For the ideal elastic material, the mechanical response is described by Hooke’s law: E where σ is the applied stress, ε is the strain, and E is… [624104]

Polymer Viscoelasticity
2.12.2014

The Ideal Elastic Response
Ideal elastic response
For the ideal elastic material, the mechanical response is described by Hooke’s
law:
E
where σ is the applied stress, ε is the strain, and E is Young’s modulus.

Pure Viscous Flow
The dominant characteristic of fluids is their viscosity, which is equivalent to elasticity in solids.
According to Newton’s law, the response of a fluid to a shearing stress  is viscous flow, given by

where η is viscosity and dγ/dt is strain rate. Thus in contrast to the ideal elastic response, strain is
a linear function of time at an applied external stress.
dtd

Rubberlike Elasticity
The response of rubbery materials to mechanical stress is a slight deviation from ideal
elastic behavior.
They show non -Hookean elastic behavior. This means that although rubbers are elastic,
their elasticity is such that stress and strain are not necessarily proportional

Viscoelasticity
Viscoelastic material such as polymers combine the
characteristics of both elastic and viscous materials.
They often exhibit elements of both Hookean elastic solid
and pure viscous flow depending on the experimental time
scale.
Application of stresses of relatively long duration may
cause some flow and irrecoverable (permanent)
deformation, while a rapid shearing will induce elastic
response in some polymeric fluids.
Otherexamples of viscoelastic response include creep and
stress relaxation.



3 1t 1
t 2 1
ds
  

tEtdtdtE Et t t
tdsd s td s Td sMechanical Models for Linear Viscoelastic Response
A. Maxwell Model
Maxwell element

Creep Experiments
In creep tests, a specimen is subjected to a constant load, and the
strain is measured as a function of time.
Creep tests are made mostly in tension, but creep experiments can also be
done in shear, torsion, flexure, or compression.
0)(
tJ(t)Compliance (J) is a time -dependent reciprocal of
modulus.
It is the ratio of the time -dependent strain to the
applied constant stress

Creep Experiment


 

t
EtEt
o
1ttcreep;for element Maxwellfor equation the,E is strain initial the that noting and equation the solving10t constant
0000

Stress Relaxation Experiments
In stress relaxation experiments, the specimen is rapidly (ideally,
instantaneously) extended a given amount, and the stress required to
maintain this constant strain is measured as a function of time. The stress
that is required to maintain the strain constant decays with time. Relaxation modulus (E( t,T)) is a function of both time
and temperature.

Stress Relaxation Experiments


t
rt
EetEt tttEtEt


 





)(exp)(E time) responseor n (relaxatio where expexp0; at t E that conditions boundary the withal differenti order first theo solution t The01 10 constant is strain the since
00
00000
Relaxation modulus Relaxation time is the time required for the
stress to decay to 1/e or 37% its initial value

B. The Voigt Element
The Voigt element has the following characteristics:

The spring and the dashpot always remain parallel. This means that the strain
in each element is the same.
The total stress supported by the Voigt element is the sum of the stresses in
the spring and the dashpot.
The Voigt Element
tETd s T



Creep Experiment
Creep and creep recovery curve for the Voigt element
tE0This is a linear differential equation with solution [ integrate between (o)=0 and ()=t]






t0Et
0
e1E)t(e1E)t(


 t
0e1J)t()t(J

Stress Relaxation Experiments
In a stress relaxation experiment (ε = constant), the rheological
equation for the Voigt element reduces to;

(t)=E

This is essentially Hooke’s law. The Voigt model is not suited for
simulating a stress relaxation experiment.

The application of an instantaneous strain induces an infinite
resistance in the dashpot. It would require an infinite stress to
overcome the resistance and get the dashpot to strain
instantaneously. This is obviously unrealistic.

voigt

Reference : Introduction to Polymers , R.J. Young and P. A. Lowell (Stanley Thornes, 2nd edition)

The Four Parameter Model
(a series combination of the Maxwell and V oigt models)
Schematic of the four parameter model Under Creep;
3 2 1)t(  


3t
30
20
10exp1Et
E)t( 

In creep recovery, say, the load is removed at time
t, the deformation, σ0 /E1, due to the spring of
modulus E1 is recovered instantaneously. This
will be followed by the retarded elastic creep
recovery due to the Voigt element given by ε3 or
3exp1
30
3 it
E

The Four Parameter Model

Materials Response Time -The Deborah Number
time) on (observati scaletime al experimenttime response materialsnD
A high Deborah number that is a long response time relative to the observation
time implies viscoelasti solid behavior, whereas a low value of Deborah number
(short response time relative to the time scale of experiment) is indicative of
viscoelastic fluid behavior .

