Fractional Viscoelastic Models Li Yan Institute of Applied Math, School of Mathematics and System Sciences, Shandong University, , P.R.China.

Tao Zhan President & Professor Analysis & Number Theory

MingYu Xu Professor Theory and Applications of the Fractional Calculus, Biofluid Mechanics, Mathematical Modeling in BME,

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Fractional Viscoelastic Models Li Yan Institute of Applied Math, School of Mathematics and System Sciences, Shandong University, , P.R.China.

Section 1. Introduction ⑴ Riemann-Liouville (R-L) Fractional Operator ⅰ Integral For p>0 In special ⅱ Derivative For p>0

ⅲ Some Properties of R-L Fractional Operator For p,q>0

For p,q ∈ R In special where c is a constant

(2) Generalized Mittag-Leffler Function When β=1 What’s more

ⅲ Phenomena of Viscoelastic Materials 1. Relaxation and Creep If the strain is held constant, the stress decreases with time (Relaxation) Relaxation Modulus is the stress response of the unit step strain. If the stress is held constant, the strain increases with time (Creep) Creep Modulus is the strain response of the unit step stress. The unit step function is

2. Hysteresis and Precondition If cyclic loading is applied, Hysteresis (a phase lag) occurs, leading to a dissipation of mechanical energy.

In strain-stress coordinates

If cyclic loading is applied, the output will experience a process of “Precondition” and then enter a relative stable condition. 3. Summary of Viscoelastic Phenomena Relaxation, Creep, Hysteresis, Precondition.

(3) Viscoelastic Models —————Spring-Dashpot models

ⅰ Spring (Hooke’s Law) where E is the Young’s Modulus. ⅱ Fractional Dashpot When α =1 where η is the viscosity.

ⅳ Fractional Viscoelastic Models 1. Fractional Maxwell Model Constitutive Equation Relaxation Modulus Creep Modulus

In Laplace Domain Constitutive Equation Relaxation Modulus Creep Modulus

2. Fractional Kelvin Model Constitutive Equation Relaxation Modulus Creep Modulus

In Laplace Domain Constitutive Equation Relaxation Modulus Creep Modulus

3. Fractional Standard Linear Solid Model Constitutive Equation Relaxation Modulus Creep Modulus

In Laplace Domain Constitutive Equation Relaxation Modulus Creep Modulus

4. Constitutive Relationship of Stress and Strain In Laplace domain where ε (0)= σ (0)=0 Moreover

Hint

5. Discussion 5-1 Fractional and Integral Maxwell model 5-2 Fractional and Integral Kelvin model

5-3 The first order derivative of relaxation and creep modulus at original point.

The unit step function

5-4 Declining Speed of Power law and M-L Functions. So declines faster than, where λ is an arbitrary positive number.

What’s more So, for large t

6. Conclusion · The fractional calculus extends the application of viscoelastic models. · The fractional models are more closely to reality than integral models. · The results of fractional viscoelastic models fit well with experimental methods.

7. In the future Description of microcosmic systems Simulation of high-speed strain

Publications Yan Li and Mingyu Xu. Hysteresis and precondition of viscoelastic solid models, Mechanics of time-dependent materials, Vol.10, No.2, June Yan Li and Mingyu Xu. Hysteresis loop and energy dissipation of viscoelastic solid models, Mechanics of time-dependent materials, Vol.11, No.1, March 2007.

Thank You !