Simple Mechanical Fractional Derivative Models 2016-07-26آ  Fractional Derivative of a periodic...

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  • Simple Mechanical Fractional Derivative Models

    1) Friedrich, C., H. Schiessel, et al. (1999). "Constitutive behavior modeling and fractional derivatives." Rheology Series: 429-466.

    2) Sokolov, I. M., J. Klafter, et al. (2002). "Fractional Kinetics." Physics Today 55(11): 48-54.

    Siddarth Srinivasan June 18th 2010

    1

  • Back to RCEs

    (1 ) ( ) E tG t

    β

    β λ

    − =  Γ −  

    For

    ( )( ) ( ) (1 )

    tE d tt dt t t dt

    β βλ γτ

    β −

    −∞

    ′ = ′ − ′ Γ − ′∫

    1 0 1 a α β β

    ∞ = − ≤ −

    < =

    1

    1

    ( )( ) d d tt E dt dt

    β β

    β

    γτ λ −

    −=

    1

    1 ( )( ) ( ) ( )

    tE d tt dt t t dt

    β

    α

    λ γτ α +−∞

    ′ = ′ Γ − − ′ ′∫

    CE for assumed form of G Definition of fractional derivative

    ( )( ) d tt E dt

    β β

    β

    γτ λ=

    *( ) ( ) ( )Gτ ω ω γ ω= 

    ( )( ) ( ) t d tt G t t

    dt γτ

    −∞

    ′ = − ′

    ′∫

    For SAOS 0( ) i tt e ωγ γ=

    ( ) ( ) ( )E i βτ ω ωλ γ ω=  *( ) ( )G E i βω ωλ=

    2

  • Fractional Derivative of a periodic function

    3

    sin( ) sin 2

    sin( ) sin 2

    nD

    D

    nx x

    x xα

    π

    απ

     = +     = +   

    The shift property for the derivative of sine carries over to fractional derivatives

    0

    0

    sin( )

    ( )( ) sin 2

    t d tt E E

    t t

    d

    α α α

    α

    γ γ

    γ απτ λ λ γ  +  

    = = 

    =

    For SAOS

    Thus, one has a viscoelastic response from fractional RCE in small amplitude oscillatory shear tests

  • Fractional mechanical element

    Fractional element is specified by the triplet ( , , )Eβ λ

    - represented via hierarchical arrangements: trees, ladders, fractals - more complex RCE’s from combinations of fractional elements

    ( )( ) d tt E dt

    β β

    β

    γτ λ=

    Images from Friedrich et al. (1999) 4

  • Hierarchical analogue to Fractional equations

    Schiessel, H. and A. Blumen (1993). "Hierarchical analogues to fractional relaxation equations." Journal of Physics A: Mathematical and General 26: 5057-5069.

    Mechanical arrangment to simulate FE

    ,, ,s d s dk k k kε ε σ σ

    1 1 0,1, , 2 d s d k k k k nε ε ε+ += + = … −

    a) Additivity of strains (series)

    0 0

    1

    d s

    d s n n

    ε ε ε

    ε ε−

    = +

    = Boundary conditions at upper and lower ends

    b) Additivity of stresses (parallel)

    1 0,1, , 1 s s d k k k k nσ σ σ+= + = … −

    0 sσ σ= Boundary conditions at left end of the ladder

    c) Structural elements 1s s

    k k k

    d d k k k

    E d dt

    ε σ

    εσ η

    =

    =

    5

  • Solution by Fourier transform

    Schiessel, H. and A. Blumen (1993). "Hierarchical analogues to fractional relaxation equations." Journal of Physics A: Mathematical and General 26: 5057-5069.

    1 1 d s d k k kε ε ε+ += +

    1s s k k

    kE ε σ=

    1 1

    1 1

    ( ) ( )( ) s d

    d k k k

    k kE E σ ω ε ωε ω + +

    + +

    = + 

    1 s s d k k kσ σ σ+= + 1( ) ( ) ( )

    s s d k k k ks i sσ σ ω η ε+= +  

    d d k k k

    d dt εσ η=

    1 1 1

    1 1

    ( ) ( ) ( ) ( )

    1 d d k k

    k ks k k

    sE E ε ω ε ω σ ω σ ω

    + + +

    + +

    = +  

     

    rearranging

    1 1 1

    21

    1 1

    ( ) / ( ) (

    1) )

    1 (

    d k k k

    k s kk d

    k

    s

    k

    EE i

    ε ω η σ ωσ ω ω

    η ε ω

    + + +

    ++

    + +

    = + +

    

    Recursion relation for k

    2 1 1 1

    1 1

    ( ) / ( )

    1 /

    d n n n

    n s n n ni

    E E

    Eε ω η σ ω ω η

    − − − −

    − −

    = + +

    Special case for k=n-2 (right end)

    * 0 0 0 0

    1

    1 1

    2 1

    2 2

    0

    /( )

    1

    ( ) 1 /( ) /

    / /1

    EG Ei

    Ei

    sE E

    E i

    s E

    ηεω ησ ω η ηω η

    ω

    = = + +

    + +

    +…

    +

    6

  • Relation between RCE and mechanical model

    Exact sum of a binomial series 1 |( ,1) | 1xx x γ γ−

  • Example: Response of gels to shear

    Schiessel, H. and A. Blumen (1995). "Mesoscopic Pictures of the Sol-Gel Transition: Ladder Models and Fractal Networks." Macromolecules 28(11): 4013-4019.

