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Energetic solutions to rate-independent large-strain elasto-plastic evolutions driven by discrete dislocation flow
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Rindler, Filip (2023) Energetic solutions to rate-independent large-strain elasto-plastic evolutions driven by discrete dislocation flow. Journal of the European Mathematical Society . doi:10.4171/jems/1424 ISSN 1435-9855. (In Press)
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Official URL: https://doi.org/10.4171/jems/1424
Abstract
This work rigorously implements a recent model of large-strain elasto-plastic evolution in single crystals where the plastic flow is driven by the movement of discrete dislocation lines. The model is geometrically and elastically nonlinear, that is, the total deformation gradient splits multiplicatively into elastic and plastic parts, and the elastic energy density is polyconvex. There are two internal variables: The system of all dislocations is modeled via one-dimensional boundaryless integral currents, whereas the history of plastic flow is encoded in a plastic distortion matrix-field. As our main result, we construct an energetic solution in the case of a rate-independent flow rule. Besides the classical stability and energy balance conditions, our notion of solution also accounts for the movement of dislocations and the resulting plastic flow. Because of the path-dependence of plastic flow, a central role is played by so-called “slip trajectories”, that is, the surfaces traced out by moving dislocations, which we represent as integral 2-currents in space-time. The proof of our main existence result further crucially rests on careful a priori estimates via a nonlinear Gronwall-type lemma and a rescaling of time. In particular, we have to account for the fact that the plastic flow may cause the coercivity of the elastic energy functional to decay along the evolution, and hence the solution may blow up in finite time.
Item Type: | Journal Article | ||||||
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Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||
Journal or Publication Title: | Journal of the European Mathematical Society | ||||||
Publisher: | European Mathematical Society Publishing House | ||||||
ISSN: | 1435-9855 | ||||||
Official Date: | 26 February 2023 | ||||||
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DOI: | 10.4171/jems/1424 | ||||||
Status: | Peer Reviewed | ||||||
Publication Status: | In Press | ||||||
Access rights to Published version: | Open Access (Creative Commons) | ||||||
Copyright Holders: | © European Mathematical Society | ||||||
Date of first compliant deposit: | 23 January 2023 | ||||||
RIOXX Funder/Project Grant: |
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