Roadmap to the Morphological Instabilities of a Stretched Twisted Ribbon

Abstract

We address the mechanics of an elastic ribbon subjected to twist and tensile load. Motivated by the classical work of Green (Proc. R. Soc. Lond. Ser. A Math. Phys. Sci. 154(882):430, 1936; 161(905):197, 1937) and a recent experiment (Chopin and Kudrolli in Phys. Rev. Lett. 111(17):174302, 2013) that discovered a plethora of morphological instabilities, we introduce a comprehensive theoretical framework through which we construct a 4D phase diagram of this basic system, spanned by the exerted twist and tension, as well as the thickness and length of the ribbon. Different types of instabilities appear in various “corners” of this 4D parameter space, and are addressed through distinct types of asymptotic methods. Our theory employs three instruments, whose concerted implementation is necessary to provide an exhaustive study of the various parameter regimes: (i) a covariant form of the Föppl–von Kármán (cFvK) equations to the helicoidal state—necessary to account for the large deflection of the highly-symmetric helicoidal shape from planarity, and the buckling instability of the ribbon in the transverse direction; (ii) a far from threshold (FT) analysis—which describes a state in which a longitudinally-wrinkled zone expands throughout the ribbon and allows it to retain a helicoidal shape with negligible compression; (iii) finally, we introduce an asymptotic isometry equation that characterizes the energetic competition between various types of states through which a twisted ribbon becomes strainless in the singular limit of zero thickness and no tension.

Appendices

Appendix A: Hookean Elasticity and Leading Order Stresses and Strains

Our theory addresses the “corner” in the 4d parameter space, defined by Eqs. (1), (2), and therefore most of the analysis in this paper employs expansions in these parameters. Why do we assume these parameters are small?

First,t≪1 and L ⁻¹≪1 stem from the definition of a ribbon. Second, we focus our discussion on the universal, material-independent behavior of elastic ribbons and therefore we consider a Hookean response, whereby the stress-strain relationship is linear. Since Hookean response is valid only for small strains, and since the exerted tension T necessarily induces strain, we must require T≪1. Finally, the assumption η≪1 is more subtle. For the unwrinkled helicoidal state, we showed in Sect. 2.3 that the components of the strain and stress tensors are proportional to positive powers of η, and therefore Hookean response is valid only for sufficiently small values of η. In contrast, for the longitudinally-wrinkled helicoidal state (i.e.$,\eta > \sqrt{24 T}$), we showed in Sect. 3.3 that the ribbon may become nearly strainless (i.e., asymptotically isometric to the undeformed ribbon) even under finite η, therefore the Hookean response for the wrinkled state is not limited to small values of η. However, even for the wrinkled state the assumption η≪1 is very useful, since it enables an easy way to compute the various components of the stress tensor and allows us to characterize the wrinkled state as a sinusoidal undulation (Eq. (54) and Appendix C). Importantly, our finding that the threshold values η _lon,η _tr vanish in the asymptotic limit t→0 (see Sect. 4.1 and Fig. 8), proves in a self-consistent manner that the basic morphological instabilities of a stretched-twisted ribbon are well described by assuming η≪1.

Our theory is thus valid at the leading order in t,L ⁻¹, η and T, and any higher order terms are ruled out from the derivations. Leading terms should be understood with respect to these expansion parameters. Namely, denoting by A a scalar, or a component of a vector or tensor (e.g., longitudinal contraction, transverse or longitudinal stress or strain), then A is expanded as

$$ A=\sum_{a_1,a_2,a_3,a_4\geq 0} T^{a_1} \eta^{a_2} t^{a_3} L^{-a_4}A_{(a_1,a_2,a_3,a_4)}, $$

(95)

and the order of A is given by the four positive integers (a _i ). Saying that the order $(a_{i}')$ is higher than the order (a _i ) means that

$$ (a_i)<\bigl(a_i'\bigr) \Longleftrightarrow \left \{\begin{array}{l} \forall i, a_i\leq a_i'\\ \hbox{and}\\ \exists i\ \hbox{such that } a_i<a_i'. \end{array} \right . $$

