Energetic formulation of multiplicative elasto-plasticity using dissipation distances

Continuum Mechanics and Thermodynamics

Volumn 15, Issue 4, 2003, Pages 351-382

  Mielke, A ^a  

^a UNIVERSITY OF STUTTGART   (Germany)

Author keywords

Dissipation potential and distance; Finite strain elasto plasticity; Incremental variational problem; Rate independent inelasticity

Indexed keywords

EID: 0042360574     PISSN: 09351175     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00161-003-0120-x     Document Type: Article

References (54)

1. [AFF97] Acerbi, E., Fonseca, I., Fusco, N.: Regularity results for equilibria in a variational model for fracture. Proc. Roy. Soc. Edinburgh Sect. A 127(5), 889-902 (1997)
2. [Ber98] Bertram, A.: An alternative approach to finite plasticity based on material isomorphisms. Int. J. Plasticity 52, 353-374 (1998)
3. [CHM02] Carstensen, C., Hackl, K., Mielke, A.: Non-convex potentials and microstructures in finite-strain plasticity. Proc. Royal Soc. London, Ser. A 458, 299-317 (2002)
4. [CP01] Carstensen, C., Plecháč, P.: Numerical analysis of a relaxed variational model of hysteresis in two-phase solids. M2AN Math. Model. Numer. Anal. 35(5), 865-878 (2001)
5. [EM03] Efendiev, M., Mielke, A.: On the rate-independent limit of systems with dry friction and small viscosity. Preprint, Uni. Stuttgart (March 2003). Submitted to J. Convex Analysis
6. [FE89] Fedelich, B., Ehrlacher, A.: Sur un principe de minimum concernant des matériaux à comportement indépendant du temps physique. C. R. Acad. Sci. Paris Sér. II Méc. Phys. Chim. Sci. Univers Sci. Terre 308(16), 1391-1394 (1989)
7. [FE97] Fedelich, B., Ehrlacher, A.: An analysis of stability of equilibrium and of quasi-static transformations on the basis of the dissipation function. European J. Mech. A Solids 16(5), 833-855 (1997)
8. 1,p; a model for the interaction between fracture and damage. Calc. Var. Partial Differential Equations 3(4), 407-446 (1995)
9. [FM93] Francfort, G. A., Mango, J.-J.: Stable damage evolution in a brittle continuous medium. European J. Mech. A Solids 12, 149-189 (1993)
10. [FM98] Francfort, G., Marigo, J.-J.: Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46, 1319-1342 (1998)
11. [GMH02] Govindjee, S., Mielke, A., Hall, G.: The free-energy of mixing for n-variant martensitic phase transformations using quasi-convex analysis. J. Mech. Physics Solids 50, 1897-1922 (2002)
12. [GNS83] Germain, P., Nguyen, Q., Suquet, P.: Continuum thermodynamics. J. Applied Mechanics 50, 1010-1020 (1983)
13. [Gur00] Gurtin, M. E.: On the plasticity of single crystals: free energy, microforces, plastic-strain gradients. J. Mech. Phys. Solids 48(5), 989-1036 (2000)
14. [Hac97] Hackl, K.: Generalized standard media and variational principles in classical and finite strain elastoplasticity. J. Mech. Phys. Solids 45(5), 667-688 (1997)
15. [HH01] Hackl, K., Hoppe, U.: On the calculation of microstructures for inelastic materials using relaxed energies. In: Miehe, C. (ed) IUTAM Symposium on Computational Mechanics of Solids at Large Strains, Kluwer (2003), pp. 77-86
16. [HMM02] Hackl, K., Mielke, A., Mittenhuber, D.: Dissipation distances in multiplicative elastoplasticity. In: Wendland, W.L., Efendiev, M. (eds) Analysis and Simulation of Multifield Problems. Springer (2003). In press
17. [HN75] Halphen, B., Nguyen, Q. S.: Sur les matériaux standards généralisés. J. Mécanique 14, 39-63 (1975)
18. [KM00] Kuczma, M., Mielke, A.: Influence of hardening and inhomogeneity on internal loops in pseudoelasticity. ZAMM Z. Angew. Math. Mech. 80(5), 291-306 (2000)
19. [KMR02] Kočvara, M., Mielke, A., Roubíček, T.: A rate-independent approach to the delamination problem. In preparation (2003)
20. [KMS99] Kuczma, M., Mielke, A., Stein, E.: Modelling of hysteresis in two-phase systems. Archive of Mechanics (Warzawa) 51, 693-715 (1999)
21. [Lee69] Lee, E.: Elastic-plastic deformation at finite strains. J. Applied Mechanics 36, 1-6 (1969)
22. [Lub73] Lubliner, J.: On the structure of rate equations of materials with internal variables. Acta Mech. 17, 109-119 (1973)
23. [Man73] Mandel, J.: Equations constitutives et directeurs dans les milieux plastiques et viscoplastiques. Int. J. Solids Structures 9, 725-740 (1973)
24. [Mau92] Maugin, G. A.: The thermomechanics of plasticity and fracture. Cambridge: Cambridge University Press (1992)
25. [Mie96] Miehe, C.: Multisurface thermoplasticity for single crystals at large strains in terms of Eulerian vector updates. Int. J. Solids Struct. 33, 1981-2004 (1996)
26. [Mie02a] Mielke, A.: Finite elastoplasticity, Lie groups and geodesies on SL(d). In: Newton, P., Weinstein, A., Holmes, P. (editors): Geometry, Dynamics, and Mechanics, pages 61-90. Springer (2002)
27. [Mie02b] Mielke, A.: Necessary and sufficient conditions for polyconvexity of isotropic functions. IADM, U. Stuttgart, preprint (December 2002)
