Regularity up to the boundary for nonlinear elliptic systems arising in time-incremental infinitesimal elasto-plasticity

SIAM Journal on Mathematical Analysis

Volumn 40, Issue 1, 2008, Pages 21-43

  Neff, Patrizio ^a   Knees, Dorothee ^b  

^a DARMSTADT UNIVERSITY OF TECHNOLOGY   (Germany)

^b WEIERSTRASS INSTITUTE FOR APPLIED ANALYSIS AND STOCHASTICS   (Germany)

Author keywords

Error estimates; Higher global regularity; Perfect plasticity; Polar materials; Quasi linear elliptic systems; Time increments

Indexed keywords

ELASTOPLASTICITY; NON NEWTONIAN FLOW; PLASTIC DEFORMATION; PLASTICITY; PLASTICITY TESTING; PLASTICS;

ERROR ESTIMATES; HIGHER GLOBAL REGULARITY; PERFECT PLASTICITY; POLAR MATERIALS; QUASI-LINEAR ELLIPTIC SYSTEMS; TIME INCREMENTS;

FUNCTIONAL POLYMERS;

EID: 61849100952     PISSN: 00361410     EISSN: None     Source Type: Journal    
DOI: 10.1137/070695824     Document Type: Article

References (68)

1. H. D. ALBER, Materials with Memory. Initial-Boundary Value Problems for Constitutive Equations with Internal Variables, Lecture Notes in Math. 1682, Springer, Berlin, 1998.
2. J. ALBERTY, C. CARSTENSEN, AND D. ZARRABI, Adaptive numerical analysis in primal elastoplasticity with hardening, Comput. Methods Appl. Mech. Engrg., 171 (1999), pp. 175-204.
3. G. ANZELLOTTI AND S. LUCKHAUS, Dynamical evolution of elasto-perfectly plastic bodies, Appl. Math. Optim., 15 (1987), pp. 121-140.
4. J. P. BARDET, Observations on the effects of particle rotations on the failure of idealized granular materials, Mech. Mater., 18 (1994), pp. 159-182.
5. 1 regularity, in Boundary Value Problems for Partial Differential Equations and Applications, J.-L. Lions and C. Baiocchi, eds., RMA Res. Notes Appl. Math. 29, Masson, Paris, 1993, pp. 3-25.
6. A. BENSOUSSAN AND J. FREHSE, Regularity Results for Nonlinear Elliptic Systems and Applications, Appl. Math. Sci. 151, Springer, Berlin, 2002.
7. M. BILDHAUER AND M. FUCHS, Smoothness of weak solutions of the Ramberg/Osgood equations on plane domains, ZAMM Z. Angew. Math. Mech., 87 (2007), pp. 70-76.
8. S. C. BRENNER AND L. R. SCOTT. The Mathematical Theory of Finite Element Methods, Springer, Heidelberg, 1994.
9. G. CAPRIZ, Continua with Microstructure, Springer, Heidelberg, 1989.
10. C. CARSTENSEN AND S. MÜLLER, Local stress regularity in scalar nonconvex variational problems, SIAM J. Math. Anal., 34 (2002), pp. 495-509
11. K. CHELMIŃSKI, Coercive approximation of viscoplasticity and plasticity, Asymptot. Anal., 26 (2001), pp. 105-133.
12. K. CHELMINSKI, Perfect plasticity as a zero relaxation limit of plasticity with isotropic hardening, Math. Methods Appl. Sci., 24 (2001), pp. 117-136.
13. K. CHELMINSKI, Global existence of weak-type solutions for models of monotone type in the theory of inelastic deformations, Math. Methods Appl. Sci., 25 (2002), pp. 1195-1230.
14. B. D. COLEMAN AND M. L. HODGDON, On shear bands in ductile materials, Arch. Rational Mech. Anal., 90 (1985), pp. 219-247.
15. E. COSSERAT AND F. COSSERAT, Théorie des Corps Déformables, Librairie Scientifique A. Hermann et Fils (Translation: Theory of Deformable Bodies, NASA TT F-11 561, Washington, DC, 1968), Paris, 1909.
16. 2,φ regularity for nonlinear elliptic systems of second order, Electron. J. Differential Equations, 2002, article 20.
17. 2 -flow theory for polar continua, Comput. Methods Appl. Mech. Engrg., 103 (1992), pp. 347-362.
18. A. DEMYANOV, Regularity in Prandtl-Reuss perfect plasticity, in Workshop: Analysis and Numerics for Rate-Independent Processes, G. Dal Maso, G. Francfort, A. Mielke, and T. Roubíček, eds., Oberwolfach Report, Vol. 4, Oberwolfach, Germany, 2007, pp. 591-666.
