Nonlocal regularization of sharp-crack energetics: a variational bridge from Griffith geometry to emergent constitutive degradation

Abstract

We develop a nonlocal diffusive crack model with displacement as the sole primary field and no independent fracture order parameter. A bounded saturation  potential regularizes a spatially averaged elastic strain-energy measure. Under the stated assumptions, an established nonlocal approximation theorem yields rigorous $\Gamma$-convergence of the corresponding unsplit reference energy to the Griffith energy as the regularization length vanishes. The small-energy slope and saturation level recover the bulk elastic energy and fracture-energy coefficient, respectively. Exact first variation gives a nonlocal stress modulated by the adjoint average of the saturation derivative. For smooth, nondecreasing, concave potentials, this derivative decreases monotonically from unity to zero, emerging as a degradation function without being prescribed independently. Potentials sharing the same Griffith limit can nevertheless produce different finite-scale stress responses and homogeneous peak strengths. The formulation thus links sharp-crack energetics and constitutive degradation through a single displacement-based energy.