Local existence and uniqueness for quasistatic finite plasticity with grain boundary relaxation
Author:
Patrizio Neff
Journal:
Quart. Appl. Math. 63 (2005), 88-116
MSC (2000):
Primary 74A35, 74C05, 74C10, 74C20, 74D10, 74E05, 74E10, 74E15, 74G30, 74G65, 74N15
DOI:
https://doi.org/10.1090/S0033-569X-05-00953-9
Published electronically:
January 20, 2005
MathSciNet review:
2126571
Full-text PDF Free Access
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Abstract: This paper is concerned with a phenomenological model of initially isotropic finite-strain multiplicative elasto-plasticity for polycrystals with grain boundary relaxation (Neff, Cont. Mech. Thermo., 2003). We prove a local in time existence and uniqueness result of the corresponding initial boundary value problem in the quasistatic rate-dependent case. Use is made of a generalized Korn first inequality (Neff, Proc. Roy. Soc. Edinb. A, 2002) taking into account the incompatibility of the plastic deformation $F_p$. This is a first result concerning classical solutions in geometrically exact nonlinear finite visco-plasticity for polycrystals. Global existence is not proved and cannot be expected due to the natural possibility of material degradation in time.
- Hans-Dieter Alber, Materials with memory, Lecture Notes in Mathematics, vol. 1682, Springer-Verlag, Berlin, 1998. Initial-boundary value problems for constitutive equations with internal variables. MR 1619546
- Karl-Heinz Anthony, Die Theorie der nichtmetrischen Spannungen in Kristallen, Arch. Rational Mech. Anal. 40 (1970/71), 50–78 (German, with English summary). MR 267823, DOI https://doi.org/10.1007/BF00281530
- A. Bertram, An alternative approach to finite plasticity based on material isomorphisms, Int. J. Plasticity 52 (1998), 353–374.
- A. Bertram and B. Svendsen, On material objectivity and reduced constitutive equations, Arch. Mech. (Arch. Mech. Stos.) 53 (2001), no. 6, 653–675. MR 1885766
- J. F. Besseling and E. van der Giessen, Mathematical modelling of inelastic deformation, Applied Mathematics and Mathematical Computation, vol. 5, Chapman & Hall, London, 1994. MR 1284037
- Kaushik Bhattacharya and Robert V. Kohn, Elastic energy minimization and the recoverable strains of polycrystalline shape-memory materials, Arch. Rational Mech. Anal. 139 (1997), no. 2, 99–180. MR 1478776, DOI https://doi.org/10.1007/s002050050049
- Frederick Bloom, Modern differential geometric techniques in the theory of continuous distributions of dislocations, Lecture Notes in Mathematics, vol. 733, Springer, Berlin, 1979. MR 546839
- S.R. Bodner and Y. Partom, Constitutive equations for elastic-viscoplastic strainhardening materials, J. Appl. Mech. 42 (1975), 385–389.
- O.T. Bruhns, H. Xiao, and A. Mayers, Some basic issues in traditional eulerian formulations of finite elastoplasticity, Int. J. Plast. 19 (2003), 2007–2026.
- C. Carstensen and K. Hackl, On microstructure occurring in a model of finite-strain elastoplasticity involving a single slip system, ZAMM (2000).
- Carsten Carstensen, Klaus Hackl, and Alexander Mielke, Non-convex potentials and microstructures in finite-strain plasticity, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 458 (2002), no. 2018, 299–317. MR 1889770, DOI https://doi.org/10.1098/rspa.2001.0864
- J. Casey and P.M. Naghdi, A remark on the use of the decomposition $F=F_e\cdot F_p$ in plasticity, J. Appl. Mech. 47 (1980), 672–675.
- P. Cermelli and M.E. Gurtin, On the characterization of geometrically necessary dislocations in finite plasticity, J. Mech. Phys. Solids 49 (2001), 1539–1568.
