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

Abstract

Aims: 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. Background: 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. Objective: The objective of this study was the deformation process of biomaterials with a fractal structure under external load. Methods: 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. Results: 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. Conclusion: 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.

Doubling Smith Method for a Class of Large-Scale Generalized Fractional Diffusion Equations

“…Another approach is to adapt the finite element method to include fractional order derivatives, which has been applied to determine the rheological properties of biomaterials that exhibit fractal structures. This method has been useful in studying the viscoelastic behavior of collagen and elastin [14]. The Galerkin method is yet another technique used to obtain numerical solutions for fractional differential equations.…”

Fractional-Order Memristive Wilson Neuron Model: Dynamical Analysis and Synchronization Patterns

“…Hence, fractional-order equations are usually more successful in modeling the nonlinear behavior of memorydependent systems [3]. For example, such fractal approaches are applicable for modeling the non-equilibrium processes [4,5]. The fractional calculus can be found in biology, electrochemistry, electromagnetism, etc.…”

Analysis of the Fractional Differential Equations Using Two Different Methods

“…Ad-ditionally, in [22], a lie group approach is implemented for solving the fractional equation. Some other applications can be seen in [23][24][25][26][27][28][29][30][31].…”