Thermodynamic Derivation and Damage Evolution for a Fractional Cohesive Zone Model

A thermodynamic derivation is presented for a fractional rate-dependent cohesive zone model recently proposed by the authors to combine damage and linear viscoelasticity. In this setting, the assumptions behind the initially proposed damage evolution law are revisited. In particular, in the original model damage evolution is driven only by the energy stored in the elastic arm of a fractional standard linear solid model and the relationship between total fracture energy and crack speed is monotonically increasing, with a sigmoidal shape. Here, physical arguments are discussed, which could support the hypothesis of allowing damage to be driven also by the remaining parts of the free energy. The implications of these different assumptions are then studied, analytically and numerically, and in both cases the assumption that damage is also driven by the remaining parts of the energy results in a nonmonotonic relationship between total fracture energy and crack speed, with a bell rather than sigmoidal shape. The analysis presented provides a novel physical interpretation of the significant differences found in the rate dependence of fracture in elastomers and glassy polymers.