Introduction
Linear viscoelasticity can be combined with time-temperature-superpositioning (TTS) in order to get a single time- and temperature-dependent material model. There are two commonly used models for performing this time-temperature scaling: (1) the Williams-Landel-Ferry (WLF) model; and (2) the Arrhenius model. In this article I will examine these two models and determine which is the most accurate. The results surprised me when I went through this exercise, I think you will be surprised too!
John D. Ferry (1912-2002)
Svante Arrhenius (1859-1927)
WLF Equation
I described the general idea of time-temperature superposition (TTS) in a previous article. In short, it describes how for thermorheologially simple materials the stress relaxation response at different temperature can be obtained from a single master curve by horizontal shifts on the logarithmic time axis: \(\log_{10}(t_{T_0}) – \log_{10}(t_T) = \log_{10}(a_T)\), 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. The Williams-Landel-Ferry (WLF) equation describes one way to perform the scaling:
\( \text{log}_{10}\,a_T(T) = \displaystyle\frac{C_1(T-T_0)}{C_2+T-T_0}. \)
This equation is available in most commercial FE solvers, e.g. Abaqus, Ansys, LS-DYNA, and they all use the same equation. Note that the model takes 3 material parameters: T0, C1, and C2. Is it often mentioned in the literature that this model is often applicable to amorphous polymers at temperatures close to the glass transition temperature. The original authors of the WLF equation has stated that common “universal constants” for the WLF equation are: \(C_1 \approx15\) and \(C_2 \approx 50\) K.
Abaqus commands:
*TRS, definition=WLF
** T0, C1, C2
293, 17.4, 51.6
Ansys APDL commands:
TB, SHIFT, matid, 1, 3, WLF
TBDATA, 1, 350 ! T0
TBDATA, 2, 17.4 ! C1
TBDATA, 3, 51.6 ! C2
Arrhenius Equation / Tool-Narayanaswamy Shift Function
The Arrhenius equation was developed by Swedish scientist Svante Arrhenius. His ideas about an energy activation barrier for flow have also been applied to polymer flow, and is the model that I will call the Arrhenius equation in this article. Abaqus writes this equation in this way:
\( \ln(a_T) = \displaystyle \frac{E_0}{R} \left[ \frac{1}{T_0-T_z} – \frac{1}{T – T_z} \right] \)
Ansys writes this equation in the following way, and calls it the Tool-Narayanaswamy model:
\( \ln(a_T) = \displaystyle \frac{H}{R} \left[ \frac{1}{T_r } – \frac{1}{T} \right]. \)
The Abaqus and Ansys models become the same if the Abaqus parameter \(T_z=0\). Also note that different FE codes use a different sign convention for the shift factors. This does not matter, since it is an arbitrary decision if a positive shift factor shifts to the left or the right.
LS-DYNA defines the Arrhenius shift function the same way as Ansys. I kind of like the Abaqus definition the best since it allows the user to define a different zero temperature point (\(T_z\)), which can be handy when using a different temperatures scale.
Also note that the WLF equation is written in terms of the log10 function, and the Arrhenius equation is using the natural logarithm.
Abaqus commands:
*TRS, definition=Arrhenius
** T0, , E0/R
290, , 2067.35
*Physical Constants, absolute zero=238.4
Ansys APDL commands:
TB, SHIFT, matid, 1, 2, TN
TBDATA, 1, 290 ! Tr
TBDATA, 2, 2067 ! H/R
Rewrite the Arrhenius Equation
If I apply some simple math manipulations of the Arrhenius equation that is used by Abaqus then I quickly get the following equation:
\( \ln(a_T) = \displaystyle \frac{1}{T_0-T_z} \cdot \frac{E_0}{R} \cdot \frac{T-T_0}{T-T_z}, \)
and if I then define \( C_2 \equiv T_0 – T_z,\) I get:
\( \ln(a_T) = \displaystyle \left[ \frac{E_0}{C_2 R} \right] \cdot \frac{T – T_0}{C_2 + T – T_0}.\)
If I then define \(C_1 \equiv E_0/(C_2 R)\), I get exactly the WLF equation. In other words, the Abaqus form of the Arrhenius equation is equivalent to the WLF equation.
The Ansys and LS-DYNA version of the Arrhenius / Tool-Narayanaswamy equation is a special case of the Abaqus version that corresponds to a WLF equation with \(C_2 = T_0\).
Use MCalibration to Check The Results
I will use MCalibration to check that the Abaqus-style Arrhenius and WLF equations are indeed mathematically identical. In the example I used a virtual tension test with an engineering strain rate of 0.1/s and the temperature was 293 K. In the first MCalibration calculation I used the following WLF parameters: T0=290 K, C1=17.4, and C2=51.6 K. In the second MCalibration simulation I used the following Arrhenius parameters: T0=290 K, Tz=238.4 K, and E0/R=2067.35. These parameters were calculated from the WLF parameters using the equations above. The results are shown in the two figures below. As expected, the two models give the same results!
Figure 1. Predictions from the WLF equation.
Figure 2. Predictions from the Arrhenius equation.
