Constitutive Models Relations between Stress and Strain
Linear Viscoelastic Materials time-dependent behavior of polymers at small
polymers show complex time-dependent behavior when subjected to a history of
stress or strain. Viscoelasticity theory
was developed to approximate this behavior in polymers that are subjected to
modest strains (less than 0.5%). A
typical application might be to model the energy dissipation during cyclic
loading of a polymeric vibration damper, or to model human tissue responding to
an electric shaver.
3.6.1 Features of
the small-strain rate dependent response of polymers
The principal features of
polymers (and some biological tissue) are summarized below
1. Polymers strongly resist volume changes at all
temperatures. The bulk modulus (the
ratio of volume change to hydrostatic component of stress) is comparable to
that of metals or covalently bonded solids;
2. The shear response of a polymer is strongly
temperature dependent. At low
temperatures (the glassy regime), the shear modulus is high, and comparable to
that of metals. At a critical
temperature (the glass transition) the modulus drops. At temperatures well
above the glass transition temperature (the rubbery regime), the shear modulus
can be as low as times that of most metals.
3. At temperatures near the glass transition, the shear
modulus is strongly time (and load history) dependent this behavior is discussed in more detail
below. The time dependent shear response can be measured in two ways: (i) by
applying a step load to a sample; or (ii) by applying a harmonic (sinusoidal)
load to the specimen.
4. The time dependent modulus of polymers is also temperature
dependent. Reducing the temperature is
qualitatively equivalent to increasing the strain rate. The equivalence of temperature and strain
rate is discussed in more detail below.
5. Most amorphous polymers are isotropic their stress-strain response is independent of
dependent response to step loading
The time dependent shear response can be measured in one of two ways:
1. Take a specimen that is free of stress at time ,
apply a constant shear stress for and measuring the resulting shear strain as a function of time. The results are generally presented by
plotting the `creep compliance’ as a function of time.
Take a specimen that is
free of stress at time ,
apply a constant shear strain for and measuring the resulting shear stress as a function of time. In this case the
results are presented by plotting the Relaxation
results of such a test depend on the degree of cross-linking in the
polymer. Heavily cross-linked materials
show `retarded elastic’ behavior, while un-cross-linked materials show
steady-state creep. A detailed
description of each type of behavior follows.
Elastic Behavior (observed in strongly crosslinked polymers):
The notable features of
this behavior are:
1. There is always an instantaneous
strain in response to a step change in stress . The instantaneous compliance is low, and only weakly dependent on temperature.
2. At temperatures significantly below
the glass transition temperature the solid is essentially elastic (there may be
a very slow increase in compliance with time). At low temperatures the
compliance is low, comparable to .
3. At temperatures significantly above
the glass transition temperature, the solid is very compliant, and the
compliance is a function of temperature.
The specimen will show an initial transient response, but will quite
quickly settle to a constant strain. The
time taken to reach steady state decreases with increasing temperature, and for
some materials the transient may be short enough to be neglected. In this case the material can be modeled
using the hyperelastic constitutive law described in the preceding section.
4. For a range of temperatures both above
and below the glass transition temperature, the solid shows a slow transient
5. The deformation is reversible if the loading is removed, the specimen will
eventually return to its original configuration, although in the transition
regime this may take a very long time.
Steady-state creep behavior (observed in uncrosslinked polymers and polymer
The notable features of
this behavior are:
1. There is always an instantaneous
strain in response to a step change in stress, exactly as in crosslinked
2. At low temperatures (well below the
glass transition temperature) the solid is essentially elastic (there may be a
very slow rate of creep), and has a very low compliance, comparable to .
3. At temperatures above the glass
transition temperature, the solid is very compliant. It may show rubbery behavior for very low
stresses, but for most practical ranges of loading the compliance will increase
more-or-less linearly with time (especially for short time intervals). The rate of change of compliance is strongly
temperature dependent, as discussed below.
