A Return Mapping Algorithm for Nonlinear Yield Criteria with the Equivalent Mohr–Coulomb Strength Parameters

This paper proposes a modified return mapping algorithm for a series of nonlinear yield criteria. The algorithm is established in the principal stress space and ignores the effect of the intermediate principal stress. Three stress return schemes are derived in this paper: return to the yield surface, return to the curve, and return to the apex point. The conditions used for determining the correct stress return type are also constructed. After the proposed algorithm is programmed in the finite element software, we merely need the equivalent Mohr–Coulomb (M-C) strength parameters, the derivatives of their functions, and the tensile strength of these nonlinear yield criteria. In addition, the Hoek–Brown (H-B) yield criterion is taken as an example to validate the proposed method. The results show that the updated stresses and the final principal stresses obtained by the proposed method are in good agreement with those obtained by other methods. Furthermore, the proposed method is more suitable for the associated plastic-flow rule.

A Phenomenological Model for Shape Memory Alloys With Uniformly Distributed Porosity

A phenomenological model is proposed for shape memory alloys considering the presence of uniformly distributed voids. The model is developed within a modified generalized standard materials framework, which considers the presence of constraints on the state variables and ensures thermodynamic consistency. Within this framework, a free energy density is first proposed for the porous material, wherein the influence of porosity is accounted for by means of scalar state variables accounting for damage and inelastic dilatation. By choosing key thermodynamic forces, derived from the expression of the energy, as sub-gradients of a pseud-potential of dissipation, loading functions are derived that govern phase transformation and martensite detwinning. Flow rules are also proposed for damage and inelastic dilatation in a way that ensures positive dissipation. The model is discretized and the integration of the time-discrete formulation is carried out using an implicit formulation, whereby a return mapping algorithm is implemented to calculate increments of dissipative variables including inelastic strains. Comparison with data from the literature is finally presented.