## Introduction

In the past I have written extensively about how to design suitable test plans for rubbers, thermoplastics, and thermosets. My proposed test plans were focused on the mechanical response at one temperature, and I simply stated “Repeat the testing at multiple temperatures as needed.”. I was recently asked to clarify this, specifically is it necessary to perform all tests at all temperatures. In this article I will answer this question!

## Proposed Strategy for Testing at Multiple Temperatures

There are many reasons for testing the mechanical response of a polymer, and in this article I will only focus on testing for the purpose of material model calibration. As I discussed in another recent article, there two ways to create a temperature-dependent material model:

- Linearly interpolate the parameters based on the temperature.
- Use a material model with an equation-based temperature dependence.

The necessary experimental test program will be different for these two cases.

#### (1) Linearly Interpolate Material Parameters

Most material models that are available in FE codes can be made temperature-dependent by providing the complete set of material parameters for a set of control temperatures. The material response at any other temperature is then found by interpolating the individual parameters with temperature. In other words, a complete material model needs to be provided at each control temperature. For this reason, **these material models require that all experiments are run at all temperatures.** The benefit of this approach is that it can used to create highly accurate temperature-dependent material models.

#### (2) Equation-Based Temperature Dependence

A small set of material models come with equation-based temperature dependence. This includes, for example, linear viscoelasticity with time-temperature-superpositioning, the PolyUMod Three Network model, and the PolyUMod TNV model.

The TNV model supports the following temperature-dependent scale factor for the shear modulus (for more info about this equation see the PolyUMod User’s Manual):

\[ f_\theta = \displaystyle\frac{1}{2} (f_g+f_r) – \frac{1}{2} (f_g-f_r) \cdot \text{tanh} \left[ \frac{\theta-\theta_g}{\Delta \theta}\right] + X_g \cdot \frac{\theta-\theta_g}{\Delta \theta}. \]

The parameters in this equation can be calibrated to a single temperature sweep DMA tests.

The PolyUMod Three Network model has a simpler temperature-dependence that can be calibrated to a single test at a second temperature. So a temperature-dependence version of this model can be calibrated by performing a complete set of experimental tests at one temperature, and one additional test at a second temperature. This is very convenient, but it also means that the predicted temperature dependence will only be valid in limited range of temperatures.

## Summary

- In most cases you need to repeat all tests at all temperatures.
- If you are using an equation-based material model (e.g. the PolyUMod TNM, the PolyUMod TNV, or linear viscoelasticity with TTS) then it is sufficient to perform full testing at one temperature, and limited tests at one or more additional temperatures. What these additional tests should be depend on the material model.