A computational model of twisted elastic ribbons Madelyn Leembruggen1Jovana Andrejevic2Arshad Kudrolli3and Chris H. Rycroft4 5 1Department of Physics Harvard University Cambridge MA 02138 USA

2025-04-30 0 0 3.04MB 19 页 10玖币

侵权投诉

A computational model of twisted elastic ribbons

Madelyn Leembruggen,

1, ∗

Jovana Andrejevic,

2, †

Arshad Kudrolli,

3, ‡

and Chris H. Rycroft

4, 5, §

Department of Physics, Harvard University, Cambridge, MA 02138, USA

Department of Physics, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA

Department of Physics, Clark University, Worcester, Massachusetts 01610, USA

Department of Mathematics, University of Wisconsin–Madison, Madison, WI 53706, USA

Computational Research Division, Lawrence Berkeley Laboratory, Berkeley, CA 94720, USA

We develop an irregular lattice mass-spring-model (MSM) to simulate and study the deformation

modes of a thin elastic ribbon as a function of applied end-to-end twist and tension. Our simulations

reproduce all reported experimentally observed modes, including transitions from helicoids to

longitudinal wrinkles, creased helicoids and loops with self-contact, and transverse wrinkles to

accordion self-folds. Our simulations also show that the twist angles at which the primary longitudinal

and transverse wrinkles appear are well described by various analyses of the F¨oppl-von K´arm´an

(FvK) equations, but the characteristic wavelength of the longitudinal wrinkles has a more complex

relationship to applied tension than previously estimated. The clamped edges are shown to suppress

longitudinal wrinkling over a distance set by the applied tension and the ribbon width, but otherwise

have no apparent eﬀect on measured wavelength. Further, by analyzing the stress proﬁle, we ﬁnd

that longitudinal wrinkling does not completely alleviate compression, but caps the magnitude of

the compression. Nonetheless, the width over which wrinkles form is observed to be wider than

the near-threshold analysis predictions– the width is more consistent with the predictions of far-

from-threshold analysis. However, the end-to-end contraction of the ribbon as a function of twist is

found to more closely follow the corresponding near-threshold prediction as tension in the ribbon is

increased, in contrast to the expectations of far-from-threshold analysis. These results point to the

need for further theoretical analysis of this rich thin elastic system, guided by our physically robust

and intuitive simulation model.

I. INTRODUCTION

Twisting a thin ribbon under tension generates com-

pression, somewhat counterintuitively, and consequently

longitudinal wrinkles in the center of the ribbon. Larger

twist angles lead to the even more whimsical creased

helicoid phase, sometimes referred to as “ribbon crys-

tals”. These nonintuitive deformation modes hint at deep

physics, and despite twisting being the basis of ancient

textile technologies—from twisting ﬁbers into yarns to

coiling ropes into piles—the mechanics of twisted mor-

phologies remain hazy [

–

]. These mysteries have un-

furled slowly over decades, beginning when Green ﬁrst

observed buckling and wrinkling patterns in ribbons and

analyzed the development of compression with twist [

It was not until nearly ﬁfty years later that the existence

of a buckling transition in twisted plates was conﬁrmed

numerically [

], and it took another twenty years to ver-

ify numerically the longitudinal wrinkling pattern Green

described [

]. Shortly after, the creased helicoid phase

was modeled geometrically in the isometric limit [

Next, experimental probes provided a full map of the

twisted ribbon phase space, revealing a sprawling zoo of

helicoids, wrinkles, and loops which are dependent on

twist angle and applied tension [

]; analytical attempts to

∗mleembruggen@g.harvard.edu

†jovana@sas.upenn.edu

‡akudrolli@clarku.edu

§chr@math.wisc.edu

characterize these post-buckling phenomena quickly fol-

lowed [

]. Surprisingly, creased helicoids still formed

in ribbons with tension, developing from the wrinkles

themselves [

] and displaying some amount of stretch-

ing in the ridges (unseating the isometric assumptions of

previous studies) [

]. Ribbons of ﬁnite-thickness are ana-

lytically slippery, requiring approximations whose regions

of validity not fully clear [

]. To fully resolve the

limitations of these approximations requires data such as

internal energy and stress distributions, quantities which

are currently diﬃcult to access by experiment or theory.

In general, thin sheets are excellent candidates for com-

putational modeling. Plenty of great work has been

done to simulate thin sheet deformations using ﬁnite

element methods (FEM) [

–

], and there are several

mass-spring-models that successfully map regular dis-

crete lattices to the bulk properties of a sheet [

–

Less work, however, has been dedicated to mapping a

microscopically random mesh’s parameters to the bulk

properties of the simulated sheet. Some models use a

random mesh to approximate a constant bulk Young’s

modulus [

], and others focus on making discretized

bending realistic [

–

]. But in some cases these models

are inconsistent with analytical descriptions of regular

meshes; what’s more, a combined stretching and bending

model has to our knowledge, not been thoroughly tested.

