Coiling of elastic rods on rigid substrates

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Coiling of Elastic Rods on Rigid Substrates

by

Mohammad Khalid Jawed

B.S.E. (Aerospace Engineering), University of Michigan (2012)

B.S.E. (Engineering Physics), University of Michigan (2012)

Submitted to the Department of Mechanical Engineering

in partial fulfillment of the requirements for the degree of

Master of Science

at the

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

September 2014

c

Massachusetts Institute of Technology 2014. All rights reserved.

Author . . . .

Department of Mechanical Engineering

August 8, 2014

Certified by . . . .

Pedro M. Reis

Associate Professor of Mechanical Engineering and Civil and

Environmental Engineering

Thesis Supervisor

Accepted by . . . .

David E. Hardt

Ralph E. and Eloise F. Cross Professor of Mechanical Engineering

Chairman, Committee on Graduate Students

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Coiling of Elastic Rods on Rigid Substrates

by

Mohammad Khalid Jawed

Submitted to the Department of Mechanical Engineering on August 8, 2014, in partial fulfillment of the

requirements for the degree of Master of Science

Abstract

We investigate the deployment of a thin elastic rod onto a rigid substrate and study the resulting coiling patterns. In our approach, we combine precision model exper-iments, scaling analyses, and computer simulations towards developing predictive understanding of the coiling process. Both cases of deposition onto static and moving substrates are considered. We construct phase diagrams for the possible coiling pat-terns, e.g. meandering, stretched coiling, alternating loops, and translated coiling, and characterize them as a function of the geometric and material properties of the rod, as well as the height and relative speeds of deployment. The various modes selected and their characteristic length-scales are found to arise from a complex in-terplay between gravitational, bending, and twisting energies of the rod, coupled to the geometric nonlinearities intrinsic to their large deformations. We give particular emphasis to the first sinusoidal mode of instability, which we find to be consistent with a Hopf bifurcation, and rationalize the meandering wavelength and amplitude. Throughout, we systematically vary natural curvature of the rod as a control param-eter, which has a qualitative and quantitative effect on the pattern formation, above a critical value that we determine. Upon establishing excellent quantitative agree-ment between experiagree-ments and simulations with no fitting parameters, we perform a numerical survey to relate the pattern size to the relevant length-scales arising from material properties and the setup geometry, and quantify the typical strain levels in the rod. The universality conferred by the prominent role of geometry in the deforma-tion modes of the rod suggests using the gained understanding as design guidelines, in the original applications that motivated the study. These include the coiling of carbon nanotubes and the deployment of submarine cables and pipelines onto the seabed.

Thesis Supervisor: Pedro M. Reis

Title: Associate Professor of Mechanical Engineering and Civil and Environmental Engineering

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Acknowledgments

The research presented in this thesis was done under the supervision of Professor Pedro Reis (MIT), and in collaboration with Professor Eitan Grinspun (Columbia University) and Fang Da (Columbia University). Prior to my joining graduate school, J. Marthelot, J. Mannet, and A. Fargette performed preliminary experiments, and Jungseock Joo developed a numerical simulation tool to study the problem presented in this thesis.

I am grateful to Basile Audoly (CNRS) for enlightening discussions on the scaling analysis to obtain meandering length-scales. This research would not be possible without the cooperation of every member of the Elasticity, Geometry and Statistics Laboratory (EGS.Lab).

I also want to thank the National Science Foundation for funding this research (CMMI - 1129894).

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Chapter 1 Introduction

Across the three eras of fast communication – telegraphic (1830s - 2013), telephonic (1876 - present), and fiber-optic (1979 - present) – submarine cables have been the backbone of international transmission of information. The laying of the first transat-lantic telegraph cable in 1850s [1] opened the path for fast long-distance communica-tion. Nowadays, submarine fiber-optic cables connect all the continents, see Fig. 1-1A, and form a key ingredient for the international communications infrastructure (e.g. the internet). Submarine cables are typically installed from a cable-laying vessel that, as it sails, pays out the cable from a spool downwards onto the seabed. The portion of suspended cable between the vessel and the contact point with the seabed takes the form of a catenary [2].

Similar procedures can also be used to deploy pipelines [3], an historical exam-ple of which is the then highly-classified Operation PLUTO (Pipe-Lines Under the Ocean) [4, 5], which provided fuel supplies across the English Channel at the end of World War II. These pipelines of 3 inch diameter were wound onto drums 30 feet in diameter weighing 250 tons (Fig. 1-1B1), and towed across the English Channel (Fig. 1-1B2). Following D-Day, two pipelines, codenamed Bambi and Dumbo, ran between the petrol storage tanks in southern England and the landing site of the Allied armies in France (Fig. 1-1B3).

One of the major challenges in the laying process of these cables and pipelines is the accurate control between the translation speed of the ship, vb, and the pay-out

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A

B1 B2 B3

Figure 1-1: (A) Network of international submarine cables, as of 2014. (B) Operation PLUTO. (B1) Pipelines wound onto cylinders. (B2) Ships towing the pipelines across the English Channel. (B3) Map showing the location of the two pipelines – Bambi, and Dumbo. Images adapted from (A) http://submarinecablemap.com/, (B1-B2) http://www.wlb-stuttgart.de, & (B3) http://worldwar2headquarters.com.

rate of the cable, v. A mismatch between the two may lead to mechanical failure due to excessive tension (if vb > v) or buckling (if vb < v), which for the case

of communication cables can cause the formation of loops and tangles resulting in undesirable signal attenuation [6, 7, 8, 9], see Fig. 1-2.

Similar issues arise in some versions of 3D-printing, e.g. direct ink writing (DIW), which can involve the deposition of flexible filamentary structure onto a substrate to fabricate ceramic materials in complex 3D shapes [10]. At the microscale, de-ployment of nanowires onto a substrate has been used to print stretchable electronic components [11], and both periodic serpentines and coils have been fabricated by the flow-directed deposition of carbon nanotubes onto a patterned substrate [12]. Other instances of engineering systems where large deformations of slender rodlike or tubu-lar structures ensue from a deployment process include inflight refueling [13], and aerial tow systems [14].

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Figure 1-2: Schematic of undersea cable deployment from a ship typically subjected to ocean currents. Loops and tangles can form in the low tension regime of the rod (colored black). Image adapted from Ref. [6].

The common thread between these engineering systems is the geometry of deposi-tion of the filamentary structure with a kinematic mismatch between the deposideposi-tion rate and the translational speed. Moreover, the suspended catenary can be treated as a thin rod [15] given that the diameter of the cable, pipe or filament can be orders of magnitude smaller than any other length-scales in the system. Defining χ = seabed depth_{rod diameter} as the slenderness parameter, the previously mentioned PLUTO has χ _{≈ 800 (pipeline of 3 inch diameter at 200 ft depth), whereas for the first} transat-lantic telegraph cable, χ _{≈ 1.7 × 10}5 _{(cable of 3/4 inch diameter at 2 miles depth).}

This provides an opportunity for us to leverage and build upon the knowledge of thin rods acquired through centuries since the seminal developments due to Muss-chenbroek in 1721, Euler in 1744 [16, 17], Kirchhoff in 1859 [18], and Cosserat in 1909 [19].

The process of pattern formation for an elastic rod coiling on a substrate, also known as the elastic sewing machine [20], has been previously studied both numeri-cally [21] and experimentally [22, 20]. However, a systematic study and a predictive understanding of the underlying mechanisms that determine the coiling modes and set the length-scale of the patterns remains aloof. Moreover, there is a need for high-fidelity numerical tools that can predictively capture the intricate geometric nonlinearities of the coiling process.

