Chapter 13: Viscoelasticity

Linear Viscoelasticity (LVE):

I have recorded a series of lectures videos to supplement this text which can be found in the playlist below:

https://www.youtube.com/playlist?lis...b5i6HX2SGPPNyA

So far we have dealt with continuum isotropic linear elasticity and anisotropic linear elasticity. All these modes of deformation are independent of time, rate, and temperature. However, today we will discuss linear viscoelasticity whose deformation is governed by time, rate, and temperature. Linear viscoelasticity is the deformation between that of an elastic solid and a viscous fluid. We can actually see this in the name linear viscoelasticity. There is still a linear relationship between stress and strain for a given time and temperature. The visco term denotes a time component and elasticity denotes reversibility. The elastic solid will again behave Hookean with the relationship

$$\begin{equation} \sigma = E \epsilon \end{equation}$$

and the viscous fluid will be governed by this new equation

$$\begin{equation} \sigma = \eta \dot{\epsilon} \end{equation}$$

Linear Viscoelasticity is typically used to describe the mechanical response for polymers, glasses, tissues, and cells.

When a viscous fluid is deformed under an applied shear stress there is no recovery when the shear stress is removed so our equation for shear stress becomes

$$\begin{equation} \tau = \eta \dot{\gamma} \end{equation}$$

where $\eta$ is the fluid viscosity (units of Pa s) and $\dot{\gamma}$ is the shear strain rate (units of inverse seconds). Again this is linear because there is a proportional increase in $\sigma$ and $\epsilon$ for some time or temperature.

We can make a couple of linear viscoelastic (LVE) models which describe some LVE behavior phenomonolgically as combinations of springs and dashpots.

Maxwell Model:

Let’s take a look at the Maxwell model which is a combination of a spring in series with a dashpot that contains a Newtonian fluid.

The spring is described by the following equation:

$$\begin{equation} \sigma_{s} = E_{s}\epsilon_{s} \end{equation}$$

where $\sigma_s$ is the stress in the spring, $E_s$ is the stiffness of the spring, and $\epsilon_s$ is the strain in the spring

The dashpot is described by:

$$\begin{equation} \sigma_{d} = \eta_{d} \dot{\epsilon_{d}} \end{equation}$$

where $\sigma_d$ is the stress in the dashpot, $\eta_d$ is the viscosity of the dashpot, and $\dot{\epsilon_{d}}$ is the strain rate of the dashpot.

Now let’s say that I stress the system (this should be reminiscent of the composites pulled transverse to the fiber direction when we get to this) the stresses should be the same for the spring and the dashpot however the strains in the dashpot and the spring will be different which gives us the following equation:

$$\begin{eqnarray} \sigma_{M} = \sigma_{s}=\sigma_{d}\\ \epsilon_{M} = \epsilon_{s} + \epsilon_{d}\\ \dot{\epsilon_{M}} = \dot{\epsilon_{s}} + \dot{\epsilon_{d}}\\ \dot{\epsilon_{M}} = \frac{\dot{\sigma_{M}}}{E} + \frac{\sigma_{M}}{\eta} \end{eqnarray}$$

With this system we can do two types of experiments: stress relaxation ( $\epsilon = \epsilon_0$ i.e. constant strain) and creep ($\sigma= \sigma_0$, i.e. constant stress).

For stress relaxation we can plug into our equation and solve for how the stress in our Maxwell model should vary over time:

$$\begin{eqnarray} 0=\frac{1}{E}\frac{d\sigma}{dt}+\frac{\sigma}{\eta}\\ \int^{\sigma}_{\sigma_{0}}\frac{d\sigma}{\sigma} = \int^{t}_{0}\frac{-E}{\eta}dt\\ \sigma(t) =\sigma_{0}\exp\bigg(\frac{-Et}{\eta}\bigg)\\ \sigma(t) =\sigma_{0}\exp\bigg(\frac{-t}{\tau}\bigg) \end{eqnarray}$$

where $\tau = \frac{E}{\eta}$ is the relaxation time. Similarly for creep:

$$\begin{eqnarray} \frac{d\epsilon}{dt} = 0 + \frac{\sigma_{0}}{\eta}\\ \int^{\epsilon}_{\epsilon_{0}}d\epsilon = \int^{t}_{0}\frac{\sigma_{0}}{\eta}dt\\ \epsilon(t) = \epsilon_{0} + \frac{\sigma_{0}t}{\eta} \end{eqnarray}$$

Kelvin-Voigt Model:

There is also the Kelvin-Voigt (KV) model has the spring and dashpot series in parallel.

