chapter 5: material functionsfmorriso/cm4650/lectures/... · 1. choose a material function 2....

CM4650 Chapter 5 Material Functions 3/5/2018

1

Chapter 5: Material Functions

© Faith A. Morrison, Michigan Tech U.

1

CM4650 Polymer Rheology

Michigan TechImposed Kinematics:

Steady Shear Flow Material Functions

Material Functions:

Viscosity

Material Stress Response:

First normal-stress coefficient

Second normal-stress coefficient

(I call these my “recipe cards”)

3/5/2018


2

What are material functions, and why do we need

them?


2

+

⋅ ⋅Momentum balance

, , ,Velocity and pressure fields

2

Constitutive

equation

predictions of:

=

+

Flow scenarios

When we know , the flow

modeling effort is expended on developing the flow scenario

and solving the math.

What are material functions and why do we need them?

NEWTONIAN

⋅ ⋅Momentum balance

, , ,Velocity and pressure fields

2

predictions of:

=

+

Flow scenarios

When we don’t know

, this approach is a

nonstarter.

?Constitutive

equation

+


NON-NEWTONIAN


3

≡ 00

Ψ ≡

Standard Velocity and pressure fields

HypotheticalConstitutive

equation

Material Functions (stress responses) ≡

Ψ ≡

+


Ability to measure stresses

predictions of:

measurements of

≡ 00

=

=


3

+material +

New approach



6

Meet here:

Material Functions




equation

Material Functions (stress responses)

+



predictions of:

measurements of

=

=

+material +

Modeling

Measuring


4


7

Meet here:

Material Functions




equation


+



predictions of:

measurements of

=

=

+material +

Modeling

Measuring

The velocity field is fixed by design (standard flows)


8



equation


+



predictions of:

measurements of

=

=

+material +

Modeling

Measuring


Approach stress/deformation investigations from two directions (modeling, measuring) to reveal the physics;

Material functions organize comparisons


5


3



equation


+



predictions of:

measurements of

=

=

+material +


• We do not know the stress/deformation relationship

• We approach stress/deformation investigations from two directions (modeling, measuring) to reveal the physics;

• Material functions organize comparisons

For non-Newtonian

fluids:


3



equation


+



predictions of:

measurements of

=

=

+material +





For non-Newtonian

fluids:

1. Choose a material function2. Predict what Newtonian fluids would do3. See what non-Newtonian fluids do4. Hypothesize a 5. Predict the material function6. Compare with what non-Newtonian fluids do7. Reflect, learn, revise model, repeat.

Let’s get started:

(1)

(2)

(3)

(4)

(5)

(6)

(7)

Investigating Stress/DeformationRelationships

(Rheology)


6

1) Choose a material function

1. Choice of flow (shear or elongation)

2. Choice of time dependence of or

3. Material functions definitions: will be based on , , in shear or , in elongational flows.

kinem

atics


11

1212

00

Investigating Stress/Deformation Relationships (Rheology)


Imposed Kinematics:

≡ 00


Material Functions:

Viscosity

© Faith A. Morrison, Michigan Tech U.12

constant

≡

Ψ ≡

Ψ ≡


0

0 0

0,

,

0



(I call these my “recipe cards”)


7

How do we predict material functions?


13



2) Predict what Newtonian fluids would do

≡ 00

Ψ ≡

Constitutive equation


Ψ ≡

+


predictions of:=


3

New Approach

How do we predict material functions?

We must know the Constitutive Equation.



8


What does the Newtonian Fluid model predict in steady shearing?

15





What does the Newtonian Fluid model predict in steady shearing?

16

You try.



9

What do we measure for these material functions?


