Summary of shear rate kinematics (part 1)pages.mtu.edu/~fmorriso/cm4650/IntrToMatFunctions2.pdf · Since we set the stress in this experiment ... The problem of change in shape is

IntroToMatFunctions2.pdf 2014 CM4650

1

© Faith A. Morrison, Michigan Tech U.

To proceed to better-designed constitutive equations, we need to know more about material behavior, i.e. we need more material functions to 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)

1

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

c. StressRelaxation 0 t

t,021

Summary of shear rate kinematics (part 1)


.

.

.

2


2


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

3

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

Summary of shear rate kinematics (part 2)


.

.

.

4


3

MASS

samples

oven

Shear Creep Flow

Constant shear stress imposed


5

123

221

0

0

)(

xt

v

0

00)(

021 t

tt

Creep Shear Flow Material FunctionsKinematics:

Material Functions:

It is unusual to prescribe stress rather than

Since we set the stress in this experiment (rather than measuring it), the material functions are related to the

deformation of the sample. We need to discuss

measurements of deformation before proceeding.

Because shear rate is not prescribed, it becomes

something we must measure.


6


4


7

Pause on Material Functions

We need to define and learn to work with strain.

Deformation (strain)


We need a way to quantify “change in shape”

t

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

8


5

Strain in Shear

The strain is related to the change of shape of the deformed particle.


H

H H

0 1 2

ΔΔ

Relative change in displacement

9

Strain in Shear

The strain is related to the change of shape of the deformed particle.

There is no unique way to measure “change of shape.”


H

H H

ΔΔ


10

H

22/

2

H

H

H

H


6

P(tref)

x1

x2

x3

r(tref)P(t)

r(t)

u(tref,t) particle path

flow

Deformation (strain))()(),( refref trtrttu

1233

2

1

)(

)(

)(

)(

ref

ref

ref

ref

tx

tx

tx

tr

1233

2

1

)(

)(

)(

)(

tx

tx

tx

tr

Shear strain

Displacement function


11

This vector keeps track of the location of a fluid particle as a function of time.

Current position compared to reference position

2

121 ),(

x

uttref


What is the strain in the standard flow steady shear?


123123

20

3

2

1

0

0

dtdxdt

dxdtdxx

v

1x

2x

)( reftr

)(tr

We can integrate this differential equation because is a

constant. We obtain .


7

Deformation in shear flow (strain)

1233

2

1

)(

)(

)(

)(

ref

ref

ref

ref

tx

tx

tx

tr

1233

2

201

1233

2

1

)(

)(

)()(

)(

)(

)(

)(

ref

ref

refref

tx

tx

xtttx

tx

tx

tx

tr


123

20

0

0

)(

)()(),(

xtt

trtrtturef

refref


13

steady

Deformation in shear flow (strain)

1 121

2 2

21 0

( , )

( , ) ( )

ref

ref ref

u dut t

x dx

t t t t

Shear strain


123

20

0

0

)(

)()(),(

xtt

trtrtturef

refref


14

(for steady shear or in unsteady shear for short

time intervals)

Our choice for measuring change in shape:

steady


8

For unsteady shear, is a function of time:

short time interval between and :

ttx

utt ppp

)(),( 1212

1121


15

1

2

3

2

123 123

( )

0

0

dxdt

dxdt

dxdt

t x

v

This integration is less

straightforward.

We can obtain the unsteady result for strain by applying the steady result over short time intervals (where may be approximated as a constant) and add up the strains.

For unsteady shear:

(short time interval)

For a long time interval, we add up the strains over short time intervals.

tttt ppp )(),( 121121

1

0

1

01211212121 )(),(),(

N

p

N

pppp tttttt

ttx

utt ppp

)(),( 1212

1121

Taking the limit as Δ → 0,

tdtttttt

t

N

pp

t

2

1

)()(lim),( 21

1

0121

02121

Strain at t2 with respect to fluid configuration at t1 in unsteady shear flow.


16

short time interval:

long time interval:


9

For shear flow (steady or unsteady):

tdtttt

t

2

1

)(),( 212121 Strain at t2 with respect to fluid configuration at t1 in shear flow (steady or unsteady).


17

Now we can continue with material functions based on strain.

Change of Shape

)(

)()()(

)(

2121

)(

21)(

2121

2121

tdt

d

tttdt

tdtdt

d

dt

d

dt

td

refdttdt

t t

t

t

ref

ref

ref

Note also, by Leibnitz rule:

Deformation rate

123

221

0

0

)(

xt

v

0

00)(

021 t

tt


Material Functions:

It is unusual to prescribe stress rather than

Since we set the stress in this experiment (rather than measuring it), the material functions are related to the

deformation of the sample..

Because shear rate is not prescribed, it becomes

something we must measure.


