Controlling fracture cascades through twisting and quenching

Ronald H. Heisser,1, ∗ Vishal P. Patil,2, † Norbert Stoop,2 and Jorn Dunkel2

1Department of Mechanical Engineering, Massachusetts Institute of Technology,77 Massachusetts Avenue, Cambridge, MA 02139, USA

2Department of Mathematics, Massachusetts Institute of Technology,77 Massachusetts Avenue, Cambridge, MA 02139, USA

(Dated: February 16, 2018)

Fracture limits the structural stability of macroscopic and microscopic materials, from beams andbones to microtubules and nanotubes. Despite recent progress, fracture control continues to presentprofound practical and theoretical challenges. A famous longstanding problem posed by Feynmanasks why brittle elastic rods appear almost always to fragment into at least three pieces when placedunder large bending stresses. Feynman’s observation raises fundamental questions about the exis-tence of protocols that can robustly induce binary fracture in brittle materials. Using experiments,simulations and analytical scaling arguments, we demonstrate controlled binary fracture of brittleelastic rods for two distinct protocols based on twisting and nonadiabatic quenching. Our experi-mental data for twist-controlled fracture agree quantitatively with a theoretically predicted phasediagram. Furthermore, we establish novel asymptotic scaling relations for quenched fracture. Dueto their generality, these results are expected to apply to torsional and kinetic fracture processes ina wide range of systems.

I. INTRODUCTION

Elastic rods (ERs) are ubiquitous in natural and man-made matter, performing important physical and bio-logical functions across a wide range of scales, fromcolumns [1], trees [2] and bones [3] to the legs of wa-ter striders [4], semi-flexible polymer [5] networks [6, 7]and carbon nanotube composites [8]. When placed underextreme stresses, the structural stability of such materi-als becomes ultimately limited by the fracture behav-iors of their individual fibrous or tubular constituents.Owing to their central practical importance in engineer-ing, ER fracture and crack propagation have been in-tensively studied for more than a century both exper-imentally [9–11] and theoretically [12–14]. Recent ad-vances in video microscopy and microscale force manipu-lation [15, 16] have extended the scope of fracture studiesto the microworld [17, 18], revealing causes and effectsof structural failure in the axonal cytoskeleton [19], fi-broblasts [20], bacterial flagellar motors [21], active liquidcrystals [6] and multi-walled carbon nanotubes [22, 23].

Although important theoretical progress [9, 10, 24, 25]has been achieved over the past two decades, even ba-sic qualitative aspects of the fracture phenomenology re-main poorly understood. Bending induced ER fragmen-tation has been thoroughly investigated in the limits ofadiabatically slow [9] and diabatically fast [10] energy in-jection, but the roles of twist and quench rate on thefracture process have yet to be clarified. These two fun-damental issues are directly linked to a famous obser-vation by Richard Feynman [26], who noted that dry

∗ Current address: Sibley School of Mechanical and Aerospace En-gineering Cornell University, 105 Upson Hall, Ithaca, New York14853† Joint first author

spaghetti, when brought to fracture by holding the endsand moving them towards each other, appears almostalways to break into at least three pieces. The phe-nomenon of non-binary ER fracture is also well knownto pole vaulters, with a notable instance occuring dur-ing the 2012 Olympic Games [27]. Below, we will revisitand generalize Feynman’s experiment, in order to inves-tigate systematically how twist and quench dynamics in-fluence the elastic fragmentation cascade[9, 10]. Specif-ically, we will demonstrate two complementary quenchprotocols for controlled binary fracture of brittle ERs,thereby identifying conditions under which Feynman’sfragmentation conjecture becomes invalid. Our experi-mental observations are in good agreement with numeri-cal predictions from a nonlinear elasticity model, and canbe rationalized through analytical scaling arguments.

