Variational phase-field models of fracture are widely used to simulate nucleation and propagation of cracks in brittle materials. They are based on the approximation of the solutions of free-discontinuity fracture energy by two smooth function: a displacement and a damage field. Their numerical implementation is typically based on the discretization of both fields by nodal $\mathbb{P}^1$ Lagrange finite elements. In this article, we propose a nonconforming approximation by discontinuous elements for the displacement and nonconforming elements, whose gradient is more isotropic, for the damage. The handling of the nonconformity is derived from that of heterogeneous diffusion problems. We illustrate the robustness and versatility of the proposed method through series of examples.

翻译：变分相场断裂模型被广泛用于模拟脆性材料中裂纹的成核与扩展。该方法通过两个光滑函数——位移场和损伤场——来近似自由间断断裂能量的解。其数值实现通常基于节点$\mathbb{P}^1$拉格朗日有限元对这两个场进行离散化。本文提出一种非协调逼近方法：对位移场采用间断元离散，对损伤场采用梯度更各向同性的非协调元离散。非协调性的处理来源于异质扩散问题的相关方法。通过一系列算例，我们验证了所提方法的鲁棒性和通用性。