Fractal solids, product measures and fractional wave equations

Abstract

This paper builds on the recently begun extension of continuum thermomechanics to fractal porous media that 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 are 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 . This procedure allows a specification of a geometry configuration of continua by ‘fractal metric’ coefficients, on which the continuum mechanics is subsequently constructed. While all the derived relations depend explicitly on D , d and R , upon setting D =3 and d =2, they reduce to conventional forms of governing equations for continuous media with Euclidean geometries. Whereas 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 fractal anisotropy. Finally, the one-, two- and three-dimensional wave equations are developed, showing that the continuum mechanics approach is consistent with that obtained via variational energy principles.