A set of highly efficient and shear lock free finite elements based on Timoshenko beam and Reissner-Mindlin plate theories has been developed for the analysis of thin and thick structures. These elements have arbitrary higher order derivatives and do not require any spcial integration scheme. All are isoparametric elements.

P.Subramanian

In collaborations with industrial engineers on the design and analysis of quartz crystal plate resonators, which involves the high frequency vibrations of piezoelectric plates with Mindlin plate equations, we have to go through the details of plate equations for every stages in the analysis. In addition, complications and bias fields are also considered. We have put together our papers and written a draft of a book detailing essential equations and methods. We now share this draft with you if you would like to read some documents to know insights on this particular subject.

The file is in Chinese and we shall come up with an English version if there are enough interests.

Hi everybody!

for plates with constant thickness and two opposite sides simply-supported B.Cs, we can write w(x,y)=f(y)*sin(m.Pi.x) and solving the problem from this viewpoin.

I wonder whether this approach is true for plates with (linearly) variable thickness?

many thanks

Many of us (including myself) have used the nonlinear equations for elastic plates, originally proposed by von Karman (1910). I recently came across a book with some interesting comments about the plate equations, which may be of interest to share with others on imechanica. The book's title is "Plates and Junctions in Elastic Multi-Structures", authored by Philippe G. Ciarlet and published by Springer-Verlag in 1990.

First, a brief comment by the author as the introduction: "The two-dimensional von Karman equations for plates, originally proposed by von Karman [1910], play a mythical role in applied mathematics. While they have been abundantly, and satisfactorily, studied from the mathematical standpoint, as regards notably various questions of existence, regularity, and bifurcation, of their solutions, their physical soundness has been often seriously questioned."

Then, the author quoted a statement by Truesdell (1978): "An analyst may regard that theory [v. Karman's theory of plates] as handed out by some higher power (a Hungarian wizard, say) and study it as a matter of pure analysis. To do so for v. Karman theory is particularly tempting because nobody can make sense out of the 'derivations'...I asked an expert, Mr. Antman, what was wrong with it [v. Karman theory]. I can do no better than paraphrase what he told me: it relies upon

1) "approximate geometry", the validity of which is assessable only in terms of some other theory.

2) assumptions about the way the stress varies over a cross-section, assumptions that could be justified only interms of some other theory.

3) commitment to some specific linear constitutive relation linear, that is, in some measure of strain, while such approximate linearity should be the outcome, not the basis, of a theory.

4) neglect of some components of strain again, something that should be proved mathematically from an overriding, self-consistent theory.

5) an apparent confusion of the referential and spatial descriptions - a confusion that is easily justified for classical linearised elasticity but here is carried over unquestioned, contrary to all recent studies of the elasticity of finite deformations."

Later in the book, I read: "In this fashion, we are able to provide an effective strategy for imbedding the von Karman equations in a rational approximation scheme that overcomes the five objections raised by C. Truesdell. More specifically, our development clearly delineates the validity of these equations, which should be used under carefully circumscribed situations."

I have not read the above-mentioned paper, as I have never been able to find it. However it is said to be "a brilliantly insightful paper which was to lay the foundations of modern elasticity." However, I believe it is also noteworthy for being one of the major contributions by a female mechanician prior to the modern era. For a great biography of Sophie Germain, including a fantastic quote from a letter from Carl Gauss on discovering that she was female--and not "Monsieur Le Blanc"--visit this site (from which the above quote, on the impact of her paper, came).

There are no female mechanicians listed on http://en.wikipedia.org/wiki/Mechanicians but I believe it could be argued that Germain deserves a mention!