A consistent finite elastoplasticity theory combining additive and multiplicative decomposition of the stretching and the deformation gradient

A phenomenological finite deformation elastoplasticity theory is proposed by consistently combining the additive decomposition of the stretching D and the multiplicative decomposition of the deformation gradient F. The proposed theory is Eulerian type and suitable for both isotropic and anisotropic elastoplastic materials with general isotropic and kinematical hardening behaviour. Within the context of the proposed theory, the Eulerian rate type constitutive formulation based on the decomposition D=De+Dep determines the total stress, the total kinematical quantities as well as the elastic part De and the coupled elastic–plastic part Dep etc. Then, the two rate quantities De and Dep are related to the elastic part Fe and the plastic part Fp in the decomposition F=FeFp in a direct and natural manner. It is found that the just-mentioned relationship between the two widely used decompositions, together with a suitable elastic relation defining the elastic stretch Ve=FeFe T, consistently and uniquely determines the elastic deformation Fe and the plastic deformation Fp and all their related kinematical quantities, without recourse to the widely used ad hoc assumption about a special form of Fe. Moreover, it is shown that for each process of purely elastic deformation the incorporated Eulerian rate type formulation intended for elastic response, which is based on the newly discovered logarithmic rate, is exactly-integrable to deliver a general hyperelastic relation with any given type of initial material symmetry, and thus the suggested theory is subjected to no self-inconsistency difficulty in the rate form characterization of elastic response, as encountered by other existing Eulerian rate type theories. In particular, it is proved that, to achieve the just-mentioned goal, the logarithmic rate is the only choice among all possible (infinitely many) objective corotational rates. Further, the proposed theory is shown to fulfill, in a full sense, the invariance requirement under the change of frame or the superposed rigid body motion. Accordingly, with the suggested theory the main fundamental discrepancies involving the decompositions of D and F disappear.