Abstract The evolution of essentially periodic crack patterns appears in many natural and technical processes. For example, typical cases of periodic crack patterns are known from drying soils, from the formation of basalt columns, from high temperature loaded ceramics, from fissure formation in coke, and from drying protein solution drops. The appearance of more or less periodic crack patterns is the result of cascades of bifurcation-instabilities of the propagation process. There exist established, mathematically rigorous criteria for the bifurcation of the crack growth process of two initially equally long parallel cracks (Nemat-Nasser et al., 1978; Bazant et al., 1979). These criteria require the determination of the mutual influence of the neighboring cracks on their growth by calculation of partial derivatives. If discretization methods are used for simulating such a growth process, the calculation of these partial derivatives may become very sensitive with respect to the density of the spatial discretization. This is especially true if strong gradients of a crack driving eigenstress field appear as, for instance, in chemical diffusion processes. In the present paper, an alternative approach is introduced, which circumvents estimations of partial derivatives and, in addition, provides a more realistic picture of the generation of crack patterns. This approach is inspired by a computationally efficient method for investigating the buckling and post-buckling behavior in structural mechanics by introducing small imperfections. As a typical example, in which the approach introduced here is especially beneficial, the simulation of self-organized crack pattern evolution in alkali feldspar, a very common brittle crystal, is presented. There, the cracks are driven by eigenstrains caused by cation exchange between the crystal and a salt melt. Comparisons between the character of the pattern formation in the simulation and in experiments show excellent agreement. Furthermore, by combining simulations and experiments the J c -value can be determined at considerably high accuracy.

中文翻译：

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模拟单晶化学诱导裂纹模式演变的替代方法

摘要 在许多自然和技术过程中出现了本质上周期性的裂纹模式的演变。例如，从干燥土壤、玄武岩柱的形成、高温加载的陶瓷、焦炭中的裂缝形成和干燥蛋白质溶液滴中已知周期性裂缝模式的典型案例。或多或少的周期性裂纹图案的出现是传播过程的分叉不稳定性级联的结果。对于两个初始等长平行裂纹的裂纹扩展过程的分岔，存在已建立的数学上严格的标准（Nemat-Nasser 等人，1978 年；Bazant 等人，1979 年）。这些标准要求通过计算偏导数来确定相邻裂纹对其生长的相互影响。如果使用离散化方法来模拟这种增长过程，则这些偏导数的计算可能对空间离散化的密度变得非常敏感。如果裂纹驱动本征应力场的强梯度出现在例如化学扩散过程中，则尤其如此。在本文中，介绍了一种替代方法，该方法绕过了偏导数的估计，此外，还提供了更真实的裂纹模式生成图。这种方法的灵感来自一种通过引入小缺陷来研究结构力学中屈曲和屈曲后行为的计算高效方法。作为一个典型的例子，这里介绍的方法特别有益，介绍了碱长石（一种非常常见的脆性晶体）中自组织裂纹模式演化的模拟。在那里，裂缝是由晶体和盐熔体之间的阳离子交换引起的本征应变驱动的。模拟和实验中图案形成的特征之间的比较显示出极好的一致性。此外，通过结合模拟和实验，可以以相当高的精度确定 J c 值。