A class of plastic constitutive equations with vertex effect. I: General theory

Abstract A class of plastic constitutive equations which shows one-to-one correspondence between plastic strain increment dϵp and stress increment do is proposed from the viewpoint of tensor algebra. It is shown that it inevitably represents the so-called vertex-hardening model. Several examples of plastic constitutive equation among this class are formulated. For plasticity, the stress time-like measure d σ ¯ = [ ( 3 / 2 ) t r ( d T 2 o ) ] 1 / 2 and the strain time-like measure d e ¯ = [ ( 2 / 3 ) t r ( d e p ) 2 ] 1 / 2 are effectively used to represent loading or straining history, where d°T is the increment of deviatoric stress in the sense of Jaumann's rate, and dep denotes the plastic deviatoric strain increment. First, material is considered to be initially isotropic but to be anisotropic with deformation. For this case, Wang's representation theorem on isotropic tensor functions is effectively used. However, anisotropy in this case is limited. Therefore the theory is extended to the case where general initial and subsequent anisotropy plays an important role. Then it becomes possible that any anisotropic rule of yielding such as kinematic hardening, kinematic-isotropic hardening and other general anisotropic hardening without vertex formation is combined with vertex hardening. Introducing the natural time-measure dt, the theory is extended to express natural time-dependent inelastic constitutive equations such as creep and/ or viscoelasticity. Furthermore, introducing the natural and internal time-measures, which are combinations of dt and d σ ¯ , or dt and d e i ¯ , the theory is also extended to the case where natural and internal time-dependent inelastic constitutive equations such as viscoplasticity and/or dynamic plasticity are required to formulate. In some cases, temperature dependency is also considered.