Implicitly constituted materials with fading memory V t Pr u sa - - PowerPoint PPT Presentation

SLIDE 1

Implicitly constituted materials with fading memory

V´ ıt Pr˚ uˇ sa prusv@karlin.mff.cuni.cz

Mathematical Institute, Charles University

31 March 2012

SLIDE 2

Incompressible simple fluid

Truesdell and Noll (1965):

T = −p I + F+∞

s=0(

Ct (t − s))

SLIDE 3

Differential type models

General form, Rivlin and Ericksen (1955):

T = −p I + f( A1, A2, A3, . . . )

where

A1 = 2 D An = d An−1

dt +

An−1 L + L⊤ An−1

Coleman and Noll (1960): These models can be understood as successive approximations of the history functional

T = −p I + F+∞

s=0(

Ct (t − s))

with “fading memory”.

SLIDE 4

Pressure dependent viscosity

T = −p I + 2µ(p) D

SLIDE 5

Stress dependent viscosity

T = −p I + µ( T) D

Seely (1964): µ(

T) = µ∞ + (µ0 − µ∞) e− | Tδ|

τ0

Blatter (1995): µ(

T) =

A

• |

Tδ|2 + τ 2

n−1

2

Matsuhisa and Bird (1965): µ(

T) =

µ0 1 + α |

Tδ|n−1

SLIDE 6

Implicit constitutive relation

Rajagopal (2003, 2006); Rajagopal and Srinivasa (2008): f (

T, D) = 0

SLIDE 7

Rate type models

T = −π I + S

Oldroyd (1958):

S + λ1

▽

S + λ3

2 (

DS + SD) + λ5

2 (Tr

S) D + λ6

2 ( S :

D) I

= −µ

• D + λ2

▽

D + λ4 D2 + λ7

2 ( D :

D) I

• Phan Thien (1978):

Y

S + λ

▽

S + λξ

2 ( DS +

SD) = −µ D

Y = e−ε λ

µ Tr

S

Notation:

▽

b♭ =def

d b♭ dt − [∇v]

b♭ − b♭ [∇v]⊤

SLIDE 8

Materials with fading memory

Implicit algebraic relation: f (

T, D) = 0

Implicit relation between the histories: H+∞

s=0 (

T(t − s), Ct (t − s)) =

Questions:

◮ Are rate type and differential type models (and other known

models) special instances or approximations of the material with fading memory?

◮ Is something like the celebrated retardation theorem

by Coleman and Noll (1960) available for implicit type materials with fading memory?

SLIDE 9

Independent variables

[. . . ] the properties of a material element may depend upon the previous rheological states through which that element has passed, but not in any way on the states of neighbouring elements and not on the motion of the element as a whole in the space. [. . . ] only those tensor quantities need to be considered which have a significance for the material element independent of its motion as a whole in space. Oldroyd (1950)

SLIDE 10

Convected coordinate system

eˆ

y

ξ3 ξ2 ξ1 τ = t − s τ = t χ X eˆ

z

eˆ

x

x = χ(X, t)

SLIDE 11

Plan

◮ Formulate the constitutive relation in the convected

coordinate system, Hencky (1925), Oldroyd (1950).

◮ Formulate the constitutive relation in an implicit form,

Rajagopal (2003).

◮ Expand the functional using an analogue of the retardation

theorem, Coleman and Noll (1960).

◮ Use representation theorems for isotropic linear and bilinear

functions, Truesdell and Noll (1965).

◮ Transform the constitutive relation to a fixed-in-space

coordinate system, Oldroyd (1950).

SLIDE 12

Constitutive assumptions

General relation: H+∞

s=0 ( (ξ, t − s),

(ξ, t − s)) = 0,

SLIDE 13

Constitutive assumptions

General relation: H+∞

s=0 ( (ξ, t − s),

(ξ, t − s)) = 0,

Special form of the general constitutive relation:

= −π I +

• 0 = G+∞

s=0 ( (ξ, t − s) −

I, (ξ, t − s))

SLIDE 14

Constitutive assumptions

General relation: H+∞

s=0 ( (ξ, t − s),

(ξ, t − s)) = 0,

Special form of the general constitutive relation:

= −π I +

• 0 = G+∞

s=0 ( (ξ, t − s) −

I, (ξ, t − s))

