Continuum mechanics models of fractal porous media: Integral relations and extremum principles

This paper continues the extension of continuum mechanics and thermodynamics 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 a theory based on dimensional regularization, in which D is also the order of fractional integrals employed to state global balance laws. Thus, we first generalize the main integral theorems of continuum mechanics to fractal media: Stokes, Reynolds, and Helmholtz- Żórawski. Then, we review balance equations and recently obtained extensions of several subfields of continuum mechanics to fractal media. This is followed by derivations of extremum and variational principles of elasticity and Hamilton's principle for fractal porous materials. In all the cases, we derive relations which depend explicitly on D, d and R, and which, upon setting D = 3 and d = 2, reduce to the conventional forms of governing equations for continuous media with Euclidean geometries.