Yi Zhang, Xiang Li, Yuanfan Dai, Bo-Hua Sun

Keywords: Curvature Brittle cracking Elastic rod Diameter-to-length ratio Spaghetti

ABSTRACT Why are pieces of spaghetti generally broken into three to ten segments instead of two as one thinks? How can one obtain the desired number of fracture segments? To answer those questions, the fracture dynamics of a strand of spaghetti is modelled by elastic rod and numerically investigated by using finite- element software ABAQUS. By data fitting, two relations are obtained: the number of fracture segments in terms of rod diameter-length ratio and fracture limit curvature with the rod diameter. Results reveal that when the length is constant, the larger the diameter and/or the smaller the diameter-length ratio D/L , the smaller the limit curvature; and the larger the diameter-length ratio D/L , the fewer the number of fractured segments. The relevant formulations can be used to obtain the desired number of broken segments of spaghetti by changing the diameter-to-length ratio.

Numerous objects can be described as elastic rods in ordinary life, such as bones [1] and trees [2,3] or other models with im- portant physical and physiological functions on various scales. Ar- tificial objects like carbon nanotube composite materials [4] , semi- flexible polymer ports [5] , and multi-walled carbon nanotubes [6,7] can all be described by the mechanical model of elastic rods as well. Elastic rods are also very common in engineering applica- tions, such as circular cross-section beams and columns in civil en- gineering [8] and automobile drive shafts [9] . During the 2012 Lon- don Olympics, Cuban Pole vaulter Lazaro Borges nearly died when the pole broke into three sections at a height of 5 m. The impor- tance of the elastic rod is evident and its ultimate curvature when it is about to break has also become a problem to urgently solve.

How to fold a piece of spaghetti in two was a scientific conun- drum. Mr. Feynman, a Nobel Prize winner, was in the kitchen one evening, picked up a piece of spaghetti, held the ends and bent them slowly. He found that no matter how he did, the spaghetti would break into three, four or more pieces, as shown in Fig. 1 . Feynman tried all night and couldn’t break the spaghetti in two pieces, and he was even more frustrated when he couldn’t ex- plain why the spaghetti was always broke into multiple pieces. Until 2005, Audoly and Neukirch [10] found that when forces was evenly applied on two ends of a piece of spaghetti, the spaghetti would bend to a critical point and break. The free end of the break would generate strong bending waves and transmit to the rest of the spaghetti, resulting in fracture cascades. Although a bending wave explains why the spaghetti breaks into multiple pieces, but still no one knows how to break the spaghetti to two pieces. 2018, Heisser [24] had broken the spaghetti into two pieces by applying a twist to each end and then controlling the speed at which it was pressed.

Fig. 1. A piece of dry spaghetti is slowly bent to its limit curvature and broken into four segments.

Fig. 2. FEM model.

The mechanism of elastic rod fracture and crack propagation has been studied [11,12] for a longtime [14–16] . It hasbeen mainly studied from the perspective of statics [17,18] . Mott’s work [19] in 1945 established the foundation for the study of the fracture phe- nomenon of elastic rods. Mahmood et al. [20] studied the stress concentration at crack tips. Wittel et al. [21] studied the fragments produced by brittle materials. Mitchell [22] studied the mecha- nism of curvature affecting the crack direction of materials. Viller- maux [23] studied the distribution of energy in brittle materials and the time of the fracture process, and discussed the number of segments and fragment lengths generated. Audoly and Neukirch [10] studied the fracture of a slender rod in the adiabatic state and the transmission of bending waves, while Gladden et al. [12] stud- ied the fracture and energy problems of a slender rod under the condition of rapid heat transfer. Fracture phenomena appear in large numbers in life and engineering, and the process of break- ing into multiple segments involves many disciplines and technical fields.