- Krzysztof Chełmiński, On self-controlling models in the theory of nonelastic material behavior of metals, Contin. Mech. Thermodyn. 10 (1998), no. 3, 121–133. MR 1634932, DOI https://doi.org/10.1007/s001610050085
- Krzysztof Chełmiński, On monotone plastic constitutive equations with polynomial growth condition, Math. Methods Appl. Sci. 22 (1999), no. 7, 547–562. MR 1682911, DOI https://doi.org/10.1002/%28SICI%291099-1476%2819990510%2922%3A7%3C547%3A%3AAID-MMA52%3E3.0.CO%3B2-T
- Philippe G. Ciarlet, Mathematical elasticity. Vol. I, Studies in Mathematics and its Applications, vol. 20, North-Holland Publishing Co., Amsterdam, 1988. Three-dimensional elasticity. MR 936420
- Y.F. Dafalias, The plastic spin concept and a simple illustration of its role in finite plastic transformations, Mechanics of Materials 3 (1984), 223–233.
- ---, Issues on the constitutive formulation at large elastoplastic deformations, Part I: Kinematics, Acta Mechanica 69 (1987), 119–138.
- ---, Plastic spin: necessity or redundancy?, Int. J. Plasticity 14 (1998), no. 9, 909–931.
- S. Ebenfeld, Aspekte der Kontinua mit Mikrostruktur, Berichte aus der Mathematik, Shaker Verlag, Aachen, 1998.
- ---, $L^2$-regularity theory of linear strongly elliptic Dirichlet systems of order $2m$ with minimal regularity in the coefficients, Preprint Nr. 2015, TU Darmstadt, (1998).
- Stefan Ebenfeld, $L^2$-regularity theory of linear strongly elliptic Dirichlet systems of order $2m$ with minimal regularity in the coefficients, Quart. Appl. Math. 60 (2002), no. 3, 547–576. MR 1914441, DOI https://doi.org/10.1090/qam/1914441
- P. Ellsiepen and S. Hartmann, Remarks on the interpretation of current non-linear finite-element-analyses as differential-algebraic equations, Int. J. Num. Meth. Engrg. 51 (2001), 679–707.
- A. E. Green and P. M. Naghdi, A general theory of an elastic-plastic continuum, Arch. Rational Mech. Anal. 18 (1965), 251–281; corrigenda, ibid. 19 (1965), 408. MR 223129, DOI https://doi.org/10.1007/BF00251666
- Morton E. Gurtin, On the plasticity of single crystals: free energy, microforces, plastic-strain gradients, J. Mech. Phys. Solids 48 (2000), no. 5, 989–1036. MR 1746552, DOI https://doi.org/10.1016/S0022-5096%2899%2900059-9
- Klaus Hackl, Generalized standard media and variational principles in classical and finite strain elastoplasticity, J. Mech. Phys. Solids 45 (1997), no. 5, 667–688. MR 1447372, DOI https://doi.org/10.1016/S0022-5096%2896%2900110-X
- Weimin Han and B. Daya Reddy, Plasticity, Interdisciplinary Applied Mathematics, vol. 9, Springer-Verlag, New York, 1999. Mathematical theory and numerical analysis. MR 1681061
- P. Haupt, On the concept of an intermediate configuration and its application to a representation of viscoelastic-plastic material behaviour, Int. J. Plasticity 1 (1985), 303–316.
- Peter Haupt, Continuum mechanics and theory of materials, Advanced Texts in Physics, Springer-Verlag, Berlin, 2000. Translated from the German by Joan A. Kurth. MR 1762655
- Ioan R. Ionescu and Mircea Sofonea, Functional and numerical methods in viscoplasticity, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1993. MR 1244578
- L.M. Kachanov, Introduction to Continuum Damage Mechanics, Martinus Nijhoff Publishers, Dordrecht, 1986.
- Memoirs of the unifying study of the basic problems in engineering sciences by means of geometry. Vol. I, Gakujutsu Bunken Fukyu-Kai, Tokyo, 1955. Kazuo Kondo, Chairman. MR 0080956
- Memoirs of the unifying study of the basic problems in engineering sciences by means of geometry. Vol. I, Gakujutsu Bunken Fukyu-Kai, Tokyo, 1955. Kazuo Kondo, Chairman. MR 0080956
- Arnold Krawietz, Materialtheorie, Springer-Verlag, Berlin, 1986 (German). Mathematische Beschreibung des phänomenologischen thermomechanischen Verhaltens. [Mathematical description of phenomenological and thermomechanic behavior]. MR 959732
- Ekkehart Kröner, Kontinuumstheorie der Versetzungen und Eigenspannungen, Ergebnisse der angewandten Mathematik. Bd. 5, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1958 (German). MR 0095615
- ---, Dislocation: a new concept in the continuum theory of plasticity, J. Math. Phys. 42 (1962), 27–37.