A discrete mesh model that compares directly to physical

materials is inﬁnitely useful, opening the door to generat-

ing large data sets useful for data-driven discoveries.

In this paper, we develop a simple, computationally

cheap, mechanical mass-spring-model (MSM) to study

twisted thin ribbons. We could equivalently use an FEM

arXiv:2210.14374v3 [cond-mat.soft] 2 Jun 2023

approach to study thin twisted ribbons. However, we are

compelled by the MSM because it is an intuitive extension

of coupled oscillators; allows local, “microscopic” control

of the mesh topology; and is not tethered to the limitations

of using a partial diﬀerential equation for the sheet, such

as describing the post-buckled shapes, or singularities in

the PDE in areas with stress-focusing. It is a simple model

with historical precedence that accurately replicates the

interesting observables of the thin elastic sheet, using only

Hookean springs with small stresses that depend linearly

on deformation. A lattice model like this, with nearest

and next-to-nearest neighbor interactions, also preserves

the possibility of learning about or from other statistical

mechanics lattice models.

Our randomly-seeded MSM maps reliably to a physical

Young’s modulus and bending rigidity, allowing us to

match experimental conditions and generate the various

modes of deformations observed as a function of ribbon

aspect ratio, applied tension and twist (see Fig. 1). We

are able to carefully analyze the onset and growth of

wrinkles in the longitudinally buckled mode using mea-

surements which are also available experimentally, such as

surface curvature. Additionally, the simulations provide

access to the invisible internal dynamics of the ribbon,

including ﬁnely resolved spatial and temporal data for

the strain and stress. These new insights are useful for

characterizing the mechanics of the buckling transition,

and probing the relevance of existing near- and far-from-

threshold approximations. Guided by recent rounds of

physical observations and analytical inquiries, we show

that numerics and simulations once again hold the key to

the next stage of discovery about twisted thin ribbons.

II. MASS SPRING MODEL

We model a thin ribbon by deﬁning a mesh with a

set of nodes, arranged either in a regular triangular lat-

tice (where each interior node has six equidistant nearest

neighbors) or a random triangular lattice in which node

coordination number and distance to nearest neighbors

vary. Nodes are connected to their neighbors via in-plane

springs and dashpots, and bending is controlled by pseudo-

springs (a quadratic penalization of bending) across each

edge between adjacent triangles, as shown in Fig. 2. A

full description of this discrete model’s relationship to

continuum elasticity is provided in Appendix A; in this

section we provide a brief history of the model and a

summary of the modiﬁcations we employ.

The regular triangular lattice is well-studied and oft-

utilized, with the attractive feature of having an analytical

mapping from the discrete stretching and bending spring

constants (

and

) to a continuous two-dimensional

Young’s modulus and bending rigidity for the sheet. The

springs

rij

connecting pairs of lattice sites,

and

dictate in-plane deformations. Hinges separating two

triangles

△ijk

and

△ikl

, with normal vectors

nijk

and

nikl

, control out-of-plane motion. Seung and Nelson [

]

showed that for the triangular lattice with unit length

springs, and potentials of the form

Es(rij ) = 1

2ks(1 − |xi−xj|)2,(1)

Eb(ˆ

nijk,ˆ

nikl) = 1

2kb|ˆ

nijk −ˆ

nikl|2,(2)

the equivalent continuous two-dimensional (2D) Young’s

modulus and bending rigidity for the sheet are

Y2D=2

√3ks, B =√3

2kb,(3)

respectively. If the rest conﬁguration of the lattice devi-

ates from hexagonal packing of equilateral triangles (such

as at the boundaries of a rectangular sheet or a collection

of randomly placed lattice points), the above relationships

are no longer correct beyond the leading order discrete

approximation of the continuum. To extend this mass-

spring-model to irregular lattice conﬁgurations, it is useful

to modify the pre-factor of each energy term.

Each in-plane spring represents an area of continuous

material that resists stretching or compression. Thus

springs adjacent to larger triangular facets should have

stiﬀer spring constants to reﬂect the greater amount of

material they represent. We modify the stretching energy

term according to Van Gelder [

] (with typos in the

original model corrected by Lloyd et al. [24]):

Es(rij ) = 1

21

ks(sij − |xi−xj|)2,

2 √3

Y2D!(sij − |xi−xj|)2,

(4)

where

Y2D

is the target Young’s modulus for the sheet,

sij

the rest length of a given spring,

the sum of the

facet areas adjacent to edge

rij

, and

the area of an

equilateral triangle with side length

sij

. When the entire

lattice is composed of equilateral triangles, this expression

reduces to the model in Eq. (1).

