Book contents
- Frontmatter
- Contents
- Preface
- Introduction
- 1 Nonlinear Theories of Elasticity of Plates and Shells
- 2 Nonlinear Theories of Doubly Curved Shells for Conventional and Advanced Materials
- 3 Introduction to Nonlinear Dynamics
- 4 Vibrations of Rectangular Plates
- 5 Vibrations of Empty and Fluid-Filled Circular Cylindrical Shells
- 6 Reduced-Order Models: Proper Orthogonal Decomposition and Nonlinear Normal Modes
- 7 Comparison of Different Shell Theories for Nonlinear Vibrations and Stability of Circular Cylindrical Shells
- 8 Effect of Boundary Conditions on Large-Amplitude Vibrations of Circular Cylindrical Shells
- 9 Vibrations of Circular Cylindrical Panels with Different Boundary Conditions
- 10 Nonlinear Vibrations and Stability of Doubly Curved Shallow-Shells: Isotropic and Laminated Materials
- 11 Meshless Discretizatization of Plates and Shells of Complex Shape by Using the R-Functions
- 12 Vibrations of Circular Plates and Rotating Disks
- 13 Nonlinear Stability of Circular Cylindrical Shells under Static and Dynamic Axial Loads
- 14 Nonlinear Stability and Vibration of Circular Shells Conveying Fluid
- 15 Nonlinear Supersonic Flutter of Circular Cylindrical Shells with Imperfections
- Index
- References
1 - Nonlinear Theories of Elasticity of Plates and Shells
Published online by Cambridge University Press: 08 January 2010
Book contents
- Frontmatter
- Contents
- Preface
- Introduction
- 1 Nonlinear Theories of Elasticity of Plates and Shells
- 2 Nonlinear Theories of Doubly Curved Shells for Conventional and Advanced Materials
- 3 Introduction to Nonlinear Dynamics
- 4 Vibrations of Rectangular Plates
- 5 Vibrations of Empty and Fluid-Filled Circular Cylindrical Shells
- 6 Reduced-Order Models: Proper Orthogonal Decomposition and Nonlinear Normal Modes
- 7 Comparison of Different Shell Theories for Nonlinear Vibrations and Stability of Circular Cylindrical Shells
- 8 Effect of Boundary Conditions on Large-Amplitude Vibrations of Circular Cylindrical Shells
- 9 Vibrations of Circular Cylindrical Panels with Different Boundary Conditions
- 10 Nonlinear Vibrations and Stability of Doubly Curved Shallow-Shells: Isotropic and Laminated Materials
- 11 Meshless Discretizatization of Plates and Shells of Complex Shape by Using the R-Functions
- 12 Vibrations of Circular Plates and Rotating Disks
- 13 Nonlinear Stability of Circular Cylindrical Shells under Static and Dynamic Axial Loads
- 14 Nonlinear Stability and Vibration of Circular Shells Conveying Fluid
- 15 Nonlinear Supersonic Flutter of Circular Cylindrical Shells with Imperfections
- Index
- References
Summary
Introduction
It is well known that certain elastic bodies may undergo large displacements while the strain at each point remains small. The classical theory of elasticity treats only problems in which displacements and their derivatives are small. Therefore, to treat such cases, it is necessary to introduce a theory of nonlinear elasticity with small strains. If the strains are small, the deformation in the neighborhood of each point can be identified with a deformation to which the linear theory is applicable. This gives a rationale for adopting Hooke's stress-strain relations, and in the resulting nonlinear theory large parts of the classical theory are preserved (Stoker 1968). However, the original and deformed configuration of a solid now cannot be assumed to be coincident, and the strains and stresses can be evaluated in the original undeformed configuration by using Lagrangian description, or in the deformed configuration by using Eulerian description (Fung 1965).
In this chapter, the classical geometrically nonlinear theories for rectangular plates, circular cylindrical shells, circular plates and spherical shells are derived, classical theories being those that neglect the shear deformation. Results are obtained in Lagrangian description, the effect of geometric imperfections is considered and the formulation of the elastic strain energy is also given. Classical theories for shells of any shape, as well as theories including shear deformation, are addressed in Chapter 2.
