Bouchbinder Eran, Hentschel H George E, Procaccia Itamar
Department of Chemical Physics, The Weizmann Institute of Science, Rehovot 76100, Israel.
Phys Rev E Stat Nonlin Soft Matter Phys. 2003 Sep;68(3 Pt 2):036601. doi: 10.1103/PhysRevE.68.036601. Epub 2003 Sep 3.
We address the theory of quasistatic crack propagation in a strip of glass that is pulled from a hot oven towards a cold bath. This problem had been carefully studied in a number of experiments that offer a wealth of data to challenge the theory. We improve upon previous theoretical treatments in a number of ways. First, we offer a technical improvement of the discussion of the instability towards the creation of a straight crack. This improvement consists in employing Padé approximants to solve the relevant Wiener-Hopf factorization problem that is associated with this transition. Next we improve the discussion of the onset of oscillatory instability towards an undulating crack. We offer a way of considering the problem as a sum of solutions of a finite strip without a crack and an infinite medium with a crack. This allows us to present a closed form solution of the stress intensity factors in the vicinity of the oscillatory instability. Most importantly we develop a dynamical description of the actual trajectory in the regime of oscillatory crack. This theory is based on the dynamical law for crack propagation proposed by Hodgdon and Sethna. We show that this dynamical law results in a solution of the actual crack trajectory in post-critical conditions; we can compute from first principles the critical value of the control parameters, and the characteristics of the solution such as the wavelength of the oscillations. We present detailed comparison with experimental measurements without any free parameters. The comparison appears quite excellent. Finally we show that the dynamical law can be translated to an equation for the amplitude of the oscillatory crack; this equation predicts correctly the scaling exponents observed in experiments.
我们研究了从热炉中拉向冷浴的玻璃条中准静态裂纹扩展的理论。这个问题已经在许多实验中得到了仔细研究，这些实验提供了大量数据来挑战该理论。我们在许多方面改进了先前的理论处理方法。首先，我们对关于产生直裂纹的不稳定性的讨论进行了技术改进。这种改进在于采用帕德近似来解决与这种转变相关的维纳 - 霍普夫因式分解问题。接下来，我们改进了对起伏裂纹的振荡不稳定性起始的讨论。我们提供了一种将该问题视为无裂纹有限条带和有裂纹无限介质的解之和的方法。这使我们能够给出振荡不稳定性附近应力强度因子的封闭形式解。最重要的是，我们在振荡裂纹区域开发了实际轨迹的动力学描述。该理论基于霍奇登和塞思纳提出的裂纹扩展动力学定律。我们表明，这个动力学定律在临界后条件下导致了实际裂纹轨迹的解；我们可以从第一原理计算控制参数的临界值以及解的特征，如振荡的波长。我们在没有任何自由参数的情况下与实验测量进行了详细比较。比较结果非常出色。最后，我们表明动力学定律可以转化为振荡裂纹振幅的方程；该方程正确地预测了实验中观察到的标度指数。