The ABAQUS finite-element model (FEM) presented in this paper is based on the experiments of Heisser et al. [24] un- der the conditions of humidity 21% ?34% and temperature 21% ?26% . The length and radius of the spaghetti isL= 240 mm andd= 0.75 mm, respectively, and its density isρ= 1.5 ±0.1 g/cm3. The elastic modulus isE= 3.8 ±0.3 GPa and Poisson’s ratio isν= 0.3 ±0.1 . The model’s data are used in this paper:ρ= 1.5 g/cm3,E= 3.8 GPa , andν= 0.3 .

To predict the breaking, the Rankine criterion is used to detect crack initiation [13] . This criterion states that a crack forms when the maximum principal tensile stress exceeds the tensile strength of the brittle material. The postfailure behavior for direct straining across cracks is specified by means of a postfailure stress-strain re- lation. The shear behavior is called shear retention model, the de- pendence is defined by expressing the postcracking shear modu- lusGc, as a fraction of the uncracked shear modulus:Gc=ρ(eck)G, whereGis the shear modulus of the uncracked material andρ(eck)is the shear retention factor,eckis the crack opening strain. Alter- natively, shear retention can be defined in the power law form:, wherepandare materialparameters andp= 1 ,= 0.0294 . This form satisfies the requirements thatρ→ 1 aseck→ 0 andρ→ 0 as.

Villermaux [23] pointed out that the mechanical elastic energy stored in the strained material plays a positive role in promoting fracture and the initial defects in the material proved to be mi- nor. Therefore, this model does not consider the initial defects. Two ends of the spaghetti were set to be hinged, and the 10 mm sur- face from each end to the reference point were coupled to simu- late the case of insertion into the support in the experiments. This model uses units of C3D8R. The schematic of the FEM of the above specimen is shown in Fig. 2 .

Fig. 3. Loading process.

Fig. 4. Dynamic fracture process of spaghetti.

Fig. 5. Relationship between limit curvature and diameter.

Heisser used the quench speed of the two ends as the vari- able in the experiments. However, directly squeezing both ends is likely to cause the stability and buckling problem [25] , discussion of which is not within the scope of this paper; Heisser’s experi- ments also exclude the relevant results.

In the present work,YandZdisplacements were first applied to both ends of the rod to reach a certain curvature, and then two ends of the rod squeezed at a certain speed to the limit curvature, as shown in Fig. 3 . Dynamic fracture process of spaghetti is shown in Fig. 4 , see the attachment for a description of the complete dy- namic fracture process.

Discussion of the limit curvature was facilitated by changing the size of the spaghetti with lengthL= 240 mm and by increasing the diameter from 1 to 3 mm in increments of 0.1 mm to achieve different diameters in order to study the relationship between the diameter-to-length ratio and limit curvature. The relationship ob- tained by the FEM is shown in Fig. 5 . It can be seen from the figure that the limit curvature exhibits an obvious trend with the change of the diameter-length ratio. These data are discussed and analyzed aided by dimensional analysis later.

Fig. 6. Using FEM to study dynamic process of spaghetti fracture.

Fig. 7. Using FEM to study dynamic process of spaghetti fracture.

Fig. 8. Relationship between fractured segment number and diameter-to-length ra- tio.

For the dynamic process after spaghetti fracture, in this paper the fracture of a rod when the length isL= 240 mm, diameter isD= 1.5 mm, and quench speed isv= 300 mm/s is taken as the object of study. From Figs. 6 and 7 , it can be seen that whent= 0 , point a is broken,κa= 7.968 m?1, whent= 0.0017 s, the curva- ture of point b isκb= 11.571 m?1. The relationship between the curvature of these two points and the dynamic process after frac- ture will be discussed in detail.

For the simulation of the number of fractured segments of spaghetti, the model is taken at the point when the lengths of the spaghetti areL= 200, 220, and 240 mm and the diameterDranges from 1 to 3 mm. The fractured segment number is used to obtain the relationship between the fractured segments. When the ratio of diameter to length is the same, the number of fractured seg- ments would be averaged, as shown in Fig. 8 . It can be seen from the figure that the number of fractured segments gradually con- verges to a constant as the diameter-to-length ratio increases. The data are discussed and fitted later.

