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A general theory of finite inelastic deformation of metals based on the concept of unified constitutive models

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Summary

First, a detailed survey of kinematics of finitely deformed inelastic solids is given. The concept is based on the multiplicative decomposition of the deformation gradient into elastic and inelastic parts. Since the intermediate configuration is chosen to be isoclinic as suggested by Mandel, this decomposition is unique. In a systematic manner, deformation tensors and strain tensors can be introduced. It is shown that a decomposition of the deformation rate tensor into elastic and inelastic parts is not unique and a kinematic relationship is established between two different definitions of the inelastic part of the deformation rate tensor. In this paper, all the kinematic quantities needed for the description of finite deformations with inelastic constitutive models are derived.

Next, a general constitutive theory for finitely deformed viscoplastic materials is given for materials which are structurally isotropic. The elastic part of the model is assumed hyperelastic. The inelastic constitutive equations are based on the concept of internal variables. They are formulated on the intermediate configuration and mapped to the current configuration. The inelastic constitutive frame is specified for a model originally proposed by Brown, Kim and Anand. Further, the implications of small elastic strains are investigated. An improved approximation for the inelastic incompressibility constraint is derived. Numerical experiments for simple shear using different elastic models demonstrate the importance of a sufficiently accurate consideration of the inelastic incompressibility constraint.

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References

  1. Bodner, S. R., Partom, Y.: Constitutive equations for elastic-viscoplastic strain-hardening materials. Trans. ASME J. Appl. Mech.42, 385–389 (1975).

    Google Scholar 

  2. Hart, E. W.: Constitutive relations for the non-elastic deformation of metals. Trans. ASME J. Eng. Mat. Tech.98, 193–202 (1976).

    Google Scholar 

  3. Krempl, E., McMahon, J., Yao, D.: Viscoplasticity based on overstress with a differential growth law for the equilibrium stress. Mech. Mat.5, 35–48 (1986).

    Google Scholar 

  4. Miller, A. K.: An inelastic constitutive model for monotonic, cyclic and creep deformation. Part 1 — Equations development and analytical procedures. Trans. ASME J. Eng. Mat. Tech.98, 97–105 (1976).

    Google Scholar 

  5. Miller, A. K.: An inelastic constitutive model for monotonic, cyclic and creep deformation. Part 2 — Application to type 304 stainless steel. Trans. ASME J. Eng. Mat. Tech.98, 106–112 (1976).

    Google Scholar 

  6. Robinson, D. H.: A candidate creep recovery model for 2−1/4 Cr-1 Mo steel and its experimental implementation. Report ORNL-TM-5110, Oak Ridge National Laboratory, Oak Ridge 1975.

    Google Scholar 

  7. Walker, K. P.: Research and development program for nonlinear structural modeling with advanced time-dependent constitutive relationships. Final Report NASA CR-165533, NASA, 1981.

  8. Anand, L.: Constitutive equations for hot-working of metals. Int. J. Plasticity1, 213–231 (1985).

    Google Scholar 

  9. Brown, S. B., Kim, K. H., Anand, L.: An internal variable constitutive model for hot working of metals. Int. J. Plasticity5, 95–130 (1989).

    Google Scholar 

  10. Nishiguchi, I., Sham, T.-L., Krempl, E.: A finite deformation theory of viscoplasticity based on overstress. Part I: Constitutive equations. Trans. ASME J. Appl. Mech.57, 548–552 (1990).

    Google Scholar 

  11. Rubin, M. B.: An elastic-viscoplastic model for large deformation. Int. J. Eng. Sci.24, 1083–1095 (1986).

    Google Scholar 

  12. Bodner, S. R., Rubin, M. B.: A unified elastic-viscoplastic theory with large deformations. In: Large deformation of solids: Physical basis and mathematical modelling (Gittus, J., Zarka, J., Nemat-Nasser, eds.), pp. 129–140. London-New York: Elsevier 1986.

    Google Scholar 

  13. Bammann, D. J.: Modelling temperature and strain rate dependent large deformations of metals. ASME, Appl. Mech. Rev.43, 312–319 (1990).

    Google Scholar 

  14. Bammann, D. J., Aifantis, E. C.: A model for finite-deformation plasticity. Acta Mech.69, 97–117 (1987).

    Google Scholar 

  15. Kröner, E.: Allgemeine Kontinuumstheorie der Versetzungen und Eigenspannungen. Arch. Rat. Mech. Anal.4, 237–334 (1960).

