Elasticity of Fractal Material by Continuum Model with Non-Integer Dimensional Space

TL;DR

This paper develops a continuum model using non-integer dimensional space to analyze the elastic properties of fractal materials, providing solutions for various geometries and conditions.

Contribution

It introduces a generalized elasticity framework for fractal materials within non-integer dimensional space, extending classical models to fractal geometries.

Findings

Solutions for equilibrium states of fractal hollow balls and pipes.

Elasticity problems for rotating and pressurized fractal structures.

Extensions to gradient elasticity and thermoelasticity in fractal materials.

Abstract

Using a generalization of vector calculus for space with non-integer dimension, we consider elastic properties of fractal materials. Fractal materials are described by continuum models with non-integer dimensional space. A generalization of elasticity equations for non-integer dimensional space, and its solutions for equilibrium case of fractal materials are suggested. Elasticity problems for fractal hollow ball and cylindrical fractal elastic pipe with inside and outside pressures, for rotating cylindrical fractal pipe, for gradient elasticity and thermoelasticity of fractal materials are solved.