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Hygrothermoelastic responses in a fractal media by extending vector calculus to non-integer dimensions

This paper generalizes vector calculus to non-integer fractal dimensions to develop a new hygrothermoelastic model incorporating Moore-Gibson-Thompson theory and multi-kernel derivatives, which is solved analytically to demonstrate that fractal dimensions, delay periods, and nonlinear kernel functions significantly influence heat and moisture transport in materials.

Original authors: Apeksha Balwir, Nagesh Dhore, Vinod Varghese

Published 2026-08-19

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CC BY 4.0

Original authors: Apeksha Balwir, Nagesh Dhore, Vinod Varghese

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a world where the rules of geometry we learn in school do not quite fit the materials around us. In standard geometry, a line has one dimension, a flat surface has two, and a solid block has three. But nature is often more intricate than these simple shapes. Think of a coastline, a cloud, or the branching of a tree; these objects are complex, self-repeating patterns that look similar whether you zoom in or out. Scientists call these shapes fractals. Because they are so tangled and detailed, they do not behave like smooth, solid blocks. To understand how heat and moisture move through such messy, natural structures, researchers have begun to treat them not as ordinary three-dimensional objects, but as spaces with dimensions that are not whole numbers. This approach allows them to describe the hidden complexity of materials like porous rocks, biological tissues, or advanced composites with much greater accuracy.

Building on this idea, a team of researchers has developed a new way to predict how these fractal materials react when they are heated and wetted at the same time. In the real world, materials rarely face just heat or just moisture; they often face both, such as a jet engine part exposed to hot, humid air or a building material absorbing rain while warming in the sun. When this happens, the material expands, contracts, and develops internal stresses that can lead to failure. The researchers created a mathematical model to simulate these conditions inside a hollow cylinder made of a fiber-reinforced composite, a material often used in high-performance engineering. They treated the material as a continuous substance existing in a space with a non-integer dimension, effectively giving the math a "fractal" shape to match the material's internal structure.

The study introduced a sophisticated way of looking at time and memory within the material. Instead of assuming that heat and moisture move instantly or follow a simple, straight path, the model accounts for the fact that the material's current state depends on its recent history. The researchers used a concept called a "memory-dependent derivative," which means the material's reaction to a change in temperature or humidity is influenced by what happened in the moments just before. They tested different mathematical functions, known as kernels, to describe how this memory works. Some functions assumed a simple, linear relationship, while others used more complex, non-linear curves. By running detailed computer simulations, they found that the non-linear functions, which allow for more complex memory effects, provided a better description of the material's behavior than the simpler linear ones.

The results of these simulations revealed several key behaviors. When the material was subjected to a changing heat source, the temperature and moisture did not spread evenly. Instead, they moved in waves that were heavily influenced by the material's fractal nature and the time delays inherent in the system. The researchers observed that the material's internal stress patterns were significantly different from what would be predicted by traditional models. In the fractal model, the stresses were generally lower in magnitude, but the distribution was more complex, with specific areas showing unique concentrations of force that standard models would miss. They also found that the delay in how the material responded to heat and moisture played a crucial role; changing the length of this delay altered the entire pattern of temperature and stress across the material.

Perhaps most importantly, the study highlighted that the way a material "remembers" its past conditions matters deeply. The simulations showed that using a non-linear memory function, which captures a more realistic and complex history of the material's state, led to more accurate predictions of how the material would deform and where it might be at risk of breaking. This suggests that for engineers designing components for aerospace, automotive, or even biomedical applications, ignoring the fractal nature of materials and their complex memory could lead to flawed designs. By incorporating these advanced mathematical tools, the researchers have provided a clearer picture of how heat and moisture travel through the intricate, non-smooth world of fractal materials, offering a more reliable guide for building safer and more efficient structures.

Technical Summary: Hygrothermoelastic Responses in a Fractal Media by Extending Vector Calculus to Non-Integer Dimensions

Problem Statement
The paper addresses the lack of detailed mathematical models that utilize vector calculus principles within non-integer dimensional spaces to examine the hygrothermoelastic response of fractal media. While fractals are ubiquitous in nature and characterized by non-integer Hausdorff dimensions, traditional continuum mechanics often relies on integer-dimensional Euclidean geometry. Furthermore, existing literature often neglects the specific coupling of heat and moisture transport under the Moore-Gibson-Thompson (MGT) theory when combined with forward memory-dependent derivatives. The study aims to generalize vector calculus to non-integer dimensions ( $D$ ) to construct a continuum model for fractal materials that accounts for higher-order time frame derivatives, phase lag delays, and memory effects.

