# Recent Developments in the Numerical Analysis of Plasticity

## Abstract

The goal of this lectures is to survey some recent developments in the numerical analysis of classical plasticity and viscoplasticity. For the infinitesimal theory, the continuum mechanics aspects of the subject are currently well understood and firmly established. Classical expositions of the basic theory can be found in the work of HILL [1950], KOITER [1960] and others. On the mathematical side, classical plasticity experienced a significant development in the 70’s and early 80’s, starting with the pioneering work of DUVAUT & LIONS [1972]. The subsequent improvement of JOHNSON [1978], MATTHIES [1979], SUQUET [1979], TEMAM & STRANG [1980] and others produced at the beginning of the 80’s a fairly complete mathematical picture of the theory.

## Keywords

Variational Inequality Finite Strain Principal Stretch Elastic Domain Multiplicative Decomposition## Preview

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## References

- J.H. Argyris and J. St. Doltsinis, [ 1979 ], “On the Large Strain Inelastic Analysis in Natural Formulation. Part I. Quasistatic Problems”,
*Computer Methods in Applied Mechanics and Engineering***20**, 213–252.CrossRefzbMATHGoogle Scholar - J.H. Argyris J.St. Doltsinis, P.M. Pimenta and H. Wüstenberg, [ 1982 ], “Thermomechanical Response of Solids at High Strains — Natural Approach”,
*Computer Methods in Applied Mechanics and Engineering***32**, 3–57.MathSciNetCrossRefzbMATHGoogle Scholar - R. Asaro, [ 1983 ], “Micromechanics of Crystals and Polycrystals” In:
*Advances in Applied Mechanics*(Ed.: T.Y. Wu, J.W. Hutchinson ),**23**, 1–115.Google Scholar - K. Burrage, and J.C. Butcher, [ 1979 ], “Stability Criteria for Implicit Runge-Kutta Methods,”
*SIAM J. Num. Math*.,**16**, 46–57.MathSciNetCrossRefzbMATHGoogle Scholar - K. Burrage, and J.C. Butcher, [ 1980 ], “Nonlinear Stability of a General Class of Differential Equation Methods,”
*BIT*,**20**, 185–203.MathSciNetCrossRefzbMATHGoogle Scholar - J.C. Butcher, [ 1975 ], “A Stability Property of Implicit Runge-Kutta Methods,”
*BIT*,**15**, 358–361.CrossRefzbMATHGoogle Scholar - F. Brezzi and M. Fortin, [ 1990 ],
*Mixed and Hibrid Finite Element Methods*, Springer-Verlag, Berlin (to appear).Google Scholar - P.G. Ciarlet, [ 1978 ],
*The Finite Element Method for Elliptic Problems*, North Holland, Amsterdam.zbMATHGoogle Scholar - P.G. Ciarlet, [ 1981 ],
*Three-dimensional Mathematical Elasticity*, North-Holland, Amsterdam.Google Scholar - B.D. Coleman, and M.E. Gurtin, [ 1967 ], “Thermodynamics with Internal State Variables”,
*J. Chemistry and Physics***47**, 597–613.CrossRefGoogle Scholar - A. Corigliano and U. Perego, [ 1991 ], “Convergent and unconditionally stable finite-step dynamic analysis of elastoplastic structures,”
*Proceedings of the European Conferemce on New Advanced in Computational Structural Mechanics*, pp. 577–584.Google Scholar - Commi, C. and G. Maier [ 1989 ], “On the Convergence of a Backward Difference Iterative Procedure in Elastoplasticity with Nonlinear Kinematic and Isotropic Hardening,”
*Computational Plasticity*,*Models*,*Software and Applications*, D.R.J. Owen, E. Hinton and E. Onate, Editors. Pineridge Press**1**, 3230334.Google Scholar - Dahlquist, G., [ 1963 ], “A Special Stability Problem for Linear Multistep Methods,”
*BIT*,**3**, 27–43.MathSciNetCrossRefzbMATHGoogle Scholar - G. Dahlquist and R, Jeltsch, [ 1979 ], “Generalized Disks of Contractivity and Implicit Runge—Kutta Methods,”
*Inst for Numerisk Analys*, Report No. TRITA—NA7906.Google Scholar - F. Demengel [ 1989 ] “Compactness Theorems for Spaces of Functions with Bounded Derivatives and Applications to Limit Analysis Problems in Plasticity,”
