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Analytical and Numerical Methods for Finite-Strain Elastoplasticity

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Multifield Problems in Solid and Fluid Mechanics

Part of the book series: Lecture Notes in Applied and Computational Mechanics ((LNACM,volume 28))

Summary

An important class of finite-strain elastoplasticity is based on the multiplicative decomposition of the strain tensor F = F el F pl and hence leads to complex geometric nonlinearities. This survey describes recent advances in the analytical treatment of time-incremental minimization problems with or without regularizing terms involving strain gradients. For a regularization controlling all of ∇F pl we provide an existence theory for the time-continuous rate-independent evolution problem, which is based on a recently developed energetic formulation for rateindependent systems in abstract topological spaces.

In systems without gradient regularization one encounters the formation of microstructures, which can be described by sequential laminates or more general gradient Young measures. We provide a mathematical framework for the evolution of such microstructures and discuss algorithms for solving the associated spacetime discretizations. In a finite-step-sized incremental setting of standard dissipative materials (also called generalized standard materials) we outline also details of relaxation-induced microstructure developments for strain softening von Mises plasticity and single-slip crystal plasticity. The numerical implementations are based on simplified assumptions concerning the complexity of the microstructures.

Project C11 “Mathematical Models of Plasticity with Finite Deformations”

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References

  1. E. Acerbi and N. Fusco. Semicontinuity problems in the calculus of variations. Archive of Rational Mechanics and Analysis, 83:125–145, 1984.

    Article  MathSciNet  Google Scholar 

  2. S. Aubry, M. Fago, and M. Ortiz. A constrained sequential-lamination algorithm for the simulation of sub-grid microstructure in martensitic materials. Computer Methods in Applied Mechanics and Engineering, 192:2823–2843, 2003.

    Article  MATH  MathSciNet  Google Scholar 

  3. J. M. Ball. Convexity conditions and existence theorems in nonlinear elasticity. Archive of Rational Mechanics and Analysis, 63:337–403, 1977.

    Article  MATH  Google Scholar 

  4. J. M. Ball. Some open problems in elasticity. In P. Newton, P. Holmes, and A. Weinstein, editors, Geometry, Mechanics, and Dynamics, pages 3–59. Springer, New York, 2002.

    Chapter  Google Scholar 

  5. S. Bartels, C. Carstensen, K. Hackl, and U. Hoppe. Effective relaxation for microstructure simulations: algorithms and applications. Computer Methods in Applied Mechanics and Engineering, 193:5143–5175, 2004.

    Article  MATH  MathSciNet  Google Scholar 

  6. M. Becker. Incompatibility and Instability Based Size Effects in Crystals and Composites at Finite Elastoplastic Strains. PhD thesis, Institut für Mechanik (Bauwesen), Report No. I-18, Universität Stuttgart, 2006.

    Google Scholar 

  7. M. A. Biot. Mechanics of Incremental Deformations. John Wiley & Sons Inc., New York, 1965.

    Google Scholar 

  8. C. Carstensen, K. Hackl, and A. Mielke. Non-convex potentials and microstructures in finite-strain plasticity. Proceedings of the Royal Society London, Series A, 458:299–317, 2002.

    MATH  MathSciNet  Google Scholar 

  9. C. Carstensen and P. Plecháč. Numerical solution of the scalar double-well problem allowing microstructures. Mathematics of Computation, 66:997–1026, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  10. P. G. Ciarlet. Mathematical Elasticity. Elsevier Science Publishers B.V., Amsterdam, 1988.

    MATH  Google Scholar 

  11. S. Conti and M. Ortiz. Dislocation microstructures and the effective behavior of single crystals. Archive of Rational Mechanics and Analysis, 176:103–147, 2005.

    Article  MATH  MathSciNet  Google Scholar 

  12. S. Conti and F. Theil. Single-slip elastoplastic microstructures. Archive of Rational Mechanics and Analysis, 178:125–148, 2005.