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1.1 Definition of the problem

In this thesis, we conduct a hybrid experimental and numerical investigation of the coiling of an elastic rod onto a moving substrate, and the resulting patterns are observed and quantified in detail. We perform precision model experiments at the desktop scale, where a custom-fabricated rod is deposited onto a conveyor belt. As the relative difference between the speeds of the injector, v, and the belt, vb, is varied, we

observe a variety of oscillatory patterns that include sinusoidal meanders, stretched coiling, alternating loops, and translated coiling. Our model experiments explore the scale-invariance of the geometric nonlinearities in the mechanics of thin elastic rods [15], thereby enabling a systematic exploration of parameter space. In parallel, we perform numerical simulations using the Discrete Elastic Rods (DER) method [23, 24, 15, 25] that we have ported for the first time from computer graphics to the engineering community, and find excellent quantitative agreement with experiments. Our investigation emphasizes (i) geometry, (ii) universality, and (iii) the signifi-cance of natural curvature. The patterns resulting from coiling of an elastic rod in our experiments have a striking resemblance to those found when deploying a viscous thread onto a moving belt [26, 27, 28, 29, 30, 24] (known as the fluid mechanical sewing machine) and in electrospinning of polystyrene fibers [31, 32]. As such, (i) this simi-larity across various systems reinforces that geometry is at the heart of the observed phenomenon, while the constitutive description plays second fiddle. Furthermore, (ii) as we investigate the first mode of instability, from straight to meandering patterns, we observe that the onset is consistent with a Hopf bifurcation [33]. Finally, (iii) we find that the natural curvature of the rod is a pivotal control parameter governing both qualitative and quantitative aspects of the coiling process. This is important given that in the engineering systems mentioned above, natural curvature may develop from the spooling of the cables and pipes for storage and transport [34]. Combined, the experiments and numerics enable us to identify the physical ingredients, pre-dictively understand the underlying characteristic length-scales, and rationalize the coiling process.

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1.2 Mechanical behavior of slender elastic rods

A structure with one dimension, L, much larger than the other two dimensions, r1, r2,

is termed as a rod. We consider a rod with circular cross-section, and isotropic linear elastic material. Isotropy implies uniform material properties in all orientations. Linear elasticity, valid only for small strain, allows for a linear relation between the stress tensor, σ, and the strain tensor, , by means of only two material parameters – Young’s modulus, E, and Poisson’s ratio, ν. This relation can be expressed in component form,

σij = λkkδij + 2Gij, (1.1)

where λ = _{(1+ν)(1+2ν)}Eν and G = _2(1+ν)E are the Lam´e constants, and δij is the Kronecker

delta. The constant G is also referred to as the shear modulus of the material. See § 6.7 for a verification of the small strain approximation for our setup.

We also assume material incompressibility implying zero volume change from de-formation, and Poisson’s ratio of ν = 0.5. This simplifies the relation between shear and Young’s moduli to G = E/3. In addition, we assume that due to the slenderness of our rods and the geometry of the setup, the rod is inextensible, i.e. it can bend but not stretch.

1.2.1 Kirchhoff rod model

Kirchhoff’s formulation of the mechanical behavior of elastic rods accounts for finite displacements and the resulting geometric nonlinearities, both of which are prevalent in our system. In the following, we provide a brief account of Kirchhoff’s rod model; further details can be found in Ref. [15].

Kinematics. The configuration of a Kirchhoff [15, 18] elastic rod is represented by an arc-length parameterized curve γ(s) in R3 and an orthonormal material frame {d1(s), d2(s), d3(s)}, see Fig. 1-3. Here, γ(s) is an arc-length parameterized curve in

R3 describing the rod’s centerline, and the material frame satisfies d3(s) = γ0(s). The

prime refers to differentiation with respect to arc-length, e.g. γ0(s) = dγ/ds. This is an adapted frame since this orthonormal frame is tangent to the curve at all points.

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A B C

κ(1)

κ(2)

Figure 1-3: A rod is defined by its centerline γ(s) as a function of its arc-length s and an adapted orthonormal material frame_{d1(s), d2(s), d3(s)}. The material frame

rotates rigidly by (a) material curvature κ1 about d1, (b) material curvature κ2 about

d2, and (c) material twist τ about d3. Image adapted from Ref. [35].

If the material frame is chosen to be orthonormal in the reference configuration, it is considered to be approximately orthonormal in the deformed state of the rod, which is known as the Euler-Bernoulli kinematical hypothesis of unshearable cross-sections. This allows for a succint description of the evolution of the material frame long the centerline parameterization s by the relation:

d0_i(s) = Ω(s)_{× d}i(s), i∈ {1, 2, 3}, (1.2)

where Ω(s) is the Darboux vector which can be interpreted as the rotational velocity vector of the material frame _{d1(s), d2(s), d3(s)} as it traverses along the arc-length

s at unit speed, and is mathematically defined as

Ω(s) = κ(1)(s)d1(s) + κ(2)(s)d2(s) + τ (s)d3(s). (1.3)

The quantities κ(1) _{and κ}(2)_{, the material curvatures, express the rotation of the}

material frame about the directions d1 and d2 of the cross-section. The thin rod

approximation implies κ(1)_{, κ}(2)_{, τ} _1/r

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The quantity τ , the twist, represents the rotation of the material frame about the tangent d3.

Constitutive equations. The constitutive equation relevant to the case studied throughout this thesis relates the internal moment M(s), a superposition of moment of bending and moment of twist, with the material properties of the rod,

M(s) = EI(1)κ(1)(s)d1(s) + EI(2)κ(2)(s)d2(s) + GJ τ (s)d3(s) (1.4)

where I(1) _{and I}(2) _{are the second moments of inertia about d}

1 and d2, and J is the

polar moment of intertia of the cross-section. For isotropic material and circular solid cross-section with radius r0, we have I(1) = I(2) = J = π₄r04.

Equilibrium. Kirchhoff equations. The static deformed shape of the rod satisfies equilibrium, where all forces and moments at each point of the rod are bal-anced. The Kirchhoff equations provide a model for equilibrium, and can be solved to obtained the deformed rod shape.

F0(s) + p(s) = 0, (1.5a)

M0(s) + d3(s)× F(s) + q(s) = 0, (1.5b)

where p(s) and q(s) are the distributed (per unit length) applied force and torque, respectively, and F(s) and M(s) are internal force and moment, respectively, at each point on the centerline of the rod. Eq. 1.5a can be interpreted as a balance of internal and external forces, whereas Eq. 1.5b gives a balance of internal and external torques.

1.2.2 Energies of a slender elastic rod

The equilibrium of an elastic rod (in fact, any physical system) corresponds to a stationary point for the energy, including the work done by external and internal forces. Kirchhoff equation (Eqs. 1.5a & 1.5b) can be deduced by considering a small perturbation (denoted by δ) on the energy of the rod Erod and equating that to the

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work done by external forces δW ,

δErod− δW = 0. (1.6)

We now introduce the energies pertinent to a slender elastic rod in the context of the rod deployment mechanism to be studied in this thesis, introduced in _{§ 1.1. The} energy per unit length of a deformed rod, _Erod, can be expressed in terms of elastic,

Ee, inertial, Ei, and gravitational, Eg, contributions,

Erod =Ee+Ei+Eg. (1.7)

The elastic energy term can itself have bending and twisting components_Ee=Eb+Et

depending on the specific deformation mode. Bending and twisting contributions can be expressed in terms of the material parameters and strains described above,

Eb = EI 2 (κ− κn) 2_, _(1.8a) Et= GJ 2 τ 2 , (1.8b)

where the actual curvature κ = _|κ(1)_d

1+ κ(2)d2|, and κn is the curvature evaluated

in the stress-free state, also known as natural curvature. The natural curvature of a straight rod (in its undeformed shape) is κn = 0. The inertial energy per unit length

for a rod moving at speed v is

Ei = πρr20v

2_, _(1.9)

and the gravitational potential per unit length is

Eg = πρgr02h, (1.10)

where h is the height from the reference plane.

