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