Two-dimensional approximations of three-dimensional eigenvalue problems in plate theory

Modeling and justification of an eigenvalue problem for a plate inserted in a three-dimensional support

“…Then P. G. Ciarlet and H. Le Dret have used formulas (6) to study the behavior of the solution of the static problem associated with the eigenvalue problem (3). But they have defined the scaled functions ü e over the set O by…”

“…Classically ( [2], [3], [11], [10]) as in the case of a single plate, we first define the open set Ö = CÜX]-1,1[ in order to deal with functions defined on sets independent of e. To avoid an overlapping over the inserted part Qfp we then introducé as in [1], [5], [6] To study the behavior of the eigenfunctions u' € of problem (3), we introducé the scaled unknowns u(e) = ( w^t) ) : Q -> R 3 , ü(e) = (M,. (e)) :S->R 3 definedby…”

“…[9]). Thus, to find stationnary solutions, we have to solve the eigenvalue probiem find (A\ u € ) e R x V € such that B e ( u Ê , v e ) = A\ u € , v e ) 6 for all v € e V e ,…”

“…The boundary condition V|^( ) = 0 being satisfied, the pair (v(e), v(e)) belongs to the space W( e ). The convergence of the séquence ( v( e ), v( e ) ) is established by P. G. Ciarlet, H. Le Dret, R. Nzengwa ( [6], [7] (13), such that the whole family (ü(e), u(e)) converges into the space H 1 (Q) xH 1 (Q) as e -» 0.…”

Modeling and optimization of non-symmetric plates

“…First fundamental contributions in this direction, any of whose basic ideas already appear in Friedrichs-Dressler [31] and Goldenveizer [32], were obtained by Ciarlet-Destuynder [15,16] with the justification of the classical bi-harmonic model of Kirchhoff-Love the bending of symmetrie plates. The application of this method to different situations (linear and nonlinear elasticity, composite and anisotropic materials, static, dynamic and thermoelastic cases, homogenization and so on) has provided important contributions, among which, without attempting to be exhaustive, we mention the works of Caillerie [8], Ciarlet [12,14], CiarletRabier [21], Destuynder [25,26,28], Raoult [42][43][44], Blanchard [4], CiarletKesavan [17], Blanchard-Ciarlet [5], Viano [46], Kohn-Vogelius [37][38][39], Cioranescu-Saint Jean Paulin [23], Davet [24], Blanchard-Francfort [6], Ciarlet-Le Dret [18], Quintela Estevez [40,41], Alvarez Vazquez-Quintela Estevez [2]. A complete analysis of plate models with exhaustive bibliographie références may be found in Ciarlet [14].…”