In this case now the strain is the same but now the stresses are different so we get:

$$\begin{eqnarray} \epsilon_{KV} = \epsilon_{s} = \epsilon_{d}\\ \sigma_{KV} = \sigma_{s} + \sigma_{d}\\ \sigma_{KV} = E_{s}\epsilon_{s}+\eta_{d}\dot{\epsilon_{d}}\\ \dot{\epsilon_{KV}} = \frac{\sigma_{KV}}{\eta_{d}} - \frac{E_{s}}{\eta_{d}}\epsilon_{KV} \end{eqnarray}$$

Now for stress relaxation we get the final equation for stress vs time as:

$$\begin{equation} \sigma(t) = E\epsilon_{0} \end{equation}$$

and for creep we get:

$$\begin{equation} \epsilon(t) = \frac{\sigma_{0}}{E}\bigg(1-\exp[-\frac{t}{\tau}]\bigg) \end{equation}$$

You can now build even more complex combinations to model viscoelastic properties like the Standard Linear Viscoelastic Solid Model. Perhaps a more useful experimental measurement for visoelastic behavior is accomplished by Dynamic Mechanical Testing.

Dynamic Mechanical Testing:

The more utilized characterization technique is typically some type of dynamic mechanical testing to probe the viscoelastic behavior of materials. In this experiment the polymer is subjected of a sinusoidal loading at variable frequencies which can be described as such as
$$\begin{equation} \sigma_{applied} = \sigma_o \sin \omega t \end{equation}$$

This is the applied stress but remember we are dealing with a viscoelastic material so there will be a phase lag, $\delta$, in the strain behavior which will also be sinusoidal

$$\begin{equation} \epsilon = \epsilon_o \sin \omega t \end{equation}$$

So what we will end up with is that the material will actually experience a stress that contains the phase lag as shown below

$$\begin{eqnarray} \sigma = \sigma_o \sin (\omega t + \delta)\\ \sigma = \sigma_o \sin \omega t \cos \delta + \sigma_o \cos \omega t \sin \delta \end{eqnarray}$$

where in the equation above we just expanded our trig functions. Notice here that the first term represents the component that is in phase with the strain or the elastic response while the second term represents the out of phase behavior or the viscous response. We can then define two elastic moduli to describe the in-phase and out of phase behavior. The storage or elastic modulus is the in-phase contribution and defined as

$$\begin{equation} E' = \frac{\sigma_o \cos \delta}{\epsilon_o} \end{equation}$$

and the loss modulus is the out of phase component is

$$\begin{equation} E'' = \frac{\sigma_o \sin \delta }{\epsilon_o} \end{equation}$$

We can now re-write our expression for the stress in the material as

$$\begin{equation} \sigma = E' \sin \omega t + E'' \cos \omega t \end{equation}$$

then by definition we have that

$$\begin{equation} \tan \delta = \frac{E''}{E'} \end{equation}$$

This is sometimes called the loss tangent and essentially represents the amount of energy lost over the energy stored[9]. You might also see these expressions written using complex variables like so

$$\begin{eqnarray} \epsilon = \epsilon_o \exp i \omega t\\ \sigma = \sigma \exp i\omega t + \delta\\ E = \frac{\sigma_o}{\epsilon_o} \exp i \delta = \frac{\sigma_o}{\epsilon_o} (\cos \delta + i \sin \delta) = E' + E'' \end{eqnarray}$$

Analysis of Dynamic Mechanical Testing Experiments:

Typically you will find, at a fixed temperature that the loss tangent and the loss modulus are typically very small at very low and and very high frequencies and they will typically peak at intermediate frequencies. The storage modulus is high at high frequencies (short times) which should make sense intuitively as polymers will typically behave glassy or elastic at high frequencies and short times (strain rate is faster than relaxation time of polymer) and at low frequencies (long time longer than relaxation time) the polymer will behave more like a vicious fluid.

More importantly from this analysis we can determine experimentally the characteristic relaxation time from the point where G’ and G” intersect.

More often we will find that typical DMA experiments where G’, G”, and tanδ are plotted as a function of temperature. Here you will typically see larger peaks and changes in these parameters as we have already seen that the modulus of polymers is temperature dependent. Alternatively, as we have already discussed at length the temperature will affect the amount of molecular motion and free volume will increase and this will affect these properties, particularly the loss tangent. So you will expect to see large changes at the glass transition temperature and melting temperature but you also might expect to see other small peaks associated with secondary transitions (note this is not referring to second order thermodynamic transitions like T_g this is just referring to the magnitude of the peaks) associated with molecular motion of the polymer such as the temperature at which backbone side group rotation is accessible.

As you can see in the figure the primary observable and measurable change in the storage modulus occurs when you pass through T_g. However, the more subtle difference in behavior can be found in the variation of the loss tangent as a function of temperature. Here we can see multiple different peaks at different temperatures with varying amplitudes. In these plots each peak is typically labeled α,β,γ, etc in order of descending temperature. In this graph the α peak or relaxation corresponds to the T_g in an amorphous polymer or it could be T_m for a semi-crystalline polymer. The β peak here for cooperative motion of segments of the main chain. The γ peak is due the phenyl group rotation around the backbone and the δ peak has been linked to some wiggling of the phenyl group most likely due to defects/tacticity differences.

For semi-crystalline polymers this analysis can become even more difficult because of the microstructure of spherulites. This makes it very difficult to separate the amorphous and crystalline behavior. One example of this can be seen for LDPE and HDPE you can see some similar transitions (i.e. γ) but other stark differences. First, the primary difference between LDPE and HDPE is the amount of branching is much larger in LDPE than in HDPE. Here the α and α⁰ peaks are associated with motion in the crystalline regions while the γ peak is associated with the amorphous region. Finally, the β relaxation is associated with the motion of the branches. Now the analysis of these peaks in polymers are a great source of debate but when analyzing them just make a reasoned argument thinking about the thermal energy required for a particular type of motion .

