A spectral method for numerical elastodynamic fracture analysis without spatial replication of the rupture event
A spectral method for numerical elastodynamic fracture analysis without spatial replication of the rupture event
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DOI：
10.1016/s0022-5096(97)00004-5
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发表时间：
1997-08-01
影响因子：
5.3
通讯作者：
Rice, JR
中科院分区：
文献类型：
--
作者：
Cochard, A;Rice, JR
Perrin等人。(1995)和Geubelle和Rice(1995)提出了一种数值求解二维和三维弹性动力断裂问题的谱方法。该方法适用于分离均匀弹性半空间的平面内的破裂。该方法将破裂面上应力和位移不连续的牵引力分量等物理变量表示为具有时变系数的空间傅立叶级数。对于每一种傅立叶模式，都找到了一个解析解，因为每个应力的傅里叶系数由对应的位移系数与特定于破裂模式的卷积核的时间卷积来表示。一旦已知该方法的2D公式，该方法很容易推广到3D问题，因为它只涉及为2D中的每个破裂模式找到的卷积核的线性组合。然而，这种概念上的简单性有一个主要的缺点：由于物理变量的傅立叶级数表示，所解决的问题实际上是断裂平面上破裂事件的无限和周期性的复制。因此，为了研究单个破裂的演化，人们必须使用足够大的空间周期，以便来自复制裂缝的波在研究的持续时间内不会进入感兴趣的区域，或者在到达时提供可以忽略不计的应力变化。我们在这里展示了如何在保持模式独立性的同时严格地弥补这一缺陷。一旦在空间域中表示，该方法就相当于在空间中截断时空卷积核，其方式是对破裂域(施加应力和位移不连续之间的本构定律)内的所有位置提供准确的评估，而不是在破裂域外。以使该方法在结构上与Perrin等人的方法相同。(1995)和Geubelle和Rice(1995)，要求傅立叶级数的周期仅为所关注的破裂区域的两倍。因此，与原始谱方法的唯一区别是，傅立叶域中的卷积核需要建立更精细的计算，但为了允许在给定域上进行模拟，这只需进行一次。(C)1997年爱思唯尔科学有限公司。
Perrin et al. (1995) and Geubelle and Rice (1995) have introduced a spectral method for numerical solution of two- and three-dimensional elastodynamic fracture problems. The method applies for ruptures confined to a plane separating homogeneous elastic half spaces. In this method, the physical variables, such as the traction components of stress and displacement discontinuity on the rupture plane, are represented as Fourier series in space with time-dependent coefficients. An analytical solution is found for each Fourier mode, in that each Fourier coefficient for stress is expressed by the time convolution of the corresponding coefficient for displacement with a convolution kernel specific to the rupture mode. Once the 2D formulation of the method is known, the method is readily generalizable to 3D problems in that it involves only linear combinations of the convolution kernels found for each rupture mode in 2D. This conceptual simplicity has, however, a major drawback: due to the Fourier series representations of the physical variables, the problem solved is in fact an infinite and periodic replication of rupture events on the fracture plane. So, in order to study the evolution of a single rupture, one has to use a spatial period large enough in order that the waves coming from the replication cracks do not enter the zone of interest during the time duration studied, or provide negligible stress alteration when they do arrive. We show here how to rigorously offset this defect while retaining the modal independence. Once expressed in the spatial domain, the method amounts to truncating in space the space-time convolution kernels, in a manner that provides an exact evaluation for all positions within the rupture domain (where the constitutive law between stress and displacement discontinuity is to be imposed), but not outside. In order for the method to be identical in structure to the method of Perrin er al. (1995) and Geubelle and Rice (1995), the period of the Fourier series is requested to be only twice as large as the rupture domain of interest. The only difference, then, to the original spectral method is that the convolution kernels in the Fourier domain require more elaborate calculations to be established, but this has to be done only once to allow simulations on a given domain. (C) 1997 Elsevier Science Ltd.