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Perturbation of a dynamic planar crack moving in a model elastic solid

机译:在模型弹性固体中移动的动态平面裂纹的摄动

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The dynamic perturbation of a crack propagating in a model elastodynamic material gov- erned by a scalar wave equation is discussed. The crack front moves with nominally constant speed through a largely homogeneous material, and is perturbed from the straight by slight heterogeneities in fracture toughness. This subject has received attention in a series of papers both numerical and theoretical the stress on the plane of the crack prior to the perturbation has the asymptotic form K/sq root2πr M/sq root at a distance r from the crack's edge. All previous work concentrated on cases where M = 0. Willis and Movchan (l995) produced an equation taking account of a possible non-zero M. This related the stress intensity factor to the position of the crack edge through a non- homogeneous integral operator. The admission of non-zero M in this equation has implications on the variance of the slope of the crack front when it encounters a region of persistent random heterogeneities. In general, it is found that with negative M, perturbations are stable, whereas with positive M, they are unstable. The case of zero M discussed in the literature forms a non- uniform limit. Explicit inversion to find the crack edge position is difficult. Instead, approximate solutions in transform space for constant and time-dependent small M are presented. This al1ows a direct numerical scheme to find solutions to problems previously only considered for the case M = 0.
机译:讨论了在由标量波动方程控制的弹性材料模型中传播的裂纹的动态扰动。裂纹前沿以名义上恒定的速度移动通过基本上均匀的材料,并由于断裂韧性的轻微不均匀性而从直线上受到干扰。该主题在一系列论文中都得到了数值和理论的关注,在扰动之前裂纹平面上的应力在距裂纹边缘的距离r处具有渐近形式K / sqroot2πrM / sq根。以前的所有工作都集中在M = 0的情况下。Willis和Movchan(l995)生成了一个考虑可能的非零M的方程。这通过非均质积分算符将应力强度因子与裂纹边缘的位置相关联。当方程中遇到持久的随机异质性区域时,该方程中非零M的取值对裂纹前沿的斜率方差有影响。通常,发现在M为负的情况下,扰动是稳定的,而M为正时的扰动是不稳定的。文献中讨论的零M的情况形成了一个非均匀极限。显式反演很难找到裂纹边缘的位置。取而代之的是,给出了变换空间中恒定且随时间变化的小M的近似解。这允许使用直接数值方案找到以前仅在情况M = 0时才考虑的问题的解决方案。

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