A framework for finite-strain viscoelasticity based on rheological representations

Abstract: This work presents a new constitutive and computational framework based on strain-like internal variables belonging to Sym(3) and two representative rheological configurations. The generalized Maxwell and generalized Kelvin-Voigt models are considered as prototypes for parallelly and serially connected rheological devices, respectively. For each configuration, distinct kinematic assumptions are introduced. The constitutive theory is derived based on thermomechanical principles, where the free energies capture recoverable elastic responses and dissipation potentials govern irreversible mechanisms. The evolution equations for the internal variables arise from the principle of maximum dissipation. A key insight is the structural distinction in the constitutive laws resulted from the two rheological architectures. In particular, the Kelvin-Voigt model leads to evolution equations with non-equilibrium processes coupled, which pose computational challenges for the constitutive integration. To address this, we exploit the Sherman-Morrison-Woodbury formula and extend it to tensorial equations to design an efficient strategy during constitutive integration. With that strategy, the integration can be performed based on an explicit update formula, and the algorithmic complexity scales linearly with the number of non-equilibrium processes. This framework offers both modeling flexibility and computational feasibility for simulating materials with multiple non-equilibrium processes and complex rheological architectures under finite strain.