(96)

The leading terms of A are the minimal orders for the relation (95) with non-zero coefficient. For example, the leading terms of the longitudinal stress in the helicoidal state are given in Eq. (3)$,\sigma^{ss}_{\mathrm{hel}}=T+\frac{\eta^{2}}{2} (r^{2}-\frac{1}{12} )$, they correspond to the orders (1,0,0,0) and (0,2,0,0). These two orders are minimal and cannot be compared. The transverse stress given in Eq. (4) has vanishing coefficients for these orders, and the minimal orders with non zero coefficients are (1,2,0,0) and (0,4,0,0) (i.e.,Tη ² and η ⁴). The s-independent transverse buckling equation (79) contains terms of order (1,2,0,0), (0,4,0,0) and (0,0,2,0) (respectively,Tη ², η ⁴ and t ²).

In a given equation, several orders may appear; in this case only the minimal ones should be considered. This happens in the computation of the stress in the helicoidal state. Equation (33) shows that σ ^rr is of higher order than σ ^ss; besides, one of the two terms in the r.h.s. of Eq. (37) has obviously the same order (0,2,0,0) as σ ^ss in Eq. (36). Thus, consistency of Eqs. (33), (36), (37) implies that the leading terms in the r.h.s. mutually cancel each other (which we call a “solvability condition”), leading to Eq. (38) for $u_{r}'(r)$. This last equation allows to compute the longitudinal stress, Eq. (39), from which we deduce the transverse stress, Eq. (40), using again Eq. (33).

Appendix B: Covariant Derivative: Definition and Application to the Helicoid

For an arbitrary surface with metric g _αβ , the covariant derivative is defined with the Christoffel symbols, that are given by

$$ \varGamma^\alpha_{\beta\gamma}= \frac{1}{2}g^{\alpha\delta} ( \partial_\beta g_{\gamma\delta}+ \partial_\gamma g_{\beta\delta}-\partial_\delta g_{\beta\gamma} ). $$

(97)

The covariant derivative of a vector u ^α is then defined as

$$ D_\alpha u^\beta=\partial_\alpha u^\beta+ \varGamma^\beta_{\alpha\gamma}u^{\gamma}. $$

(98)

The strain tensor has two indices, so that its covariant derivative is

$$ D_\alpha \sigma^{\beta\gamma}=\partial_\alpha \sigma^{\beta\gamma}+\varGamma^\beta_{\alpha\delta}\sigma^{\delta\gamma}+ \varGamma^\gamma_{\alpha\delta}\sigma^{\beta\delta}. $$

(99)

For the helicoid with metric (30)

$$ g_{\alpha\beta}=\begin{pmatrix} 1+\eta^2 r^2 - 2\chi + 2\eta^2 r u_r(r) & 0 \\ 0 & 1+2 u_r'(r) \end{pmatrix}, $$

(100)

the non-zero Christoffel symbols are (to the leading order):

$$\begin{aligned} \varGamma^r_{ss} & = -\eta^2 r, \end{aligned}$$

(101)

$$\begin{aligned} \varGamma^s_{sr} & = \eta^2 r, \end{aligned}$$

(102)

$$\begin{aligned} \varGamma^r_{rr} & = u_r''(r). \end{aligned}$$

(103)

Appendix C: Shape of the Longitudinally Wrinkled Helicoid Far from Threshold

Far from threshold, longitudinal wrinkles relax the longitudinal compression. In the main text, we propose the following form for the wrinkles:

(104)

where the longitudinal contraction is given by . The longitudinal strain in this configuration is

$$ \varepsilon_{ss}(s,r)=\frac{\eta^2}{2} \bigl(r^2-r_\mathrm {wr}^2 \bigr) +\frac{1}{4}k^2f(r)^2- \eta k rf(r)\sin(ks)-\frac{1}{4}k^2f(r)^2\cos(2ks). $$

