Analytical and numerical aspects of time-dependent models with internal variables

ZAMM Zeitschrift fur Angewandte Mathematik und Mechanik

Volumn 90, Issue 10-11, 2010, Pages 861-902

  Gruber, P ^a   Knees, D ^b   Nesenenko, S ^c,d   Thomas, M ^b  

^a JOHANNES KEPLER UNIVERSITY LINZ   (Austria)

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

^c DARMSTADT UNIVERSITY OF TECHNOLOGY   (Germany)

^d UNIVERSITÉ PIERRE ET MARIE CURIE   (France)

Author keywords

Elasto plasticity; Energetic formulation of rate independent processes; Existence of solutions; Models of monotone type; Monotone operator method; Slant newton method; Spatial regularity; Temporal regularity.; Visco plasticity

Indexed keywords

EID: 78349241342     PISSN: 00442267     EISSN: 15214001     Source Type: Journal    
DOI: 10.1002/zamm.200900387     Document Type: Article

References (114)

1. H.D. Alber, Materials with Memory. Initial-Boundary Value Problems for Constitutive Equations with Internal Variables, Lecture Notes in Mathematics, Vol. 1682 (Springer, Berlin, 1998).
2. H.D. Alber and K. Chełmiński, Quasistatic Problems in Viscoplasticity Theory I: Models with linear Hardening, in: Theoretical Methods and Applications to Mathematical Physics, edited by I. Gohberg, A. dos Santos, F. O. Speck, F. Teixeira, and W. Wendland, Operator Theory. Advances and Applications Vol. 147 (Birkhäuser, Basel, 2004), pp. 105-129.
3. H.D. Alber and K. Chełmiński, Quasistatic problems in viscoplasticity theory. II. Models with nonlinear hardening, Math. Models Methods Appl. Sci. 17(2), 189-213 (2007).
4. H.D. Alber and S. Nesenenko, Local and Global Regularity in Time Dependent Viscoplasticity, in: IUTAM Symposium on Theoretical, Computational and Modelling Aspects of Inelastic Media, edited by B. D. Reddy, IUTAM Bookseries Vol. 11 (Springer, Netherlands, 2008), pp. 363-372.
5. 1/3-δ-regularity up to the boundary in time dependent viscoplasticity, Asymtotic Anal. 63(3), 151-187 (2009).
6. J. Alberty, C. Carstensen, and D. Zarrabi, Adaptive numerical analysis in primal elastoplasticity with hardening, Comput. Methods Appl. Mech. Eng. 171(3-4), 175-204 (1999).
7. L. Ambrosio, N. Fusco, and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems (Oxford University Press, Oxford, 2005).
8. G. Anzellotti, On the existence of the rates of stress and displacement for Prandtl- Reuss plasticity, Q. Appl. Math. 41, 181-208 (1983).
9. H. Attouch and M. Thera, A general duality principle for the sum of two operators, J. Convex Anal. 3(1), 1-24 (1996).
10. F. Auricchio, A robust integration algorithm for a finite-strain shape-memory-alloy superelastic model, Int. J. Plast. 17, 971-990 (2001).
11. V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces (Editura Academiei, Bucharest, 1976).
12. 1 Regularity, in: Boundary value problems for partial differential equations and applications, edited by J. L. Lions and C. Baiocchi, RMA Vol. 29 (Masson, Paris, 1993), pp. 3-25.
13. loc regularity, Comment. Math. Univ. Carolinae 37(2), 285-304 (1996).
14. M. Bildhauer and M. Fuchs, Continuity properties of the stress tensor in the 3-dimensional Ramberg/Osgood model, J. Appl. Anal. 13(2), 209-233 (2007).
15. H. Blum and J. Frehse, Boundary Differentiability for the Solution to Hencky's Law of Elastic Plastic Plane Stress, Preprint SFB 611 435 (University Bonn, Bonn, 2008).
16. B. Bourdin, G. Francfort, and J.J. Marigo, The variational approach to fracture, J. Elast. 91, 5-148 (2008).
17. H. Brézis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert (North-Holland Mathematics Studies, Amsterdam, London, 1973).