Depending on the relative magnitude of the various energies, the following coiling regimes can be identified [22]: i) elastic (_Ee Eg ∼ Ei), ii) gravitational (Eg ∼

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Ee Ei), and iii) inertial (Ei ∼ Ee Eg). Given the properties of our rod and slow

injection speeds 0.5cm/s < v < 6.0cm/s , the ratio between inertial and gravitational energies yields a Froude number, F r = v2_{/[gR] (R is the typical radius of curvature}

of deformation of the rod) that lies within 10−4 < F r < 10−2. Inertial effects can therefore be neglected. The elastic and bending energies are of the same order of magnitude entailing the mechanics studied in this thesis to lie in the gravito-bending regime.

1.2.3 Gravito-bending length, L

gb

We now identify the gravito-bending length, Lgb, as the primary characteristic

length-scale of our system, by balancing gravitational and bending energies. We start by assuming the case of a planar, twist-free deformation (_Et = 0) of a straight rod

(κn = 0) that is deformed into a configuration with curvature κ. The corresponding

bending energy is _Eb = EIκ2/2, where I = πr40/4, and the gravitational energy is

Eg = πρgr02R, where the radius of curvature, R = κ−1, was taken as the relevant

length-scale for height (h _{∼ R). Equating the bending and gravitational energies,} yields the characteristic length,

Lgb = r2 0E 8ρg 1₃ . (1.11)

This gravito-bending length [36] sets the radius of curvature of the rod near its contact point with the substrate, see Fig. 1-4, and will be shown to be crucial in setting the various features of the coiling patterns throughout this thesis. Hereafter, overbar represents nondimensionalization of length by Lgband time by Lgb/v, e.g. ¯κn= κnLgb

denotes the dimensionless natural curvature.

Appendix A details the role of Lgb in setting the shape of a rod between the

injector and the contact point, where we find that the heel radius, R∗, a measure of the radius of curvature at the contact point, is given by

R∗ = Lgb 0.20 log H Lgb + 1.15 , (1.12)

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for a substrate to injector height of H.

L

_gb

L_gb Lgb

Figure 1-4: A schematic of gravito-bending length, Lgb, showing that it sets the radius

of curvature near the contact point with the substrate. The green rod (left) has a greater Lgb than the purple rod (middle). The bead chain (right) has the smallest

value of Lgb of these three examples.

1.3 Literature review

1.3.1 Fluid-mechanical sewing machine

When a viscous jet is deposited onto a static substrate, the fluid coils in a nearly circular pattern [37]. If the substrate is set in motion, a number of additional complex patterns of the fluid thread arise, see Figs. 1-5A & B, which have motivated the naming of this mechanism as fluid mechanical sewing machine. In 1958, Barnes and Woodcock [26]were the first to make the analogy between coiling of elastic ropes, and that of viscous jets. Since then, this phenomenon has been studied extensively experimentally [37, 38, 39, 27, 40, 41], theoretically [42, 43], and numerically [28, 29, 24, 30].

What follows is a scaling analysis to retrieve the characteristic radius of coil re-sulting from deposition of viscous fluid on a static substrate (a more detailed account can be found in Refs. [30, 37]), which shows similarities to that presented above, in § 1.2.2, for an elastic rod.

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A

B

Figure 1-5: Fluid mechanical sewing machine. (A) A viscous fluid is injected at a volumetric rate Q through a nozzle with diameter d which sits at a height H from the belt. The belt is moving at a speed v. The contact point is denoted by ζ. The Langrangian coordinate along the centerline is S, and each point on the thread (inset) has an associated orthonormal trial of basis vectors_{d1, d2, d3}. (B) Patterns

formed by the sewing machine – numerical simulations (black lines) and experiments (photographs). Image adapted from Ref. [29].

The motion of a coiling jet is controlled by a balance between three forces – viscous (Fv), gravitational (Fg), and inertial (Fi). Consider a fluid with volumetric density,

ρ, and kinematic viscosity, ν, is injected at a volumetric rate, Q, through a nozzle with diameter d, onto a substrate from a height, H, see Fig. 1-5A. The resulting characteristic jet radius is r0, radius of coil RC, and axial jet speed v = Q/πr02. The

associated energies per unit length then scale as

Ev ∼ ρνr₀4v R3 C , (1.13a) Eg ∼ ρgr02RC, (1.13b) Ei ∼ ρr20v 2 . (1.13c)

Three different regimes emerge, in a very similar fashion to the elastic sewing machine as discussed in _{§ 1.2.2, based on how these energies scale in the coils formed} by the fluid – (i) “viscous” coiling when gravity and inertia are negligible compared to viscosity (_Ev Eg,Ei), (ii) “gravitational” coiling with gravity and viscosity balancing

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with inertia balancing viscosity (_Ei ∼ Ev Eg). This balance of energies provides a

scaling for the coiling radius RC as

RC ∼ H [viscous], (1.14a) RC ∼ νQ g 1/4 [gravitational], (1.14b) RC ∼ νr4 0 Q 1/3 [inertial]. (1.14c)

An additional degree of complexity is introduced when the belt is set to move at a constant speed, which leads to nonlinear pattern formation depending on the material properties of the fluid, and physical parameters of the setup. Discrete Viscous Threads (DVT) method [24] has been developed and successfully employed to simulate this pattern formation process [29]; representative examples of nonlinear patterns are given in Fig. 1-5B. DVT is similar in spirit to the Discrete Elastic Rods (DER) simulation tool used throughout this thesis. Some of these patterns can also be found in electrospinning, to be reviewed next in _{§ 1.3.2, and the elastic sewing machine, to} be explored in detail in this thesis.

1.3.2 Electrospinning of fibers

Electrospinning is a fabrication technique to produce fine fibers from polymer solu-tions or melts by applying electrostatic forces [44]. Applicasolu-tions of this process range from manufacturing of non-woven garments in the textile industry to artificial organs and implants in the medical industry.

Two types of instabilities are noted – bending and buckling, and an experimental study of the second type is presented in Ref. [31]. In their setup shown in Fig. 1-6A, an electric field creates a charged jet of polymer solution emanating from a pipette. As the jet travels in air, the solvent evaporates, thereby leaving behind a charged fiber that is deposited onto a moving collector tilted at an angle θ. Electrical bending instability refers to the situation when the jet of viscoelastic polymer solution bends

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A

B

C

g

Figure 1-6: Electrospinning of fibers. (A) Schematic diagram of the experimental setup that is typically employed to study patterns formed in electrospinning. De-pending on the length of the electrified jet, it may stay (B) straight, or (C) curved with loops due to electrical bending instability. Image adapted from [31].

due to the mutual repulsion of the excess electric charges in the jet. Depending on the inter-electrode distance, either straight jets (at short distance, Fig. 1-6B), or bent jets (at long distance, Fig. 1-6C) can form. When these jets impact the collector surface, a buckling instability can arise, independently of the electrical bending insta-bility, due to compression of the jet at impingement onto the substrate, which results in reproducible periodic patterns. The experimentally measured frequencies of the patterns compared reasonably well with theoretical predictions for an uncharged fluid jet, i.e. fluid-mechanical sewing machine [31].

Xin and Reneker [32] have further investigated the formation of patterns at the micron scale through the buckling instability of electrified jet from a polystyrene solution. The patterns emerging on the collector were found to depend on the applied voltage during electrospinning, the distance between the tip of the pipette and the collector, as well as the collector speed.