(105)

Setting $k^{2}f(r)^{2}=2\eta^{2} (r^{2}-r_{\mathrm{wr}}^{2} )$ (Eq. (54)) allows to cancel the s-independent part. However, since this equation implies that the product kf does not vanish in the limit t→0, we find that the s-dependent terms in the above expression for ε _ss remain finite as t→0, in apparent contradiction to our assumption that the wrinkled state becomes asymptotically strainless in the limit t,T→0. This shortcoming can be fixed, however, by adding to the deformation (104) a longitudinal displacement term,u _s (s,r),

(106)

leading to the longitudinal strain

$$ \varepsilon_{ss}(s,r)=-\eta k r f(r)\sin(ks)- \frac{1}{4}k^2f(r)^2\cos(2ks)+\partial_s u_s(s,r). $$

(107)

Setting

$$ u_s(s,r)=-\eta rf(r)\cos(ks)+\frac{1}{8}kf(r)^2 \sin(2 ks), $$

(108)

we find that both s-dependent and s-independent terms of the longitudinal strain ε _ss in the wrinkled zone vanish for t→0 (up to higher order terms in η).

The configuration given by Eqs. (106), (108) has also transverse and shear strains, given by

$$\begin{aligned} \varepsilon_{rr}(s,r) &= \frac{1}{2}f'(r)^2 \cos(ks)^2, \end{aligned}$$

(109)

$$\begin{aligned} \varepsilon_{sr}(s,r) &= -\eta f(r)\cos(ks) - \frac{1}{8}kf(r)f'(r) \sin(2ks). \end{aligned}$$

(110)

However, in contrast to the individual terms in Eq. (105), which are proportional to the product kf (that remains finite as t→0), all terms on the r.h.s. of Eqs. (109), (110) vanish in the limit of small thickness (since, while kf(r) is finite, the amplitude f(r)→0 in this limit).

Appendix D: The Stability Analysis of Kirchoff Rod Equations

Here we translate the relevant results of [9], which addressed the stability analysis of a Kirchoff rod with non-symmetric cross section to the terminology of our paper.

The relevant results for us pertain to the linear stability analysis of the helicoidal (“straight”) state of the ribbon. This is summarized in Eqs. (58) and (59) that provide the threshold for the two types of instabilities of the centerline (“tapelike” = TL, and “thick” = th). We explain the meaning of the parameters a,b and ρ.

The parameter a (Eq. (9) of [9]) is the ratio between the two principal moments of inertia of the rod (I ₁<I ₂). The parameter b (also Eq. (9) of [9]) involves also the Poisson ratio (denoted σ in [9], and ν in our manuscript), and the “mixed” moment of inertia J. In the limit t≪1: I ₁∼Wt ³,I ₂∼tW³ and J∼tW³, with some numerical coefficients that depend on the exact shape of the cross section. In Eq. (12), both a and b are evaluated for an ellipsoidal cross section, but we assume that the same expressions (i.e., the exact respective ratios between J,I ₁,I ₂) hold also for a rectangular cross section, from which we can translate to our terminology:

$$ a \to t^2,\qquad b \to \frac{2 t^2}{1+\nu}, $$

(111)

where we assume already the limit t≪1 and expanded b to lowest order in t.

Now, let us consider the parameter $\rho=F_{3}^{(0)}/{\kappa_{3}^{(0)}}^{2}$ (Eq. (35) of [9]), where $F_{3}^{(0)}$ is the normalized force exerted along the centerline and $\kappa_{3}^{(0)}$ is the exerted “torsion” of the centerline. We will show that the translation to our terminology is:

$$ \rho \to \frac{t^2 T}{\eta^2}. $$

(112)

To see this, first note that $F_{3}^{(0)}$ and $\kappa_{3}^{(0)}$ are defined as the tension and the twist density in the sentence after Eq. (14) of [9]. In order to understand the normalization, we need the normalization of lengths and forces, given, respectively in Eqs. (6) and (7b). Note that lengths are measured in units of t (since I ₁∼t ³ W and A∼tW). The expressions of κ ₃ and F ₃ in our parameters is therefore$:\kappa_{3}^{(0)} = (\theta/L)/t = \eta/ t$, and $F_{3}^{(0)} = \mathrm{force}/(E tW) = \mathrm{force}/(Y W) = T$. Substituting this expression for $F_{3}^{(0)}$ and $\kappa_{3}^{(0)}$ in Eq. (35) of [9], we find the above transformation of the parameter ρ to our parameters. Importantly,ρ of [9] is inversely proportional to the ratio η ²/T, and hence the unstable range of the helicoidal state (gray zones in Fig. 4 of [9]) corresponds to large value of twist/tension (namely η ²/T above some threshold).

Let us turn now to Eqs. (58), (59), and express them in our terminology. From Eq. (58) we obtain the threshold for the “tapelike” mode to be:

$$ \biggl(\frac{\eta^2}{T} \biggr)_\mathrm{TL} \approx \frac{1+\nu}{1-\nu}, $$

(113)

in the limit t≪1, which ranges from 1 to 3 as ν ranges from 0 to 1/2. Equation (59) leads the threshold for the “thick” mode,

$$ \biggl(\frac{\eta^2}{T} \biggr)_\mathrm{th} \approx \frac{1+\nu}{2 (1+2\nu-2\sqrt{\nu(1+\nu)} )}, $$

(114)

in the limit t≪1. This expression ranges from 1/2 to 2.8 as ν ranges from 0 to 1/2 and it is smaller than the first threshold for any value of ν.

Appendix E: Estimating the Clamping-Induced Energy

The transverse displacement u _r must vanish near the clamped edges (s=±L/2), and is expected to approach u _r ≈−νTr/2 beyond a characteristic length ℓ from the clamped edges. In the region s∈(−L/2+ℓ,L/2−ℓ) the Poisson contraction applies, such that the strain can be approximated as:

$$ \varepsilon_{ss} = T,\qquad \varepsilon_{rr} = -\nu T,\qquad \varepsilon_{xy} = 0, $$

(115)

and the corresponding energy per length is:

$$ \biggl(1-\frac{2\ell}{L} \biggr) T^2. $$

(116)

In the near-boundary zones s∈±(L/2−ℓ,L/2), where u _r is not determined by the Poisson effect, we may express the strain field as:

$$\begin{aligned} \varepsilon_{ss} =& \frac{1}{1-\nu^2}T + \frac{\nu T}{\ell} f_1, \\ \varepsilon_{rr} =& \frac{\nu T}{\ell} f_2, \qquad \varepsilon_{sr} = \frac{\nu T}{\ell} f_3, \end{aligned}$$

(117)

where f _i (s/ℓ,r) are O(1) functions that characterize the variation of the displacement field from the clamped edge to its bulk value. Note that the ℓ-independent component of ε _ss is derived from the Hookean stress-strain relationship by assuming σ ^ss≈T and ε _rr ≈0. Integrating over the boundary zones s∈±(L/2−ℓ,L/2), the energy per length associated with the strain field is estimated as:

$$ \frac{1}{L} \biggl(\frac{T^2\ell}{1 -\nu^2} + \frac{\nu^2 T^2}{\ell} \biggr) $$

(118)

(where some unknown numerical constants, which are independent on ℓ and ν, multiply each of the two terms in the above expression). Combining the two energies, Eqs. (116), (118), and minimizing over ℓ, we obtain:

$$ \Delta U_\mathrm{clamp} \sim \frac{\nu F(\nu) T^2}{L}, \qquad \ell \sim \nu $$

(119)

where F(ν) is some smooth function of ν that satisfies F(ν)→cst for ν→0.