A Corrected Sadowsky Functional for Inextensible Elastic Ribbons

Abstract

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 and width w, depending on the form of the isometric deformation and the size of the half-len...

One-dimensional von Kármán models for elastic ribbons

“…It incorporates into its definition the non-linear constraint (1.1) that appears into the two-dimensional model (CvK) ε . The energy density Q agrees with the corrected Sadowsky energy density found in [9] in the isotropic case, and with that found in [10] for the general anisotropic case. To the best of our knowledge, the model (CvK) is new.…”

Dimension reduction via $$\Gamma $$ Γ -convergence for soft active materials

“…An adaptation of the theoretical results of [17] allows to prove in [6] a compactness result for sequences {u ε } such that E ε (u ε ) is uniformly bounded, and a Γ-convergence result for the functionals E ε . We gather in Theorem 2.3 below the most important consequences of these results.…”

“…Building upon our recent work [3,6], which has been in turn inspired by [17,18,24], we discuss in this paper the derivation of models capable of describing non-trivial shapes that can be spontaneously exhibited by thin sheets of nematic elastomers. In particular, we focus on the twist director geometry, and we present a derivation of reduced models for wide and narrow ribbons (namely, rods whose cross-section is a thin rectangle) by following a sequential dimension reduction technique, namely, a 3D→2D→1D procedure, see Section 2.…”

Forceless Sadowsky strips are spherical

“…The class of functionals with this property may be wider. However, it does not include the corrected Sadowsky functional constructed in [8] (although this correction only affects solutions where |η| > 1, so solutions for which |η| ≤ 1 everywhere are still spherical). Nor does it include the narrow limit (w → 0) of the functional for annular strips derived in [6], nor, seemingly, the functional for narrow residually-stressed strips derived in [7].…”