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

Abstract

Key words disordered materials, failure process, size effects, fractal geometry, fractional calculus, principle of virtual work MSC (2000) 74A60Fractal 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.In solid mechanics, with the term size effect we mean the dependence of one or more material parameters on the size of the structure made by that material. In other words, we speak of size effect when geometrically similar structures show a different structural behavior. The first observations about size effect in solid mechanics date back to Galileo. For instance, in his "Discorsi e dimostrazioni matematiche intorno a due nuove scienze attenenti alla meccanica e i movimenti locali" (1638), he observed that the bones of small animals are more slender than the bones of big animals. In fact, increasing the size, the growth of the load prevails on the growth of the strength, since the first increases with the bulk, the latter with the area of the fracture surface. In the last century, fracture mechanics allowed a deeper insight in the size effect phenomenon. Nowadays, the most used model to describe damage localization in materials with disordered microstructure (also called quasi-brittle materials) is the cohesive crack model, introduced by Hillerborg et al. [1].According to Hillerborg's model, the material is characterized by a stress-strain relationship (σ-ε), valid for the undamaged zones, and by a stress-crack opening displacement relationship (σ-w, the cohesive law), describing how the stress decreases from its maximum value σ u to zero as the distance between the crack lips increases from zero to the critical displacement w c . The area below the cohesive law represents the energy G F spent to create the unit crack surface. The cohesive crack model is able to simulate tests where high stress gradients are present, e.g. tests on pre-notched specimens; in particular, it captures the ductile-brittle transition occurring by increasing the structural size. On the other hand, relevant scale effect...

Fractal solids, product measures and fractional wave equations

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Diffusion problems in fractal media defined on Cantor sets

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