Representation of a Smooth Isometric Deformation of a Planar Material Region into a Curved Surface

We consider the problem of characterizing the smooth, isometric deformations of a planar material region identified with an open, connected subset ({mathcal{D}}) of two-dimensional Euclidean point space (mathbb{E}^{2}) into a surface ({mathcal{S}}) in three-dimensional Euclidean point space (mathbb{E}^{3}). To be isometric, such a deformation must preserve the length of every possible arc of material points on ({mathcal{D}}). Characterizing the curves of zero principal curvature of ({mathcal{S}}) is of major importance. After establishing this characterization, we introduce a special curvilinear coordinate system in (mathbb{E}^{2}), based upon an à priori chosen pre-image form of the curves of zero principal curvature in ({mathcal{D}}), and use that coordinate system to construct the most general isometric deformation of ({mathcal{D}}) to a smooth surface ({mathcal{S}}). A necessary and sufficient condition for the deformation to be isometric is noted and alternative representations are given. Expressions for the curvature tensor and potentially nonvanishing principal curvature of ({mathcal{S}}) are derived. A general cylindrical deformation is developed and two examples of circular cylindrical and spiral cylindrical form are constructed. A strategy for determining any smooth isometric deformation is outlined and that strategy is employed to determine the general isometric deformation of a rectangular material strip to a ribbon on a conical surface. Finally, it is shown that the representation established here is equivalent to an alternative previously established by Chen, Fosdick and Fried (J. Elast. 119:335–350, 2015).