“Wunderlich, Meet Kirchhoff”: A General and Unified Description of Elastic Ribbons and Thin Rods

Abstract

The equations for the equilibrium of a thin elastic ribbon are derived by adapting the classical theory of thin elastic rods. Previously established ribbon models are extended to handle geodesic curvature, natural out-ofplane curvature, and a variable width. Both the case of a finite width (Wunderlich's model) and the limit of small width (Sadowksky's model) are recovered. The ribbon is assumed to remain developable as it deforms, and the direction of the generatrices is used as an internal variable. Internal constraints expressing inextensibility are identified. The equilibrium of the ribbon is found to be governed by an equation of equilibrium for the internal variable involving its second-gradient, by the classical Kirchhoff equations for thin rods, and by specific, thin-rod-like constitutive laws; this extends the results of Starostin and van der Heijden (2007) to a general ribbon model. Our equations are applicable in particular to ribbons having geodesic curvature, such as an annulus cut out in a piece of paper. Other examples of application are discussed. By making use of a material frame rather than the Frenet-Serret frame, the present work unifies the description of thin ribbons and thin rods.

Buckling of Naturally Curved Elastic Strips: The Ribbon Model Makes a Difference

“…In another paper in this special issue [3], Dias and Audoly extend Sadowsky's model to handle natural curvature. They also derive equilibrium equations for ribbons by adapting the classical approach, i.e., by writing a principle of virtual work for elastic ribbons analogous to elastic rods.…”

“…Following [3], we refrain from eliminating the parameter η, and view instead η and ω 3 as two variables linked by a kinematical constraint. The ribbons we consider here have zero geodesic curvature, i.e., ω 2 = 0 in the reference configuration, and their midsurface is inextensible, as previously noted.…”

“…The strain energy stored in the deformed ribbon is derived in Eq. (34) of reference [3] from the energy of an inextensible cylindrical shell as…”

“…To this end, we use the ribbon model introduced by Dias and Audoly in another paper of this special issue [3], which extends both the energy proposed by Sadowsky [20] and the equations of equilibrium derived by Starostin and van der Heijden [22] for a flat ribbon, to a naturally curved ribbon. This curved ribbon model is formulated mathematically as a constrained elastic rod: like rods, but unlike other ribbon models, it makes use of a frame of directors; in this model, the inextensibility 1 of the midsurface of the ribbon is taken into account by specifying kinematical constraints, which gives rise to an effectively nonlinear constitutive law [3].…”

“…This curved ribbon model is formulated mathematically as a constrained elastic rod: like rods, but unlike other ribbon models, it makes use of a frame of directors; in this model, the inextensibility 1 of the midsurface of the ribbon is taken into account by specifying kinematical constraints, which gives rise to an effectively nonlinear constitutive law [3]. Being expressed in the same language as an elastic rod model, the ribbon model of [3] enables us in the present paper (i) to set up a stability analysis for a one-dimensional ribbon model for the first time, by a straightforward adaptation of the classical stability methods used for rods, and (ii) to carry out a detailed comparison of the predictions of the rod and ribbon models.…”