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# Discussion of fracture paper #38 - Fracture of the thinnest of sheets - Graphene

The Nobel laureate Andre Geim made graphene by playing with pencil leads and Scotch tape and coauthored a paper on how to get the Nobel prize the fun way. Before that, he co-authored with his hamster, Ter Tisha, a paper on diamagnetic levitation and demonstrated it on a frog. He was honoured with the Ig Nobel prize for the paper and later became the only person so far who got both the Harvard Ig version and the real Alfred version of the Nobel prize. Geim is one of my favourite scientists, which led me to read the paper

The paper turned out to be well-written and really interesting. The subject itself is indeed exciting. Several obstacles arise. Already the dimensions of the object that normally never would be disputed in conventional continuum mechanical analyses pop up as problematic. Length and width are easy but what is the thickness?

Graphene, being a single carbon atom thick sheet of graphite is as thin as it can be. Still, measuring the thickness itself poses a challenge. A surface is a geometric object, that is between what is inside and outside. How do we define the position of the surface of a single atom or a sheet with a single atom thickness? Perhaps, a known mass density of graphite, the total mass of the graphene sheet and the atomic ditto of carbon could define the thickness. We could go for the distance between the crystal planes in a graphite crystal. However, I am sure that another Nobel laureate, Lev Landau, would have said that the chemical potential that makes the graphite form a solid, also narrows the crystal planes in the graphite. I guess it means, at least theoretically, that the space between the atoms increases with a decreasing number of atomic layers.

The thickness is a paramount quantity. As long as tension is applied across a crack in a strip there will be compression along the crack surface. The stress initiating crack growth is independent of the thickness while the stress causing buckling is proportional to the square of (sheet thickness/crack length). Once the sheet buckles and bends the region around the crack will get the shape of a finch beak. The stress expansion will not include the square root singularity the stress intensity factor, as I see it, loses its meaning.

This is not diminishing the importance of the present paper but it would be an interesting continuation to see what happens if the present model includes out-of-plane motion. Either the buckling comes before the initiation of crack growth. If not the crack may grow until the remote stress, which decreases proportionally to the inversed square root of the crack length meets the buckling stress, which decreases much faster and proportionally to the inversed squared crack length.

It would be interesting to hear from the author or anyone else who would like to discuss or provide comments or thoughts, regarding the subject, the method, or anything related. If you do not have an iMechanica account and fail to register, please email me at **per.stahle@solid.lth.se** and I will post your comments in your name. If the paper is not open access it will be that in a couple of days.

Per Ståhle

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## Comments

## Dear Per,

Dear Per, I read your questions in iMechanica, but I really don't understand why we should have compression and buckling on the crack faces. Here we have a single layer of atoms. Anyway, the EFM paper seems very simple to me, with a transition from strength criterion to LEFM.

Regards

Mike

(engineeringchallenges@gmail.com)

## Dear Mike, Thanks for the

Dear Mike, Thanks for the comment. I will try to explain. Let's say that a crack occupies the region |

x| <a,y= 0 in a large sheet exposed to a remote biaxial stress equal in bothxandydirections. If the sheet is large enough it can be treated as infinite and allow us to use the solution by Inglis from 1913 or the more versatile version by Westergaard from 1939. According to their exact elastic solutions, the normal stresses σxand σyare equal in the entire crack plane (y=0). Along the crack surfaces, both stresses vanish. If the remote stress in thex-direction is removed, the stress will drop just as much everywhere. Therefore, σxbecomes compressive while σyremains zero at the crack surfaces.At small loads, the crack opens and along the crack surfaces the σ

xis compressive but may be too small to cause buckling. Increased remote tension increases the crack surface compression and it will eventually surpass the buckling load.The buckling reduces the compression along the crack surfaces. Somewhere along the line of events, the crack may grow.

Andre Geim would probably have suggested a less boring way to understand this. Perhaps a Friday afternoon experiment on strips of normal copy machine paper could work for you. A strip with a 1 cm central crack in a 7 times 27cm piece of paper worked for me.

I hope this helped, with best regards, Per

## I see your point, has anybody studied that?

I see that for thin sheet there may be out-of-plane motion, and in a simple experiment this seems to happen. Has anybody studied that? It sounds to me of the effect of T-stress on crack propagation, but I have never heard of that in terms of buckling load. I agree it is worth investigation. The paper you mention of course has no possibility of buckling.