Thermo-poromechanics of fractal media

Abstract

This article advances continuum-type mechanics of porous media having a generally anisotropic, product-like fractal geometry. Relying on a fractal derivative, the approach leads to global balance laws in terms of fractal integrals based on product measures and, then, converting them to integer-order integrals in conventional (Euclidean) space. Proposed is a new line transformation coefficient that is frame invariant, has no bias with respect to the coordinate origin and captures the differences between two fractal media having the same fractal dimension but different density distributions. A continuum localization procedure then allows the development of local balance laws of fractal media: conservation of mass, microinertia, linear momentum, angular momentum and energy, as well as the second law of thermodynamics. The product measure formulation, together with the angular momentum balance, directly leads to a generally asymmetric Cauchy stress and, hence, to a micropolar (rather than classical) mechanics of fractal media. The resulting micropolar model allowing for conservative and/or dissipative effects is applied to diffusion in fractal thermoelastic media. First, a mechanical formulation of Fick’s Law in fractal media is given. Then, a complete system of equations governing displacement, microrotation, temperature and concentration fields is developed. As a special case, an isothermal model is worked out. This article is part of the theme issue ‘Advanced materials modelling via fractional calculus: challenges and perspectives’.

Impact force and moment problems on random mass density fields with fractal and Hurst effects

“…The last few decades have seen much activity in partial differential equations where temporal and/or spatial derivatives are assumed to be fractional. The article [33] discusses the formulation of a continuum mechanics model smoothing a fractal porous microstructure. Here, we note that there exists no solution to a Lamb-type problem with space fractional derivatives, which would directly correspond to (at least) the anti-plane elastodynamics problem in a random, linear elastic medium [5].…”

Advanced materials modelling via fractional calculus: challenges and perspectives

“…In the theme issue, the mechanics of fractal media is addressed by Li & Ostoja-Starzewski [119] with focus on porous media. They propose a continuum model for anisotropic porous media of fractal type, which expresses the global balance laws in terms of fractal integrals based on proper product measures and converts them to integer-order integrals in conventional (Euclidean) space.…”

Applications of Distributed-Order Fractional Operators: A Review

“…Unlike integer-order operators, the intrinsic multiscale nature of fractional operators enabled a very unique and effective approach to model historically challenging physical processes involving, as an example, nonlocality or memory effects. Indeed, many of the early applications of FC to physical modeling included viscoelastic effects [ 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 ], nonlocal behavior [ 8 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 ], anomalous and hybrid transport [ 9 , 10 , 11 , 24 , 25 , 26 , 27 , 28 , 29 , 30 ], fractal media [ 12 , 31 , 32 , 33 , 34 , 35 ], and even control theory [ 36 , 37 , 38 , 39 ]. The interested reader is referred to the work in [ 40 ] for a detailed account of the birth and evolution of fractional calculus.…”