Waves in Fractal Media

Abstract

The term fractal was coined by Benoît Mandelbrot to denote an object that is broken or fractured in space or time. Fractals provide appropriate models for many media for some finite range of length scales with lower and upper cutoffs. Fractal geometric structures with cutoffs are called pre-fractals. By fractal media, we mean media with prefractal geometric structures. The basis of this study is the recently formulated extension of continuum thermomechanics to such media. The continuum theory is based on dimensional regularization, in which we employ fractional integrals to state global balance laws. The global forms of governing equations are cast in forms involving conventional (integerorder) integrals, while the local forms are expressed through partial differential equations with derivatives of integer order. Using Hamilton's principle, we derive the equations of motion of a fractal elastic solid under finite strains. Next, we consider one-dimensional models and obtain equations governing nonlinear waves in such a solid. Finally, we study shock fronts in linear viscoelastic solids under small strains. In all the cases, the derived equations for fractal media depend explicitly on fractal dimensions and reduce to conventional forms for continuous media with Euclidean geometries upon setting the dimensions to integers.

Comment on “Hydrodynamics of fractal continuum flow” and “Map of fluid flow in fractal porous medium into fractal continuum flow”

“…The product measure we previously proposed allows the treatment of fractal anisotropy by decoupling coordinate * junli@caltech.edu † martinos@illinois.edu variables, and this further simplifies the Green-Gauss theorem and induces the definition of a fractal derivative [3][4][5]. To see it clearly, the Green-Gauss theorem is generalized as…”

“…The expression of c 1 (x k ,α k ) should reflect the physical scaling law of fractal mass m ∼ x α k k . As such, various forms could be proposed [1][2][3][4][5][6] and, for example, Tarasov [6] …”

“…[1] incorrectly commented on our derivatives in Refs. [3][4][5] as being nonlocal and unrecoverable from Jumarie's fractional derivative. To clarify it, observe the following:…”

“…(2) The form [3][4][5] removes the singularity at the origin and the distance to the boundary position satisfies the frame-independence requirement (i.e., objectivity), while it introduces singularities on the boundaries. On the other hand, the form of (x k /l 0 + 1) α k −1 [1,2] removes singularities both at the origin and the boundaries, while the objectivity (frame independence) is violated since a different choice of the origin results in the change of the coordinate value x k for a fixed point and, further, a different c 1 (x k ,α k ) and fractal derivative associated with that point.…”

“…Two recent papers [1,2] studied the fluid flow in the fractal pore space employing the local fractional calculus. In this Comment we discuss the relation of those operators to the product measure and fractal derivative employed in a previous formulation of continuum mechanics of fractal media [3][4][5]. For reference, the product measure is based on a dimensional regularization of fractal mass to a fractal continuum, [3][4][5], (1) with the length element along each coordinate axis given by a line coefficient c…”