II. RESULTS

Model. We describe an ER at time t by its arclength parametrized centerline x(s, t), s ∈ [0, L], and anorthonormal frame {d1(s, t),d2(s, t),d3(s, t)} such thatd3 = x′, where primes denote s-derivatives and dots de-note t-derivatives. We further assume the rod is uniformwith density ρ, naturally straight and inextensible withcircular cross sectional area A = πr2. Its moment ofinertia I and moment of twist J are then given by[28]J = 2I = πr4/2. The rod’s dynamics are governed bythe damped Kirchhoff equations [28, 29] (SI)

F′′ = ρAd3 (1a)

M′ + d3 × F = L + 4bρIω3d3 (1b)

where F(s, t) is the force, M(s, t) = EIκ1d1 +EIκ2d2 +µJκ3d3 is the internal moment, L(s, t) = ρIω1d1 +ρIω2d2 + 2ρIω3d3 is the cross sectional angular mo-mentum, and the vectors κ = κidi, ω = ωidi satisfy

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2 pieces3 pieces4 pieces5 pieces

ν = 0.3

ν = 0.5

ν = 0.1

óc = 1.9 x 107 Nm-2

ócVM = 1.9 x 107 Nm-2

ócp = 5.3 x 107 Nm-2

Eff

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ress

(N

m-2)

x107

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FIG. 1. Using twist to break Feynman’s fragmentation bound. (a) High-speed images from an experiment withsubcritical twist angle showing fragmentation into more than two pieces, in agreement with Feynman’s conjecture (Movie 1).Time t = 0 (left) is defined as the moment (last frame) before fracture (b) Simulations also predict fracture in more than twopieces for parameters corresponding to the experiment in (a). Due to perfectly symmetric initial conditions, our simulationsgenerally produce even fragment numbers (Movie 1). Red line illustrates twist. (c) At supercritical twist angles, the maximumcurvature before fracture is significantly lowered enabling twist-controlled binary fracture (Movie 2). (d) Simulations forthe experimental parameters in (c) also confirm binary fracture (Movie 2). (e) Phase diagram showing that binary fracturedominates for twist angles larger than ∼ 250 degrees (73 data points in total). The theoretically predicted region (purple) inwhich an ideal rod is expected to exhibit binary fracture depends only weakly on the Poisson’s ratio ν and agrees well withthe data. (f) Experimental data from (e), averaged over 10 sectors defined by the rays (µ2JL−1 cos (nπ/2) , E2I sin (nπ/2)) forn = 0, 1 . . . 11, follow the theoretically predicted critical ellipse (solid curve) from Eq. (3). The dashed curve shows the vonMises ellipse from Eq. (4), and the dash-dotted curve shows the parabola of constant maximum principal stress from Eq. (5).The data in Fig. 2 yield σc = σVM

c = 1.9 × 107 N/m2 and σPc = 5.3 × 107 N/m2 at zero twist. Error bars show standard

deviations. The critical curvatures for simulations at different twist in (b) and (d) are chosen according to the critical stressellipse in (f), and the minimum fragment length λ = 3 cm estimated from the data in Fig. 2. Diameter of rod in (b,d) enhancedfor visualization. Scale bars in (a-d) 3 cm.

di′ = κ× di, di = ω × di. The Young’s modulus E and

the shear modulus µ are related by E/µ = 2(1 + ν),where ν is the Poisson’s ratio. Most materials have0.2 < ν < 0.5. The final term on the rhs. of Eq. 1(b) de-notes damping of twist modes with damping parameter b.Our measurements of this parameter using a torsion pen-dulum (Appendix) indicate that twist is approximatelycritically damped (SI). Since the timescale for the entirefracture cascade is an order of magnitude smaller thanthe time period of the fundamental bending mode, we donot need to include bending damping terms in our anal-ysis. The average material properties of our experimen-tal samples are 2r = 1.4 ± 0.1 mm, ρ = 1.5 ± 0.1 g/cm

3,

E = 3.8±0.3 GPa, µ = 1.5±0.2 GPa, and by consideringmean values for E,µ we obtain ν = 0.3±0.1 (Appendix).Finally, we note that the Kirkhhoff equations do not ac-count for certain shear effects described by Timoshenkobeam theory. Indeed, the Timoshenko theory does pro-vide a more accurate description of bending waves with

large wavenumber compared to rod radius. However, todescribe fracture, we will only need to consider wavenum-bers k with kr/2π < 0.1. In this regime, the differencebetween the Timoshenko and Kirkhhoff beam theories isnegligible [30].