    Article  MATH  MathSciNet  Google Scholar 

  13. B. Dacorogna. Direct Methods in the Calculus of Variations. Springer-Verlag, Berlin Heidelberg, 1989.

    MATH  Google Scholar 

  14. A. DeSimone and G. Dolzmann. Material instabilities in nematic elastomers. Physica D, 136:175–191, 2000.

    Article  MATH  MathSciNet  Google Scholar 

  15. G. Dolzmann. Variational Methods for Crystalline Microstructure-Analysis and Computation. Springer-Verlag, Berlin Heidelberg, 2003.

    MATH  Google Scholar 

  16. M. Efendiev and A. Mielke. On the rate-independent limit of systems with dry friction and small viscosity. Journal of Convex Analysis, 13(1):151–167, 2006.

    MathSciNet  MATH  Google Scholar 

  17. N. A. Fleck, G. M. Müller, M. F. Ashby, and J. Hutchinson. Strain gradient plasticity: theory and experiment. Acta Materialia, 42:475–487, 1994.

    Article  Google Scholar 

  18. G. Francfort and A. Mielke. Existence results for a class of rate-independent material models with nonconvex elastic energies. J. Reine Angew. Math., 2006. In print.

    Google Scholar 

  19. P. Germain. Cours de Mécanique des Milieux Continus. Masson et Cie, Paris, 1973.

    MATH  Google Scholar 

  20. S. Govindjee, A. Mielke, and G. J. Hall. The free-energy of mixing for n-variant martensitic phase transformations using quasi-convex analysis. Journal of the Mechanics and Physics of Solids, 50:1897–1922, 2002.

    Article  MATH  MathSciNet  Google Scholar 

  21. E. Gürses and C. Miehe. Relaxation analysis of material instabilities in damage mechanics based on incremental convexification techniques. Submitted to International Journal of Solids and Structures, 2006.

    Google Scholar 

  22. M. E. Gurtin. A gradient theory of single-crystal viscoplasticity that accounts for geometrically necessary dislocations. Journal of the Mechanics and Physics of Solids, 50:5–32, 2002.

    Article  MATH  MathSciNet  Google Scholar 

  23. K. Hackl. Generalized standard media and variational principles in classical and finite strain elastoplasticity. Journal of the Mechanics and Physics of Solids, 45(5):667–688, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  24. K. Hackl and U. Hoppe. On the calculation of microstructures for inelastic materials using relaxed energies. In C. Miehe, editor, IUTAM Symposium on Computational Mechanics of Solids at Large Strains, pages 77–86. Kluwer, 2003.

    Google Scholar 

  25. K. Hackl, A. Mielke, and D. Mittenhuber. Dissipation distances in multiplicative elastoplasticity. In W. Wendland and M. Efendiev, editors, Analysis and Simulation of Multifield Problems, pages 87–100. Springer-Verlag, 2003.

    Google Scholar 

  26. J. Hadamard. Leçons sur la propagation des ondes et les équations de l’hydrodynamique. Hermann, Paris, 1903.

    MATH  Google Scholar 

  27. R. Hill. Acceleration waves in solids. Journal of the Mechanics and Physics of Solids, 10:1–16, 1962.

    Article  MATH  MathSciNet  Google Scholar 

  28. D. Knees and A. Mielke. Energy release rate for cracks in finite-strain elasticity. Mathematical Methods in the Applied Sciences, 2006. Submitted (WIAS Preprint 1100).

    Google Scholar 

  29. R. V. Kohn. The relaxation of a double-well problem. Continuum Mechanics and Thermodynamics, 3:193–236, 1991.

    Article  MATH  MathSciNet  Google Scholar 

  30. R. V. Kohn and G. Strang. Optimal design and relaxation of variational problems i, ii, iii. Communications on Pure and Applied Mathematics, 39:113–137, 139–182, 353–377, 1986.

    MATH  MathSciNet  Google Scholar 

  31. A. Krawietz. Materialtheorie: Mathematische Beschreibung des phänomenologischen thermomechanischen Verhaltens. Springer-Verlag, Berlin, 1986.

    MATH  Google Scholar 

  32. M. Kružík, A. Mielke, and T. Roubíček. Modelling of microstructure and its evolution in shape-memory-alloy single-crystals, in particular in CuAlNi. Meccanica, 40:389–418, 2005.

