Formulation and implementation of strain rate-dependent fracture toughness in context of the phase-field method

The phase-field approach is a promising technique for the realistic simulation of brittle fracture processes, both in quasi-static and transient analysis. Considering fast loading, experimental evidence indicates a strong relationship between the rate of strain and the material's resistance against fracture, which can be considered by a dynamic increase factor for the strength of the material. The paper at hand presents a novel approach within the framework of phase-field models for brittle fracture. A rate-dependent fracture toughness is formulated as a function of the rate of crack driving strain components, which results in higher strength for faster loading. Beside the increased amount of energy necessary to evolve a crack at a high strain rate loading situation, the model incorporates quasi-viscous stress-type quantities that are not directly related to the formation of the crack and exist only in the phase-field transition zone between broken and sound material. The governing strong form equations for a transient simulation are derived and the relevant information for an implementation of the model into a finite element code is outlined in detail. The performance of the model is demonstrated for static and dynamic benchmark simulations and for a comparison to experimental findings.