*Archi ve for Rational Mechanics and Analysis*, 105, No. 2, pp. 123–161.MathSciNetzbMATHGoogle Scholar - G. Duvaut, and J.L. Lions, [ 1972 ],
*Les Inequations en Mecanique et en Physique*, Dunot, Paris.zbMATHGoogle Scholar - A.L. Eterovich and K.J. Bathe, [ 1990 ], “A hyperelastic-based large strain elastoplastic constitutive formulation with combined isotropic-kinematic hardening using logarithmic stresses and strain measures,”
*International Journal for Numerical Methods in Engineering*, 30, No. 6, 1099–1115.CrossRefGoogle Scholar - R.A. Eve, B.D. Reddy
*and*R.T. Rocckafellar, [1991], “An internal variable theory of elastoplasticity based on the maximum work inequality,”*Quarterly J. Applied Mathematics*,to appear.Google Scholar - J.O. Hallquist and D.J. Benson, [1987],
*DYNASD User’s Manual*,Report No. UCID19592, Rev.3, Lawrence Livermore National Laboratory.Google Scholar - H.M. Hilber [ 1976 ],
*Analysis and design of numerical integration methods in structural dynamics*, Report No. E.E.R.C. 76–29. Earthquake Eng. Research Center. University of California, Berkeley, CA.Google Scholar - H.M. Hilber, T.J.R. Hughes
*and*R.L. Taylor, [1977], “Improved numerical dissipation for time integration algorithms in structural dynamics,”*Earthquake Engng. 8 Struct. Dyn*.,**5**, 283–292.Google Scholar - R. Hill, [ 1950 ],
*The Mathematical Theory of Plasticity*, Clarendon, Oxford.zbMATHGoogle Scholar - T.J.R. Hughes, [ 1980 ], “Generalization of Selective Integration Procedures to Aniso tropic and Nonlinear Media,”
*International J. Numerical Methods in Engineering*,**15**, 1413–1418.CrossRefzbMATHGoogle Scholar - T.J.R. Hughes, [ 1984 ], “Numerical Implementation of Constitutive Models: Rate Independent Deviatoric Plasticity,” in
*Theoretical Foundations for Large Scale Computations of Nonlinear Material Behavior*, Editors S. Nemat-Nasser, R. Asaro and G. Hegemier, Martinus Nijhoff Publishers, The Netherlands.Google Scholar - T.J.R. Hughes, [ 1983 ], “Analysis of Transient Algorithms with Special Reference to Stability Behavior,” in
*Computational Methods for Transient Analysis*,*Volume**1*, T. Belytschko and T.J.R. Hugues Editors, Noth—Holland; Amsterdam.Google Scholar - T.J.R. Hughes and R.L. Taylor, [ 1978 ], “Unconditionally Stable Algorithms for Quasi Static Elasto/Viscoplastic Finite Element Analysis,”
*Computers and Structures*,**8**, 169–173.CrossRefzbMATHGoogle Scholar - C. Johnson, [ 1976a ], “Existency Theorems for Plasticity Problems,”
*Journal De Mathematiques Pures et Appliques*, 55, 431–444.zbMATHGoogle Scholar - C. Johnson, [ 1978 ], “On Plasticity with Hardening,”
*Journal of Applied Mathematical Analysis*,**62**, 325–336.CrossRefzbMATHGoogle Scholar - T. Kato, [ 1974 ], “On the Trotter-Lie Product Formula,”
*Proceedings Japan Academy*,**50**, 694–698.CrossRefzbMATHGoogle Scholar - W.T. Koiter, [1960], “General Theorems for Elastic—Plastic Solids, Chapter IV in
*Progress in Solid Mechanics*,**1**, 167–221.Google Scholar - R.D. Krieg and S.W. Key, [ 1976 ], “Implementation of a Time Dependent Plasticity Theory into Structural Computer Programs,”
*Constitutive Equations in Viscoplasticity: Computational and Engineering Aspects*, Editors J.A. Stricklin and K.J. Saczlski, AMD-20, ASME, New York.Google Scholar - R.D. Krieg and D.B. Krieg, [ 1977 ], “Accuracies of Numerical Solution Methods for the Elastic-Perfectly Plastic Model,”
*Journal of Pressure Vessel Technology*, ASME, Volume**99**.Google Scholar - S.J. Kim and J.T. Oden. [ 1990 ], “Finite element analysis of a class of problems in finite strain elastoplasticity based on the thermodynamical theory of materials of type N,”