    Article  MathSciNet  MATH  Google Scholar 

  33. M. Lambrecht, C. Miehe, and J. Dettmar. Energy relaxation of non-convex incremental stress potentials in a strain-softening elastic-plastic bar. International Journal of Solids and Structures, 40:1369–1391, 2003.

    Article  MATH  Google Scholar 

  34. M. Luskin. On the computation of crystalline microstructure. Acta Numerica, 36:191–257, 1996.

    Article  MathSciNet  Google Scholar 

  35. A. Mainik and A. Mielke. Existence results for energetic models for rateindependent systems. Calc. Var. PDEs, 22:73–99, 2005.

    Article  MATH  MathSciNet  Google Scholar 

  36. A. Mainik and A. Mielke. An existence result for rate-independent straingradient plasticity at finite strains. In preparation, 2006.

    Google Scholar 

  37. J. E. Marsden and T. J. R. Hughes. Mathematical Foundations of Elasticity. Dover Publications Inc., New York, 1994.

    MATH  Google Scholar 

  38. J. B. Martin. Plasticity. Fundamentals and General Results. MIT press, Cambridge, Massachusetts, 1975.

    Google Scholar 

  39. C. Miehe. Strain-driven homogenization of inelastic microstructures and composites based on an incremental variational formulation. International Journal for Numerical Methods in Engineering, 55:1285–1322, 2002.

    Article  MATH  MathSciNet  Google Scholar 

  40. C. Miehe and N. Apel. Anisotropic elastic-plastic analysis of shells at large strains. A comparison of multiplicative and additive approaches to enhanced finite element design and constitutive modeling. International Journal for Numerical Methods in Engineering, 61:2067–2113, 2004.

    Article  MATH  MathSciNet  Google Scholar 

  41. C. Miehe, N. Apel, and M. Lambrecht. Anisotropic additive plasticity in the logarithmic strain space: Modular kinematic formulation and implementation based on incremental minimization principles for standard materials. Computer Methods in Applied Mechanics and Engineering, 191:5383–5425, 2002.

    Article  MATH  MathSciNet  Google Scholar 

  42. C. Miehe and M. Becker. Incompatibility based strain gradient crystal plasticity. Submitted to Computer Methods in Applied Mechanics and Engineering, 2006.

    Google Scholar 

  43. C. Miehe and E. Gürses. A robust algorithm for configurational-force-driven brittle crack propagation with r-adaptive mesh alignment. Submitted to International Journal for Numerical Methods in Engineering, 2006.

    Google Scholar 

  44. C. Miehe and M. Lambrecht. 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, 58:1–41, 2003.

    Article  MATH  MathSciNet  Google Scholar 

  45. C. Miehe and M. Lambrecht. A two-scale finite element relaxation analysis of shear bands in non-convex inelastic solids: Small-strain theory for standard dissipative materials. Computer Methods in Applied Mechanics and Engineering, 192:473–508, 2003.

    Article  MATH  MathSciNet  Google Scholar 

  46. C. Miehe, M. Lambrecht, and E. Gürses. Analysis of material instabilities in inelastic solids by incremental energy minimization and relaxation methods: Evolving deformation microstructures in finite plasticity. Journal of the Mechanics and Physics of Solids, 52:2725–2769, 2004.

    Article  MATH  MathSciNet  Google Scholar 

  47. C. Miehe and J. Schotte. Anisotropic finite elastoplastic analysis of shells: Simulation of earing in deep-drawing of single-and polycrystalline sheets by taylortype micro-to-macro transitions. Computer Methods in Applied Mechanics and Engineering, 193:25–57, 2004.

    Article  MATH  MathSciNet  Google Scholar 

  48. C. Miehe and J. Schotte. Crystal plasticity and evolution of polycrystalline microstructure. In E. Stein, R. de Borst, and J. R. Hughes, editors, Encyclopedia of Computational Mechanics, chapter 8, pages 267–289. John Wiley & Sons, 2004.

    Google Scholar 

  49. C. Miehe, J. Schotte, and M. Lambrecht. Homogenization of inelastic solid materials at finite strains based on incremental minimization principles. Application to the texture analysis of polycrystals. Journal of the Mechanics and Physics of Solids, 50:2123–2167, 2002.