Constitutive assumption: (ξ, t − s)S = O (

(ξ, t − s) − IΓ) as (ξ, t − s) − IΓ → 0+,

SLIDE 15

Norm

[...] the deformations that occurred in the distant past should have less influence in determining the present stress than those that occurred in the recent past. Truesdell and Noll (1965) L2

h =def

+∞

s=0

|

(s)|2 h(s) ds

1

2

• [

, ]⊤

• L2

h×L2 h

=def

• 2

L2

h + 2

L2

h

1

2

SLIDE 16

Taylor series for the functional

= −π I +

• 0 = G+∞

s=0 ( (ξ, t − s) −

I, (ξ, t − s))

G+∞

s=0

• (ξ, t − s) −

I (ξ, t − s)

• = A+B+C+o
• (ξ, t − s) −

I (ξ, t − s)

• 2

L2

h×L2 h

• A = G+∞

s=0

• B = δG+∞

s=0

• (ξ, t − s) −

I (ξ, t − s)

• C =
• (ξ, t − s) −

I, (ξ, t − s)

⊤ δ2G+∞

s=0

• (ξ, t − s) −

I (ξ, t − s)

SLIDE 17

Slow history

t − s t − α¯ s

(x, t − s)

t

SLIDE 18

Taylor series for the metric tensor–stress tensor history

Informal expansion:

• (ξ, t − α¯

s) −

I (ξ, t − α¯

s)

• =
• (ξ, t) −

I (ξ, t)

• − α¯

s d

• dt (ξ, t)

d

• dt (ξ, t)
• + 1

2α2¯ s2

• d2
• dt2 (ξ, t)

d2

• dt2 (ξ, t)
• + o
• α2

=

• (ξ, t)
• (0)

g

−α d

• dt (ξ, t)

d

• dt (ξ, t)
• (1)

g

¯ s + 1 2α2

• d2
• dt2 (ξ, t)

d2

• dt2 (ξ, t)
• (2)

g

¯ s2 + o

• α2

Rigorous result: lim

α→0+

1 α2

• (ξ, t − α¯

s) −

I (ξ, t − α¯

s)

• −
• (

)

g − α

( 1 )

g¯ s + 1 2α2(

2 )

g¯ s2

• L2

h×L2 h

= 0

SLIDE 19

Approximation for slow histories

Use constitutive assumption (ξ, t − s)S = O (

(ξ, t − s) − IΓ) as (ξ, t − s) − IΓ → 0+,

and substitute to the Taylor formula for the functional. First order: G+∞

s=0

• (ξ, t − α¯

s) −

I (ξ, t − α¯

s) −

(ξ, t)

• = f0

(

)

g

• + f1

(

1 )

g

• + o (α)

Second order: G+∞

s=0

• (ξ, t − α¯

s) −

I (ξ, t − α¯

s) −

(ξ, t)

• = f0

(

)

g

• + f1

(

1 )

g

• + f2

(

2 )

g

• + g00

(

)

g,

( )

g

• + g10

(

1 )

g,

( )

g

• + g01

(

)

g,

( 1 )

g

• + g11

(

1 )

g,

( 1 )

g

• + o
• α2

SLIDE 20

Linear and bilinear tensor functions

Representation for isotropic linear functions: h(

A) = a1 (Tr A) I + a2 A

Representation for isotropic bilinear functions: h ( A,

B) = (c1 Tr A Tr B + c2 Tr ( AB)) I

+ c3 (Tr

A) B + c4 (Tr B) A + c5 ( AB + BA)

SLIDE 21

Time derivatives with respect to fixed-in-space coordinate system

Oldroyd (1950): dbk

i

dt =def ∂bk

i

∂t + vmbk

i|m

• − vk|mbm

i + bk mvm|i,

dbki dt =def ∂bki ∂t + vmbki |m

• + vm|kbmi + bkmvm|i,

dbki dt =def ∂bki ∂t + vmbki |m

• − vk|mbmi − bkmvi|m,

SLIDE 22

Time derivatives with respect to fixed-in-space coordinate system

d

b

dt = d b dt − [∇v]

b + b [∇v] ,

d

b♯

dt = d b♯ dt + [∇v]⊤

b♯ + b♯ [∇v] ,

d

b♭

dt = d b♭ dt − [∇v]

b♭ − b♭ [∇v]⊤ ,

SLIDE 23

Identification of the derivatives

(ξ, t) → S(x, t)

d dt (ξ, t) →

D(x, t)

d2

• dt2 (ξ, t) →

▽

D(x, t)

d dt (ξ, t) →

▽

S(x, t)

d2

• dt2 (ξ, t) →

▽ ▽

S(x, t)

SLIDE 24

Approximation formulae

First order: G+∞

s=0

• (ξ, t − α¯

s) −

I (ξ, t − α¯

s) −

(ξ, t)

