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
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