18. M. Bulíĉek, J. Frehse, and J. Málek, On Boundary Regularity for the Stress in Problems of Linearized Elasto-plasticity, Preprint SFB 611 443 (University Bonn, Bonn, 2009).
19. C. Carstensen and S. Müller, Local stress regularity in scalar nonconvex variational problems, SIAM J. Math. Anal. 34(2), 495-509 (2002).
20. C. Carstensen, Domain decomposition for a non-smooth convex minimization problem and its application to plasticity, Numer. Linear Algebr. Appl. 4(3), 177-190 (1997).
21. K. Chełmiński, Global existence of weak-type solutions for models of monotone type in the theory of inelastic deformations, Math. Methods Appl. Sci. 25, 1195-1230 (2002).
22. K. Chełmiński and P. Gwiazda, Nonhomogenious initial boundary value problems for coercive and self-controlling models of monotone type, Contin. Mech. Thermodyn. 12, 217-234 (2000).
23. K. Chełmiński and Z. Naniewicz, Coercive limits for constitutive equations of monotone-gradient type, Nonlinear Anal. 48, 1197-1214 (2002).
24. loc-stress and strain regularity in Cosserat plasticity, Z. Angew. Math. Mech. 89(4), 257-266 (2009).
25. X. Chen, Z. Nashed, and L. Qi, Smoothing methods and semismooth methods for nondifferentiable operator equations, SIAM J. Numer. Anal. 38(4), 1200-1216 (2000).
26. P. Colli and A. Visintin, On a class of doubly nonlinear evolution equations, Commun. Partial Differ. Equ. 15(5), 737-756 (1990).
27. G. Dal Maso, A. De Simone, and M.G. Mora, Quasistatic evolution problems for linearly elastic - perfectly plastic materials, Arch. Ration. Mech. Anal. 180, 237-291 (2006).
28. G. Dal Maso, G. Francfort, and R. Toader, Quasistatic crack growth in nonlinear elasticity, Arch. Ration. Mech. Anal. 176(2), 165-225 (2004).
29. A. De Simone, N. Grunewald, and F. Otto, A new model for contact angle hysteresis, Netw. Heterog. Media (electronic), 2(2), 211-225 (2007).
30. A. Demyanov, Regularity of stresses in Prandtl-Reuss perfect plasticity, Calc. Var. Partial Differ. Equ. 34(1), 23-72 (2009).
31. G. Duvaut and J. Lions, Inequalities in Mechanics and Physics, A Series of Comprehensive Studies in Mathematics, Vol. 219 (Springer, Berlin, 1976).
32. C. Ebmeyer, Mixed boundary value problems for nonlinear elliptic systems with p-structure in polyhedral domains, Math. Nachr. 236, 91-108 (2002).
33. I. Ekeland and R. Témam, Convex Analysis and Variational Problems, English edition, Classics in Applied Mathematics, Vol. 28, translated from the French (Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1999)
34. J. Fernández, W. Han, M. Sofonea, and J. Viaño, Variational and numerical analysis of a frictionless contact problem for elastic-viscoplastic materials with internal state variables, Q.J. Mech. Appl. Math. 54(4), 501-522 (2001).
35. G. Francfort and A. Mielke, Existence results for a class of rate-independent material models with nonconvex elastic energies, J. Reine Angew. Math. 595, 55-91 (2006).
36. J. Frehse and D. Löbach, Hölder continuity for the displacements in isotropic and kinematic hardening with von Mises yield criterion, Z. Angew. Math. Mech. 88(8), 617-629 (2008).
37. J. Frehse and D. Löbach, Regularity results for three-dimensional isotropic and kinematic hardening including boundary differentiability, Math. Models Methods Appl. Sci. 19(12), 2231-2262 (2009).
38. M. Fuchs, On a class of variational problems related to plasticity with polynomial hardening, Appl. Anal. 60, 269-275 (1996).
39. M. Fuchs and G. Seregin, Variational Methods for Problems from Plasticity Theory and for Generalized Newtonian Fluids, Lecture Notes in Mathematics, Vol. 1749 (Springer, Berlin, 2000).
40. A. Giacomini, Ambrosio-Tortorelli approximation of quasi-static evolution of brittle fracture, Calc. Var. Partial Differ. Equ. 22, 129-172 (2005).
