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Periodic folding of a falling viscoelastic sheet

Kui Pan, A. Srikantha Phani, and Sheldon Green
Phys. Rev. E 101, 013002 – Published 7 January 2020
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  • INTRODUCTION
  • THEORETICAL MODEL
  • RESULTS AND DISCUSSION
  • CONCLUSION
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    A viscoelastic solid sheet fed from a certain height towards a rigid horizontal plane folds on itself provided that there is no slip. This phenomenon commonly occurs in the manufacturing process of textile and paper products. In this paper we apply a particle dynamics model to investigate this phenomenon. At a low feeding velocity and low viscosity, the inertial effect and the viscous dissipation within the sheet are negligible, and our model successfully reproduces the existing quasistatic results in the gravitational regime. As the feeding velocity and the viscosity of the sheet increase, the folding process changes significantly. The length of the folds decrease and the “rolling back” motion of the sheet vanishes. In the inertial regime, a scaling law between the fold length and the feeding velocity is derived by balancing the kinetic energy and the elastic bending energy involved in folding, which is verified by the simulation. It is found that above a critical feeding velocity, the folding morphology transforms from line contact into point contact with the sheet exhibiting a lemniscate-like pattern. Finally, a phase diagram for the folding morphology is constructed. The results presented in this work may offer some insights into the high-speed manufacturing of paper and fabric sheets.

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    • Received 23 February 2019
    • Revised 12 October 2019

    DOI:https://doi.org/10.1103/PhysRevE.101.013002

    ©2020 American Physical Society

    Physics Subject Headings (PhySH)

    1. Physical Systems
    Dynamical systemsMultiple time scale dynamics
    1. Techniques
    GeometryNumerical approximation & analysis
    General PhysicsNonlinear Dynamics

    Authors & Affiliations

    Kui Pan, A. Srikantha Phani*, and Sheldon Green

    • Department of Mechanical Engineering, University of British Columbia, Vancouver, British Columbia V6T 1Z4, Canada

    • *Corresponding author: skpa@mail.ubc.ca

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    Issue

    Vol. 101, Iss. 1 — January 2020

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    Images

    • Figure 1
      Figure 1

      (a) Edge-on view of a buckled thin viscoelastic sheet when it touches ground. V0 and H represent the feeding velocity and feeding height, respectively. (b) Schematic of the particle system:ka represents the axial stiffness; kb represents the stiffness of the rotational spring; ca represents the axial damping coefficient; cb represents the rotational damping coefficient. (c) Schematic of elastic bending forces in the local bending system formed by particles (i−1)−(i)−(i+1). (d) Schematic of viscous bending forces: fv1,i is due to the angular velocity ω1,i, fv2,i is due to the angular velocity ω2,i, and fv,i is the negative sum of fv1,i and fv2,i.

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    • Figure 2
      Figure 2

      The equilibrium configurations of the sheet at different stages during the formation of one fold. The falling height is H=9lc and feeding velocity is V0=0.01gH. wf and hf are defined as the width and the height of the final fold, respectively, corresponding to line 6.

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    • Figure 3
      Figure 3

      Simulation results of the fold width wf (solid line) and fold height hf (dashed line) as a function of the characteristic length lc.

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    • Figure 4
      Figure 4

      The evolution of the sheet configuration during the formation of the first fold under dynamic case. The parameters are chosen as H=9lc, V0/gH=0.5 and μ/μ*=2, where μ* is defined as μ*=ρgH3.

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    • Figure 5
      Figure 5

      The configuration of the sheet at steady state after a sufficiently long simulation under the dynamic case. The parameters are the same as in Fig. 4, and the self-contact of the sheet is neglected here.

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    • Figure 6
      Figure 6

      Normalized fold length lf/lc vs normalized velocity V0/gH under different viscosity. The solid lines with symbols represent the simulation results, and the dashed lines represent the fitted scaling law.

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    • Figure 7
      Figure 7

      (a) Normalized fold height hf/lc vs. normalized fold length lf/lc. lf_c is the critical folding length below which the folding pattern becomes point contact. (b) Normalized fold width wf/lc vs normalized fold length lf/lc. In both figures the discrete symbols represent the simulation results under various viscosity and velocity. The solid lines are the linear fitting according to Eqs. (13) and (14). The inserted figures in (a) show the steady-state fold patterns at different values of lf/lc.

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    • Figure 8
      Figure 8

      Phase diagram of the folding morphology. The solid line with symbol represents the exact simulation results of Vc, and the dashed line represents the solution according to Eq. (16).

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    • Figure 9
      Figure 9

      Schematic of a beam segment of length a0, width w, and thickness h undergoing bending deformation by an angle θ such that the section A deforms to A′, the radius of curvature after deformation is R.

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    • Figure 10
      Figure 10

      Schematic of an elastica under axial load with clamping boundary condition before buckling (dashed line) and after buckling (solid line). O and Q are the endpoints, L0 is the original length, and ΔL is the displacement of end point Q after deformation.

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    • Figure 11
      Figure 11

      Normalized deflection ζ/h vs normalized end displacement ΔL/L0. The solid line represents the solution of extensible elastica model obtained by shooting method, and the dashed line with open symbols represents the simulation results.

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    • Figure 12
      Figure 12

      Time response of the deflection of the mid-point of an excited simply supported beam.

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