A nonlocal and fully nonlineardegenerate parabolic system from strain-gradient plasticity

Michiel Bertsch; Roberta Dal Passo; Lorenzo Giacomelli; Giuseppe Tomassetti

doi:10.3934/dcdsb.2011.15.15

Article Contents

2011, Volume 15, Issue 1: 15-43. Doi: 10.3934/dcdsb.2011.15.15

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A nonlocal and fully nonlinear degenerate parabolic system from strain-gradient plasticity

1.
Istituto per le Applicazioni del Calcolo "M. Picone”, Consiglio Nazionale delle Ricerche, Via dei Taurini 19, 00185 Roma
2.
Dipartimento di Matematica, Università di Roma "Tor Vergata”, Via della Ricerca Scientifica 1, 00133 Roma
3.
Dipartimento di Scienze di Base e Applicate per l’Ingegneria, Università di Roma "La Sapienza”, Via Scarpa 16, 00161 Roma
4.
Dipartimento di Ingegneria Civile, Università di Roma "Tor Vergata”, Via Politecnico 1, 00133 Roma

Received: May 31, 2009

Revised: July 31, 2010

Published: June 2011

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Abstract

We consider a system of partial differential equations which describes anti-plane shear in the context of a strain-gradient theory of plasticity proposed by Gurtin in [6]. The problem couples a fully nonlinear degenerate parabolic system and an elliptic equation. It features two types of degeneracies: the first one is caused by the nonlinear structure, the second one by the dependence of the principal part on twice the curl of a planar vector field. Furthermore, the elliptic equation depends on the divergence of such vector field - which is not controlled by twice the curl - and the boundary conditions suggested in [6] are of mixed type.
To overcome the latter complications we use a suitable, time-dependent representation of a divergence-free vector field which plays the role of the elastic stress. To handle the nonlinearities, by a suitable reformulation of the problem we transform the original system into one satisfying a monotonicity property which is more "robust" than the gradient flow structure inherited as an intrinsic feature of the mechanical model. These two insights make it possible to prove existence and uniqueness of a solution to the original system.

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Mathematics Subject Classification: 35K51, 35K65, 35K99, 74C10.

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