# Is Uniaxial Compression the Same as Biaxial Tension?

## 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

Assume incompressible:

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

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
Assume incompressible:
$$\mathbf{F} = \begin{bmatrix} 1/\lambda^2 & 0 & 0 \\ 0 & \lambda & 0 \\ 0 & 0 & \lambda \end{bmatrix}$$

Take $$\lambda = \sqrt{2}$$, giving:

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

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

Incompressible biaxial tension:

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

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.

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.

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