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This contribution presents a comparison between a discrete and a smeared approach to approximate a crack in finite element simulations including the contribution of inertia to the behavior of brittle material under transient loading in the case of fracture. The discrete approximation of a crack is based in this case on a node duplication technique triggered by the evaluation of the so-called "material force" at the crack tip. The smeared approximation of a crack bases on the diffuse description of the crack by a phase-field approach. The governing equations under consideration of transient contributions are shown and the procedure for the finite element implementation is outlined. Numerical simulations investigate the capabilities and limitations of both methods. Firstly, the procedure to introduce initial cracks in a structure and the setup necessary to make them interact with stress waves properly, are under investigation. Moreover, this study deals with the evaluation of the velocity of the crack propagation and its comparison to experimental data. Finally, the phenomenon of crack branching is studied. The presentation and discussion of the results of the simulations provide an overview on the potential of both approaches with respect to an efficient and a realistic simulation of fracture processes in dynamic problems.

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Summary 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.

Modell der Schädigungsträgheit Standard-Maxwell-Modell der Visko-Elastizität Kraft Zeit Dehnung Spannung 2G 0 2G 1 2η 1 Bild 1 Rheologische Modellvorstellungen für Beton bei niedriger und hoher Dehnrate Rheological models of concrete at low and high strain rate