17

(for polymer solutions, for example)



3) See what non-Newtonian fluids do

1.E+02

1.E+03

1.E+04

1.E+05

0.01 0.1 1 10 100

, Poise

1,

(dyn/cm2)s2

1, s

o1

o

1

Figure 6.1, p. 170 Menzes and Graessley conc. PB solution

Steady shear viscosity and first normal stress coefficient


18



10

Steady shear viscosity and first normal stress coefficient

1.E-01

1.E+00

1.E+01

1.E+02

1.E+03

1.E+04

1.E+05

1.E+06

0.01 0.1 1 10 100

, P

ois

e

1.E+02

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+08

1.E+09

1,

dyn

es/

cm2

1, s

1

Figure 6.2, p. 171 Menzes and Graessley conc. PB solution; c=0.0676 g/cm3

813 kg/mol517 kg/mol350 kg/mol200 kg/mol


19



Steady shear viscosity for linear and branched PDMS

Figure 6.3, p. 172 Piau et al., linear and branched PDMS

+ linear 131 kg/molebranched 156 kg/molelinear 418 kg/molbranched 428 kg/mol

20



11

0.1

1

10

100

1000

0.01 0.1 1 10 100

stea

dy

she

ar m

ater

ial f

un

ctio

ns

1 (Pa s2)

(Pa)

2 (Pa s2)

PS/TCP squaresPS/DOP circles

1, s

Steady shear viscosity and first and secondnormal stress coefficient

Figure 6.6, p. 174 Magda et al.; Polystyrene solns


21


(Boger fluid)


22



4) Hypothesize a(how do we

do that, exactly?)


12


What have the investigations of the steady shear material functions

taught us so far?

•Newtonian constitutive equation is inadequate

1. Predicts constant shear viscosity (does not predict rate dependence)

2. Predicts no shear normal stresses (a nonlinear effect; these stresses are generated for many fluids)

•Behavior depends on the material (chemical structure, molecular weight, concentration)

23

4) Hypothesize a

Imposed Kinematics:


Material Functions:

Viscosity





Can we fix the Newtonian Constitutive Equation?

Tvv

Let’s replace with a function of shear rate because we want to predict a rate dependence

24

4) Hypothesize a

New hypothesis for

functionof


13

100

1000

10000

100000

0.01 0.1 1 10 100

viscosity, Poise

shear rate, 1/s

Menzes and Graessley, conc. PB solution; 350 kg/mol

viscosity poise

Predicted by Fake‐O model


25

1

0

67.0

24.0

000,12

000,18

s

n

m

poiseM

c

4) Hypothesize a

≡ 00

Ψ ≡

Constitutive equation


Ψ ≡

+


predictions of:=


3

We are guessing

this



14


What does this model predict for steady shear viscosity?

27



5) Predict the material function (with new )


28

You try.


5) Predict the material function


15


Answer:

29




If we choose:

logslope = (n-1)

Problem solved!

30

(we are forcing the observed rate dependence)


Fake-O Model®

The model and the experiments for match.

loglog


16


But what about the normal stresses?

31


0 00 0

0 0 0

⇒ Ψ Ψ 0

0 0 00 0

0 0 0

0 00 0

0 0 0


32



6) Compare with what non-Newtonian fluids do

?


17

100

1000

10000

100000

0.01 0.1 1 10 100

viscosity, Poise

shear rate, 1/s


viscosity poise



33


1

0

67.0

24.0

000,12

000,18

s

n

m

poiseM

c


34


1.E+02

1.E+03

1.E+04

1.E+05

0.01 0.1 1 10 100

, Poise

1,

(dyn/cm2)s2

1, s

o1

o

1

Ψ 0Ψ 0

Fake-O Model Predictions: Steady shear:

Ψ 0Ψ 0

Polymer Behavior:


18


35


1.E+02

1.E+03

1.E+04

1.E+05

0.01 0.1 1 10 100

, Poise

1,

(dyn/cm2)s2

1, s

o1

o

1

Ψ 0Ψ 0

Fake-O Model Predictions: Steady shear:

Ψ 0Ψ 0

Polymer Behavior:


Why did we get 0 for the normal stresses?

It appears that should not be

simply proportional to

Need to try something else . . .

36


0 00 0

0 0 0

⇒ Ψ Ψ 0

⋅

⋅…


19


37



7) Reflect, learn, revise model, repeat



equation


+



predictions of:

measurements of

=

=

+material +





For non-Newtonian

fluids:

(1)

(2)

(3)

(4)

(5)

(6)

(7) Reflect Learn Revise model Propose new Repeat cycle….


What shall we guess next?

To sort out how to fix the Newtonian equation, we need more observations (to give us ideas).

Let’s try another material function that’s not a steady flow (but stick to shear).

