Analysis of microstructure development in shearbands by energy relaxation of incremental stress potentials: Large-strain theory for standard dissipative solids

International Journal for Numerical Methods in Engineering

Volumn 58, Issue 1, 2003, Pages 1-41

  Miehe, Christian ^a   Lambrecht, Matthias ^a  

^a UNIVERSITY OF STUTTGART   (Germany)

Author keywords

Energy relaxation methods; Inelasticity; Material instabilities; Quasi convexification; Rank one convexification; Shearband localization; Variational methods

Indexed keywords

COMPUTER SIMULATION; DEFORMATION; ELASTOPLASTICITY; MICROSTRUCTURE; STRESS ANALYSIS;

ENERGY RELAXATION;

SOLIDS;

DEFORMATION; ELASTIC PLASTIC ANALYSIS; MATHEMATICAL MODELING; MICROSTRUCTURE; SHEAR STRESS;

EID: 0042902231     PISSN: 00295981     EISSN: None     Source Type: Journal    
DOI: 10.1002/nme.726     Document Type: Article

References (66)

1. Hadamard J. Leçons sur la propagation des ondes et les équations de l'hydrodynamique. Hermann: Paris, 1903.
2. Nadai A. Theory of Flow and Fracture of Solids, vol. 1. McGraw-Hill: New York, 1950.
3. Vardoulakis I. Scherfugenbildung in Sandkörpern als Verzweigungsproblem. Veröffentlichungen des Instituts für Bodenmechanik und Felsmechanik der Universität Karlsruhe, Heft 70, Universität Karlsruhe, 1977.
4. Needleman A. Material rate dependence and mesh sensitivity in localization problems. Computer Methods in Applied Mechanics and Engineering 1988; 67:69-85.
5. Baźant ZP, Lin FB. Non-local yield limit degradation. International Journal for Numerical Methods in Engineering 1988; 26:1805 1823.
6. de Borst R, Sluys LJ. Localization in a cosserat continuum under static and dynamic loading. Computer Methods in Applied Mechanics and Engineering 1991; 90:805-827.
7. Mühlhaus HB, Aifantis EC. A variational principle for gradient plasticity. International Journal of Solids and Structures 1991; 28:845-858.
8. Ortiz M, Leroy Y, Needleman A. A finite-element method for localized failure analysis. Computer Methods in Applied Mechanics and Engineering 1987; 61:189-214.
9. Belytschko T, Fish J, Engelmann BE. A finite-element with embedded localization zones. Computer Methods in Applied Mechanics and Engineering 1988; 70:59-89.
10. Larsson R, Runesson K, Ottosen NS. Discontinuous displacement approximation for capturing plastic localization. International Journal for Numerical Methods in Engineering 1993; 36:2087-2105.
11. Miehe C, Schröder J. Post-critical discontinuous localization analysis of small-strain softening elastoplastic solids. Archive of Applied Mechanics 1994; 64:267-285.
12. Simó JC, Oliver J, Armero G. An analysis of strong discontinuities induced by strain-softening. Computational Mechanics 1993; 12:277-296.
13. Armero F, Garikipati K. An analysis of strong discontinuities in multiplicative finite strain plasticity and their relation with the numerical simulation of strain localization in solids. International Journal of Solids and Structures 1996; 33:2863-2885.
14. Garikipati K, Hughes TJR. A variational multiscale approach to strain localization-formulation for multidimensional problems. Computer Methods in Applied Mechanics and Engineering 2000; 188:39-60.
15. Mühlhaus HB (ed.). Continuum Models for Material with Microstructure. Wiley; Chichester, 1995.
16. de Borst R, van der Giessen E (eds). Material Instabilities in Solids. Wiley: Chichester, 1998.
17. Miehe C. Strain-driven homogenization of inelastic microstructures and composites based on an incremental variational formulation. International Journal for Numerical Methods in Engineering 2002; 55:1285-1322.
