## Introduction

Siméon Denis Poisson was a brilliant scientist who lived about 200 years ago. He made many important contributions to the physical sciences, including: Poisson’s equation, potential theory, optics, electricity and magnetism, mathematics, and mechanics, including the Poisson’s ratio:

\( \nu = \displaystyle -\frac{\varepsilon_{transverse}}{\varepsilon_{axial}}\).

This equation shows that ν is 0 if the bulk deformation resistance is 0, and that ν is 0.5 if the material is incompressible.

**In this article I will explain 3 issues with this definition of the Poisson’s ratio, and how to overcome those issues.**

## Poisson's Ratio - Issue #1

The definition of the Poisson’s ratio is the ratio of 2 strains, but **what strain should you use? **Engineering strain, true strain, Biot strain, etc? The traditional definition is only well-defined for infinitesimal strains. That is clearly not good enough!

To overcome this issue we need to refine the definition. Here are 2 new alternative definitions (for true and engineering) Poisson’s ratio:

\( \nu^{true} = \displaystyle -\frac{\varepsilon^{true}_{transverse}}{\varepsilon^{true}_{axial}}\)

\( \nu^{eng} = \displaystyle -\frac{\varepsilon^{eng}_{transverse}}{\varepsilon^{eng}_{axial}}\)

In MCalibration you can specify both engineering and true Poisson’s ratio when plotting experimental and predicted results.

*Figure 1. MCalibration can plot both true and engineering Poisson’s Ratio.*

## Poisson's Ratio - Issue #2

The Poisson’s Ratio is often treated as a constant even though it is a function of the applied strain. If a test lab is providing you with experimental data, then make sure they don’t just give a single number, but instead provide a whole column with the transverse strain in the experimental data file. Using digital image correlation (DIC), it is really simple to measure the transverse strain at the same time as the axial strain. In MCalibration you can read in a data file containing not only the traditional time-strain-stress data, but also the transverse strain (see Figure 2).

*Figure 2. When loading the experimental data make sure you also read in the transverse strain (if it is available).*

After you have loaded the experimental data, also instruct MCalibration to assign a fitness weight for the transverse strain (see Figure 3). This way MCalibration will be able to use that information during the material model calibration.

*Figure 3. Assign a fitness weight to the transverse strain data.*

## Poisson's Ratio - Issue #3

Due to its definition, the Poisson’s ratio can get crazy values during cyclic loading. Figure 4 shows the predicted results from the PolyUMod TN model during cyclic loading. The Poisson’s ratio initially increases from about 0.47 to 0.49 during the tension step, but then during unloading the Poisson’s ratio continues to increase, and will eventually become very large (way higher than 0.5!). This is to be expected, and is not because of a problem with the material model. To understand this, recall that the Poisson’s ratio is defined as the ratio of the transverse strain to the axial strain. During unloading it is absolutely possible for the axial to reach 0, with a non-zero value for the transverse strain. This corresponds to \(\nu \rightarrow \infty\). Note that this is the reason MCalibration does not allow you to directly input the Poisson’s ratio, but instead uses the transverse strain.

*Figure 4. Predicted stress-strain and true Poisson’s ratio response.*

## Summary

- The traditional definition of Poisson’s ratio need to include the type of strain used.
- The Poisson’s ratio is mainly used as a substitute for measuring the bulk modulus.
- A Poisson’s ratio function can be used in MCalibration during material model calibration.

Fun fact: As I mentioned in my article on the Prony series, the French scientist Gaspard de Prony is one of the 72 names that are engraved in the Eiffel tower. Apparently Poisson’s name is also on the Eiffel tower.