Eigenfracture approximation of quasi-static crack growth in brittle materials

TL;DR

The paper develops a rigorous variational framework for quasi-static crack growth using a two-field eigenfracture approximation $E_\varepsilon(u,\gamma)$ with a nonlocal neighborhood, and proves the existence of irreversible quasi-static evolutions for fixed $\varepsilon$ followed by a convergence as $\varepsilon\to0$ to an irreversible Griffith crack evolution in the antiplane setting. It establishes energy convergence and detailed compactness, showing that the limit $(u(t),\Gamma(t))$ satisfies irreversibility, global stability, and energy balance for the Griffith energy, with $\gamma_\varepsilon(t)$ converging to the singular part $D^s u(t)$. The core technical contribution lies in adapting a BV jump-transfer strategy to the eigenfracture setting, including density results, Besicovitch coverings, and careful handling of good/bad cubes and simplices to control nonlocal neighborhoods. The work also discusses simultaneous time-discretization and space-discretization limits and situates the eigenfracture approach relative to phase-field and Ambrosio–Tortorelli frameworks, providing a solid link between nonlocal two-field approximations and classical fracture theory. Overall, it offers a rigorous bridge between eigenfracture approximations and Griffith fracture evolutions, with implications for numerical schemes and broader variational fracture analyses.

Abstract

We study an approximation scheme for a variational theory of quasi-static crack growth based on an eigendeformation approach. We consider a family of energy functionals depending on a small parameter $\varepsilon$ and on two fields, the displacement field and an eigendeformation field that approximates the crack in the material. By imposing a suitable irreversibility condition and adopting an incremental minimization scheme, we define a notion of quasi-static evolution for this model. We then show that, as $\varepsilon \to 0$, these evolutions converge to a quasi-static crack evolution for the Griffith energy of brittle fracture, characterized by irreversibility, global stability, and an energy balance.

Figures (3)

• Figure 1: First Picture: $Q_i$ with the blue triangles belonging to $\Gamma_n(t)$. Second picture: a possible $\partial^*E^n_{t_i}$. Third picture: the resulting $\Phi^n_i$.
• Figure 2: For both figures: blue curve $\Psi^n_{i,{\rm pre}}$, magenta curve $\Phi^n_i$, blue triangles $\triangle^n_i \cap \Gamma_n(t)$, orange triangles $\triangle^n_i \setminus \color{black} \Gamma_n(t)$, and green zones $U_{l_n}(\Gamma_n(t))$. First figure: continuation of Figure \ref{['figur1']}, where $U_{l_n}(\Gamma_n(t))$ completely covers $\triangle^n_i$, i.e., $\triangle^n_i = \triangle^{n, {\rm cur}}_i$. Second figure: there are orange triangles that are not covered by $U_{l_n}(\Gamma_n(t))$, i.e., $\triangle^{n, {\rm new}}_i \neq \emptyset$.
• Figure 3: An example in $d=3$, where \ref{['toprov']} does not hold for $\partial^*E^n_{t_i}\setminus J_{y_n}$ in place of $\Phi^n_i$ since $\partial^*E^n_{t_i}\setminus J_{y_n}$ (in green) has a small surface. Yet, the set has big diameter, and therefore the volume of $U^\mathcal{T}_{n}(\partial^*E^n_{t_i}\setminus J_{y_n})$ (in orange) is of order $\varepsilon_n$. Consequently, in order to guarantee the estimate in Corollary \ref{['corollary:minimalseparator']}, we need to replace $\partial^*E^n_{t_i}\setminus J_{y_n}$ by $\Phi^n_i$ (in red), as defined in \ref{['Psi']}.

Theorems & Definitions (40)

• Theorem 2.1: Quasi-static eigenfracture evolution
• Definition 2.2
• Theorem 2.3: Approximation of quasi-static crack growth
• Theorem 2.4: Simultaneous limit
• Theorem 2.5: A version of Theorem \ref{['theorem:griffithmin']} for the continuous eigenfracture approximation
• Lemma 3.1: Eigenstrain supports
• proof
• Lemma 3.2: Convergence of strains
• Lemma 3.3: Stability
• proof