41. D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, No. 224 in Grundlehren der Mathematischen Wissenschaften (Springer Verlag, Berlin, Heidelberg, 1977).
42. P. Grisvard, Elliptic Problems in Nonsmooth Domains (Pitman Publishing Inc., Boston, 1985).
43. P.G. Gruber, Adaptive Strategies for High Order FEM in Elastoplasticity, Tech. Rep. 02, DK W1214, Computational Mathematics (Johannes Kepler University Linz, Linz, 2010).
44. P.G. Gruber and J. Valdman, Solution of one-time-step problems in elastoplasticity by a Slant Newton Method, SIAM J. Sci. Comput. 31(2), 1558-1580 (2009).
45. P.G. Gruber and J. Valdman, Solution of Elastoplastic Problems Based on the Moreau-Yosida Theorem, SFB Report 5, SFB F013, Numerical and Symbolic Scientific Computing (Johannes Kepler University Linz, Linz, 2006).
46. P.G. Gruber and J. Valdman, Implementation of an elastoplastic solver based on the Moreau-Yosida Theorem, Math. Comput. Simul. 76(1-3), 73-81 (2007).
47. P.G. Gruber and J. Valdman, Newton-like Solver for Elastoplastic Problems with Hardening and its Local Super-linear Convergence, SFB Report 6, SFB F013, Numerical and Symbolic Scientific Computing (Johannes Kepler University Linz, Linz, 2007).
48. W. Han and B.D. Reddy, Plasticity, Mathematical Theory and Numerical Analysis (Springer Verlag, New York, 1999).
49. A. Haraux, Nonlinear Evolution Equation - Global Behavior of Solutions, Lecture Notes in Mathematics, Vol. 841 (Springer, Berlin, 1981).
50. M. Hintermüller, K. Ito, and K. Kunisch, The primal-dual active set strategy as a semismooth Newton method, SIAM J. Optim. 13(3), 865-888 (2003).
51. S. Hu and N.S. Papageorgiou, Handbook of Multivalued Analysis, Vol. II. Applications, Mathematics and its Applications (Kluwer, Dordrecht, 2000).
52. A. Ioffe and V. Tichomirov, Theorie der Extremalaufgaben (VEB Deutscher Verlag der Wissenschaften, Berlin, 1979).
53. I. Ionescu and M. Sofonea, Functional and Numerical Methods in Viscoplasticity (Oxford University Press, Oxford, 1993).
54. C. Johnson, Existence theorems for plasticity problems, J. Math. Pures Appl., IX. Ser. 55, 431-444 (1976).
55. C. Johnson, On plasticity with hardening, J. Math. Anal. Appl. 62(2), 325-336 (1978).
56. D. Knees, Regularity Results for Quasilinear Elliptic Systems of Power-law Growth in Nonsmooth Domains: Boundary, Transmission and Crack Problems, PhD thesis, Universität Stuttgart, 2005, http://elib.uni-stuttgart.de/opus/volltexte/2005/2191/.
57. D. Knees, Global regularity of the elastic fields of a power-law model on Lipschitz domains, Math. Methods Appl. Sci. 29, 1363-1391 (2006).
58. D. Knees, Global stress regularity of convex and some nonconvex variational problems, Ann. Mat. Pura Appl. 187, 157-184 (2008).
59. D. Knees, Global spatial regularity for a regularized elasto-plastic model, GAMM-Mitt. (2010), submitted.
60. D. Knees, On global spatial regularity and convergence rates for time dependent elasto-plasticity, Math. Models Methods Appl. Sci. accepted (2010)
61. D. Knees and P. Neff, Regularity up to the boundary for nonlinear elliptic systems arising in time-incremental infinitesimal elasto-plasticity, SIAM J. Math. Anal. 40(1), 21-43 (2008).
62. D. Knees, A. Mielke, and C. Zanini, On the inviscid limit of a model for crack propagation, Math. Models Methods Appl. Sci. (M3AS) 18, 1529-1569 (2008).
63. V. Korneev and U. Langer, The regularity of solutions of problems from plastic-flow theory, Differ. Uravn. 20(4), 597-615 (1984).
64. M. Krasnosel'skij, P. Zabrejko, E. Pustylnik, and P. Sobolevskij, Integral Operators in Spaces of Summable Functions (Monographs and Textbooks on Mechanics of Solids and Fluids (Noordhoff International Publishing XV, Leyden, 1976).