38




equation


+



predictions of:

measurements of

=

=

+material +





For non-Newtonian

fluids:

(1)

(2)

(3)

(4)

(5)

(6)

(7)

Reflect Learn Revise model Propose new Repeat cycle….



20





kinem

atics


39

1212

00



≡ 00

Start-up of Steady Shear Flow Material Functions

Shear stress growth

function


0 00

Ψ , ≡

Ψ , ≡

0 0

0,

0 0

First normal-stress growth coefficient

Second normal-stress growth coefficient

,

,

,

,

,

,

Imposed Kinematics:

Material Functions:


, ≡


21


What does the Newtonian Fluid model predict in start-up of steady shearing?

Again, since we know , we can just substitute it into the constitutive equation

and calculate the stresses. 41




Tvv


What does the Newtonian Fluid constitutive equation predict in start-up of steady shearing?

42

You try.

2) Predict what Newtonian fluids would do (round 2)

Tvv


22

)(t

t

Material functions predicted for start-up of steady shearing of a Newtonian fluid

0

00)(

t

tt


Do these predictions match observations?

43

2) Predict what Newtonian fluids would do (round 2)

Ψ ≡ 0

Ψ ≡ 0

What do we measure for these material functions?


44

(for polymer solutions, for example)





23

Startup of Steady Shearing


Figures 6.49, 6.50, p. 208 Menezes and Graessley, Polybutadiene soln

45

3) See what non-Newtonian fluids do (round 2)

0 00

≡ 00

0

00)(

t

tt

Ψ ≡


What about other non-steady flows?

46





24

≡ 00

Cessation of Steady Shear Flow Material Functions

Shear stress decay

function


47

00 0

Ψ , ≡

Ψ , ≡

0

0,

0 0

First normal-stress decay coefficient

Second normal-stress decay coefficient

,

,

,

0

,,,

Imposed Kinematics:

Material Functions:


, ≡

Cessation of Steady Shearing

© Faith A. Morrison, Michigan Tech U.Figures 6.51, 6.52, p. 209 Menezes and Graessley, PB soln

48

3) See what non-Newtonian fluids do (round 3)

≡ 00

00 0


25


What does the Fake-O model® predict for start-up and cessation of shear?

49





What does the Fake-O model® predict for start-up and cessation of shear?

50

You try.

5) Predict the material functions (rounds 2 & 3)

2018: Homework 3


26


51




?

100

1000

10000

100000

0.01 0.1 1 10 100

viscosity, Poise

shear rate, 1/s


viscosity poise



52

6) Compare with what non-Newtonian fluids do (rounds 2&3)

1

0

67.0

24.0

000,12

000,18

s

n

m

poiseM

c


27

• captures rate dependence, but,

• Also, response is not instantaneous © Faith A. Morrison, Michigan Tech U.

53


Ψ 0Ψ 0

Fake-O Model Shear Material Function Predictions:

Start-up of steady shear:

0 00

where

Ψ 0

Ψ 0

Polymer Behavior:

Cessation of steady shear:

00 0

where

Ψ 0

Ψ 0


Observations

•The Fake-O model® predicts an instantaneous stress response, and this is not what is observed for polymers

•The predicted unsteady material functions depend on the shear rate, which is observed for polymers

•No shear normal stresses are predicted

Progress here

54

,



28


Observations

•The Fake-O model® predicts an instantaneous stress response, and this is not what is observed for polymers

•The predicted unsteady material functions depend on the shear rate, which is observed for polymers

•No shear normal stresses are predicted

Progress here

55

,


Lacks memory

Related to nonlinearities


56






equation


+



predictions of:

measurements of

=

=

+material +





For non-Newtonian

fluids:

(1)

(2)

(3)

(4)

(5)

(6)

(7) Reflect Learn Revise model Propose new Repeat cycle….


29


57




equation


+



predictions of:

measurements of

=

=

+material +





For non-Newtonian

fluids:

(1)

(2)

(3)

(4)

(5)

(6)

(7)To proceed to better-designed constitutive equations, we need to know more about material behavior, i.e. we need more material functionsto predict, and we need measurements of these material functions.