To compare individual experiments with theoreticalpredictions, we solve the Kirchhoff equations (1) nu-merically with a discrete differential geometry algo-rithm [31, 32] (Appendix), adopting a stress-based frac-ture criterion defined as follows: We define the twistof the rod, θ(s, t), from the twist density, κ3(s, t), byθ′ = κ3. The effective stress at a point, σ(s), is ob-tained by integrating a scalar invariant of the full stresstensor, S, over a cross section of the rod

σ(s, t)2 =1

2πr2

∫tr(S>S)dA

=1

4E2r2κ(s, t)2 +

1

2µ2r2θ′(s, t)2

(2)

3

where κ =(κ21 + κ22

)1/2is the geometrical curvature of

the centerline and tr denotes the trace. If the rod is ina steady state, the twist density is constant, θ′ = Tw/L,where Tw is the total applied twist. We posit that the rodfractures at a point s along the curve if the effective stressσ(s) exceeds a critical value σc, and that no two fracturescan occur within a minimal fracture distance[33] λ of eachother. For a uniform twist distribution, the critical stressimposes a critical yield curvature κc by

σ2c =

1

4E2r2κ2c +

1

2µ2r2

(Tw

L

)2

. (3)

For comparison, by integrating the classical von Misesstress criterion over a cross section, we obtain a criticallocal stress ellipse given by (SI)

(σVMc

)2=

1

4E2r2κ2c +

3

4µ2r2

(Tw

L

)2

(4)

Another common criterion comes from considering themaximum eigenvalue of the stress tensor, or maximumprincipal stress, on the boundary of the rod. This givesa critical stress parabola

(σpc )2

= µ2r2(Tw

L

)2

+ E2r2κ2c+

+ Er2κc

√E2κ2c + 2µ2

(Tw

L

)2

(5)

All three curves are qualitatively consistent with ourdata, with Eq. (3) yielding the best quantitative agree-ment (Fig. 1f).

Our model contains exactly one free parameter, λ,which we introduce to account for the fact that the Kirch-hoff equations become invalid over the small length scalesand time scales near the fracture tip. As shown by Au-doly and Neukirch[9], when an initially uniformly curvedER is released from one end, its local curvature increasesat the free end. When a rod fractures at a point of max-imum curvature, the Kirchhoff model possesses solutionsin which the curvature near the fracture tip increaseseven further; in the case of λ = 0, this would triggeradditional fractures arbitrarily close to the first fracture,which is not observed experimentally. Following stan-dard fragmentation theory[33, 34], we therefore assume afinite minimum fragment length λ > 0, which allows oneto model accurately the fragmentation of the whole rodwithin the Kirchhoff theory while avoiding the many dif-ficulties associated with the small scale behavior arounda fracture. Below, we present measurements and scal-ing arguments that show how λ depends on the end-to-end bending speed. To describe the near-adiabatic twistexperiments, we adopt the empirical value λ0 ≈ 3.0 cmmeasured at speed ∼ 3 mm/s and zero twist.

Twist controlled fracture. The first protocol ex-plores the role of twist in bending-induced ER frac-

ture. Twisting modes are known to cause many counter-intuitive phenomena in ER morphology [35, 36], includ-ing Michell’s instability [37] and supercoiling [38]. Themotivation for combining twisting and bending to achievecontrolled binary fracture is based on the idea that tor-sional modes can contribute to the first stress-inducedfracture but may dissipate sufficiently fast to preventsubsequent fractures. To test this hypothesis, we built acustom device consisting of a linear stage with two freelypivoting manual rotary stages placed on both sides (Ap-pendix & SI Fig.1a,b). Aluminum gripping elements wereattached to each rotary stage to constrain samples closeto the torsional and bending axes of rotation (Appendix).As in Feynman’s original experiment [26], we used com-mercially available spaghetti as test rods. To ensure re-producibility, individual rods were cut to the same fixedlength L = 24 cm, and experiments were performed ina narrow temperature and humidity range (Appendix).The rods’ ends were coated with epoxy to increase thefrictional contact with the gripping elements, enablingus to twist samples to the point of purely torsional fail-ure, which occurred at ∼360 degrees for our ERs. Ineach individual twist experiment, a rod was loaded intothe device, twisted to a predetermined angle, and thenbent near-adiabatically (end-to-end speed < 3 mm/s) un-til fracture occurred. Select trials were recorded with ahigh-speed camera at 1972 fps (Appendix).

