110
4
viscoelastic constants. Most of the examples given here
involves creep phenomena but can be applied to stress
relaxation.
Several models have been developed to describe the viscoelastic nature of polymers. The models consist of Hookean
springs and viscous Newtonian dashpots. A Hookean spring
elongates instantaneously to an equilibrium elongation ε, which
is directly proportional to the applied stress σ (i.e. ε = σ/E where
E is Young’s modulus) and instantaneously recovers when the
stress is removed. A dashpot is similar to a shock absorber in
that an equilibrium elongation is never reached and it does not
recover elongation already incurred. Mathematically, the
applied stress σ is equal to the rate of elongation through a proportionally constant η known as the viscosity effect of an applied
and removed stress as a function of time, i.e.
ε σ η
= ( ) t
(4.34)
There are three relatively simple models, viz. Maxwell, Kelvin–
Voigt, and the four-parameter. The creep behaviour of the
Maxwell model adequately represents the instantaneous elongation or recovery and permanent deformation due to the stress in
a polymer. However, Maxwell model does not characterize the
exponential decrease in the elongation rate or the recovery
demonstrated by most polymers. The Kelvin–Voigt model is
deficient in near instantaneous elongation or recovery. Also, this
model reaches an equilibrium elongation that is not indicative
of a true polymer and shows upon the permanent set.
The best features of both the Maxwell and Kelvin–Voigt models are combined into the four-parameter model. The creep
response of this model adequately matches the creep behaviour of
the distribution in molecular weight and other complex morphological structures associated with polymers. Thus, several models
consisting of many Maxwell or Kelvin–Voigt elements with a distribution of relaxation times have been introduced, although the
equation is complicated, as shown for a Kelvin–Voigt series:
J t
L
t
d
a
( ) = ( ) −
−
(
)
∫
0
1
1
λ
λ
λ
exp
/
(4.35)
where J is the compliance and L (λ) is the continuous relaxation
time distribution.
2. Dynamic (Oscillatory) Shear Rheology—Dynamic measurements are very useful in terms of understanding the structure
of delicate materials over a short and medium period of time.
Dynamic measurements deal with the state of the material due
to the quiescent structure at small deformations. These
measurements provide valuable information regarding the
microstructure of the sample(s) under investigation as well as
their processability. Dynamic melt rheology usually involves
imposing a small amplitude sinusoidal strain equation as
Chapter 4 · Rheology in Processing of Polymeric Composites
4
viscoelastic constants. Most of the examples given here
involves creep phenomena but can be applied to stress
relaxation.
Several models have been developed to describe the viscoelastic nature of polymers. The models consist of Hookean
springs and viscous Newtonian dashpots. A Hookean spring
elongates instantaneously to an equilibrium elongation ε, which
is directly proportional to the applied stress σ (i.e. ε = σ/E where
E is Young’s modulus) and instantaneously recovers when the
stress is removed. A dashpot is similar to a shock absorber in
that an equilibrium elongation is never reached and it does not
recover elongation already incurred. Mathematically, the
applied stress σ is equal to the rate of elongation through a proportionally constant η known as the viscosity effect of an applied
and removed stress as a function of time, i.e.
ε σ η
= ( ) t
(4.34)
There are three relatively simple models, viz. Maxwell, Kelvin–
Voigt, and the four-parameter. The creep behaviour of the
Maxwell model adequately represents the instantaneous elongation or recovery and permanent deformation due to the stress in
a polymer. However, Maxwell model does not characterize the
exponential decrease in the elongation rate or the recovery
demonstrated by most polymers. The Kelvin–Voigt model is
deficient in near instantaneous elongation or recovery. Also, this
model reaches an equilibrium elongation that is not indicative
of a true polymer and shows upon the permanent set.
The best features of both the Maxwell and Kelvin–Voigt models are combined into the four-parameter model. The creep
response of this model adequately matches the creep behaviour of
the distribution in molecular weight and other complex morphological structures associated with polymers. Thus, several models
consisting of many Maxwell or Kelvin–Voigt elements with a distribution of relaxation times have been introduced, although the
equation is complicated, as shown for a Kelvin–Voigt series:
J t
L
t
d
a
( ) = ( ) −
−
(
)
∫
0
1
1
λ
λ
λ
exp
/
(4.35)
where J is the compliance and L (λ) is the continuous relaxation
time distribution.
2. Dynamic (Oscillatory) Shear Rheology—Dynamic measurements are very useful in terms of understanding the structure
of delicate materials over a short and medium period of time.
Dynamic measurements deal with the state of the material due
to the quiescent structure at small deformations. These
measurements provide valuable information regarding the
microstructure of the sample(s) under investigation as well as
their processability. Dynamic melt rheology usually involves
imposing a small amplitude sinusoidal strain equation as
Chapter 4 · Rheology in Processing of Polymeric Composites