Scaling laws for strength and toughness of disordered materials: A unified theory based on fractal geometry

Abstract

Direct tensile testing of concrete was carried out in a stable manner and the deformation eccentricity was adjusted continuously during the loading process. In this way, the secondary flexural stresses were minimized so that it was possible to obtain nominal tensile strength and fracture energy of the material. Both such properties appear to vary monotonically with the diameter of the cylindrical specimens: the tensile strength decreases, whereas the fracture energy increases. For disordered materials it is impossible to measure constant material properties, unless we depart from integer dimensions of the material ligament at peak stress and of the fracture surface at final rupture, in the framework of fractal geometry. In this way we can define new tensile properties with physical dimensions depending on the fractal dimension of the damaged microstructure, which turn out to be scale-invariant material constants. This represents the so-called renormalization procedure, already proposed in the statistical physics of random processes. As a limit case, when the material is extremely disordered, both tensile strength and fracture energy present the physical dimension characteristic of the stress-intensity factor, [ F] [L] minus; 3 2 .