Continuum Fractal Mechanics of the Earth’s Crust

In the cases when the Earth’s crust possesses self-similar structure its mechanical behaviour can be modelled by a continuous sequence of continua each determined by the size of the averaging volume element. It is shown that tensorial properties and integral state variables scale by power laws with exponents common for all components of the tensors. Thus the scaling is always isotropic with anisotropy accounted for by the prefactors. As an example, scaling laws for effective moduli of the Earth’s crust with self-similar cracking are derived for the cases of isotropic distribution of disk-like cracks and two mutually orthogonal sets of 2-D cracks. Real systems are not self-similar therefore the proposed approach is based on their approximation by self-similar systems. A necessary condition is formulated for such an approximation.