viscoelastic properties polymers have both ideal elastic and viscous behavior depending on time and...
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Viscoelastic properties
Polymers have both ideal elastic and viscous behavior depending on time
and temperature.
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Ideal (elastic) Solid
E Hooks Law
response is independent of time and the deformation is dependent on the spring constant.
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Ideal Solid
E
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Ideal Liquid
dt
de h= viscosity
de/dt = strain rate
The viscous response is generally time- and rate-dependent.
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Ideal Liquid
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• The behaviour of linear elastic were given by Hooke’s law:
Ee
E= Elastic modulus
s= Stress
e=strain
de/dt = strain rate
ds/dt = stress rate
h= viscosity
ordt
deE
dt
d
• The behaviour of linear viscous were given by Newton’s Law:
dt
de
** This equation only applicable at low strain
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• Behaviour of most polymer is in between behaviour of elastic and viscous materials.
1. At low temperature & high strain rate, Polymer demonstrate elastic behaviour,
2. At high temperature & low strain rate, Polymer demonstrate viscous behaviour
3. At intermediate temperatures & rate of strain Polymer demonstrate visco-elastic behaviour
Viscoelastic behavior
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• Polymer is called visco- elastic because: • Showing both behaviour elastic & viscous behaviour
• Instantaneously elastic strain followed by viscous time dependent strain
Load added
Load released
elastic
elastic
viscousviscous
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Maxwell Model
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Kelvin Voigt Model
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Burger Model
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Static Modulus of Amorphous PS
Glassy
Leathery
Rubbery
Viscous
Polystyrene
Stress applied at x and removed at y
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Dynamic Mechanical Analysis
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Spring Modelg = g0⋅sin (ω⋅t)
g0=maximum strainw = angular velocity
Since stress, ,t is
=t Gg
=t Gg0sin(wt)
And t and g are in phase
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Dashpot ModelWhenever the strain in a dashpot is at its maximum, the rate of change of the strain is zero ( g = 0).Whenever the strain changes from positive values to negative ones and then passes through zero, the rate of strain change is highest and this leads to the maximum resulting stress.
)tcos(odashpot ωωγ=γ=τ &
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Kelvin-Voigt Model
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Dynamic (Oscillatory) TestingIn the general case when the sample is deformed sinusoidally, as a response the stress will also oscillate sinusoidally at the same frequency, but in general will be shifted by a phase angle d with respect to the strain wave. The phase angle will depend on the nature of the material (viscous, elastic or viscoelastic)
)tsin(o ωγ=γ
Input
Response
)tsin(o δ+ωτ=τwhere 0°<d<90°
3.29stress strain viscosity Gmodulus
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Dynamic (Oscillatory) TestingBy using trigonometry:
)tcos()tsin()tsin( ooo ωτ′′+ωτ′=δ+ωτ=τ
Let’s define: oooo G and G γ′′=τ ′′γ′=τ′
In-phase component of the stress, representing solid-like behavior
Out-of-phase component of the stress, representing liquid-like behavior
Modulusor Loss Viscous ,strain maximum
stress phaseofout)(G
ModulusStorageor Elastic, strain maximum
stress phasein)(G
o
o
o
o
γτ′′
=−−
=ω′′
γτ′
=−
=ω′where:
(3-1)
3.30
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Physical Meaning of G’, G”[ ])tcos()("G)tsin()(Go ωω+ωω′γ=τEquation (3-1) becomes:
G
Gtan
′′′
=δWe can also define the loss tangent:
)tsin(GG ospring ωγ=γ=τFor solid-like response:
°=δ=δ=′′=′∴ 0 0, tan0,G ,GG
For liquid-like response:
)tcos(odashpot ωωγ=γ=τ &
°=δ∞=δω=′′=′∴ 09 , tan,G ,0G
G’storage modulus G’’loss modulus
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Real Visco-Elastic Samples
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Typical Oscillatory Data
Rubbers – Viscoelastic solid response:G’ > G” over the whole range of frequencies
G’
G’’
log G
log
Rubber
G’storage modulus
G’’loss modulus
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Typical Oscillatory Data
Polymeric liquids (solutions or melts) Viscoelastic liquid response:G” > G’ at low frequenciesResponse becomes solid-like at high frequenciesG’ shows a plateau modulus and decreases with w-2 in the limit of low
frequency (terminal region)G” decreases with w-1 in the limit of low frequency
G’G’’
log G
log
Melt or solution
G0
G’storage modulus
G’’loss modulus
Less liquid like
More liquid like
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Blend
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Epoxy
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Nylon-6 as a function of humidity
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Polylactic acid
E’storage modulus
E’’loss modulus
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Tg 87 °C
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Tg -123 °C (-190 F)Tm 135 °C (275 F)
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Polyurethane foam (Tg 160 C)
G’storage modulus
G’’loss modulus
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These data show the difference between the behaviour of un-aged and aged samples of rubber, and were collected in shear mode on the DMTA at 1 Hz. The aged sample has a lower modulus than the un-aged, and is weaker. The loss peak is also much smaller for the aged sample.
G’storage modulus
G’’loss modulus
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Tan d of paint as it dries
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Epoxy and epoxy with clay filler
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Dynamic test of a Voigt solid
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Benefits of Dynamic Testing