4. Above the glass transition
temperature, the deformation is irrreversible if the loading is removed, the specimen will
not return to its original shape.
Response to harmonic loading
In addition to measuring the response
of a material to a step change in load, one can subject it to cyclic strain,
e.g. with strains that vary sinusoidally with time ,
where Re(z) denotes the real part of a complex number z. The stress history will also be harmonic, and
could be expressed as ,
where is the stress amplitude, and is a phase shift. Both and δ depend on ω.
One can define a complex modulus as
Experimental data is usually presented by plotting the real
part of the complex modulus against the inverse of frequency, where
The variation of the modulus with frequency is illustrated in
(WLF) Time/temperature equivalence
You may have noticed that the figures showing the
variation of modulus with temperature and frequency are remarkably similar. Of
course these are just sketches, but in fact the connection between temperature
and loading rate is more than just a qualitative trend. This can be demonstrated by means of the
1. At temperature ,
subject a specimen to a step change in
shear strain and measure the relaxation modulus
2. Repeat the experiment at several progressively higher
temperatures to obtain a series of relaxation modulus
3. Plot -v- for the raw data. The results will look like a
complicated mess something like the picture on the right.
4. However, you will find that if you simply shift the
modulus curves for the higher temperatures to the right, you can make the data
collapse onto a single master-curve, as shown.
5. This observation can be expressed mathematically as where the function f represents the master curve, and represents the horizontal shift from
temperature T1 to T2. is known as the
WLF shift function.
6. If you measure at a series of
temperatures, and plot as a function of temperature T, you will find that the data can be
well approximated by a function of the form .
The scaling holds for any two temperatures, but of course and must depend on the choice of .
In practice it is convenient (and conventional) to use the glass transition
temperature as the reference temperature.
The scaling law can then be written as
The values of and vary slightly (but surprisingly little) from
one polymer to another: typical ranges are and .
The expression works (again surprisingly) for both above and below - but of course the expression blows up if . For temperatures below this critical value,
the material is perfectly elastic (with constant elastic moduli).
7. Note that because ,
the constants ,
are related by ,
. This means that if you measure a time
dependent modulus at temperature ,
and know the values of ,
for the material, you can immediately
calculate the modulus at any other temperature as ,
3.6.2 General constitutive equations for linear viscoelastic solids
The general stress-strain law for a linear
viscoelastic solid is constructed as follows:
Assume that the material experiences small
shape changes and rotations. The deformation
can then be characterized using the infinitesimal strain tensor defined in Section 2.1.7.
For small strains, all stress measures are
equal. We can use the Cauchy stress as the stress measure.
Assume that for time t<0, the solid is stress free, and .
For small strains/stresses, we can assume that
the stress and strain are related through linear equations. (This doesn’t mean that stress is
proportional to strain, of course instead stress, strain and their rates are
related by a time dependent linear ODE, as discussed below)
Assume that the material is isotropic.
In most practical applications we can assume
that material response to a pure volumetric strain ( with all other ) is perfectly elastic (with no time dependent
behavior). The volumetric strain will
induce a state of hydrostatic tension with all other . The stress is related to the strain by where K
is the bulk modulus.
Viscoelastic response most commonly
characterized by the shear relaxation modulus measured at some reference temperature . (Recall that the shear relaxation modulus can
be measured by subjecting a specimen to a step increase in shear strain ,
and measuring the resulting shear stress . The relaxation modulus follows as )
The temperature dependence of the modulus is
characterized by the WLF constants ,
and the glass transition temperature ,
through the WLF shift function defined in the preceding section.
Since the stress is linearly related to
strain, the stress history resulting from an arbitrary strain history can be computed by appropriately superposing
the step response. The result is
Here, the temperature T is assumed to be constant up to time t.
It is not hard to extend the formula to account for time varying
temperatures but the result looks messy and is difficult to visualize.
