Discrete approximation of dynamic phase-field fracture in visco-elastic materials
- Thomas, Marita
- Tornquist, Sven
2010 Mathematics Subject Classification
- 74H10 74H20 74H30 35M86 35Q74
- Visco-elastodynamic damage, Ambrosio-Tortorelli model for phase-field fracture, viscous evolution, rate-independent limit, temporal regularity of solutions
This contribution deals with the analysis of models for phase-field fracture in visco-elastic materials with dynamic effects. The evolution of damage is handled in two different ways: As a viscous evolution with a quadratic dissipation potential and as a rate-independent law with a positively 1-homogeneous dissipation potential. Both evolution laws encode a non-smooth constraint that ensures the unidirectionality of damage, so that the material cannot heal. Suitable notions of solutions are introduced in both settings. Existence of solutions is obtained using a discrete approximation scheme both in space and time. Based on the convexity properties of the energy functional and on the regularity of the displacements thanks to their viscous evolution, also improved regularity results with respect to time are obtained for the internal variable: It is shown that the damage variable is continuous in time with values in the state space that guarantees finite values of the energy functional.
- Discrete Contin. Dyn. Syst. Ser. S, 14 (2021), pp. 3865--3924, DOI 10.3934/dcdss.2021067 .