A New Derivation of the Equations for the Deformation of Elastic Shells

The Classical Field Theories

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

Vibration analysis of rectangular plates coupled with fluid

Accurate equations for laminated composite deep thick shells

We consider two of the simplest problems associated with the packing of a naturally flat thin elastic sheet. Both problems involve packing the sheet into a hollow cylinder; the first considers the partial contact of a cylindrically curved sheet with a cylindrical surface, while the second considers the partial contact of a conically curved sheet with the edge of a cylindrical surface. In each case, we solve the free-boundary problems to determine the shape, response and stability of the confined surfaces. In particular, we show that an exact description of both the cylindrical and conical structures is given by solutions of the Elastica equation, allowing us to present a unified description of a large class of elastic developable surfaces. This includes what is possibly the simplest example of strain localization, occurring at a point and forming one of the constituent elements of a crumpled elastic sheet.

A New Derivation of the Equations for the Deformation of Elastic Shells

Citations

The Classical Field Theories

The Theory of Shells and Plates

Vibration analysis of rectangular plates coupled with fluid

Accurate equations for laminated composite deep thick shells

Confined developable elastic surfaces: cylinders, cones and the Elastica

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