Viscoelastic Model With Weak Singularity Kernel And Its Application

A series of viscoelastic model with weakly singular kernel are constructed on the basis of the function with weak singularity. The classical model theory with constitutive equation has the advantages of simple, straightforward form of clear physical concept and so on, we can use the function of a large number of containing weak singularities, such as the use of classical model based on a series of the Mittag-Leffler function to replace Newton dashpot in viscoelastic model, and then the relaxation modulus and creep compliance of constitutive equations are obtained. Weakly singular kernel with viscoelastic model can overcome the classical model which is not fitting well with the experimental data in the beginning of material relaxation and creep. The main contents of this paper are as follows:1. The fractional derivative is defined as Riemann-Liouville, Grunwald-Letnikov, Weyl,Caputo and other types, and each kind of different definitions have their corresponding expression and properties, we discussed two definitions of fractional derivative,Riemann-Liouville definition and Caputo definition. And the fractional derivative theory of Laplasse transform and its inverse change are elaborated on this basis. The definition of Mittag-Leffler function and the Laplasse transform of generalized Mittag-Leffler functions and common weak singularity kernel are given.2. Several kinds of description method of traditional viscoelastic materials(component model, the relaxation modulus and creep compliance and describe the dynamic characteristics of the complex of relaxation modulus and creep compliance) was given. With weak singularity function instead of the classical model of Newton dashpot, the standard linear model with Abel, Nel kernel, Mel kernel is established. Analysis shows that: the standard linear model with a weakly singular kernel is fitting better with the experimental data compared to the classic body and they can overcome the classical model which can’t fit well in the early stage of stress relaxation test.3. By using the Abel kernel to replace the Newton dashpot in classical Nishihara model,we get a different salt rock creep model with a weakly singular kernel. The new model with a weakly singular kernel is better than the original model in the fitting correlation coefficient with the experimental data.4. By Imitating the fractional viscoelastic element model,we can construct the fractional Poisson’s ratio model, fractional Poisson’s ratio of Kelvin model and the fractional Poisson’s ratio of standard linear mode of viscoelastic materials. In the relaxation test and creep test data fitting analysis, the data fitting correlation coefficient is high, and the model can describe the variation of Poisson’s ratio effectively.