Open AccessDiscrete approximation of dynamic phase-field fracture in visco-elastic materials
Author(s) -
Marita Thomas,
Sven Tornquist
Publication year - 2021
Publication title -
discrete and continuous dynamical systems. series s
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.481
H-Index - 34
eISSN - 1937-1632
pISSN - 1937-1179
DOI - 10.3934/dcdss.2021067
Subject(s) - dissipation , convexity , fracture (geology) , mathematics , quadratic equation , phase space , space (punctuation) , field (mathematics) , homogeneous , constraint (computer aided design) , viscoelasticity , mathematical analysis , physics , statistical physics , computer science , pure mathematics , geometry , materials science , quantum mechanics , thermodynamics , financial economics , economics , composite material , operating system
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 \begin{document}$ 1 $\end{document} -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.