- Ekkehart Kröner and Alfred Seeger, Nicht-lineare Elastizitätstheorie der Versetzungen und Eigenspannungen, Arch. Rational Mech. Anal. 3 (1959), 97–119 (German). MR 106587, DOI https://doi.org/10.1007/BF00284168
- E.H. Lee, Elastic-plastic deformation at finite strain, J. Appl. Mech. 36 (1969), 1–6.
- J. Lemaitre and J.L. Chaboche, Mécanique des matériaux solides, Dunod, Paris, 1985.
- Massimiliano Lucchesi and Paolo Podio-Guidugli, Materials with elastic range: a theory with a view toward applications. I, Arch. Rational Mech. Anal. 102 (1988), no. 1, 23–43. MR 938382, DOI https://doi.org/10.1007/BF00250922
- J. Mandel, Equations constitutives et directeurs dans les milieux plastiques et viscoplastiques, Int. J. Solids Structures 9 (1973), 725–740.
- Jerrold E. Marsden and Thomas J. R. Hughes, Mathematical foundations of elasticity, Dover Publications, Inc., New York, 1994. Corrected reprint of the 1983 original. MR 1262126
- Gérard A. Maugin, Material inhomogeneities in elasticity, Applied Mathematics and Mathematical Computation, vol. 3, Chapman & Hall, London, 1993. MR 1250832
- ---, The Thermomechanics of Nonlinear Irreversible Behaviors, Nonlinear Science, no. 27, World Scientific, Singapore, 1999.
- G.A. Maugin and M. Epstein, Geometrical material structure of elastoplasticity, Int. J. Plasticity 14 (1998), 109–115.
- A. Mayers, P. Schiebe, and O.T. Bruhns, Some comments on objective rates of symmetric eulerian tensors with application to eulerian strain rates, Acta Mech. 139 (2000), 91–103.
- C. Miehe, A theory of large-strain isotropic thermoplasticity based on metric transformation tensors, Archive Appl. Mech. 66 (1995), 45–64.
- A. Mielke, Finite elastoplasticity, Lie groups and geodesics on SL(d), Sonderforschungsbereich 404 Bericht 2000/26, University of Stuttgart, 2000.
- A. I. Murdoch, Objectivity in classical continuum physics: a rationale for discarding the ‘principle of invariance under superposed rigid body motions’ in favour of purely objective considerations, Contin. Mech. Thermodyn. 15 (2003), no. 3, 309–320. MR 1986708, DOI https://doi.org/10.1007/s00161-003-0121-9
- W. Muschik and L. Restuccia, Changing the observer and moving materials in continuum physics: objectivity and frame-indifference, Technische Mechanik 22 (2002), 152–160.
- P. M. Naghdi, A critical review of the state of finite plasticity, Z. Angew. Math. Phys. 41 (1990), no. 3, 315–394. MR 1058818, DOI https://doi.org/10.1007/BF00959986
- A. Needleman, Material rate dependence and mesh sensitivity in localization problems, Comp. Meth. Appl. Mech. Engrg. 67 (1988), 69–85.
- P. Neff, Formulation of visco-plastic approximations in finite plasticity for a model of small elastic strains, Part I: Modelling, Preprint 2127, TU Darmstadt, 2000.
- ---, Mathematische Analyse multiplikativer Viskoplastizität, Ph.D. Thesis, TU Darmstadt, Shaker Verlag, ISBN:3-8265-7560-1, Aachen, 2000.
- ---, Formulation of visco-plastic approximations in finite plasticity for a model of small elastic strains, Part IIa: Local existence and uniqueness results, Preprint 2138, TU Darmstadt, 2001.
