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Roadmap to the Morphological Instabilities of a Stretched Twisted Ribbon

  • Julien Chopin
  • Vincent Démery
  • Benny DavidovitchEmail author
Chapter
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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.

Keywords

Buckling and wrinkling Far from threshold Isometry Helicoid 
ssFvK equations

“small-slope” (standard) Föppl–von Kármán equations

cFvK equations

covariant Föppl–von Kármán equations

t, W, L

thickness, width and length of the ribbon (non-italicized quantities are dimensional)

\(t\), \(W=1\), \(L\)

thickness, width and length normalized by the width

\(\nu\)

Poisson ratio

\(\mathrm{E}, \mathrm {Y},\mathrm {B}=\frac{\mathrm {Y}\mathrm {t}^{2}}{12(1-\nu^{2})}\)

Young, stretching and bending modulus

\(Y=1\), \(B=\frac{t^{2}}{12(1-\nu^{2})}\)

stretching and bending modulus, normalized by the stretching modulus

\(T=\mathrm {T}/\mathrm {Y}\)

tensile strain (tensile load normalized by stretching modulus)

\(\theta\), \(\eta=\theta/L\)

twist angle and normalized twist

\((\hat{\boldsymbol {x}},\hat{\boldsymbol {y}},\hat{\boldsymbol {z}})\)

Cartesian basis

\(s\), \(r\)

material coordinates (longitudinal and transverse)

\(z(s,r)\)

out of plane displacement (of the helicoid) in the small-slope approximation

\(\boldsymbol {X}(s,r)\)

surface vector

\(\hat{\boldsymbol {n}}\)

unit normal to the surface

\(\sigma^{\alpha \beta}\)

stress tensor

\(\varepsilon_{\alpha \beta}\)

strain tensor

\(g_{\alpha \beta}\)

metric tensor

\(c_{\alpha \beta}\)

curvature tensor

\(\mathcal{A}^{\alpha\beta\gamma\delta}\)

elastic tensor

\(\partial_{\alpha}\), \(D_{\alpha}\)

partial and covariant derivatives

\(H\), \(K\)

mean and Gaussian curvatures

\(\zeta\)

infinitesimal amplitude of the perturbation in linear stability analysis

\(z_{1}(s,r)\)

normal component of an infinitesimal perturbation to the helicoidal shape

\(\eta_{\mathrm{lon}}\), \(\lambda_{\mathrm{lon}}\)

longitudinal instability threshold and wavelength

\(\eta_{\mathrm{tr}}\), \(\lambda_{\mathrm{tr}}\)

transverse instability threshold and wavelength

\(\alpha=\eta^{2}/T\)

confinement parameter

\(\alpha_{\mathrm{lon}}\)

threshold confinement for the longitudinal instability

\(r_{\mathrm{wr}}\)

(half the) width of the longitudinally wrinkled zone

\(\Delta \alpha=\alpha-24\)

distance to the threshold confinement

\(f(r)\)

amplitude of the longitudinal wrinkles

\(U_{\mathrm{hel}}\), \(U_{\mathrm{FT}}\)

elastic energies (per length) of the helicoid and the far from threshold longitudinally wrinkled state

\(U_{\mathrm{dom}}\), \(U_{\mathrm{sub}}\)

dominant and subdominant (with respect to \(t\)) parts of \(U_{\mathrm{FT}}\)

\({\boldsymbol {X}_{\mathrm{cl}}}(s)\)

ribbon centerline

\(\hat{t} = d\boldsymbol {X}_{\mathrm{cl}}(s)/ds\)

tangent vector in the ribbon midplane

\(\hat{\boldsymbol {r}}(s)\)

normal to the tangent vector

\(\hat{\boldsymbol{b}}(s)\)

Frenet binormal to the curve \(\boldsymbol {X}_{\mathrm{cl}}(s)\)

\(\tau(s), \kappa(s)\)

torsion and curvature of \(\boldsymbol {X}_{\mathrm{cl}}(s)\)

Mathematics Subject Classification (2010)

74K20 53Z05 35Q74 74K35 

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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Julien Chopin
    • 1
    • 2
  • Vincent Démery
    • 3
  • Benny Davidovitch
    • 3
    Email author
  1. 1.Civil Engineering Department, COPPEUniversidade Federal do Rio de JaneiroRio de JaneiroBrazil
  2. 2.Department of PhysicsClark UniversityWorcesterUSA
  3. 3.Physics DepartmentUniversity of MassachusettsAmherstUSA

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