    Google Scholar 

  16. Lee, E. H.: Elastic-plastic deformation at finite strains. Trans. ASME J. Appl. Mech.36, 1–6 (1969).

    Google Scholar 

  17. Haupt, P.: On the concept of an intermediate configuration and its application to a representation of viscoelastic-plastic material behavior. Int. J. Plasticity1, 303–316 (1988).

    Google Scholar 

  18. Casey, J., Naghdi, P. M.: A correct definition of elastic and plastic deformation and its computational significance. Trans. ASME J. Appl. Mech.48, 983–985 (1981).

    Google Scholar 

  19. Casey, J., Naghdi, P. M.: A remark on the use of the polar decompositionF=F e F p in plasticity. Trans. ASME J. Appl. Mech.47, 672–674 (1980).

    Google Scholar 

  20. Dafalias, Y. F.: The plastic spin. Trans. ASME J. Appl. Mech.52, 865–871 (1985).

    Google Scholar 

  21. Dafalias, Y. F.: The plastic spin concept and a simple illustration of its role in finite plastic transformations. Mech. Mat.3, 223–233 (1984).

    Google Scholar 

  22. Mandel, J.: Equations constitutives et directeurs dans les milieux plastiques et viscoplastiques. Int. J. Solids Struct.9, 725–740 (1973).

    Google Scholar 

  23. Mandel, J.: Thermodynamics and plasticity. In: Foundations of continuum thermodynamics (Delgado, J. J. et al., eds.), pp. 283–304. New York: MacMillan 1974.

    Google Scholar 

  24. Zbib, H. M., Aifantis, E. C.: On the concept of relative and plastic spins and its implications to large deformation theories. Part II: anisotropic hardening plasticity. Acta Mech.75, 35–56 (1988).

    Google Scholar 

  25. Cleja-Tigoiu, S., Soós, E.: Elastoviscoplastic models with relaxed configurations and internal state variables. ASME Appl. Mech. Rev.43, 131–151 (1990).

    Google Scholar 

  26. Van der Giessen: Micromechanical and thermodynamic aspects of the plastic spin. Int. J. Plasticity7, 365–386 (1991).

    Google Scholar 

  27. Van der Giessen, E., Van Houtte, P.: A 2D analytical multiple slip model for continuum texture development and plastic spin. Mech. Mat.13, 93–115 (1992).

    Google Scholar 

  28. Marsden, E., Hughes, T. J. R.: Mathematical foundations of elasticity. Englewood Cliffs. Prentice-Hall 1983.

    Google Scholar 

  29. Simo, J. C., Ortiz, M.: A unified approach to finite deformation elastoplastic analysis based on the use of hyperelastic constitutive equations. Comp. Meth. Appl. Mech. Eng.49, 221–245 (1985).

    Google Scholar 

  30. Wriggers, P.: Konsistente Linearisierung in der Kontinuumsmechanik und ihre Anwendungen auf die Finite-Element-Methode. Habilitation thesis, Universität Hannover 1986.

  31. Wriggers, P., Stein, E.: Die Verwendung von Lie Ableitungen bei der Angabe von Spannungsflüssen für große Deformationen. Unpublished report, Universität Hannover 1987.

  32. Miehe, C.: Zur numerischen Behandlung thermomechanischer Prozesse. PhD thesis, Universität Hannover 1988.

  33. Müller-Hoeppe, N.: Beiträge zur Theorie und Numerik finiter inelastischer Deformationen. PhD thesis, Universität Hannover 1990.

  34. Hackenberg, H.-P.: Über die Anwendung inelastischer Stoffgesetze auf finite Deformationen mit der Methode der Finiten Elemente. PhD thesis, Technische Hochschule Darmstadt 1992.

  35. Hackenberg, H.-P., Kollmann, F. G.: A methodology for formulating large strain viscoplastic constitutive equations with applications to simple shear. In: Finite inelastic deformations — Theory and applications (Besdo, D., Stein, E., eds.), pp. 81–91, IUTAM Symposium, Hannover, Germany 1991. Berlin-Heidelberg-New York: Springer 1992.

    Google Scholar 

  36. Argyris, J. H., Doltsinis, J. St.: On the large strain inelastic analysis in natural formulation. Part I. Quasi-static problems. Comp. Meths. Appl. Mech. Eng.20, 213–251 (1979).

    Google Scholar 

  37. Kollmann, F. G., Hackenberg, H.-P.: On the algebra of two-point tensors on manifolds with applications in nonlinear solid mechanics. ZAMM73, 307–314 (1993).

    Google Scholar 

  38. Nemat-Nasser, S.: Decomposition of strain measures and their rates in finite deformation plasticity. Int. J. Solids Struct.15, 155–166 (1979).

    Google Scholar 

  39. Abraham, R., Marsden, J. R., Ratiu, T.: Manifolds, tensor analysis and applications, 2nd ed. Reading: Addison-Wesley 1983.

    Google Scholar 

  40. Truesdell, C., Noll, W.: The non-linear field theories of mechanics, Vol. III/3. Berlin-Heidelberg-New York: Springer 1965.

    Google Scholar 

  41. Kollmann, F. G., Hackenberg, H.-P.: Kinematics and rate-type elastic constitutive equations for finite rate dependent inelastic deformation of materials with small elastic strains. Unpubl. report MuM-Report 90/2, Techn. Hochschule Darmstadt 1990.

  42. Haupt, P., Tsakmakis, Ch.: On the application of dual variables in continuum mechanics. Contin. Mech. Thermodyn.1, 165–196 (1989).

    Google Scholar 

  43. Simo, J. C.: A framework for finite strain elastoplasticity based on maximum plastic dissipation and the multiplicative decomposition. Part I. Continuum formulation. Comp. Meths. Appl. Mech. Eng.66, 199–219 (1988).

    Google Scholar 

  44. Hart, E. W.: A micromechanical basis for constitutive equations with internal state variables. Trans. ASME J. Eng. Tech.106, 322–325 (1984).

    Google Scholar 

  45. Onat, E. T.: Representation of inelastic behavior in the presence of anisotropy and finite deformations. In: Plasticity of metals: Theory, computations and experiment (Lee, E. H., Mallet, R. L., eds.), pp. 519–544. Stanford Univ. and Rensselaer Polytech. Inst. 1982.

  46. Simo, J. C., Taylor, R. L., Pister, K. S.: Variational and projection methods for the volume constraint in finite deformation elastoplasticity. Comp. Meths. Appl. Mech. Eng.51, 177–208 (1985).

    Google Scholar 

  47. Simo, J. C.: A framework for finite strain elastoplasticity based on maximum plastic dissipation and the multiplicative decomposition. Part II. Computational aspects. Comp. Meths. Appl. Mech. Eng.68, 1–31 (1988).

    Google Scholar 

  48. Flory, P. J.: Thermodynamic relations for high elastic materials. Trans. Faraday Soc.57, 829–838 (1961).

    Google Scholar 

  49. Kratochvil, J.: On a finite strain theory of elastic-plastic materials. Acta Mech.16, 127–142 (1973).

    Google Scholar 

  50. Cordts, D., Kollmann, F. G.: An implicit time integration scheme for inelastic constitutive equations with internal state variables. Int. J. Num Meth. Eng.23, 533–554 (1986).

    Google Scholar 

  51. Hornberger, K., Stamm, H.: An implicit integration algorithm with a projection method for viscoplastic constitutive equations. Int. J. Num. Meth. Eng.28, 2397–2421.

  52. Hughes, T. J. R., Taylor, R. L.: Unconditionally stable algorithms for quasistatic elasto/visco-plastic finite element analysis. Comp. Struct.8, 169–173 (1978).

    Google Scholar 

  53. Lush, A. M., Weber, G., Anand, L.: An implicit time integration procedure for a set of internal variable constitutive equations for isotropic elasto-viscoplasticity. Int. J. Plasticity5, 521–549 (1989).

    Google Scholar 

  54. Weber, G. G.: Computational procedures for a new class of finite deformation elastic-plastic constitutive equations. PhD thesis, Massachusetts Institute of Technology 1988.

  55. Simo, J. C., Miehe, C.: Associated coupled thermoplasticity at finite strains: formulation, numerical analysis and implementation. Comp. Meth. Appl. Mech. Eng.98, 41–104 (1992).

    Google Scholar 

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  1. H. -P. Hackenberg & F. G. Kollmann

    Present address: MTU Motoren- und Turbinen-Union GmbH, Postfach 500640, D-80976, München, Germany

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  1. Technische Hochschule Darmstadt, Hochschulstrasse 1, D-64289, Darmstadt, Germany

    F. G. Kollmann

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Hackenberg, H.P., Kollmann, F.G. A general theory of finite inelastic deformation of metals based on the concept of unified constitutive models. Acta Mechanica 110, 217–239 (1995). https://doi.org/10.1007/BF01215426

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