Parameters Estimation Studies Of The Fractional Constitutive Equation For The Viscoelastic Materials

We focus on parameters estimation to the fractional constitutive equation of viscoelastic materials in this paper. This paper consists of three chapter-s:In the first chapter, we make a brief introduction about origin, devel-opment, definition and properties of fractional calculus as well as some inte-gral transforms and several special functions; In chapter 2, we introduce the fractional constitutive equation and its analytical solutions to the human root dentin. We also estimate the parameters of this equation with Bayesian method and Levenberg-Marquardt method; In chapter 3, we introduce generalized frac-tional element (GFE) network Zener model and estimate the parameters of the model with Bayesian method and Levenberg-Marquardt method. Besides, we find the advantages and the disadvantages between the above two methods.In the first section of the first chapter, we make a brief introduction about origin and development of fractional calculus; In the second section, we give the definition of Riemann-Liouville fractional calculus, Caputo fractional calculus and Grunwald-Letnikov fractional calculus; In the third section, we introduce the properties of fractional calculus; In the fourth section, we introduce Laplace transform; We introduce Mittag-Lefler function and Gamma function in the last section.In the second chapter, we introduce the model about human root dentin of M. P. Ljubomir et al., and they gave the fractional constitutive equation and its analytical solutions we estimate the three parameters a, a and b in the above equation with Bayesian method, and compare the results with the experimental data. And then we study what effects the three parameters have on the stress relax-ation and we also check if the initial guess have effects on the estimated re-sults. Besides, we also estimate the three parameters of the same equation with Levenberg-Marquardt method, and then we verify the effectiveness of Levenberg-Marquardt method.In the last chapter, we introduce the GFE network Zener model σ(t)+τα-β0Dtα-βσ(t)= E0τα0Dtαε(t)+Eτλ0Dtλε(t)+Eτλ+α-β0Dtλ+α+βε(t), and its analytical solutions we use the experimental data of ultrahigh molecular weight polyethylene(UHMWPE) to estimate the six parameters α、β、τ、λ、Eo和E in the above equa-tion with Bayesian method, and compare the results with the experimental data. And then we study what effects the six parameters have on the stress relaxation and we also check if the initial guess have effects on the estimated results. Besides, we estimate the four parameters α、β、τ、λ A of UHMW-PE with Levenberg-Marquardt method and we also verify the stability of this method. In the last, we compare the advantage and disadvantage between Bayesian method and Levenberg-Marquardt method.