To apply this stress-strain relation in practice, it is
necessary to find a convenient way to fit the relaxation modulus . Various approaches to doing this are
described in the next two sections.
3.6.3 Spring-Damper approximations to the relaxation modulus
Spring-damper models are
often used as a simple, approximate model of the behavior of a viscoelastic
solid. The figures illustrate the
general idea: in each case the force
applied to the spring-dashpot system represents shear stress, while the
extension represents shear strain. It is
straightforward to show that they are related by
For a material with time independent bulk modulus K, these can be generalized to
multi-axial loading as
Qualitatively, these models describe the behavior of a
typical polymer. The Kelvin-Voigt model
gives retarded elastic behavior, and represents a crosslinked polymer. The Maxwell model gives steady state creep,
and would represent an uncrosslinked polymer.
With an appropriate choice of and ,
the 3 parameter model can describe both types of behavior.
For hand calculations it is often more convenient to
use the differential equations relating stress to strain than the integral
integral form given in the preceding section.
However, it is straigthforward to calculate the relaxation modulus for
the Maxwell and 3 parameter models
Kelvin-Voigt model does not have a well defined relaxation modulus.
3.6.4 Prony series representation for
the relaxation modulus
The models described in the preceding section are too
simple to give a good quantitative fit to any polymer over an extended period
of time. We can make a more versatile model
by connecting a bunch of Maxwell elements in series, and adding a spring in parallel
with the whole array. The relaxation
modulus for this material has the form
where is the steady-state stiffness (represented by the
parallel spring), and are the time constants and stiffnesses of the
Maxwell elements. These parameters are
used directly as the properties of the material. The sum of exponentials is
known as the `Prony series.’
3.6.5 Calibrating the constitutive laws
for linear viscoelastic solids
data for the time dependent behavior of polymers can be presented in several
1. The Young’s modulus or shear modulus as a function of time t,at various temperatures
2. The tensile compliance or shear compliance as a function of time, at various
3. The complex modulus ,
or, more usually, just the real part of the complex modulus as a function of frequency and temperature.
4. The complex compliance or the real part of the complex compliance as a function of frequency and temperature.
The material parameters must be fit to this data. For each data set, the first step is to
combine data from tests at various temperatures into a master-curve of ,
or at a single
reference temperature, using the WLF scaling procedure described in 3.6.1. The parameters should then be chosen to give
the best fit to this master curve. (A simple way to fit the parameters is to
choose to be spaced at exponentially increasing time
intervals, and then choose to minimize the square of the difference
between the predicted and measured values ).
To do the fit, it is helpful to find formulas for or in terms of material properties. It is
straightforward to show that
It is slightly more cumbersome to fit the Prony series
parameters to compliance measurements.
The compliances can be expressed in terms of as follows
where denotes an inverse Laplace
transform (which can be calculated using a symbolic manipulation program), and were defined above.
If you are given experimental data for Young’s modulus
or tensile compliance ,
you will need to estimate or . Precise values can’t be found without knowing
the bulk modulus or Poisson’s ratio of the material, but for most practical
applications you can assume that the bulk modulus is very large, in which case and .
3.6.6 Representative values for
viscoelastic properties of polymers
The properties of polymers are very sensitive to their
molecular structure, so for accurate predictions you will need to obtain data
for the particular material you intend to use.
As a rough guide to typical values, data for the Young’s modulus of
polyisobutylene (from McCrum, Buckley, Bucknall “Principles of polymer
engineering” OUP New York 1988, p. 112) is shown in the picture on the
right. They give the glass transition
temperature for this material as .
The master-curve of and the WLF shift function can be deduced from their data, following the
procedure discussed in Section 3.5.1.
The results are plotted below. The resulting WLF parameters, together with
moduli and time constants for a 7-term Prony series fit to the data are listed
in the table. The shear modulus
predicted by the Prony series is shown on the modulus-v-time plot for
comparison with the experimental data.
properties of polyisobytylene