•More non-steady material functions (material functions that tell us about memory)

•Material functions that tell us about nonlinearity (strain)

•Material functions for a different flow

Back to step 1





kinem

atics


58

1212

00




30

Imposed Kinematics:

Steady Elongational Flow Material Functions

Material Functions: Elongational

Viscosity


constant

≡


0

0

0,

0

Alternatively,

≡


60


2) Predict what Newtonian fluids would do.


?


31


61




?

You try.


62




?


32

Steady State Elongation Viscosity


Figure 6.60, p. 215 Munstedt.; PS melt

Both tension thinning and thickening are observed.

0

TrTrouton ratio:

63



What does the Fake-O®-model predict for steady elongational viscosity?

64



5) Predict the material function (with new )


33


65


?


66


?(What is doing in an elongational

calculation?


34


What if we make the following replacement?

This at least can be written for any flow and it is equal to the shear rate in shear flow.

67



→


Observations

•The model contains parameters that are specific to shear flow – makes it impossible to adapt for elongational or mixed flows

•Also, the model should only contain quantities that are independent of coordinate system (i.e. invariant)

We will try to salvage the model by replacing the flow-specific kinetic parameter with

something that is frame-invariant and not flow-specific.

68

4) Hypothesize a constitutive equation


35


We will take out the shear rate and replace with

the magnitude of the rate-of-deformation tensor

(which is related to the second invariant of that tensor).

69

4) Hypothesize a constitutive equation

(Hold that thought; finish the chapter)

Elongational stress growth

Elongational stress cessation (nearly impossible)

Elongational creep

Step elongational strain

Small-amplitude Oscillatory Elongation (SAOE)

The other elongational experiments are analogous to shear experiments (see text)


70

1) Choose a material function – elongational flow


(Redundant with SAOS)


36

Imposed Kinematics:



Viscosity


Alternatively,

Elongation Material Functions

© Faith A. Morrison, Michigan Tech U.3/5/2018 71

a) Steady b) Start‐up

f) Creepd) Step strain e) SAOE

“Recipe cards”

(exists, but less often discussed)

(exists, but easily converted to SAOS so is redundant)

(exists)

c) Cessation

(currently unobservable)

Part II‐A. Continuum versus molecular modeling

≡21, xx

3x

Imposed Kinematics:



Viscosity


constant

≡


0

0

0,

0

Alternatively,

≡


37


73




?


Figure 6.63, p. 217 Inkson et al.; LDPE

Figure 6.64, p. 218 Kurzbeck et al.; PP

Start-up of Steady Elongation

Strain-hardening

Fit to an advanced constitutive equation (12 mode pom-pom model)

74



38


75




equation


+



predictions of:

measurements of

=

=

+material +





For non-Newtonian

fluids:

(1)

(2)

(3)

(4)

(5)

(6)

(7)To proceed to better-designed constitutive equations, we need to know more about material behavior, i.e. we need more material functionsto predict, and we need measurements of these material functions.



•Material functions for a different flow

Back to step 1

What now?

0 t

a. Steady

b. StressGrowth

c. StressRelaxation 0 t

0 t

Summary of shear rate kinematics (part 1)


76

1) Choose a material function – Rate based

0 t

0 t

0 t 0 t

0 t

0 t

0,


39

0 t

t

0 t

t

0 t

t

0 t

t21

0 t

t,021

0 t

t,021

0 t

t21

0 t

t21

a. Steady

b. StressGrowth


t,021

Summary of shear rate kinematics (part 1, rate-based)


.

.

.

77

Strain-rate based

1) Choose a material function – Rate based


The next three families of material functions incorporate the concept of strain.

78





equation


+



predictions of:

measurements of

=

=

+material +





For non-Newtonian

fluids:

(1)

(2)

(3)

(4)

(5)

(6)

(7)

To proceed to better-designed constitutive equations, we need to know more about material behavior, i.e. we need more material functionsto predict, and we need measurements of these material functions.



Back to step 1


40

0 t

t

0 t

t

0 t

t

0 t

t21

0 t

t,021

0 t

t,021

0 t

t21

0 t

t21

a. Steady

b. StressGrowth


t,021

.

.

.


79

Strain-ratebased

0 t 0 t

0 t

0 t

t

t t0

d. Creep

e. StepStrain

f. SAOS

0

t

t

The first three recipe cards were strain-rate based.

