Fractal Solids, Product Measures and Continuum Mechanics

Abstract

This paper builds on the recently begun extension of continuum thermomechanics to fractal porous media which are specified by a mass (or spatial) fractal dimension D, a surface fractal dimension d, and a resolution length scale R. The focus is on pre-fractal media (i.e., those with lower and upper cut-offs) through a theory based on a dimensional regularization, in which D is also the order of fractional integrals employed to state global balance laws. In effect, the governing equations may be cast in forms involving conventional (integer-order) integrals, while the local forms are expressed through partial differential equations with derivatives of integer order but containing coefficients involving D, d and R. The formulation allows a generalization of the principles of virtual work, and virtual stresses, which, in turn, allow us to extend the extremum and variational theorems of elasticity and plasticity, as well as handle flows in fractal porous media. In all the cases, the derived relations depend explicitly on D, d and R, and, upon setting D=3 and d=2, reduce to conventional forms of governing equations for continuous media with Euclidean geometries. While the original formulation was based on a Riesz measure—and thus more suited to isotropic media—the new model is based on a product measure making it capable of grasping local material anisotropy. The product measure allows one to grasp the anisotropy of fractal dimensions on a mesoscale and the ensuing lack of symmetry of the Cauchy stress, as noted in condensed matter physics. On this basis, a framework of micropolar mechanics of fractal media is developed.