Linear Shell Theory. Three-dimensional linearized elasticity and Korn's inequalities in curvilinear coordinates.
Mathematical Elasticity: Volume 27 : Volume II: Theory of Plates
Inequalities of Korn's type on surfaces. Asymptotic analysis of linearly elastic shells: Preliminaries and outline. Linearly elastic elliptic membrane shells. Linearly elastic generalized membrane shells. Linearly elastic flexural shells.
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Koiter's equations and other linear shell theories. Part B. Nonlinear Shell Theory.
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Asymptotic analysis of nonlinearly elastic shells: Preliminaries. Nonlinearly elastic membrane shells.
Nonlinearly elastic flexural shells. Koiter's equations and other nonlinear shell theories. The objective of Volume III is to lay down the proper mathematical foundations of the two-dimensional theory of shells. To this end, it provides, without any recourse to any a priori assumptions of a geometrical or mechanical nature, a mathematical justification of two-dimensional nonlinear and linear shell theories, by means of asymptotic methods, with the thickness as the "small" parameter.
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Mathematical Elasticity: Volume II: Theory of Plates - Google книги
View on ScienceDirect. Authors: Philippe Ciarlet. The article covers several mathematical problems relating to elastic stability of thin shells in view of inconsistencies that have been recently identified between the experimental data and the predictions based on the shallow- shell theory. The new calculations often find the low critical stress close to zero. Therefore, the low critical stress cannot be used as a safety factor for the buckling analysis of the thinwalled structure, and the equations of the shallow-shell theory need to be replaced with other differential equations.
The new theory also requires a buckling criterion ensuring the match between calculations and experimental data. The article demonstrates that the contradiction with the new experiments can be resolved within the dynamic nonlinear three-dimensional theory of elasticity.
Mathematical Elasticity by Ciarlet
The stress when bifurcation of dynamic modes occurs shall be taken as a buckling criterion. The nonlinear form of original equations causes solitary solitonic waves that match non-smooth displacements patterns, dents of the shells. It is essential that the solitons make an impact at all stages of loading and significantly increase closer to bifurcation. The solitonic solutions are illustrated based on the thin cylindrical momentless shell when its three-dimensional volume is simulated with twodimensional surface of the set thickness.
It is noted that the pattern-generating waves can be detected and their amplitudes can by identified with acoustic or electromagnetic devices. Thus, it is technically possible to reduce the risk of failure of the thin shells by monitoring the shape of the surface with acoustic devices.