19. A. DIETSCHE, P. STEINMANN, AND K. WILLAM, Micropolar elastoplasticity and its role in localization, Int. J. Plasticity, 9 (1993), pp. 813-831.
20. C. EBMEYER, Mixed boundary value problems for nonlinear elliptic systems with p-structure in polyhedral domains, Math. Nachr., 236 (2002), pp. 91-108.
21. C. EBMEYER AND J. FREHSE, Mixed boundary value problems for nonlinear elliptic equations in multidimensional non-smooth domains, Math. Nachr., 203 (1999), pp. 47-74.
22. A. C. ERINGEN, Microcontinuum Field Theories, Springer, Heidelberg, 1999.
23. S. FOREST, G. CAILLETAUD, AND R. SIEVERT, A Cosserat theory for elastoviscoplastic single crystals at finite deformation, Arch. Mech., 49 (1997), pp. 705-736.
24. M. FUCHS AND G. SEREGIN, Variational Methods for Problems from Plasticity Theory and for Generalized Newtonian Fluids, Ann. Univ. Sarav. Ser. Math. 10, FB Mathematik, University of Saarbrücken, Saarbrücken, Germany, 1999.
25. M. FUCHS AND G. SEREGIN, Variational Methods for Problems from Plasticity Theory and for Generalized Newtonian Fluids, Lecture Notes in Math. 1749, Springer, Berlin, 2000.
26. M. GIAQULNTA AND S. HILDEBRANDT, Calculus of Variations I, Springer, Berlin, Heidelberg, 1996.
27. D. GILBARG AND N. S. TRUDINGER, Elliptic Partial Differential Equations of Second Order, Grundlehren Math. Wiss. 224, Springer, Berlin, 1977.
28. V. GIRAULT AND P. A. RAVIART, Finite Element Approximation of the Navier-Stokes Equations, Lecture Notes in Math. 749, Springer, Heidelberg, 1979.
29. W. HAN AND B. D. REDDY, Plasticity. Mathematical Theory and Numerical Analysis, Springer, Berlin, 1999.
30. I. R. IONESCU AND M. SOFONEA, Functional and Numerical Methods in Viscoplasticity, 1st ed., Oxford University Press, Oxford, UK, 1993.
31. M. M. IORDACHE AND K. WILLAM, Localized failure analysis in elastoplastic Cosserat continua, Comput. Methods Appl. Mech. Engrg., 151 (1998), pp. 559-586.
32. D. KNEES, Global regularity of the elastic fields of a power-law model on Lipschitz domains, Math. Methods Appl. Sci., 29 (2006), pp. 1363-1391.
33. D. KNEES, Global stress regularity of convex and some nonconvex variational problems, Ann. Mat. Pura Appl. (4), 187 (2008), pp. 157-184.
34. D. KNEES AND A. MIELKE, Energy release rate for cracks in finite-strain elasticity, Math. Methods Appl. Sci., 31 (2008), pp. 501-528.
35. R. V. KOHN AND R. TEMAM, Dual spaces of stresses and strains, with applications to Hencky plasticity, Appl. Math. Optim., 10 (1983), pp. 1-35.
36. A. MIELKE, Deriving new evolution equations for microstructures via relaxation of variational incremental problems, Comput. Methods Appl. Mech. Engrg., 193 (2004), pp. 5095-5127.
37. A. MIELKE, Existence of minimizers in incremental elasto-plasticity with finite strains, SIAM J. Math. Anal., 36 (2004), pp. 384-404.
38. A. MIELKE AND S. MÜLLER, Lower semicontinuity and existence of minimizers in incremental finite-strain elastoplasticity, ZAMM Z. Angew. Math. Mech., 86 (2006), pp. 233-250.
39. C. B. MORREY, Multiple Integrals in the Calculus of Variations, Springer, New York, 1966.
40. H. B. MÜHLHAUS, Shear band analysis for granular materials within the framework of Cosserat theory, Ing. Archiv., 56 (1989), pp. 389-399.
41. J. NEČAS, Les Méthodes Directes en Théorie des Équations Elliptiques, Masson et Cie, Éditeurs, Paris, 1967.
42. J. NEČAS, Introduction to the Theory of Nonlinear Elliptic Equations, Teubner Verlagsge-sellschaft, Leipzig, 1983.
43. P. NEFF, Existence of minimizers for a finite-strain micromorphic elastic solid, Proc. Roy. Soc. Edinburgh Sect. A, 136 (2006), pp. 997-1012.
