Recent theoretical and computational progress has led to unprecedented understanding of symmetry-breaking instabilities in 2D dynamic fracture. At the heart of this progress resides the identification of two intrinsic, near crack tip length scales — a nonlinear elastic length scale $\ell$ and a dissipation length scale $\xi$ — that do not exist in Linear Elastic Fracture Mechanics (LEFM), the classical theory of cracks. In particular, it has been shown that at a propagation velocity $v$ of about 90% of the shear wave-speed, cracks in 2D brittle materials undergo an oscillatory instability whose wavelength varies linearly with $\ell$, and at larger loading levels (corresponding to yet higher propagation velocities), a tip-splitting instability emerges, both in agreements with experiments. In this paper, using phase-field models of brittle fracture, we demonstrate the following properties of the oscillatory instability: (i) It exists also in the absence of near-tip elastic nonlinearity, i.e. in the limit $\ell \to 0$, with a wavelength determined by the dissipation length scale $\xi$. This result shows that the instability crucially depends on the existence of an intrinsic length scale associated with the breakdown of linear elasticity near crack tips, independently of whether the latter is related to nonlinear elasticity or to dissipation. (ii) It is a supercritical Hopf bifurcation, featuring a vanishing oscillations amplitude at onset. (iii) It is largely independent of the phenomenological forms of the degradation functions assumed in the phase-field framework to describe the cohesive zone, and of the velocity-dependence of the fracture energy $\Gamma \left(v\right)$ that is controlled by the dissipation time scale in the Ginzburg–Landau-type evolution equation for the phase-field. These results substantiate the universal nature of the oscillatory instability in 2D. In addition, we provide evidence indicating that the tip-splitting instability is controlled by the limiting rate of elastic energy transport inside the crack tip region. The latter is sensitive to the wave-speed inside the dissipation zone, which can be systematically varied within the phase-field approach. Finally, we describe in detail the numerical implementation scheme of the employed phase-field fracture approach, allowing its application in a broad range of materials failure problems.

最近的理论和计算进展已导致对二维动态裂缝中对称破坏不稳定性的空前了解。这一进展的核心在于确定两个固有的，接近裂纹尖端的长度尺度—非线性弹性长度尺度。$\ell$ 和耗散长度标度 $\xi$-在经典的裂纹理论线性弹性断裂力学（LEFM）中不存在。特别地，已经表明，在传播速度下$v$ 在大约90％的剪切波速下，二维脆性材料中的裂纹经历了振荡不稳定，其波长随波长线性变化 $\ell$，并且在较大的负载水平（对应更高的传播速度）下，都会出现尖端分裂不稳定性，这与实验一致。在本文中，使用脆性断裂的相场模型，我们证明了振荡不稳定性的以下特性：（i）它也存在于不存在尖端弹性非线性的情况下，即在极限范围内。$\ell \to 0$，其波长由耗散长度标度决定 $\xi$。该结果表明，不稳定性至关重要地取决于与裂纹尖端附近的线性弹性破坏相关的固有长度尺度的存在，而与裂纹尖端与非线性弹性或耗散有关无关。（ii）这是一个超临界霍普夫分叉，其起振点振幅消失。（iii）在很大程度上不依赖于描述相干区的相场框架中假定的退化函数的现象学形式，以及断裂能的速度依赖性。$\Gamma （v）$它由金兹堡-兰道型相场演化方程中的耗散时间尺度控制。这些结果证实了二维振动不稳定的普遍性。此外，我们提供的证据表明，尖端断裂不稳定性受裂纹尖端区域内部弹性能量传输的限制速率控制。后者对耗散区内的波速敏感，可以在相场方法内系统地改变波速。最后，我们详细描述了所采用的相场断裂方法的数值实现方案，从而将其应用在广泛的材料失效问题中。