An existence result for the fractional Kelvin-Voigt's model on time-dependent cracked domains

TL;DR

This work addresses existence for the dynamic viscoelastic system with Caputo fractional derivatives on time-dependent cracked domains, modeled by $\ddot u -\mathrm{div}(\mathbb{C}eu+\mathbb{B}D_t^\alpha eu)=f$ in $\Omega\setminus\Gamma_t$. The authors regularize the memory kernel with a smooth kernel $\mathbb G$ and prove existence for the regularized problem via time discretization, accompanied by energy-dissipation inequalities and uniform energy estimates. They then pass to the limit, using compactness, to obtain a generalized solution $u^*$ to the original fractional Kelvin-Voigt system that satisfies an energy-dissipation inequality; a separate construction shows that the weak formulation is attained. Finally, uniqueness is proved in the special case of a stationary crack. The results provide a mathematically rigorous foundation for fractional viscoelastic modeling of dynamic fracture with evolving crack fronts and establish an energy-dissipation framework essential for subsequent analyses of fracture evolution and numerical schemes.

Abstract

We prove an existence result for the fractional Kelvin-Voigt's model involving Caputo's derivative on time-dependent cracked domains. We first show the existence of a solution to a regularized version of this problem. Then, we use a compactness argument to derive that the fractional Kelvin-Voigt's model admits a solution which satisfies an energy-dissipation inequality. Finally, we prove that when the crack is not moving, the solution is unique.

Theorems & Definitions (45)

• Remark 2.1
• Definition 2.2: Generalized solution
• Remark 2.3
• Theorem 2.4
• Definition 3.1: Generalized solution
• Definition 3.2: Weak solution
• Proposition 3.3
• proof
• Lemma 3.4
• proof