0 t

t

0 t

t

0 t

t

0 t

t21

0 t

t,021

0 t

t,021

0 t

t21

0 t

t21

a. Steady

b. StressGrowth


t,021

.

.

.

The first three recipe cards were strain-rate based.


80

Strain-ratebased

0 t 0 t

0 t

0 t

t

t t0

d. Creep

e. StepStrain

f. SAOS

0

t

t

Strainbased

The second three recipe cards are strain based.


41

d. Creep

e. StepStrain

f. SAOS

Summary of shear rate kinematics (part 2; strain-based)


81

t

cos

0 t

0 t 0 t0 t

0,

t0

0,

t

sin

0,

t0

t

sin

1) Choose a material function – Strain based


82

What is strain?

Strain is a measure of deformation (change in shape)

Investigating Stress/Deformation Relationships (Rheology)1) Choose a material function – Strain based

Initial or reference

state

Final state

(shape change relative to some other state)


42

Deformation (=strain)


We need a way to quantify “change in shape” due to flow.

t

The problem of change in shape is a difficult, 3-dimensional problem; we can start simple with unidirectional flow (shear).

83

There must be an initial (reference) shape and a final shape (at time of interest)

fluid particle shape at

shape at

What is strain? Answer: change in shape


NOTE:

Strain can be conceptually complicated

84

Deformation (=Strain)

We need a way to quantify “change in shape” due to flow.

t

The problem of change in shape is a difficult, 3-dimensional problem; we can start simple with unidirectional flow (shear).

There must be an initial (reference) shape and a final shape (at time of interest)

fluid particle shape at

shape at



Let’s begin with the answer


43

For shear flow (steady or unsteady, ):


85

,

What is shear strain? Summary

Strain is the integral of strain

rate

The time derivative of

strain is the strain rate

Strain is our measure of deformation (change of shape)

Strain accumulates as

the flow progresses

The strain rate is the rate of

instantaneous shape change Deformation rate

Shear strain tensor:

(see next page)


86

Strain Tensor—small strains


Infinitesimal strain tensor:

,

′ ′ ′

′ ′ ′

′ ′ ′

More on this later.


44

For uniaxial elongational flow (steady or unsteady, ):


87

What is elongational strain? Summary

Strain is the integral of strain

rate


Strain accumulates as

the flow progresses

, ′ ′

The time

derivative of strain is the strain

rate

The strain rate is the rate of

instantaneous shape change Deformation rate

?Now we can continue

with material functions based on strain.

(finite-strain tensor is

complicated)

(see Ch9)


88

What is the strain in start-up of steady shear?

Practice with strain

What is the strain in cessation of steady shear?


Shear stress growth

function



Imposed Kinematics:

Material Functions:



Shear stress decay

function



Imposed Kinematics:

Material Functions:


(let ∞

(let 0


45

≡ 00


Shear stress growth

function


89

0 00

Ψ , ≡

Ψ , ≡

0 0

0,

0 0



,

,

,

,

,

,

Imposed Kinematics:

Material Functions:


, ≡

≡ 00


Shear stress decay

function


90

00 0

Ψ , ≡

Ψ , ≡

0

0,

0 0



,

,

,

0

,,,

Imposed Kinematics:

Material Functions:


, ≡


46


What shall we do next?

To sort out how to model non-Newtonian behavior, we need more observations (to give us ideas).

We want to try another material function.

91



equation


+



predictions of:

measurements of

=

=

+material +





For non-Newtonian

fluids:

(1)

(2)

(3)

(4)

(5)

(6)

(7)

Reflect Learn Revise model Propose new Repeat cycle….







kinem

atics


92

1212

00




47

d. Creep

e. StepStrain

f. SAOS



93

t

cos

0 t

0 t 0 t0 t

0,

t0

0,

t

sin

0,

t0

t

sin


Step Strain Shear Flow Material Functions

Relaxation modulus


94

≡ 00

lim→

0 0/ 00

, ≡ , ,

, ≡

, ≡

0

0,

0 0

First normal-stress relaxation modulus

Second normal-stress relaxation modulus

,

,

,

,,,

.

.

.

0

Imposed Kinematics:

Material Functions:



48


95

?

You try.

What is the strain in the step-strain flow?


What is the strain in the step-strain flow?

