A study of two finite strain plasticity models: An internal time theory using Mandel's director concept, and a general isotropic/kinematic-hardening theory

Abstract

Using Mandel's director concept, the internal time theory for small deformation deviatoric plasticity (as given by Valanis and by Watanabe and Atluri) is extended to the case of finite elastoplasticity. The introduction of the material director triad together with an isoclinic configuration makes possible an integral representation of the induced stress tensor in terms of the deformation history [referred to the relaxed intermediate configuration.] For a class of deformations in which elastic strains are small, this leads to rate-type constitutive equations similar to those in the small deformation theory. The model retains the characteristics of the original internal time theory, such as the existence of a von Mises type of yield surface that undergoes translation (nonlinear kinematic hardening), as well as uniform expansion in the stress space, and of the normality of the tensor of plastic rate of deformation to the yield hypersurface. In modelling finite uniaxial compression and finite torsion test data, the present internal time theory is found to lead to results very similar to those from a phenomenological theory with a priori postulated evolution equation for an arbitrary objective rate (say, the Jaumann rate) of the back stress using the general representation for the symmetric tensor function. This latter phenomenological theory is a slight modification of the one proposed earlier by Reed and Atluri and accounts for large plastic rotations in finite strain deformations. Both models are shown (1) to lead to nonoscillatory shear stresses for all magnitudes of shear in a simple shear deformation, (2) to predict the correct magnitude (as found in experiments) of axial stresses/strains in a finite-strain torsion test, and (3) to correctly model the experimentally noted differences in the von Mises equivalent Cauchy stress (or J 2) values in uniaxial compression test and torsion test, respectively, at large values of equivalent strains. Both these models may be interpreted as theories wherein arbitrary, say, the Jaumann rates of Cauchy stress and Cauchy back stress are represented as objective functions of an appropriate number of different types of internal variables. The present results (1) indicate that concerns about “appropriate stress rates” in current literature are vacuous, (2) point to the need for a deeper understanding of the underlying micromechanics of finite strain plasticity and the need for appropriate additional internal variables that may be necessary to model the test data properly, and (3) indicate that the “plastic-spin” is not an indispensible internal variable to correctly predict not only the nonoscillatory shear stress but also the correct magnitude of axial strain in a finite torsion test.