A fractional calculus approach to the description of stress and strain localization in fractal media

Evidence of fractal patterns in materials with disordered microstructure under tensile loads is undeniable. Unfortunately fractal functions cannot be solution of classical differential equations. Hence a new calculus must be developed to handle fractal processes. In this paper, we use the local fractional calculus operators recently introduced by K.M. Kolwankar [Studies of fractal structures and processes using methods of fractional calculus. PhD thesis, University of Pune, India, 1998]. Through these new mathematical tools we get the static and kinematic equations that model the uniaxial tensile behavior of heterogeneous materials. The fractional operators respect the non-integer (fractal) physical dimensions of the quantities involved in the governing equations, while the virtual work principle highlights the static-kinematic duality among them. The solutions obtained from the model are fractal and yield to scaling power laws characteristic of the nominal quantities, i.e., they reproduce the size effects due to stress and strain localization.