The strain imposed is a constant


96


∞, ′

lim→

0 0/ 00

′

lim→

′


49


97




?Step Strain Shear Flow Material Functions

Relaxation modulus



.

.

.

Imposed Kinematics:

Material Functions:



98





Relaxation modulus



.

.

.

Imposed Kinematics:

Material Functions:


You try.


50


99




Answer: mostly zero


Relaxation modulus



.

.

.

Imposed Kinematics:

Material Functions:



100




?


51

0

1

10

100

1,000

10,000

1 10 100 1000 10000

time, s

G(t), Pa

<1.87

3.34

5.22

6.68

10

13.4

18.7

25.4

Step shear strain -strain dependence

Figure 6.57, p. 212 Einaga et al.; PS soln


0

101


Linear viscoelastic limit

At small strains the relaxation modulus is independent of strain.

Damping function, :

The damping function summarizes the non-linear effects as a function of strain amplitude.


102

The polystyrene solutions on the previous slide show time-strain independence, i.e. the curves have the same shape at different strains.


lim→

,

≡,


52

1

10

100

1000

10000

100000

1 10 100 1000 1000

time, t

shif

ted

G(t

), P

a

Step shear strain - Damping Function

Figure 6.58, p. 213 Einaga et al.; PS soln

0.01

0.1

1

10

0.1 1 10 100

strain

dam

pin

g f

un

ctio

n,

h


103

A special time?

One thing is happening

here.

Something else is happening here?


2

121 x

uG

1u 21 xv

2x

initial state,no flow,no forces

deformed state,

Hooke's law forelastic solids

spring restoring force

11 xkf

initial state,no forceinitial state,no force

deformed state,

Hooke's law forlinear springs

f

1x1x

Hooke’s Law for elastic solids:

Similar to the linear spring law

What types of materials generate stress in proportion to the strain imposed? Answer: elastic solids


104


0,(shear flow)


53


105




?

Hookean solids

0,Hookean Solid

Constitutive Equation


106




Answer: much more interesting.

Hookean solids

0,Hookean Solid

Constitutive Equation

Solution:• Recipe card gives kinematics• Calculate • Calculate material functions


54





kinem

atics


107

1212

00



d. Creep

e. StepStrain

f. SAOS



108

t

cos

0 t

0 t 0 t0 t

0,

t0

0,

t

sin

0,

t0

t

sin



55

Small-Amplitude Oscillatory Shear Material Functions

SAOS stress


109

≡ 00

cos

, , sin sin cos

′ ≡cos ′′ ≡

sin

0

0,

0

0(linear viscoelastic regime)

Storage modulus

Loss modulus

0

cos sin

sin

phase difference between stress and strain waves

Imposed Kinematics:

Material Functions:


What is the strain in small-amplitude oscillatory shear?



Answer: 0, sin

(let 0


56

In SAOS the strain amplitude is small, and a sinusoidal imposed strain induces a sinusoidal measured stress.

tttt

tt

cossinsincossincoscossin

)sin()(

00

00

021

portion in-phase with strain

portion in-phase with strain-rate


111


sin


-3

-2

-1

0

1

2

3

0 2 4 6 8 10

is the phase difference between the stress wave and the strain wave

112


0, sin


57

SAOS Material Functions

ttt

cossin

sincos)(

0

0

0

0

0

21



G G

For Newtonian fluids, stress is proportional to strain rate:

G” is thus known as the viscous loss modulus. It characterizes the viscous contribution to the stress response.


113


SAOS Material Functions

ttt

cossin

sincos)(

0

0

0

0

0

21



G G

For Hookean solids, stress is proportional to strain :

G’ is thus known as the elastic storage modulus. It characterizes the elastic contribution to the stress response.

(note: SAOS material functions may also be expressed in complex notation. See pp. 156-159 of Morrison, 2001)


114



58

Answer: ; ′′© Faith A. Morrison, Michigan Tech U.

115




?


SAOS stress

(linear viscoelastic regime)

Storage modulus

Loss modulus


Imposed Kinematics:

Material Functions:



116



2) Predict what Newtonian fluids would do.Hookean solids

0,

Hookean Solid Constitutive

Equation

?


SAOS stress

(linear viscoelastic regime)

Storage modulus

Loss modulus


Imposed Kinematics:

Material Functions:


Answer: 0;


59


117




?


