On the Elastic-Plastic Constitutive Equations in Incremental Form

Abstract

It is emphasized that when the rate-type constitutive equation of finite elasticity is converted into the up-dated form, it is not equivalent to the original constitutive equation any more, and therefore such conversion is not effective except the special cases where the current rates of stress and strain are completely independent of deformation state and history in a tensorial sense, for non-elastic materials as well. It is demonstrated that large simple shear gives a shape of material element different from that subjected to large pure shear at the same equivalent strain even though the commonly used elastoplastic constitutive equation with J2-flow theory \(\mathop \sigma \limits^{\rm O} = \hat D:D\) takes the same stress history (i.e. only shear stress acts), where \(\mathop \sigma \limits^ \circ \) = Jaumann rate of Cauchy stress σ, D = stretching. A new plastic constitutive equation is proposed which involves a material parameter ρ with 0 ≦ ρ ≦ 1. For ρ = 0 it reduces to J2-flow theory, and for ρ = 1 it resembles deformation theory. Its stress response to simple shear is discussed with various value of ρ, and the well-known axial extension of a cylinder subjected to torsion is explained using an appropriate value of ρ. The spin to be used in a constitutive equation and evolution equation of internal variables is also discussed.