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# Controlling fracture cascades through twisting and quenching

Edited by Alain Karma, Northeastern University, Boston, MA, and accepted by Editorial Board Member Herbert Levine July 22, 2018 (received for review February 15, 2018)

## Significance

Fracture processes are ubiquitous in nature, from earthquakes to broken trees and bones. Understanding and controlling fracture dynamics remain one of the foremost theoretical and practical challenges in material science and physics. A well-known problem with direct implications for the fracture behavior of elongated brittle objects, such as vaulting poles or long fibers, goes back to the famous physicist Richard Feynman who observed that dry spaghetti almost always breaks into three or more pieces when exposed to large bending stresses. While bending-induced fracture is fairly well understood nowadays, much less is known about the effects of twist. Our experimental and theoretical results demonstrate that twisting enables remarkable fracture control by using the different propagation speeds of twist and bending waves.

## Abstract

Fracture fundamentally limits the structural stability of macroscopic and microscopic matter, from beams and bones to microtubules and nanotubes. Despite substantial recent experimental and theoretical progress, fracture control continues to present profound practical and theoretical challenges. While bending-induced fracture of elongated rod-like objects has been intensely studied, the effects of twist and quench dynamics have yet to be explored systematically. Here, we show how twist and quench protocols may be used to control such fracture processes, by revisiting Feynman’s observation that dry spaghetti typically breaks into three or more pieces when exposed to large pure bending stresses. Combining theory and experiment, we demonstrate controlled binary fracture of brittle elastic rods for two distinct protocols based on twisting and nonadiabatic quenching. Our experimental data for twist-controlled fracture agree quantitatively with a theoretically predicted phase diagram, and we establish asymptotic scaling relations for quenched fracture. Due to their general character, these results are expected to apply to torsional and kinetic fracture processes in a wide range of systems.

Elastic rods (ERs) are ubiquitous in natural and man-made matter, performing important physical and biological functions across a wide range of scales, from columns (1), trees (2⇓–4), and bones (5) to the legs of water striders (6), semiflexible polymer (7) networks (8, 9), and carbon nanotube composites (10). When placed under extreme stresses, the structural stability of such materials becomes ultimately limited by the fracture behaviors of their individual fibrous or tubular constituents. Owing to their central practical importance in engineering, ER fracture and crack propagation have been intensively studied for more than a century both experimentally (11⇓–13) and theoretically (14⇓–16). Recent advances in video microscopy and microscale force manipulation (17, 18) have extended the scope of fracture studies to the microworld (19, 20), revealing causes and effects of structural failure in the axonal cytoskeleton (21), fibroblasts (22), bacterial flagellar motors (23), active liquid crystals (8), and multiwalled carbon nanotubes (24, 25).

Built on Sir Neville Mott’s foundational studies on the fragmentation of ring-shaped explosives (26), theoretical work on ER fracture has flourished over the past two decades (12, 13, 27⇓⇓–30). However, many basic aspects of the ER fracture phenomenology remain poorly understood. Bending-induced ER fragmentation has been thoroughly investigated in the limits of adiabatically slow (12) and diabatically fast (13) energy injection, but the roles of twist and quench rate on the fracture process have yet to be clarified. These two fundamental issues are directly linked to a famous observation by Richard Feynman (31), who noted that dry spaghetti, when brought to fracture by holding the ends and moving them toward each other, appears almost always to break into at least three pieces. The phenomenon of nonbinary ER fracture is also well known to pole vaulters, with a notable instance occurring during the 2012 Olympic Games (32). Below, we revisit and generalize Feynman’s experiment to investigate systematically how twist and quench dynamics influence the elastic fragmentation cascade (12, 13). Specifically, we demonstrate two complementary quench protocols for controlled binary fracture of brittle ERs. Our experimental observations are in agreement with numerical predictions from a nonlinear elasticity model and can be rationalized through analytical scaling arguments.