As the first main result, our experiments demonstratethat supercritical twist angles give rise to binary frac-ture (Fig. 1). By contrast, for small twist angles, rodsare found to fragment typically into three or more pieces(Fig. 1a), in agreement with Feynman’s conjecture andsupporting recent experimental and theoretical results [9]for the zero-twist case. For large twist angles, however,the maximum curvature before the first fracture is sub-stantially lowered and binary fracture becomes favored(Fig. 1c). Although sample inhomogeneities lead to adistribution of fragment numbers at the same twist an-gles, the average number of fragments exhibits a robusttrend towards binary fracture for twist angles larger than∼ 250 degrees (Fig. 1e,f). In particular, the experimen-tal data follows a von Mises-type ellipsoidal curve whenplotted in the plane spanned by the limit curvature andtwist angle (Fig. 1e,f). We next rationalize these obser-vations by performing mode analysis using the nonlinearelasticity model.

We consider the dynamics after the first fracture, start-ing from the fact that twist enables the rod to store itsenergy in more than one mode. We assume the first frac-ture occurs at t = 0 at the midpoint of the rod, whenthe curvature exceeds the critical value κc determined byEq. (3). Our experiments and simulations show that atlarge twists, the rod breaks with low curvature (Fig. 1c-f). Focusing on this limit, we may assume that the rodis approximately planar, and that the bending is small.Under these assumptions, the twist density and bendingmodes uncouple (SI), and the dynamical equation for θreduces to a damped wave equation (µ/ρ)θss = θtt+2bθt

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ë ~ v-1/4

FIG. 2. Dynamically quenched fracture in brittle elastic rods. (a) Experiment at low quench speed v showing binaryfracture (Movie 3). (b) Experiment at high quench speed v showing fracture into multiple fragments (Movie 4), even though thelimit curvature before the first fracture is similar to that in (a). (c) Distributions of the limit curvature (mean values highlightedin pink) are not significantly affected by the quench speed v, but the mean number fragments increases with v. (d) The meanlength of the smallest fragments follows the theoretically predicted power law scaling. (e) The number of fragments approachesan asymptotic power law as expected from (d). At the lowest quench speed (v = 1 mm/s) the rod breaks into fewer than threepieces on average. Scale bars in (a,b) 3 cm. Error bars in (d,e) show standard error.

for s ∈ [0, L/2]. Similarly, the small bending assumptionallows us to describe the bending dynamics via the Euler-Bernoulli beam equation, EIyssss+ρAytt = 0, where thecenterline is now given by y(s, t). Scaling arguments sim-plify the analysis of these equations. The speed of thetwist waves is determined by shear modulus and density,cθ =

√µ/ρ. Enforcing the free-end boundary condition,

θ′(L/2) = 0, for the undamped twist equation yields asolution with a region of zero twist stress (θ′ = 0) grow-ing at speed cθ from the s = L/2 endpoint (SI). Withnon-zero damping, this picture is valid for propagationover small distances. In particular, the time taken for thezero twist stress front to travel distance λ0, the minimumfragment length, is T 0

θ = λ0/cθ. Over longer lengthscales` > λ0, the damping term becomes important. The zerotwist front travels distance ` in time T `θ = b`2/c2θ. Sincetwist modes are approximately critically damped (SI),with b ≈ πcθ/L, we find T `θ = π`2/cθL, implying thattwist modes dissipate after a time T diss

θ = L/cθ. We anal-yse the speed of bending modes in a similar way. Overshort timescales, we consider a wavepacket of bendingwaves. The speed of a bending wavepacket peaked atwavenumber k is given by cb = 2k

√EI/ρA. The small-

est relevant length scale is the minimum fragment lengthλ0, yielding the maximum allowed k0 = 2π/λ0. Thus thetime taken for the bending wavepacket to travel distance

λ0 is T 0b ≈ λ0

[(4π/λ0)

√EI/ρA

]−1= (λ20/2πr)

√ρ/E.

Similarly, the time taken for the bending wave withwavenumber k0 to travel a longer distance ` is given byT `k0 = (λ0`/2πr)

√ρ/E. However, if the time taken for

the location of maximum bending stress to travel dis-

tance ` is T `b , then owing to dispersive effects, T `b > T `k0 .

