2016-module 2-l5

8/17/2019 2016-module 2-L5

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CHEE3301 –

Polymer Engineering –

1st Semester 2016

Module 2: Lecture 5

Viscoelastic models – complex modulus

8/17/2019 2016-module 2-L5


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CHEE3301 –


1st Semester 2016

Models of linear viscoelasticity

• Elements:

– Spring, modulus E

– Dashpot, viscosity h d

dt

h h

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CHEE3301 –


1st Semester 2016

Components of viscoelastic

behaviour

Recall:

• polymers fall on a spectrum of behaviours from the

extremes of linear elastic behaviour to Newtonian viscosity

• the relative importance of the two behaviours will depend

on the time frame and the temperature

• time relative to molecular relaxations

To model viscoelastic behaviour we will need to combine

these elements of elastic behaviour and flow behaviour.

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CHEE3301 –


1st Semester 2016

Maxwell (series) model

• Spring and dashpot in series – Stress is the same

– Strain is additive

• Consider stress relaxation

– Model held at constant strain, so

– So

– Integrate and impose IC t=0, 0

2, ,h

1 2

0d

dt

1 2 1d d d d

dt dt dt E dt

h

d E dt

h

0 0exp exp Et t h where is Relaxation time

This has right form for stress relaxation in polymers

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CHEE3301 –


1st Semester 2016

Maxwell model (2)

• Consider creep, where stress is constant

– Leads to

– Which is Newtonian viscous flow

0d

dt

1 2 1d d d d

dt dt dt E dt

h

d

dt

h

Not the right form for creep in polymers

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CHEE3301 –


1st Semester 2016

Kelvin-Voigt (parallel) Model

• Spring and Dashpot in parallel

– Strain is the same

– Stresses are additive

• Consider stress relaxation

– Leads to – Which is Hookean elastic behaviour

0d

dt

E

d E

dt

h

Not the right form for stress relaxation in polymers

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1st Semester 2016

Kelvin-Voigt Model (2)

• Consider creep, constant stress, so (dividing through by h and

rearranging):

– This standard differential equation has solution

0d E

dt

h h

1 exp 1 exp Et

t E E

h

where is Retardation time

This does has right form for creep in polymers

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1st Semester 2016

Standard Linear Viscoelastic model

• Maxwell model (series) describes stress relaxation, but doesn’t fit creep

• Kelvin-Voigt Model (parallel) describes creep, but not stress relaxation

• Combine two:

1 1, , E

2 2 2, , E

2 3, ,h

1 2

3 2

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1st Semester 2016

Standard Linear Model (2)

• It can be shown (see tutorial question) that this model reduces to the

Kelvin-Voigt model for creep and to the Maxwell model for stress

relaxation

• The time scale of both creep and stress relaxation (the relaxation orretardation time) is the same

• Note though that for most polymers we have

– a spectrum of relaxation times

– non-linear effects

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10101010CHEE3301 –


1st Semester 2016

Standard Linear Model (3)

• Apply an oscillating strain to the Standard Linear Model

• Viscoelastic response is

0 sin t

0

0 0

0

sin

sin cos cos sin

sin cos

t

t t

E t E t

What would pure elastic

or pure viscous response

look like?

time

time

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11111111CHEE3301 –


1st Semester 2016

• Response to sinusoidal strain:

– In phase component, E ′ , –

Storage modulus

(stored energy returned onremoval of load)

– Out of phase, E ″ , loss modulus

(energy lost in a cycle)

• Define:tan = E ″ /E ′ (damping)

Complex modulus

E'

E"

E *

.

2

0

2

0 0

0

2

2 2

0

0

2

0

sin cos cos

sin cos cos

d W d dt

dt

E t E t t dt

E t t E t dt

E

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12121212CHEE3301 –


1st Semester 2016

tan

• Can measure E ′ and E ″ in DynamicMechanical Thermal Analysis (DMTA)

– Torsional pendulum

– 3 pt bending

– Tension

– Etc

– Can also use dielectric measurements

• Constant frequency and ramp temperature

• Constant temperature and vary frequency

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13131313CHEE3301 –


1st Semester 2016

Torsional pendulum:

Dynamic Mechanical

Thermal Analysis

(DMTA)

• Polymer sample set oscillating at setfrequency

• Measure decrease in amplitude as forcedoscilation ‘damped’ out

• Can calculate G’ and G’’ from measuringthe ratio of the amplitude of the motionfrom two successive cycles

Ref: Young and Lovell p333

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14141414CHEE3301 –


1st Semester 2016

t a n

Tg

b

g

1

2

Note: molecular transitions

designated by Greek alphabet

Highest temperature relaxation – a;

lower T transitions b, g, , etc

In semi-crystalline

polymers a , usually

related to crystalline

region, can split intotwo, a1 and a2

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15151515CHEE3301 –


1st Semester 2016

DMTA examples:

polyethylene

• Peaks in tan sensitive to

molecular structure and

microstructure

a peak present in both samples - splits

in 2 for HDPE

related to crystalline regions

b peak

absent from HDPE –

amorphous regions

g peak – present in both

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16161616CHEE3301 –


1st Semester 2016

DMTA example:

Nylon 6.6

• Quenched sample amorphous –

large Tg (a peak)

• Thermal energy allows it to

crystallise after Tg – increasein G’

2016-module 2-l5

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