Chapter 3

Constitutive Models $–$ Relations between Stress and Strain

3.6 Linear Viscoelastic Materials $–$ time-dependent behavior of polymers at small strains

Amorphous 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 ${10}^{-5}$ 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 material orientation.

The time dependent shear response can be measured in one of two ways:

1.      Take a specimen that is free of stress at time $t=0$, apply a constant shear stress $\Delta \tau$ for $t>0$ and measuring the resulting shear strain $\epsilon$ as a function of time.  The results are generally presented by plotting the creep compliance’ $J\left(t\right)=\epsilon \left(t\right)/\Delta \tau$ as a function of time.

2.      Take a specimen that is free of stress at time $t=0$, apply a constant shear strain $\epsilon$ for $t>0$ and measuring the resulting shear stress $\tau \left(t\right)$ as a function of time. In this case the results are presented by plotting the Relaxation Modulus: $G\left(t\right)=\tau \left(t\right)/\Delta \epsilon$

The 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.

Retarded Elastic Behavior (observed in strongly crosslinked polymers):

The notable features of this behavior are:

1.      There is always an instantaneous strain $\Delta \epsilon$ in response to a step change in stress $\Delta \tau$.  The instantaneous compliance ${J}_{g}=\Delta \epsilon /\Delta \tau$ 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 ${J}_{g}$.

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 response.

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.

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 polymers.

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 ${J}_{g}$.

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.

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 $\epsilon \left(t\right)={\epsilon }_{0}\mathrm{cos}\left(\omega t\right)={\epsilon }_{0}\mathrm{Re}\left[\mathrm{exp}\left(i\omega t\right)\right]$, where Re(z) denotes the real part of a complex number z.  The stress history will also be harmonic, and could be expressed as $\tau ={\tau }_{0}{\epsilon }_{0}\mathrm{Re}\left[\mathrm{exp}\left(i\omega t\right)\mathrm{exp}\left(i\delta \right)\right]$, where ${\tau }_{0}$ is the stress amplitude, and $\delta$ is a phase shift. Both ${\tau }_{0}$ and δ depend on ω.  One can define a complex modulus as

${G}^{*}\left(\omega ,T\right)={\tau }_{0}\mathrm{exp}\left(i\delta \right)/{\epsilon }_{0}$

Experimental data is usually presented by plotting the real part $G\text{'}\left(\omega ,T\right)$ of the complex modulus ${G}^{*}\left(\omega ,T\right)$ against the inverse of frequency, where

${G}^{*}=G\text{'}+iG\text{'}\text{'}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}G\text{'}={\tau }_{0}\mathrm{cos}\left(\delta \right)/{\epsilon }_{0}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}G\text{'}\text{'}={\tau }_{0}\mathrm{sin}\left(\delta \right)/{\epsilon }_{0}$

The variation of the modulus with frequency is illustrated in the picture.

Williams-Landell-Ferry (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 following experiment:

1.      At temperature ${T}_{1}$, subject a specimen to a step change  in shear strain $\Delta \epsilon$ and measure the relaxation modulus $G\left(t,{T}_{1}\right)=\tau \left(t\right)/\left(2\Delta \epsilon \right)$

2.      Repeat the experiment at several progressively higher temperatures ${T}_{2},{T}_{3}...{T}_{n}$ to obtain a series of relaxation modulus curves $G\left(t,{T}_{2}\right),G\left(t,{T}_{3}\right)...$

3.      Plot $\mathrm{log}\left(G\left(t\right)\right)$ -v- $\mathrm{log}\left(t\right)$ 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 $\mathrm{log}\left(G\right)=f\left\{\mathrm{log}\left(t\right)+\mathrm{log}\left[A\left(T;{T}_{1}\right)\right]\right\}$ where the function f represents the master curve, and $\mathrm{log}A\left(T;{T}_{1}\right)$ represents the horizontal shift from temperature T1 to T2$A\left(T;{T}_{1}\right)$ is known as the WLF shift function.