With similar physical motivations as given above—

namely that an area of continuous material distributed

over a longer length scale should be ﬂoppier, or easier to

bend—we modify the bending energy term to be

Eb(ˆ

nijk,ˆ

nikl) = 1

22A0

Akb|ˆ

nijk −ˆ

nikl|2

24

√3

AB|ˆ

nijk −ˆ

nikl|2.

(5)

The area dependence of the coeﬃcient is inspired by

Grinspun et al. [

] and adapted such that it reduces

to the Seung and Nelson bending energy (Eq.

(2)

) when

all triangles are equilateral. Although this quantity is

presented as a ratio of areas, it implicitly accounts for

the shapes of the adjacent triangles through the quantity

FIG. 1. Deformation modes of twisted thin sheets, replicated by our simulation whose details are summarized in Fig. 2. Ribbon

(a) is labeled with its dimensions and quantities relevant to the twisting-under-tension procedure.

is the scaled longitudinal

tension applied, which is total applied force

normalized by the width, Young’s modulus, and thickness of the ribbon, and

a scaled twist angle: the end-to-end angle

normalized by the ribbon’s aspect ratio. These quantities are deﬁned in Table I. (a)

The helicoid phase initially present before any buckling transitions occur. (b) Longitudinal buckling occurs at an angle

ηlon

and has a ﬁxed wavelength

λlon

set by the tension

and thickness

. (c) Creased helicoids develop from the longitudinally

buckled ribbons as the buckle ridges “turn” to form triangular facets. (d) At tensions below the crossover tension

T∗

, ribbons

will snap-through to form a loop at an angle

ηtran

. If twisted far enough, ribbons will develop self-contact at an angle

ηsc

. (e)

Transverse buckling occurs at tensions greater than

T∗

, transitioning at angle

ηtran

. (f) The wavelength of transverse buckling

λtran

is set by the aspect ratio of the sheet; sheets with smaller aspect ratios can display several wavelengths of transverse

buckling. (g) At high twists, low aspect ratio sheets display “accordion folding” and approach the yarning transition [

Snapshots (a)–(e) fall within the phase diagram presented in Fig. 3.

FIG. 2. Nodes are connected by in-plane springs with stiﬀness

and dashpot damping

bint

. The

of each in-plane spring is

set by the local geometry of the mesh and the target Young’s

modulus for the sheet,

. Out-of-plane stiﬀness

is similarly

dictated by the local geometry and the target bending rigidity,

. Both sets of springs ensure quadratic energy penalization

for stretching and misalignment of the vectors normal to the

triangular mesh facets, Eqs. (4) and (5).

. Grinspun et al. [

] provide a full explanation of this

shape consideration.

Although our mesh is 2D, the bending rigidity imparts

an eﬀective thickness to the sheet. The bending rigidity

Bis related to the Young’s modulus Yby [29]

B=Y h3

12(1 −ν2),(6)

where

is the thickness of the material and

is Poisson’s

ratio. For triangular lattices

cannot be independently

tuned, so

= 1

3 always [

], and

Y2D/h

. Thus

we can rearrange Eq.

(6)

and use the relationships in

Eq. (3) to deﬁne an eﬀective thickness:

heﬀ =r8kb

.(7)

This length scale is used to set the sheet’s self-avoidance:

the sheet is not allowed to come within

heﬀ

of itself. Self-

avoidance is enforced by introducing a repulsive force

between contact sites within an interaction range

heﬀ

each other [

]. In general, the contact sites are spaced

more closely than the mesh nodes. For example, the

thinnest sheet we consider here (

= 127

µm

) has a mesh

spacing about 8 times coarser than

heﬀ

. Thus we imple-

ment “level 3” reﬁnement (iterative bisection of triangle

edges three times, placing additional contact sites at the

midpoints) such that the spacing of contact sites is on the

order of the sheet’s thickness. Forces between the reﬁned

contact sites are distributed to the nodes of the mesh

nearest to the sites, weighted by proximity to the site.

The site reﬁnement applies only during contact detection

and allows the mesh to remain coarse in all other calcu-

lations. From now on, mentions of the simulated sheet’s

thickness hare in reference to this eﬀective thickness.

Property Symbol Formula

Length L−L/2< y < L/2

Width W−W/2< x < W/2

Thickness h

Young’s modulus Y Y2D/h

Poisson’s ratio ν1/3

Bending rigidity BY h3

12(1−ν2)

End-to-end twist angle θ

Scaled twist angle η θ W

Scaled applied tension TF

Y hW

Conﬁnement parameter α η2/T

TABLE I. Deﬁnitions of variables and physical parameters.

