We analyze a finite-difference approximation of a functional of Ambrosio–Tortorelli type in brittle fracture, in the discrete-to-continuum limit. In a suitable regime between the competing scales, namely if the discretization step \(\delta \) is smaller than the ellipticity parameter \(\varepsilon \), we show the \(\varGamma \)-convergence of the model to the Griffith functional, containing only a term enforcing Dirichlet boundary conditions and no \(L^p\) fidelity term. Restricting to two dimensions, we also address the case in which a (linearized) constraint of non-interpenetration of matter is added in the limit functional, in the spirit of a recent work by Chambolle, Conti and Francfort.

我们在离散到连续极限范围内分析了脆性骨折中Ambrosio-Tortorelli型功能的有限差分近似。在竞争标度之间的一个合适体制中，即，如果离散化步骤\（\ delta \）小于椭圆度参数\（\ varepsilon \），我们将模型的（（varGamma \）-收敛到格里菲斯功能性的，仅包含执行Dirichlet边界条件的项，没有\（L ^ p \）保真度项。限于二维，我们还根据Chambolle，Conti和Francfort最近的工作精神，解决了在极限函数中添加物质的非互穿性（线性化）约束的情况。