Abstract
Chapter 13 develops dual variational formulations for the two dimensional equations of the nonlinear elastic Kirchhoff-Love plate model. We obtain a convex dual variational formulation which allows non positive definite membrane forces. In the third section, similar to the triality criterion introduced in [36], we obtain sufficient conditions of optimality for the present case. Again the results are based on fundamental tools of Convex Analysis and the Legendre Transform, which can easily be analytically expressed for the model in question.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
R.A. Adams, Sobolev Spaces (Academic Press, New York, 1975)
F. Botelho, Dual variational formulations for a non-linear model of plates. J. Convex Anal. 17(1), 131–158 (2010)
F. Botelho, Topics on Functional Analysis, Calculus of Variations and Duality (Academic Publications, Sofia, 2011)
F. Botelho, On duality principles for scalar and vectorial multi-well variational problems. Nonlinear Anal. 75, 1904–1918 (2012)
P. Ciarlet, Mathematical Elasticity, vol. I, Three Dimensional Elasticity (North Holland, Elsevier, 1988)
P. Ciarlet, Mathematical Elasticity, vol. II Theory of Plates (North Holland, Elsevier, 1997)
P. Ciarlet, Mathematical Elasticity, vol. III – Theory of Shells (North Holland, Elsevier, 2000)
I. Ekeland, R. Temam, Convex Analysis and Variational Problems (Elsevier-North Holland, Amsterdam 1976).
A. Galka, J.J. Telega, Duality and the complementary energy principle for a class of geometrically non-linear structures. Part I. Five parameter shell model Arch. Mech. 47, 677–698; Part II. Anomalous dual variational principles for compressed elastic beams. Arch. Mech. 47, 699–724 (1995)
D.Y. Gao, On the extreme variational principles for non-linear elastic plates. Q. Appl. Math. XLVIII(2), 361–370 (1990)
D.Y. Gao, Pure complementary energy principle and triality theory in finite elasticity. Mech. Res. Comm. 26(1), 31–37 (1999)
D.Y. Gao, Finite deformation beam models and triality theory in dynamical post-buckling analysis. Int. J. Non Linear Mech. 35, 103–131 (2000)
D.Y. Gao, Duality Principles in Nonconvex Systems, Theory, Methods and Applications (Kluwer, Dordrecht, 2000)
R.T. Rockafellar, Convex Analysis (Princeton University Press, Princeton, New Jersey, 1970)
J.J. Telega, On the complementary energy principle in non-linear elasticity. Part I: Von Karman plates and three dimensional solids, C.R. Acad. Sci. Paris, Serie II 308, 1193–1198; Part II: Linear elastic solid and non-convex boundary condition. Minimax approach, C.R. Acad. Sci. Paris, Serie II 308, 1313–1317 (1989)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Botelho, F. (2014). Duality Applied to a Plate Model. In: Functional Analysis and Applied Optimization in Banach Spaces. Springer, Cham. https://doi.org/10.1007/978-3-319-06074-3_13
Download citation
DOI: https://doi.org/10.1007/978-3-319-06074-3_13
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-06073-6
Online ISBN: 978-3-319-06074-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)