- Patrizio Neff, On Korn’s first inequality with non-constant coefficients, Proc. Roy. Soc. Edinburgh Sect. A 132 (2002), no. 1, 221–243. MR 1884478, DOI https://doi.org/10.1017/S0308210500001591
- Patrizio Neff, Finite multiplicative plasticity for small elastic strains with linear balance equations and grain boundary relaxation, Contin. Mech. Thermodyn. 15 (2003), no. 2, 161–195. MR 1970291, DOI https://doi.org/10.1007/s00161-002-0109-x
- Patrizio Neff and Christian Wieners, Comparison of models for finite plasticity: a numerical study, Comput. Vis. Sci. 6 (2003), no. 1, 23–35. MR 1985199, DOI https://doi.org/10.1007/s00791-003-0104-1
- M. Ortiz and E.A. Repetto, Nonconvex energy minimization and dislocation structures in ductile single crystals, manuscript, 1993.
- M. Ortiz and E. A. Repetto, Nonconvex energy minimization and dislocation structures in ductile single crystals, J. Mech. Phys. Solids 47 (1999), no. 2, 397–462. MR 1674064, DOI https://doi.org/10.1016/S0022-5096%2897%2900096-3
- M. Ortiz, E. A. Repetto, and L. Stainier, A theory of subgrain dislocation structures, J. Mech. Phys. Solids 48 (2000), no. 10, 2077–2114. MR 1778727, DOI https://doi.org/10.1016/S0022-5096%2899%2900104-0
- M. Ortiz and L. Stainier, The variational formulation of viscoplastic constitutive updates, Comput. Methods Appl. Mech. Engrg. 171 (1999), no. 3-4, 419–444. MR 1685716, DOI https://doi.org/10.1016/S0045-7825%2898%2900219-9
- Waldemar Pompe, Korn’s first inequality with variable coefficients and its generalization, Comment. Math. Univ. Carolin. 44 (2003), no. 1, 57–70. MR 2045845
- Annie Raoult, Nonpolyconvexity of the stored energy function of a Saint-Venant-Kirchhoff material, Apl. Mat. 31 (1986), no. 6, 417–419 (English, with Russian and Czech summaries). MR 870478
- M.B. Rubin, Plasticity theory formulated in terms of physically based microstructrural variables– Part I. Theory, Int. J. Solids Structures 31 (1994), no. 19, 2615–2634.
- C. Sansour and F.G. Kollmann, On theory and numerics of large viscoplastic deformation, Comp. Meth. Appl. Mech. Engrg. 146 (1996), 351–369.
- Miroslav Šilhavý, On transformation laws for plastic deformations of materials with elastic range, Arch. Rational Mech. Anal. 63 (1976), no. 2, 169–182. MR 669403, DOI https://doi.org/10.1007/BF00280603
- J. C. Simo, Numerical analysis and simulation of plasticity, Handbook of numerical analysis, Vol. VI, Handb. Numer. Anal., VI, North-Holland, Amsterdam, 1998, pp. 183–499. MR 1665428
- J. C. Simo and T. J. R. Hughes, Computational inelasticity, Interdisciplinary Applied Mathematics, vol. 7, Springer-Verlag, New York, 1998. MR 1642789
- J.C. Simo and M. Ortiz, A unified approach to finite deformation elastoplastic analysis based on the use of hyperelastic constitutive equations, Comp. Meth. Appl. Mech. Engrg. 49 (1985), 221–245.
- P. Steinmann, Views on multiplicative elastoplasticity and the continuum theory of dislocations, Int. J. Engrg. Sci. 34 (1996), 1717–1735.
- B. Svendsen and A. Bertram, On frame-indifference and form-invariance in constitutive theory, Acta Mech. 132 (1999), no. 1-4, 195–207. MR 1659588, DOI https://doi.org/10.1007/BF01186967
- Tullio Valent, Boundary value problems of finite elasticity, Springer Tracts in Natural Philosophy, vol. 31, Springer-Verlag, New York, 1988. Local theorems on existence, uniqueness, and analytic dependence on data. MR 917733
- C. Wieners, M. Amman, S. Diebels, and W. Ehlers, Parallel 3-d simulations for porous media models in soil mechanics, Comp. Mech. 29 (2002), 75–87.
- 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.
- K.H. Anthony, Die Theorie der nichtmetrischen Spannungen in Kristallen, Arch. Rat. Mech. Anal. 40 (1971), 50–78.
- A. Bertram, An alternative approach to finite plasticity based on material isomorphisms, Int. J. Plasticity 52 (1998), 353–374.
