A disordered microstructure material model based on fractal geometry and fractional calculus

Fractal patterns often arise in the failure process of materials with a disordered microstructure. It is shown that they are responsible of the size effects on the parameters characterizing the material behaviour in tensile tests (i.e. the strength, the fracture energy, and the critical displacement). Based on fractal geometry, a simple model of a generic disordered material is set. The physical quantities describing the stress‐strain state of such fractal medium are pointed out. They show anomalous (non integer) physical dimensions. In terms of these fractal quantities, it is possible to define a fractal cohesive law, i.e. a constitutive law describing the tensile failure of an heterogeneous material, which is scale invariant. Then we propose new mathematical operators from fractional calculus to handle the fractal quantities previously introduced. In this way, the static and kinematic (fractional) differential equations of the model are pointed out. These equations form the basis of the mechanics of fractal media. In this framework, the principle of virtual work is also obtained.