Using the estimate for T 0b and the lower bound for T `b , we

can compare the time scales on which twist and bendingoperate:

T 0b =

λ02πr

[1

2(1 + ν)

]1/2T 0θ (6a)

and

T `b >Lλ0

2π2`r

[1

2(1 + ν)

]1/2T `θ (6b)

Using the measured value λ0 ≈ 3.0 cm, we find T 0b > 4T 0

θand T `b > 2T `θ for all ` < L/2, indicating that twist dissi-pates before bending waves can trigger another fracture.In addition, we find that T diss

θ ≈ 2T 0b , which further sug-

gests that twist plays no role in future fracture events.The above difference in propagation times is a robustresult. For example, Timoshenko theory predicts evenslower bending waves than Euler-Bernoulli theory [30],although both beam models agree very closely in ourparameter regime. To complete the argument, we ob-serve that all the fractures occur before reflection of thebending waves at s = 0 becomes important. Anotherfracture will then be triggered if and only if σ(s, t) > σcfor any s satisfying the minimum fragment length cri-terion and t ∈ [0, t0], where t0 is the time for the highenergy bending waves to reach s = 0. Since twist dissi-pates before the bending waves become relevant, we havemaxt∈[0,t0] σ

2 = E2I maxt∈[0,t0] κ2. Let C be such that

maxt∈[0,t0] κ = Cκc. We note that even though twistdissipates quickly, the initial twist still determines the

5

ë =

2m

m3

30

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es

(a)

(b)

Eff

. st

ress

(N

m-2)

x107

0.0

1.0

1.5

0.5

0.11 ms

50

0 m

m/s

0.22 ms

(c)

0.01 ms 0.03 ms 0.06 ms

0.03 ms 0.10 ms 0.20 ms

2.0

FIG. 3. Simulations explain the role of twist in the fracture cascade. (a) Bending waves originating at the point of firstfracture create additional fractures separated by at least the minimum fragment length λ. In simulations, fragments are frozenafter fracture and do not evolve further. (b) Twist waves propagate much faster than bending waves while simultaneouslylowering the critical curvature for fracture. At supercritical twist, the resulting bending waves are not strong enough to causesubsequent fractures at a spacing allowed by typical values of λ (here λ = 3 cm). (c) Fracture cascade as observed in quenchexperiments at zero twist. The timings of secondary fractures (circled in blue) are in agreement with the simulations in (a).Diameter of rod and gaps between fragments enhanced for visualization in (a,b). Scale bars in (a,b) 3 cm. Scale bar in (c) 2 cm.

shape of the rod at t = 0, so C is a function of κcL, Twand possibly other parameters. We calculate C numeri-cally from the Kirchhoff equations for our experimentalparameters, and find to one decimal place C = 1.5 for allrelevant values of κcL and Tw (SI). The criterion thatthe rod only breaks into two pieces then takes the formE2IC2κ2c < σ2

c . Using (3) to eliminate σc, the criterionfor binary fracture becomes:

κc <Tw√

2L(1 + ν)√C2 − 1

. (7)

The right hand side of this inequality describes a weaklyν-dependent straight line in the curvature-twist plane(Fig. 1e). Ideal ERs that undergo their first fractureat values κc and Tw satisfying (7) lie below this line(purple region in Fig. 1e) and are expected to breakinto exactly two pieces. This prediction agrees well withthe mean number of fragments measured in our experi-ments (Fig. 1f). We note that while binary fracture doesoccur outside the critical region, it is a very low probabil-ity event. This is consistent with results of Audoly andNeukirch [9] who showed that non-binary fracture occursat zero twist. By contrast, we have demonstrated thatbinary fracture is almost certain in the critical region.

Quench controlled fracture. Twist fracture exper-iments are carried out for a fixed speed v = 3 mm/s inthe near-adiabatic regime. To systematically explore howquenching affects ER fracture, we built a second fracturedevice coupling a DC stepper motor to a linear stage(Appendix & SI Fig. 1c). By adjusting the motor veloc-ity, we can vary the quench speed v, defined as absoluterelative velocity of the ends, by more than two orders ofmagnitude (Fig. 2). Our nonadiabatic quench protocolallows the rod to bend before fracturing, in contrast toultra-fast diabatic protocols [10] that cause fracture byexciting buckling modes in the unbent state. Previousstudies have shown that the fractal nature of fragmenta-tion [39] and the effects of disorder [40] can give rise to

universal power laws. Here, we will see that nonadiabaticquenching leads to a new class of asymptotic power lawrelations that involve the quench parameter v and can berationalized through scaling arguments.

