The interaction of crack fronts with asperities is central to fracture criteria in heterogeneous materials and for predicting fracture surface formation. It is known how dynamic crack fronts respond to small, 1st-order, perturbations. However, large and localized disturbances to crack motion induce dynamic and geometric nonlinear effects beyond the existing linear theories. Because the determination of the 3D elastic fields surrounding perturbed crack fronts is a necessary step toward any theoretical study of crack front dynamics, we develop a 2nd-order perturbation theory for the asymptotic fields of planar crack fronts. Based on previous work, we consider two models of fracture: (1) Fracture in a scalar elastic solid which is an analog of antiplane shear fracture (Mode III). In this model, the near-crack fields are obtained via matched asymptotic expansions. (2) Tensile Mode I fracture, in which a self-consistent expansion is used to resolve the fields near the crack front. These methods can be readily extended to higher perturbation orders. The main results of this work are the 2nd-order expressions of the dynamic energy-release-rates for arbitrary perturbations of straight fronts. The formulae recover the known energy-release-rates of curved quasi-static fronts and of simple 2D cracks. We show that the expressions are separable as a product of a dynamical prefactor that only depends on the instantaneous local normal front velocity, and a history functional that integrates past front configurations. To gain insight, the energy-release-rates in the two models are computed for a traveling wave perturbation. While similar at low wave velocities, the two theories behave differently for fast waves. In scalar elasticity, the 2nd-order contributions are always sub-dominant. However, in the Mode I theory, the 2nd-order correction becomes the dominant term at the crack front wave velocity, where the 1st-order term is zero. We discuss employing the energy-release-rate expressions to predict crack front dynamics via energy balance with dissipation.

中文翻译：

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平面裂纹前沿动力学非线性扰动的综合研究


裂纹前沿与粗糙体的相互作用是异质材料断裂准则和预测断裂表面形成的核心。众所周知，动态裂纹前沿如何响应小的一阶扰动。然而，裂纹运动的大的局部扰动会引起超出现有线性理论的动态和几何非线性效应。由于确定扰动裂纹前沿周围的 3D 弹性场是任何裂纹前沿动力学理论研究的必要步骤，因此我们开发了平面裂纹前沿渐近场的二阶扰动理论。基于以前的工作，我们考虑两种断裂模型：（1）标量弹性固体中的断裂，它是反平面剪切断裂的类似物（模式III）。在该模型中，近裂纹场是通过匹配渐近展开获得的。 (2) 拉伸 I 型断裂，其中使用自洽膨胀来解决裂纹前沿附近的场。这些方法可以很容易地扩展到更高的扰动阶。这项工作的主要结果是直线锋面任意扰动的动态能量释放率的二阶表达式。这些公式恢复了弯曲准静态前沿和简单二维裂缝的已知能量释放率。我们证明这些表达式是可分离的，作为仅取决于瞬时局部法向锋面速度的动力学前因子和集成过去锋面配置的历史泛函的乘积。为了深入了解，两个模型中的能量释放率是针对行波扰动进行计算的。虽然在低波速下相似，但这两种理论对于快波的表现不同。在标量弹性中，二阶贡献始终是次主导的。 然而，在 I 型理论中，二阶校正成为裂纹前波速度处的主导项，其中一阶项为零。我们讨论使用能量释放率表达式通过能量平衡与耗散来预测裂纹前沿动力学。