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von Karman plate equations

Rui Huang's picture

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."


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