A continuum framework for mechanics of fractal materials I: from fractional space to continuum with fractal metric

Abstract

This paper is devoted to the mechanics of fractal materials. A continuum framework accounting for the topological and metric properties of fractal domains in heterogeneous media is developed. The kinematics of deformations is elucidated and the symmetry of the Cauchy stress tensor is established. The mapping of mechanical problems for fractal materials into the corresponding problems for the fractal continuum is discussed. Stress and strain distributions in elastic fractal bars are analyzed. Some features of acoustic wave propagation and localization in scale-invariant media are briefly discussed. The effect of fractal correlations in the material microstructure on the crack mechanics is revealed. It is shown that the fractal nature of heterogeneity can either delay or assist the crack initiation and propagation, depending on the interplay between metric and topological properties of the fractal domain.

Anomalous diffusion of fluid momentum and Darcy-like law for laminar flow in media with fractal porosity

“…[43,[47][48][49][50][51][52]). Furthermore, the mapping F ⊂ E 3 → F ν ⊆ E 3 implies the use of metric partial derivatives ∂/∂ i instead of conventional partial derivatives ∂/∂x i [43].…”

“…The development of vector differential calculus in the fractional dimensional space with ν < 3 is not so straightforward (see Refs. [7,43,[50][51][52] and references therein). Fortunately, in the special case of one-dimensional flow through a fractally permeable medium, as studied in our experiments (see Fig.…”

“…(4a). Consequently, the gradient of function f (x j ) in the flow direction i = 1 reads as ∂ f /∂ 1 = (∂ 1 /∂x 1 ) −1 ∂ f /∂x 1 [43,[47][48][49][50][51][52]. Accordingly, for the one-dimensional laminar flow through a fractal the Darcy-like equation can be obtained by changing the spatial variable (x 1 → 1 ) and gradient operator (∂/∂ 1 → ∂/∂x 1 ) in the equation q = (k/μ)∂ P /∂x 1 derived in Ref.…”

“…For such materials classical homogenization methods become inapplicable because heterogeneities play an important role on almost all scales [7]. Even so, it is expected that a linear relation between the laminar flux of a Newtonian fluid and the pressure drop holds [8][9][10].…”

Fractional-dimensional Child-Langmuir law for a rough cathode

“…24,25 The axiomatic basis of spaces with fractional dimension had been introduced by Stillinger, 24 where he described the integration on a space with non-integer dimension, and provided a generalization of second order Laplace operators. This approach has been widely applied in quantum field theory, 18,26,27 general relativity, 28 thermodynamics, 29 mechanics, [30][31][32] hydrodynamics, 33 and electrodynamics. 20,[34][35][36][37][38][39][40][41][42][43] To expand the range of possible applications of models with fractional-dimensional spaces, a complete generalization of vector calculus operators has been reported recently.…”