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- Publisher: Cambridge University PressPrint publication year: 2008
References
2003 Journal of Sound and Vibration 264, 1091–1125. Comparison of shell theories for large-amplitude vibrations of circular cylindrical shells: Lagrangian approach.CrossRefGoogle Scholar
and 2003 Applied Mechanics Reviews 56, 349–381. Review of studies on geometrically nonlinear vibrations and dynamics of circular cylindrical shells and panels, with and without fluid-structure interaction.CrossRefGoogle Scholar
1968 Journal of Applied Mechanics 35, 393–401. Notes on nonlinear shell theory.CrossRefGoogle Scholar
1828 Exercises de Mathematique 3, 328–355. Sur l'équilibre et le mouvement d'une plaque solid.Google Scholar
and 1956 Journal of Applied Mechanics 23, 532–540. Influence of large amplitude on free flexural vibrations of rectangular elastic plates.Google Scholar
and 1990 International Journal of Non-Linear Mechanics 25, 67–85. Large displacement and rotation formulation for laminated shells including parabolic transverse shear.CrossRefGoogle Scholar
1934 Transactions of the ASME 56, 795–806. A new theory for the buckling of thin cylinders under axial compression and bending.Google Scholar
1973 ASME Journal of Applied Mechanics 40, 471–477. Large-amplitude forced vibrations of simply supported thin cylindrical shells.Google Scholar
and 1999 AIAA Journal 37, 1489–1494. Nonlinear dynamic finite element analysis of composite cylindrical shells considering large rotations.CrossRefGoogle Scholar
T. von Kármán 1910 Festigkeitsprobleme im Maschinenbau. Encyklopadie der Mathematischen Wissenschaften. Vol. 4, Heft 4, 311–385.
and 1941 Journal of the Aeronautical Sciences 8, 303–312. The buckling of thin cylindrical shells under axial compression.CrossRefGoogle Scholar
1850 Journal für die Reine und Angewandte Mathematik (Crelle's) 40, 51–88. Uber das gleichgewicht und die bewegung einer elastischen scheibe.CrossRefGoogle Scholar
and 1995 International Journal of Non-Linear Mechanics 30, 57–66. Large-amplitude free vibration of thick shallow shells supported by shear diaphragms.CrossRefGoogle Scholar
1966 Proceedings Koninklijke Nederlandse Akademie van Wetenschappen B 69, 1–54. On the nonlinear theory of thin elastic shells. I, II, III.Google Scholar
and 1988 The Nonlinear Theory of Elastic Shells, 2nd edition 1998. Academic Press, London, UK.Google Scholar
1987 Quarterly of Applied Mathematics 45, 1–22. Refined geometrically nonlinear theories of anisotropic laminated shells.CrossRefGoogle Scholar
and 1957 Non-Linear Theory of Thin Elastic Shells. Academy of Sciences (Nauka), Kazan'; English version, NASA-TT-F62 in 1961.Google Scholar
and 1963 Quarterly of Applied Mathematics 21, 49–59. On the nonlinear theory of elastic shells under the Kirchhoff hypothesis.CrossRefGoogle Scholar
1953 Foundations of the Nonlinear Theory of Elasticity. Graylock Press, Rochester, NY, USA (now available from Dover, NY, USA).Google Scholar
and 1994 Nonlinear Dynamics 6, 459–500. A unified nonlinear formulation for plate and shell theories.Google Scholar
1829 Mémoires de l'Académie Royale des Sciences de l'Institut 8, 357–570. Mémoire sur l'eguilibre et le mouvement des corp élastique.Google Scholar
1984 Journal of Engineering Mechanics 110, 794–809. Exact solutions of moderately thick laminated shells.CrossRefGoogle Scholar
1990 International Journal of Non-Linear Mechanics 25, 677–686. A general non-linear third-order theory of plates with moderate thickness.CrossRefGoogle Scholar
2003 Mechanics of Laminated Composite Plates and Shells: Theory and Analysis, 2nd edition. CRC Press, Boca Raton, FL, USA.Google Scholar
and 1985 International Journal of Non-Linear Mechanics 20, 79–90. Geometrically non-linear transient analysis of laminated, doubly curved shells.CrossRefGoogle Scholar
and 1985 International Journal of Engineering Science 23, 319–330. A higher-order shear deformation theory of laminated elastic shells.CrossRefGoogle Scholar
1963 Quarterly of Applied Mathematics 21, 21–36. Nonlinear theories for thin shells.CrossRefGoogle Scholar
, and 1997 Nonlinear Dynamics 13, 279–305. Nonlinear dynamics of shells: theory, finite element formulation, and integration schemes.CrossRefGoogle Scholar
1992 Journal of Pressure Vessel Technology 114, 105–109. Nonlinear analysis of transverse shear deformable laminated composite cylindrical shells. Part I: derivation of governing equations.CrossRefGoogle Scholar
and 1991 International Journal of Non-Linear Mechanics 26, 379–388. On the finite element analysis of non-linear vibration for cylindrical shells with high-order shear deformation theory.CrossRefGoogle Scholar
and 2000 International Journal of Solids and Structures 37, 1663–1677. Active control of nonlinear piezoelectric circular shallow spherical shells.CrossRefGoogle Scholar
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