Viscoelastic materials
• Alfrey (1957) listed 3 methods that use
experimental curves to map out the
viscoelastic character of a material:
– Creep curve: function of time
– Ralaxation curve: function of time
– Dynamic modulus curve: dynamic modulus as
a function of frequency of the sinusoidal strain
• All of them should be independent of the
magnitude of the imposed stress or strain.
(linear viscoelastic materials)
Dynamic tests
Dynamic testing
• Rapid test with minimal chemical and
physical changes.
• There are 4 types (Morrow and
Mohsenin, 1968):
– Direct measurement of stress and strain
– Resonance methods
– Wave propagation methods
– Transducer methods
Dynamic tests
• There are 3 criteria for dynamic tests
– L/ < 1 : direct measurement of
sinusoidally varying  and  (use in
most foods)
– L/ = 1 : resonance vibration
– L/ > 1 : pulsed wave propagation
(ultrasonic, sound wave: high frequency
or low )
L = length of sample
 = wave length
Dynamic-Mechanical Analysis
(DMA)
direct measurement of
sinusoidally varying  and 
Dynamic or oscillatory tests
Dynamic or oscillatory tests are performed to
study the viscoelastic properties of a sample.
The tests are called microscale experiments
compared to macroscale tests like rotational or
viscometry tests.
Viscoelastic samples have both elastic (solid)
and viscous (liquid) properties, the extreme
described by Hooke’s law of elasticity and
Newton’s law of viscosity.
Parallel-plate geometry for
shearing of viscous materials
(DSR instrument).
Rheometrics RFS II
• Dynamic mechanical analysis (DMA), dynamic
mechanical thermal analysis (DMTA) or dynamic
thermomechanical analysis is a technique used to
study and characterize materials.
• It is most useful for observing the viscoelastic
nature of polymers. An oscillating force is applied to
a sample of material and the resulting displacement
of the sample is measured.
• From this the stiffness of the sample can be
determined, and the sample modulus can be
calculated. By measuring the time lag in the
displacement compared to the applied force it is
possible to determine the damping properties of the
material.
• Viscoelastic materials such as polymers
typically exist in two distinct states. They
exhibit the properties of a glass (high
modulus) at low temperatures and those
of a rubber (low modulus) at higher
temperatures.
By
scanning
the
temperature during a DMA experiment this
change of state, the glass transition or
alpha relaxation, can be observed.
Dynamic Mechanical Testing
 An oscillatory (sinusoidal)
Deformation
deformation (stress or strain)
is applied to a sample.
The material response
Response
(strain or stress) is measured.
The phase angle d, or phase
shift, between the deformation
and response is measured.
Phase angle
d
Dynamic Mechanical Testing:
Response for Classical Extremes
Purely Elastic Response
(Hookean Solid)
Purely Viscous
Response
(Newtonian Liquid)
d = 90°
d = 0°
Stress
Stress
Strain
Strain
Dynamic Mechanical Testing:
Viscoelastic Material Response
Phase angle
Strain
Stress
0° < d < 90°
Dynamic (Oscillatory) Rheometry
A.The ideal elastic solid
A rigid solid incapable of viscous dissipation
of energy follows Hooke’s Law, wherein
stress and strain are proportional (=E).
Therefore, the imposed strain function:
(w)=o sin(wt)
generates the stress response
(w)=E osin(wt) =  o sin(wt)
and the phase angle, d, equals zero.
Where o = maximum amplitude
w = 2¶f = angular frequency
f = frequency in Hz or cycle/s
Dynamic (Oscillatory) Rheometry
B. The ideal viscous liquid
A viscous liquid is incapable of storing
inputted energy, the result being that the
stress is 90 degrees out of phase with the
strain. An input of:
(w)=o sin(wt)
generates the stress response
(w)=  o sin(wt+p/2)
and the phase angle, d, p/2.
Where o = maximum amplitude
w = 2¶f = angular frequency
f = frequency in Hz or cycle/s
DMA Viscoelastic Parameters:
The Complex, Elastic, & Viscous Stress
The stress in a dynamic experiment is referred to as the
complex stress *
The complex stress can be separated into two components:
1) An elastic stress in phase with the strain. ' = *cosd
' is the degree to which material behaves like an elastic solid.
2) A viscous stress in phase with the strain rate. " = *sind
" is the degree to which material behaves like an ideal liquid.
Phase angle d
Complex Stress, *
Strain, 
* = ' + i"
16
Generally, measurements for visco.
materials are represented as a complex
modulus E* to capture both viscous and
elastic behavior:
E* = E’ + iE”
E*2 = E’2
+ E”2
In dynamic mechanical analysis (DMA,
aka oscillatory shear or viscometry), a
sinusoidal  or  applied.
0 = peak stress
E’ = 0 cosd/0 = E* cosd
E” = 0 sind/0 = E* sind
Schematic of stress  as a function of t with
dynamic (sinusoidal) loading (strain).
COMPLEX MODULUS:
E*=E’ + iE”
I E* I = Peak Stress / Peak Strain
o

STRESS
STRAIN
o
0
d/ w
2p / w
STORAGE ( Elastic) MODULUS
I E' I = I E* I cos
d
t
LOSS MODULUS
I E" I = I E* I sin
d
The “E”s (Young’s moduli) can all be
replaced with “G”s (rigidity or shear
moduli), when appropriate. Therefore:
G* = G’ + iG"
where the shearing stress is  and the
deformation (strain) is  or . Theory
SAME.
