A plate and more generally a shell is a special three-dimensional body whose boundary surface has special features. Although we defer defining a shell-like body in precise terms until Sect. 4, for the purpose of these preliminary remarks consider a surface—called a reference surface—and imagine material filaments from above and below surrounding the surface along the normal at each point of the reference surface. Suppose further that the bounding surfaces formed by the end points of the material filaments are equidistant from the reference surface. Such a three-dimensional body is called a shell if the dimension of the body along the normals, called the thickness, is small. A shell is said to be thin if its thickness is much smaller than a certain characteristic length of the reference surface, e.g., the minimum radius of the curvature of the reference surface for initially curved shells.2

The Theory of Shells and Plates

A theory for three–dimensional finite deformations of elastic solids with conforming elastic films attached to their bounding surfaces is described. The Gurtin–Murdoch theory incorporating elastic resistance of the film to strain is generalized to account for the effects of intrinsic flexural resistance. This modification yields a model that can be used to describe equilibrium deformations in the presence of compressive–surface stress fields. An associated variational theory is given and material symmetry considerations are discussed. The theory is illustrated by examples.

Elastic surface—substrate interactions

The effect of interfacial bending stiffness on the deformation of liquid capsules enclosed by elastic membranes is discussed and investigated by numerical simulation. Flow-induced deformation causes the development of in-plane elastic tensions and bending moments accompanied by transverse shear tensions due to the non-infinitesimal membrane thickness or to a preferred configuration of an interfacial molecular network. To facilitate the implementation of the interfacial force and torque balance equations involving the hydrodynamic traction exerted on either side of the interface and the interfacial tensions and bending moments developing in the plane of the interface, a formulation in global Cartesian coordinates is developed. The balance equations involve the Cartesian curvature tensor defined in terms of the gradient of the normal vector extended off the plane of the interface in an appropriate fashion. The elastic tensions are related to the surface deformation gradient by constitutive equations derived by previous authors, and the bending moments for membranes whose unstressed shape has uniform curvature, including the sphere and a planar sheet, arise from a constitutive equation that involves the instantaneous Cartesian curvature tensor and the curvature of the resting configuration. A numerical procedure is developed for computing the capsule deformation in Stokes flow based on standard boundary-element methods. Results for spherical and biconcave resting shapes resembling red blood cells illustrate the effect of the bending modulus on the transient and asymptotic capsule deformation and on the membrane tank-treading motion.

Effect of membrane bending stiffness on the deformation of capsules in simple shear flow

Stability of relative equilibria. Part I: The reduced energy-momentum method

This review article summarizes the development of higher order theories for the dynamic analysis of piezoelectric plates, and describes their applications, especially for crystal resonators. The theories are categorized according to how the displacement components and electric potential are assumed to vary through the plate thickness. The greatest attention has been given to ordinary polynomial variations, especially the efforts of RD Mindlin and HF Tiersten and their coworkers. Considerable progress has also been made using trigonometric series representations, especially by PCY Lee and his coworkers. Legendre polynomial and normal mode expansions have also been developed. Both analytical and finite element methods of dealing with problems are described. The article contains 174 references.

Higher-Order Theories of Piezoelectric Plates and Applications

Elastic shell nonlinear theory, introducing nonsymmetric stress tensor and principles of virtual work and objectivity

On a nonlinear theory of elastic shells.

1 Prolegomena to the Theory.- (i) Historical Perspectives.- (ii) Toward the Rational Mechanics of Pseudo-rigid Bodies.- (iii) Toward the Study of Cross-theory Comparison.- 2 Foundations of the Theory of Pseudo-rigid Bodies.- (i) Event Worlds and Body Manifolds.- (ii) Kinematics.- (iii) Equations of Balance.- (iv) Forces and Force-moments.- (v) Rigid-body Mechanics: The Principle of Balance of Angular Momentum.- (vi) Changes of Frame.- (vii) Constitutive Relations.- 3 Consistency with Other Continuum Theories.- (i) The General Notion of Consistency.- (ii) Elasticity Theory. Subtheories.- (iii) The Subtheory of Pseudo-rigid Motions.- (iv) Symmetry. Uniqueness.- (v) Approximation of the Coarsifier.- 4 Explicit Motions of Pseudo-rigid Bodies.- (i) Rigid Motions.- (ii) Roto-deformations.- (iii) Pure Stretch Motions.- (iv) Shearing Motions.- (v) Plane Motions.- 5 Variational Formulation of the Theory.- (i) Lagrangian and Hamiltonian Formulations.- (ii) Symmetry and Conserved Quantities.- (iii) The Stability of Steady Motions: A Special Case.- (iv) Ignorable Coordinates: Remarks on the General Analysis of Stability.- 6 Approximations for Almost-rigid Bodies.- (i) The Problem of Transition to Rigidity.- (ii) Derivation of the Asymptotic Expansion.- (iii) Proof of the Spectral Theorem.- (iv) Special Results for Axially Symmetric Bodies.- (v) Gyroscopic Motions.- References.

The theory of pseudo-rigid bodies

Etude d'une theorie decrivant de maniere assez grossiere les deformations de corps elastiques en termes d'equations differentielles ordinaires. On etudie la structure generale de ce systeme ainsi que ses solutions exactes les plus simples. On montre les avantages pratiques d'une telle theorie sur celle basees sur des equations aux derivees partielles

The dynamics of pseudo-rigid bodies: general structure and exact solutions

Impulsive Motions of Elastic Pseudo-Rigid Bodies

We consider an unbalanced shaft rotating in bearings where there is coulomb friction acting. We show that for certain values of the moments and prodcuts of inertia, motion becomes impossible

H. Cohen

Papers

On a nonlinear theory of elastic shells.

The theory of pseudo-rigid bodies

The dynamics of pseudo-rigid bodies: general structure and exact solutions

Impulsive Motions of Elastic Pseudo-Rigid Bodies

The occurrence of Painleve's paradox in the motion of a rotating shaft