Power scaling laws and dimensional transitions in solid mechanics

Physicists have often observed a scaling behaviour of the main physical quantities during experiments on systems exhibiting a phase transition. The main assumption of a scaling theory is that these characteristic quantities are self-similar functions of the independent variables of the phenomenon and, therefore, such a scaling can be interpreted be means of power-laws. Since a characteristic feature of phase transitions is a catastrophic change of the macroscopic parameters of the system undergoing a continuous variation in the system state variables, the phenomenon of fracture of disordered materials can be set into the wide framework of critical phenomena. In this paper new mechanical properties are defined, with non integer physical dimensions depending on the scaling exponents of the phenomenon (i.e. the fractal dimension of the damaged microstructure, or the exponent of a power-constitutive relation), which turn out to be scale-invariant material constants. This represents the so-called renormalization procedure, already proposed in the statistical physics of random process.