Thermodynamically Consistent Phase-Field Models for Investigating Self-Heating Induced by Crack Growth in Linear Thermoviscoelastic Solids

One indicator of crack formation is a rise in temperature around the deformed region. At the same time, temperature plays a crucial role in influencing deformation behavior and crack growth in many materials. While this phenomenon is well understood in plastic materials, it remains less explored in viscoelastic solids such as polymers and soft matter. This study investigates temperature surges and their consequences for thermal softening in viscoelastic solids during crack growth using a phase-field model. To capture this behavior, we propose two new phase-field models for crack growth in Maxwell-type thermoviscoelastic materials, derived from rigorous physical and mathematical principles through the microforce balance approach. To ensure thermodynamic consistency, we establish an energy dissipation identity that incorporates viscoelastic energy, surface energy, and thermal energy, based on energy conservation and the second law of thermodynamics, specifically the Clausius--Duhem inequality. For the numerical evaluation, we employ the anisotropic adaptive finite element method, which is effective for solving phase-field models of crack propagation. All simulations are performed using FreeFEM, a finite element software for solving partial differential equations. The results reveal three main findings: (1) heat generation during crack propagation originates from mechanical energy dissipation, (2) thermal softening occurs more rapidly in materials with higher viscosity due to enhanced self-heating, and (3) self-heating and thermal softening are strongly correlated, each influencing the other. Overall, the numerical results are consistent with established studies.