Dear Dmitry,

your question is very interesting, and I hope to can give you a good answer. I have a main question about your experiment, does the cylinder is hallow or it is full?

If it is hallow there is the high possibility that in your compressive test, you are not measuring the compressive stiffness of your material, but the critical buckling load and post buckling behavior of your cylinder.

In that case the load that cause the instability is related to the geometry ( second moment of inertia lenght^4) and to the Young's modulus of the material:

Fbuckling=(4*pi^2*E*J)/L^2 for a both fixed end case

If you have the Force,  you can measure the length, and calculate the second moment of inertia J, then obtain E. Of course this equation is quite simple and do not consider no linear behavior of the geometry, boundary conditions and load. However, is a good starting point I think to have an estimate of the E.

Kind regards

Daniele

PS

A good idea is also to perform some Finite Element Analyses simulating your experiment using the Riks approach to find out exactly the critical load, and matching numerically your experiment. I used to solve buckling and post buckling problems years ago, if you think this is a good route let me know.

Hello, Daniele,

Thank you very much for your answer!

I work with cylinders which have h~4 mm and D=25 mm, so it is a disk rather than a rod. My samples do not show buckling behaviour. I think they are too soft, I get typical E~5-10 kPa.

Hello Dmitry,

I think this technique which uses Atomic Force Microscopy instrument, is very good for your case of study. This is adopted by biologist to study the effect of drugs on cell's membrane.

Kind regards

https://www.bruker.com/fileadmin/user_upload/8-PDF-Docs/SurfaceAnalysis/AFM/Brochures/PeakForce_Quantitative_Nanomechanical_Property_Mapping_b.pdf

Hello!

PeakForce is absolutely great! I use AFM a lot and I know that PeakForce opens limitless investigation opportunities in nanomechanics and imaging.

Unfortunately I can't use this technique in my case. First, I do not have accsess to a device with PeakForce module. Second, my samples are highly porous hydrogels. As many other porous samples, they are poorly suitable for AFM.

I dare give an explanation but I am not sure it is the right one. When you compress - in air- the disc there a radial expansion which could generate a suction in the center and a negative force. When you test in the fluid the porosity allows a radial centripetal flow which compensates so that you do not have same results. Due to the fluid presence the loaded surfaces can slide with almost no friction on the compression plates. You should make the test with different velocities to see if the negative force is depending on velocity or not.

According to me, If it is only the youngs modulus that you are interested, then you can disregard the negative force region and just measure the slope of the linear part of curve of the compression region. the slope gives you the measure of stiffness. 

You can convert the force displacement curve into stress-strain curve using your sample dimensions and the slope of that curve will give you young's modulus.

The negative force region means there is a force of attraction between your sample and the upper plate you are using. this attraction may be present in air and is not there in water media.

Question to precedent comment: It would be interesting to find an explanation for this "attraction force" and why it is present in air and not in water.

I want to enlarge a bit my previous attempt to find a reason for it. The superficial tension manifests itself when the sample is in air and not when in water, this could be an explanation for the inverted trend at low deformations.

The slope of the curve is only valid if there are no changes in transverse section of sample.

Hello!

I am very glad that there are so many researchers who join the discussion!

I should emphasize that the measurements in air are carried out on a wet sample, since it is a hydrogel. As every hydrogel, it contains water in it, the water is held inside the polymer network by the osmotic pressure. There might be misunderstanding, since in my case "in air"  does not mean "dry"!

So I suppose that the attractive force is due to the capillary effect. When I immerse (submerge) the sample into liquid, there is no capillary effect and hence no force. It is absolutely the same in AFM measurements.

Nick, why "The slope of the curve is only valid if there are no changes in transverse section of sample"? I thought that in case of transverse contraction we measure a combination of Young's modulus and Poisson's ratio, as described in "On the elastic equilibrium of circular cylinders under certain practical systems of load" Phil. Trans. R. Soc. Lond. A 1902 198, 147-233.

you can measure directly by rheometer  

hi anyone can suggest accurate ASTM reference for hydrogels sample? thank you