## Introduction

I frequently meet people that have opinions about whether uniaxial compression can be used as a substitute for a biaxial tension test. If you think about, it would certainly be great if it was true, since biaxial tension experiments are both difficult to perform and expensive. In this article I will try to answer, once and for all, if this is true or an urban legend.

### Uniaxial Compression

- Easy to perform
- Friction issues

\( \mathbf{F} = \begin{bmatrix} \lambda & 0 & 0 \\ 0 & 1/\sqrt{\lambda} & 0 \\ 0 & 0 & 1/\sqrt{\lambda} \end{bmatrix} \)

Take \( \lambda=0.5\), giving:

\( \mathbf{F} = \begin{bmatrix} 0.5 & 0 & 0 \\ 0 & \sqrt{2} & 0 \\ 0 & 0 & \sqrt{2} \end{bmatrix} \)

### Biaxial Tension

- Difficult to perform
- No friction
- Useful for calibrating an I2-based hyperelastic model

Take \( \lambda = \sqrt{2} \), giving:

\( \mathbf{F} = \begin{bmatrix} 0.5 & 0 & 0 \\ 0 & \sqrt{2} & 0 \\ 0 & 0 & \sqrt{2} \end{bmatrix} \)

## What about the Stresses?

Incompressible uniaxial loading:

\( \boldsymbol{\sigma} = \begin{bmatrix} \sigma_u & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} \)

Incompressible biaxial tension:

\( \boldsymbol{\sigma} = \begin{bmatrix} 0 & 0 & 0 \\ 0 & \sigma_b & 0 \\ 0 & 0 & \sigma_b \end{bmatrix} = \begin{bmatrix} -\sigma_b & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} + \sigma_b \mathbf{I} \)

## What about Compressible Materials?

The following table is from my book. It shows that one can convert from uniaxial compression data to biaxial tension data **with good accuracy** if the Poisson’s ratio is close to 0.5. That is, if the material is a rubber. For thermoplastics, which have a Poisson’s ratio around 0.4, you cannot accurately convert uniaxial compression data to biaxial tension data.

## What about Friction?

In a previous study of friction in compression experiments, I showed that friction can significantly reduce the accuracy of a uniaxial compression experiment. In a common compression experiment the measured stress is at least 5% too high due to friction. This further compounds with the error from the conversion between compression and biaxial tension.

## Conclusions

You can convert a uniaxial compression stress to a biaxial tension stress if: (1) the material is an almost incompressible rubber material (this technique does not work for thermoplastics); (2) the compression test had very low friction effects.

If you can run an accurate biaxial test (which is not easy), then that will always be more accurate.