A dynamic model for viscoelastic materials with prescribed growing cracks
Maicol Caponi, Francesco Sapio
TL;DR
This work addresses dynamic fracture in viscoelastic solids with time-evolving cracks and possibly degenerate viscosity, introducing a model $\ddot u - \mathrm{div}(\mathbb C E u) - \mathrm{div}(\Psi^2 \mathbb B E \dot u) = f$ on $\Omega \setminus \Gamma_t$. The authors prove the existence of weak solutions via a time-discretization scheme, derive an energy–dissipation inequality, and establish partial uniqueness in 2D under stronger regularity and a vanishing viscosity region near the crack tip. They also provide a moving-crack example showing that Griffith’s dynamic energy–dissipation balance can hold in the presence of a vanishing-viscosity zone, and, in a broader context, demonstrate that crack growth can contribute extra dissipation when viscosity does not vanish near the tip. Collectively, the results give a rigorous framework for elastodynamics with growing cracks in viscoelastic media, clarifying when energy balance can be achieved and how crack motion interacts with dissipation. The work has implications for dynamic fracture modeling where material damping may degenerate spatially or temporally.
Abstract
In this paper we prove the existence of solutions for a class of viscoelastic dynamic systems on time--dependent cracked domains, with possibly degenerate viscosity coefficients. Under stronger regularity assumptions we also show a uniqueness result. Finally, we exhibit an example where the energy--dissipation balance is not satisfied, showing there is an additional dissipation due to the crack growth.