2016-module 2-L5
Transcript of 2016-module 2-L5
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CHEE3301 –
Polymer Engineering –
1st Semester 2016
Module 2: Lecture 5
Viscoelastic models – complex modulus
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CHEE3301 –
Polymer Engineering –
1st Semester 2016
Models of linear viscoelasticity
• Elements:
– Spring, modulus E
– Dashpot, viscosity h d
dt
h h
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CHEE3301 –
Polymer Engineering –
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 –
Polymer Engineering –
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 –
Polymer Engineering –
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 –
Polymer Engineering –
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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7777CHEE3301 –
Polymer Engineering –
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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8888CHEE3301 –
Polymer Engineering –
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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9999CHEE3301 –
Polymer Engineering –
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 –
Polymer Engineering –
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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Polymer Engineering –
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 –
Polymer Engineering –
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 –
Polymer Engineering –
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 –
Polymer Engineering –
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 –
Polymer Engineering –
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 –
Polymer Engineering –
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’