The equations of motion for a node iin the sheet are

xi=vi,

mai=Fi,(8)

where

is the three-dimensional position of a node,

its velocity,

its acceleration, and

the sum of

the forces applied to node i, which is given by

Fi=−X

j∈Ni

∇xi(Es(rij )) −X

(ijk)∈Ti

(ikl)∈Ti

∇xi(Eb(ˆ

nijk,ˆ

nikl))

j∈Ni

Fint

d(xi,xj,vi,vj) + Fiso

d(vi)

Fc(xi,xj) + Fext.(9)

Here

are the neighbors of node

, and

are the

adjacent triangles. The damping term

Fint

is due to the

dashpots pictured in Fig. 2, and

Fiso

is an additional

isotropic, global damping.

is the contact force, which

is repulsive and turns on when two nodes come within

distance of one another [

]. Finally, any external applied

forces are included in Fext.

High-frequency elastic waves dissipate quickly in materi-

als that are of interest to us (e.g. paper, Mylar, aluminum

foil). Instead, wrinkling is characterized by slower defor-

mations on the scale of the system size, with occasional

hysteretic, snap-through events. We therefore assume

the sheet is mostly in a quasistatic regime where the

acceleration of a node is close to zero:

ai≈

0. The

quasistatic equations of motion constitute a diﬀerential-

algebraic system which is well-suited to an implicit numer-

ical integration scheme in time [

]. When the quasistatic

approximation is violated, such as near a snap-through

event accompanied by large changes in node velocities,

the full dynamic equations of motion given in Eq.

(8)

are integrated explicitly instead. Further details on the

numerical integration schemes and switching criteria are

provided in Appendix B, and a thorough explanation is

given in Ref. [30].

We note that plasticity can be added to this model

by allowing the rest angle (length) of a hinge (spring) to

change if the hinge (spring) is deformed past a speciﬁed

yield threshold. The plastic damage can then accumulate

according to a purely plastic, strain hardening, or even

strain weakening model [

]. In this work, however, we

will discuss only purely elastic sheets which do not fatigue

or accumulate damage.

III. END-TO-END TWISTING

A. Ribbon Setup and Boundary Conditions

Our simulated ribbons correspond to length

and width

= 2

. Three diﬀerent thick-

nesses are used throughout this paper: 127

µm

, 254

µm

and 508

µm

. A single, randomly-seeded ribbon mesh was

used across all simulations, with an average node spacing

= 1

. The random lattice is generated by seeding

the ribbon with nodes using the Voro++ library [

then Lloyd’s algorithm [

] is iteratively applied 100 times

to make the mesh more regular. The mesh conﬁguration

at this point is taken as its rest conﬁguration, and all the

springs and facets have diﬀering lengths and areas. The

spring and hinge stiﬀnesses are set according to the model

in Eqs.

(4)

and

(5)

, and tuned such that the Young’s

modulus of the sheet is approximately

= 3

GPa

and

the bending rigidity is approximately

= 0

653

mPa ·m3

= 5

mPa ·m3

, or

= 41

mPa ·m3

, respectively,

for the three thicknesses. Properties of the thinnest ribbon

are given in physical units in Table II.

For both the regular and randomly seeded meshes we

performed a series of three modulus convergence tests:

stretch, shear, and bend. The average node spacing had

the range

d∈

mm,

]. In Appendix C we see

that the model in Eqs.

(4)

(5)

converges to the expected

values of Young’s modulus (

), shear modulus (

), and

bending rigidity (

). While there is some amount of

error in these moduli, it is within the range of variation

expected, for example, from physical material samples.

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

AcomputationalmodeloftwistedelasticribbonsMadelynLeembruggen,1,∗JovanaAndrejevic,2,†ArshadKudrolli,3,‡andChrisH.Rycroft4,5,§1DepartmentofPhysics,HarvardUniversity,Cambridge,MA02138,USA2DepartmentofPhysics,UniversityofPennsylvania,Philadelphia,Pennsylvania19104,USA3DepartmentofPhysics,ClarkUniversity...

展开>> 收起<<

A computational model of twisted elastic ribbons Madelyn Leembruggen1Jovana Andrejevic2Arshad Kudrolli3and Chris H. Rycroft4 5 1Department of Physics Harvard University Cambridge MA 02138 USA.pdf

共19页,预览4页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

A computational model of twisted elastic ribbons Madelyn Leembruggen1Jovana Andrejevic2Arshad Kudrolli3and Chris H. Rycroft4 5 1Department of Physics Harvard University Cambridge MA 02138 USA

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: