Eigenfracture approximation of quasi-static crack growth in brittle materials

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.

💡 Research Summary

The paper presents a rigorous approximation framework for the variational theory of quasi‑static crack growth in brittle materials, based on an eigendeformation (eigenfracture) approach. The authors introduce a family of energy functionals depending on a small regularization parameter ε>0 and on two fields: the displacement u and an auxiliary eigendeformation field γ. The functional reads

Eε(u,γ)=∫Ω Q(∇u−γ) dx + (κ/2ε) ℒd(UTε({γ≠0})),

where Q is a positive‑definite quadratic form, κ>0 is the fracture toughness, and UTε(A) denotes the ε‑neighbourhood of a set A (or its discrete analogue on a triangulation). The term involving γ allows the displacement to develop jumps at no elastic cost, while the second term penalizes the volume of the ε‑neighbourhood of the support of γ, thus mimicking the surface energy of a crack.

The work proceeds in three main stages. First, under the scaling assumption that the mesh size h(ε) satisfies h(ε)/ε→0, the authors recall from previous work that the Γ‑limit of Eε as ε→0 is the classical Griffith energy

E(u)=∫Ω Q(∇u) dx + κ ℋd‑1(Ju),

with u∈SBV2(Ω) and Ju the jump set. This establishes that the eigenfracture functional is a valid static approximation of Griffith’s model.

Second, a time‑discrete incremental minimization scheme is built. A dense set of time points I∞⊂