*Computer Methods in Applied Mechanics and Engineering*,**53**, 277–302.MathSciNetCrossRefGoogle Scholar - E.H. Lee, [ 1969 ], “Elastic-plastic Deformation at Finite Strains”,
*Journal of Applied Mechanics*, 1–6.Google Scholar - D.G. Luenberger, [ 1972 ],
*Optimization By Vector Space Methods*, John Wiley and Sons Inc., New York.Google Scholar - G. Maier, [ 1970 ], “A Matrix Structural Theory of Piecewise Linear Elastoplasticity with Interacting Yield Planes,”
*Meccanica*, pp. 54–66.Google Scholar - G. Maenchen and S. Sacks, [ 1964 ], “The tensor code,” in
*Methods of Computational Physics*, Volume 3, 181–210. B. Alder, S. Fernback and M. Rotenberg Editors. Academic Press, New York.Google Scholar - J.B. Martin, [ 1988 ], “Convergence and Shakedwon for Discrete Load Steps in Statically Loaded Elastic—Plastic Bodies,”
*Mech*.,*Struct. 6 Mach*.,**16**(1), 1–16.CrossRefGoogle Scholar - J.B. Martin and Caddemi [ 1990 ], “Suffient Conditions for the Convergence of the Newton—Raphson Iterative Algorithm in Incremental Elastic Plastic Analysis,” (preprint).Google Scholar
- J. Mandel, [ 1974 ], “Thermodynamics and Plasticity”, In:
*Foundations of Continuum Thermodynamics*(Ed.: J.J. Delgado Domingers, N.R. Nina, J.H. Whitelaw ), Macmillan, London, 283–304.Google Scholar - H. Matthies, G. Strang, and E. Christiansen, [ 1979 ], “The saddle point of a differential,” in
*Energy Methods in Finite Element Analysis*, Glowinski, Rodin and Zienkiewicz Edts., J. Wiley and Sons.Google Scholar - H. Matthies, [ 1978 ], “Problems in Plasticity and Their Finite Element Approximation”, Ph.D. Thesis, Department of Mathematics, Massachusetts Institute of Technology, Cambridge Massachusetts.Google Scholar
- Matthies, H. [ 1979 ], “Existence theorems in thermoplasticity,”
*J. de Mechanique*,**18**, No. 4, pp. 695–711.MathSciNetGoogle Scholar - I. Miranda, R.M. Ferencz and T.J.R. Hughes, [ 1989 ], “An improved Implicit-Explicit Time Integration Method for Structural Dynamics,”
*Earthquake Engineering and Structural Dynamics*,**18**, 643–653.CrossRefGoogle Scholar - B. Moran, M. Ortiz and F. Shi [ 1990 ], “Formulation of implicit finite element methods for multiplicative plasticity,”
*International J. Numerical Methods in Engineering*,**29**, 483–514.CrossRefzbMATHGoogle Scholar - Moreau, J.J., [ 1976 ], “Applications of Convex Analysis of the Treatment of Elastoplastic Systems,” in
*Applications of Functional Analysis to Problems in Mechanics*, Lecture Notes in Mathematics, Vol.**503**, 56–89, Springer—Verlag, Berlin.Google Scholar - Moreau, J.J., [ 1977 ], “Evolution Problem Associated with a Moving Convex Set in a Hilbert Space”,
*Journal of Differential Equations*,**26**, 347.MathSciNetCrossRefzbMATHGoogle Scholar - J.C. Nagtegaal, D.M. Parks and J.R. Rice, [ 1974 ], “On numerically accurate finite ’ element solutions in the fully plastic range,”
*Computer Methods Applied Mechanics Engineering*,**4**, 153–177.MathSciNetCrossRefzbMATHGoogle Scholar - J.C. Nagtegaal and J.E. de Jong, [ 1981 ], “Some aspects of non-isotropic work hardening in in finite strain plasticity,” in
*Plasticity of of Metals at Finite Strains*, Proceedings Research Workshop, Stanford University, E.H. Lee and R.L. Mallet Editors, pp. 65–102.Google Scholar - A. Needleman and V. Tvergaard, [ 1984 ], “Finite Element Analysis of Localization Plasticity,” in
*Finite elements*,*Vol V: Special problems in solid mechanics*, Editors J.T. Oden and G.F. Carey, Prentice-Hall, Englewood Cliffs, New Jersey.Google Scholar - Nguyen, Q.S., [ 1977 ], “On the elastic-plastic initial boundary value problem and its numerical integration,”
