An implicit functional theory of viscoplasticity

Abstract

A constitutive theory suitable for the description of metallic material behavior has been developed within the framework of a thermodynamical continuum theory. The theoretical development is firmly rooted in the concepts initiated recently by Professor E. Krempl and Professor K.O. Valanis.

The theory is developed to handle rate-dependence, finite deformation and temperature variations. In its infinitesimal form the theory is used to predict in a qualitative manner the effect of strain rate on stress-strain diagrams; loading-unloading-reloading behavior; cross-effects in tension-torsion experiments; softening of tensile response due to initial shear stress (corresponding to the formation of blunt vertices on the yield surface in classical plasticity); cyclic hardening and softening of hysteresis loops; cyclic creep and relaxation; deformation-induced anisotropy and the Bauschinger effect (corresponding to expansion and translation of the yield surface in classical plasticity); and viscous creep experiments in which creep and plastic effects interact.

The effects of plasticity which are reflected in the changing kernels are assimilated to a scalar structure parameter R. This function may be thought of as a measure of the damage accrued by the material during deformation. With this identification of the structure parameter a constitutive equation for the growth of R with deformation is proposed in the form of an implicit integral equation. When this integral equation is solved it is found that R grows slowly with deformation in the "elastic" regions immediately succeeding loading and unloading, whilst its growth with deformation in the "plastic" regions is large; and this desireable property is achieved without the use of a yield surface. It is also found that the dissipation due to plastic deformation is small in the "elastic" regions and large in the "plastic" regions because of the fact that the growth of R "switches on and off", as it were, during deformation. We have, in effect, achieved an intrinsic time scale which switches off when the deformation is elastic and switches on when the deformation is plastic.

In order to to decscribe plastic material behavior the constitutive functional must have the capability of changing its form during plastic deformation. This means that in a multiple integral formulation the kernels or integrands must possess the capability of changing with deformation. The changes which occur in the kernels reflect the plastic nature of the deformation, and to complete the theoretical description it is necessary to specify how such changes depend on the deformation history.

A functional theory is proposed in which the stress depends on the strain history through a multiple integral expansion. Such multiple integral expansions have, in conjunction with fading memory assumptions, been widely used to represent viscoelastic material behavior; but they are not suited for the description of metallic behavior.