28. [Mie02c] Mielke, A.: Relaxation via Young measures of material models for rate-independent inelasticity. Preprint (February 2002). Submitted to J. Convex Analysis
29. [Mie03] Mielke, A.: Existence results in incremental elasto-plasticty with finite strains. In preparation (2003)
30. [ML01] Miehe, C., Lambrecht, M.: Analysis of microstructure development in shearbands by energy relaxation of incremental stress potentials: large-strain theory for generalized standard solids. Int. J. Numer. Meth. Eng. (2003), in press
31. [Mor83] Moreau, J.-J.: Application of convex analysis to the treatment of elastoplastic systems. In: Germain, P., Nayrolles, B. (editors): Applications of Methods of Functional Analysis to Problems in Mechanics. Springer (1983), pp. 56-89
32. [MS92] Miehe, C., Stein, E.: A canonical model of multiplicative elasto-plasticity. Formulation and aspects of numerical implementation. Europ. J. Mech. A/Solids 11, 25-43 (1992)
33. [MSL01] Miehe, C., Schotte, J., Lambrecht, M.: Homogenization of inelastic solid materials at finite strain based on incremental minimization princicples. Application to texture analysis of poly crystals. J. Mech. Physics Solids 50, 2123-2167 (2002)
34. [MSS99] Miehe, C., Schröder, J., Schotte, J.: Computational homogenization analysis in finite plasticity. Simulation of texture development in polycrystalline materials. Comput. Methods Appl. Mech. Engrg. 171, 387-418 (1999)
35. [MT99] Mielke, A., Theil, F.: A mathematical model for rate-independent phase transformations with hysteresis. In: Alber, H.-D., Balean, R., Farwig, R. (editors): Proceedings of the Workshop on "Models of Continuum Mechanics in Analysis and Engineering", pages 117-129. Shaker-Verlag (1999)
36. [MT01] Mielke, A., Theil, F.: On rate-independent hysteresis models. Nonl. Diff. Eqns. Appl. (NoDEA). To appear (2001)
37. [MTL98] Mielke, A., Theil, F., Levitas, V.: Mathematical formulation of quasistatic phase transformations with friction using an extremum principle. Preprint A8, Hannover (Sept. 1998)
38. [MTL02] Mielke, A., Theil, F., Levitas, V.: A variational formulation of rate-independent phase transformations using an extremum principle. Arch. Rational Mech. Anal. 162, 137-177 (2002)
39. [Nef00] Neff, P.: Visco-plastic approximations in finite plasticity for a model of small elastic strains. Part I: Modelling. TU Darmstadt, Preprint 2127 (October 2000)
40. [Nef01] Neff, P.: Visco-plastic approximations in finite plasticity for a model of small elastic strains. Part IIa: Local existence and uniqueness results. TU Darmstadt, Preprint 2138 (January 2001)
41. [OR99] Ortiz, M., Repetto, E.: Nonconvex energy minimization and dislocation structures in ductile single crystals. J. Mech. Phys. Solids 47(2), 397-462 (1999)
42. [ORS00] Ortiz, M., Repetto, E., Stainier, L.: A theory of subgrain dislocation structures. J. Mech. Physics Solids 48, 2077-2114 (2000)
43. [OS99] Ortiz, M., Stainier, L.: The variational formulation of viscoplastic constitutive updates. Comput. Methods Appl. Mech. Engrg. 171(3-4), 419-444 (1999)
44. [Ric71] Rice, J.: Inelastic constitutive relations for solids: an internal-variable theory and its applications to metal plasticity. J. Mech. Phys. Solids 19, 203-240 (1971)
45. [RK02a] Roubíček, T., Kružík, M.: Mesoscopical model for ferromagnetism with isotropic hardening. Caesar Preprint 37. Submitted to ZAMP (2002)
46. [RK02b] Roubíček, T., Kružík, M.: Mircrostructure evolution model in micromagnetics. Zeits. angew. math. Physik. In print (2002)
47. [Rou02a] Roubíček, T.: Evolution model for martensitic phase transformation in shape-memory alloys. Interfaces Free Bound. 4, 111-136 (2002)
48. [Rou02b] Roubíček, T.: Microstructure in ferromagnetics and its steady-state and evolution models. In Ruffing, A., Robnik, M. (editors): Bexbach Colloq. on Sciences. Shaker Verlag (2002)
49. [Sil01a] Šilhavý, M.: Dissipation postulates in finite-deformation plasticity. In Proc. of 15th AIMETA Congress of Theoretical and Applied Mechanics, page 10 pp (2001)
50. [Sil01b] Šilhaý, M.: On the nonconvex potentials for elastic-plastic materials. Preprint (Sept. 2001)
51. [Sim88] Simo, J.: A framework for finite strain elastoplasticity based on maximum plastic dissipation and the multiplicative decomposition. I. Continuum formulation. Comput. Methods Appl. Mech. Engrg. 66(2), 199-219 (1988)
52. [SO85] Simo, J., Ortiz, M.: A unified approach to finite deformation elastoplastic analysis based on the use of hyperelastic constitutive relations. Comput. Methods Appl. Mech. Engrg. 49, 221-245 (1985)
53. [TS76] Teodosiu, C., Sidoroff: A theory of finite elastoviscoplasticity of single crystals. Int. J. Eng. Science 14, 165-176 (1976)
54. [ZW87] Ziegler, H., Wehrli, C.: The derivation of constitutive relations from the free energy and the dissipation function. In: Advances in Applied Mechanics, Vol. 25, pages 183-237. Orlando, FL: Academic Press (1987)