Ebobisse, François; Neff, Patrizio Existence and uniqueness for rate-independent infinitesimal gradient plasticity with isotropic hardening and plastic spin. (English) Zbl 1257.74023 Math. Mech. Solids 15, No. 6, 691-703 (2010). Summary: Existence and uniqueness for infinitesimal dislocation based rate-independent gradient plasticity with linear isotropic hardening and plastic spin are shown using convex analysis and variational inequality methods. The dissipation potential is extended non-uniquely from symmetric plastic rates to non-symmetric plastic rates and three qualitatively different formats for the dissipation potential are distinguished. Cited in 20 Documents MSC: 74C05 Small-strain, rate-independent theories of plasticity (including rigid-plastic and elasto-plastic materials) 74G25 Global existence of solutions for equilibrium problems in solid mechanics (MSC2010) 74G30 Uniqueness of solutions of equilibrium problems in solid mechanics 35Q74 PDEs in connection with mechanics of deformable solids Keywords:plasticity; gradient plasticity; dislocations; plastic spin; quasistatic evolution; rate-independent process; variational inequality PDFBibTeX XMLCite \textit{F. Ebobisse} and \textit{P. Neff}, Math. Mech. Solids 15, No. 6, 691--703 (2010; Zbl 1257.74023) Full Text: DOI References: [1] Alber, H.D., Materials with Memory. Initial-Boundary Value Problems for Constitutive Equations with Internal Variables, Lecture Notes in Mathematics, Vol. 1682 (1998) · Zbl 0977.35001 [2] Han, W., Mathematical Theory and Numerical Analysis (1999) · Zbl 0926.74001 [3] Simo, J.C., Interdisciplinary Applied Mathematics, Vol. 7 (1998) [4] Alber, H.D., Math. Models Methods Appl. Sci. 17 (2) pp 189– (2007) · Zbl 1114.74007 [5] Menzel, A., J. Mech. Phys. Solids 48 pp 1777– (2000) · Zbl 0999.74029 [6] Gurtin, M.E., J. Mech. Phys. Solids 48 pp 989– (2000) · Zbl 0988.74021 [7] Gurtin, M.E., J. Mech. Phys. Solids 53 pp 1– (2005) · Zbl 1084.74009 [8] Gurtin, M.E., Part I: Small deformations. J. Mech. Phys. Solids 53 pp 1624– (2005) [9] Aifantis, E.C., Mechanics of Materials 35 pp 259– (2003) [10] Svendsen, B., J. Mech. Phys. Solids 50 (25) pp 1297– (2002) · Zbl 1071.74554 [11] Bardella, L., J. Mech. Phys. Solids 54 pp 128– (2006) · Zbl 1120.74387 [12] Reddy, B.D., Int. J. Plasticity 24 pp 55– (2008) · Zbl 1139.74009 [13] Ebobisse, F. , McBride, A.T. and Reddy, B.D. On the mathematical formulations of a model of gradient plasticity, in ed. B.D. Reddy, IUTAM-Symposium on Theoretical, Modelling and Computational Aspects of Inelastic Media (in Cape Town, 2008), pp. 117-128. Springer , Berlin, 2008. · Zbl 1209.74018 [14] Mielke, A., Math. Meth. Appl. Sci. 29 pp 1393– (2006) · Zbl 1096.74017 [15] Mielke, A. A mathematical framework for generalized standard materials in the rate-independent case, in ed. R. Helmig, A. Mielke, and B. Wohlmuth , Multifield Problems in Solid and Fluid Mechanics, Lecture Notes in Applied and Computational Mechanics, Vol. 28, pp. 399-428. Springer , Heidelberg, 2006. · Zbl 1298.74006 [16] Neff, P., Math. Mod. Meth. Appl. Sci. 19 (2) pp 1– [17] Neff, P. Uniqueness of strong solutions in infinitesimal perfect gradient plasticity with plastic spin, in ed. B.D. Reddy, IUTAM-Symposium on Theoretical, Modelling and Computational Aspects of Inelastic Media (in Cape Town, 2008), pp. 129-140. Springer, Berlin, 2008. · Zbl 1209.74020 [18] Neff, P., Int. J. Num. Meth. Engrg. 77 (3) pp 414– (2009) · Zbl 1155.74316 [19] Svendsen, B., Z. Angew. Math. Mech [20] Neff, P., Technische Mechanik 28 (1) pp 13– (2008) [21] Gudmundson, P., J. Mech. Phys. Solids 52 pp 1379– (2004) · Zbl 1114.74366 [22] Bardella, L., Int. J. Plasticity 23 pp 296– (2007) · Zbl 1127.74319 [23] Djoko, J.K., Comp. Meth. Appl. Mech. Engrg. 196 (37) pp 3881– (2007) · Zbl 1173.74410 [24] Djoko, J.K., Comp. Meth. Appl. Mech. Engrg. 197 (1) pp 1– (2007) · Zbl 1169.74595 [25] Nesenenko, S., SIAM J. Math. Anal. [26] Nye, J.F., Acta Metall. 1 pp 153– (1953) [27] Kröner, E., Arch. Rat. Mech. Anal. 3 pp 97– (1959) · Zbl 0085.38601 [28] Gupta, A., Math. Mech. Solids 12 (6) pp 583– (2007) · Zbl 1133.74009 [29] Mielke, A. Finite elastoplasticity, Lie groups and geodesics on SL(d), in ed. P. Newton, Geometry, Mechanics and Dynamics. Volume in Honour of the 60th Birthday of J.E. Marsden, pp. 61-90. Springer-Verlag , Berlin, 2002. · Zbl 1146.74309 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.