By Jeffrey M. Lemm

The target of quantity II is to teach how asymptotic tools, with the thickness because the small parameter, certainly offer a robust technique of justifying two-dimensional plate theories. extra in particular, with none recourse to any a priori assumptions of a geometric or mechanical nature, it truly is proven that during the linear case, the three-d displacements, as soon as correctly scaled, converge in H1 in the direction of a restrict that satisfies the well known two-dimensional equations of the linear Kirchhoff-Love thought; the convergence of pressure is usually established.In the nonlinear case, back after advert hoc scalings were played, it really is proven that the major time period of a proper asymptotic enlargement of the third-dimensional resolution satisfies recognized two-dimensional equations, akin to these of the nonlinear Kirchhoff-Love thought, or the von Kármán equations. detailed consciousness can be given to the 1st convergence consequence got as a result, which ends up in two-dimensional huge deformation, frame-indifferent, nonlinear membrane theories. it's also established that asymptotic equipment can likewise be used for justifying different lower-dimensional equations of elastic shallow shells, and the coupled pluri-dimensional equations of elastic multi-structures, i.e., constructions with junctions. In every one case, the lifestyles, area of expertise or multiplicity, and regularity of suggestions to the restrict equations bought during this style also are studied.

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**Example text**

Minimization problem: ~ E V(a~) and j~(~) = inf j~(r/), nEV(~) where V(w) -- {~ -- (~i) E HI(w) • HI(w) x H2(w); rli- Ourl3 - 0 on 7o}, -- lL{ea a~ 0o, - - Variational problem: r = L } (r/H)e~(r/~/) daJ + e 1 - ( L Pi~rli da~ - ] i % ~0,ga dco) + - (~[) - (r E V(a~) and for all e m~c~0c~r/3 da~+ L n~zOz~ da~ - L P~rl~da~ - L q~Oa~3da~, C3 1All notations and definitions used in this section are recalled in the section "Main notations and definitions" under the heading "Plates". liii liv Plate equations at a glance Boundary value problem" -O~m~ - p~ + cO~q~ in w, ~ --p~a i n ~ , - O~naz C = 0~r = 0 on 70, e ~'aL'~ - 0 on 71.

378 The method of formal asymptotic expansions: displacement-stress approach . . . . . . . . 381 The Identification of the leading term u~ the limit scaled "displacement" two-dimensional problem . . . . 382 Identification of the leading term E0; explicit forms of the limit scaled stresses . . . . . . . . . 6. Equivalence of the limit scaled “displacement” problem with the scaled von Kbrmbn equations . . . 7. Justification of the von KBrman equations of a nonlinearly elastic plate; commentary and bibliographical notes .

3. The method of formal asymptotic expansions: The displscement approach . . . . . . . . . . . 4. Cancellation of the factors of @,-4 5 q 5 0, in the scaled three-dimensional problem . . . . . . . 5. Identification of the leading term uo in the displacement approach . . . . . . . . . . . . . 6. 7. Table of contents The limit scaled two-dimensional problem: Existence and regularity of solutions, formulation as a boundary value problem . . . . . . . . . . . .