Shell Mathematical Models

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.

[1]  E. Reissner.  The effect of transverse shear deformation on the bending of elastic plates , 1945 .

[2]  Adel Blouza,et al.  Existence and uniqueness for the linear Koiter model for shells with little regularity , 1999 .

[3]  Carlo Sansour,et al.  A theory and finite element formulation of shells at finite deformations involving thickness change: Circumventing the use of a rotation tensor , 1995, Archive of Applied Mechanics.

[4]  P. M. Naghdi,et al.  FOUNDATIONS OF ELASTIC SHELL THEORY , 1962 .

[5]  Klaus-Jürgen Bathe,et al.  A 4-node 3D-shell element to model shell surface tractions and incompressible behavior , 2008 .

[6]  H. Hencky,et al.  Über die Berücksichtigung der Schubverzerrung in ebenen Platten , 1947 .

[7]  P. Ciarlet,et al.  Mathematical elasticity, volume I: Three-dimensional elasticity , 1989 .

[8]  R. Glowinski,et al.  Computing Methods in Applied Sciences and Engineering , 1974 .

[9]  R. D. Mindlin,et al.  Influence of rotary inertia and shear on flexural motions of isotropic, elastic plates , 1951 .

[10]  R. Valid.  The Nonlinear Theory of Shells Through Variational Principles: From Elementary Algebra to Differential Geometry , 1996 .

[11]  N. Coutris.  Théorème d'existence et d'unicité pour un problème de coque élastique dans le cas d'un modèle linéaire de P. M. Naghdi , 1978 .

[12]  E. Reissner,et al.  Stress Strain Relations in the Theory of Thin Elastic Shells , 1952 .

[13]  V. V. Novozhilov,et al.  Thin shell theory , 1964 .

[14]  I. S. Sokolnikoff.  Mathematical theory of elasticity , 1946 .

[15]  Dr.-Ing. C. Sansour.  A Theory and finite element formulation of shells at finite deformations involving thickness change , 1995 .

[16]  Dominique Chapelle,et al.  MODELING OF THE INCLUSION OF A REINFORCING SHEET WITHIN A 3D MEDIUM , 2003 .

[17]  S. Timoshenko,et al.  THEORY OF PLATES AND SHELLS , 1959 .

[18]  M. Delfour.  Intrinsic P (2, 1) Thin Shell Model and Naghdi's Models without A Priori Assumption on the Stress Tensor , 1999 .

[19]  Walter D. Pilkey,et al.  Structural Mechanics Computer Programs , 1975 .

[20]  P. G. Ciarlet,et al.  Sur L’Ellipticite du Modele Lineaire de coques de W.T. Koiter , 1976 .

[21]  W. Flügge.  Stresses in Shells , 1960 .

[22]  Douglas N. Arnold,et al.  Derivation and Justification of Plate Models by Variational Methods , 2000 .

[23]  T. Charlton.  Progress in Solid Mechanics , 1962, Nature.

[24]  K. Bathe.  Finite Element Procedures , 1995 .

[25]  C. R. Calladine,et al.  Theory of Shell Structures , 1983 .