SCALING LAWS AND RENORMALIZATION-GROUPS FOR STRENGTH AND TOUGHNESS OF DISORDERED MATERIALS

The abundant literature on tensile strength and fracture energy size effects as well as the newly-introduced fractal and renormalization group theories would appear to indicate the need for a dramatic change in our conceptual framework, if we want to consider and measure material constants in Strength of Materials as well as in Fracture Mechanics. For disordered materials, such as for example concrete and rocks, the renormalized tensile strength is given by a force acting on a surface having a fractal dimension lower than 2, just as the renormalized fracture energy is represented by a dissipation over a surface with a dimension higher than 2. In the case of tensile strength, the dimensional decrement represents self-similar weakening of the reacting cross section or ligament, due to pores, voids, defects, cracks, aggregates, inclusions, etc. Likewise, in the case of fracture energy, the dimensional increment represents self-similar tortuosity of the fracture surface, due to aggregates and inclusions, as well as self-similar microcrack overlapping and distribution also in the direction orthogonal to that of the forming macrocrack. It can be demonstrated that the sum of the dimensional decrement (for material ligament) and the dimensional increment (for fracture surface) must be lower than unity.