Dynamo-Mechanical Analysis of
Materials (Polymers)
Instructor: Ioan I. Negulescu
CHEM 4010
Tuesday,
October 29, 2002
Viscoelasticity
According to rheology (the science of
flow), viscous flow and elasticity are only
two extreme forms of the possible types of
behavior of matter. It is appropriate to
consider the entropic-elastic (or rubberelastic), viscoelastic, and plastic bodies as
other special cases.
All materials have viscoelasticity, which is a
combination of viscosity and elasticity in
varying amounts. When this viscoelasticity
is measured dynamically, there is a phase
shift between the force applied (stress) and
the deformation (strain) which occurs in
response.
The tensile stress  and the deformation
(strain)  are related via the elasticity
modulus E as follows:
=E
Generally
the
measurements
are
represented as a complex modulus E* to
insure an accurate expression:
E* = E’ + iE”
where: i2 = -1
In dynamic mechanical analysis, DMA, a
sinusoidal strain or stress is applied to a
sample and the response is monitored as a
function of frequency.
Polymers are not ideal energy elastic
bodies; they are viscoelastic materials. In
such cases the deformation (strain) lags
behind the applied stress.
Schematic representation of the stress  as a function
of time t with dynamic (sinusoidal) loading (strain).
COMPLEX MODULUS:
E*=E’ + iE”
I E* I = Peak Stress / Peak Strain
o

STRESS
STRAIN
o
0
/ 
2 / 
STORAGE ( Elastic) MODULUS
I E' I = I E* I cos

t
LOSS MODULUS
I E" I = I E* I sin 
Parallel-plate geometry for shearing of viscous
materials (DSR instrument).
The “E”s above (Young modulus) can all be
replaced with “G”s (rigidity modulus).
Therefore one may write an equation similar
to that for E*:
G* = G’ + iG"
where the shearing stress  and the
deformation (strain)  are related via the
rigidity modulus G as follows:
=G
Definition of elastic and viscous materials under shear.
In analyzing the polymeric materials, G*, the
ratio of the peak stress to the peak strain,
reflects the total stiffness. The in-phase
component of IG*I, i. e. the shear storage
modulus G', represents the part of the input
energy which is not lost to heat (the elastic
portion).
The out-of-phase component, i. e. the shear
loss modulus G" represents viscous
component of G*, viz., it reflects the loss of
useful
mechanical
energy
through
dissipation as heat.
The complex dynamic shear viscosity *
can be obtained from G* divided by the
frequency, while the dynamic viscosity is
 = G"/ or  = G"/2f
For purely elastic materials, the phase angle
will be zero, whereas for purely viscous
materials, the phase angle will be 90. Thus,
the phase angle, expressed as tangent, is an
important parameter for describing the
viscoelastic properties of a material. The loss
tangent is calculated simply as the tangent of
the phase angle, or alternatively, as the ratio of
the loss, to storage moduli: tan  = G"/ G',
because G" = G*sin and G' = G*cos.
There is a general tendency in dynamic
viscoelasticity measurements for the
detected transition temperature to shift when
the measurement frequency is changed.
This phenomenon is based on a principle
called the time-temperature superposition
principle, where the transition temperature
(for example the top peak temperature of
tan) tends to rise when the measurement
frequency is increased.
Dynamic mechanical analysis of a viscous polymer
solution. Dependence of tan  on frequency.
• DMA techniques are very sensitive to
temperature changes.
• Secondary transitions, observed with
difficulty by DSC or DTA techniques, are
clearly evidenced by DMA.
• Any thermal transition in polymers will
generate a peak for tan and E" or G".
• The peak maxima for G" (or E") and tan
do not occur at the same temperature.
Dynamic mechanical analysis of recyclable HDPE.
Dependence of tan  on frequency. The  transition is
seen at 62C and  transition occurs at -117C.
Dependence of G", G' and of their ratio (tan) on frequency
for a sample of HDPE analyzed at constant temperature
(180C).
Data obtained with 2C/min showing the glass transition at
about -40C (read as tan or E" maxima) and a false
transition at 15.5C due to the nonlinear increase of
temperature versus time.
1.0
E'
m
Te
80
E"
tan
0.6
40
tan
0.4
0.0
15.5oC
tan
-40
E'
E"
Temperature
-80
0.2
0.0
-10
0
10
20
30
40
50
60
Time, min
70
80
90
100
110
Temperature, oC
Continental Carbon
Sample A-97058
0.8
re
tu
a
r
pe
Dynamo-mechanical analysis of low crystallinity poly(lactic
acid): Dependence of tan upon the temperature and
frequency for the 1st heating run
Tg
0.8
o
62 C
0.6
o
69 C
Crystallization
tan 
o
75 C
Crystallization
0.4
1.0 Hz
50 Hz
Tg
o
66 C
0.2
0.0
PLALC
30
40
50
60
70
80
o
Temperature, C
90
100
Dynamo-mechanical analysis of the Low Crystallinity
Poly(lactic acid). Dependence of the storage modulus upon
the thermal history.
2G
Storage Modulus (Pa)
st
E' @ 10 Hz (1 h)
nd
E' @ 10 Hz (2 h)
st
1 heating
Crystallization
(Stiffening)
1G
Glass
Transition
Tg
T CR
2
70
80
nd
heating
10 Hz
PLALC
0
40
50
60
o
Temperature, C
90
100