Discrete self-similarity of multiscale materials and systems. Universality of scaling exponents

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.