@article{oai:oist.repo.nii.ac.jp:00000830,
 author = {Chen, Yi-Chao and Fosdick, Roger and Fried, Eliot},
 journal = {Journal of the Mechanics and Physics of Solids},
 month = {Mar},
 note = {The stored energy of an unstretchable material surface is assumed to depend only upon the curvature tensor. By control of its edge(s), the surface is deformed isometrically from its planar undistorted reference configuration into an equilibrium shape. That shape is to be determined from a suitably constrained variational problem as a state of relative minimal potential energy. We pose the variational problem as one of relative minimum potential energy in a spatial form, wherein the deformation of a flat, undistorted region D in E-2 to its distorted form S in E-3 is assumed specified. We then apply the principle that the first variation of the potential energy, expressed as a functional over S boolean OR partial derivative S, must vanish for all admissible variations that correspond to isometric deformations from the distorted configuration S and that also contain the essence of flatness that characterizes the reference configuration 1:), but is not covered by the single statement that the variation of S correspond to an isometric deformation. We emphasize the commonly overlooked condition that the spatial expression of the variational problem requires an additional variational constraint of zero Gaussian curvature to ensure that variations from S that are isometric deformations also contain the notion of flatness. In this context, it is particularly revealing to observe that the two constraints produce distinct, but essential and complementary, conditions on the first variation of S. The resulting first variation integral condition, together with the constraints, may be applied, for example, to the case of a flat, undistorted, rectangular strip D that is deformed isometrically into a closed ring S by connecting its short edges and specifying that its long edges are free of loading and, therefore, subject to zero traction and couple traction. The elementary example of a closed ring without twist as a state of relative minimum potential energy is discussed in detail, and the bending of the strip by opposing specific bending moments on its short edges is treated as a particular case. Finally, the constrained variational problem, with the introduction of appropriate constraint reactions as Lagrangian multipliers to account for the requirements that the deformation from D to S is isometric and that D is flat, is formulated in the spatial form, and the associated Euler-Lagrange equations are derived. We then solve the Euler-Lagrange equations for two representative problems in which a planar undistorted rectangular material strip is isometrically deformed by applied edge tractions and couple tractions (i.e., specific edge moments) into (i) a bent and twisted circular cylindrical helical state, and (ii) a state conformal with the surface of a right circular conical form.},
 pages = {290--322},
 title = {Isometric deformations of unstretchable material surfaces, a spatial variational treatment},
 volume = {116},
 year = {2018}
}