Time-Temperature Equivalence of Polymeric Materials

There is a well-defined difference in the Young’s modulus between glassy, amorphous polymers and rubbery elastic polymers. Furthermore, we know that there is a temperature dependence for the Young’s modulus. Below the T_g the modulus is very high, while at higher temperatures the modulus falls to that of a rubber. This relationship is typically drawn on a log (E) vs. log (T) plot where the several order of magnitude difference between the glassy and rubbery moduli is clear.

One other thing that must be noted is that at high temperatures, strictly amorphous polymers will have moduli that fall to zero; only crosslinked rubbers will maintain a non-zero modulus at high temperatures that is relatively invariant compared to the change between the amorphous and rubbery moduli. If there are no crosslinks, the polymer will simply flow irreversibly.

Now, if we remember back to the discussion of the glass transition, we said that the onset of glassy behavior is really due to the change in the characteristic relaxation

Figure 9. Crystalline, Amorphous, and Cross-Linked Polymeric Stiffness vs. Temperature.

time of a polymer, and how that relaxation time compares to the experimental time associated with a given stress effect. The glass transition would then be interpreted as changing the relaxation time with temperature, which molecularly could be understood as providing more (or less) thermal energy to overcome barriers to flow. However, we also said that if you change the timescale of the perturbation (e.g. deform a polymer really really fast) you could obtain glassy behavior even without changing the temperature. This information leads us to believe that the modulus of a polymer is also time dependent, a fact that is born out empirically. In fact, the graph of log (E) vs log (T) is qualitatively the same as the plot of log (E) vs. log (time), reflecting the origin of the glass transition.

In addition to changing the value of the Young’s modulus, changing the timescale or temperature of an experiment can qualitatively change the shape of a stressstrain curve, as well. For example, heating up a brittle, glassy polymer sample will decrease the Young’s modulus but also increase the ductility of the sample, allowing longer elongations and more plastic deformation. There is thus a ductile-to-brittle transition where the polymer switches from undergoing stable plastic deformation (necking) to brittle failure; the exact temperature depends on the rate of testing and vice versa.

Finally, note that semi-crystalline polymers are characterized by a degree of crystallinity, which reflects the relative proportion of amorphous to crystalline regions. We would imagine that these regions could have very different moduli, especially above the glass transition temperature, and thus would expect the modulus depends highly on the degree of crystallinity. This is indeed the case, and the modulus of semi-crystalline polymers has been observed to increase by a factor of over 100 as the degree of crystallinity increases. At low degrees of crystallinity, this change can be regarded as an increase in an effective crosslinking density for the amorphous regions, as the crystalline regions will likely bear very little load themselves. However,as the degree of crystallinity gets higher the material can be more accurately regarded as a composite of a low modulus material and high modulus material, a problem that is often treated in materials science.

Time-temperature Equivalence

For viscoelastic polymers there is a theoretical equivalence of time and temperature. We have seen previously that a polymer can behave glassy or rubbery behavior by changing either the temperature or the strain rate, i.e. time. So we can establish the principle of time-temperature superposition, or the theoretical equivalence between changing the timescale or temperature of an experiment. An experimentally-derived equation relating these two quantities, known as the WLF equation was found by noticing that you can superpose curves by keeping one curve fixed and shifting all the others by different amounts horizontally parallel to the logarithmic time axis. We can take a reference temperature point, T_s, and a temperature point, τ_s, in order to fix one curve. We then have to find the τ for a new curve with the same compliance as our reference curve at T_s. The amount of shift will then simply be logτ_s − logτ which we will define as our shift factor

$$\log a_T = \log \tau_s - \log \tau = \log \frac{\omega_s}{\omega} \nonumber$$

Well with this Williams-Landel-Ferry (WLF) empirically fit these polymer curves and found the following empirical equation:

$$\log a_T = \frac{-C_1 (T - T_s)}{C_2 + (T - T_s)} \nonumber$$

where C₁ and C₂ are empirical fitting constants, 17.44 and 51.6K respectively for the two constants if one chooses T_s = T_g. This equation was later re-derived using theory based on free volume, akin to the theory for the glass transition, reflecting similar physical origins. The derivation is beyond the scope (and you all are probably sick of derivations at this point but you can read in Young and Lovell Chapter 5) so the resulting equation is

$$\log a_T = \frac{-(B/2.303f_g) (T-T_g)}{f_g/\alpha + (T-T_g)} \nonumber$$

where f_g is the fractional free volume at the glass transition temperature and α is the volumetric thermal expansion coefficient. Typically for polymers f_g = 0. − 25 and α = 4.8 × 10⁻⁴K⁻¹. And B is another fitting constant.

Limitations of WLF

The WLF equation is applicable to homogeneous, linear viscoelastic materials, that are isotropic and amorphous. They must be in the temperature range T_g −T_g + 100^oC.