Complex modulus - G*
The complex modulus describes the total resistance
of the sample to oscillatory shear,
 = G* 
Similar is he resistance to flow in rotational tests,
.
=h
The complex modulus is determined in an oscillatory
test at small angles of deformation. The viscosity is,
on the other hand, calculated in rotational tests at
varying shear rates (large deformation rates)
In analyzing polymeric materials:
G* = (0)/(0), ~ total stiffness.
In-phase component of IG*I = shear storage
modulus G‘ ~ elastic portion of input energy
= G*cosd
The
out-of-phase
component,
G"
represents the viscous component of G*,
the loss of useful mechanical energy as
heat
= G*sind = loss modulus
The complex dynamic shear viscosity h*
is G*/w, while the dynamic viscosity is
h = G"/w or h = G"/2pf
For purely elastic materials, the phase
angle d = 0, for purely viscous
materials, 90.
The tan(d) is an important parameter
for
describing
the
viscoelastic
properties; it is the ratio of the loss to
storage moduli:
tan d = G"/ G',
Complex modulus G*
G’’
G*
d
G’
G* = G’+iG’’= (G’2+G’’2)1/2
tan d = G” / G’
G’ = elastic modulus or
storage modulus
G’’ = viscosity modulus or
loss modulus
tan d = phase angle or
loss angle
Tests
Dynamic Oscillatory Shear Test
• Plate oscillates at increasing
frequencies
• Strain and stress are
measured to determine G’
and G’’
– G’ represents the elastic
(storage) modulus
– G’’ represents the viscous (loss)
modulus
• When G’ > G’’ the fluid
behaves more elastic
• When G’ < G’’ the fluid
behaves more viscous
Phase angle - tan d = damping factor
Phase angle tan d is associated with the degree of
viscoelsticity of the sample. A low value in tan d
or d indicates a higher degree of viscoelasticity
(more solidlike). The phase angle d can be used
to describe the properties of a sample.
d = 90  G*= G´´ and G´= 0  viscous sample
d = 0  G*= G´ and G´´= 0  elastic sample
0 < d < 90   viscoelastic sample
d > 45   G´´> G´  semi liquid sample
d < 45   G´> G´´  semi solid sample
Complex viscosity - h*
Complex viscosity describes the flow
resistance of the sample in the
structured state, originating as viscous
or elastic flow resistance to the
oscillating movement.
h* = G* / w
w = 2pf
A high value for the complex viscosity the
greater is the resistance to flow in the
structured state.
DMA Viscoelastic Parameters
The Modulus: Measure of
materials overall resistance to
deformation.
G = Stress/Strain
The Elastic (Storage) Modulus:
Measure of elasticity of material.
The ability of the material to store
energy.
G' = (stress/strain)cosd
The Viscous (loss) Modulus:
The ability of the material to
dissipate energy. Energy lost as
heat.
G" = (stress/strain)sind
Tan Delta:
Measure of material damping - such
as vibration or sound damping.
Tan d =
G"/G'
Storage and Loss
of a Viscoelastic Material
SUPER BALL
LOSS
X
TENNIS
BALL
STORAGE
DMA Viscoelastic Parameters:
Damping, tan d
Dynamic measurement
represented as a vector
G*
G"
Phase angle d
G'
The tangent of the phase angle is the ratio of the
loss modulus to the storage modulus.
tan d = G"/G'
"TAN DELTA" (tan d) is a measure of the damping
ability of the material (damping properties).
Viscoelasticity in Crosslinked,
Amorphous Polymers
•
Plots of log G’, log G” and tand
against log angular frequency
(in radians per second) for a
typical elastomer above its Tg;
•
Poly(styrene-co-butadiene)
lightly vulcanized with a
peroxide cure.
Storage modulus
Loss modulus
•
•
Note that at low frequencies the
material has a low modulus and
behaves elastically.
As frequency is increased, the
material becomes stiffer, and
less capable of storing inputted
energy (generates heat upon
deformation).
tan d = G” / G’
Other methods
L/ = 1
Resonance Vibration Method
• In physics, resonance is the tendency
of a system to oscillate at maximum
amplitude at a certain frequency.
This frequency is known as the
system's
natural
frequency
of
vibration, resonant frequency, or
eigenfrequency.
38.3  L4 f r
E =
d2
f r = frequency( Hz)
2
 = density ( g / cm3 )
L = Length (cm)
d = diam eter(cm)
Amplitude
ratio
am plitude
am plituderatio =
max am plitude
1
0.5
w0.5
frequency
damping=internalfriction= tand
w0.5 w0.707
tand =
=
wr
wr 3
Wr = frequency at
ratio equal to 1.0
Free vibration method
• Make an object vibrate freely. Vibration
will stop with time.
• Due to internal friction or viscosity, the
dead of amplitude after time occurs.
A1
A2
Force acts here
A1 , A 2 = successive am plitudein each cycle
1
A1
tand = ln
p A2
L/ > 1
The use of ultrasonic or sound
wave for properties determination
• Pulse Method:
– Pulse = short duration wave (discontinuous)
• Ultrasonic: high frequency
• Transducer will produce ultrasonic wave.
Input and output data can be obtained.
• Time for sound travel through specimen
can be calculate from length by time
(L/T=velocity).
Determination of water content in
crude oil
t *  tc
volum e fraction of water =
t d  tc
t * = travel tim e in m ixture
tc = travel tim e in continuousphase(oil)
t d = travel tim e in dispersed phase( water)
Instrument
receiver
transducer
Test
cell
Pulse
generator
amplifier
oscilloscope