18


10

2

2021 0

0

0

0

)(

tt

tt

t

t

123

221

0

0

)(

xt

v


Material Functions:

0

21

)(0

),0(),(

2

ttJ

t

Shear creep compliance

00

),~(0

)~(),~(),~(

22

ttRtJ r

tttttr

Recoverable creep

compliance


19

0, 0,

Creep Recovery

-After creep, stop pulling forward and allow the flow to reverse

-In linear-viscoelastic materials, we can calculate the recovery material function from creep measurements

),0(),0()~( 21221 tttr Recoverable strainRecoil strain

Strain at the end of the forward motion

Strain at the end of the recovery

00

)~(),~(

t

tJ rr

Recoverable creep

compliance


20


11

Material functions predicted for creep of a Newtonian fluid


21

?),()(0

2

t

tJ Shear creep compliance

?),~( 0 tJ rRecoverable creep compliance

Tvvt )(Newtonian:

Material functions predicted for creep of a Newtonian fluid

)(tJ

t

1


remove stress (t2) 22

0)~(

)( 1

tJ

ttJ

r

No recovery in Newtonian fluids


12

Shear Creep of a Viscoelastic liquid


Figure 6.53, p. 210 Plazek; PS melt

0

00)(

021 t

tt

123

221

0

0

)(

xt

v

0

210

),0(),(

t

tJ

refrefp T

TTJJ

)(

Data have been corrected for vertical shift.

23

)(

1

1

0

0

21

t

t

statesteady dt

d

dt

dJ

•At long times the creep compliance J(t,0) becomes a straight line (steady flow).

The slope at steady state is the inverse of the steady state viscosity

CttJt

statesteady

)(

1)(

)( 0sJ

Steady-state compliance


24

Characteristics of a Creep Curve

•We can define a steady-state compliance


13

Shear creep material functions

t0

constant slope =

osJ

Steady-state compliance


25

regime of steady flow at shear rate

)(

1

t

t

, recovery0

0 2t

2t

Ultimate recoil function


26

2~

~

ttt

t

Characteristics of a Creep Recovery Curve

•At long recoil times we can define an ultimate recoil function

),~(),~( 00 tRtJr

)( 0R

)( 0R


14

In the Linear Viscoelastic (LVE) Limit, it is easy to relate the two shear creep material functions

),( otJ

creept,, recovery0

0 2t

2t


27

2~

~

ttt

t

),~( 0tJ r

Linear Viscoelastic Creep (no dependence on 0)

0

0

00

)()(

)()(

ttt

ttt

r

tr

total strain

recoverable strain

non-recoverable strain

0

0

)()(

)()(

t

tJtJ

ttJtJ

r

r


28

For LVE materials, we can obtain R(t) without a recovery

experiment


15


Figures 6.54, 6.55, p. 211 Plazek; PS melt

Shear Creep - Recoverable Compliance

)~(tJ r

29

Time-temperature shifted

creept,

)(tJ

0

0

constant slope =o

1

Shear creep material functionsLinear-viscoelastic (LVE) limit

2t2t

Ultimate recoil function

LVE steady-state compliance


30

)~(tJ r

0

2

t Non-recoverable

strain

00sJR

, recovery

2~

~

ttt

t

0

2

t

0SJ


16

Step Shear Strain Material Functions

00

00

constant

0

0

0

0

lim)(

t

t

t

t

Kinematics:

123

2

0

0

)(

xt

v

Material Functions:

0

0210

),(),(

t

tG

20

22111

G

20

33222

GRelaxation

modulus

First normal-stress relaxation modulus

Second normal-stress relaxation

modulus


31

What is the strain in this flow?

0

0

0

0

0

2121

lim

0

0

0

0

lim

)(),(

0

td

td

t

t

t

tdtt

t

t

The strain imposed is a constant


32


17

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

33

Linear viscoelastic limit

)(),(lim 000

tGtG

At small strains the relaxation modulus is independent of strain.

Damping function, h

)(

),()( 0

0 tG

tGh

The damping function

summarizes the non-linear effects as a function of

strain amplitude.


34

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


18

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

2121 G


35

Small-Amplitude Oscillatory Shear Material Functions

0

0

0 cos)(

tt

Kinematics:

123

2

0

0

)(

xt

v

Material Functions:

tGtGt

cossin),(

0

021

Storage modulus

cos)(

0

0G sin)(

0

0G

Loss modulus

is the phase difference between

stress and strain waves)


36


19


t

tdt

tdtt

t

t

sin

cos

)(),0(

0

0

0

0 2121

The strain imposed is sinusoidal.


The strain amplitude is

00

37

steady shear

h

1x

2x constantV

small-amplitudeoscillatory shear

h

1x

2x

thVttb o)(

th

thtb

o

o

sin

sin)(

periodicV

Generating Small Amplitude Oscillatory Shear (SAOS)


38


20

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

sinusoidal measured stress.

