Fractal Scale Effect in Quasi-Brittle Materials Using a Version of the Discrete Element Method

In engineering applications, it is crucial to consider the size dependence of a material’s mechanical properties and its overall behavior. One of the theories that quantifies this phenomenon in quasi-brittle materials is the cohesive fractal theory (CFT) introduced by Carpinteri and his collaborators. This theory describes the behavior of materials using fractal dimensions. To investigate whether the scale effect can be analyzed using the CFT, a version of the Lattice Discrete Element Method (LDEM) is employed. The accuracy of the LDEM in capturing the scale effect is evaluated through simulations of three primary tests. Specifically, rock specimens are subjected to tensile, compressive, and bending loads to determine their mechanical properties. The influence of material heterogeneity and boundary conditions is also examined. In scenarios involving tensile and bending loads, the localization of a significant crack leads to failure. According to the CFT, the sum of the fractal exponents is close to unity, with values of 1.0 (mean value) for tensile loading and 0.97 for bending loading. However, the compressive loading results do not exhibit this characteristic, as no single prominent crack leads to failure. Overall, the LDEM results are consistent with the CFT, effectively quantifying the scale effect without modifying the elementary constitutive law.