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

Abstract

We consider the problem of characterizing the smooth, isometric deformations of a planar material region identified with an open, connected subset D of two-dimensional Euclidean point space E2 into a surface S in three-dimensional Euclidean point space E3. To be isometric, such a deformation must preserve the length of every possible arc of material points on D. Characterizing the curves of zero principal curvature of S is of major importance. After establishing this characterization, we introduce a special curvilinear coordinate system in E2, based upon an à priori chosen pre-image form of the curves of zero principal curvature in D, and use that coordinate system to construct the most general isometric deformation of D to a smooth surface 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 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).