)sin()( 021 tt

tttt

tt

cossinsincossincoscossin

)sin()(

00

00

021

portion in-phase with strain

portion in-phase with strain-rate


39


-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

)(21 t

)(),0( 2121 tt

40


21

SAOS Material Functions

ttt

cossin

sincos)(

0

0

0

0

0

21



G G

For Newtonian fluids, stress is proportional to strain rate: 2121

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


41

SAOS Material Functions

ttt

cossin

sincos)(

0

0

0

0

0

21



G G

For Hookean solids, stress is proportional to strain : 2121 G

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)


42


22


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, G’(), elastic

character

Loss Modulus, G”(); viscous

character

The SAOS moduli as a function of frequency

may be correlated with material composition and

used like a mechanical spectroscopy.

We have discussed six shear material

functions;

Now, the equivalent elongational material

functions


44

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

c. StressRelaxation 0 t

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

.

.

.


23

Steady Elongational Flow Material Functions

constant)( 0 t

Kinematics:

Material Functions:

0

11331

oror

PB

Uniaxial or Biaxial or First Planar Elongational Viscosity

123

3

2

1

)(

)1)((2

1

)1)((2

1

xt

xbt

xbt

v

Elongational flow: b=0,

Biaxial stretching: b=0,

Planar elongation: b=1,

0)( t0)( t0)( t

0

11222

p

Second PlanarElongational Viscosity


45



(to answer, review how strain was developed/defined for previous flows. . . )


24


P(tref)

x1

x2

x3

r(tref)P(t)

r(t)

u(tref,t) particle path

flow

Deformation (strain))()(),( refref trtrttu

1233

2

1

)(

)(

)(

)(

ref

ref

ref

ref

tx

tx

tx

tr

1233

2

1

)(

)(

)(

)(

tx

tx

tx

tr

Shear strain


This vector keeps track of the location of a fluid particle as a function of time.

Current position compared to reference position

2

121 ),(

x

uttref


1233

2

201

1233

2

1

)(

)(

)()(

)(

)(

)(

)(

ref

ref

refref

tx

tx

xtttx

tx

tx

tx

tr


123

20

0

0

)(

)()(),(

xtt

trtrtturef

refref

021

2

121

)(),(

),(

refref

ref

tttt

x

utt

Shear strain(for steady shear or in

unsteady shear for short time intervals)

Our choice for measuring change in shape:

Recall, for shear . ..

How did we do that before . . ?


Path to strain for shear:

Try to follow for elongation.

1233

2

1

)(

)(

)(

)(

ref

ref

ref

ref

tx

tx

tx

tr ?

)(

)(

)(

)(

1233

2

1

tx

tx

tx

tr

?

?)()(),(

T

refref

uu

trtrttu

, → → → ,


25


1 0 2

10 2

1 0 2

1 1,0 0 2

1 1,00

2

( )

v x

dxx

dtdx x dt

x x t x

x xt

x

tx

x

dtx

dx

xdt

dx

xv

00,3

3

03

3

303

303

ln

Shear Elongation

constantx

v

Piece of deformation over time interval Δ

•The way we quantified deformation for shear, du1/dx2, is not so appropriate for elongation.

•Velocity gradient constant in both flows (but not the same gradient)

Notes:

•(Velocity gradient) (t) is a measure of deformation that accumulates linearly with flow


Press on:

tdtttt

t

2

1

)(),( 212121

strain = dtvelocity gradient

tdtttt

t

2

1

)(),( 21

homogeneous flows (velocity gradient the same

everywhere in the flow)

Need a better definition of strain for the general case

Note:

Shear:

Elongation:


26

0

0

ln

)(),(

l

l

t

tdtttt

trefref

The strain imposed is proportional to time.

The ratio of current length to initial length is

exponential in time.

Hencky strain


(choose tref=0)

51


What does the Newtonian Fluid model predict in uniaxial steady elongational flow?

Tvv

Again, since we know v, we can just plug it in to the constitutive equation

and calculate the stresses.

52


27

Steady State Elongation Viscosity


Figure 6.60, p. 215 Munstedt.; PS melt

Both tension thinning and

thickening are observed.

0

TrTrouton ratio:

53


What does the model we guessed at predict for steady uniaxial elongational flow?

c

nc

m

MM

0

10

000

TvvM 0

54


28


What if we make the following replacement?

2

10 x

v

This at least can be written for any flow and it is equal

to the shear rate in shear flow.

55


Observations

c

nc

m

MM

0

10

000

TvvM 0

•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.

56


29


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).

c

n

c

m

MM

1

0

TvvM

57

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)


58


30


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

Figure 6.64, p. 218 Kurzbeck et al.; PPStart-up of

Steady Elongation

Strain-hardening

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

59

What’s next?


60

Underlying physics (mass, momentum balances, stress tensor)

Standard flows

Material functions

?

•Constitutive equations

•Model flows/solve engineering problems

We want to design constitutive equations based on the material behavior of real non-

Newtonian fluids.

What is the behavior of non-Newtonian fluids?

Documents

Summary of shear rate kinematics (part 1)pages.mtu.edu/~fmorriso/cm4650/IntrToMatFunctions2.pdf · Since we set the stress in this experiment ... The problem of change in shape is