By Jean Alexandre Dieudonné
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Extra info for Calcul infinitesimal (Collection Methodes)
Of the exact solu- tion whose difference from the exact solution is exponentially small as (h/a) -> 0 in the region n 0 < r < a -n , |z|sh for any fixed positive η0 and TU. The fact that this difference is exponentially small as (h/a) -> 0 is of a great significance; it means that our theory is the best possible "thick plate" theory in the sense that it is accurate to 0([h/a] ) for arbitrarily large N. , (see p. 475), which claims to be correct to 0(h 2 /a 2 ) are not correct to this order and are even inferior to the Kirchhoff "thin plate" solution.
1) we obtain : 3 σ. (v°) LL v. dy = /d J where G G we note that the integral over ij (v°) n. ds -f - σ. (v ) e. 7), and. 8) gives : J J a.... e. , e, (v )n. v.
10)) we search for a two-scale asymptotic expansion in the variables (cf. l) or x-,x , by considering y = X l l/£ ; y 2 = Χ 2/ε ( 3 · 3 ) and searching for an inner expansion u e (x) for = y- v (x 1 ,x 2 ,y 1 ,y 2 ,y 3 ) = χ^/ε , y 2 - + ε v ( x ^ x ^ y ^ y ^ y ^ + ε2... x /ε), with vJ ω-periodic in y_,y . 4) 59 where it is to be noticed that the dependence on nature than the dependence on but not on the outer one variable y1,y is of a very different depend on the inner The functions ; oppositely, they depend on x..