*International Journal for Numerical Methods in Engineering***11**, 817.MathSciNetCrossRefzbMATHGoogle Scholar - R.W. Ogden, [ 1984 ],
*Nonlinear Elastic Deformations*, Ellis Hardwood Limited Chichester, U.K.zbMATHGoogle Scholar - M. Ortiz and E.P. Popov, [ 1985 ], “Accuracy and stability of integration algorithms for elastoplastic constitutive relations,”
*International J. Numerical Methods in Engineering*,**21**, 1561–1576.MathSciNetCrossRefzbMATHGoogle Scholar - Fj. Perk, D.R.J. Owen and M.E. Honnor, [ 1989 ], “A model for finite strain elastoplasticity,” in
*Proceedings of the 2nd International Conference on Computational Plasticity*, R. Owen, E. Hinton and E. Onate Editors, 111–126, Pineridge Press, U.K.Google Scholar - R.D. Richtmyer and K.W. Morton [ 1967 ],
*Difference Methods for Initial Value Problems*, 2nd Edition, Interscience, New YorkzbMATHGoogle Scholar - J.R. Rice and D.M. Tracey, [ 1973 ], “Computational fracture mechanics, ” in
*Proceedings of the Symposium on Numerical Methods in Structural Mechanics*, Editor S.J. Fenves, Urbana Illinois, Academic Press.Google Scholar - W.D. Rolph III and K.J. Bathe, [ 1984 ], “On a large strain finite element formulation for elastoplastic analysis,” in
*Constitutive Equations: Macro and Computational Aspects*, K.J. Willan editor, Winter Anual Meeting, 131–147, ASME.Google Scholar - L.R. Scott and M. Vogelius, [ 1985 ], “Conforming Finite Element Methods for Incompressible and Nearly Incompressible Continua,”
*Lectures in Applied Mathematics*,**22**, 221–244.MathSciNetGoogle Scholar - J.C. Simo, [ 1985 ], “On the computational significance of the intermediate configuration and hyperelastic stress relations in finite deformation elastoplasticity,”
*Mechanics of Materials*,**4**, 439–451.CrossRefGoogle Scholar - J.C. Simo and M. Ortiz, [ 1985 ], “A unified Approach to Finite Deformation Elastoplastic Analysis based on the use of Hyperelastic Constitutive Equations”,
*Computer Methods in Applied Mechanics and Engineering***49**, 221–245.CrossRefzbMATHGoogle Scholar - J.C. Simo R.L. Taylor, and K.S. Pister, [ 1985 ], “Variational and Projection Methods for the Volume Constraint in Finite Deformation Elasto-plasticity”,
*Computer Methods in Applied Mechanics and Engineering***51**, 177–208.MathSciNetCrossRefzbMATHGoogle Scholar - J.C. Simo and R.L. Taylor, [ 1985 ], “Consistent Tangent Operators for Rate Independent Elastoplasticity”,
*Computer Methods in Applied Mechanics and Engineering***48**, 101–118.CrossRefzbMATHGoogle Scholar - J.C. Simo and R.L. Taylor, [ 1986 ], “A return mapping algorithm for plane stress elastoplasticity,”
*Internation J. Numerical Methods in Engineering*,**22**, 649670.Google Scholar - J.C. Simo, [1988a,ó], “A Framework for Finite Strain Elastoplasticity Based on Maximum Plastic Dissipation and Multiplicative Decomposition: Part L Continuum Formulation; Part II.; Computational Aspects”,
*Computer Methods in Applied Mechanics and Engineering***66**199–219 and**68**1–31.Google Scholar - J.C. Simo, J.G.Kennedy and S.Govindjee, [ 1988 ], “General Return Mapping Algorithms for Multisurface Plasticity and Viscoplasticity,”
*International Journal of Numerical Methods in Engineering*,**26**, No. 2, 2161–2185.MathSciNetCrossRefzbMATHGoogle Scholar - Simo, J.C, and S. Govindjee, [1989], “B—Stability and Symmetry Preserving Return Mapping Algorithms for Plasticity and Viscoplasticity,”
*International Journal for Numerical Methods in Engineering*,in pressGoogle Scholar - J.C. Simo, [ 1990 ], “Nonlinear Stability of the Time Discrete Variational Problem in Nonlinear Heat Conduction and Elastoplasticity,”