- A. Bertram and B. Svendsen, On material objectivity and reduced constitutive equations, Arch. Mech. 53 (2001), 653–675.
- J.F. Besseling and E. van der Giessen, Mathematical Modelling of Inelastic Deformation, Applied Mathematics and Mathematical Computation, vol. 5, Chapman Hall, London, 1994.
- K. Bhattacharya and R.V. Kohn, Elastic energy minimization and the recoverable strains of polycrystalline shape memory materials, Arch. Rat. Mech. Anal. 139 (1997), 99–180.
- F. Bloom, Modern Differential Geometric Techniques in the Theory of Continuous Distributions of Dislocations, Lecture Notes in Mathematics, vol. 733, Springer, Berlin, 1979.
- S.R. Bodner and Y. Partom, Constitutive equations for elastic-viscoplastic strainhardening materials, J. Appl. Mech. 42 (1975), 385–389.
- O.T. Bruhns, H. Xiao, and A. Mayers, Some basic issues in traditional eulerian formulations of finite elastoplasticity, Int. J. Plast. 19 (2003), 2007–2026.
- C. Carstensen and K. Hackl, On microstructure occurring in a model of finite-strain elastoplasticity involving a single slip system, ZAMM (2000).
- C. Carstensen, K. Hackl, and A. Mielke, Non-convex potentials and microstructure in finite strain plasticity, Proc. Roy. Soc. London, Ser. A 458 (2002), 299–317.
- J. Casey and P.M. Naghdi, A remark on the use of the decomposition $F=F_e\cdot F_p$ in plasticity, J. Appl. Mech. 47 (1980), 672–675.
- P. Cermelli and M.E. Gurtin, On the characterization of geometrically necessary dislocations in finite plasticity, J. Mech. Phys. Solids 49 (2001), 1539–1568.
- K. Chelminski, On self-controlling models in the theory of inelastic material behaviour of metals, Cont. Mech. Thermodyn. 10 (1998), 121–133.
- ---, On monotone plastic constitutive equations with polynomial growth condition, Math. Meth. App. Sc. 22 (1999), 547–562.
- P.G. Ciarlet, Three-Dimensional Elasticity, first ed., Studies in Mathematics and its Applications, vol. 1, Elsevier, Amsterdam, 1988.
- Y.F. Dafalias, The plastic spin concept and a simple illustration of its role in finite plastic transformations, Mechanics of Materials 3 (1984), 223–233.
- ---, Issues on the constitutive formulation at large elastoplastic deformations, Part I: Kinematics, Acta Mechanica 69 (1987), 119–138.
- ---, Plastic spin: necessity or redundancy?, Int. J. Plasticity 14 (1998), no. 9, 909–931.
- S. Ebenfeld, Aspekte der Kontinua mit Mikrostruktur, Berichte aus der Mathematik, Shaker Verlag, Aachen, 1998.
- ---, $L^2$-regularity theory of linear strongly elliptic Dirichlet systems of order $2m$ with minimal regularity in the coefficients, Preprint Nr. 2015, TU Darmstadt, (1998).
- ---, $L^2$-regularity theory of linear strongly elliptic Dirichlet systems of order $2m$ with minimal regularity in the coefficients, Quart. Appl. Math. 60 (2002), no. 3, 547–576.
- P. Ellsiepen and S. Hartmann, Remarks on the interpretation of current non-linear finite-element-analyses as differential-algebraic equations, Int. J. Num. Meth. Engrg. 51 (2001), 679–707.
- A.E. Green and P.M. Naghdi, A general theory of an elastic-plastic continuum, Arch. Rat. Mech. Anal. 18 (1965), 251–281.
- M.E. Gurtin, On the plasticity of single crystals: free energy, microforces, plastic-strain gradients, J. Mech. Phys. Solids 48 (2000), 989–1036.
- K. Hackl, Generalized standard media and variational principles in classical and finite strain elastoplasticity, J. Mech. Phys. Solids 45 (1997), 667–688.
- W. Han and B.D. Reddy, Plasticity. Mathematical Theory and Numerical Analysis, Springer, Berlin, 1999.
- P. Haupt, On the concept of an intermediate configuration and its application to a representation of viscoelastic-plastic material behaviour, Int. J. Plasticity 1 (1985), 303–316.
