TY - JOUR
T1 - From the classical to the generalized von Kármán and Marguerre-von Kármán equations
AU - Ciarlet, Philippe G.
AU - Gratie, Liliana
PY - 2006/6/1
Y1 - 2006/6/1
N2 - In this work, we describe and analyze two models that were recently proposed for modeling generalized von Kármán plates and generalized Marguerre-von Kármán shallow shells. First, we briefly review the "classical" von Kármán and Marguerre-von Kármán equations, their physical meaning, and their mathematical justification. We then consider the more general situation where only a portion of the lateral face of a nonlinearly elastic plate or shallow shell is subjected to boundary conditions of von Kármán type, while the remaining portion is free. Using techniques from formal asymptotic analysis, we obtain in each case a two-dimensional boundary value problem that is analogous to, but is more general than, the classical equations. In particular, it is remarkable that the boundary conditions for the Airy function can still be determined on the entire boundary of the nonlinearly elastic plate or shallow shell solely from the data. Following recent joint works, we then reduce these more general equations to a single "cubic" operator equation, which generalizes an equation introduced by Berger and Fife, and whose sole unknown is the vertical displacement of the shell. We next adapt an elegant compactness method due to Lions for establishing the existence of a solution to this operator equation. © 2005 Elsevier B.V. All rights reserved.
AB - In this work, we describe and analyze two models that were recently proposed for modeling generalized von Kármán plates and generalized Marguerre-von Kármán shallow shells. First, we briefly review the "classical" von Kármán and Marguerre-von Kármán equations, their physical meaning, and their mathematical justification. We then consider the more general situation where only a portion of the lateral face of a nonlinearly elastic plate or shallow shell is subjected to boundary conditions of von Kármán type, while the remaining portion is free. Using techniques from formal asymptotic analysis, we obtain in each case a two-dimensional boundary value problem that is analogous to, but is more general than, the classical equations. In particular, it is remarkable that the boundary conditions for the Airy function can still be determined on the entire boundary of the nonlinearly elastic plate or shallow shell solely from the data. Following recent joint works, we then reduce these more general equations to a single "cubic" operator equation, which generalizes an equation introduced by Berger and Fife, and whose sole unknown is the vertical displacement of the shell. We next adapt an elegant compactness method due to Lions for establishing the existence of a solution to this operator equation. © 2005 Elsevier B.V. All rights reserved.
KW - Compactness method
KW - Formal asymptotic analysis
KW - Nonlinear plate theory
KW - Nonlinear shallow shell theory
KW - Von Kármán boundary conditions
UR - http://www.scopus.com/inward/record.url?scp=31944439941&partnerID=8YFLogxK
UR - https://www.scopus.com/record/pubmetrics.uri?eid=2-s2.0-31944439941&origin=recordpage
U2 - 10.1016/j.cam.2005.04.008
DO - 10.1016/j.cam.2005.04.008
M3 - RGC 21 - Publication in refereed journal
VL - 190
SP - 470
EP - 486
JO - Journal of Computational and Applied Mathematics
JF - Journal of Computational and Applied Mathematics
SN - 0377-0427
IS - 1-2
T2 - International Conference on Mathematics and its Application
Y2 - 28 May 2004 through 31 May 2004
ER -