Abstract
In this chapter we describe and analyse the linear shell models that we consider in this book. We first describe the fundamental shell kinematics used. Then we discuss the “basic shell model” which is implicitly employed in general finite element solutions and from which other classical shell and plate models can be derived. We summarize the shell models that we call the “shear-membrane-bending model” and the “membrane-bending model”, and introduce the proper mathematical framework in which they define well-posed problems. As special cases of these shell models we obtain well-known plate models.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Bibliography
Ciarlet, P.G. (2000). Mathematical Elasticity - Volume III: Theory of Shells. Amsterdam: North-Holland.
Kim, D.N., & Bathe, K.J. (2008). A 4-node 3D-shell element to model shell surface tractions and incompressible behavior. Comput. & Structures, 86(21-22), 2027-2041.
Reissner, E. (1945). The effect of transverse shear deformation on the bending of elastic plates. J. Appl. Mech., 67, A69-A77.
Bathe, K.J., & Wilson, E.L. (1974). Thick shells. In W. Pilkey, K. Saczalski, & H. Schaeér (Eds.) Structural Mechanics Computer Programs.
Sansour, C. (1995). A theory and finite element formulation of shells at finite deformations involving thickness change: circumventing the use of a rotation tensor. Arch. Appl. Mech., 65, 194-216.
Flügge, W. (1973). Stresses in Shells. New York, Heidelberg: Springer-Verlag, 2nd ed.
Coutris, N. (1978). Théorème d’existence et d’unicité pour un problème de coque élastique dans le cas d’un modèle linéaire de P.M. Naghdi. R.A.I.R.O. Anal. Numér., 12, 51-57.
Green, A.E., & Zerna, W. (1968). Theoretical Elasticity. Oxford: Clarendon Press, 2nd ed.
Timoshenko, S., & Woinowsky-Krieger, S. (1959). Theory of Plates and Shells. New York: McGraw-Hill.
Mindlin, R.D. (1951). Inuence of rotary inertia and shear on exural motion of isotropic elastic plates. J. Appl. Mech., 18, 31-38.
Reissner, E. (1952). Stress strain relations in the theory of thin elastic shells. J. Math. Phys., 31, 109-119.
Ciarlet, P.G. (1988). Mathematical Elasticity - Volume I: Three-Dimensional Elasticity. Amsterdam: North-Holland.
Blouza, A., & Le Dret, H. (1999). Existence and uniqueness for the linear Koiter model for shells with little regularity. Quart. Appl. Math., 57, 317-337.
Valid, R. (1995). The Nonlinear Theory of Shells through Variational Principles. Chichester: John Wiley & Sons.
Bathe, K.J. (1996). Finite Element Procedures. Englewood Cliffs: Prentice Hall.
Bernadou, M., & Ciarlet, P.G. (1975). Sur l’ellipticité du modéle linéaire de coques de W.T. Koiter. In R. Glowinski, & J. Lions (Eds.) Computing Methods in Applied Sciences and Engineering. Heidelberg: Springer-Verlag.
Delfour, M.C. (2000). Tangential diérential calculus and functional analysis on a C1,1 submanifold. In R. Gulliver, W. Littman, & R. Triggiani (Eds.) Diérential-Geometric Methods in the Control of Partial Differential Equations, (pp. 83-115). Providence: AMS.
Calladine, C.R. (1983). Theory of Shell Structures. Cambridge: Cambridge University Press.
Wunderlich, W. (1980). On a consistent shell theory in mixed tensor formulation. In Proc. 3rd IUTAM Symposium on Shell Theory: Theory of Shells. Amsterdam: North Holland.
Bathe, K.J., Sussman, T., & Walczak, J. (201x). Crash and crush shell analyses with implicit integration. In preparation.
Love, A.E.H. (1927). Mathematical Theory of Elasticity. 4th ed.
Alessandrini, S.M., Arnold, D.N., Falk, R.S., & Madureira, A. L. (1999). Derivation and justification of plate models by variational methods. In Plates and shells (Québec, QC, 1996), vol. 21 of CRM Proc. Lecture Notes, (pp. 1-20). Providence: Amer. Math. Soc.
Novozhilov, V.V. (1970). Thin Shell Theory. Groningen: Wolters-Noordhoff Publishing, 2nd ed.
Naghdi, P.M. (1963). Foundations of elastic shell theory. In Progress in Solid Mechanics, vol. 4, (pp. 1-90). Amsterdam: North-Holland.
Koiter, W.T. (1965). On the nonlinear theory of thin elastic shells. Proc. Kon. Ned. Akad. Wetensch., B69, 1-54.
Chapelle, D., & Ferent, A. (2003). Modeling of the inclusion of a reinforcing sheet within a 3D medium. Math. Models Methods Appl. Sci., 13, 573-595.
Delfour, M.C. (1999). Intrinsic P(2,1) thin shell model and Naghdi’s models without a priori assumption on the stress tensor. In K. Hoffmann, G. Leugering, & F. Tröltzsch (Eds.) Optimal Control of Partial Diérential Equations, (pp. 99-113). Basel: Birkhäuser.
Hencky, H. (1947). Über die Berücksichtigung der Schubverzerrung in ebenen Platten. Ingenieur Archiv, 16, 72-76.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Chapelle, D., Bathe, KJ. (2011). Shell Mathematical Models. In: The Finite Element Analysis of Shells - Fundamentals. Computational Fluid and Solid Mechanics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-16408-8_4
Download citation
DOI: https://doi.org/10.1007/978-3-642-16408-8_4
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-16407-1
Online ISBN: 978-3-642-16408-8
eBook Packages: EngineeringEngineering (R0)