ABSTRACT
A special case of elasticity with two independent variables that is not plane elasticity (Chapter 4) is a membrane. If the boundary curve is a plane, the membrane remains a plane under tension, unless a transverse force is applied, for example, its own weight; the transverse force deflects the membrane until it is balanced by the transverse component of the tension, so that the deflection is linear (nonlinear) for small (large) slope [Section 6.1 (Section 6.2)]. The surface tension acts as a membrane, since it is associated with pressure or stress differences across an interface between two fluids (Section 6.3); examples are capillary effects (Section 6.4) such as a liquid wetting a wall at an angle or the rise in the free surface of a liquid between walls. A third less obvious analogy is the torsion of a slender straight rod: the cross section is similar to a membrane (Section 6.5), and the moment applied at the ends of the rod causes a relative rotation of successive cross sections (Section 6.6). The cross section of the bar under torsion can be singly or multiply connected; for example, a triangular prism (Section 6.8) has a simply connected cross section, whereas a hollow ellipse (Section 6.7) has a doubly connected cross section. A fourth analogous problem is a potential flow in a rotating vessel (Section 6.9) where the centrifugal energy adds a rotational component that affects the trajectories of fluid particles. Eight more analogies (Notes 6.6 through 6.9) lead to a total of 12.
