Summary
A formulation of isotropic thermoplasticity for arbitrary large elastic and plastic strains is presented. The underlying concept is the introduction of a metric transformation tensor which maps a locally defined six-dimensional plastic metric onto the metric of the current configuration. This mixed-variant tensor field provides a basis for the definition of a local isotropic hyperelastic stress response in the thermoplastic solid. Following this fundamental assumption, we derive a consistent internal variable formulation of thermoplasticity in a Lagrangian as well as a Eulerian geometric setting. On the numerical side, we discuss in detail an objective integration algorithm for the mixed-variant plastic flow rule. The special feature here is a new representation of the stress return and the algorithmic elastoplastic moduli in the eigenvalue space of the Eulerian plastic metric for plane problems. Furthermore, an algorithm for the solution of the coupled problem is formulated based on an operator split of the global field equations of thermoplasticity. The paper concludes with two representative numerical simulations of thermoplastic deformation processes.
Similar content being viewed by others
References
Anand, L.: Constitutive equations for hot-working of metals. Int. J. Plasticity 1 (1985) 213–231
Argyris, J. H.;Doltsinis, J. St.: On the natural formulation and analysis of large deformation coupled thermomechanical problems. Comp. Meth. Appl. Mech. Eng. 25 (1981) 195–253
Besdo, D.: Zur Formulierung von Stoffgesetzen der Plastomechanik im Dehnungsraum nach Ilyushins Postulat. Ing. Arch. 51 (1981) 1–8
Bever, M. B.;Holt, D. L.;Titchener, A. L.: The stored energy of cold work. Oxford: Pergamon Press 1973
Coleman, B. D.;Gurtin, M. E.: Thermodynamics with internal state variables. J. Chemistry and Physics 47 (1967) 597–613
Cuitiño, A. M.;Ortiz, M.: A material — independent method for extending stress update algorithms from small-strain plasticity to finite plasticity with multiplicative kinematics. Eng. Comput. 9 (1992) 437–451
Dillon, O. W.: Coupled thermoplasticity. J. Mech. Phys. Solids 11 (1963) 21–33
Doltsinis, J. St.: Aspects of modelling and computation in the analysis of metal forming. Eng. Comput. 7 (1990) 2–20
Drucker, D. C.: A more fundamental approach to plastic stress-strain relations. Proc. first U.S. National Congress of Applied Mechanics, Chicago, June 11–16, 1951, pp. 487–491
Eckart, E.: The thermodynamics of irreversible processes: IV. The theory of elasticity and anelasticity. Phys. Rev. 73 (1948) 337–382
Germain, P.;Nguyen, Q. S.;Suquet, P.: Continuum thermodynamics. Trans. ASME/J. Appl. Mech. 50 (1983) 1010–1020
Green, A. E.;Naghdi, P. M.: A general theory of an elasto-plastic continuum. Arch. Rat. Mech. Analysis 18 (1965) 251–281
Green, A. E.;Naghdi, P. M.: Some remarks on elastic-plastic deformation at finite strains. Int. J. Eng. Science 9 (1971) 1219–1229
Kratochvil, J.: On a finite strain theory of elastic-inelastic Materials. Acta Mechanica 16 (1973) 127–142
Krawietz, A.: Passivität, Konvexität und Normalität bei elastisch-plastischen Material. Ing. Arch. 51 (1981) 257–274
Lee, E. H.: Elastic-plastic deformation at finite strains. Trans. ASME Ser. E: J. Appl. Mech. 36 (1969) 1–6
Lehmann, T.: Einige Bemerkungen zu einer Klasse von Stoffgesetzen für große elasto-plastische Formänderungen. Ing. Arch. 41 (1972) 297–310
Lehmann, T.: General frame for the definition of constitutive laws for large non-isothermal elastic-plastic and elastic-viscoplastic deformations. In: Lehmann, T. (ed.) The Constitutive Law in Thermoplasticity, CISM Courses and Lectures No. 281. Berlin: Springer 1984
Lemonds, J.;Needleman, A.: Finite element analysis of shear localization in rate and temperature dependent solids. Mech. Mater. 5 (1986) 339–361
Lubliner, J.: Normal rules in large-deformation plasticity. Mech. Mater. 5 (1986) 29–34
Lubliner, J.: Non-isothermal generalized plasticity. In: Bui, H. D.; Nguyen, Q. S. (eds.) Thermomechanical Coupling in Solids pp. 121–131
Mandel, J.: Plasticité classique et viscoplasticité. CISM Courses and Lectures No. 97. Berlin: Springer 1972
Mandel, J.: Thermodynamics and plasticity. In: Delgado Domingers, J. J.; Nina, N. R.; Whitelaw, J. H. (eds.) Foundation of Continuum Thermodynamics, pp. 283–304. London: Macmillan 1974
Miehe, C.: On the representation of Prandtl-Reuss tensors within the framework of multiplicative elasto-plasticity. Int. J. Plasticity 10 (1994) 609–621
Miehe, C.: Computation of isotropic tensor functions. Commun. Appl. Numer. Methods 9 (1983) 889–896
Miehe, C.: Aspects of the formulation and finite element implementation of large strain isotropic elasticity. Int. J. Num. Meth. Eng. 37 (1994) 1981–2004
Miehe, C.: Entropic thermoelasticity at finite strains. Aspects of the formulation and numerical implementation. Comp. Meth. Appl. Mech. Eng. 120 (1995) 243–269
Miehe, C.: A generalization of Melan-Prager-Type kinematic hardening to large-strain elastoplasticity based on hyperelastic internal micro-stress response. Acta Mechanica (in press)
Miehe, C.;Stein, E.: A canonical model of multiplicative elastoplasticity: Formulation and aspects of the numerical implementation. European J. Mech. A/Solids 11 (1992) 25–43
Rice, J. R.: Inelastic constitutive relations for solids: An internal-variable theory and its application to metal plasticity. J. Mech. Phys. Solids 19 (1971) 433–455
Simo, J. C.;Miehe, C.: Associative coupled thermoplasticity at finite strains: Formulation, numerical analysis and implementation. Comp. Meth. Appl. Mech. Eng. 98 (1992) 41–104
Simo, J. C.;Armero, F.: Geometrically nonlinear enhanced strain mixed methods and the method of incompatible modes. Int. J. Num. Meth. Eng. 33 (1992) 1413–1449
Simo, J. C.: Algorithms for static and dynamic multiplicative plasticity that preserve the classical return mapping schemes of the infinitesimal theory. Comp. Meth. Appl. Mech. Eng. 99 (1992) 61–112
Truesdell, C.;Noll, W.: The nonlinear field theories of mechanics. In: Handbuch der Physik Bd. III/3. Berlin: Springer 1965
Weber, G.;Anand, L.: Finite deformation constitutive equations and a time integration procedure for isotropic, hyperelastic-viscoplastic solids. Comp. Meth. Appl. Mech. Eng. 79 (1990) 173–202
Wriggers, P.;Miehe, C.;Kleiber, M.;Simo, J. C.: On the coupled thermomechanical treatment of necking problems via finite element methods. Int. J. Num. Meth. Eng. 33 (1992) 869–883
Zienkiewicz, O. C.;Taylor, R. L.: The Finite Element Method. London: McGraw-Hill 1989
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Miehe, C. A theory of large-strain isotropic thermoplasticity based on metric transformation tensors. Arch. Appl. Mech. 66, 45–64 (1995). https://doi.org/10.1007/BF00786688
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00786688