Anisotropic Fractal Media by Vector Calculus in Non-Integer Dimensional Space

TL;DR

This paper reviews and develops a generalized vector calculus framework for modeling anisotropic fractal media using non-integer and multi-fractional dimensional spaces, enabling continuum models of complex fractal materials.

Contribution

It introduces a product measure-based generalization of vector calculus for non-integer dimensional spaces to describe anisotropic fractal media.

Findings

Defined differential operators in non-integer dimensional spaces.

Applied the framework to Poisson's, Euler-Bernoulli, and Timoshenko equations.

Enabled continuum modeling of anisotropic fractal materials.

Abstract

A review of different approaches to describe anisotropic fractal media is proposed. In this paper differentiation and integration non-integer dimensional and multi-fractional spaces are considered as tools to describe anisotropic fractal materials and media. We suggest a generalization of vector calculus for non-integer dimensional space by using a product measure method. The product of fractional and non-integer dimensional spaces allows us to take into account the anisotropy of the fractal media in the framework of continuum models. The integration over non-integer-dimensional spaces is considered. In this paper differential operators of first and second orders for fractional space and non-integer dimensional space are suggested. The differential operators are defined as inverse operations to integration in spaces with non-integer dimensions. Non-integer dimensional space that is product of spaces with different dimensions allows us to give continuum models for anisotropic type of the media. The Poisson's equation for fractal medium, the Euler-Bernoulli fractal beam, and the Timoshenko beam equations for fractal material are considered as examples of application of suggested generalization of vector calculus for anisotropic fractal materials and media.