    Article  MATH  MathSciNet  Google Scholar 

  50. A. Mielke. Finite elastoplasticity, Lie groups and geodesics on SL(d). In P. Newton, A. Weinstein, and P. J. Holmes, editors, Geometry, Dynamics, and Mechanics, pages 61–90. Springer-Verlag, 2002.

    Google Scholar 

  51. A. Mielke. Energetic formulation of multiplicative elasto-plasticity using dissipation distances. Continuum Mechanics and Thermodynamics, 15:351–382, 2003.

    Article  MATH  MathSciNet  Google Scholar 

  52. A. Mielke. Evolution of rate-independent inelasticity with microstructure using relaxation and Young measures. In C. Miehe, editor, IUTAM Symposium on Computational Mechanics of Solid Materials at Large Strains (Stuttgart Aug. 2001), pages 33–44. Kluwer, 2003.

    Google Scholar 

  53. A. Mielke. Deriving new evolution equations for microstructures via relaxation of variational incremental problems. Computer Methods in Applied Mechanics and Engineering, 193:5095–5127, 2004.

    Article  MATH  MathSciNet  Google Scholar 

  54. A. Mielke. Existence of minimizers in incremental elasto-plasticity with finite strains. SIAM Journal on Mathematical Analysis, 36:384–404, 2004.

    Article  MATH  MathSciNet  Google Scholar 

  55. A. Mielke. Evolution in rate-independent systems (ch. 6). In C. Dafermos and E. Feireisl, editors, Handbook of Differential Equations, Evolutionary Equations, vol. 2, pages 461–559. Elsevier B.V., 2005.

    Google Scholar 

  56. A. Mielke. Necessary and sufficient conditions for polyconvexity of isotropic functions. Journal of Convex Analysis, 12:291–314, 2005.

    MathSciNet  MATH  Google Scholar 

  57. A. Mielke. A mathematical framework for generalized standard materials in the rate-independent case. In SFB404-Abschlussband. Springer-Verlag, 2006.

    Google Scholar 

  58. A. Mielke. Temperature-induced phase transformations in shape-memory alloys. In preparation, 2006.

    Google Scholar 

  59. A. Mielke and S. Müller. Lower semicontinuity and existence of minimizers for a functional in elastoplasticity. ZAMM Zeitschrift für Angewandte Mathematik und Mechanik, 86:233–250, 2006.

    Article  MATH  Google Scholar 

  60. A. Mielke and M. Ortiz. A class of minimum principles for characterizing the trajectories of dissipative systems. In preparation, 2006.

    Google Scholar 

  61. A. Mielke, R. Rossi, and G. Savaré. On a metric approach to a class of rateindependent problems. In preparation, 2006.

    Google Scholar 

  62. A. Mielke and F. Theil. On rate-independent hysteresis models. Nonlinear Partial Differential Equations and their Applications (NoDEA), 11:151–189, 2004. (Accepted July 2001).

    MATH  MathSciNet  Google Scholar 

  63. A. Mielke, F. Theil, and V. I. Levitas. A variational formulation of rateindependent phase transformations using an extremum principle. Archive of Rational Mechanics and Analysis, 162:137–177, 2002.

    Article  MATH  MathSciNet  Google Scholar 

  64. C. B. Morrey. Quasiconvexity and the semicontinuity of multiple integrands. Pacific Journal of Mathematics, 2:25–53, 1952.

    MATH  MathSciNet  Google Scholar 

  65. S. Müller. Variational models for microstructure and phase transisitions. In S. Hilderbrandt and M. Struwe, editors, Calculus of Variation and Geometric Evolution Problems, Lecture Notes in Mathematics 1713, pages 85–210. Springer Verlag, Berlin Heidelberg, 1999.

    Google Scholar 

  66. M. Ortiz and E. Repetto. Nonconvex energy minimization and dislocation structures in ductile single crystals. Journal of the Mechanics and Physics of Solids, 47(2):397–462, 1999.