Fig. 9. Red dotted line represents average limit curvature of Heisser’s experimental results and black dotted line represents average limit curvature of FEM used in the present paper.

Fig. 10. Obtaining fitting curve of limit curvature through dimensional analysis.

By changing the relative quench speed at loading both ends as follows, 1, 2, 3, 5, 10, 20, 30, 50, 100, 200, 300, and 500 mm/s, different limit curvatures and fracture segment numbers are ob- tained. After collating the data and comparing them with those from Heisser’s experiment, the result is shown in Fig. 9 .

The difference between average limit curvature obtained by Heisser and FE simulation being 6.9% . The difference may be caused by the imperfect support conditions in the experiments and the non-uniformity of the spaghetti. The trend of the number of fractured segments is the same as that in the experiments, but the number of fractured segments obtained by the experiment tends to three while that obtained by the FEM is between three and four. This may be caused by the boundary conditions imposed by the FEM being ideal and completely symmetrical, so the number of fractured segments tends to be even, so that there are more results that tend to produce four segments [24] . Different quench speeds have little effect on the limit curvature, which is consistent with Heisser’s experimental results and conclusions, which verifies the reliability of the FEM.

To discuss the limit curvature and fit the data in Fig. 5 , through the theory of dimensional analysis [26,27] , the relationship be- tween the radius and limit curvature is thus obtained as:

As shown in Fig. 10 :

The relationship between the radius and limit curvature when the length of the spaghetti isL= 240 mm is thus obtained.

Audoly believes that the multi-segment fracture of spaghetti is due to the cascading effect caused by the transmission of bend- ing waves after the first fracture. Taking Fig. 11 as an example, the fractured spaghetti is divided into three parts designated a, b, and c. Sections a and c can be regarded as the cantilever cases, which is shown in Fig. 12 , and section b denotes the spaghetti segments with two free ends. Only the cantilever situation is discussed here, as shown in Fig. 11 a and 11c. Section a is taken as an example.

Fig. 11. Sections a and c are cantilever beams; section a is taken as an example.

Fig. 12. Spaghetti model after first spaghetti break.

The Euler-Bernoulli beam equation [28] can be applied to the spaghetti after breaking:

whereyis the deflection,sthe arc length,tthe time,Ithe mo- ment of inertia of the section, andAthe cross-sectional area of the rod. The above governing equation must be modified to get fractal scaling law, detail please referred to Yang and Liu [29] and Yang [30] .

Taking the double derivative of the arc length s on both sides of Eq. (2), one obtains= 0. Using the def- inition of curvature,, Eq. (2) can be rewritten as

The self-similarity solution of Eq. (3) is

whereC1 ,C2 ,C3 , andC4 are undetermined coefficients. The places= 0 is the free end,s=Lis the fixed end, and the boundary conditions are brought in asκ(0,t)= 0 ,κ(s,0)=κ0,(0,t)= 0 , and(0,t)= 0 (κ0is the initial curve), after which one obtainsC1=C3=C4= 0 ,C2= 2κ0.

SubstitutingC1 =C3 =C4 = 0 ,C2 = 2κ0into Eq. (4) , and then re- gardingξ=as a self-similarity variable, Eq. (4) is rewrit-

ten as

Equation (5) is the expression of the curvature of spaghetti after breaking. Figure 13 shows that, whenξ=FresnelSreached the maximum value; at this time, the curvatureκis 1.428 times the initial curvatureκ0. The numerical solution of Eq. (5) un- der the boundary condition is showed in Fig. 14 , thusθandycan be calculator by integrating Eq. (5) as shown in Figs. 15 and 16 .

Fig. 13. Obtaining function image = 2 FresnelSwhich is easy to know whencould reach the maximus of 1.428.

Fig. 14. Numerical solution of Eq. (5) under the boundary condition that one end is fixed and the other is free. It can be seen that the curvature of the free end κ tends to zero in a short time.

