Brittle solids typically fail by growth and propagation of a crack from a surface flaw. This process is modelled using linear elastic fracture mechanics, which parameterizes the toughness of a material by the critical stress intensity factor, or the prefactor of the singular stress field. This widely used theory applies for cracks that are planar, but cracks typically are not planar, and instead are geometrically complex, violating core tenets of linear elastic fracture mechanics. Here we characterize the crack tip kinematics of complex crack fronts in three dimensions using optical microscopy of several transparent, brittle materials, including hydrogels of four different chemistries and an elastomer. We find that the critical strain energy required to drive the crack is directly proportional to the geodesic length of the crack, which makes the sample effectively tougher. The connection between crack front geometry and toughness has repercussions for the theoretical modelling of three-dimensional cracks, from engineering testing of materials to ab-initio development of novel materials, and highlights an important gap in the current theory for three-dimensional cracks.

脆性固体通常会因表面缺陷的裂纹生长和传播而失效。该过程使用线弹性断裂力学进行建模，该力学通过临界应力强度因子或奇异应力场的预因子来参数化材料的韧性。这种广泛使用的理论适用于平面裂纹，但裂纹通常不是平面的，而是几何复杂的，违反了线弹性断裂力学的核心原则。在这里，我们使用几种透明脆性材料（包括四种不同化学成分的水凝胶和弹性体）的光学显微镜，在三个维度上表征复杂裂纹前沿的裂纹尖端运动学。我们发现驱动裂纹所需的临界应变能与裂纹的测地线长度成正比，这使得样品实际上更坚韧。裂纹前沿几何形状与韧性之间的联系对三维裂纹的理论建模（从材料的工程测试到新型材料的从头开始开发）产生了影响，并凸显了当前三维裂纹理论中的一个重要差距。