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 tdt
ββλ γτ
β−
−∞
′= ′ − ′Γ − ′∫
1 0 1aα β β
∞= − ≤−
<=
1
1
( )( ) d d tt Edt dt
ββ
β
γτ λ−
−=
1
1 ( )( )( ) ( )
tE d tt dtt t dt
β
α
λ γτα +−∞
′= ′Γ − − ′ ′∫
CE for assumed form of G Definition of fractional derivative
( )( ) d tt Edt
ββ
β
γτ λ=
*( ) ( ) ( )Gτ ω ω γ ω=
( )( ) ( )t d tt G t t
dtγτ
−∞
′= − ′
′∫
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( ) sin2
sin( ) sin2
nD
D
nx x
x xα
π
απ
= + = +
The shift property for the derivative of sine carries over to fractional derivatives
0
0
sin( )
( )( ) sin2
td tt E E
tt
d
αα 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 Edt
ββ
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, , 2d s dk k k k nε ε ε+ += + = … −
a) Additivity of strains (series)
0 0
1
d s
d sn n
ε ε ε
ε ε−
= +
=Boundary conditions at upper and lower ends
b) Additivity of stresses (parallel)
1 0,1, , 1s s dk k k k nσ σ σ+= + = … −
0sσ σ= Boundary conditions at left
end of the ladder
c) Structural elements1s s
k kk
dd kk k
Eddt
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 1d s dk k kε ε ε+ += +
1s sk k
kEε σ=
1 1
1 1
( ) ( )( )s d
d k kk
k kE Eσ ω ε ωε ω + +
+ +
= +
1s s dk k kσ σ σ+= + 1( ) ( ) ( )s s d
k k k ks i sσ σ ω η ε+= +
dd kk k
ddtεσ η=
11 1
1 1
( ) ( )( ) ( )
1d dk k
k ksk k
sE Eε ω ε ωσ ω σ ω
++ +
+ +
= +
rearranging
1 11
21
1 1
( ) /( )(
1))
1(
dk k k
k skkd
k
s
k
EEi
ε ω ησ ωσ ω ω
η ε ω
+ ++
++
+ +
= ++
Recursion relation for k
2 1 11
1 1
( ) /( )
1/
dn n n
n sn n ni
EE
Eε ω ησ ω ω η
− − −−
− −
= ++
Special case for k=n-2 (right end)
* 0 00 0
1
1 1
2 1
2 2
0
/( )
1
( ) 1 /( )/
//1
EG Ei
Ei
sE E
Ei
sE
ηεω ησ ω ηηω η
ω
= = ++
++
+…
+
6
Relation between RCE and mechanical model
Exact sum of a binomial series1 |( ,1) | 1xx x γ γ− <+ ∈
0 0 0
1 0 0
1 1 0
(/
/ )1·(01
)/1.2
EcE
c
c
E
ηη γ
γη
=+
−
=
=
1* 0 0
0 ( ) 1 1c c ni i
E Gγ
ωω ω
− =
+ + →∞simplifies as
for small ie, ω 1/0 1c γω δ δ<
*0 0 0
( )) ( / )( )
( E c iE G γε ωω ωσ ω
= =
0 0 0 0( ) ( / ) ( ) ( ) ( )E i E E iγ γσ ω ωη ε ω ωλ ε ω= =
0( )( ) d tt E
dt
γγ
γ
εσ λ=
IFTWithin certain limits, hierarchical mechanical model reproduces fractional RCE!
0( , , )Eγ λ
3 parameter FE
7
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 pre gel and 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 1X
X
X
i
X
E
E iωη
ωλ
= + + +
=
FE with parameters 0 )1/ 2,( ,E λ
For arbitrary fractional order?
1 1 14 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 XX i EX 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 − ( ) ( , ) (( , ) , )ij i ij j ji iw P t w PdP 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 timesTauberian theorems
~wλ
0.317β =
is the spectral dimensionsd
10
Fractional Maxwell & Kelvin-Voigt
2 2
11 1 1
12
( )( )
( )( )
d tt Edt
d tt Edt
αα
α
ββ
β
τγ λ
τγ λ
−− −
−
−− −
−
=
=
1 2( ) ( ) ( )t t tγ γ γ= +
1 11 1
2 2
( )( ) E d tt EE dt
α αα
β α
λ γτ λλ
+ =
( ) ( )( ) d t d tt Edt dt
α β αα β α
α β α
τ γτ λ λ−
−−+ =
Fractional Maxwell Model (FMM) Fractional Kelvin-Voigt Model (FKVM)
11 1 1
2 21
2
( )( )
( )( )
d tt Edt
d tt Edt
αα
α
ββ
β
γτ λ
γτ λ
=
=
1 2( ) ( ) ( )t t tτ τ τ= +
1 21 1
1 2( ) ( )( ) d t d tt E E
dt dt
α βα β
α β
γ γτ λ λ= +
( ) ( )( ) d t d tt E Edt dt
α βα 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
zzk
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
dt dt dt dt
α β α γ γ α βα β α γ γ α β
α α γ γ α β
τ γ γ γτ λ λ λ λ− + −
− + −+ −+ = + +
0 0
( ) ( ) ( ) ( )( ) d ddt
E t E t d t d tt E EE E dtdt dt
α γ β γ α βα γ β γ α β
α γ β γ α β
τ τ γ γτ λ λ λ λ− −
− −− −+ + = +
FCM RCE
FPTM RCE
λ 1 , E0 are parameter groupings14