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Limit Problems for Plates Surrounded by Soft Material

Summary: Limit Problems for Plates Surrounded
by Soft Material
Communicated by E. GIUST!
I. In~oducfion
Consider an inhomogeneous clamped plate D, submitted to an external force
g(x). The (small) vertical displacement u(x) solves the minimum problem
min {/[.1 'E(x) }-g~x) (l~ul2- 2(1 - cr(x)) detD2u) -r g(x) u] dx: u EHI(D) ,
where E and a are the Young modulus and the Poisson coefficient respectively,
and D2u denotes the 2 · 2 matrix of second derivatives of u. We study a plate
having a central part/2 surrounded by an increasingly narrow annulus Z', made
of an increasingly soft material (i.e. the Young modulus E, tends to zero in X8)-
The free energy of the plate is then
(1.1) F,(u) = _f l E--ff--~(]Aul 2 - 2(1 - a) det Dzu)dx
+ 2 - 2Cl-, ,)det , u dx
We study in particular the behavior as e--> 0 of the solution u~ of
(1.2) min {F~(u)§ fg(x) udx:uEH~(~kJSe)}.
-QV~ e
If re is the width of Se, we may have different limit problems depending on the


Source: Acerbi, Emilio - Dipartimento di Matematica, Universitą degli Studi di Parma


Collections: Mathematics