[Paper Review] Existence of strong minimizers for the Griffith static fracture model in dimension two

Sergio Conti, Matteo Focardi+1 more|arXiv (Cornell University)|Nov 10, 2016

Numerical methods in engineeringEngineering47 references36 citations

TL;DR

This paper establishes the existence of strong minimizers for the Griffith static fracture model in two dimensions, proving that minimizers have closed jump sets and continuously differentiable deformations. The key innovation is a vectorial Poincaré-type inequality for SBDp functions in two dimensions, enabling compactness and regularity via approximation by Sobolev functions, which overcomes the failure of the coarea formula in the vectorial case.

ABSTRACT

We consider the Griffith fracture model in two spatial dimensions, and prove existence of strong minimizers, with closed jump set and continuously differentiable deformation fields. One key ingredient, which is the object of the present paper, is a generalization of the decay estimate by De Giorgi, Carriero, and Leaci to the vectorial situation. This is based on replacing the coarea formula by a method to approximate $SBD^p$ functions with small jump set by Sobolev functions and is restricted to two dimensions. The other two ingredients are contained in companion papers and consist respectively in regularity results for vectorial elliptic problems of the elasticity type and in a method to approximate in energy $GSBD^p$ functions by $SBV^p$ ones.

Motivation & Objective

• To establish the existence of strong minimizers for the Griffith fracture model in two spatial dimensions.
• To overcome the lack of a coarea formula in the vectorial setting by constructing a Poincaré-type inequality for SBDp functions with small jump sets.
• To prove that minimizers of the relaxed functional have closed jump sets and are C1 outside the fracture set.
• To extend the De Giorgi–Carriero–Leaci approach to the vectorial elasticity setting in dimension two.
• To provide a foundation for existence theory in brittle fracture with linear elasticity and surface energy.

Proposed method

• Develops a new approximation result for SBDp functions with small jump sets by W1,p functions, valid only in two dimensions.
• Uses this approximation to derive a Poincaré-type inequality for SBDp functions, replacing the coarea formula used in the scalar case.
• Applies the approximation result to control the energy of GSBDp functions via SBVp functions, leveraging a companion result on GSBDp approximation.
• Combines the new Poincaré inequality with elliptic regularity theory for linear elasticity systems to deduce C1 regularity of minimizers away from the fracture set.
• Uses a decay estimate argument based on energy comparison and scaling to prove a lower density bound for the jump set.
• Relies on the fact that in two dimensions, the jump set of a minimizer is essentially closed, allowing the definition of a closed fracture set Γ = cl(Ju).

Experimental results

Research questions

• RQ1 Does the Griffith fracture model in two dimensions admit strong minimizers with C1 deformations and closed fracture sets?
• RQ2 Can the De Giorgi–Carriero–Leaci decay estimate be extended to the vectorial case where the energy depends on the symmetrized gradient?
• RQ3 Is there a Poincaré-type inequality for SBDp functions in two dimensions that avoids the coarea formula?
• RQ4 Can GSBDp functions be approximated in energy by SBVp functions with bounded traces in two dimensions?
• RQ5 What conditions ensure that the jump set of a minimizer is closed and the deformation is C1 outside the fracture?

Key findings

• The functional (1.1) admits a minimizer in the class of closed fracture sets and C1 deformations in two dimensions.
• The jump set of any minimizer is essentially closed, meaning H1(Ω ∩ Ju) = H1(Ω ∩ cl(Ju)) holds.
• Minimizers are C1-smooth on Ω \ Ju, the complement of the fracture set.
• A lower density bound H1(Ju ∩ Bρ(x)) ≥ ϑ1ρ holds for all x ∈ Ω ∩ Ju and small ρ, with ϑ1 depending only on problem parameters.
• The proof relies crucially on a new Poincaré-type inequality for SBDp functions in two dimensions, which replaces the coarea formula.
• The result extends to p-growth energies with fµ(ξ) = 1/p((Cξ·ξ + µ)^{p/2} - µ^{p/2}) and h(x,z) = κ|z - g(x)|^p, with g ∈ L∞(Ω; R2) for p ≤ 2 and g ∈ W1,p(Ω; R2) for p > 2.