1.3.3 Elastic sewing machine

The problem of deployment of a flexible elastic rod on a rigid substrate was first studied by Mahadevan et al. [21] in 1996. The numerical solutions presented are

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par-tially incorrect due to a sign error involving inertial terms, as pointed out in Ref. [22]. However, the solution procedure based on dynamic Kirchhoff-Love equations is in-formative, and is reviewed in more detail next. Their numerical work exmained the deployment of a naturally straight rod from a fixed height onto a static rigid substrate. The motion of the rod eventually reaches a steady state and forms a circular coil of uniform radius, RC, which is determined as a function of six parameters – (1)

injec-tion speed, v, (2) density of the rod , ρ, (3) Young’s modulus, E, (4) cross-secinjec-tional area, A (related to the cross-sectional radius r0 by A = πr02 assuming circular) and

moment of inertia, I = πr4₀/4, (5) deployment height, H, and (6) acceleration due to gravity, g. These parameters can be reduced to three dimensionless parameters upon applying Buckingham-Pi theorem,

F r = v 2 gH, (1.15a) Γ = ρAgH 3 EI , (1.15b) ζ = A H2. (1.15c)

Physically, these dimensionless group can be interpreted as follows. The Froude number, F r, measures the ratio of the inertial and gravitational potential energies. The dimensionless gravity, Γ, is the ratio between gravitational potential and elastic energies. The dimensionless area, ζ, quantifies the rotary inertia of the rod. For thin rods deployed from a height H _r0, ζ 1 and rotary inertia can be neglected to

arrive at

RC = Hf (F r, Γ, ζ)∼ Hf(F r, Γ, 0). (1.16)

We can consider two asymptotic forms of this solution to recover the gravito-bending and the elastic regimes, discussed in _{§ 1.2.2.}

Gravito-bending regime: For slow injection speeds, inertia can be neglected (F r_{1) compared to elastic and gravitational energies simplifying Eq. 1.16 to}

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Using the physical intuition that an infinite deployment height H _{→ ∞ should result} in a finite coiling radius RC, Eq. 1.17 can be written as

RC ∼ HΓ−1/3= C r2 0E 8ρg 1/3 , (1.18)

where C is a numerical constant.

Elastic regime: If the elastic energy is large compared to the gravitational potential energy (Γ_{1) and the injection speed is moderate such that F r ∼ 1, Eq. 1.16 reduces} to

RC = Hg2(F r). (1.19)

A boundary value problem for the shape of the rod and its coiling radius is formu-lated, and a numerical solution using a continuation method is suggested, to finally arrive at asymptotic solutions in the regimes described above [21].

The scenario of a rod coiling onto a static surface, in all three regimes defined in § 1.2.2, was examined experimentally by Habibi et al. [22] in 2007, and the obser-vations were compared against numerical simulations. The observed coiling radius, RC, for the gravito-bending and elastic regimes follow Eqs. 1.17 & 1.19, respectively,

whereas in the inertial regime it scales as

RC ∼ 1 v r2 0E ρ 1₂ . (1.20)

The first detailed study on the deployment of elastic rods onto a moving substrate was performed by Ribe et al. [20]. Their experimental setup comprises of a silicone thread wound onto a wheel, which was rotated by an electric motor to feed the thread onto a moving belt. The experiments, accompanied by a numerical model, span across the gravito-bending and the inertial regimes, and realize a multitude of patterns – meandering, biperiodic meanders, W patterns, figures of 8, ampersands, W-by-8 patterns, “two by one”, “two by two”, double coiling, translated coiling, and several transient and semirandom states. These patterns are presented in the form of phase diagrams on (H, vb) and (v, vb) parameter spaces, where H is the deployment

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height, vb is the belt speed, and v is the injection speed. The meandering state, where

the rod makes a sinusoidal pattern, is quantified through measurements of amplitude, A, and wavelength, λ, as functions of belt speed, vb. The amplitude is found to follow

the scaling A_∼√, where = (v_{− v}b)/v. The onset of meandering is found to follow

a Hopf bifurcation [33].

A physical understanding of the pattern formation process, as well as the combi-nation of material and geometric parametric parameters that set the pattern length-scales, is yet to be achieved. The primary goal of this thesis is to explain the nonlinear pattern formation in gravito-bending regime, and realize the underlying physical pa-rameters with a strong emphasis on the natural curvature of the rod.

1.4 Outline of the thesis

The problem description, underlying motivation, and a brief introduction to the me-chanics of slender elastic rods were stated in the current chapter. Chapter 2 describes the physical experiments and numerical tools used in our investigation on the deploy-ment of rods onto substrates. We divide the scenario into two cases – static coiling or deployment onto a static substrate, and dynamic coiling or deployment onto a moving substrate. Static coiling is studied in Chapter 3, followed by dynamic coiling in Chapter 4, where we present phase diagrams of pattern formation process along multiple parameter spaces. The first mode of instability, the meandering patterns, is discussed in detail in Chapter 5. Throughout Chapters 3-5, experiments and simula-tions go hand in hand informing and validating each other. Assured by the validity of our simulation tool, we apply it to survey the patterns beyond meandering in Chapter 6. The outcomes of the thesis are summarized in Chapter 7, and possible venues for research are suggested.

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Chapter 2 Physical and numerical

experiments

In this chapter we present the methodology that we have followed to explore the dynamics of the deployment of slender elastic rods on rigid substrates. We consider a model setup where rod is deployed at a controlled rate onto a conveyor belt moving at a constant speed, and perform a synergistic combination of precision desktop-scale experiments and numerical simulations ported from computer graphics. It is important to note that throughout the thesis, the comparisons between experiments and simulations are shown with no fitting parameters. All geometric, material, and control parameters were measured independently prior to any simulation.

In _{§ 2.1, we first list the assumptions made in order to simplify our study. We} detail our experiments in _{§ 2.2 and the numerical procedure in § 2.3 respectively.}

2.1 Our model system: simplifying assumptions

We seek a minimal model setup to understand the mechanics that underlie the geo-metrically nonlinear pattern formation resulting from the deployment of a slender rod. In order to focus on the key ingredients that govern the mechanics of this process, we make a series of simplifying assumptions stated below.

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are homogeneous, and identical in all directions. We also consider a material that is linear elastic, and exclude any effect of plasticity in the rod, which might a priori have been considered relevant for the engineering applications that motivated our study. Due to the slenderness of the rods, the resulting strain levels are sufficiently low that no plastic deformation occurs (details in § 6.7). The material is assumed incompressible, and the change in volume with deformation can be neglected.

• Planar substrate. The substrate is assumed to be perfectly smooth and planar. In engineering applications (e.g. undersea cable deployment), however, the substrate can have inhomogeneous topography. Even a plane substrate presents a wealth of nonlinear patterns to explore, and the additional complexity of handling nonplanar substrate is left for future study.

• No inertia. The rate of rod deployment and the speed of the belt are kept low enough to ensure that inertial effects can be neglected. This was rationalized formally in _{§ 1.2.2.}

• Rod-surface adhesion and friction. We skip a detailed analysis of frictional effects on rod deployment. A simplified model for contact between the rod and the substrate is implemented, as outlined in _{§ 2.3.}

• Hydrodynamic effects. The fluid forces that can arise from ocean currents, viscosity, and buoyancy are ignored, and left for future study.

2.2 Desktop-scale physical experiments

We have custom fabricated a desktop-scale apparatus for experimentation, a photo-graph of which is presented in Fig. 2-1 (more details below). An elastomeric rod was deployed at a controlled injection speed, v, onto a conveyor belt moving with speed, vb. The rod was custom-fabricated out of a silicone-based rubber (Vinylpolysiloxane,

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the circular cross-sectional radius, r0, density, ρ, Young’s modulus, E, Poisson’s

ra-tio ν _{≈ 0.5, and natural curvature, which can be varied systematically in the range} 0 < κn< 0.4cm−1. Depending on the relative difference between v and vb, a variety

of coiling patterns can be attained when the rod becomes in contact with the belt and the process is imaged by a digital camera.

