Addressing viscosity-driven singularities: accurate development of thermo-elasto-visco-plastic constitutive models

A novel analytical-mathematical formulation for the multi-physics thermo-elasto-visco-plastic (TEVP) behavior of materials with nonlinear combined hardening is proposed. New closed-form expressions for the incremental visco-plastic multiplier (IVPM) and the consistent tangent operator (CTO) were derived. Specifically, all stiffness, hardening, and viscous coefficients were treated as temperature-dependent, and their temperature derivatives were explicitly included in the analytical solution. A UMAT (User Material) subroutine was programmed and implemented to compute the IVPM, CTO, and isotropic, kinematic, and viscous stresses for TEVP modeling. Finite element (FE) models were created and compared for the Abaqus^® built-in material model and the developed UMAT subroutine. The IVPM and CTO equations were successfully validated and the influence of the initial IVPM value on the accuracy of the results and the run time of simulations was examined for the first time. It was found that, in the Newton-Raphson method, the initial IVPM value must not only be nonzero to avoid singularity issues, but also be less than or equal to \(10^{-8}\) to ensure accurate results. In addition, the initial IVPM value did not influence computational efficiency. Ultimately, based on a comparative study of analytical solutions, UMAT-driven simulations, and standard Abaqus simulations, the developed formulation enables accurate prediction of strains, stresses, and temperatures in TEVP problems, providing a solid foundation for modeling industrial manufacturing processes such as quenching.