11 problems

Let $(u,K)$ be a local minimizer of the Griffith energy in an open set $\Omega$, and let $\nabla u$ denote the gradient of its displacement field. For $p>2$, write…

Stiff-fracture limit conjecture. The stiff case should lead, in the limit $\varepsilon\rightarrow 0$, to an elastic plate-type model.

Global minimality conjecture. This minimality extends to all increases in the crack set, not only to regular increments.

Anisotropic surface-term conjecture. Adding the Ciarlet–Nečas condition to the nonlinear Griffith functional should produce an additional anisotropic surface term, possibly dependi…

Let $K_+$ and $K_-$ denote the upper and lower fracture thresholds, respectively, for a fixed domain radius $R$. Let $a_1$ and $a_2$ be the model parameters, let $R_*$ denote the r…

Convergence conjecture. Nonlocal elastodynamics converge to weak solutions of the wave equation for growing cracks described by closed countably rectifiable subsets of $D$ with bou…

Uniform boundedness conjecture. For $\bar q(u)$ given by Eq.,

The numerical examples use a nonlocal elastodynamic model for brittle fracture, comparing the energy consumed by a numerically simulated crack with the Griffith prediction. The two…

Let $\Omega\subset\mathbb{R}^n$ be the domain, and let $SBD^p(\Omega)$ denote the space of special functions of bounded deformation with strain in $L^p$ and jump set of finite…

The sharp-interface limit is described by a displacement–crack set pair $u_0(t,x)$ and $S_{u^0(t)}$, with the wave equation coupling elastic waves to the evolving fracture path. Th…

Higher-gradient extension conjecture. The present theory should be extended by adding higher surface gradient effects to the model for stress-deformation behavior within the fractu…