Constitutive modelling in finite thermoviscoplasticity: a physical approach based on nonlinear rheological models

Abstract

Abstract This essay deals with a physical approach to formulate constitutive laws of finite thermoviscoplasticity. As proposed, for example, by Besdo (Besdo D., 1980. Zur Formulierung von Stoffgesetzen fur plastisch anistrope/elastisch isotrope Medien im Dehnungsraum, Zeitschr. angew. Math. Mech., 60, 101–103) or Negahban (Negahban, M., 1995. A study of thermodynamic restrictions, constraint conditions and material symmetry in fully strain-space theories of plasticity, Int. J. Plast., 11, 679–724) the whole theory is formulated in the strain space. For the sake of clarity and owing to the stringency required when choosing appropriate internal variables and evolution laws, the layout of the theory is dictated by rheological models. Combined with the concept of dual variables proposed by Haupt and Tsakmakis (Haupt, P., Tsakmakis, C., 1989. On the application of dual variables in continuum mechanics, Continuum Mech. Thermodyn. 1, 165–196), this method ensures the compatibility of the constitutive theory with the second law of thermodynamics. To illustrate the train of thought, we begin with the formulation of a uniaxial model of thermoviscoplasticity and restrict ourselves to kinematic hardening. In order to take thermal strains into account, we dissect the total strain into a thermal and a mechanical part. The mechanical deformation is the driving force for the stress and the thermal strain is a function of the temperature. In addition, we divide the mechanical strain into an elastic and an inelastic part. The stress depends only on the elastic strain, whereas the inelastic deformation is a functional of the process history. It corresponds to that part of strain which remains if the stress is reduced to zero. For the purpose of describing kinematic hardening with internal variables of strain type we introduce a further decomposition and dissect the inelastic deformation into two parts which has a motivation on the microscopic scale. The first part can be interpreted as the spatial average of local lattice deformations caused by dislocation fields (cf. Bruhns, O.T., Lehmann, T., Pape, A., 1992. On the description of transient cyclic hardening behaviour of mild steel CK15. Int. J. Plast. 8, 331–359) and the second can be attributed to inelastic slip processes on the microscale. Based on these ideas, it is straightforward to specify the free energy and to satisfy the dissipation principle in the form of the Clausius–Duhem inequality. The potential relations for the stress, the kinematic hardening variable and the entropy as well as the evolution laws for the internal variables are sufficient conditions for the non-negativity of the entropy production. Based on simplifying assumptions, we find that the Armstrong–Frederick model of hardening is incorporated as a special case. Subsequently, we transfer the structure of the theory to finite non-isothermal deformations. To this end we apply the thermomechanical decomposition of the deformation gradient as proposed by Lu and Pister (Lu, S.C.H., Pister, K.D., 1975. Decomposition of deformation and representation of the free energy function for isotropic thermoelastic solids. Int. J. Solids Struc. 11, 927–934). As prompted above, we define two further decompositions: the first one dissects the mechanical part of the deformation gradient into an elastic and an inelastic part and the second one splits the inelastic part into two further sections. Its first part can be interpreted as an averaged elastic lattice deformation which is caused by dislocations and its second part as an averaged plastic strain due to local plastic slip effects. To develop the constitutive relations for the free energy, the stress, the internal variables and the entropy we consider the rheological model in combination with the concept of dual variables and evaluate the Clausius–Duhem inequality.