Research On The Theory Of Fractional Derivative Model

Macromolecule polymer materials are widely used in our various engineering field, the scientific researchers also pay more and more attention to the mechanical properties of the polymer materials, especially the viscoelastic properties. In the deformation process, polymer materials depend much on the time and temperature, generally the classical viscoelastic models are not very suitable for describing the deformation behavior. Experiments show that in the creep and relaxation process, the classical theory of viscoelastic constitutive models can’t inosculate the experiment data, especially in the initial stage. However, the fractional derivative models can well describe the mechanical behaviors of actual materials.In this paper,the main contents and conclusions are as follows:l.Two kinds of common definitions of the fractional derivative are discussed, those are Riemann-Liouville and Caputo. Laplace transform, inverse Laplace transform, Fourier transform, inverse Fourier transform and their related properties are outlined. The Mittag-Leffler function with single parameter and double parameters is introduced.2.Analyse the fractional derivative models that base on the Scott-Blair pot. Introduing the complex shear modulus, by the Laplace transform and inverse Laplace transform, the relaxation modulus and creep compliance of the fractional derivative models of forms Kelvin-Voigt, Maxwell, Zener, anti-Zener and Burgers are derived. Through the dimensionless processing, the dimensionless expression for the relaxation modulus and creep compliance are abtained. Through the material function images, we can very intuitive see the effects of fractional orders to the fractional derivative models, whcih can help us to choose appropriate fractional derivative models.3. On the basis of summarizing the fractional derivative models with Abel kernel, another DJ function which also has a weak singularity kernel is discussed. A fractional exponential viscoelastic model including the weak singularity kernel function is proposed, the practical analytical expressions of relaxation modulus and creep compliance are provided. The analysis show that a decides the up and down movements of the curve, β affects the rotation of the corner, y determines the corner angle.By comparison with the Standard linear body model, we found that the DJ model can better describe the mechanics behaviors of the viscoelastic materials.4. Studying the Mittag-Leffler function which has the weak singularity kernel, giving fixed form of Mittag-Leffler kernel funciton(called MJ kernel). Replacing the Abel kernel with MJ kernel function, constructing a new fractional derivative viscoelastic model. Through the experiment we can see MJ model can be well agreement with the experimental data.