Background Mathematics and Linear Shell Theory

We propose a new approach to the existence theory for quadratic minimization problems that arise in linear shell theory. The novelty consists in considering the linearized change of metric and change of curvature tensors as the new unknowns, instead of the displacement vector field as is customary. Such an approach naturally yields a constrained minimization problem, the constraints being ad hoc compatibility relations that these new unknowns must satisfy in order that they indeed correspond to a displacement vector field. Our major objective is thus to specify and justify such compatibility relations in appropriate function spaces. Interestingly, this result provides as a corollary a new proof of Korn's inequality on a surface. While the classical proof of this fundamental inequality essentially relies on a basic lemma of J. L. Lions, the keystone in the proposed approach is instead an appropriate weak version of a classical theorem of Poincaré. The existence of a solution to the above constrained minimization problem is then established, also providing as a simple corollary a new existence proof for the original quadratic minimization problem.

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Linear shell theory for rapid variation solutions

Meccanica ◽

10.1007/bf02129051 ◽

1971 ◽

Vol 6 (1) ◽

pp. 59-64

Author(s):

Placido Cicala

Keyword(s):

Shell Theory ◽

Rapid Variation ◽

Linear Shell Theory

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On Finite Symmetrical Deflections of Thin Shells of Revolution

Journal of Applied Mechanics ◽

10.1115/1.3564619 ◽

1969 ◽

Vol 36 (2) ◽

pp. 267-270 ◽

Cited By ~ 44

Author(s):

Eric Reissner

Keyword(s):

Differential Equations ◽

Nonlinear Theory ◽

Transverse Shear ◽

Shell Theory ◽

Middle Surface ◽

Thin Shells ◽

Shells Of Revolution ◽

Shear Deformations ◽

Linear Shell Theory ◽

Classical Subject

Recent simplifications of linear shell theory through consideration of transverse shear deformations and stress moments with axes normal to the shell middle surface suggest analogous approaches to the corresponding problem of nonlinear theory. As a first step in this direction consideration is given here to the classical subject of finite symmetrical deformations of shells of revolution. The principal new results of the present analysis concern the form of strain-displacement and compatibility differential equations.

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Blending Functions for Vibration Analysis of a Cylindrical Shell with an Oblique End

International Journal of Structural Stability and Dynamics ◽

10.1142/s0219455403000987 ◽

2003 ◽

Vol 03 (03) ◽

pp. 405-418 ◽

Cited By ~ 1

Author(s):

X. J. Hu ◽

D. Redekop

Keyword(s):

Cylindrical Shell ◽

Vibration Analysis ◽

Shell Theory ◽

Differential Quadrature Method ◽

Quadrature Method ◽

Irregular Domain ◽

Parametric Studies ◽

Blending Functions ◽

Linear Shell Theory ◽

Parent Domain

The free vibration problem of a cylindrical shell with an oblique end is considered. A theoretical solution based on the Sanders–Budiansky linear shell theory, and the differential quadrature method, is presented. The surface of the shell is first developed onto a plane, and the resulting irregular domain is then mapped, using blending functions, onto a square parent domain. The analysis is finally carried out in the parent domain. Two solutions are derived, using either trigonometric or polynomial trial functions in the circumferential direction of the domain. Convergence, validation and parametric studies are carried out. Results from the two solutions are compared with each other and with finite element results. The paper ends with an appropriate set of conclusions.

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