6.      If you measure $A\left(T;{T}_{1}\right)$ at a series of temperatures, and plot $\mathrm{log}\left(A\left(T;{T}_{1}\right)\right)$ as a function of temperature T, you will find that the data can be well approximated by a function of the form $\mathrm{log}\left[A\left(T;{T}_{1}\right)\right]=-{C}_{1}\left(T-{T}_{1}\right)/\left\{{C}_{2}+\left(T-{T}_{1}\right)\right\}$. The scaling holds for any two temperatures, but of course ${C}_{1}$ and ${C}_{2}$ must depend on the choice of ${T}_{1}$. 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

$A\left(T;{T}_{g}\right)=\mathrm{exp}\left\{-\frac{{C}_{1}^{g}\left(T-{T}_{g}\right)}{{C}_{2}^{g}+\left(T-{T}_{g}\right)}\right\}$

The values of ${C}_{1}^{g}$ and ${C}_{2}^{g}$ vary slightly (but surprisingly little) from one polymer to another: typical ranges are  ${C}_{1}^{g}\approx 10-40$ and ${C}_{2}^{g}\approx 50\text{\hspace{0.17em}}-100\text{\hspace{0.17em}}\text{Kelvin}$. The expression works (again surprisingly) for $T$ both above and below ${T}_{g}$ - but of course the expression blows up if $T<{T}_{g}-{C}_{2}^{g}$.  For temperatures below this critical value, the material is perfectly elastic (with constant elastic moduli).

7.      Note that because $A\left(T;{T}_{1}\right)=A\left({T}_{g};{T}_{1}\right)A\left(T;{T}_{g}\right)$, the constants ${C}_{1}$, ${C}_{2}$, ${T}_{1}$ and ${C}_{1}^{g}$, ${C}_{2}^{g}$, ${T}_{g}$ are related by ${C}_{1}={C}_{1}^{g}{C}_{2}^{g}/\left({C}_{2}^{g}+{T}_{1}-{T}_{g}\right)$, ${C}_{2}={C}_{2}^{g}+{T}_{1}-{T}_{g}$.  This means that if you measure a time dependent modulus $G\left(t,{T}_{1}\right)$ at temperature ${T}_{1}$, and know the values of ${C}_{1}^{g}$, ${C}_{2}^{g}$, ${T}_{g}$ for the material, you can immediately calculate the modulus at any other temperature as $G\left(t,T\right)=G\left(A\left(T;{T}_{1}\right)t,{T}_{1}\right)$, where

$A\left(T;{T}_{1}\right)=\mathrm{exp}\left(\frac{-{C}_{1}^{g}{C}_{2}^{g}\left(T-{T}_{1}\right)}{\left[{C}_{2}^{g}+{T}_{1}-{T}_{g}\right]\left[{C}_{2}^{g}+T-{T}_{g}\right]}\right)$

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 ${\epsilon }_{ij}=\left(\partial {u}_{i}/\partial {x}_{j}+\partial {u}_{j}/\partial {x}_{i}\right)/2$ defined in Section 2.1.7.

For small strains, all stress measures are equal.  We can use the Cauchy stress ${\sigma }_{ij}$ as the stress measure.

Assume that for time t<0, the solid is stress free, and ${\epsilon }_{ij}=0$.

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 ( ${\epsilon }_{11}={\epsilon }_{22}={\epsilon }_{33}=\left(V-{V}_{0}\right)/\left(3V\right)$ with all other ${\epsilon }_{ij}=0$ ) is perfectly elastic (with no time dependent behavior).  The volumetric strain will induce a state of hydrostatic tension ${\sigma }_{11}={\sigma }_{22}={\sigma }_{33}=\sigma$ with all other ${\sigma }_{ij}=0$.  The stress is related to the strain by $\sigma =3K\delta V/V$ where K is the bulk modulus.