SAOS Moduli of a Polymer Melt

Figure 8.8, p. 284 data from Vinogradov, PS melt

1.E+01

1.E+02

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07

aT, rad/s

G' (Pa)

G'' (Pa)

k k (s) gk(kPa) 1 2.3E-3 16 2 3.0E-4 140 3 3.0E-5 90 4 3.0E-6 400 5 3.0E-7 4000

Storage Modulus, ’ ,elastic

character

Loss Modulus, ′′ ;viscous

character

The SAOS moduli as a function of frequency may be correlated with material composition and used like a mechanical spectroscopy.



60


SAOS Moduli of a Polymer Melt

Figure 8.8, p. 284 data from Vinogradov, PS melt

1.E+01

1.E+02

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07

aT, rad/s

G' (Pa)

G'' (Pa)

k k (s) gk(kPa) 1 2.3E-3 16 2 3.0E-4 140 3 3.0E-5 90 4 3.0E-6 400 5 3.0E-7 4000

Storage Modulus, ’ ,elastic

character

Loss Modulus, ′′ ;viscous

character

The SAOS moduli as a function of frequency may be correlated with material composition and used like a mechanical spectroscopy.


We’ll talk more about this frequency dependence in

Chapter 6.

We have discussed six shear material functions;

Now, the equivalent elongationalmaterial functions


120

0 t

o

t

0 t

o

t

0 t

o

t

0 t

o

t21

0 t

o

t,021

0 t

o

t,021

0 t

t21

0 t

t21

a. Steady

b. StressGrowth


t,021

.

.

.

t21

0 t 0 t

t,021

0 t

o

t

0 t

o

t21

t

tto cos

t t

t21

0

d. Creep

e. StepStrain

f. SAOS

0

o

t,021

t

t21)sin( to

t,021

t

to sin

.

.

.

≡


61

Imposed Kinematics:



Viscosity


121

constant

≡


0

0

0,

0

Alternatively,

≡

Elongational stress growth

Elongational stress cessation (nearly impossible)

Elongational creep

Step elongational strain

Small-amplitude Oscillatory Elongation (SAOE)

The other elongational experiments are analogous to shear experiments (see text)


122

1) Choose a material function – elongational flow


(Redundant with SAOS)


62

Imposed Kinematics:

Start-up of Steady Elongation Material Functions


Start-up Function


, ≡


0

0

0,

Alternatively, ,

0

0 00

≡





?


63


Figure 6.63, p. 217 Inkson et al.; LDPE

Figure 6.64, p. 218 Kurzbeck et al.; PP

Start-up of Steady Elongation

Strain-hardening

Fit to an advanced constitutive equation (12 mode pom-pom model)

125


Imposed Kinematics:



Viscosity


Alternatively,

Elongation Material Functions

© Faith A. Morrison, Michigan Tech U.3/5/2018 126

a) Steady b) Start‐up

f) Creepd) Step strain e) SAOE

“Recipe cards”

(exists, but less often discussed)

(exists, but easily converted to SAOS so is redundant)

(exists)

c) Cessation

(currently unobservable)

Part II‐A. Continuum versus molecular modeling

≡21, xx

3x


64

What’s next?


127

Make better constitutive equations

1. Add invariants (replace , which is flow-specific).2. Make constitutive equations that reference flow in the past

(not purely instantaneous)3. Investigate strain

Be inspired by material behavior

1. Become informed on more rheological behavior2. Get more of a feel for what is observed, and when


128

Done with Material Functions.

Let’s move on to Experimental Data


65

Chapter 6: Experimental Data


129

CM4650 Polymer Rheology

Michigan Tech

10

10000

10000000

10000000000

1E-13 1E-10 1E-07 0.0001 0.1 100 100000

G'

G''107

1010

104

modulus, Pa

oNG

10-4 10-1 102 10510-7

10-1010-13

frequency, rad/s

Small-Amplitude Oscillatory Shear - Storage and Loss Moduli

Figure 6.30, p. 192 Plazek and O’Rourke; PS

Terminal zone

Glassy zone

Rubbery zone

chapter 5: material functionsfmorriso/cm4650/lectures/... · 1. choose a material function 2....

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