## Results and Discussion

### Timescales in Fragmentation.

Revisiting Feynman’s experiment, we monitor the fracture dynamics of dry spaghetti, using high-speed imaging at frame rates ranging from 1,972 frames per second (fps) to 1,000,000 fps (Fig. 1 and *Materials and Methods*). The highest time-resolution data show that already a basic fracture event involves several timescales, from initial crack nucleation and growth to catastrophic failure (Fig. 1*A* and Movie S1). The initial nucleation phase is relatively slow, lasting *B* and *C*), as shown by Audoly and Neukirch (12). Our goal is to control the fragmentation dynamics on this slower elastic timescale, which can be treated accurately within the Kirchhoff theory (Fig. 1 *B* and *C*).

### Damped Kirchhoff Model.

We describe an ER at time t by its arc-length–parametrized centerline *SI Appendix*, *Kirchhoff Model*)**1b** denotes damping of twist modes with damping parameter b. Our measurements of this parameter using a torsion pendulum indicate that twist is approximately critically damped (*SI Appendix*, *Dissipation of Twist*). Since the timescale for the entire fracture cascade is an order of magnitude smaller than the time period of the fundamental bending mode, we do not need to include bending damping terms in our analysis. The average material properties of our experimental samples are *SI Appendix*, *Sample Characterization*). We present additional data for rods of radius *SI Appendix*, Fig. S2. Finally, we note that the Kirchhoff equations do not account for certain shear effects described by Timoshenko beam theory. Indeed, the Timoshenko theory does provide a more accurate description of bending waves with large wavenumber compared with rod radius. However, to describe fracture, we will need only to consider wavenumbers k with

### Minimum Fragment Length.

As shown by Audoly and Neukirch (12), when an initially uniformly curved ER is released from one end, its local curvature increases at the free end. However, when a rod fractures at a point of maximum curvature, the Kirchhoff model possesses solutions in which the curvature near the fracture tip increases even further. Assuming a curvature-based fracture criterion, this would trigger additional fractures arbitrarily close to the first fracture, which is not observed experimentally (Fig. 1 *B* and *D*). In agreement with standard fragmentation theory (36, 37), our data show the existence of a finite minimum fragment length *D*). This observation, along with the separation of bending wave and crack propagation timescales (Fig. 1*A*), confirms the applicability of the Kirchhoff theory (Fig. 1 *B* and *C*).

### Fracture Criteria.

To compare individual experiments with theoretical predictions, we solve the Kirchhoff Eqs. **1a** and **1b** numerically with a discrete differential geometry algorithm (38, 39) (*Materials and Methods*), adopting a stress-based fracture criterion defined as follows: We define the twist of the rod,

We posit that the rod fractures at the point s if *D*). For a uniform twist distribution, the critical stress imposes a critical yield curvature

We can now rewrite the above expression Eq. **2** for

For comparison, by integrating the classical von Mises stress criterion over a cross-section, we obtain a critical local stress ellipse given by

Another common criterion comes from considering the maximum eigenvalue of the stress tensor, or maximum principal stress, on the boundary of the rod; this gives

All three curves are qualitatively consistent with our data, with Eq. **4** yielding the best quantitative agreement (Fig. 2*F*). For samples of different length and radius, and hence a different value **4** is still found to agree well with the data (*SI Appendix*, *Robustness Under Parameter Variations*). In the limit of zero twist, all three fracture criteria predict a critical curvature, which has been successfully used to rationalize aspects of twist-free ER fracture (12). While the origin of this critical curvature requires a deeper theory (30), the above fracture criterion Eq. **4** suffices for our purposes.

### Twist-Controlled Fracture.

The first protocol explores the role of twist in bending-induced ER fracture. Twisting modes are known to cause many counterintuitive phenomena in ER morphology (40⇓⇓–43), including Michell’s instability (44) and supercoiling (45). The motivation for combining twisting and bending to achieve controlled binary fracture is based on the idea that torsional modes can contribute to the first stress-induced fracture but may dissipate sufficiently fast to prevent subsequent fractures. To test this hypothesis, we built a custom device consisting of a linear stage with two freely pivoting manual rotary stages placed on both sides (*SI Appendix*, Fig. S1). Aluminum gripping elements were attached to each rotary stage to constrain samples close to the torsional and bending axes of rotation (*SI Appendix*, *Twist Tests*). As in Feynman’s original experiment (31), we used commercially available spaghetti as test rods. To ensure reproducibility, individual rods were cut to the same fixed length *SI Appendix*, *Preparation of Experimental Samples*). The rods’ ends were coated with epoxy to increase the frictional contact with the gripping elements, enabling us to twist samples to the point of purely torsional failure, which occurred at ∼360° for our ERs. In each individual twist experiment, a rod was loaded into the device, twisted to a predetermined angle, and then bent near adiabatically (end-to-end speed *SI Appendix*, *Twist Tests*).