- ---, Continuum Mechanics and Theory of Materials, Springer, Heidelberg, 2000.
- I.R. Ionescu and M. Sofonea, Functional and Numerical Methods in Viscoplasticity, first ed., Oxford Mathematical Monographs, Oxford University Press, Oxford, 1993.
- L.M. Kachanov, Introduction to Continuum Damage Mechanics, Martinus Nijhoff Publishers, Dordrecht, 1986.
- K. Kondo, Geometry of elastic deformation and incompatibility, Memoirs of the Unifying Study of the Basic Problems in Engineering Science by Means of Geometry (K. Kondo, ed.), vol. 1, Division C, Gakujutsu Bunken Fukyo-Kai, 1955, pp. 5–17 (361–373).
- ---, Non-Riemannien geometry of imperfect crystals from a macroscopic viewpoint, Memoirs of the Unifying Study of the Basic Problems in Engineering Science by Means of Geometry (K. Kondo, ed.), vol. 1, Division D-I, Gakujutsu Bunken Fukyo-Kai, 1955, pp. 6–17 (458–469).
- A. Krawietz, Materialtheorie, Springer, Berlin, 1986.
- E. Kröner, Kontinuumstheorie der Versetzungen und Eigenspannungen, Ergebnisse der Angewandten Mathematik, vol. 5, Springer, Berlin, 1958.
- ---, Dislocation: a new concept in the continuum theory of plasticity, J. Math. Phys. 42 (1962), 27–37.
- E. Kröner and A. Seeger, Nichtlineare Elastizitätstheorie der Versetzungen und Eigenspannungen, Arch. Rat. Mech. Anal. 3 (1959), 97–119.
- E.H. Lee, Elastic-plastic deformation at finite strain, J. Appl. Mech. 36 (1969), 1–6.
- J. Lemaitre and J.L. Chaboche, Mécanique des matériaux solides, Dunod, Paris, 1985.
- M. Lucchesi and P. Podio-Guidugli, Materials with elastic range: a theory with a view toward applications. Part I, Arch. Rat. Mech. Anal. 102 (1988), 23–43.
- J. Mandel, Equations constitutives et directeurs dans les milieux plastiques et viscoplastiques, Int. J. Solids Structures 9 (1973), 725–740.
- J.E. Marsden and J.R. Hughes, Mathematical Foundations of Elasticity, Prentice-Hall, Englewood Cliffs, New Jersey, 1983.
- G. Maugin, Material Inhomogeneities in Elasticity, Applied Mathematics and Mathematical Computations, Chapman-Hall, London, 1993.
- ---, The Thermomechanics of Nonlinear Irreversible Behaviors, Nonlinear Science, no. 27, World Scientific, Singapore, 1999.
- G.A. Maugin and M. Epstein, Geometrical material structure of elastoplasticity, Int. J. Plasticity 14 (1998), 109–115.
- A. Mayers, P. Schiebe, and O.T. Bruhns, Some comments on objective rates of symmetric eulerian tensors with application to eulerian strain rates, Acta Mech. 139 (2000), 91–103.
- C. Miehe, A theory of large-strain isotropic thermoplasticity based on metric transformation tensors, Archive Appl. Mech. 66 (1995), 45–64.
- A. Mielke, Finite elastoplasticity, Lie groups and geodesics on SL(d), Sonderforschungsbereich 404 Bericht 2000/26, University of Stuttgart, 2000.
- A.I. Murdoch, Objectivity in classical continuum physics: a rationale for discarding the ‘principle of invariance under superposed rigid body motions’ in favour of purely objective considerations, Cont. Mech. Thermo. 15 (2003), 309–320.
- W. Muschik and L. Restuccia, Changing the observer and moving materials in continuum physics: objectivity and frame-indifference, Technische Mechanik 22 (2002), 152–160.
- P.M. Naghdi, A critical review of the state of finite plasticity, J. Appl. Math. Phys.(ZAMP) 41 (1990), 315–394.
- A. Needleman, Material rate dependence and mesh sensitivity in localization problems, Comp. Meth. Appl. Mech. Engrg. 67 (1988), 69–85.
- P. Neff, Formulation of visco-plastic approximations in finite plasticity for a model of small elastic strains, Part I: Modelling, Preprint 2127, TU Darmstadt, 2000.
