Introduction
I will consider a simple stress relaxation experiment in order to explain Time-Temperature Superposition (TTS). Remember, in a stress relaxation experiment you deform a specimen to a target strain, and then hold that strain for some time. If the material is viscoelastic, the stress will then gradually relax to a lower magnitude. The following figure shows a made-up example:
Figure 1. Stress relaxation results at a base temperature.
Not that it is important for this article, but in case you are curios, I used the following Python code to generate the figure…
import numpy as np
import matplotlib.pyplot as plt
x1 = np.logspace(-3,3)
y1 = 3.0*np.exp(-0.4*x1)+1.0
fig,ax = plt.subplots(figsize=(6,4))
plt.semilogx(x1, y1, label='Relaxation Temp 1', color='blue')
plt.xlabel('Log Time')
plt.ylabel('Stress')
plt.legend()
plt.grid(True)
ax.set_ylim(0,4)
plt.savefig('Blog_TTS_Fig1.png', dpi=200)
plt.show()
There is really nothing exciting about this stress-time curve 😐. But now assume that you repeat that experimental stress relaxation test first at a higher temperature and then at a lower temperature, and assume that you get the curves in Figure 2. This time the results are VERY interesting. Do yo see that the shape of the curves are exactly the same, they are just shifted horizontally? This does not always happen when you test a material, but when it does, the material (under the conditions tested) is called thermorheologically simple. Note also that the time has to be on a logarithmic scale when plotting the data.
Figure 2. Stress relaxation results at 3 different temperatures.
If you can then determine an equation that specifies how to horizontally shift the curves with temperature, you would have a viscoelastic material model that can predict the response at any temperature (as long as the material is still thermorheologically simple). This idea is the time-temperature superposition (TTS).
WLF Equation
In general, the horizontal shift can be quantified as follows: \(log_{10}(t_{T_0}) – log_{10}(t_T) = log_{10}(a_T)\), which is equivalent to \(t_T = t_{T_0} / a_T\). This equation shows that the effective material time is scaled by the factor \(1/a_T\) which depends on the temperature. Remember that in the theory of relativity, Einstein figured out that time depends on velocity. For this class of polymers, time depends on the temperature.
One common way to mathematically describe how to horizontally shift the stress relaxation curves with temperature is the WLF equation (Williams-Landel-Ferry):
\( \text{log}_{10}\,a_T(T) = \displaystyle\frac{C_1(T-T_0)}{C_2+T-T_0} \)
Examine TTS using MCalibration
It is easy to explore how Time-Temperature Superposition (TTS) works using MCalibration. First select a material model that supports the WLF equation. In the following example I used an Abaqus Neo-Hookean mateiral model with a single Prony series term, and the WLF equation.
Figure 3. WLF parameters in MCalibration.
Once the material model has been specified, you can calibrate both the viscoelastic and WLF parameters to experimental data. Or if you like, you can explore how the WLF equation works. The following figure shows the MCalibration calculated modulus as a function of temperature for the parameters listed in Figure 3.
Figure 4. MCalibration predicted modulus vs temperature response.
Summary
- Time-Temperature Superposition (TTS) is a very useful tool when a material is thermorheologically simple.
- You should always experimentally verify that a material is thermorheologically simple before applying a TTS equation. You can do this, for example, by measuring the stress relaxation response at different temperatures.
- I frequently see people apply TTS to DMA data in order to establish a master curve. You should only do this once you have established that horizontally shifting curves can be used (i.e. make sure the material is thermorheologically simple).
- One common TTS equation is the WLF model. Another common TTS model is the Arrhenius model. I discuss that model in another article.
