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

Li, Jun; Ostoja–Starzewski, Martin

doi:10.1103/physreve.88.057001

Cited by 21 publications

(5 citation statements)

References 25 publications

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“…In [7,27], we set up the fractal strain on the principle that the field equations derived using the mechanical approach have to be the same as those derived using the variational approach. To this end, consider a one-dimensional (1D) fractal (i.e.…”

Section: Kinematics Of Fractalsmentioning

confidence: 99%

Micropolar mechanics of product fractal media

Ostoja–Starzewski

2022

Proc. R. Soc. A.

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Motivated by the abundance of fractals in the natural world, this paper further develops continuum-type models for product-like fractals. The theory is based on a version of the non-integer dimensional space approach, in which global balance laws written for fractal media are expressed in terms of conventional (integer-order) integrals. Key relations of calculus for finite strain kinematics of fractal media are obtained, especially clarifying the fractal Jacobian and the fractal Reynolds transport theorem. The local forms are then written in terms of partial differential equations with derivatives of integer order. Hence, fractal versions of local continuity, linear and angular momenta and energy balance are derived. The angular momentum balance implies that the approximating continuum is micropolar rather than classical. Accordingly, the Cauchy postulate, lemma and theorem for Cauchy force-stress and couple-stress are re-formulated. The corresponding partial differential equations for finite as well as infinitesimal elasticity are given explicitly in both the displacement and stress formulations. The invariance of the stress field in planar fractal elastic media is shown to hold just like the one in planar micropolar elasticity.

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Section: Kinematics Of Fractalsmentioning

confidence: 99%

Micropolar mechanics of product fractal media

Ostoja–Starzewski

2022

Proc. R. Soc. A.

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“…In order to do any analysis involving B α , in the vein of Tarasov (2005Tarasov ( , 2006, we employ the integration in non-integer dimensional space. That is, we take the infinitesimal element dB α of B α according to (Li and Ostoja-Starzewski, 2013):…”

Section: Mass Densitiesmentioning

confidence: 99%

Saturn rings: fractal structure and random field model

Malyarenko,

Ostoja-Starzewski

2016

Preprint

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This study is motivated by the observation, based on photographs from the Cassini mission, that Saturn's rings have a fractal structure in radial direction. Accordingly, two questions are considered: (1) What Newtonian mechanics argument in support of that fractal structure is possible? (2) What kinematics model of such fractal rings can be formulated? Both challenges are based on taking Saturn's rings' spatial structure as being statistically stationarity in time and statistically isotropic in space, but statistically nonstationary in space. An answer to the first challenge is given through the calculus in non-integer dimensional spaces and basic mechanics arguments (Tarasov (2006) Celest. Mech. Dyn. Astron. 94). The second issue is approached in Section 3 by taking the random field of angular velocity vector of a rotating particle of the ring as a random section of a special vector bundle. Using the theory of group representations, we prove that such a field is completely determined by a sequence of continuous positive-definite matrix-valued functions defined on the Cartesian square F 2 of the radial cross-section F of the rings, where F is a fat fractal.

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“…However, some fundamental questions regarding to the definition of fractal continuum still remain under debate (see Refs. [43,44]).…”

Section: Introductionmentioning

confidence: 99%

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

Balankin

2015

Eur. Phys. J. B

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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.

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

Cited by 21 publications

References 25 publications

Micropolar mechanics of product fractal media

Micropolar mechanics of product fractal media

Saturn rings: fractal structure and random field model

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

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