2.2.1 Experimental apparatus

The apparatus comprised of a conveyor belt with a vinyl surface, and an injection system to deploy the rod. The belt was wrapped around two cylinders, one of which was driven by a stepper motor (MDrive) computer controlled using LabView. The injection system ‘sandwiched’ the elastic rod in between two neoprene rollers, one of them being rotated using another stepper motor that was also controlled by LabView. This rotating roller fed the rod into a groove with height 1 cm to orient it along the vertical direction before deployment onto the belt. Acrylic plates were laser-cut to a precise shape and size to mount the rollers and create the groove of the injection system. For consistency, we ensured that the rod was aligned at the injection nozzle

d1) e1) f1) H 1 ₂ 4 5 3 47 cm 4 24 cm H 5 cm 2 3 5 1

A

B

2.2 cm

Figure 2-1: Experimental apparatus: (A) photograph, and (B) schematic. The thin elastic rod (3) is deposited by an injector (2) onto a conveyor belt (4), which is driven by a stepper motor (5). The patterns that form on the belt are recorded by a digital video camera (1).

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in a way that the orientation of its natural curvature was fixed throughout injection. The conveyor belt was calibrated by marking each 10 cm interval of the belt, and noting the time of each calibration mark passing a laser sheet using a digital camera. The calibration of the injection system was done by marking each 20 cm interval of a naturally straight rod, and recording the time to inject each interval.

Image processing: Our study on meandering presented in Ch. 5 requires precise measurement of the trace of the rod onto the belt. A digital video camera (Sony Handycam HDR-CX100) was used to capture the motion of the rod on the belt. The recorded video of each experimental trial was analyzed in MATLAB, see Fig. 2-2A for one frame of a representative video without any post-processing. This frame is converted to a binary image, where the bright pixels (value 1) correspond to the rod, and the dark ones (value 0) represent the conveyor belt, see Fig. 2-2B for the binary form of Fig. 2-2A. However, due to the presence of bright objects and reflective surfaces in the setup, not all the pixels with a value of 1 belong to the rod, as evident

0 1000 2000 3000 4000 5000 6000 7000 8000 0 100 200 300 4000 50 100 150 200 250 0 5 10 15 20 t [cm] t [pixels] y [pixels] y [cm]

D

A

B

C

Figure 2-2: Image processing. (A) one frame of a video capturing sinusoidal meaders (top view). (B) Binary version of the frame. (C) 50_{×50 pixels region at x}c = 400, yc=

275. All the bright pixels in this region represent the rod. The y-coordinates of the bright pixels along the vertical dashed line was averaged to get one point on the rod centerline. (D) The trace of the rod on the belt obtained upon converting the video of 4 min length.

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from Fig. 2-2B. This required an additional phase of post-processing to extract the rod from the image. Provided an initial guess for the location of any point on the trace of the rod, e.g. xc= 400, yc = 275 in Fig. 2-2B where x is along the motion of

the belt, our code processes a small rectangular region around that (xc, yc) location

and takes the mean of the y-coordinates of the bright pixels at xc = 400 to be a

point on the rod centerline. This rectangular region, taken as 50_{× 50 pixels in a} 720_{× 480 frame, is sufficiently small that no other bright object other than the rod is} present, see Fig. 2-2C. Approximating the bright pixels (red in Fig. 2-2C) to lie on a straight line, the yclocation of the rod at xc= 400 in the next frame can be predicted.

Centering around the predicted (xc, yc) location in each subsequent frame, a 50× 50

pixels region is processed to obtain points on the rod centerline and to update the prediction for yc in the next frame. One point from each frame thus contributes one

point to the trace which are then stacked together to construct the motion of the rod at xc = 400 with time, t, represented in pixels.

Fig. 2-2D shows the trace of the rod extracted by this method where a rod with Lgb = 3.3 cm was deployed from a height of ¯H = 15 (see § 2.2.2 for detailed rod

properties), and the belt was moving at vb = 1 cm/s. Knowing the value of frames

per second (fps) of the camera (29.97 in Fig. 2-2C), 1 pixel corresponds to a length of vb/f ps along the time axis. To determine pixel to the length conversion along the

y-axis, an object of known dimensions was filmed for calibration.

This simple predictor-corrector method needs processing of a 50_{× 50 pixel region} instead of the entire frame of size 720_{×480 pixels reducing the processing time.} How-ever, this method does not support detection of loops and coils where the trace of the rod self-intersects. We can also compute the arc-length of the injected rod by ana-lyzing the pattern, which can be used in conjunction with the belt speed to compute the injection speed. Since calibration of the injection system is unreliable owing to the slippage between the roller and the rod and the non-uniformity of cross-sectional radii across multiple rods, this method allows us to forgo injector calibration and use this direct measurement when high precision is necessary, e.g. for the quantification of the amplitude and wavelength of the meandering patterns.

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2.2.2 Rod manufacturing

Two types of rods were used in our experiments presented in this thesis. The first (Elite Double 32, Zhermack) had radius, r0 = 0.16 cm, density, ρ = 1.18 g/cm3,

and Young’s modulus, E = 1.3 MPa. This gives a gravito-bending length, Lgb =

_r2 0E

8ρg

1₃

= 3.3 cm (defined in Eq. 1.11). The second type (Elite Double 8, Zhermack) had radius, r0 = 0.16 cm, density, ρ = 1.02 g/cm3, and Young’s modulus, E =

0.18 MPa, resulting in Lgb= 1.8 cm.

The rod manufacturing process is illustrated in Fig. 2-3. The rods used in the ex-periments were cast with silicone-based rubbers (Vinylpolysiloxane, Elite Zhermack) using PVC tubes (VWR International) as molds. To impart natural curvature to the rod, the tubes were first wrapped around cylindrical objects with the desired radii. The fluid mixture of polymer and catalyst was injected into each tube, which was carefully cut after the curing process to extract the soft elastic rod. The natural cur-vature, 0 < κn < 0.4cm−1, of such a rod is set by the inverse of the sum of the radius

of cylindrical object, thickness of PVC tube (1/8 inch) and radius of rod cross section. For the fabrication of a straight rod, the mold was attached to a rigid straight bar. Details on how κn is measured is provided below.

36.5m−1 16.5m−1 21.0m−1 5 cm

A B C

Figure 2-3: Rod manufacturing process. (A) Elite Double 32 base (green) and catalyst (white). Image adapted from http://en.zhermack.com/. (B) Mixture of base and catalyst injected in a hollow PVC tube, and wrapped around a cylinder to induce natural curvature. (C) Demolding of elastomeric rod through removal of the tube.

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2.2.3 Measurement of material properties

Our experiments required measurement of the material properties of the manufac-tured rods, e.g. cross-sectional radius, r0, density, ρ, Young’s modulus, E, and natural

curvature, κn.

Cross-sectional radius, r0, and density, ρ. Thin slices (∼ 1mm in height) were

cut from a rod specimen using a sharp scalpel. Those slices were scanned at high resolution using a desktop digital photograph scanner. The scans were analyzed using ImageJ1, a public domain Java-based image processing program, to obtain the radius, r0. Upon obtaining the radius, a rod specimen with length L (measured using slide

calipers) was weighed in a digital balance to get its mass, m, and finally compute the density, ρ = _πrm2

0L .

Young’s modulus, E. The Young’s modulus was computed from the measured natural frequency of a rod specimen. A clamped rod specimen (length _{∼ 3.5 cm)} held in an vertical arrangement was displaced laterally by_{∼ 1 cm, and subsequently} released. The resulting oscillation was captured using a digital video camera (Sony Handycam HDR-CX100, fps=29.97), and the the natural frequency of the first mode of oscillation, fn, was calculated. The Young’s modulus can then be calculated from

fn = β₁2 2πL2 s EI ρA, (2.1)

where β1 = 0.597π for clamped-free beam, L is the length of the specimen, A = πr02

is the cross-sectional area, and I = πr2₀/4 is the area moment of inertia for circular cross-section.

Natural curvature, κn. If the rod was wrapped around a cylinder of radius, rc,

and the hollow PVC tube used as a mold has thickness rP, the natural curvature of

the manufactured rod is κn= 1/(rc+ rP + r0).

