational Mechanics (Grant No. 47584)

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Reddy, B.D. Modelling, Analysis and Computation in Plasticity. Acta. Math. Sin.-English Ser. 35, 64–82 (2019). https://doi.org/10.1007/s10114-018-7477-z

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  • DOI: https://doi.org/10.1007/s10114-018-7477-z

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