On the existence of minimisers for strain-gradient single-crystal plasticity

0481468 - ÚTIA 2019 RIV DE eng J - Článek v odborném periodiku
Anguige, K. - Dondl, P. - Kružík, Martin
On the existence of minimisers for strain-gradient single-crystal plasticity.
ZAMM-Zeitschrift fur Angewandte Mathematik und Mechanik. Roč. 98, č. 3 (2018), s. 431-447. ISSN 0044-2267. E-ISSN 1521-4001
Grant CEP: GA ČR GA14-15264S; GA ČR(CZ) GF16-34894L
Institucionální podpora: RVO:67985556
Klíčová slova: existence of minimizers * plasticity
Obor OECD: Pure mathematics
Impakt faktor: 1.467, rok: 2018
http://library.utia.cas.cz/separaty/2017/MTR/kruzik-0481468.pdf

We prove the existence of minimisers for a family of models related to the single-slip-to-single-plane relaxation of single-crystal, strain-gradient elastoplasticity with L p -hardening penalty. In these relaxed models, where only one slip-plane normal can be activated at each material point, the main challenge is to show that the energy of geometrically necessary dislocations is lower-semicontinuous along bounded-energy sequences which satisfy the single-plane condition, meaning precisely that this side condition should be preserved in the weak L p -limit. This is done with the aid of an ‘exclusion’ lemma of Conti & Ortiz, which essentially allows one to put a lower bound on the dislocation energy at interfaces of (single-plane) slip patches, thus precluding fine phase-mixing in the limit. Furthermore, using div-curl techniques in the spirit of Mielke & Müller, we are able to show that the usual multiplicative decomposition of the deformation gradient into plastic and elastic parts interacts with weak convergence and the single-plane constraint in such a way as to guarantee lower-semicontinuityo fthe(polyconvex)elasticenergy,andhencethetotalelasto-plasticenergy, givensufficient(p > 2) hardening, thus delivering the desired result.
Trvalý link: http://hdl.handle.net/11104/0277042