Viscoelastic response most commonly characterized by the shear relaxation modulus $G\left(t,{T}_{1}\right)$ measured at some reference temperature ${T}_{1}$.  (Recall that the shear relaxation modulus can be measured by subjecting a specimen to a step increase in shear strain $\Delta \epsilon$, and measuring the resulting shear stress $\tau \left(t\right)$.  The relaxation modulus follows as  $G\left(t,{T}_{1}\right)=\tau \left(t\right)/\left(2\Delta \epsilon \right)$ )

The temperature dependence of the modulus is characterized by the WLF constants ${C}_{1}^{g}$, ${C}_{2}^{g}$ and the glass transition temperature ${T}_{g}$, through the WLF shift function defined in the preceding section.

Since the stress is linearly related to strain, the stress history ${\sigma }_{ij}\left(t\right)$ resulting from an arbitrary strain history ${\epsilon }_{ij}\left(t\right)$ can be computed by appropriately superposing the step response.  The result is

${\sigma }_{ij}\left(t\right)=\underset{0}{\overset{t}{\int }}2G\left(A\left(T;{T}_{1}\right)\left(t-\xi \right),{T}_{1}\right)\left[{\stackrel{˙}{\epsilon }}_{ij}\left(\xi \right)-{\stackrel{˙}{\epsilon }}_{kk}\left(\xi \right){\delta }_{ij}\right]d\xi +K{\epsilon }_{kk}{\delta }_{ij}$

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 $G\left(t,{T}_{1}\right)$.  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

$\begin{array}{l}k\sigma +\eta \frac{d\sigma }{dt}=k\eta \frac{d\epsilon }{dt}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{Maxwell}\\ \sigma =k\epsilon +\eta \frac{d\epsilon }{dt}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{Kelvin-Voigt}\\ {k}_{1}\sigma +\eta \frac{d\sigma }{dt}={k}_{1}{k}_{2}\epsilon +\left({k}_{1}+{k}_{2}\right)\eta \frac{d\epsilon }{dt}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}3\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{Parameter}\end{array}$

For a material with time independent bulk modulus K, these can be generalized to multi-axial loading as

$\begin{array}{l}{e}_{ij}={\epsilon }_{ij}-{\epsilon }_{kk}{\delta }_{ij}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\sigma }_{ij}={S}_{ij}+K{\epsilon }_{kk}{\delta }_{ij}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\\ k{S}_{ij}+\eta \frac{d{S}_{ij}}{dt}=k\eta \frac{d{e}_{ij}}{dt}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{Maxwell}\\ {S}_{ij}=k{e}_{ij}+\eta \frac{d{e}_{ij}}{dt}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{Kelvin-Voigt}\\ {k}_{1}{S}_{ij}+\eta \frac{d{S}_{ij}}{dt}={k}_{1}{k}_{2}{e}_{ij}+\left({k}_{1}+{k}_{2}\right)\eta \frac{d{e}_{ij}}{dt}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}3\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{Parameter}\end{array}$

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 ${k}_{1}$ and ${k}_{2}$, 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

$\begin{array}{l}G\left(t\right)=k{e}^{-kt/\eta }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{Maxwell}\\ G\left(t\right)={k}_{2}+{k}_{1}{e}^{-{k}_{1}t/\eta }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}3\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{Parameter}\end{array}$

The 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

$G\left(t\right)={G}_{\infty }+\sum _{i=1}^{N}{G}_{i}{e}^{-t/{t}_{i}}$

where ${G}_{\infty }$ is the steady-state stiffness (represented by the parallel spring), and ${G}_{i},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{t}_{i}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=1...N$ 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

Experimental data for the time dependent behavior of polymers can be presented in several different ways:

1.      The Young’s modulus $E\left(t,{T}_{i}\right)$ or shear modulus $G\left(t,{T}_{i}\right)$ as a function of time t,at various temperatures ${T}_{i}$

