“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-of-plane 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 (Nat. Mater. 6(8):563–567, 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.

Appendix:  Curvature Tensor of a Developable Surface

Here, we use the notation introduced in Sect. 2.1 to prove the geometrical identities announced in Sects. 2.3 to 2.5, relevant to developable surfaces. We consider a general ruled surface, enforce the condition of developability, and derive the expression of the curvature tensor at an arbitrary point on the surface. By doing so, we extend the expressions obtained by Wunderlich [28] and by Starostin and van der Heijden [25] to account for the geodesic curvature κ _g of the center-line.

Let us first calculate the tangent vectors at an arbitrary point y(S,V) of the surface:

$$\begin{aligned} \mathbf{y}_{,S}(S,V) & = \mathbf{d}_{3}(S) + V \mathbf{q}'(S), \end{aligned}$$

(35a)

$$\begin{aligned} \mathbf{y}_{,V}(S,V) & = \mathbf{q}(S) , \end{aligned}$$

(35b)

where q′(S) denotes the total derivative of q defined in Eq. (4) as q=η d ₃+d ₁.

Using Eq. (6), we write the following equation

$$ \mathbf{q}' = \eta \omega_{2} \mathbf{d}_{1} + (\omega_{3} - \eta \omega_{1}) \mathbf{d}_{2} + \bigl(\eta'-\omega_{2}\bigr) \mathbf{d}_{3} , $$

(36)

which implies

$$ \mathbf{q} \times \mathbf{q}' = (\eta \omega_{1}- \omega_{3}) \mathbf{q}^\perp + \frac{1}{V_{\mathrm{c}}} \mathbf{d}_{2}, $$

(37)

where V _c is the quantity defined by Eq. (13*), and q ^⊥ is the vector

$$ \mathbf{q}^\perp = \mathbf{d}_{2} \times \mathbf{q} = - \mathbf{d}_{3} + \eta \mathbf{d}_{1} . $$

(38)

Later on, we shall show that q ^⊥ is a vector perpendicular to q lying in the plane tangent to the surface; hence the notation.

The classical condition for a ruled surface to be developable [23, Sect. 3.II] is that the following three vectors are linearly dependent: the tangent d ₃ to the center-line (called the directrix in the context of the geometry of surfaces), the vector q spanning the generatrices, and its derivative with respect to the arc-length along the center-line. This is expressed by (q×q′)⋅d ₃=0. In view of Eq. (37), this yields ηω ₁=ω ₃, which is the constraint of developability announced in Eq. (11a*).

As a result, q′⋅d ₂=0, and so y _,S ⋅d ₂=0. On the other hand, Eq. (35b) shows that y _,V ⋅d ₂=0. The director d ₂(S) is orthogonal to both tangents: d ₂ is a unit normal at any point of the developable surface.

The element of area on the surface reads

$$ \mathrm{d} a = \vert \mathbf{y}_{,S}\times \mathbf{y}_{,V} \vert \, \mathrm{d}S \,\mathrm{d}V = \bigl\vert \mathbf{d}_{3}\times \mathbf{q} + V \mathbf{q}'\times \mathbf{q}\bigr\vert \,\mathrm{d}S\, \mathrm{d}V = \biggl\vert \biggl(1-\frac{V}{V_{\mathrm{c}}} \biggr) \mathbf{d}_{2} \biggr\vert \,\mathrm{d}S \,\mathrm{d}V. $$

(39)

Noting that 1−V/V _c>0 by the inequality (14b), we arrive at the result announced in Eq. (12).

To compute the curvature tensor K(S,V), we note that the direction of the generatrix is a principal direction of zero curvature, since the surface is developable. Therefore, there exists some scalar field k(S,V) such that

$$ \mathbf{K} = k \mathbf{q}^\perp \otimes \mathbf{q}^\perp . $$

(40)

The quantity k in equation above can be found by contracting with y _,S on both sides of the equation to give:

$$ \mathbf{y}_{,S}\cdot \mathbf{K}\cdot \mathbf{y}_{,S} =k \bigl(\mathbf{q}^\perp \cdot \bigl(\mathbf{d}_{3}+V \mathbf{q}' \bigr) \bigr)^2 = k \biggl(-1+ \frac{V}{V_{\mathrm{c}}} \biggr)^2 . $$

(41)

By the definition of the curvature tensor (second fundamental form) [23], the left-hand side of the resulting identity is the normal projection of the second derivative y _,SS :

$$ \mathbf{y}_{,S}\cdot \mathbf{K}\cdot \mathbf{y}_{,S} = \mathbf{y}_{,SS}\cdot \mathbf{d}_{2} = \bigl( \mathbf{d}_{3}' + V \mathbf{q}'' \bigr)\cdot \mathbf{d}_{2} = -\omega_{1} +V \biggl( \frac{\mathrm{d}( \mathbf{q}'\cdot \mathbf{d}_{2})}{\mathrm{d}S} - \mathbf{q'}\cdot \mathbf{d}_{2}' \biggr). $$

(42)

In this equation, q′⋅d ₂=ω ₃−ηω ₁=0 by the developability condition, and $\mathbf{q}'\cdot \mathbf{d}_{2}' = \mathbf{q}'\cdot (\boldsymbol{\omega}\times \mathbf{d}_{2}) = \mathbf{q}'\cdot (\omega_{1} \mathbf{q}\times \mathbf{d}_{2}) = - \omega_{1} \mathbf{d}_{2}\cdot(\mathbf{q}\times \mathbf{q}') = -\omega_{1} / V_{\mathrm{c}}$. Therefore,

$$ \mathbf{y}_{,S}\cdot \mathbf{K}\cdot \mathbf{y}_{,S} = - \omega_{1} \biggl(1-\frac{V}{V_{\mathrm{c}}} \biggr). $$

(43)

From Eqs. (41) and (43), we can solve for k, giving k=−ω ₁/(1−V/V _c). Inserting this result into Eq. (40) yields the expression of curvature tensor announced in Eqs. (15) and (16*). The curvature tensor keeps the same form as in the case of zero geodesic curvature [25, 28] provided that the proper definition of V _c in terms of κ _g is used; see Eq. (13*).