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See Ref. [35] for a more detailed account of the rod manufacturing procedure, and the protocol to measure material property.

2.2.4 Experimental protocol

After manufacturing a rod, with a typical length of 4_{∼ 5.5 m, its material properties} were measured following the protocols in 2.2.3. The injector and the conveyor belt were calibrated for speed, whereas the video camera was calibrated for pixel to length conversion. During each experimental run, the rod was injected at a controlled speed, v, onto the belt moving at speed vb. The pattern produced by the rod on the belt

was video recorded, see Fig. 2-1 for the position of the camera. The video was then processed automatically (details of image processing in _{§ 2.2.1), or inspected visually} when only a qualitative description, e.g. pattern type, was necessary.

2.3 Numerics ported from computer graphics

In parallel with experiments, we used a numerical simulation tool developed by Fang Da and Eitan Grinspun at Columbia University, who were collaborators in the re-search project presented in this thesis. This tool, written in C++, is based on Discrete Elastic Rods (DER) formulation [23, 24] introduced in _{§ 2.3.1. We implemented the} boundary conditions described in_{§ 2.3.2 that are specific to our model setup. To} per-form a parameter sweep, e.g. to quantify the meandering wavelength as a function of the natural curvature, a series of simulations were launched from python scripts, each simulation with a different set of physical parameters, e.g. natural curvature. Since the simulation tool employs a single threaded process, parallel computing was implemented to run multiple simulations simultaneously, and thus optimize the use of computing resources.

The DER code was originally developed to serve the visual special effects and animated feature film industries’ pursuit of visually dramatic (i.e. nonlinear, finite deformations) dynamics of hair, fur, and other rod-like structures. This method is based on discrete differential geometry (DDG), a budding field of mathematics

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that is particularly well suited for the formulation of robust, efficient, and geometri-cally nonlinear numerical treatments of elasticity [47]. This method allows for rods with arbitrary (i.e. curved) undeformed configurations, arbitrary cross sections (i.e. non-circular), dynamics, and contact. This generality along with its robustness and efficiency allows us to explore the nontrivial mechanics and geometric nonlinearity involved in our specific problem.

2.3.1 Discrete Elastic Rods

A brief introduction to Discrete Elastic Rods (DER) follows. Ref. [24] provides a detailed account of this method.

The configuration of a Kirchhoff [15, 18] elastic rod is represented by an arc-length parameterized curve γ(s) in R3_{and an orthonormal material frame}_{{t(s), m}

1(s), m1(s)},

see Fig. 2-4A1 for visual depiction of the material frame. Here, γ(s) is an arc-length parameterized curve in R3_{describing the centerline of the rod, and the material frame}

satisfies t(s) = γ0(s). The prime refers to differentiation with respect to arc-length, e.g. γ0(s) = dγ/ds. The centerline’s curvature vector is related to the tangent

x

_i₋₁

x

_i

x

i+1 mi₁ mi₂ π− φi ti ti−1 ui vi ui vi mi₁ mi₂

γ(s)

A1

A2

B

θi

e

_i

e

_i₋₁

Figure 2-4: (A1) The centerline of an elastic rod is represented by a curve γ(s) and a material frame _{{t(s) = γ}0(s), m1(s), m1(s)}. (A2) The natural (twist-free) Bishop

frame _{{t(s) = γ}0(s), u(s), v(s)_{} assigned to the same centerline. (B) Discrete rod} described by the vertices xi (red dots) and edges ei (dashed arrows). Images (A1-A2)

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through the relation κ(s) = t0(s). It is also an adapted frame since this orthonormal frame is tangent to the curve at all points. For a fixed centerline, the geometrically most natural frame is the Bishop frame_{{t(s) = γ}0(s), u(s), v(s)_{}, an adapted frame} with zero twist showed schematically in Fig. 2-4A2. This twist-free Bishop frame al-lows for a succinct representation of the rod by an arc-length parameterized adapted framed curve _{{γ(s), θ(s)}, where γ(s) describes the rod’s centerline, and θ(s)} mea-sures the rotation about the tangent of the material frame relative to the Bishop frame [48]. Upon deformation of the rod, the local strains are captured by the twist, τ = θ0(s) = (m1)0· m2, and curvature, κ(s) =|t0(s)|. Although the reference

configu-ration of the rod is assumed to be stress-free, we consider rods that possess a natural curvature κn, which we will take as a primary parameter in our study. Stretching

can also be captured by the deformation of the centerline, however, we neglect it on grounds of the inextensibility of the rod.

In the discrete setting (see Fig. 2-4B), the frame curve Γ is composed of (n+2) vertices x0, . . . , xn+1, and (n+1) edges e0, . . . , en such that ei = xi+1− xi. Each edge

has an associated material frame _{ti_{, m}i

1, mi2}, and Bishop frame {ti, ui, vi}. If φi is

the turning angle between two consecutive edges, the discrete curvature is

κi = 2 tan(φi/2), (2.2)

and the curvature binormal at a vertex, defined as the cross product of tangent and curvature vectors, can be written as

(κb)i =

2ei−1_{× e}i

|ei−1_||ei_{| + e}i−1_{· e}i−1. (2.3)

Hereafter, quantities with an under-bar denote evaluation in rest state.

Energy formulation. Upon inextensibility assumption, a rod has twofold elastic energies – bending and twisting. For an isotropic rod, following _{§ 1.2.2, the bending} energy takes the form Eb = 1₂R EI (κ − κn)2ds, whereas the twisting energy can be

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The discrete formulation of bending energy is Eb = n X i=1 1 2l_i i X j=i−1 ωj_i _{− ω}j_iT Bj ωj_i _{− ω}j_i , (2.4)

where li =|ei−1| + |ei|, Bj = EII for an isotropic rod (I is a 2× 2 identity matrix),

and the material curvature

ωj_i = (κb)i· mj2,−(κb)i· mj1

T

, j _{∈ {i − 1, i},} (2.5)

Note that ωj_i = 0 for a naturally straight rod. The twisting energy of an discrete rod is

Et= n X i=1 GJ(θ i_{− θ}i−1₎2 l_i . (2.6)

At each time step in DER, first the twist-free frame is updated using parallel transport which is a way of transporting geometrical data along a smooth curve. A vector x is parallel transported along the rod certerline by integrating the equation x0 = κb _{× x. Subsequently, under the quasistatic material frame postulation, the} material frame is updated by energy minimization _dθdEj = 0 with relevant boundary conditions. The next step is to compute the centerline forces by taking the gradient of energy formulated above, and then update the centerline based on those forces.

2.3.2 Implementation of the boundary conditions

The study presented in this thesis requires implementation of boundary conditions specific to the problem under study detailed below.

Classification. Simulation nodes and edges (jointly “degrees of freedom”) are classified as either constrained (under the direct influence of boundary conditions) or free (under the influence of the equations of motion for the dynamics of a Kirchhoff elastic rod, absent boundary conditions). In the following we describe the modifica-tions required to integrate the dynamics of constrained degrees of freedom, and to

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transition between the two states.

Injector boundary condition. At any point during the simulation, three nodes and two edges are constrained to model the injector boundary condition. Consider the initial conditions in which constrained nodes x1 through x3 and edges e1 through e2

constitute the injector boundary conditions (Fig. 2-5 A). Under the conditions, each nodal position is prescribed to move downward with fixed downward velocity v; each edge material frame is prescribed to remain fixed, such that the natural curvature is aligned to the belt velocity vector vb.

