| Issue |
ESAIM: COCV
Volume 27, 2021
Regular articles published in advance of the transition of the journal to Subscribe to Open (S2O). Free supplement sponsored by the Fonds National pour la Science Ouverte
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|---|---|---|
| Article Number | S1 | |
| Number of page(s) | 32 | |
| DOI | https://doi.org/10.1051/cocv/2018004 | |
| Published online | 01 March 2021 | |
Duality results and regularization schemes for Prandtl–Reuss perfect plasticity*
Department of Mathematics, Humboldt-Universität zu Berlin,
Unter d. Linden 6,
D-10099
Berlin, Germany.
** Corresponding author: hint@math.hu-berlin.de
Received:
27
January
2017
Accepted:
1
January
2018
We consider the time-discretized problem of the quasi-static evolution problem in perfect plasticity posed in a non-reflexive Banach space. Based on a novel equivalent reformulation in a reflexive Banach space, the primal problem is characterized as a Fenchel dual problem of the classical incremental stress problem. This allows to obtain necessary and sufficient optimality conditions for the time-discrete problems of perfect plasticity. Furthermore, the consistency of a primal-dual stabilization scheme is proven. As a consequence, not only stresses, but also displacements and strains are shown to converge to a solution of the original problem in a suitable topology. The corresponding dual problem has a simpler structure and turns out to be well-suited for numerical purposes. For the resulting subproblems an efficient algorithmic approach in the infinite-dimensional setting based on the semismooth Newton method is proposed.
Mathematics Subject Classification: 74C05 / 49M15 / 49K20 / 49M29
Key words: Perfect plasticity / Prandtl–Reuss plasticity / small-strain / Fenchel duality / semismooth Newton
This research was carried out in the framework of Matheon supported by the Einstein Foundation Berlin within the ECMath projects OT1, SE5 and SE15 as well as project A-AP24. The authors further gratefully acknowledge the support of the DFG through the DFG-SPP 1962: Priority Programme “Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization” within Projects 10, 11 and 13.
© EDP Sciences, SMAI 2021
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