Small-strain heterogeneous elastoplasticity revisited. (English) Zbl 1396.74036
Summary: The elastoplastic quasi-static evolution of a multiphase material – a material with a pointwise varying yield surface and elasticity tensor, together with interfaces between the phases – is revisited in the context of conservative globally minimizing movements. Existence is shown, and classical evolutions are recovered under natural constraints on the plastic dissipation potential. Special attention is paid to the interfaces where the correct dissipation has to be enforced on the interfaces. Further, the evolution is shown to be a limit of that obtained for a model with linear isotropic hardening as the hardening becomes vanishingly small. The duality between plastic strains and admissible stresses is also revisited for Lipschitz boundaries, and its role in deriving a classical evolution is circumscribed.
MSC:
| 74C05 | Small-strain, rate-independent theories of plasticity (including rigid-plastic and elasto-plastic materials) |
| 74E05 | Inhomogeneity in solid mechanics |
| 74H20 | Existence of solutions of dynamical problems in solid mechanics |
| 74A40 | Random materials and composite materials |
| 35Q74 | PDEs in connection with mechanics of deformable solids |
Keywords:
interface dissipation law; existence; quasi-static evolution; multiphase material; plastic dissipation potential; linear isotropic hardeningReferences:
| [1] | Ambrosio, Fine properties of functions with bounded deformation, Arch. Rational Mech. Anal. 139 pp 201– (1997) · Zbl 0890.49019 · doi:10.1007/s002050050051 |
| [2] | Ambrosio, Oxford Mathematical Monographs, in: Functions of bounded variation and free discontinuity problems (2000) |
| [3] | Anzellotti, Dynamical evolution of elasto-perfectly plastic bodies, Appl. Math. Optim. 15 pp 121– (1987) · Zbl 0616.73047 · doi:10.1007/BF01442650 |
| [4] | Dal Maso, Quasistatic evolution problems for linearly elastic-perfectly plastic materials, Arch. Ration. Mech. Anal. 180 pp 237– (2006) · Zbl 1093.74007 · doi:10.1007/s00205-005-0407-0 |
| [5] | Demengel, Savoirs Actuels (Les Ulis), in: Espaces fonctionnels. Utilisation dans la résolution des équations aux dérivées partielles (2007) |
| [6] | Francfort , G. A. Giacomini , A. |
| [7] | Hill, The mathematical theory of plasticity (1950) · Zbl 0041.10802 |
| [8] | Johnson, Existence theorems for plasticity problems, J. Math. Pures Appl. (9) 55 pp 431– (1976) · Zbl 0351.73049 |
| [9] | Johnson, On plasticity with hardening, J. Math. Anal. Appl. 62 pp 325– (1978) · Zbl 0373.73049 · doi:10.1016/0022-247X(78)90129-4 |
| [10] | Kohn, Dual spaces of stresses and strains, with applications to Hencky plasticity, Appl. Math. Optim. 10 pp 1– (1983) · Zbl 0532.73039 · doi:10.1007/BF01448377 |
| [11] | Lubliner, Plasticity theory (1990) |
| [12] | Mainik, Existence results for energetic models for rate-independent systems, Calc. Var. Partial Differential Equations 22 pp 73– (2005) · Zbl 1161.74387 · doi:10.1007/s00526-004-0267-8 |
| [13] | Mielke , A. Evolution of rate-independent systems Handbook of differential equations: Evolutionary equations II 461 559 North-Holland Amsterdam 2005 |
| [14] | Nečas, Séminaire de mathématiques supérieures, Équations aux derivées partielles pp 102– (1966) |
| [15] | Solombrino, Quasistatic evolution problems for nonhomogeneous elastic plastic materials, J. Convex Anal. 16 pp 89– (2009) · Zbl 1166.74006 |
| [16] | Spector, Simple proofs of some results of Reshetnyak, Proc. Amer. Math. Soc. 139 pp 1681– (2011) · Zbl 1215.49025 · doi:10.1090/S0002-9939-2010-10593-2 |
| [17] | Suquet, Un espace fonctionnel pour les équations de la plasticité, Ann. Fac. Sci. Toulouse Math. (5) 1 pp 77– (1979) · Zbl 0405.46027 · doi:10.5802/afst.531 |
| [18] | Suquet, Sur les équations de la plasticité: existence et régularité des solutions, J. Mécanique 20 pp 3– (1981) |
| [19] | Suquet, International Center for Mechanical Sciences, Courses and Lectures IV, 302, in: Nonsmooth mechanics and applications pp 278– (1988) |
| [20] | Temam, Méthodes Mathématiques de l’Informatique, 12, in: Problèmes mathématiques en plasticité (1983) |
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