Finite Element Calculation of the Linear Elasticity Problem for Biomaterials with Fractal Structure

Q3 Computer Science

Open Bioinformatics Journal Pub Date : 2021-11-19 DOI:10.2174/18750362021140100114

V. Shymanskyi, Yaroslav Sokolovskyy

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引用次数: 14

Abstract

The aim of this study was to develop the mathematical models of the linear elasticity theory of biomaterials by taking into account their fractal structure. This study further aimed to construct a variational formulation of the problem, obtain the main relationships of the finite element method to calculate the rheological characteristics of a biomaterial with a fractal structure, and develop application software for calculating the components of the stress-strain state of biomaterials while considering their fractal structure. The obtained results were analyzed. The development of adequate mathematical models of the linear elasticity theory for biomaterials with a fractal structure is an urgent scientific task. Finding its solution will make it possible to analyze the rheological behavior of biomaterials exposed to external loads by taking into account the existing effects of memory, spatial non-locality, self-organization, and deterministic chaos in the material. The objective of this study was the deformation process of biomaterials with a fractal structure under external load. The equations of the linear elasticity theory for the construction of the mathematical models of the deformation process of biomaterials under external load were used. Mathematical apparatus of integro-differentiation of fractional order to take into account the fractal structure of the biomaterial was used. A variational formulation of the linear elasticity problem while taking into account the fractal structure of the biomaterial was formulated. The finite element method with a piecewise linear basis for finding an approximate solution to the problem was used. The main relations of the linear elasticity problem, which takes into account the fractal structure of the biomaterial, were obtained. A variational formulation of the problem was constructed. The main relations of the finite-element calculation of the linear elasticity problem of a biomaterial with a fractal structure using a piecewise-linear basis are found. The main components of the stress-strain state of the biomaterial exposed to external loads are found. Using the mathematical apparatus of integro-differentiation of fractional order in the construction of the mathematical models of the deformation process of biomaterials with a fractal structure makes it possible to take into account the existing effects of memory, spatial non-locality, self-organization, and deterministic chaos in the material. Also, this approach makes it possible to determine the residual stresses in the biomaterial, which play an important role in the appearance of stresses during repeated loads.

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分形结构生物材料线弹性问题的有限元计算

本研究的目的是通过考虑生物材料的分形结构，建立生物材料线性弹性理论的数学模型。本研究进一步旨在构建该问题的变分公式，获得计算具有分形结构的生物材料流变特性的有限元法的主要关系，并开发考虑分形结构的生物材料应力-应变状态分量计算的应用软件。对所得结果进行了分析。为具有分形结构的生物材料的线性弹性理论建立适当的数学模型是一项紧迫的科学任务。找到它的解决方案将使分析暴露于外部载荷下的生物材料的流变行为成为可能，考虑到材料中存在的记忆、空间非局部性、自组织和确定性混沌的影响。本研究旨在研究具有分形结构的生物材料在外加载荷作用下的变形过程。利用线性弹性理论建立了生物材料在外加载荷作用下变形过程的数学模型。采用分数阶积分-微分数学方法考虑生物材料的分形结构。提出了考虑生物材料分形结构的线性弹性问题的变分公式。采用分段线性基有限元法求问题的近似解。得到了考虑生物材料分形结构的线性弹性问题的主要关系式。构造了该问题的变分形式。给出了分形结构生物材料线性弹性问题的分段线性有限元计算的主要关系。得到了生物材料在外部载荷作用下应力-应变状态的主要组成部分。利用分数阶积分微分的数学装置，构建具有分形结构的生物材料变形过程的数学模型，可以考虑材料中存在的记忆效应、空间非定域效应、自组织效应和确定性混沌效应。此外，这种方法使得确定生物材料中的残余应力成为可能，残余应力在重复载荷期间的应力外观中起重要作用。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Open Bioinformatics Journal Computer Science-Computer Science (miscellaneous)

CiteScore

2.40

自引率

0.00%

发文量

期刊介绍： The Open Bioinformatics Journal is an Open Access online journal, which publishes research articles, reviews/mini-reviews, letters, clinical trial studies and guest edited single topic issues in all areas of bioinformatics and computational biology. The coverage includes biomedicine, focusing on large data acquisition, analysis and curation, computational and statistical methods for the modeling and analysis of biological data, and descriptions of new algorithms and databases. The Open Bioinformatics Journal, a peer reviewed journal, is an important and reliable source of current information on the developments in the field. The emphasis will be on publishing quality articles rapidly and freely available worldwide.