Phase-field models have the advantage in that no geometric descriptions of cracks are required, which means that crack coalescence and branching can be treated without additional effort. Miehe et al. [1] introduced a rate-independent phase-field damage model for finite strains in which a viscous damage regularization was proposed. We extend the model to depend on the loading rate and time by incorporating rubber's strain-rate dependency in the constitutive description of the bulk, as well as in the damage driving force. The parameters of the model are identified using experiments at different strain rates. Local strain fields near the crack tip, obtained with digital image correlation (DIC), are used to help identify the length scale parameter. Three different degradation functions are assessed for their accuracy to model the rubber's rate-dependent fracture. An adaptive time-stepping approach with a corrector scheme is furthermore employed to increase the computational efficiency with a factor of six, whereas an active set method guarantees the irreversibility of damage. Results detailing the energy storage and dissipation of the different model constituents are included, as well as validation results that show promising capabilities of rate-dependent phase-field modeling.

Fatigue phase-field damage modeling of rubber using viscous dissipation: Crack nucleation and propagation, Mechanics of Materials (2019), doi:

Phase-field damage models are able to describe crack nucleation as well as crack propagation and coalescence without additional technicalities, because cracks are treated in a continuous, spatially finite manner. Previously, we have developed a phase-field model to capture the rate-dependent failure of rubber, and we have further enhanced it to describe failure due to cyclic loading. Although the model accurately describes fatigue failure, the associated cyclic simulations are slow. Therefore, this contribution presents an acceleration scheme for cyclic simulations of our previously introduced phase-field damage model so that the simulation speed is increased to facilitate large-scale simulations of industrially relevant problems. We formulate an explicit and implicit cycle jump method, which, depending on the selected jump size, reduces the calculation time up to 99.5%. To circumvent the manual tuning of the jump size, we also present an adaptive jump size selection procedure. Thanks to the implicit adaptive scheme, all material parameters are identified from experiments, which include fatigue crack nucleation and crack growth. Finally, the model and its parameters are validated with additional measurements of the fatigue crack growth rate.

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