    - Leaf springs and inelastic blocks

    a) pre-gel: finite clusters, terminate ladder with inelastic block

    b) critical gel, infinite network

    c) post gel: more crosslinks, terminate with leaf springs

    η=E=1, n=1000. numerically computed β =1/2 for critical gel. Can modify E, η to tune β 1/2 post gel. β represents structural parameter (connectivity)

    8

  • Fractal Tree Representation of FE

    Heymans, N. and J. C. Bauwens (1994). "Fractal rheological models and fractional differential equations for viscoelastic behavior." Rheologica Acta 33(3): 210-219.

    1 1 1 1 1 X

    X

    X

    i

    X

    E

    E iωη

    ωλ

      = + + +       

    =

    FE with parameters 0 )1/ 2,( ,E λ

    For arbitrary fractional order?

    1 1 1 4 2 4( )

    X ZE

    Z i X

    X i E X

    ωη

    ωη

    =

    =

    =

    (1/2 ) (1/2 ) 1 (1/2 ) (1/2 )

    1

    ( ) )

    (1/ 2 ) (1/ 2 ) ( 2

    (

    1/ )

    ji i j

    i

    i j

    E X X i E X i

    β β

    ωη

    ωη

    β

    Σ Σ −Σ −Σ

    −=

    =

    Σ = Σ +Σ

    1 2X Z Z=

    iterate

    9

  • Connectivity of mechanical networks

    jrnodes in the networks - each connected to neighboring by equal spring (E)ir jr

    node linked to planar ground with site-dependent viscosity where is the coordination number

    ( )i izη η= r )( iz r

    ( )( ) ( ) ( )i i j i j it E t tη γ γ γ = Σ −  ( ) ( , ) ( ( , ) , )i j i ij j ji iw P t w P

    dP r t t dt

    = − Σ  r r

    Stress balance on node i Master diffusion equation

    ( ) ~P(r , )i i it tη γ

    2( ) sd

    tP t −

    ∝1 / 2sdβ = −

    Heymans, N. and J. C. Bauwens (1994). "Fractal rheological models and fractional differential equations for viscoelastic behavior." Rheologica Acta 33(3): 210-219.

    At large times Tauberian theorems

    ~wλ

    0.317β =

    is the spectral dimensionsd

    10

  • Fractional Maxwell & Kelvin-Voigt

    2 2

    1 1 1 1

    1 2

    ( )( )

    ( )( )

    d tt E dt

    d tt E dt

    α α

    α

    β β

    β

    τγ λ

    τγ λ

    − − −

    − − −

    =

    =

    1 2( ) ( ) ( )t t tγ γ γ= +

    1 1 1 1

    2 2

    ( )( ) E d tt E E dt

    α α α

    β α

    λ γτ λ λ

    + =

    ( ) ( )( ) d t d tt E dt dt

    α β α α β α

    α β α

    τ γτ λ λ −

    − −+ =

    Fractional Maxwell Model (FMM) Fractional Kelvin-Voigt Model (FKVM)

    1 1 1 1

    2 2 1

    2

    ( )( )

    ( )( )

    d tt E dt

    d tt E dt

    α α

    α

    β β

    β

    γτ λ

    γτ λ

    =

    =

    1 2( ) ( ) ( )t t tτ τ τ= +

    1 2 1 1

    1 2 ( ) ( )( ) d t d tt E E

    dt dt

    α β α β

    α β

    γ γτ λ λ= +

    ( ) ( )( ) d t d tt E E dt dt

    α β α β

    α β

    γ γτ λ λ= +

    Friedrich et al. (1999) 11

  • Material Functions of FMM and FKVM

    Table from Friedrich et al. (1999)

    Generalized Mittag-Lefler function of argument (z)

    ( ), 0 ( )

    k

    k

    zz k

    Eα β α β

    =

    = Γ +∑

    -FMM and FKVM : two power law regions intersecting at t=λ

    - For G(t) of FMM has flat decrease of slope (-β) followed by faster decrease (-α)

    - For G(t) of FKVM, fast decrease followed by flat decrease 12

  • FMM and FKVM

    - polyisobutylene: FKVM unable to fit in glassy and transitional zones

    - modified polybutadiene’s terminal relaxation zone described by FMM

    - 4 fitting parameters?

    and GG′ ′′

    Images from Friedrich et al. (1999) 13

  • 3 element models

    The FMM and FKVM cannot describe ‘S’ shaped transitions in G(t)

    3-Fractional Element models: the Fractional Zerner Model (FZM) and the Fractional Poynting-Thompson Model (FPTM)

    0 0 ( ) ( ) ( ) ( )( ) d t d t d t d tt E E E