As the first main result, our experiments demonstrate that supercritical twist angles give rise to binary fracture (Fig. 2). By contrast, for small twist angles, rods are found to fragment typically into three or more pieces (Fig. 2*A*), in agreement with Feynman’s conjecture and supporting recent experimental and theoretical results (12) for the zero-twist case. For large twist angles, however, the maximum curvature before the first fracture is substantially lowered and binary fracture becomes favored (Fig. 2*C*). Although sample inhomogeneities lead to a distribution of fragment numbers at the same twist angles, the average number of fragments exhibits a robust trend toward binary fracture for twist angles larger than *E* and *F*). In particular, the experimental data follow a von Mises-type ellipsoidal curve when plotted in the plane spanned by the limit curvature and twist angle (Fig. 2 *E* and *F*). We next rationalize these observations by performing mode analysis using the nonlinear elasticity model.

We consider the dynamics after the first fracture, starting from the fact that twist enables the rod to store its energy in more than one mode. We assume the first fracture occurs at **4**. Our experiments and simulations show that at large twists, the rod breaks with low curvature (Fig. 2 *C*–*F*). Focusing on this limit, we may assume that the rod is approximately planar and that the bending is small. Under these assumptions, the twist density and bending modes uncouple (*SI Appendix*, *Small Deflections*), and the dynamical equation for θ reduces to a damped-wave equation *SI Appendix*, *Dissipation of Twist*). With nonzero damping, this picture is valid for propagation over small distances. In particular, the time taken for the zero twist stress front to travel distance *SI Appendix*, *Dissipation of Twist*), with *D*). Let *SI Appendix*, *Euler–Bernoulli Equation*). This may be understood by observing that the speed of a bending wavepacket peaked at wavenumber k is given by *SI Appendix*, *Euler–Bernoulli Equation*). The criterion that the rod breaks into only two pieces then takes the form **4** to eliminate

In terms of the dimensionless Li number from Eq. **3**, this condition is approximately *E*). Ideal ERs that undergo their first fracture at values **7** lie below this line (purple region in Fig. 2*E*) and are expected to break into exactly two pieces. This prediction agrees well with the mean number of fragments measured in our experiments (Fig. 2*F*). Additional data for samples of a different length and radius also show good agreement with Eq. **7** (*SI Appendix*, Fig. S2). The raw data show that binary fracture events can occur with low probability outside the critical region (Fig. 2*E*), which could be caused by sample defects and inhomogeneities. The distinct transition from binary to nonbinary fracture in the averaged data (Fig. 2*F*) indicates, however, that defects do not dominate the fracture statistics. Our results for the low-twist regime are consistent with those of Audoly and Neukirch (12) who reported nonbinary fracture at zero twist. By contrast, binary fracture becomes almost certain in the high-twist regime.

### Quench-Controlled Fracture.

Twist fracture experiments are carried out for a fixed speed *SI Appendix*, Fig. S1). By adjusting the motor velocity, we can vary v, defined as absolute relative velocity of the ends, by more than two orders of magnitude (Fig. 3). Our nonadiabatic quench protocol allows the rod to bend before fracturing, in contrast to ultrafast diabatic protocols (13) that cause fracture by exciting buckling modes in the unbent state. Previous studies have shown that the fractal nature of fragmentation (46) and disorder (47) can give rise to universal power laws. Here, we will see that nonadiabatic quenching leads to asymptotic power-law relations that involve the quench parameter v.

