Modeling and algorithms for two-scale time homogenization of viscoelastic-viscoplastic solids under large numbers of cycles

• A VE-VP constitutive model is formulated within the framework of the thermodynamics of irreversible processes.
• Computational and fully implicit-time integration algorithms for time homogenization of VE–VP solids are proposed.
• Numerical simulations for several loading cases and a comparison with experimental data are presented.
• Numerical simulations illustrate the accuracy of the method and a reduction in the amount of computation time is obtained.

A two-scale time homogenization formulation and the corresponding algorithms are proposed for coupled viscoelastic–viscoplastic (VE–VP) homogeneous solids subjected to large numbers of cycles. The main aim is to predict the long time response while reducing the computational cost considerably. The method is based on the definition of macro and micro-chronological time scales, and on asymptotic expansions of the unknown variables. First, the VE–VP constitutive model is formulated based on a thermodynamical framework. Next, the original VE–VP initial-boundary value problem is decomposed into coupled micro-chronological (fast time scale) and macro-chronological (slow time-scale) problems. The former is purely VE, and solved once for each macro time step, whereas the latter problem is nonlinear and solved iteratively using fully implicit time integration. For micro-scale time averaging, one-point and multi-point integration algorithms are developed. Several numerical simulations on uniaxial and multiaxial cyclic loadings illustrate the computational efficiency and the accuracy of the proposed methods.