To investigate how quenched bending dynamics af-fects fracture, we performed 240 fracture experiments dis-tributed over 12 different quench speeds v ranging from1 mm/s to 500 mm/s. As in the twist experiments, rodswere of length L = 24 cm and temperature and humid-ity were controlled to minimize variance due to environ-mental effects (Appendix). Select trials were recorded at9000 fps (Fig. 2a,b). Generally, our experiments showthat an increase in the quench speed v has no signifi-cant effects on the curvature prior to fracture (Fig. 2c),in stark contrast to the effects of twist discussed above.Changing v does however affect strongly both the min-imal size of the fragments (Fig. 2d) and the number offragments (Fig. 2e,f).

To understand why quench speed (at zero twist) doesnot affect the limit curvature, note that the critical cur-vature of the samples at first fracture is of the orderof 10 m−1 across all experiments (Fig. 2c). This meansthat the potential energy density at the first fracture isEP ≈ EIκ2 ≈ 10−1 J/m. For comparison, for a hypo-thetical quench speed of v = 1 m/s, considerably higherthan realized in our experiments, the kinetic energy den-sity is EK ≈ ρAv2/2 ≈ 10−3 J/m� EP . Hence, kineticenergy is negligible in our experiments, explaining whythere is no change in the curvature at which the firstfracture occurs (Fig. 2e).

Yet, higher quench speeds v lead to higher fragmentnumbers (Fig. 2a,b). This effect can be traced back tothe fact that the minimum fragment length λ decays withv (Fig. 2d). We rationalize this using dimensional analy-sis. The dynamics of the rod are overdamped, so as therod is quenched the force on any element scales as F ∼ v.In one dimension, force has units of energy density sowe will balance F against the other fundamental energydensity of the system, namely the potential energy den-

6

sity. From the Kirkhhoff equations, the energy densityof the k’th bending mode scales as Ek ∼ k4. Dimen-sional arguments therefore imply the scaling k ∼ v1/4.The minimal fragment length λ imposes an energy cut-off, k ∼ λ−1. Thus we obtain the scaling λ ∼ v−1/4 asobserved in our experiments (Fig. 2d). We note that forvery high impact energies over short timescales, the ki-netic energy density will become relevant again. SinceEK ∼ v2, this implies the scaling k ∼ v1/2. The typicalfragment size λ′ will then scale as λ′ ∼ v−1/2. This is in-deed observed for diabatically fast energy injection [10].In one-dimensional fragmentation the number of piecesN and the minimal fragment length λ are predicted [33]to scale as λ ∼ N−1, however deviations from pure powerlaw scaling for large N are expected due to the finite totallength of our samples. Combining these scaling results,we obtain the prediction N ∼ v1/4 at small v, in agree-ment with our data (Fig. 2f). In particular, this alsoexplains why rods can undergo binary fracture when thequench velocity is very small (Fig. 2a).

III. SUMMARY

We have demonstrated two distinct protocols forachieving controlled binary fracture in brittle elastic rods.By generalizing classical fracture arguments [9] to ac-count for twisting and quenching, we were able to ra-tionalize the experimentally observed fragmentation pat-terns (Fig. 3). Due to their generic nature, the abovetheoretical considerations can be expected to apply totorsional and kinetic fracture processes in a wide range ofone-dimensional structures, from construction beams [1]to the intracellular cytoskeleton [19, 20]. Indeed theappearance of a power law response to quenching is acommon feature of many results in condensed matterphysics [41, 42]. While our results demonstrate two con-crete loopholes for violating Feynman’s conjecture, theyalso suggest several directions for future research. Newtheory beyond the Kirchhoff model is needed to clarifythe microscopic origin of the minimum fragment lengthand to explain the nonplanar geometry of the fractureinterfaces. From a practical perspective, it will be in-teresting to explore whether, and how, twist can be uti-lized to control the fracture behavior of two- and three-dimensional materials.

