Constitutive relations, 2-D vs. 3-D. The starting point for modeling cellular membranes is the constitutive relations in 2-D space. It is important to set up the corresponding equations directly in two dimensions rather than to consider them as an asymptotic limit of three-dimensional relationships, like it is done in the shell theory. The main reason for the direct 2-D relations is that 3-D continuum approaches are not applicable to membranes whose thickness in on the order of magnitude of the dimension of a single molecule.

A thin film subject to compressive strain can either bend (for large strain gradient) or wrinkle (for small strain gradient). The bending is traditionally used in thermostats (bimetal stripes), but couple of years ago, it was extended to the nanoscale thin films which can bend and roll-up to tubes with defined number of rotations. The wrinkles are also rather common in macro- and microscale thin films.
Here, we developed an equilibrium phase diagram for the shape of
compressively strained free-hanging films by total strain energy
minimization.

The following is a (relatively minor) question which had occurred to me more than two decades ago. By now I have forgotten precisely when it was... It could have been when I was in my TE (third year engineering) at COEP. ... Or, perhaps, it was later on, when I as at IIT Madras (studying stress analysis on my own). ... I don't remember precisely when it occurred to me, only *how* it did---it was when I was poring over the first part of Dieter's book.

IMHO, a matter like this should have been explicitly dealt with by the undergraduate texts on solid mechanics / elasticity. But, none does. Without straining your curiosity any further, let me tell you what that (minor) problem is:

Thanks to Marino, I have found the reason for the difference in our bending stiffness calculation. The original discussion is here:
http://imechanica.org/node/4029

The reason why we have a higher bending stiffness is due to the dihedral term. This dihedral term does have a significant contribution to the bending stiffness. However, in Ref. [26], apparently, this dihedral term was ignored.
I have written a short document showing the contribution of the dihedral term to the bending stiffness. Please take a look at the attachment.
I received great help from Dr. Huang and Marino. Thank you very much.

Qiang Lu and Rui Huang

Department of Aerospace Engineering and Engineering mechanics, University of Texas, Austin,
TX 78712, USA

(J Appl Phys, 100, 114327, 2006)