After a duration ∆L/v elapses (Fig. 2-5 B), we lift the constraints on node x1

and edge e1, allowing them to simulate as free degrees of freedom under the equations

of motion. Simultaneously, we introduce the new constrained edge e3 and vertex

x4, corresponding to the injection of additional rod. The lifting of constraints on x1

and e1, combined with the introduction of constrained degrees of freedom e3 and x4,

maintain the invariant that four nodes are constrained to model the injector boundary condition.

t

=

0 t

=

ΔL/

v

A

Figure 2-5: The injector boundary condition. (A) The beginning of the simulation. (B) After injecting ∆L of the rod. Blue nodes and edges are constrained; green nodes and edges are free.

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Belt contact boundary condition. Once the free end of the rod contacts the belt (the plane of z = 0), a new constraint must be introduced to model the contact. We must first detect the contact, and then respond. Correspondingly, each simulation step begins with a predictor step, optionally followed by a corrector step.

The predictor step is performed by ignoring the direct effect of the belt on free degrees of freedom. At the end of the predictor step, we determine whether any free nodes have made contact with belt (see Fig. 2-6 B). Such nodes form the new contact set Ξ. Contact is defined as zi <= zbelt+ r0, where zi is the z coordinate of the node,

zbelt is the z coordinate of the belt, and r0 is the rod radius.

If new contacts have been identified, a corrector step is required (Fig. 2-6 C). We rewind the simulation to the start of the time step. The vertical coordinate z of each node in the new contact set Ξ is transitioned from the free state to the constrained state, with vertical position prescribed to lie in contact with the belt, i.e. z = zbelt+r0;

for these vertically-constrained nodes, the horizontal coordinate degrees of freedom (x, y) continue to be governed by the equations of motion. After introducing the vertical constraints, we integrate forward in time again, this time assured that the nodes previously in the new contact set cannot penetrate through the belt. At the end of the corrector step, each vertically-constrained node is transitioned either to the free or the (fully) constrained state: if the relative velocity between the node and the belt is below a threshold (we use 10−6 cm/s), we permanently constrain the node

belt

v _contnew_act set

nodesin the new contactsetare vertically constrained

A B C D

Figure 2-6: The belt boundary condition. (A) Beginning of the time step. (B) The predictor step, where the new contact set is detected. (C) The corrector step, where the nodes in the new contact set are vertically constrained. (D) The vertically constrained nodes become fully constrained as its horizontal velocity becomes small.

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to follow the belt, i.e. to maintain the belt velocity (Fig. 2-6 D). Otherwise, we free the node. Because we focus on the small Froude number regime, where inertial effects are negligible, we observe that typically vertically-constrained nodes are fully belt-constrained after the corrector step.

We implemented a simple contact model by applying Dirichlet boundary condi-tions (pinned nodes) at the points of rod–substrate contact. However, the edges (we recall that an edge connects two consecutive nodes) on the deposited rod that are within a certain distance (measured along the arc-length of the rod) of the contact point can rotate about the rod centerline. We take this distance to be αLgb, and all

the edges that are more than αLgb away from the contact point node into the fixed

portion are prevented from rotating by a constraint on the material frame angle. We observe that as long as 5 _{≤ α ≤ 50, it has negligible influence on our quantities of} interest, e.g. the coiling radius, as well as the meandering amplitude and wavelength. In our numerical experiments, we used α = 5.

It is interesting to note that in the experiments, as the rod accumulates on the surface, it coils over existing segments, i.e. a non-uniform surface, whereas this effect was not considered in the simulations (past the contact point the rod is frozen on the belt and assumed to be flat). The excellent quantitative agreements between exper-iments and numerics throughout this thesis supports our assumption for neglecting self-contact.

Source code. The source code of our numerical simulation tools has been made available at http://www.cs.columbia.edu/cg/elastic_coiling/. Instructions on how to use the program are also provided on the webpage.

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Chapter 3 Static Coiling

In this chapter, we consider the scenario when a slender elastic rod is deployed onto a steady substrate (belt speed vb = 0), which we refer to as static coiling. We investigate

the periodic nonlinear patterns that emerge as a result of this deployment through ex-periments and simulations. In particular, we examine the effect of natural curvature on the pattern formation process. For rods with high values of their natural curva-ture, we observe a periodic change in coiling direction, which is referred to as coiling inversion. Upon achieving excellent agreement between experiments and numerics in capturing the patterns, we rationalize our observations through a scaling analysis of the energies involved: elastic (bending and twisting) and gravity. Throughout our analysis, the relatively slow injection speed allows us to neglect inertial effects, and remain in the gravito-bending regime introduced in _{§ 1.2.2. Even though inertia can} change the coiling pattern (see _{§ 1.3.3), this is outside the scope of our study.}

In _{§3.1 & §3.2, we draw a qualitative picture of our observations on the static} coiling of naturally straight rods (κn = 0) and naturally curved rods (κn > 0)

re-spectively, and demonstrate that natural curvature fundamentally alters the coiling pattern and can induce coiling inversions. Next, we quantify the effect of natural curvature of the rod as a systematic parameter. In _{§3.3, we rationalize how the} ra-dius of the deposited coil varies with the natural curvature of the rod. In _{§3.4, we} discuss how the interplay between curvature, twist, and bending leads to the periodic inversion in the coiling direction for naturally curved rods.

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3.1 Static coiling of straight rods

When a naturally straight rod (natural curvature, ¯κn = 0) is deployed onto a steady

substrate, after undergoing transient dynamics, it coils in a circular fashion, see Fig. 3-1A. The coiling orientation is set once by the breaking of symmetry during the initial contact. In Fig. 3-2A, we present the top view of the coil formed by a straight rod. In Fig. 3-2B, we present the trace of the rod deposited on the substrate obtained from DER simulations, and observe that it eventually takes a perfectly circular shape with normalized radius, ¯RC = RC/Lgb. In our coordinate system, ¯x is oriented along the

belt motion, and ¯y is perpendicular to ¯x on the plane of the substrate. The projection of the nozzle on the substrate is taken as the origin.

In Fig. 3-2C, we plot the normalized coiling radius, ¯RC =p ¯x2+ ¯y2, as a function

of time, and find that ¯RC asymptotes to a value of 2. The steady state coiling radius

from our experiments using a rod with Lgb = 3.3 cm (detail material properties are in

§ 2.2.2) is ¯RC = 1.96± 0.15, confirming the simulation result. Both experiments and

simulations are in agreement with the dimensional and scaling analysis reported in § 1.3. The transient dynamics is outside the scope of this study, and we concentrate on the steady state coiling radius. Hereafter, dimensionless coiling radius, ¯RC, implies

its steady state measurement.

A1 A2 B1 B2

¯

κn = 0.69

¯

κn = 0 κ¯n = 0 κ¯n = 0.69

Figure 3-1: Static coiling (belt is stopped, vb = 0, and speed of injection is v =

2cm/s). (A) Coiling of a straight rod (¯κn= 0, Lgb = 3.3cm). (B) Coiling of a curved

rod (¯κn = 0.69, Lgb = 3.3cm) with an imminent reversal in coiling direction. (A1)

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í í 0 í í í 0

¯

x

¯y

0 50 100 150 0 1 2 3 4 Norm. time,

Norm. coiling radius,

¯ R

C

¯

t

A

B

C

Figure 3-2: Evolution of coiling pattern for a naturally straight rod (¯κn = 0) with

time. The rod was deployed from ¯H = 15 at an injection speed v = 1.0 cm/s. (A) Top view of static coiling of a straight rod. (B) Trace of the rod centerline deposited on the substrate. The arrows, drawn at 2 seconds interval, indicate deposition with increasing time. (C) Normalized coiling radius, ¯RC, as a function of dimensionless

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To understand the result ¯RC = 2, we recall the gravito-bending length, Lgb,

introduced in _{§ 1.2.2. Referring back to Fig. 3-2A, the radius of curvature of the} suspended portion of the rod, near the contact point with the substrate, is set by the balance between bending and gravitational potential energies_Eb ∼ Eg, and, therefore,

is of order Lgb. By inspection, the dashed circle in Fig. 3-2 has radius Lgb, and

together with the above observation the radius of coiling for ¯κn = 0 is RC = 2Lgb,

which matches our experimental and numerical results.