18. Miehe C, Schotte J, Lambrecht M. Homogenization of inelastic solid materials at finite strains based on incremental minimization principles. Journal of the Mechanics and Physics of Solids 2001; 50:2123-2167.
19. Ortiz M, Repetto EA. Nonconvex energy minimization and dislocation structure in ductile single crystals. Journal of the Mechanics and Physics of Solids 1999; 47:397-462.
20. Ortiz M, Stainier L. The variational formulation of viscoplastic constitutive updates. Computer Methods in Applied Mechanics and Engineering 1999; 171:419-444.
21. Carstensen C, Hackl K, Mielke A. Nonconvex potentials and microstructures in finite-strain plasticity. Max-Planck-Institut für Mathematik in den Naturwissenschaften. Preprint No. 72, 2000.
22. Biot MA. Mechanics of Incremental Deformations. Wiley: New York, 1965.
23. Ziegler H. Some extremum principles in irreversible thermodynamics with application to continuum mechanics. In Progress in Solid Mechanics, vol. IV, Sneddon IN, Hill R (eds). North-Holland: Amsterdam, 1963.
24. Germain P. Cours de Mécanique des Milieux Continus. Masson et Cie: Paris, 1973.
25. Halphen B, Nguyen QS. Sur les Matéraux Standards Généralisés. Journal de Mécanique 1975; 40:39-63.
26. Maugin GA. The Thermodynamics of Plasticity, and Fracture. Cambridge University Press: Cambridge, 1992.
27. Nguyen A. Stability and Nonlinear Solid Mechanics. Wiley: New York, 2000.
28. Martin JB. Plasticity, Fundamentals and General Results. MIT Press: Cambridge, MA, 1975.
29. Thomas TY. Plastic Flow and Fracture in Solids. Academic Press: London, 1961.
30. Hill R. Acceleration waves in solids. Journal of the Mechanics and Physics of Solids 1962; 6:1 16.
31. Rice JR. The localization of plastic deformation. In Theoretical and Applied Mechanics Koiter WT (ed.). North-Holland: Amsterdam, 1976; 207-220.
32. Dacorogna B. Direct Methods in the Calculus of Variations. Springer: Berlin, Heidelberg.
33. Krawietz A. Materialtheorie: Mathematische Beschreibung des phänomenologischen thermomechanischen Verhaltens. Springer: Berlin, 1996.
34. Ciarlet PG. Mathematical Elasticity. Elsevier: Amsterdam, 1988.
35. Marsden JE, Hughes TJR. Mathematical Foundations of Elasticity. Dover, Inc.: New York, 1994.
36. Şilhavý M. The Mechanics and Thermodynamics of Continuous Media. Springer: Berlin, Heidelberg, New York, 1997.
37. Morrey CB. Quasiconvexity and the semicontinuity of multiple integrands. Pacific Journal of Mathematics 1952; 2:25-53.
38. Kohn RV, Strang G. Explicit relaxation of a variational problem in optimal design. Bulletin of the American Mathematical Society 1983; 9:211-214.
39. Kohn RV, Strang G. Optimal design and relaxation of variational problems I, II, III. Communications on Pure and Applied Mathematics 1986; 39:113-137, 139-182, 353-377.
40. Müller S. Variational models for microstructure and phase transitions. In Calculus of Variations and Geometric Evolution Problems, Bethuel F, Huisken G, Müller S, Steffen K, Hildebrandt S, Struwe M (eds), Springer, Berlin, 1998.
41. Kohn RV. The relaxation of a double-well problem. Continuum Mechanics and Thermodynamics 1991; 3: 193-236.
42. Luskin M. On the computation of crystalline microstructure. Acta Numerica 1996; 36:191-257.
43. Carstensen C. Plecháč P. Numerical solution of the scalar double-well problem allowing microstructures. Mathematics of Computation 1997; 66:997-1026.
44. Govindjee, S, Mielke A, Hall GJ. The free-energy of mixing for n-variant martensitic phase transformations using quasi-convex analysis. 2000, preprint.
