Recent strategy to study fractal-order viscoelastic polymer materials using an ancient Chinese algorithm and He’s formulation

Elı́as-Zúñiga, Alex; Palacios-Pineda, Luis Manuel; Olvera-Trejo, Daniel; Martínez-Romero, Óscar

doi:10.1177/14613484221085413

Cited by 10 publications

(2 citation statements)

References 82 publications

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“…The two-scale theory has become an effective tool to various discontinuous problems [36][37][38][39][40].…”

Section: Fractal Dimensionsmentioning

confidence: 99%

Fractal Dimensions of a Porous Concrete and Its Effect on the Concrete’s Strength

He,

Liu

2023

FU Mech Eng

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All mechanical properties of a porous medium depend upon its fractal dimensions, however, how to measure the fractal dimensions is still an open issue. This paper adopts the two-scale fractal theory to calculate fast and effectively the fractal dimensions of a porous concrete. Of the concrete's properties that have been fascinating engineers and scientists, by far the most perplexing is the effects of its porosity and pore size on concrete's strength. Though there were many ad hoc empirical formulae for predicting the strength, much deviation arose for practical applications. Here a dimensionless model and the fractal theory are adopted to insight theoretically into the effects, and for the first time ever, some physically relative and mathematically reliable formulations are proposed. Additionally nano/micro particles’ size and distribution can also be used for theoretical prediction of the concrete’s strength, it shows that the boundary-induced force occurs when the particles tend to micro/nanoscales. The present theory sheds new light on the optimal design of various functional concretes.

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“…The two-scale theory has become an effective tool to various discontinuous problems [36][37][38][39][40].…”

Section: Fractal Dimensionsmentioning

confidence: 99%

Fractal Dimensions of a Porous Concrete and Its Effect on the Concrete’s Strength

He,

Liu

2023

FU Mech Eng

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“…When we observe a motion at a large scale, it may be a continuous change, while at a small scale, it may become discontinuous. Therefore, the two-scale fractal theory is a powerful mathematical tool to study the world with greater precision [30].…”

Section: Two-scale Fractal Theorymentioning

confidence: 99%

A simple frequency formulation for fractal–fractional non-linear oscillators: A promising tool and its future challenge

2023

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This paper proposes a simple frequency formula developed from He’s frequency formulation for fractal systems. In this approach, the initial guess can be judiciously chosen. Even the simplest initial guess leads to a highly accurate approximate solution. A detailed theoretical development is elucidated, and the solving process is given step by step. The simple calculation and reliable results have been merged into an effective tool for deeply studying fractal vibration systems, and the present approach offers a completely new angle for the fast insight into the physical properties of a non-linear vibration system in a fractal space.

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Dynamic Response of a Fractal Cushioning Packaging System

Elı́as-Zúñiga¹,

Palacios-Pineda²,

Trejo³

et al. 2022

Fractals

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This paper focuses on predicting the shape, duration, and peak magnitude of the displacement, velocity, and acceleration curves while dropping weight over a viscoelastic fractal cushioning packaging system. Furthermore, to capture high-frequency harmonic components observed during the impact time span, the approximate frequency–amplitude expression of the governing equation of motion will be obtained via an ancient Chinese algorithm and He’s formulation by assuming initial trial solutions based on Jacobi elliptic functions. Numerical simulations confirmed the ability of the Jacobi elliptic functions to capture high-frequency harmonics observed during the cushioning packaging system dynamic response with a root mean square error (RMSE) value that does not exceed 0.0218, which is an indication of the great accuracy attained from our derived solution when compared to the exact numerical one. Furthermore, our derived approximate solution predicts that when a weight is dropped over the cushioning packaging, the cushion material with smaller porosity will absorb the produced kinetic energy faster.

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Recent strategy to study fractal-order viscoelastic polymer materials using an ancient Chinese algorithm and He’s formulation

Cited by 10 publications

References 82 publications

Fractal Dimensions of a Porous Concrete and Its Effect on the Concrete’s Strength

Fractal Dimensions of a Porous Concrete and Its Effect on the Concrete’s Strength

A simple frequency formulation for fractal–fractional non-linear oscillators: A promising tool and its future challenge

Dynamic Response of a Fractal Cushioning Packaging System

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