65. P. Krejĉí, Evolution Variational Inequalities and Multidimensional Hysteresis Operators, edited by Drábek, Pavel et al, Nonlinear Differential Equations. Proceedings of Talks Given at the Seminar in Differential Equations, Chvalatice, Czech Republic, June 29-July 3, 1998. (Chapman & Hall/CRC Boca Raton, 1998); Res. Notes Math. 404, 47-110, (1999).
66. K.L. Kuttler and M. Shillor, Quasistatic evolution of damage in an elastic body, Nonlinear Anal., Real World Appl. 7, 674-699 (2006).
67. J. Lemaitre and R. Desmorat, Engineering Damage Mechanics (Springer, Berlin, Heidelberg, New York, 2005).
68. A. Mainik and A. Mielke, Existence results for energetic models for rate-independent systems, Calc. Var. Partial Differ. Equ. 22, 73-99 (2005).
69. A. Mainik and A. Mielke, Global existence for rate-independent gradient plasticity at finite strain, J. Nonlinear Sci. 19(3), 221-248 (2009).
70. H. Matthies, Existence theorems in thermo-plasticity, J. Mec. 18, 659-712 (1979).
71. A. Mielke and T. Roubíĉek, Rate-independent damage processes in nonlinear elasticity, Math. Models Methods Appl. Sci. (M3AS) 16, 177-209 (2006).
72. A. Mielke, T. Roubíĉek, and U. Stefanelli, Γ-limits and relaxations for rate-independent evolutionary problems, Calc. Var. Partial Differ. Equ. 31, 387-416 (2008).
73. A. Mielke and F. Theil, On rate-independent hysteresis models, Nonlinear Differ. Equ. Appl. (NoDEA) 11, 151-189 (2004).
74. A. Mielke, Evolution in Rate-independent Systems, Chap. 6, in: Handbook of Differential Equations, Evolutionary Equations, Vol. 2, edited by C. Dafermos and E. Feireisl (Elsevier B.V, Amsterdam, 2005), pp. 461-559.
75. A. Mielke, L. Paoli, and A. Petrov, On the existence and approximation for a 3D model of thermally induced phase transformations in shape-memory alloys, SIAM J. Math. Anal. 41(4), 1388-1414 (2009).
76. A. Mielke and A. Petrov, Thermally driven phase transformation in shape-memory alloys, Adv. Math. Sci. Appl. 17, 667-685 (2007).
77. A. Mielke and R. Rossi, Existence and uniqueness results for a class of rate-independent hysteresis problems, Math. Models Methods Appl. Sci. (M3AS) 17, 81-123 (2007).
78. E. Miersemann, Zur Regularität der quasistatischen elasto-viskoplastischen Verschiebungen und Spannungen, Math. Nachr. 96, 293-299 (1980).
79. J.J. Moreau, Sur l'evolution d'un systeme elasto-visco-plastique, C.R. Acad. Sci. A, B (France) 273, 118-121 (1971).
80. J.J. Moreau, Proximité et dualité dans un espace Hilbertien, Bull. Soc. Math. (France) 93, 273-299 (1965).
81. J. Moreau, Application of Convex Analysis to the Treatment of Elastoplastic Systems, Appl. Methods Funct. Anal. Probl. Mech. (IUTAM/IMU-Symp., Marseille, 1975; Lect. Notes Math. 503, 56-89 (1976).
82. J. Naumann and J. Wolf, Interior differentiability of weak solutions to the equations of stationary motion of a class of non-Newtonian fluids, J. Math. Fluid Mech. 7(2), 298-313 (2005).
83. P. Neff and K. Chełmiński, Infinitesimal elastic-plastic Cosserat micropolar theory. Modelling and global existence in the rate independent case, Proc. R. Soc. Edinb. A 135, 1017-1039 (2005).
84. q-Almost Solvability of Visco-plastic Problems of Monotone Type, submitted.
85. S. Nesenenko, A note on existence result for viscoplastic models with nonlinear hardening, Math. Mech. Solids, first published on March 20, 2009 as doi:10.1177/1081286509103818.
86. D. Pascali and S. Sburlan, Nonlinear Mappings of Monotone Type (Editura Academiei, Bucharest, 1978).
87. W. Pompe, Existence Theorems in the Viscoplasticity Theory, Dissertation, (Darmstadt University of Technology, Darmstadt, 2003).
