On Preserving Variational and Thermodynamic Consistency in Modeling Fracture of Zener-Type Viscoelastic Solids under Antiplane Loading

This work develops a variationally based and thermodynamically consistent phase-field model for fracture in Zener-type linear viscoelastic solids under antiplane loading. The formulation is cast as a rate-dependent dynamical system with internal variables, using a microforce balance structure to couple elastic, viscous, and fracture processes within a unified energetic framework. This guarantees that the governing equations and evolution laws satisfy the first and second laws of thermodynamics, yielding a dissipative and well-posed time evolution. The model captures the interaction between viscoelastic relaxation and crack growth in time-dependent materials. Numerical simulations based on an adaptive finite element method resolve the crack path and its temporal evolution efficiently. The results demonstrate the ability of the proposed framework to represent viscoelastic effects on fracture while preserving full variational and thermodynamic consistency. MSC Classification: 49S05 , 65M08 , 74A15 , 74A45