3.2 Static coiling of naturally curved rods

We now extend our study to consider static coiling of a naturally curved rod (¯κn > 0),

and, both in experiments and simulations, find that coiling of a naturally curved rod can be qualitatively distinct from the case of a straight rod, discussed above. In Fig. 3-3, using DER simulations, we plot the trace of the deposited part of a naturally curved rod (¯κn = 0.69) with the red arrows marking the occurrence of

−3 −2 −1 0 1 2 −3 −2 −1 0 1 2 3

¯

x

¯y

Figure 3-3: Trajectory of the contact point between the rod and the substrate during coiling of a naturally curved (¯κn = 0.69) rod. The rod was deployed from a height

¯

H = 15 at an injection speed v = 1.0 cm/s. The same coordinate system from Fig. 3-2B has been used. The arrows, drawn at 2 seconds interval, indicate the direction of rod deposition. Red arrows correspond to the inversion event described in the text.

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coiling inversion, when the rod changes its coiling orientation (from anti-clockwise to clockwise). We also observe the normalized coiling radius, ¯RC, is lower than 2 (the

result for a straight rod). In the following sections, we examine in more detail the aforementioned twofold fundamental effects of natural curvature: (i) change in coiling radius, and (ii) inversion in coiling direction.

3.3 Coiling radius of naturally curved rods

In Fig. 3-4, we plot the normalized coiling radius, ¯RC, as a function of the normalized

natural radius of the rod, ¯Rn = ¯κ−1n , at a fixed height of ¯H = 15, and find excellent

quantitative agreement between experiments and simulations. We can decompose the scenario into two regimes: (i) for low values of the natural radius, the coiling radius first scales as ¯RC ∼ ¯Rn (red solid line), but then (ii) for higher natural radius, ¯RC

levels off and eventually in the limit of ¯Rn→ ∞ (for nearly straight rods) asymptotes

to ¯RC ∼ 2 (orange solid line).

0 2 4 6 8 10 0 0.5 1 1.5 2

Normalized natural radius,

Normalized coiling radius,

Exp. DER Eq. 3.3

¯ R

C

¯

R

n (i) (ii) ¯ RC= 2 ¯ RC = ¯Rn

Figure 3-4: Variation of the normalized coiling radius, ¯RC, with the normalized

natu-ral radius of the rod, ¯Rn= 1/¯κn. The rods had a gravito-bending length of Lgb = 3.3

cm. Dimensionless deployment height, ¯H = 15, and injection speed, v = 2.0 cm/s, were kept fixed. The two regimes – (i) ¯RC ∼ ¯Rn (red solid line corresponds to

¯

RC = ¯Rn), and (ii) ¯RC ∼ 2 (orange solid line corresponds to ¯RC = 2), are separated

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In _{§ 1.2.3, we showed that for a straight rod with κ}n = 0 the balance between

bending and gravitational energies results in the characteristic length-scale Lgb, which

we now find to also set the coiling radius. To understand the results in Fig. 3-4, we extend the previous scaling analysis that lead to Lgb (see § 1.2.3) to consider a rod

with natural curvature. When a rod that has a curvature of κn in its stress-free (or

natural) state is deformed to coil at radius RC = κ−1C , the bending energy per unit

length scales as_Eb ∼ (EI/2)(κC− κn)2. Balancing this with the gravitational energy

per unit length, _Eb ∼ Eg yields

(EI/2)(κC− κn)2 ∼ ρπr02gRC. (3.1)

Rearranging Eq. 3.1 and introducing c as the constant of proportionality, we obtain

κC(κC − κn)2 = c2ρπr 2 0g EI 3 = c Lgb 3 , (3.2)

which upon nondimensionalization by Lgb reduces to

¯

κC(¯κC− ¯κn)2 = c3. (3.3)

From our observation on a straight rod in _{§ 3.1, R}C|κn=0= 2Lgb, the value of c = 0.5 can be determined. Replacing this value in Eq. 3.3 yields the solid line plotted in Fig. 3-4, which is in good agreement with both experiments and simulations. Note that the prediction from Eq. 3.3 comes solely from scaling analysis without rigor-ous calculation for the proportionality constant, yet it closely follows the data from experiments and simulations.

3.4 Coiling inversion

In this section, we examine the coiling inversion in detail, and study how the coupled effect of natural curvature, twist, and bending is responsible for this phenomenon.

(46)

3.4.1 Coiling inversion: interplay between curvature & twist

An important parameter to quantify the coiling inversions is the length of rod deployed between subsequent coiling direction reversals, which we refer to as the inversion length, Linv. In Fig. 3-5A, we plot the normalized inversion length, ¯Linv, as a function

of normalized deployment height, ¯H, and observe that deployment height modifies the inversion length as ¯Linv ∼ ¯H, with a slope monotonic in ¯Rn.

0 50 100 150 í í í 0 1 1RUPWZLVWDORQJKHHO >UDGLDQ@

¯ θ

1RUPWLPH

_t

¯

¯ κn ¯ κn ¯ κn ¯ κn 0 5 10 15 20 25 0 14 28 42 56 70

Norm. deployment height,

Norm. inversion length,

Exp.: = 0.69 DER: = 0.69 DER: = 0.83 DER: = 0.99 Exp.: = 1.22 DER: = 1.22 ¯ κn ¯ κn ¯ κn ¯ κn ¯ κn ¯ κn ¯ Linv

¯

H

A B

Figure 3-5: (A) Normalized inversion length, ¯Linv, as a function of normalized

deployment height, ¯H, at different values of ¯κn. (B) Time series of twist, ¯θ0(¯t), of the

suspended heel (0 < s < sc). The dashed line represents the average value within

0 < s < sc and the thin solid lines correspond to the minima and maxima within

this interval. All other values of ¯θ0 are within the shaded region. The curves with ¯

κn ={0, 0.17} correspond to cases where no coiling inversion is observed. The curves

with ¯κn = {0.43, 0.69} correspond to the scenario exhibiting torsional buckling and

Coiling of elastic rods on rigid substrates

Coiling of Elastic Rods on Rigid Substrates

by

Mohammad Khalid Jawed

B.S.E. (Aerospace Engineering), University of Michigan (2012)

B.S.E. (Engineering Physics), University of Michigan (2012)

Submitted to the Department of Mechanical Engineering

in partial fulfillment of the requirements for the degree of

Master of Science

at the

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

September 2014

c

Massachusetts Institute of Technology 2014. All rights reserved.

Author . . . .

Department of Mechanical Engineering

August 8, 2014

Certified by . . . .

Pedro M. Reis

Associate Professor of Mechanical Engineering and Civil and

Environmental Engineering

Thesis Supervisor

Accepted by . . . .

David E. Hardt

Ralph E. and Eloise F. Cross Professor of Mechanical Engineering

Chairman, Committee on Graduate Students

Coiling of Elastic Rods on Rigid Substrates

by

Mohammad Khalid Jawed

Abstract

Acknowledgments

Contents

Chapter 1

Introduction

1.1

Definition of the problem

1.2

Mechanical behavior of slender elastic rods

1.2.1

Kirchhoff rod model

1.2.2

Energies of a slender elastic rod

1.2.3

Gravito-bending length, L

L

1.3

Literature review

1.3.1

Fluid-mechanical sewing machine

A

B

1.3.2

Electrospinning of fibers

A

B

C

g

1.3.3

Elastic sewing machine

1.4

Outline of the thesis

Chapter 2

Physical and numerical

experiments

2.1

Our model system: simplifying assumptions

2.2

Desktop-scale physical experiments

2.2.1

Experimental apparatus

A

B

D

A

B

C

2.2.2

Rod manufacturing

2.2.3

Measurement of material properties