bending

After having taught graduate structural mechanics for several years, I am finally
able to write down my lecture notes (attached) for teaching the beam theory. In
the notes, we formulated the complete classical beam model
(extension/torsion/bending in two directions), which is also called
Euler-Bernoulli-Saint beam theory, in three ways: Newtonian method, variational
method, and variational asymptotic method, using 3D elasticity theory as the
starting point. Many self-contradictions of the various assumptions used in both
Newtonian method and variational method are clearly pointed out. The

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.

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)