Page 53 - Engineering Plastics Handbook

Products and Design 27

Poisson’s ratio can be viewed, in simple terms, as directly related to the
modulus of elasticity E and inversely related to the shear modulus G:

21 + ν ) = E
(
G

The Boltzmann Superposition Principle

This superposition principle states that the response of a viscoelastic
plastic to a load is independent of any other load already applied to the
plastic. Further, strain is directly proportional to applied stress when
the strains are observed at equal time intervals. The Boltzmann super-
position principle quantifies creep strain as a function of stress and
time at a given temperature. Constitutive equations express the rela-
tionships among stress, strain, and time [12].
ε(t) = D(t −τ )σ + D(t – τ )(σ −σ ) + D(t – τ )(σ −σ n − 1 )
1
1
2
2
1
n
n
where D(t) is a compliance function of a polymer at initial stress and a
specific temperature. Long-term stress and strain, creep, and stress
relaxation at different temperatures are described by Ferry [13].
Williams-Landel-Ferry (WLF) Equation
This equation is the WLF time-temperature superposition principle.
When shear compliance versus frequency is plotted for a given fre-
quency, the curves can be superposed at different temperatures by keep-
ing one curve fixed and shifting all other curves along the frequency axis.
This superposition creates a shift factor, expressed as [14, 15]
− C T − T )
(
Log aT = 1 r
+
C (T − T )
2 r
where aT = shift factor =ηT/(ηT )
r
C = constant
1
T = temperature for plotting shear compliance versus
frequency
T = reference temperature
r
= constant
C 2
ηT = viscosity at temperature T
ηT = viscosity at reference temperature T r
r
This is a standard empirical equation of the WLF time-temperature
superposition principle. By using the shift factor to create curves along
the frequency axis, a master curve can be produced. The master curve