The evolution of brittle fracture in a material can be conveniently investigated by means of the phase-field technique introducing a smooth crack density functional. Following Borden et al. (2014), two distinct types of phase-field functional are considered: (i) a second-order model and (ii) a fourth-order one. The latter approach involves the bi-Laplacian of the phase field and therefore the resulting Galerkin form requires continuously differentiable basis functions: a condition we easily fulfill via Isogeometric Analysis. In this work, we provide an extensive comparison of the considered formulations performing several tests that progressively increase the complexity of the crack patterns. To measure the fracture length necessary in our accuracy evaluations, we propose an image-based algorithm that features an automatic skeletonization technique able to track complex fracture patterns. In all numerical results, damage irreversibility is handled in a straightforward and rigorous manner using the Projected Successive Over-Relaxation algorithm that is suitable to be adopted for both phase-field formulations since it can be used in combination with higher continuity isogeometric discretizations. Based on our results, the fourth-order approach provides higher rates of convergence and a greater accuracy. Moreover, we observe that fourth- and second-order models exhibit a comparable accuracy when the former methods employ a mesh-size approximately two times larger, entailing a substantial reduction of the computational effort.

通过引入平滑裂纹密度泛函的相场技术，可以方便地研究材料中脆性断裂的演变。继博登等人之后。(2014)，考虑了两种不同类型的相场泛函：(i) 二阶模型和 (ii) 四阶模型。后一种方法涉及相场的双拉普拉斯，因此所得的伽辽金形式需要连续可微的基函数：我们可以通过等几何分析轻松满足这一条件。在这项工作中，我们对所考虑的配方进行了广泛的比较，并进行了多项测试，这些测试逐渐增加了裂纹模式的复杂性。为了测量准确性评估中所需的骨折长度，我们提出了一种基于图像的算法，该算法具有能够跟踪复杂骨折模式的自动骨架化技术。在所有数值结果中，使用投影连续过松弛算法以直接而严格的方式处理损伤不可逆性，该算法适合两种相场公式采用，因为它可以与更高连续性等几何离散化结合使用。根据我们的结果，四阶方法提供了更高的收敛速度和更高的精度。此外，我们观察到，当前一种方法采用大约两倍大的网格尺寸时，四阶和二阶模型表现出相当的精度，从而大大减少了计算量。