Fig. 15. After the first break (released as a free end) at t = 0 . 0 0 01 , 0 . 0 0 09 , and 0.0017 s, the relationship between θand s is obtained.

Fig. 16. After the first break (released as a free end) at t = 0 . 0 0 01 , 0 . 0 0 09 , and 0 . 0017 s, the relationship between y and s is obtained.

Fig. 17. Fitting-curve diagram of fracture segment number and diameter-to-length ratio. D/L varies continuously, but the corresponding number of fracture segments Nis a rational number. Mathematically speaking, it will inevitably appear in a cer- tain interval of D/L , which can only correspond to a number of segments N, so the number of segment platforms in the graph appears.

Audoly and Neukirch [10] believes that the coefficient 1.428 is uni- versal, which is double the maximum value of the Fresnel sine in- tegral. The coefficient of= 1.428 obtained by the mathematical model through simulation was also verified in this article, and the corresponding finite element is shown in Figs. 6 and 7 . For a piece of spaghetti that has been bent to the limit curvature, the end that was broken could be regarded as the free end that is released, and the other end is 10 mm long for plug-in articulation in this pa- per. The inserted part can be regarded as a fixed end relative to the other parts of the spaghetti, so it meets the boundary and ini- tial conditions; this process does not involving any material prop- erties. The curvatureκaof the breaking point a att= 0 relaxed in a short period of time. This sudden relaxation produces an explo- sive stress wave, strong enough to break the remaining part at this time,κa= 7.968 m?1. Whent= 0.0017 s ,κb= 11.571 m?1at point b, and point b breaks whent= 0.0018 s. In this process,κbis 1.452 timesκa. The difference between 1.452 and 1.428 is 1.7% . This also supports Audoly’s= 1.428 coefficient.

Audoly’s work [10] may be able to give the specific fracture time and location without using any specific fracture criteria, but to obtain a simple and universal law that could be convenient in engineering applications, the following lengths of the spaghetti model are taken,L= 200, 220, and 240 mm, when the diameterDis in the range of 1 to 3 mm. The relationship between the frac- tured segment number and the diameter-to-length ratio are then discussed. When the diameter-to-length ratios are the same, the number of fractured segments is averaged, as shown in Fig. 17 . It can be seen that the number of fracture segments gradually con- verges to a value with increasing diameter-to-length ratio. To study the relationship between them, polynomial fitting is performed on the FE data, and the formulation is+δ. Fitting to obtainα= 1760.41 ,β= ?214.25 ,γ= ?49.05 , andδ= 9.88 gives a scaling law:

Drawing the fitting curve together with the FE data, the plot shown in Fig. 17 is obtained. It can be seen that the number of fractured segments converges to 3.6 with diameter-to-length ratio. This is an easy rule to apply.

In terms of fracture limit curvature, the relationship between fracture limit curvature and diameter is obtained through dimen- sional analysis and after fitting the FE simulation data, it was found that when the spaghetti lengthLis constant, the limit curvature is inversely proportional to the diameter of the elastic rod, and the relationship between them is expressed by Eq. (1) .

In terms of the dynamic mathematical model after fracture, the Euler-Bernoulli beam equation and FE method are used in this pa- per to confirm the conclusion of Audoly that= 1.428 . This con- clusion can be used to theoretically predict the time and location of the next cantilever break based on the initial conditions after the first break.

In terms of the number of fractures, the occurrence of frac- ture segment numbers when the spaghetti lengths areL= 200, 220, and 240 mm and the diametersDare in the range of 1 to 3 mm was simulated. The FE simulation results show that the fracture segment number gradually decreases from eight with increasing diameter-to-length ratio, and converges to 3.6. The relationship be- tween them is expressed by Eq. (6) .

Declaration of Competing Interest

The authors declare that they have no known competing finan- cial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work was supported by Xi’an University of Architecture and Technology (Grant No. 002/2040221134).

Supplementary material

Supplementary material associated with this article can be found, in the online version, at doi: 10.1016/j.taml.2022.100347 .