Fractional Viscoelastic Models

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Fractional Viscoelastic Models

• Li Yan
• Institute of Applied Math,
• School of Mathematics and System Sciences,
• Shandong University, 250100, P.R.China.
• E-mail yan_at_cc.usu.edu

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• Tao Zhan
• President Professor
• Analysis Number Theory

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• MingYu Xu
• Professor
• Theory and Applications of the Fractional
  Calculus,
• Biofluid Mechanics,
• Mathematical Modeling in BME,
• xumingyu_at_sdu.edu.cn

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Fractional Viscoelastic Models

• Li Yan
• Institute of Applied Math,
• School of Mathematics and System Sciences,
• Shandong University, 250100, P.R.China.
• E-mail yan_at_cc.usu.edu

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Section 1. Introduction

• ? Riemann-Liouville (R-L) Fractional Operator
• ? Integral
• For pgt0
• In special
• ? Derivative
• For pgt0

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• ? Some Properties of R-L Fractional Operator
• For p,qgt0

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

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• (2) Generalized Mittag-Leffler Function
• When ß1
• Whats more

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• ? 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

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

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• In strain-stress coordinates

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• 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.

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• (3) Viscoelastic Models
• Spring-Dashpot models

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• ? Spring (Hookes Law)
• where E is the Youngs Modulus.
• ? Fractional Dashpot
• When a1
• where ? is the viscosity.

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

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• In Laplace Domain
• Constitutive Equation
• Relaxation Modulus
• Creep Modulus

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• 2. Fractional Kelvin Model
• Constitutive Equation
• Relaxation Modulus
• Creep Modulus

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• In Laplace Domain
• Constitutive Equation
• Relaxation Modulus
• Creep Modulus

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

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• In Laplace Domain
• Constitutive Equation
• Relaxation Modulus
• Creep Modulus

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• 4. Constitutive Relationship of Stress and Strain
• In Laplace domain
• where e(0)s(0)0
• Moreover

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• Hint

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

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

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• The unit step function

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

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• Whats more
• So, for large t

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• 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.

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

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• Publications
• Yan Li and Mingyu Xu. Hysteresis and precondition
  of viscoelastic solid models, Mechanics of
  time-dependent materials, Vol.10, No.2, June
  2006.
• 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.

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• Thank You !