45. DeSimone, Dolzmann. Material instabilities in nematic elastomers. Physica D 2000; 136:175-191.
46. Ortiz M, Repetto EA, Stainier L. A theory of subgrain dislocation structures. Journal of the Mechanics and Physics of Solids 2000; 48(10):2077-2114.
47. Young LC. Lectures on the Calculus of Variations and Optimal Control Theory. Saunders: London, 1969.
48. Belytschko T, Moës N, Usui S, Parimi C. Arbitrary discontinuities in finite elements. International Journal for Numerical Methods in Engineering 2001; 50:993-1013.
49. Ziegler H, Wehrli C. The derivation of constitutive relations from the free energy and the dissipation function. In Advances in Applied Mechanics, vol. 25, Wu ThY, Hutchinson JW (eds). Academic Press Inc.: New York, 1987.
50. Rockafellar RT. Convex Analysis. Princeton University Press: Princeton, NJ, 1970.
51. Moreau JJ. On unilateral constraints, friction and plasticity. In New Variational Techniques in Mathematical Physics, Capriz G, Stampacchia G (eds). CISM Course. Springer: Berlin, 1974.
52. Moreau JJ. Application of convex analysis to the treatment of elastoplastic systems. In Application of Methods of Functional Analysis to Problems in Mechanics, Germain P, Nayroles B (eds). Springer: Berlin, 1976.
53. Mielke A. Finite Elastoplasticity, Lie Groups and Geodesics on SL(d). Bericht 2000/26, Sonderforschungsbereich 404 Mehrfeldprobleme in der Kontinuumsmechanik, Universität Stuttgart, 2000.
54. Perié D. On a class of constitutive equations in viscoplasticity: formulation and computational issues. International Journal for Numerical Methods in Engineering 1993; 36:1365-1393.
55. Perzyna P. Fundamental problems in viscoplasticity. Advances in Applied Mechanics 1996; 9:243-377.
56. Luenberger DG. Linear and Nonlinear Programming. Addison-Wesley: Reading, MA, 1984.
57. Bertsekas DP. Nonlinear Programming. Athena Scientific: Belmont, MA, 1995.
58. Acerbi E, Fusco N. Semicontinuity problems in the calculus of variations. Archive for Rational Mechanics and Analysis 1984; 83:125-145.
59. Ball JM. Convexity conditions and existence theorems in nonlinear elasticity. Archive for Rational Mechanics and Analysis 1977; 63:337-403.
60. Truesdell C, Noll W. The nonlinear field theories of mechanics. In Handbuch der Physik Bd. III13, Flügge S (ed.). Springer: Berlin, 1965.
61. Carstensen C, Roubiček T. Numerical approximation of young measures in non-convex variational problems. Numerische Mathematik 2000; 84:395-415.
62. Lambrecht M, Miehe C, Dettmar J. Energy relaxation of non-convex incremental stress potentials in a strain-softening elastic-plastic bar. International Journal of Solids and Structures 2003; 40:1369-1391.
63. Miehe C. A constitutive frame of elastoplasticity at large strains based on the notion of a plastic metric. International Journal of Solids and Structures 1998; 35:3859-3897.
64. Miehe C. A formulation of finite elastoplasticity based on dual co- and contra-variant eigenvector triads. normalized with respect to a plastic metric. Computer Methods in Applied Mechanics and Engineering 1998; 159:223-260.
65. Miehe C, Keck J. Superimposed finite elastic-viscoelastic-plastoelastic stress response with damage in filled rubbery polymers. Experiments, modelling and algorithmic implementation. Journal of the Mechanics and Physics of Solids 2000; 48:323-365.
66. Simó JC, Armero F. Geometrically non-linear enhanced strain mixed methods and the method of incompatible modes. International Journal for Numerical Methods in Engineering 1992; 33:1413-1449.