Uniformly strained anisotropic elastoplastic rods: Determination of elastoplastic constitutive relations and yield surface in terms of rod’s variables

Abstract

This work presents a formulation for the elastoplastic deformation of the rod’s cross-section wherein the rod material can obey an arbitrary three-dimensional stored energy density model together with a rate-independent arbitrary phenomenological plasticity model. In order to do so, we impose a uniform strain field along the rod’s arc-length which reduces the three-dimensional nonlinear elastoplastic rod problem to just its cross-section. To solve this elastoplastic cross-sectional deformation problem, a three-dimensional elastoplastic moduli tensor is first derived along with a return map algorithm for the update of relevant internal variables at every point in the cross-section. A finite element formulation is then presented to solve the aforementioned cross-sectional problem. The elastoplastic stress resultants (internal contact force and moment in the rod’s cross-section) and elastoplastic stiffnesses (such as bending, twisting, extensional and shearing) of an elastically isotropic but plastically transversely-isotropic rod are then obtained numerically along with the warped state of the plastified cross-section. Several deformations (increment/decrement in combinations of extension, torsion, bending and shearing) are studied to observe the evolution of plasticity at each quadrature point in the rod’s cross-section. We observe that off-axis anisotropy during plasticity leads to unusual out-of-plane warping even during torsion of circular cross-sections. It also leads to coupling between various modes such as between bending and twisting. We also demonstrate that the rod’s various elastoplastic stiffnesses are not only dependent on the current strain state of the rod but also on the current direction of strain rate. Finally, we integrate the plastic work over the rod’s cross-section and use the criterion of limiting plastic work to obtain discrete points on the rod’s yield surface in terms of stress resultants. The obtained yield surface turns out to be different from the one obtained for purely isotropic material model.