2.      The tensile compliance $C\left(t,{T}_{i}\right)$ or shear compliance $J\left(t,{T}_{i}\right)$ as a function of time, at various temperatures;

3.      The complex modulus $G*\left(\omega ,{T}_{i}\right)=G\text{'}\left(\omega ,{T}_{i}\right)+iG\text{'}\text{'}\left(\omega ,{T}_{i}\right)$, or, more usually, just the real part of the complex modulus $G\text{'}\left(\omega ,{T}_{i}\right)$ as a function of frequency $\omega$ and temperature.

4.      The complex compliance $J*\left(\omega ,T\right)$ or the real part of the complex compliance $J\text{'}\left(\omega ,T\right)$ as a function of frequency and temperature.

The material parameters ${G}_{\infty },{G}_{i},{t}_{i}$ 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 $G\left(t,T\right)$, $G\text{'}\left(\omega ,T\right)$ or $J\left(t,T\right)$$J\text{'}\left(\omega ,T\right)$ 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 ${t}_{i}$ to be spaced at exponentially increasing time intervals, and then choose ${G}_{\infty },{G}_{i}$  to minimize the square of the difference between the predicted and measured values $\mathrm{log}\left(G\left(t\right)\right)$ ).

To do the fit, it is helpful to find formulas for $G\left(t,T\right)$ or $G\text{'}\left(\omega ,T\right)$ in terms of material properties. It is straightforward to show that

$\begin{array}{l}G\left(t,T\right)={G}_{\infty }+\sum _{i=1}^{N}{G}_{i}\mathrm{exp}\left(-t/{t}_{i}\right)\\ G\text{'}\left(\omega ,T\right)={G}_{\infty }+\sum _{i=1}^{N}\frac{{G}_{i}{\omega }^{2}{t}_{i}^{2}}{1+{\omega }^{2}{t}_{i}^{2}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}G\text{'}\text{'}\left(\omega ,T\right)=\sum _{i=1}^{N}\frac{{G}_{i}\omega {t}_{i}}{1+{\omega }^{2}{t}_{i}^{2}}\end{array}$

It is slightly more cumbersome to fit the Prony series parameters to compliance measurements.  The compliances can be expressed in terms of  ${G}_{\infty },{G}_{i},{t}_{i}$ as follows

$J\left(t,T\right)={L}^{-1}\left\{\frac{1}{{s}^{2}}{\left(\frac{{G}_{\infty }}{s}+\sum _{i=1}^{N}\frac{{t}_{i}{G}_{i}}{{t}_{i}s+1}\right)}^{-1}\right\}$

$J\text{'}\left(\omega ,T\right)=\frac{G\text{'}\left(\omega ,T\right)}{G\text{'}{\left(\omega ,T\right)}^{2}+G\text{'}\text{'}{\left(\omega ,T\right)}^{2}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}J\text{'}\text{'}\left(\omega ,T\right)=\frac{G\text{'}\text{'}\left(\omega ,T\right)}{G\text{'}{\left(\omega ,T\right)}^{2}+G\text{'}\text{'}{\left(\omega ,T\right)}^{2}}$

where ${L}^{-1}$ denotes an inverse Laplace transform (which can be calculated using a symbolic manipulation program), and $G\text{'},G\text{'}\text{'}$ were defined above.

If you are given experimental data for Young’s modulus $E\left(t\right)$ or tensile compliance $C\left(t\right)$, you will need to estimate $G\left(t\right)$ or $J\left(t\right)$.  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 $G\left(t\right)=E\left(t\right)/3$ and $J\left(t\right)=3C\left(t\right)$.

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 ${T}_{g}=193K$.

The master-curve of $G\left(t,{T}_{g}\right)$ and the WLF shift function $A\left(T,{T}_{g}\right)$ 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.

 Viscoelastic properties of polyisobytylene WLF parameters: