Existence of solutions to a phase-field model of dynamic fracture with a crack-dependent dissipation
Maicol Caponi
TL;DR
This paper develops a dynamic phase-field model for brittle fracture using the Ambrosio–Tortorelli regularization with crack-tip velocity–dependent dissipation. It proves the existence of a dynamic evolution $(u(t),v(t))$ that satisfies irreversibility, crack stability, and Griffith’s dynamic energy–dissipation balance by a time-discretization scheme that alternates between elastodynamics and phase-field minimization and then passes to the limit using compactness and lower semicontinuity tools. It also analyzes a no-dissipation variant, establishing an energy–dissipation inequality rather than a full balance, and shows that with a nonnegative phase-field and appropriate $b$, $v\ge0$ can be maintained. Overall, the work provides a rigorous variational framework for dynamic fracture with crack-tip–velocity dissipation and connects to prior static and dynamic phase-field formulations.
Abstract
We propose a phase-field model of dynamic fracture based on the Ambrosio--Tortorelli's approximation, which takes into account dissipative effects due to the speed of the crack tips. By adapting the time discretization scheme contained in [C.J. Larsen, C. Ortner, and E. Süli, Math. Models Methods Appl. Sci. (2010)], we show the existence of a dynamic crack evolution satisfying an energy-dissipation balance, according to Griffith's criterion. Finally, we analyze the dynamic phase-field model of [B. Bourdin, C.J. Larsen, and C.L. Richardson, Int. J. Fracture (2011)] and [C.J. Larsen, IUTAM Symposium on Variational Concepts with Applications to the Mechanics of Materials (2010)] with no dissipative terms.