Journal of Convex Analysis 28 (2021), No. 2, 395--428
Copyright Heldermann Verlag 2021
Regularity Results for Two Standard Models in Elasto-Perfect-Plasticity Theory with Hardening
Miroslav Bulícek
Mathematical Institute, Charles University, 186 75 Prague, Czech Republic
mbul8060@karlin.mff.cuni.cz
Jens Frehse
Institute of Applied Mathematics, University of Bonn, 53121 Bonn, Germany
aaa@iam.uni-bonn.de
Maria Specovius-Neugebauer
Fachbereich Mathematik, Universität Kassel, 34132 Kassel, Germany
specovi@mathematik.uni-kassel.de
We consider two most studied standard models in the theory of elasto-plasticity with hardening in arbitrary dimension d ≥ 2, namely, the kinematic hardening and the isotropic hardening problem. While the existence and uniqueness of the solution is very well known, the optimal regularity up to the boundary remains an open problem. Here, we show that in the interior we have Sobolev regularity for the stress and hardening while for their time derivatives we have the "half" derivative with the spatial and time variable. This was well known for the limiting problem but we show that these estimates are uniform and independent of the order of approximation. The main novelty consist of estimates near the boundary. We show that for the stress and the hardening parameter, we control tangential derivative in the Lebesgue space L2, and for time derivative of the stress and the hardening we control the "half" time derivative and also spatial tangential derivative. Last, for the normal derivative, we show that the stress and the hardening have the 3/5 derivative with respect to the normal and for the time derivative of the stress and the hardening we show they have the 1/5 derivative with respect to the normal direction, provided we consider the kinematic hardening or near the Dirichlet boundary. These estimates are independent of the dimension. In case, we consider the isotropic hardening near the Neumann boundary we shall obtain Wα,2 regularity for the stress and the hardening with some α > 1/2 depending on the dimension and Wβ,2 with some β > 1/6 for the time derivative of the stress and the hardening. Finally, in case of kinematic hardening the same regularity estimate holds true also for the velocity gradient.
Keywords: Elasto-perfect-plasticity with hardening, Cauchy stress, boundary regularity, fractional regularity.
MSC: 74G40, 35Q72, 74C05, 74G10.
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