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In reply to radius of carbon atom
This is not the definition of the physical thickness I refered to. Please read the paper by
Wang LF, Zheng QS et al, PRL 2005.
In reply to what is physical thickness?
I wrote in my summary. It appears that you did not acknowledge the difference between the nominal and physical thickness. It makes sense to me.
In reply to Dear Rui,
I guess the physical thickness by Sulin has something to do with the atomic radius of carbon. So I checked and found the radius of carbon to be about 0.07 nm. With this radius, you may argue that the "physical" thickness of a monolayer graphene is twice of the radius, about 0.14 nm, which is close to the C-C bond length (unsurprisingly). But again, this thickness should never be used in Eh^3 to calculate the bending modulus of graphene.
Researchers have seeded cells on graphene and the cells can survive. But whether graphene is biologically toxic was inclusive, based on my reading several years back.
In reply to Dear Rui,
Where do you get 0.06 nm as the "physical" thickness? Sorry, but it is irritating to see people do this nonsense. You know where Eh^3 comes from if you teach undergraduate mechanics.
In reply to not far from scaling?
I used the physical thickness for the scaling.
Bending rigidity = 1.1eV = 1.76x10^(-19) J
E = 1.0 TPa = 10^12 N/m^2
h = 0.06nm; h^3 =2.16x10^(-31) m^3
Eh^3 = 2.16x10^(-19)
The generally accepted bending modulus of monolayer graphene is about 1.5 eV, which is around 60 k_b*T for T = 300 K. This is about four times the 15 k_b*T condition for biocompaibility as noted above. It is same order of magnitude, but the difference may be significant from biocompatibility point of view.
In reply to Rui,
Sulin, what do you mean by "graphene is not far from this scaling"? You can always fit the bending modulus with a thickness, but that does not give you a physical thickness. It is not physical at all! The point is, there is no reason for this scaling to hold for a monolayer of atoms. Can stress be distributed across the thickness of an atom? On the other hand, the nominal thickness is meaningful only for multilayers and depends on the van der Waals interactions between the layers (thus not a fundamental property of the monolayer). However, using the nominal thickness for graphene (~0.335 nm), the B ~ E*h^3 scaling works pretty well down to bilayer graphene, but is two orders of magnitude off for monolayer. A simple formula for the bending modulus of monolayer to multilayer graphene was given as Eq. (6) of another short paper (W. Gao and R. Huang, Effect of surface roughness on adhesion of graphene membranes. J. Phys. D: Appl. Phys. 44, 452001, 2011).
Thank you for your excellent question.
Convergency is always a problem in numerical simulations. For dynamics simulations, the problem is less servere - at least you get some results (sometimes are not fully converged though). For static simulations, optimizer may fail in the middle of the simulation.
The difficulty for simulating shape evolution of 2D crystals arises from the fact the energy landscape is highly irregular, and minimizer goes through many valleys, owing to the ease of bending into the 3rd dimension. If your initial perturbation is far from the targeted valley, it may take a lot of time because of the kinetic barriers along the long path.
I have limited experience in dealing with the convergency problem. Instead of using NR method, we use conjugate gradient methods as static minimizer, which generally works quite well. Preconditioned CG works even better.
In reply to thickness and bending modulus of graphene
Your point is well received, and I have read your paper before. As I stated, thin shell theory has been frequently "borrowed" to simulate these 2D crystals, including graphene. Then we would ask wether this connection between the bending modulus and thickness still holds phenomelogically. It turns out that graphene is not far from this scaling.
Dear Prof. Zhang and Hongyan,
Thank you very much for this excellent summary on the mechanics of 2D crystal. As you discussed, the 2D crystalline structures are generally very flexible and often exhibit complicated shapes. Therefore, the conventional numerical solver such as Newton-Raphson (NR) method may have convergence problems. In fact, we found that the damped dynamics methods (i.e. fire http://dx.doi.org/10.1103/PhysRevLett.97.170201) usually work better than the NR solver based on our studies on defect controlled wrinkles in graphene (doi:10.1016/j.jmps.2014.02.005). However, it still takes a quite long time and is sensitive to the initial perturbation. For example, the symmetry of the final configuration may be broken when using random perturbations.
What's your experience and suggestion on the numerical solvers for the wrinkle/fold deformation of the 2D crystalline structures?
In reply to 3D FEA of stress concentration
Regarding question1: or at least some kind of approximation method which I can use as a reference to get a sense how correct my FE estimation of stress concentration factor is.
Apparently you have done a lot of works on a variety of 2D materials. I learnd a lot from this summary. However, I am surprised that you still use B ~ E*h^3 in the discussion of thickness of graphene. The bending modulus of graphene has nothing to do with its thickness! Thinking about how B ~ E*h^3 is derived for a beam, you would understand why this relation does not apply for an atomic monolayer. The physical origin of bending modulus of an atomic monolayer (graphene or not) is the multibody interactions between the atoms. For a more detailed discussion, please take a look at the following short paper.
Q. Lu, M. Arroyo, R. Huang, Elastic bending modulus of monolayer graphene. J. Phys. D: Appl. Phys. 42, 102002 (2009).
The critical moment will be your LTB moment. So, depending on the load you are inputing the LTB moment will be able to be calculated. Of course, you need to remember that for I girders the moment gradient is also important for the determination of the LTB critical loads.
In reply to Creating a path in python for Abaqus
Thank you for the hint. In this special case, the path is a straight edge, so I can use what you just suggested. Thank u very much!
You may use,
.SetFromNodeLabels(...) for the first problem, and
.getElements() for the second.
Why not use a new step from the point sine wave ends, and 'vanishing' starts.
In reply to Creating a path in python for Abaqus
One way is to first create a node set containing your path. Then iterate within this node set, looking for nodes based on coordinate information (x,y or z) from beginning to end of desired path points, and appropriately appending the found nodes in to a list.
Of course this only works if your desired path geometry is linear or programmable.
Session 11: NANOMECHANICS (http://nscj.co.uk/ecm4/sessions/session11.html)
Mechanical properties are crucial to better understand the final properties of applicable materials. From this point of view Nanomechanics is an important Nanoscience branch, which allows to study fundamental mechanical properties at the nanoscale. This knowledge is key point for the application of novel materials.
All research investigations done recently in the field of Nanomechanics are welcome to be discussed in this session in order to better understand mechanical behavior of the materials at the nanoscale.
Thanks for sharing this excellent work!
I have a quick question to Figure 4(a) of your paper. Is the interlayer potential shown here a statistical/time-averaged value?
Look forward to your feedback!