A Variational Model for Anisotropic and Naturally Twisted Ribbons

Abstract

We consider thin plates whose energy density is a quadratic function of the difference between the second fundamental form of the deformed configuration and a "natural" curvature tensor. This tensor either denotes the second fundamental form of the stress-free configuration, if it exists, or a target curvature tensor. In the latter case, residual stress arises from the geometrical frustration involved in the attempt to achieve the target curvature: as a result, the plate is naturally twisted, even in the absence of external forces or prescribed boundary conditions. Here, starting from this kind of plate energies, we derive a new variational one-dimensional model for naturally twisted ribbons by means of Γ-convergence. Our result generalizes, and corrects, the classical Sadowsky energy to geometrically frustrated anisotropic ribbons with a narrow, possibly curved, reference configuration.

One-dimensional von Kármán models for elastic ribbons

“…Arguing as in [10,Lemma 12], we see that for every ε ≤ ε j the matrix ∇Φ j (ξ 1 , ξ 2 ) is invertible for |ξ 2 | ≤ ε, and the map (Φ j ) −1 : S ε → R 2 is well defined. For such ε define z j : S ε → R by setting…”

“…Wunderlich's technique is quite ingenious, but it leads to a singular energy density; we refer to [17] for a rigorous analysis of the so-called Wunderlich energy. A corrected Sadowsky type of energy was derived in [9] and generalized in [1,10].…”

Asymptotically exact strain-gradient models for nonlinear slender elastic structures: A systematic derivation method

“…There exists a variety of models for slender structures, going much beyond the traditional models for the stretching of bars and the bending of beams. The applicability of classical models being limited to materials having linear, homogeneous and isotropic elastic properties, a number of extensions have been considered to account for different elastic behaviors such as hyperelastic materials (Cimetière et al, 1988) or more specifically nematic elastomers (Agostiniani et al, 2016), for inhomogeneous elastic properties in the crosssection, for the presence of natural curvature or twist (Freddi et al, 2016) or more generally for the existence of inhomogeneous pre-stress in the cross-section (Lestringant and Audoly, 2017). As the classical rod models are inapplicable if the cross-section itself is a slender 2d domain, specific models have been derived, e.g., to address inextensible ribbons (Sadowsky, 1930;Wunderlich, 1962), as well as thin walled beams having a flat (Freddi et al, 2004) or curved (Hamdouni and Millet, 2006) cross-section.…”

Capturing the helical to spiral transitions in thin ribbons of nematic elastomers

“…Our result confirms that the switching from one configuration to another is the result of the competition between stretching energy and bending energy. A possible developments of this approach might be a further dimension reduction where the width of the strip tends to null, using a variational approach as in [10,2] or a deductive approach based on reduced kinematics in the spirit of [13,14].…”