• → b0 (Tr

S) I + b1 S + 2b3 D + b4

• Tr

▽

S

• I + b5

▽

S + o (α)

SLIDE 25

Approximation formulae

Second order: G+∞

s=0

• (ξ, t − α¯

s) −

I (ξ, t − α¯

s) −

(ξ, t)

• →
• b0 (Tr

S) + b4

• Tr

▽

S

• + b6
• Tr

▽ ▽

S

• + (b15 − 2b8) Tr ( D)2

+

• b10 (Tr

S)2 + b11 Tr ( S)2

+

• b18
• Tr

▽

S

2 + b19 Tr ▽

S

2 +b23 Tr

• D

▽

S

• +
• b27 Tr

S Tr

▽

S + b28 Tr

• S

▽

S

• + b33 Tr ( SD)
• I

+

• b1 + b12 (Tr

S) + b30

• Tr

▽

S

• S+
• b3 + b25
• Tr

▽

S

• + b34 (Tr

S)

• D

+ b13 (

S)2 + b17 ( D)2 + b36 ( SD + DS)

+ b9

▽

D +

• b5 + b20
• Tr

▽

S

• + b29 (Tr

S)

▽

S + b21

▽

S

2 + b26

• D

▽

S +

▽

SD

• + b31
• S

▽

S +

▽

SS

• + b7

▽ ▽

S + o

• α2

SLIDE 26

Rate type models

T = −π I + S

Oldroyd (1958):

S + λ1

▽

S + λ3

2 (

DS + SD) + λ5

2 (Tr

S) D + λ6

2 ( S :

D) I

= −µ

• D + λ2

▽

D + λ4 D2 + λ7

2 ( D :

D) I

• Phan Thien (1978):

Y

S + λ

▽

S + λξ

2 ( DS +

SD) = −µ D

Y = e−ε λ

µ Tr

S

Notation:

▽

b♭ =def

d b♭ dt − [∇v]

b♭ − b♭ [∇v]⊤

SLIDE 27

Conclusion

◮ Implicit constitutive relations provide a general concept that

can accommodate both differential and rate type models. Generalization of the concept of simple fluid.

◮ Rate type models for viscoelastic materials can be seen as

special instances of a general material with fading memory. (After a rigorous approximation procedure.)

◮ This is only a proof of concept—there are better ways how to

derive implicit type constitutive relations that are consistent with the laws of thermodynamics.

SLIDE 28

References

Blatter, H. (1995). Velocity and stress-fields in grounded glaciers – a simple algorithm for including deviatoric stress gradients. J. Glaciol. 41(138), 333–344. Coleman, B. D. and W. Noll (1960). An approximation theorem for functionals, with applications in continuum

• mechanics. Arch. Ration. Mech. Anal. 6, 355–370.

Hencky, H. (1925). Die Bewegungsgleichungen beim nichtstation¨ aren Fließen plastischer massen. Z. Angew. Math.

• Mech. 5, 144–146.

Matsuhisa, S. and R. B. Bird (1965). Analytical and numerical solutions for laminar flow of the non-Newtonian Ellis fluid. AIChE J. 11(4), 588–595. Oldroyd, J. G. (1950). On the formulation of rheological equations of state. Proc. R. Soc. A-Math. Phys. Eng.

• Sci. 200(1063), 523–541.

Oldroyd, J. G. (1958). Non-newtonian effects in steady motion of some idealized elastico-viscous liquids. Proc. R.

• Soc. A-Math. Phys. Eng. Sci. 245(1241), 278–297.

Phan Thien, N. (1978). Non-linear network viscoelastic model. J. Rheol. 22(3), 259–283. Rajagopal, K. R. (2003). On implicit constitutive theories. Appl. Math., Praha 48(4), 279–319. Rajagopal, K. R. (2006). On implicit constitutive theories for fluids. J. Fluid Mech. 550, 243–249. Rajagopal, K. R. and A. R. Srinivasa (2008). On the thermodynamics of fluids defined by implicit constitutive

• relations. Z. angew. Math. Phys. 59(4), 715–729.

Rivlin, R. S. and J. L. Ericksen (1955). Stress-deformation relations for isotropic materials. J. Ration. Mech.

• Anal. 4, 323–425.

Seely, G. R. (1964). Non-newtonian viscosity of polybutadiene solutions. AIChE J. 10(1), 56–60. Truesdell, C. and W. Noll (1965). The non-linear field theories of mechanics. In S. Fl¨ uge (Ed.), Handbuch der Physik, Volume III/3. Berlin-Heidelberg-New York: Springer.