44. P. NEFF, A finite-strain elastic-plastic Cosserat theory for polycrystals with grain rotations, Internat. J. Engrg. Sci., 44 (2006), pp. 574-594.
45. P. NEFF, Finite Multiplicative Elastic-Viscoplastic Cosserat Micropolar Theory for Polycrystals with Grain Rotations. Modelling and Mathematical Analysis, Preprint 2297; http://wwwbib.mathematik.tu-darmstadt. de/Math-Net/Preprints/Listen/pp03.html.
46. P. NEFF AND K. CHEŁMIŃSKI, Infinitesimal elastic-plastic Cosserat micropolar theory. Modelling and global existence in the rate independent case, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), pp. 1017-1039.
47. P. NEFF AND K. CHEŁMIŃSKI, A note on approximation of Prandtl-Reuss plasticity through Cosserat plasticity, Quart. Appl. Math., to appear.
48. P. NEFF AND K. CHEŁMIŃSKI, Well-posedness of dynamic Cosserat plasticity, Appl. Math. Optim., 56 (2007), pp. 19-35.
49. P. NEFF, K. CHEŁMINSKI, W. MÜLLER, AND C. WIENERS, Numerical solution method for an infinitesimal elastic-plastic Cosserat model, Math. Models Methods Appl. Sci., 17 (2007), pp. 1211-1239.
50. P. NEFF AND C. WIENERS, Comparison of models for finite plasticity. A numerical study, Comput. Vis. Sci., 6 (2003), pp. 23-35.
51. S. NESENENKO, Homogenization and Regularity in Viscoplasticity. Ph.D. thesis, TU Darmstadt, Logos Verlag, Berlin, 2006.
52. M. ORTIZ AND L. STAINIER, The variational formulation of viscoplastic constitutive updates, Comput. Methods Appl. Mech. Engrg., 171 (1999), pp. 419-444.
53. J.-P. RAYMOND, Régularité globale des solutions de systèmes elliptiques non linéaires, Rev. Mat. Univ. Complut. Madrid, 2 (1989), pp. 241-270.
54. S. I. REPIN, Errors of finite element method for perfectly elasto-plastic problems, Math. Models Methods Appl. Sci., 6 (1996), pp. 587-604.
55. M. RISTINMAA AND M. VECCHI, Use of couple-stress theory in elasto-plasticity, Comput. Methods Appl. Mech. Engrg., 136 (1996), pp. 205-224.
56. C. SANSOUR, A theory of the elastic-viscoplastic Cosserat continuum, Arch. Mech., 50 (1998), pp. 577-597.
57. G. SAVARÉ, Regularity results for elliptic equations in Lipschitz domains, J. Funct. Anal., 152 (1998), pp. 176-201.
58. G. A. SEREGIN AND T. N. SHILKIN, Regularity for minimizers of some variational problems in plasticity theory, J. Math. Sci., 99 (2000), pp. 969-988.
59. G. A. SEREGIN, Differentiability of extremals of variational problems in the mechanics of elastic perfectly plastic media, Differential Equations, 23 (1987), pp. 1349-1358.
60. J. C. SIMO AND J. R. HUGHES, Computational Inelasticity, Interdiscip. Appl. Math. 7, Springer, Berlin, 1998.
61. J. C. SIMO AND R. L. TAYLOR, Consistent tangent operators for rate-independent elastoplasticity, Comput. Methods Appl. Mech. Engrg., 48 (1985), pp. 101-118.
62. P. STEINMANN, A micropolar theory of finite deformation and finite rotation multiplicative elastoplasticity, Internat. J. Solids Structures, 31 (1994), pp. 1063-1084.
63. R. TEMAM, Mathematical Problems in Plasticity, Gauthier-Villars, New York, 1985.
64. R. TEMAM, A generalized Norton-Hoff model and the Prandtl-Reuss law of plasticity, Arch. Rational Mech. Anal., 95 (1986), pp. 137-183.
65. T. VALENT, Boundary Value Problems of Finite Elasticity, Springer, Berlin, 1988.
66. C. WIENERS, Multigrid methods for Prandtl-Reuss plasticity, Numer. Linear Algebra Appl., 6 (1999), pp. 457-478.
67. C. WIENERS, Efficient elasto-plastic simulation, in Multifield Problems. State of the Art, A. M. Sandig, W. Schiehlen, and W. L. Wendland, eds., Springer, Berlin, 2000, pp. 209-216.
68. J. WLOKA, Partielle Differentialgleichungen, Teubner Verlag, Stuttgart, 1982.