From a conceptual standpoint, the Deborah number is related to the time one
must wait to observe the onset of flow or creep .

For example, the Deborah number of a wooden beam at 30% moisture is much
smaller than that at 10% moisture content .

Relaxation and Retardation Spectra
A. Maxwell -Weichert Model (Relaxation )-The generalized
model consists of an arbitrary number of Maxwell elements in a parallel arrangement
n 1n 3 2 1 …….   
Stress relaxation

iit
i0t
0 iEeE e ti i    
i where

Relaxation and Retardation Spectra
A. Maxwell -Weichert Model (Relaxation) 1-The generalized
model consists of an arbitrary number of Maxwell elements in a parallel arrangement
Stress relaxation
For the Maxwell -Reichert model under a constant strain, ε0,

i it n
1iit n
1ii 0 eE )t(Eor eE )t( 

   
If n is large, the summation in the equation may be approximated by the integral of a continuous distribution of
relaxation times E(r).
 de)(E tEt
0

If one of the Maxwell elements in the Maxwell –Weichert model is
replaced with a spring or a dashpot of infinite viscosity, then the stress
in such a model would decay to a finite value rather than zero. This
would approximate the behavior of a cross -linked polymer .

B. Voigt -Kelvin (Creep Model) -The generalized Voigt element or
the Voigt –Kelvin model is a series arrangement of an arbitrary number of Voigt
elements
Creep
For a large value of n (i.e., n → ∞), the discrete summation
may be replaced by an integration over all the retardation
times:
If the generalized Voigt model is to represent a linear
polymer (viscoelastic liquid), then the modulus of one
of the springs must be zero. This element has infinite
compliance and represents a simple dashpot in series
with all the other Voigt elements.

B.TIME -TEMPERATURE SUPERPOSITION PRINCIPLE

The time –temperature superposition principle may be expressed mathematically for a stress relaxation
experiment as;

This means that the effect on the modulus of changing the temperature from T1 to T2 is equivalent to
multiplying the time scale by a shift factor ar which is given by;

taTEtTET r r , ,2 1
0TT
0TT
0TT
Ttta

where tT is the time required to reach a particular mechanical response (modulus in this case) at
temperature T, and tT0 is the time required to produce the same response at the reference
temperature T0.

Time –temperature superposition for polyisobutylene
Reference : Polymer Science and Technology, Robert O Ebewele (CRC press, New York,2000)

http://openlearn.open.ac.uk/mod/oucontent/view.php?id=397829

An important empirical relation correlating the shift factor with temperature changes has been
developed by Williams, Landel, and Ferry, the so -called WLF. The WLF equation, which is valid
between Tg and Tg + 100 °C, is given by the general expression

0 20 1
T 10TT CTTCa log
where T0 is the reference temperature and C1 and C2 are constants to be determined experimentally. The
temperatures are in degrees Kelvin. It is common practice to choose the glass transition temperature,Tg,
as the reference temperature. In this case, the WLF equation is given by;

gg
T 10TT6.51TT44.17a log

Example 1: For some viscoelastic polymers that are subjected to stress relaxation tests, the
stress decays with time according to

whe re σ(t) and σ(0) represent the time -dependent and initial (i.e., time = 0) stresses,
respectively, and t and τ denote elapsed time and relaxation time, respectively ; τ is a time –
dependent constant characteristic of the material .

A specimen of some viscoelastic polymer, the stress relaxation of which obeys this
equation, was suddenly pulled in tension to a measured strain of 0.69; the stress necessary
to maintain this constant strain was measured as function of time . Determine E r (10) for
this material (in MPa ) if the initial stress level was 2.77 MPa which dropped to 1.68 MPa
after 38 s.


texp0

Example 2: In a stress relaxation experiment conducted at 25 °C, it took 107 h for
the modulus of the polymer to decay to 105 N/m2. Using the WLF equation,
estimate how long will it take for the modulus to decay to the same value if the
experiment were conducted at 100 °C. Assume that 25 °C is the Tg of the polymer.