Acknowledgments We thank Dr. Jim Bales (MIT)for providing the high speed cameras. This work was sup-ported by an Alfred P. Sloan Research Fellowship (J.D.)and a Complex Systems Scholar Award from the JamesS. McDonnell Foundation (J.D.).

Appendix A: Experiments

All experiments used Barilla no. 3 raw spaghetti oflength L = 24 cm. High air humidity and large tem-perature fluctuations can affect bending and fracture be-

havior of the samples. Throughout our experiments, airhumidity was kept low in the range 21%-34%. Temper-ature during the twist experiments was kept constant at22.5 ± 1.5 C and during the kinetic quench experimentsat 25.5 ± 0.5 C. The rod diameter was measured usingcalipers for 5 samples to give 2r = 1.4±0.1 mm. The den-sity was obtained by weighing 10 samples cut to 24 cm.Treating the samples as cylinders of radius r we obtainedρ = 1.5± 0.1 g/cm

3. The Young’s modulus, E, was mea-

sured by applying a slowly increasing longitudinal com-pression force to samples positioned upright upon a scale.The sample length ` and the mass m shown on the scalesat the point of buckling were recorded. We repeated thisfor 20 samples of varying lengths, and in each case calcu-lated E from the Euler buckling criteria, mg = π2EI/`2,to find E = 3.8 ± 0.3 GPa. The shear modulus wasmeasured by attaching a mass of known moment of iner-tia, I0, to samples of varying length to create a torsionpendulum. The angular frequency, ω2 = µJ/I0L, wasobtained for 5 samples, by filming the pendulum withan Edgertronic SC2 at 1972 fps. The Poisson’s ratio,ν = (E/2µ) − 1 was calculated from the mean valuesof E and µ. Using standard error of these mean valuesto quantify uncertainty gives ν = 0.3 ± 0.1. The twistdamping parameter was obtained from the decay rate ofthe torsion pendulum (SI).

Twist experiments were performed using a custom-built device comprising of a manual linear stage withtwo freely pivoting manual rotary stages placed on bothsides. Aluminum gripping elements were attached to eachrotary stage to constrain samples close to the torsionaland bending axes of rotation (SI). We completed 73 trialsat various twist angles up to 360 degrees, correspondingto the approximate pure torsion yield stress of the sam-ples. To ensure proper reproducible twisting of the sam-ples within the desired range, the ends of each rod werecoated with Devcon R© 5 minute epoxy gel. The epoxyincreased friction between each sample and the grippingelement in our testing device, enabling us to twist sam-ples to the point of torsional failure. Each sample wasthen loaded into the device, twisted to the chosen an-gle, and bent until fracture occurred. The ends weremoved together slowly (< 3 mm/s) to ensure a quasi-static regime. The end-to-end distance at the onset offracture was recorded for each trial. Select trials wererecorded with an Edgertronic SC2 at 1972 fps.

Kinetic quench experiments consisted of 20 trials eachfor 12 speeds ranging from 1 mm/s to 500 mm/s, usinga custom single-axis linear stage controlled by a NEMA17 Bipolar DC Stepper Motor (SI). The device movedthe ends of each sample towards each other at a fixedspeed v while allowing them to pivot freely as bendingoccurred. The end-to-end distance at the onset of frac-ture was obtained by recording each trial with a PhotronFASTCAM Mini AX200 at 9000 fps and examining theplayback. Select trials were stored permanently.

7

Appendix B: Experiments

Numerical results in Fig. 1b,d were obtained by sim-ulating the Kirchhoff equations (1) using a discrete dif-ferential geometry algorithm [31, 32]. Each rod was dis-cretized into 50 elements and one time-step of simulationtime corresponded to 1µs of real time. Time-stepping wasperformed with a Verlet scheme. Fracture was simulatedby disconnecting the rod in 1 time-step wherever theabove fracture criterion was satisfied. The radius of the

rod has been enhanced in simulation images (Fig. 1b,dand Movies 1,2) for visualization purposes. The curva-tures in Fig. 1e,f and Fig. 2c were obtained numericallyfrom the observed end-to-end distance by initializing arod with the appropriate boundary conditions and twist,and allowing it to relax to its lowest energy state via gra-dient descent. In the case of zero twist, there is a closedform relationship between end-to-end distance and max-imum curvature, which we used to validate the code (SI).The Matlab code is available on request.

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