Methodology
The research employs a continuum model framework extended to non-integer dimensional spaces. The methodology proceeds through the following steps:

Mathematical Generalization: The authors extend vector calculus operators (gradient, divergence, and Laplacian) from integer dimensions to non-integer dimensions ( $D$ ). For a radially symmetric vector field in a non-integer dimensional space, specific forms for the divergence, gradient, scalar Laplacian, and vector Laplacian are derived.
Constitutive Modeling: The study integrates the Moore-Gibson-Thompson (MGT) theory with forward memory-dependent derivatives (fMDD). The classical Fourier and Fick laws are modified to include relaxation time and memory effects over a sliding interval. The heat and moisture flux equations are coupled with the conservation of mass and energy, leading to a system of linearly coupled differential equations.
Problem Formulation: The physical problem is defined as a hollow cylinder occupying a non-integer dimensional space ( $r_1 \leq r \leq r_2$ ). The governing equations for temperature ( $T$ ), moisture concentration ( $C$ ), and displacement ( $u$ ) are non-dimensionalized.
Analytical Solution: The system of coupled partial differential equations is solved using the Laplace transform method. A potential function is introduced to decouple the displacement equation. The resulting characteristic equation is a sixth-order differential equation, the roots of which are determined analytically. The general solution in the Laplace domain is expressed using modified Bessel functions of the first and second kinds.
Numerical Inversion: To obtain time-domain solutions, the Laplace inverse transform is computed using a Fourier series expansion method. The authors utilize the "Korrektur" method to reduce discretization error and the $\epsilon$ -algorithm to accelerate the convergence of the series.
Numerical Simulation: The theoretical framework is applied to a fiber-reinforced composite hollow cylinder (Thornel T300 graphite fibers and Narmco 5208 epoxy resin). Numerical results are generated to analyze the effects of fractal dimensions, time delays, and different kernel functions (linear vs. nonlinear) on temperature, moisture, displacement, and stress distributions.

Key Contributions

Extension of Vector Calculus: The paper successfully generalizes gradient, divergence, and Laplacian operators to non-integer dimensional spaces, providing the necessary mathematical tools to model fractal media as continua.
Novel Hygrothermoelastic Model: It presents a new hygrothermal model for fractal media that incorporates the MGT theory and forward memory-dependent derivatives. This is the first study, according to the authors, to apply forward memory-dependent derivatives within a hygrothermoelastic framework for fractal materials.
Analytical Solution: The derivation of exact solutions for the coupled hygrothermoelastic equations in the Laplace domain for a hollow cylinder geometry in non-integer dimensions.
Comparative Analysis of Kernels: The study investigates the performance of various kernel functions, specifically comparing linear and nonlinear kernels within the memory-dependent derivative framework.

Results
The computational outcomes and graphical illustrations reveal several key behaviors:

Impact of Fractal Dimensions: The fractal dimension significantly influences the distribution of temperature, moisture, displacement, and stress. The study notes that non-fractal scenarios generally exhibit higher temperatures along the radius compared to fractal scenarios under harmonically varying heat sources.
Memory and Delay Effects: The delay period (phase lag) and the memory-dependent derivative parameters are critical. The magnitude of temperature and moisture responses varies with the delay time.
Kernel Function Performance: The analysis demonstrates that nonlinear kernel functions perform better within this theoretical framework than linear ones. Specifically, the choice of kernel function alters the magnitude and distribution of displacement and stress.
Stress Behavior: In the steady state, radial stress transitions from tensile to compressive. The fractal structure leads to stress concentrations and variations not observed in traditional integer-dimensional materials. The peak compressive stress magnitude increases over time but follows distinct patterns dictated by the fractal nature of the material.
Coupling: There is a direct relationship between moisture content and heat; increasing moisture adds heat to the system, while decreasing moisture removes it. The interplay between temperature and humidity is time-dependent, with moisture spreading inward from the boundary over time.

Significance and Claims
The paper claims that this generalized hygrothermal model offers a valuable theoretical framework for understanding heat and moisture transport in various materials and devices, particularly those with complex, fractal-like microstructures. The authors assert that the model provides insights into how delay periods and fractal dimensions act as significant parameters affecting material response.

The study highlights the superiority of nonlinear kernels over linear ones in this specific theoretical context. While the paper suggests that these findings could guide nanostructure design and inform the design of implants, prosthetics, and energy storage technologies, it maintains a modest tone regarding immediate experimental verification. The authors conclude that the model is widely applicable but emphasize that future work should focus on developing more comprehensive models for multi-phase systems and obtaining experimental verification to tailor material properties for specific applications. The research positions itself as a contribution to mathematical physics with practical implications for materials science and engineering, particularly in fields requiring precise control over thermal and mechanical properties under combined stresses.

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