In this work, we make a distinction between the differential geometric notion of an isometry relationship among two dimensional surfaces embedded in three-dimensional point space and the continuum mechanical notion of an isometric deformation of a two dimensional material surface. We illustrate the importance of separating the abstract theory of surfaces in differential geometry and their related differential geometric features from the physical notion of a material surface which is subject to a deformation from a given reference configuration. In differential geometry, while two surfaces may be isometric, the mapping between them that characterizes the isometry is simply a mapping between the points of the surfaces and not necessarily between corresponding material particles of a single deformed material surface.
We review two equivalent characterizations of a smooth isometric deformation of a flat material surface into a curved surface, and emphasize the requirement that the referential directrix and rulings, and their deformed counterparts, must provide a basis for establishing a complete curvilinear coordinate covering of the material surface in both the reference and deformed states. Because this covering requirement has been overlooked in recent publications concerning the isometric bending of ribbons, we illustrate its importance in properly defining the deformation of a ribbon in the two examples of a flat rectangular material strip that is isometrically deformed into either (i) a portion of a circular cylindrical surface, or (ii) a portion of a circular conical surface. We then show how the accurate calculation of the bending energy in these two examples is influenced by this oversight. In example (i), the curvature along the generators of the deformed surface, generally helical in form, is constant. In this special circumstance overlooking the covering requirement, as has been done in the literature by integrating the specific bending energy, dependent only on the curvature, over a domain on the supporting circular cylindrical surface equal in area, though not equal in geometric form, to that of the deformed ribbon, gives the correct bending energy result. In example (ii), the curvature along a generator of the cone is not constant and the calculation of the bending energy is, indeed, compromised by this oversight.
The historically important dimensional reductions that Sadowsky and Wunderlich introduced to study the bending energy and the equilibrium configurations of isometrically deformed rectangular strips have gained classical notoriety within the subject of elastic ribbons and Möbius bands. We show that the Sadowsky and Wunderlich functionals also overlook the covering requirement and that the exact bending energy is underestimated by these functionals, the Sadowsky functional being the lowest. We then show that the error in using these functionals can be great for a rectangular strip of given length ℓ and width w, depending on the form of the isometric deformation and the size of the half-length-to-width ratio w/2ℓ. The Sadowsky functional is meant to apply to strips for which w/2ℓ is sufficiently small, in which case the covering requirement is of little consequence, and for such strips it yields an acceptable approximation of the actual bending energy. In such cases the Wunderlich functional shows only an incremental improvement over the Sadowsky calculation. While the Wunderlich functional is meant to apply accurately for all strips, without restricting the size of w/2ℓ, we show that in overlooking the covering requirement it greatly underestimates the actual bending energy for many isometrically deformed ribbons. In particular, we show relative errors between the exact, the Wunderlich, and the Sadowsky calculations of the bending energy as a function of w/2ℓ for the case of a rectangular strip which is isometrically deformed into a portion of a right circular conical surface, and we observe that the error in approximating the exact bending energy by the Wunderlich functional for reasonable ratios w/2ℓ is large and unacceptable. We then give an example of the isometric deformation of a rectangular strip whose Wunderlich functional predicts zero bending energy but for which the exact bending energy can be as large as one pleases.
Finally, contrary to suggestions in the literature, we argue that Kirchhoff rod theory does not generally apply to the study of the isometric deformation of a thin rectangular strip because for this class of problems the through thickness dimension of the strip is assumed to be infinitesimal as compared to its width w. For Kirchhoff rod theory to apply, these dimensions must be comparable.