# theoretical explanation for smaller Young's modulus of silicone foam

I measured the Young's modulus of a rectangular silicone foam sample which is equivalent in material constitution to silicone rubber except that it is 80% empty. The Young's modulus for this foam was measured to be about $E_f= .02 MPA$ which is much smaller than the $E_s=1 MPA$ expected for silicone rubber. The internal geometry of the foam can be approximately described by spherical holes that are randomly distributed across space with sizes ranging from 0.0008 - 0.005 microns.

My first attempt at an explanation involves the following:

1. Assuming that we have a sample of silicone rubber with equivalent dimensions it would be 80% smaller in volume. Given that Volume has dimensions $L^3$ I would need to divide the Young's Modulus of Silicone Foam by $C=.2^{2/3}\approx .34$ in order to take into account that the silicone foam is mostly empty. This gives me $E_f \approx .02/.34 \approx .06$ MPA

2. Now, let's suppose that a rectangular sample of silicone rubber stretched along its major axis can be modeled as large number of rubber bands joined together with negligible bending. Given that the mass of the silicone rubber and silicone foam are equal the number of 'bands' in each one is also equal where these bands are taken to be of very small cross-sectional diameter. Then we must take into account that unlike the silicone rubber sample, the silicone foam sample has the strands almost always at an angle. Now, taking all of these angles $0 \leq \theta \leq \frac{\pi}{2}$ to be equally likely a force through any particular strand of the silicone foam would on average need to be divided by $cos(\frac{\pi}{4})=\frac{\sqrt2}{2}$ in order to correct for this difference. So $E_f \approx .06*\sqrt{2} \approx .08 MPA$

This still means that I am off by a factor of approximately $12$. My question is whether I can come up with a better theoretical explanation for the difference in Young's Modulus without knowing more about the internal geometry than the information I have given using first principles. Otherwise, I am also interested in research done on the elastic properties of silicone foam but I haven't found any so far.

• In general, I would have a look at Coussy: Mechanics of Porous Continua (Wiley 1995) if you can get a hold of it. I however think that a factor of ~10 for a rather handwaiving explanation is no serious problem ... Jul 5 '16 at 8:06

After reading 'The properties of Foams and Lattices' by M.F. Ashby I found a much better explanation. Apparently, experiments and numerical modeling demonstrate the following proportionality:

$\frac{E_f}{E_s} \propto (\frac{\rho_f}{\rho_s})^2$ where $E_f$ is the young's modulus of the foam and $E_s$ is the young's modulus of the solid it is made of. Now, given that this ratio approaches one as $\rho_f \rightarrow \rho_s$ the proportionality constant is approximately 1. This means that for my particular calculation we have:

$E_f \approx \frac{E_s}{25}=\frac{1}{25}=0.04 MPA$ since we know that the foam is 80% empty. And we are now off by a factor of 2 but this is not surprising as $E_f$ is a mixture of silicon and a fraction of some other material which is supposed to lower its tensile strength.

How do you get the factor of 12x? $\frac{0.06}{0.02}=3$ and $\frac{0.08}{0.02}=4$. As Sanya points out, why do you expect to do better from such an ad hoc model?

Altering your model to get a value closer to experiment does not validate your model. It is fine for predicting future measurements, but not for confirming your choice of model.

• I get the factor of 12x from $\frac{1 MPA}{0.08 MPA} \approx 12$
– user29305
Jul 5 '16 at 23:15