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

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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 18 th 2010 1

Transcript of Simple Mechanical Fractional Derivative Models · 2016-07-26 · Fractional Derivative of a...

Page 1: Simple Mechanical Fractional Derivative Models · 2016-07-26 · Fractional Derivative of a periodic function. 3. sin( ) sin 2 sin( ) sin 2. D. n D n. x x. α x x. π απ = + = +

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

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Page 2: Simple Mechanical Fractional Derivative Models · 2016-07-26 · Fractional Derivative of a periodic function. 3. sin( ) sin 2 sin( ) sin 2. D. n D n. x x. α x x. π απ = + = +

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

Page 3: Simple Mechanical Fractional Derivative Models · 2016-07-26 · Fractional Derivative of a periodic function. 3. sin( ) sin 2 sin( ) sin 2. D. n D n. x x. α x x. π απ = + = +

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

Page 4: Simple Mechanical Fractional Derivative Models · 2016-07-26 · Fractional Derivative of a periodic function. 3. sin( ) sin 2 sin( ) sin 2. D. n D n. x x. α x x. π απ = + = +

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

Page 5: Simple Mechanical Fractional Derivative Models · 2016-07-26 · Fractional Derivative of a periodic function. 3. sin( ) sin 2 sin( ) sin 2. D. n D n. x x. α x x. π απ = + = +

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

=

=

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Page 6: Simple Mechanical Fractional Derivative Models · 2016-07-26 · Fractional Derivative of a periodic function. 3. sin( ) sin 2 sin( ) sin 2. D. n D n. x x. α x x. π απ = + = +

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

= = ++

++

+…

+

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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

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Page 8: Simple Mechanical Fractional Derivative Models · 2016-07-26 · Fractional Derivative of a periodic function. 3. sin( ) sin 2 sin( ) sin 2. D. n D n. x x. α x x. π απ = + = +

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)

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Page 9: Simple Mechanical Fractional Derivative Models · 2016-07-26 · Fractional Derivative of a periodic function. 3. sin( ) sin 2 sin( ) sin 2. D. n D n. x x. α x x. π απ = + = +

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

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Page 10: Simple Mechanical Fractional Derivative Models · 2016-07-26 · Fractional Derivative of a periodic function. 3. sin( ) sin 2 sin( ) sin 2. D. n D n. x x. α x x. π απ = + = +

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

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Page 11: Simple Mechanical Fractional Derivative Models · 2016-07-26 · Fractional Derivative of a periodic function. 3. sin( ) sin 2 sin( ) sin 2. D. n D n. x x. α x x. π απ = + = +

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

Page 12: Simple Mechanical Fractional Derivative Models · 2016-07-26 · Fractional Derivative of a periodic function. 3. sin( ) sin 2 sin( ) sin 2. D. n D n. x x. α x x. π απ = + = +

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

Page 13: Simple Mechanical Fractional Derivative Models · 2016-07-26 · Fractional Derivative of a periodic function. 3. sin( ) sin 2 sin( ) sin 2. D. n D n. x x. α x x. π απ = + = +

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

Page 14: Simple Mechanical Fractional Derivative Models · 2016-07-26 · Fractional Derivative of a periodic function. 3. sin( ) sin 2 sin( ) sin 2. D. n D n. x x. α x x. π απ = + = +

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