Rate-independent evolutions in phase-field fracture generated by equilibrium configurations

Abstract: In the phase-field approach, and in other regularized fracture models, rate-independent (quasi-static) evolutions are usually obtained by means of time-discrete schemes, providing at each time an equilibrium configuration of the system. Numerically, equilibria are computed by descent methods for the free energy (e.g. staggered, monolithic etc.) endowed with a suitable irreversibility constraint on phase-field parameter (e.g. monotonicity in time). The time-continuous limit evolution is characterized in terms of evolutionary variational inequalities and in terms of Griffith's criterion. The study of the limit evolution reveals a couple of propagation regimes: stable (or steady state) and unstable (or catastrophic). In the stable regime the evolution is simultaneous (in displacement and phase field parameter) and satisfies Griffith's criterion in terms of toughness and phase field energy release rate. In the unstable regime the evolution is not necessarily simultaneous and may not satisfy Griffith's criterion. Energy identity holds in general with an instantaneous drop of energy in unstable times. The monotonicity constraint turns out to be fully thermodynamically consistent along the whole evolution. Technically, the proof relies on: (a) the strong convergence of the phase field parameter, (b) Kuratowski convergence of parametrized discrete times, (c) the limit of the power identity. The result is very general since it works independently of the scheme employed in the incremental problem, it holds for instance for AT1 and AT2 energies, in plane strain and plane stress settings. Numerically, we compare energies, energy release rates and evolutions computed in the sharp crack and in the phase field setting for a couple of reference benchmarks: the double cantilever beam (for steady state propagation) and the single edge notch under tension (for unstable propagation).

MathematicsPhysics

Audience: researchers in the topic

Nečas Seminar on Continuum Mechanics

Series comments: This seminar was founded on December 14, 1966.

Faculty of Mathematics and Physics, Charles University, Sokolovská 83, Prague 8. If not written otherwise, we will meet on Mondays at 15:40 in lecture hall K3 (2nd floor)