To investigate how quenched bending dynamics affect fracture, we performed 350 fracture experiments distributed over 12 different quench speeds v ranging from *SI Appendix*, Fig. S2). Select trials were recorded at 75,000 fps (Fig. 3 *A* and *B*). Generally, our experiments show that an increase in the quench speed v has only a weak effect on the curvature before fracture (Fig. 3*C*), in stark contrast to the effects of twist discussed above. Changing v does, however, affect strongly both the minimal size of the fragments (Fig. 3*D*) and the number of fragments (Fig. 3 *E* and *F*). To understand why quench speed (at zero twist) only weakly affects the limit curvature, note that the critical curvature of the samples at first fracture is of the order of *C*). This means that the potential energy density at the first fracture is

However, higher quench speeds v lead to higher fragment numbers (Fig. 3 *A* and *B*), reflecting the fact that the minimum fragment length λ decays with v (Fig. 3*D*). We can rationalize this using dimensional analysis. The dynamics of the rod are overdamped (Movies S4 and S5), so during a quench the force on any element scales as *D*). The same scaling is implied by the following more detailed argument. Assume the rod is approximately flat at time *SI Appendix*, *Euler–Bernoulli Equation*) as *SI Appendix*, Fig. S3), which suggests a role for acoustic waves triggered by the equilibration wavepacket. These acoustic waves will weaken the rod at their antinodal high-strain sites. At early times, the dominant acoustic wavenumber can be expected to match the dominant triggering bending wavenumber, yielding for the minimum fragment length *E*). In particular, this also explains why rods can undergo binary fracture when the quench velocity is very small (Fig. 3*A*). At sufficiently high quench speeds, the number of fracture sites, and hence the number of fragments, saturates (Fig. 3*E*). This can be understood in terms of stochastic fracture theory (*SI Appendix*, *Stochastic Fracture Theory*). The picture of quenched fragmentation which emerges is as follows: The quench speed releases bending and acoustic waves of particular wavelengths which weaken the rod in certain patterns. At low speeds the number of fracture sites simply increases as a power law, whereas at high speeds this growth is limited by the rate of subcritical crack growth.

## Conclusions

We have demonstrated two distinct protocols for achieving controlled binary fracture in brittle elastic rods. By generalizing classical fracture arguments (12) to account for twisting and quenching, we were able to rationalize the experimentally observed fragmentation patterns. Due to their generic nature, the above theoretical considerations can be expected to apply to torsional and kinetic fracture processes in a wide range of engineered (1) and natural (21, 22) 1D structures. Our experimental results suggest several directions for future research. New theory beyond the Kirchhoff model is needed to clarify conclusively the microscopic origin of the minimum fragment length. Moreover, a detailed experimental analysis of the crack propagation dynamics will require going beyond megahertz time resolution (Fig. 1). An interesting practical question is whether, and how, twist can be used to control the fracture behavior of 2D and 3D materials.

## Materials and Methods

A comprehensive description of all experimental procedures and details of derivations is provided in *SI Appendix*.

### Experiments.

Rods were Barilla no. 1, 3, or 5 raw spaghetti of length

### Simulations.

The Kirchhoff equations were simulated with a discrete differential geometry algorithm (38, 39), using a Verlet time-stepping scheme. Each rod was discretized into 50 elements. One simulation time step corresponded to

## Acknowledgments

We thank Dr. Jim Bales [Massachusetts Institute of Technology (MIT)] and the MIT Edgerton Center for providing the high-speed cameras. This work was supported by an Alfred P. Sloan Research Fellowship (to J.D.) and a Complex Systems Scholar Award from the James S. McDonnell Foundation (to J.D.).

## Footnotes

↵

^{1}R.H.H. and V.P.P. contributed equally to this work.- ↵
^{2}To whom correspondence should be addressed. Email: dunkel{at}mit.edu.

Author contributions: R.H.H., V.P.P., and J.D. designed research; R.H.H., V.P.P., N.S., E.V., and J.D. performed research; R.H.H. and V.P.P. analyzed data; and R.H.H., V.P.P., N.S., E.V., and J.D. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission. A.K. is a guest editor invited by the Editorial Board.

Data deposition: The Matlab code can be downloaded from https://github.com/vppatil28/elastic_rod_fracture_simulation.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1802831115/-/DCSupplemental.

Published under the PNAS license.

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