    Article  MATH  MathSciNet  Google Scholar 

  67. M. Ortiz, E. A. Repetto, and L. Stainier. A theory of subgrain dislocation structures. Journal of the Mechanics and Physics of Solids, 48:2077–2114, 2000.

    Article  MATH  MathSciNet  Google Scholar 

  68. M. Ortiz and L. Stainier. The variational formulation of viscoplastic constitutive updates. Computer Methods in Applied Mechanics and Engineering, 171:419–444, 1999.

    Article  MATH  MathSciNet  Google Scholar 

  69. J. R. Rice. The localization of plastic deformation. In W. T. Koiter, editor, Theoretical and Applied Mechanics, pages 207–220. North-Holland, Amsterdam, 1976.

    Google Scholar 

  70. M. Šilhavý. The Mechanics and Thermodynamics of Continuous Media. Springer-Verlag, Berlin Heidelberg New York, 1997.

    MATH  Google Scholar 

  71. P. Steinmann. Views on multiplicative elastoplasticity and the continuum theory of dislocations. International Journal of Engineering Science, 34:1717–1735, 1996.

    Article  MATH  Google Scholar 

  72. B. Svendsen. Continuum thermodynamic models for crystal plasticity including the effects of geometrically-necessary dislocations. Journal of the Mechanics and Physics of Solids, 50:1297–1329, 2002.

    Article  MATH  MathSciNet  Google Scholar 

  73. T. Y. Thomas. Plastic Flow and Fracture in Solids. Academic Press, London, 1961.

    MATH  Google Scholar 

  74. C. Truesdell and W. Noll. The nonlinear field theories of mechanics. In S. Flügge, editor, Handbuch der Physik, chapter Bd. III/3. Springer-Verlag, Berlin, 1965.

    Google Scholar 

  75. L. C. Young. Lectures on the Calculus of Variations and Optimal Control Theory. Saunders, London, 1969.

    MATH  Google Scholar 

  76. H. Ziegler. Some extremum principles in irreversible thermodynamics with application to continuum mechanics. In I. N. Sneddon and H. R., editors, Progress in Solid Mechanics, Vol. IV. Springer Verlag, Berlin Heidelberg, 1963.

    Google Scholar 

  77. H. Ziegler and C. Wehrli. The derivation of constitutive relations from the free energy and the dissipation function. In Advances in applied mechanics, Vol. 25, pages 183–237. Academic Press, Orlando, FL, 1987.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Institute of Applied Mechanics, Faculty of Civil and Environmental Engineering, University of Stuttgart, Pfaffenwaldring 7, 70569, Stuttgart, Germany

    Ercan Gürses & Christian Miehe

  2. Institute of Analysis, Dynamics and Modeling, University of Stuttgart, Pfaffenwaldring 57, 70569, Stuttgart

    Andreas Mainik

  3. Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstraße 39, 10117, Berlin, Germany

    Alexander Mielke

  4. Department of Mathematics, Humboldt-Universität zu Berlin, Rudower Chaussee 25, 12489, Berlin-Adlershof, Germany

    Alexander Mielke

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  1. Ercan Gürses
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  2. Andreas Mainik
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  3. Christian Miehe
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  4. Alexander Mielke
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Editors and Affiliations

  1. Institut für Wasserbau, Hydromechanik und Hydrosystemmodellierung, Universität Stuttgart, Pfaffenwaldring 61, 70569, Stuttgart, Germany

    Rainer Helmig

  2. Weierstraß-Institut für Angewandte Analysis und Stochastik, Mohrenstraße 39, 10117, Berlin, Germany

    Alexander Mielke

  3. Institut für Angewandte Analysis und Numerische Simulation, Universität Stuttgart, Pfaffenwaldring 57, 70569, Stuttgart, Germany

    Barbara I. Wohlmuth

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Gürses, E., Mainik, A., Miehe, C., Mielke, A. (2006). Analytical and Numerical Methods for Finite-Strain Elastoplasticity. In: Helmig, R., Mielke, A., Wohlmuth, B.I. (eds) Multifield Problems in Solid and Fluid Mechanics. Lecture Notes in Applied and Computational Mechanics, vol 28. Springer, Berlin, Heidelberg . https://doi.org/10.1007/978-3-540-34961-7_15

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