- ---, Mathematische Analyse multiplikativer Viskoplastizität, Ph.D. Thesis, TU Darmstadt, Shaker Verlag, ISBN:3-8265-7560-1, Aachen, 2000.
- ---, Formulation of visco-plastic approximations in finite plasticity for a model of small elastic strains, Part IIa: Local existence and uniqueness results, Preprint 2138, TU Darmstadt, 2001.
- ---, On Korn’s first inequality with nonconstant coefficients, Proc. Roy. Soc. Edinb. 132A (2002), 221–243.
- ---, Finite multiplicative plasticity for small elastic strains with linear balance equations and grain boundary relaxation, Cont. Mech. Thermodynamics 15(2) (2003), no. DOI 10.1007/s00161-002-0190-x, 161–195.
- P. Neff and C. Wieners, Comparison of models for finite plasticity. A numerical study, Comput. Visual. Sci. 6 (2003), 23–35.
- M. Ortiz and E.A. Repetto, Nonconvex energy minimization and dislocation structures in ductile single crystals, manuscript, 1993.
- ---, Nonconvex energy minimization and dislocation structures in ductile single crystals, J. Mech. Phys. Solids 47 (1999), 397–462.
- M. Ortiz, E.A. Repetto, and L. Stainier, A theory of subgrain dislocation structures, J. Mech. Phys. Solids 48 (2000), 2077–2114.
- M. Ortiz and L. Stainier, The variational formulation of viscoplastic constitutive updates, Comp. Meth. Appl. Mech. Engrg. 171 (1999), 419–444.
- W. Pompe, Korn’s first inequality with variable coefficients and its generalizations, Comment. Math. Univ. Carolinae 44,1 (2003), 57–70.
- A. Raoult, Non-polyconvexity of the stored energy function of a St.Venant-Kirchhoff material, Aplikace Matematiky 6 (1986), 417–419.
- M.B. Rubin, Plasticity theory formulated in terms of physically based microstructrural variables– Part I. Theory, Int. J. Solids Structures 31 (1994), no. 19, 2615–2634.
- C. Sansour and F.G. Kollmann, On theory and numerics of large viscoplastic deformation, Comp. Meth. Appl. Mech. Engrg. 146 (1996), 351–369.
- M. Silhavy, On transformation laws for plastic deformations of materials with elastic range, Arch. Rat. Mech. Anal. 63 (1976), 169–182.
- J.C. Simo, Numerical analysis and simulation of plasticity, Handbook of Numerical Analysis (P.G. Ciarlet and J.L. Lions, eds.), vol. VI, Elsevier, Amsterdam, 1998.
- J.C. Simo and J.R. Hughes, Computational Inelasticity, Interdisciplinary Applied Mathematics, vol. 7, Springer, Berlin, 1998.
- J.C. Simo and M. Ortiz, A unified approach to finite deformation elastoplastic analysis based on the use of hyperelastic constitutive equations, Comp. Meth. Appl. Mech. Engrg. 49 (1985), 221–245.
- P. Steinmann, Views on multiplicative elastoplasticity and the continuum theory of dislocations, Int. J. Engrg. Sci. 34 (1996), 1717–1735.
- B. Svendsen and A. Bertram, On frame-indifference and form-invariance in constitutive theory, Acta Mechanica 132 (1997), 195–207.
- T. Valent, Boundary Value Problems of Finite Elasticity, Springer, Berlin, 1988.
- C. Wieners, M. Amman, S. Diebels, and W. Ehlers, Parallel 3-d simulations for porous media models in soil mechanics, Comp. Mech. 29 (2002), 75–87.
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Additional Information
Patrizio Neff
Affiliation:
AG6, Fachbereich Mathematik, Darmstadt University of Technology, Schlossgartenstrasse 7, 64289 Darmstadt, Germany
Address at time of publication:
Department of Mathematics, University of Technology, Darmstadt
Email:
neff@mathematik.tu-darmstadt.de
Keywords:
Plasticity,
visco-plasticity,
solid mechanics,
elliptic systems,
variational methods.
Received by editor(s):
May 13, 2004
Published electronically:
January 20, 2005
Article copyright:
© Copyright 2005
Brown University