Incremental constitutive models for elastoplastic materials undergoing finite deformations by using a four-dimensional formalism

When constructing incremental constitutive models of elastoplasticity for materials undergoing finite deformations, the tensors and their rates should respect the principle of frame-indifference. Instead of classical 3D approaches in which different objective transports may be arbitrarily used in the constitutive equations, we propose to model the constitutive equations using the four-dimensional formalism of the theory of Relativity. This formalism ensures that any 4D tensor is frame-indifferent thanks to the principle of covariance. It is further possible to define 4D rate operators that are all, by construction, frame-indifferent. Among these covariant rates, the 4D Lie derivative is chosen to construct incremental constitutive relations because it is invariant to the superposition of rigid body motion. A 4D rate type model of elastoplasticity with isotropic hardening is thus developed and compared with existing classical 3D constitutive models of elastoplasticity established in the context of finite deformations.