By P. G. Ciarlet

The 1st goal of this monograph is to teach that the strategy of asymptotic expansions, with the thickness because the parameter, offers a really powerful software for justifying two-dimensional plate theories, in either the nonlinear and the linear case. with out resorting to any a priori assumption of a geometric or mechanical nature, it really is proven that, the displacements and stresses similar to the best time period of the growth of the third-dimensional resolution do certainly clear up the classical equations of 2-dimensional nonlinear plate theories reminiscent of the von Kármán equations. the second one goal is to increase this research to the mathematical modeling of junctions in elastic multi-structures, e.g. in most cases a constitution comprising a "3-dimensional" half, and a "2-dimensional" half. those might be folded plates, H-shaped beams, plates with stiffeners, plates held through rods as in a sun panel, and so on. an identical asymptotic research presents a scientific method of discovering the versions for such multi-structures, because the "thin" half method. apparently, the restrict difficulties present in this manner are coupled, multi-dimensional, difficulties of a brand new variety supplying new situations of stiff difficulties. The e-book written by way of one of many prime specialists the world over within the box of numerical equipment utilized to strong mechanics provides an updated document on an lively study subject, and should be an invaluable reference for utilized mathematicians and engineers operating with elastic multi-structures.

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**Extra info for Plates and junctions in elastic multi-structures : an asymptotic analysis**

**Sample text**

5-3. ) G = {y = i M v s ) e U \ n ) X H\il);diV3-\-dsVi = 0 in Ü, V = o on To} such that / Jq and -f- / alp dpu ldgvsdx = / fiVidx -f / QiVida Jq Jr+uTfor all V = (vi) e V/cl ( ü ), where Jq == 1 2 + datlsd^Uj). ). Then problem least one solution^ and all its solutions are obtained by Brst solving the two-dimensional variational problem P(0) (which always has at least one solution ; cf. ”. •*<»:= j ^ yzfadyz + g t - 9 a ^ and, secondly, by letting — Cor ^3 — Cs* Consequently, one has (a eH \ co ), e H \ ^ ), (c) C s G ii» , ul e H ^ a ) .

2), the tensor-valued mappings E® and E^, and the fourthorder tensors are all independent o f e. The notation used here is self-explanatory : = ( 4 K " ) ) ) , {^ ^ 1 1 - E ^ (u (e )) = ( 4 - ( u ( e ) ) ) , o\ a 7 7 ^J 2 / i(3^A + 2 ^ )^ B °E = i 2/i ‘’’22 ^ 0 for an arbitrary matrix E = 2/j(3A+2/t)‘’ Tf7 [ 0 ^ 0 0/ etc. ; (ii) Equate to zero the factors o f the successive powers o f e, arranged by in creasing values o f the exponentSy found in the scaled principle of virtual work PVW (e) and in the scaled inverted constitutive equation ICE(e) when u(e) and E(e) are replaced by their form al expansions ; (iii) Assuming ad-hoc properties on the successive terms found in both fo r mal expansionSy pursue this procedure until the leading terms u®,E^ can be fully identified.

Juf Since {T>(lo)}~ = H q(uj), we thus conclude that - j if C3,T]3 € Ho{u>), which proves assertion (b). 3 € ifo(‘^). thus proving assertion (c). Again it suffices to establish this relation when Cz^Pz G P(u;), since both sides are continuous when ^3 and 773 vary in the space H q(oj) ; but in this case, we can write / nafidaCzd^pzdiO = / na^dfi{{daC z)r]z}^ ~ / {'T^a^dafiCzjrjzdiO, Juj JiO Juf 28 Chapter 1 = 0 since (9aC3)^3 ^ V{ lo). follows. 4-3. In what follows, (i/^) denotes the unit outer normal vector along 7 ; ( tc^) denotes the unit tangential vector along 7 oriented in such a way that the angle from i^a) to i'J'a) is + f ; and dr denote the outer normal and tangential derivative d^ d^ operators along 7 , respectively ; := denotes the (two-dimensional) biharmonic operator.