# Poisson's controversial stretch-squeeze ratio

Before reading this, you might want to check out the previous post about Siméon Denis Poisson's life and career. Then come back here...

Physicists and mathematicians knew about Poisson's ratio well before Poisson got involved with it. Thomas Young described it in his 1807 Lectures on Natural Philosophy and the Mechanical Arts:

We may easily observe that if we compress a piece of elastic gum in any direction, it extends itself in other directions: if we extend it in length, its breadth and thickness are diminished.

Young didn't venture into a rigorous formal definition, and it was referred to simply as the 'stretch-squeeze ratio'.

### A new elastic constant?

Twenty years later, at a time when France's scientific muscle was fading along with the reign of Napoleon, Poisson published a paper attempting to restore his slightly bruised (by his standards) reputation in the mechanics of physical materials. In it, he stated that for a solid composed of molecules tightly held together by central forces on a crystalline lattice, the stretch squeeze ratio should equal 1/2 (which is equivalent to what we now call a Poisson's ratio of 1/4). In other words, Poisson regarded the stretch-squeeze ratio as a physical constant: the same value for all solids, claiming, 'This result agrees perfectly' with an experiment that one of his colleagues, Charles Cagniard de la Tour, recently performed on brass.

Poisson's whole-hearted subscription to the corpuscular school certainly prejudiced his work. But the notion of discovering of a new physical constant, like Newton did for gravity, or Einstein would eventually do for light, must have been a powerful driving force. A would-be singular elastic constant could unify calculations for materials soft or stiff — in contrast to elastic moduli which vary over several orders of magnitude.

### Poisson's (silly) ratio

Later, between 1850 and 1870, the physics community acquired more evidence that the stretch-squeeze ratio was different for different materials, as other materials were deformed with more reliable measurements. Worse still, de la Tour's experiments on the elasticity of brass, upon which Poisson had hung his hat, turned out to be flawed. The stretch-squeeze ratio became known as Poisson's ratio not as a tribute to Poisson, but as a way of labeling a flawed theory. Indeed, the falsehood became so apparent that it drove the scientific community towards treating elastic materials as continuous media, as opposed to an ensemble of particles.

Today we define Poisson's ratio in terms of strain (deformation), or Lamé's parameters, or the speed $$V$$ of P- and S-waves:

Interestingly, if Poisson turned out to be correct, and Poisson's ratio was in fact a constant, that would mean that the number of elastic constants it would take to describe an isotropic material would be one instead of two. It wasn't until Augustin Louis Cauchy used the notion of a stress tensor to describe the state of stress at a point within a material, with its three normal stresses and three shear stresses, did the need for two elastic constants become apparent. Tensors gave the mathematical framework to define Hooke's law in three dimensions. Found in the opening chapter in any modern textbook on seismology or mechanical engineering, continuum mechanics represents a unique advancement in science set out to undo Poisson's famously false deductions backed by insufficient data.

References

Greaves, N (2013). Poisson's ratio over two centuries: challenging hypothesis. Notes & Records of the Royal Society 67, 37-58. DOI: 10.1098/rsnr.2012.0021

Editorial (2011). Poisson's ratio at 200, Nature Materials10 (11) Available online.

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# All the elastic moduli

An elastic modulus is the ratio of stress (pressure) to strain (deformation) in an isotropic, homogeneous elastic material:

$$\mathrm{modulus} = \frac{\mathrm{stress}}{\mathrm{strain}}$$

### OK, what does that mean?

Elastic means what you think it means: you can deform it, and it springs back when you let go. Imagine stretching a block of rubber, like the picture here. If you measure the stress $$F/W^2$$ (i.e. the pressure is force per unit of cross-sectional area) and strain $$\Delta L/L$$ (the stretch as a proportion) along the direction of stretch ('longitudinally'), then the stress/strain ratio gives you Young's modulus, $$E$$.

Since strain is unitless, all the elastic moduli have units of pressure (pascals, Pa), and is usually on the order of tens of GPa (billions of pascals) for rocks.

The other elastic moduli are:

There's another quantity that doesn't fit our definition of a modulus, and doesn't have units of pressure — in fact it's unitless —  but is always lumped in with the others:

### What does this have to do with my data?

Interestingly, and usefully, the elastic properties of isotropic materials are described completely by any two moduli. This means that, given any two, we can compute all of the others. More usefully still, we can also relate them to $$V_\mathrm{P}$$, $$V_\mathrm{S}$$, and $$\rho$$. This is great because we can get at those properties easily via well logs and less easily via seismic data. So we have a direct path from routine data to the full suite of elastic properties.

The only way to measure the elastic moduli themselves is on a mechanical press in the laboratory. The rock sample can be subjected to confining pressures, then squeezed or stretched along one or more axes. There are two ways to get at the moduli:

1. Directly, via measurements of stress and strain, so called static conditions.
2. Indirectly, via sonic measurements and the density of the sample. Because of the oscillatory and transient nature of the sonic pulses, we call these dynamic measurements. In principle, these should be the most comparable to the measurements we make from well logs or seismic data.

### Let's see the equations then

The elegance of the relationships varies quite a bit. Shear modulus $$\mu$$ is just $$\rho V_\mathrm{S}^2$$, but Young's modulus is not so pretty:

$$E = \frac{\rho V_\mathrm{S}^2 (3 V_\mathrm{P}^2 - 4 V_\mathrm{S}^2 }{V_\mathrm{P}^2 - V_\mathrm{S}^2}$$

You can see most of the other relationships in this big giant grid I've been slowly chipping away at for ages. Some of it is shown below. It doesn't have most of the P-wave modulus expressions, because no-one seems too bothered about P-wave modulus, despite its obvious resemblance to acoustic impedance. They are in the version on Wikipedia, however (but it lacks the $$V_\mathrm{P}$$ and $$V_\mathrm{S}$$ expressions).

Some of the expressions for the elastic moduli and velocities — click the image to see them all in SubSurfWiki.

In this table, the mysterious quantity $$X$$ is given by:

$$X = \sqrt{9\lambda^2 + 2E\lambda + E^2}$$

In the next post, I'll come back to this grid and tell you how I've been deriving all these equations using Python.

Top tip... To find more posts on rock physics, click the Rock Physics tag below!

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### Matt Hall

Matt is a geoscientist in Nova Scotia, Canada. Founder of Agile Scientific, co-founder of The HUB South Shore. Matt is into geology, geophysics, and machine learning.