Discrete self-similarity of multiscale materials and systems. Universality of scaling exponents
Discrete self-similarity of multiscale materials and systems. Universality of scaling exponents
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DOI：
10.1016/j.ijengsci.2020.103244
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发表时间：
2020-04-01
影响因子：
6.6
通讯作者：
Pasternak, E.
中科院分区：
文献类型：
--
作者：
Dyskin, A., V;Pasternak, E.
材料通常具有复杂的微观结构，显示出覆盖广泛尺度的结构层次。具有多尺度结构的物质的一个例子是地壳。这种材料结构的最简单模型是自相似性，使得尺度相关量由幂律表示。更现实的模型是离散自相似性，即物质结构仅在离散（和自相似）尺度集上自相似。我们表明，在离散自相似的幂律依赖被保存，但只在离散集的尺度。这项工作的主要结果是，只要离散标度量是线性关系，它们必须以相同的指数标度或消失;我们称之为标度指数的普适性。特别是，同一张量的非零分量必须以相同的指数缩放。这导致弹性模量的相同比例，因此，导致波速的相同比例。平均应力和应变及其较高的统计矩也按幂律缩放;简单的关系被确定之间的标度指数的平均应变，应力，其较高的时刻和弹性模量。由于平均应变的标度可以从观测中确定，标度模量可以从测量的波速中推断，因此导出的关系可以用于推断应力分布的统计参数的标度。所提出的概念将是有用的，在表征属性的Meta和混合材料和岩土材料。(C)2020爱思唯尔有限公司保留所有权利。
Materials often possess complex microstructure showing structural hierarchy covering a wide range of scales. An example of material with multiscale structure is the Earth's crust. The simplest model of such material structure is self-similarity making the scale dependent quantities to be expressed by the power law. Somewhat more realistic model is the discrete self-similarity that is the material structure is self-similar only over a discrete (and self-similar) set of scales. We show that in discrete self-similarity the power law dependence is preserved but only over the discrete sets of scales. The main result of this work is that as long as discrete-scaling quantities are in a linear relationship, they must either scale with the same exponent or vanish; we call this property the Universality of scaling exponents. In particular, non-zero components of the same tensor must scale with the same exponent. This leads to the same scaling of the elastic moduli and, consequently, to the same scaling of the wave velocities. Average stress and strain and their higher statistical moments also scale by power law; simple relationships are identified between the scaling exponents of average strain, stress, their higher moments and the elastic moduli. As the scaling of average strain could be determined from observations and scaling moduli can be inferred from the measured wave velocities, the derived relationships can be used to infer the scaling of the statistical parameters of stress distributions. The presented concept will be useful in characterising properties of both the meta- and hybrid materials and geomaterials. (C) 2020 Elsevier Ltd. All rights reserved.