A true shell has double curvature: If undeformed, its Gaussian curvature vanishes nowhere, except possibly along certain curves (such as the crown of a toroid) or at isolated points (such as the apex of a very flat dome). Shells of revolution, with the exception of conical and cylindrical shells, are the simplest, useful class of true shells. It is a corollary of Gauss’ Theorema egregium (Struik, 1961) that if a surface is bent, its Gaussian curvature will change only if the surface is simultaneously stretched. Thus, a shell will be relatively stiff if, by virtue of symmetry, loading, or boundary conditions, inextensional deformation is impossible. Such is the case with a shell of revolution that undergoes torsionless, axisymmetric deformation without rigid body motion. This feature of axishells is in contrast to the behavior of beamshells where near inextensional bending is typical.

In a beamshell there is one extensional strain, one shearing strain, and one bending strain; an arbitrary prescription of these three fields always yields some displacement field. However, in an axishell there are six strains: two extensional strains, one shearing strain, two bending strains, and one normal bending strain; these fields cannot be prescribed independently but must satisfy compatibility conditions if an associated displacement field is to exist. There is one differential compatibility condition if the rotation is introduced as a dependent variable, but three—two are differential and one is algebraic—if the strains are the basic kinematic unknowns.