Static Viscoelastic Constitutive Model Based On Fractional Derivative And Its Application

Polymer materials are widely used; viscoelasticity is one of the most important mechanical properties. According to time dependence of the applied load ,viscoelasticity can be classified into two types: static viscoelasticity and dynamic viscoelasticity. Based on the fractional calculus, the static viscoelasticity of polymer is investigated in this paper. The contents of the work are outlined in the following:1. First, the definition of fractional calculus is discussed in detail and the expression of Riemann-Liouville is derived. And then Laplace transform, inverse Laplace transform,Fourier transform, inverse Fourier transform of the fractional calculus and the applicability of Mittag-Leffler are introduced. The fractional derivative Maxwell model, fractional derivative Kelvin model, and fractional derivative linear solid model, self-similar fractional derivative linear solid model are developed, and the expression of storage modulus, loss modulus, storage compliance, loss compliance, loss factor, creep compliance, relaxation modulus, etc are derived.2. The physical meanings of the parameters in the expressions of creep compliances of different models are analyzed. Employing the fractional derivative Maxwell model to study the experimental data, the results reveal that the elastic modulus E and relaxation timeτdecrease with the increase in temperature, nevertheless, the fractional orderαincreases with the increasing temperature. Under a certain range of time, E andτlinearly decrease with the increases in logarithmic ageing time, andαis almost the constant. The long-term creep compliances, expressed by master curve, was determined by time-temperature superposition and modeled with the fractional derivative Maxwell model. The model prediction coincides with the master curve very well; therefore, the fractional derivative Maxwell model can be used to predict the long-term creep behavior of polymers accurately.3. The physical meanings of the parameters in the expressions of relaxation modulus of different models are analyzed. According to the transformation properties of the Mittag-Leffler function, two different kinds of expression of relaxation modulus of the fractional order derivative linear solid model are obtained. The investigation reveals increase number of the term of one of the expressions can characterize the stress relaxation behavior well, and the error of the expression with varied number of term are analyzed. Comparing the fractional order derivative linear solid model with the classical linear solid model, the results show that the fractional order derivative model can characterize viscoelastic behavior of polymer better.This work was supported by National Natural Science Foundation of China (10772156) , Program for New Century Excellent Talents in University (No.NCET-08-0685), Key Project of Chinese Ministry of Education (No.209085) and Scientific Research Fund of Hunan provincial Education Department (No.08A069).