*Computer Methods in Applied Mechanics and Engineering*,**88**, p. 111–121.MathSciNetCrossRefGoogle Scholar - J.C. Simo and R.L. Taylor, [ 1991 ], “Finite Elasticity in Principal stretches; Formulation and Finite element implementation,”
*Computer Methods in Applied Mechanics and Engineering*,**85**, 273–310.MathSciNetCrossRefzbMATHGoogle Scholar - J.C. Simo and C.Miehe [1990], “Coupled associative thermoplasticity at finite strains. Formulation, numerical analysis and implementation,
*Computer Methods in Applied Mechanics and Engineering*,(in press)Google Scholar - J.C. Simo and F. Armero [ 1992 ], “Geometrically nonlinear enhanced strain mixed methods and the method of incompatible modes,”
*International J. Numerical Methods in Engineering*,**33**, 1413–1449.MathSciNetCrossRefzbMATHGoogle Scholar - J.C. Simo, [1992], “Algorithms for Multiplicative Plasticity that Preserve the Form of the Return Mappings of the Infinitesimal Theory,” Computer Methods in Applied Mechanics and Engineering, in press.Google Scholar
- J.C. Simo and G. Meschke, [1992], “A New Class of Algorithms for Classical Plasticity Extended to Finite Strains. Application to Geomaterials,” International J. Computational Mechanics, in press.Google Scholar
- J.C. Simo, N. Tarnow and K. Wong [1991], “Exact energy—momentum conserving algorithms and symplectic schemes for nonlinear dynamics,” Computer Methods in Applied Mechanics and Engineering
*in press*.Google Scholar - J.C. Simo and N. Tarnow [1992], “Conserving Algoritms for Nonlinear Elastodynamics: The Energy-Momentum Method,”
*ZAMM*,in press.Google Scholar - Strang, H., H. Matthies, and R. Temam, [ 1980 ], “Mathematical and Computational Methods in Plasticity,” in
*Variational Methods in the Mechanics of Solids*, S. Nemat—Nasser Edt. Vold**3**, Pergamon Press, Oxford.Google Scholar - Suquet, P., [ 1979 ], “Sur les Equations de la Plasticite;”
*Ann. Fac. Sciences Tolouse*,**1**, pp. 77–87.MathSciNetCrossRefzbMATHGoogle Scholar - Temam, R., and G. Strang [ 1980 ], “Functions of bounded deformation,”
*Archive for Rational Mechanics and Analysis*,**75**, pp. 7–21.MathSciNetCrossRefzbMATHGoogle Scholar - T. Sussman, and K.J. Bathe, [ 1987 ], “A finite element formulation for nonlinear incompressible elastic and inelastic analysis,”
*Computers and Structures*,**26**, Nol/2, 357–109.Google Scholar - R.L.Taylor, P.J. Beresford and E.L. Wilson, [ 1976 ], “A non-conforming element for stress analysis,”
*International Journal for Numerical Methods in Engineering*,**10**, No. 6, 1211–1219.CrossRefGoogle Scholar - Temam, R., and G. Strang [ 1980 ], “Functions of bounded deformation,”
*Archive for Rational Mechanics and Analysis„***75**, pp. 7–21.MathSciNetzbMATHGoogle Scholar - C. Truesdell and W. Noll, [ 1965 ], “The Nonlinear Field Theories of Mechanics”, In:
*Handbuch der Physik Bd*.*111/3*(Ed.: S. Fluegge ), Springer—Verlag, Berlin.Google Scholar - G. Wanner, [ 1976 ], “A short Proof of Nonlinear A—Stability,”
*BIT*,**16**, 226–227.MathSciNetCrossRefzbMATHGoogle Scholar - G. Weber and L. Anand, [ 1990 ], “Finite Deformation Constitutive Equations and Time Integration Procedure for Isotropic Hiperelastic-Viscoelastic Solids,”
*Computer Methods in Applied Mechanics and Engineering*,**79**, 173–202.CrossRefzbMATHGoogle Scholar - M.L. Wilkins, [1964], “Calculation of Elastic-plastic Flow, in
*Methods of Computational Physics*, Volume**3**, 211–272.Google Scholar - B. Alder, S. Fernback and M. Rotenberg Editors. Academic Press, New York.Google Scholar
- Zeidler, E. [ 1985 ], Nonlinear Functional Analysis and its Applications I II: Variational Methods and Optimization, Springer—Verlag, Berlin.CrossRefzbMATHGoogle Scholar
- O.C. Zienkiewicz and R.L.Taylor, [ 1989 ], The Finite Element Method, Volume 1, McGraw-Hill, London.Google Scholar