Small-horizon limit of the regularized bond-breakage model

Identify the limit of the solutions of the regularized non-linear bond-based peridynamic model with irreversible damage as the horizon parameter δ decreases to zero, in particular determining whether and how the solutions converge to a sharp discontinuity or free-fracture model.

Background

The paper proves convergence only with respect to the spatial discretization parameter at fixed horizon δ>0. Its regularity result shows that, for fixed δ, the displacement remains continuous or locally Lipschitz continuous wherever the input data has the corresponding regularity, while the derived bounds diverge as δ decreases to zero.

The authors interpret the fixed-horizon model as a regular approximation of a sharp fracture model, but explicitly leave unresolved the mathematical behavior of the solutions in the small-horizon limit.