Oscillatory and tip-splitting instabilities in 2D dynamic fracture: The roles of intrinsic material length and time scales
Oscillatory and tip-splitting instabilities in 2D dynamic fracture: The roles of intrinsic material length and time scales
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DOI：
10.1016/j.jmps.2021.104372
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发表时间：
2020-12
影响因子：
5.3
通讯作者：
A. Vasudevan;Yuri Lubomirsky;Chih-Hung Chen;Eran Bouchbinder;A. Karma
中科院分区：
文献类型：
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作者：
A. Vasudevan;Yuri Lubomirsky;Chih-Hung Chen;Eran Bouchbinder;A. Karma
最近的理论和计算进展使人们对二维动态断裂中的对称破缺不稳定性有了前所未有的了解。这一进展的核心在于确定了两个固有的、接近裂纹尖端的长度标度--非线性弹性长度标度ℓ和耗散长度标度ξ--这是经典的裂纹理论线弹性断裂力学中不存在的。特别地，当传播速度v约为剪切波速的90%时，二维脆性材料中的裂纹经历了波长随ℓ线性变化的振荡不稳定性，而在较大的加载水平(对应于更高的传播速度)，出现了尖端劈裂不稳定性，这与实验结果一致。本文利用脆性断裂的相场模型，证明了振荡不稳定性的下列性质：(I)在不存在近端弹性非线性的情况下，也存在振荡不稳定性，即在极限ℓ→0中，其波长由耗散长度尺度ξ决定。这一结果表明，这种不稳定性主要取决于与裂尖附近线弹性断裂有关的本征长度尺度的存在，而与后者是与非线性弹性有关还是与耗散有关。(Ii)它是超临界霍普夫分叉，其特征是起始处的振荡幅度为零。(Iii)它在很大程度上与相场框架中用来描述粘聚区的退化函数的唯象形式无关，也与相场的金兹堡-朗道型演化方程中由耗散时间尺度控制的断裂能Γ(V)的速度依赖性无关。这些结果证实了二维振荡不稳定性的普遍性质。此外，我们提供的证据表明，裂尖不稳定性是由裂尖区域内弹性能量传输的极限速率控制的。后者对耗散区内的波速很敏感，可以在相场法中系统地改变耗散区内的波速。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，使其可以应用于广泛的材料破坏问题。
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 ℓ and a dissipation length scale ξ—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 ℓ, 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, ie in the limit ℓ→ 0, with a wavelength determined by the dissipation length scale ξ. 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 Γ (v) 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.