88. W. Pompe, Linear functionals on nonlinear spaces and applications to problems from viscoplasticity theory, Math. Methods Appl. Sci. 31(7), 775-792 (2008).
89. S. Repin, Errors of finite element method for perfectly elasto-plastic problems, Math. Models Methods Appl. Sci. 6(5), 587-604 (1996).
90. T. Roubíĉek, L. Scardia, and C. Zanini, Quasistatic delamination problem, Contin. Mech. Thermodyn. (2009), submitted.
91. T. Roubíĉek, Rate independent processes in viscous solids at small strains, Math. Methods Appl. Sci. 32(7), 825-862 (2008).
92. G. Savaré, Regularity results for elliptic equations in Lipschitz domains, J. Funct. Anal. 152, 176-201 (1998).
93. J. Schöberl, Netgen - an advancing front 2d/3d-mesh generator based on abstract rules, Comput. Vis. Sci 1, 41-52 (1997).
94. G. Seregin, On differential properties of weak solutions of nonlinear elliptic systems arising in plasticity theory, Math. USSR Sbornik 58(2), 289-309 (1987).
95. G. Seregin, On differential properties of extremals of variational problems arising in the theory of plasticity, J. Differ. Equations 26(6), 756-766 (1990).
96. G. Seregin, Differential properties of solutions of evolutionary variational inequalities in plasticity theory, Probl. Mat. Anal. 12, 153-173 (1992).
97. G. Seregin, On the regularity of the minimizers of some variational problems of plasticity theory, St. Petersburg Math. J. 4(5), 989-1020 (1993).
98. G. Seregin, Remarks on the regularity up to the boundary for solutions to variational problems in plasticity theory, J. Math. Sci. 93(5), 779-783 (1999).
99. P. Shi, Interior regularity of solutions to a dynamic cyclic plasticity model in higher dimensions, Adv. Math. Sci. Appl. 9(2), 817-837 (1999).
100. E. Showalter, Monotone Operators in Banach Spaces and Nonlinear Partial Differential Equations, Mathematical Surveys and Monographs, Vol. 49 (AMS, Providence, 1997).
101. J. Simo and J. Hughes, Computational Inelasticity, Interdisciplinary Applied Mathematics, Vol. 7 (Springer, Berlin, 1998).
102. J. Simo and R.L. Taylor, Consistent tangent operators for rate-independent elasto-plasticity, Comput. Methods Appl. Mech. Eng. 48, 101-118 (1985).
103. E. Stein, Error-controlled Adaptive Finite Elements in Solid Mechanics (Wiley, Chichester, 2003).
104. M. Steinhauer, On Analysis of Some Nonlinear Systems of Partial Differential Equations of Continuum Mechanics, PhD thesis (Universität Bonn, Mathematisch-Naturwissenschaftliche Fakultät, Bonn, 2003).
105. P.M. Suquet, Evolution problems for a class of dissipative materials, Q. Appl. Math. 38(4), 391-414 (1980/81).
106. L. Tartar, Interpolation non linéaire et régularité, J. Funct. Anal. 9, 469-489 (1972).
107. R. Temam, Mathematical Problems in Plasticity (Bordas, Paris, 1983).
108. R. Temam, A generelized Norton-Hoff model and the Prandtl-Reuss law of plasticity, Arch. Ration. Mech. Anal. 95(2), 137-183 (1986).
109. M. Thomas and A. Mielke, Damage of nonlinearly elastic materials at small strain: existence and regularity results, Z. Angew. Math. Mech. 90(2), 88-112 (2010).
110. H. Triebel, Interpolation Theory, Function Spaces, Differential Operators (North Holland Publishing Company, Amsterdam, New York, Oxford, 1978).
111. T. Valent, Boundary Value Problems of Finite Elasticity (Springer, Berlin, 1988).
112. A. Visintin, Strong convergence results related to strict convexity, Commun. Partial Differ. Equ. 9(5), 439-466 (1984).
113. E. Zeidler, Nonlinear Functional Analysis and its Applications. III Variational Methods and Optimization (Springer-Verlag, New York, 1986).
114. E. Zeidler, Nonlinear Functional Analysis and its Applications. II/B: Nonlinear Monotone Operators (Springer-Verlag, Berlin, Heidelberg, New York, 1990).