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A Variational Model for Plastic Slip and Its Regularization via Γ-Convergence

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Abstract

A variational model is presented able to interpret the onset of plastic deformations, here modeled as displacement jumps occurring along slip surfaces at constant yielding stress. The corresponding strain energy functional, leading to a free-discontinuity problem set in the space of SBV functions, is then approximated by a sequence of regularized elliptic functionals following the seminal work by Ambrosio and Tortorelli (Commun. Pure Appl. Math. 43, 999–1036, 1990) within the framework of Γ-convergence. Comparisons between the results obtainable with the free-discontinuity model and its regularized approximation, in terms of stability of the pure elastic phase, irreversibility of plastic slip and response under unloading, are presented, in general, for the 2-D case of antiplane shear and exemplified, in particular, for the 1-D case.

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Notes

  1. Later on, the Γ-convergence result was demonstrated also in the vectorial case for the elastic energy by Chambolle [15] and for a certain class of general integrands by Focardi [28]. An exhaustive list of references for this kind of problems can be found in [5].

  2. The interest is not limited to fracture mechanics, but similar regularizations can be found in front capturing methods in gas dynamics (level set method) or phase field methods in the theory of defects.

  3. To this respect, we should mention the almost unknown work by Nakanishi, and [43] in particular.

  4. Here, to be rigorous, we should refer to the relaxed minimization problem (3.3), but for the sake of simplicity we will drop this distinction that does not produce substantial differences.

  5. To be precise, one should define also for the 1-D case the counterpart of the relaxed problem (3.3), but for simplicity of analysis we will assume that the distinction is understood.

  6. In the context of the pure Ambrosio-Tortorelli functional, it has been shown in [29] that critical points of the 1d-Ambrosio-Tortorelli model with Dirichlet data do converge to (adequately defined) critical points of the free discontinuity functional. However, our functional is substantially different; the mathematical proof of convergence of local minimizers is delicate and goes beyond the scope of this paper.

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Acknowledgements

L.Am. gratefully acknowledges the support of the MIUR PRIN06 grant. G.R.C. would like to acknowledge the partial support of the Italian MURST under the PRIN-2008 program “Structural use of glass”. This work was done while A.Le. was affiliated to the mathematical research center E. De Giorgi and he wishes to thank all the members of the center and the Scuola Normale Superiore of Pisa for their kind hospitality.

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Authors and Affiliations

  1. Scuola Normale Superiore, Piazza dei Cavalieri 7, 56127, Pisa, Italy

    Luigi Ambrosio

  2. LJLL, Université Denis Diderot (Paris 7), Rue Thomas-Mann 5, 75205, Paris cedex 13, France

    Antoine Lemenant

  3. Università degli Studi di Parma, Viale G. Usberti 181/A, 43100, Parma, Italy

    Gianni Royer-Carfagni

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  1. Luigi Ambrosio
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Correspondence to Gianni Royer-Carfagni.

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Ambrosio, L., Lemenant, A. & Royer-Carfagni, G. A Variational Model for Plastic Slip and Its Regularization via Γ-Convergence. J Elast 110, 201–235 (2013). https://doi.org/10.1007/s10659-012-9390-5

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