Adaptive finite element approximation for quasi-static crack growth

TL;DR

This work develops an adaptive finite element framework to approximate quasi-static crack growth in 2D using a displacement-void formulation. By combining a time-discrete, history-dependent minimization with an a priori sharp void-modification analysis, the authors pass to the simultaneous limits $\delta\to 0$ and $\varepsilon\to 0$ to obtain an irreversible Griffith evolution $(u(t),K(t))$ in GSBD$^2(\Omega)$, with energy balance and unilateral minimality. A key novelty is the displacement-void representation and the void-healing techniques, which yield a sharp lower bound for the surface energy and enable a robust Γ-convergence argument alongside a stability analysis. The main result provides a rigorous link between adaptive finite element approximations and continuum Griffith fracture evolution, including convergence of crack sets in the $\sigma$-sense and strong convergence of linear strains, with implications for numerical treatment of brittle fracture models.

Abstract

We provide an adaptive finite element approximation for a model of quasi-static crack growth in dimension two. The discrete setting consists of integral functionals that are defined on continuous, piecewise affine functions, where the triangulation is a part of the unknown of the problem and adaptive in each minimization step. The limit passage is conducted simultaneously in the vanishing mesh size and discretized time step, and results in an evolution for the continuum Griffith model of brittle fracture with isotropic surface energy [FriedrichSolombrino16] which is characterized by an irreversibility condition, a global stability, and an energy balance. Our result corresponds to an evolutionary counterpart of the static Gamma-convergence result in [BabadjianBonhomme23] for which, as a byproduct, we provide an alternative proof.

Figures (8)

• Figure 1: An example for $H$ consisting of three components and the corresponding graph $(\mathcal{V}(H),\mathcal{E}(H))$. Note that $\overline{H}$, instead, is connected. The triangles highlighted in blue are part of $\mathbf{T}^{\rm ex}_{H} = \mathbf{T}_H^{\rm ex, good} \cup \mathbf{T}_H^{\rm ex, bad} \color{black}$. The vertices depicted in red are part of $\mathcal{V}_{4}(H)$ whereas the vertices depicted in black are all part of $\mathcal{V}_{2}(H)$. $\mathcal{E}(H)$ is the set of edges of triangles contained in $\partial H$. Here and in the following figures, we always use subsets of a regular triangular lattice for illustration purposes.
• Figure 2: The first graphic depicts an example for a self-intersecting component $H_i\in \mathcal{D}_2(H)$ and the corresponding graph. Note that the vertex $\tilde{v}$ is contained twice in the corresponding tuple $(v_i^{1}, ..., v_i^{J})$ and that $\mathbb{R}^2 \setminus \overline{H_i}$ consists of two components. In the second example, the two components $H_1,H_2 \in \mathcal{D}_1(H)$ have boundaries $\partial H_1, \partial H_2$ that consist of two connected components each.
• Figure 3: Triangles in the sets $\mathbf{M}_{j}(H)$. If we suppose that in all four pictures the other triangles which are not depicted are contained in $\Omega \setminus H$, the green vertices correspond to the vertices given in \ref{['defMheal']}.
• Figure 4: The sets $Z$ (violet), $Y$ (green), $N_Z\setminus Y$ (red), and $\{p, \, q\}=\overline{Y}\cap \overline{Z}$.
• Figure 5: Schematic illustration of different components of $\overline{B}$. Note that $\overline{B}$ is connected and $\# \mathcal{C}(B) = 12$. The red dots depict the separating vertices. In this example the green parts correspond to the $7$ elements in $\mathcal{G}_{\rm sep}^{\rm small}$. The component with magenta border depicts a possible component in $\mathcal{D}_1^{\rm small}(B_{\rm sep})$.

Theorems & Definitions (54)

• Theorem 2.1: Void modification
• Theorem 2.2
• proof
• Corollary 3.1
• Lemma 3.2
• proof
• Lemma 3.3: Healing of triangles
• Lemma 3.4: Healing of entire components
• proof : Proof of Theorem \ref{['generaltheorem']}
• Remark 3.5