Spaghetti Breaking Dynamics

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? For these problems, in this paper a strand of spaghetti is considered an elastic rod, and the ﬁnite-element software ABAQUS is used to simulate the detailed fracture dynamics of the elastic rod. By changing the size (length and diameter) of the rod, the relevant data on the fracture limit curvature and the number of fractured segments of the elastic rod are obtained. The ABAQUS simulation results conﬁrm the scientiﬁc judgment of B. Audoly and S. Neukirch (Fragmentation of rods by cascading cracks: Why spaghetti does not break in half. Phys Rev Lett 95: 095505). Using dimensional analysis to ﬁt the ﬁnite-element data, two relations of the elastic rod fracture dynamics are obtained: (1) the relationship between the fracture limit curvature and the diameter, and (2) the relationship between the number of fracture segments and the diameter-length ratio. Results reveal that when the length is constant, the larger the diameter (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. ,


INTRODUCTION
Numerous objects can be described as elastic rods in ordinary life, such as bones [1] and trees [2−4] , and even fiber cells [5] or other models with important physical and physiological functions on various scales. Artificial objects like carbon nanotube composite materials [6] , semi-flexible polymer ports [7] , and multi-walled carbon nanotubes [8,9] can all be described by the mechanical model of elastic rods as well. Elastic rods are also very common in engineering applications, such as circular cross-section beams and columns in civil engineering [10] and automobile drive shafts [28] . During the 2012 London Olympics, Cuban Pole vaulter Lazaro Borges nearly died when the pole broke into three sections at a height of 5 m. The importance of the elastic rod is evident and its ultimate curvature when it is about to break has also become a problem to urgently solve.
Thus, a quasi-static experiment is considered: a piece of dry spaghetti is slowly bent to its limit curvature. In most cases, a strand of spaghetti, which can be regarded as an elastic rod, will not break into two pieces as intuitively thought, but into three to ten pieces. This simple and interesting experiment created Feynman's doubts, and the issue was not resolved at that time [12] .
The mechanism of elastic rod fracture and crack propagation has been studied for a long time [11−16] . It has been mainly studied from the perspective of statics [17,18] . Mott's work [19] in 1945 established the foundation for the study of the fracture phenomenon of elastic rods. Shockey [20] took the transmission of stress waves as the starting point, and Wittel [21] studied the fragments produced by brittle materials. Mitchell [22] studied the mechanism of curvature affecting the crack direction of materials. Villermaux [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 [12] studied the fracture of a slender rod in the adiabatic state and the transmission of bending waves, while Gladden [13] studied 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 breaking into multiple segments involves many disciplines and technical fields.
In the first part of this paper, simulation of spaghetti curvature and the number of fractured sections in ABAQUS finite-element software are described; in addition, the experiments of Heisser [24] are reproduced and his results verified. In the second part, the data of limit curvature are fitted through dimensional analysis. In the third part, a mathematical model of the fracture dynamics of spaghetti is given, and the coefficient of Audoly is verified as κ κ0 = 1.428 [12] . In the fourth part, the data of the number of fractures are processed in ABAQUS and the fitting equation given.

Finite-element-model analysis
The ABAQUS finite-element model (FEM) presented in this paper is based on the experiments of Heisser [24] under the conditions of humidity 21%−34% and temperature 21% − 26%. The length and radius of the spaghetti is L = 240mm and d = 0.75mm, respectively, and its density is ρ = 1.5 ± 0.1 g/cm 3 . The elastic modulus is E = 3.8±0.3GPa and Poisson's ratio is ν = 0.3±0.1. The model's data here are ρ = 1.5 g/cm 3 , E = 3.8GPa, and ν = 0.3. 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 minor. Therefore, this model does not consider the initial defects. Two ends of the spaghetti strand were set to be hinged, and the 10mm surface from each end to the reference point were coupled to simulate 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.  Heisser's experiments used the quench speed of the two ends as the variable in the experiments. However, directly squeezing both ends is likely to cause the stability and buckling problem [29] , discussion of which is not within the scope of this paper; Heisser's experiments also exclude the relevant results.
In the present work, Y and Z displacements 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. See the attachment for a description of the complete dynamic fracture process.

Simulation of limit curvature
Discussion of the limit curvature was facilitated by changing the size of the spaghetti with length L = 240mm and by increasing the diameter from 1 to 3 mm in increments of 0.1mm to achieve different diameters in order to study the relationship between the diameter-tolength ratio and limit curvature. The relationship obtained 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 in Section 2.

Simulation of dynamic process after fracture
For the dynamic process after spaghetti fracture, in this paper the fracture of a rod when the length is L = 240mm, diameter is D = 1.5mm, and quench speed is v = 300mm/s is taken as the object of study. From Figs. 6 and 7, it can be seen that when t = 0, point a is broken, κ a = 7.968m −1 , when t = 0.0017s, the curvature of point b is κ b = 11.571m −1 . In Section 3, the relationship between the curvature of these two points and the dynamic process after fracture is discussed in detail.

Simulation of number of fractured sections
For the simulation of the number of fractured segments of spaghetti, the model is taken at the point when the lengths of the spaghetti are L = 200, 220, and 240mm and the diameter D ranges from 1 to 3mm. 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 segments would be averaged, as shown in Fig. 8. It can be seen from the figure that the number of fractured segments gradually converges to a constant as the diameterto-length ratio increases. The data are discussed and fitted in Section 4. Simulation verification of experiment 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 obtained. 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.

DIMENSIONAL ANALYSIS AND CURVATURE FITTING
To discuss the limit curvature and fit the data in Fig.  5, through the theory of dimensional analysis [26,27] , six parameters of the elastic rod can be listed, all six of which have three dimensions (L, M, and T), as shown in Table  1.
As a result, this problem produces 6-3=3 dimensionless quantities Π, as follows:  a,b,c, a 1 ,b 1 ,c 1 , and a 2 ,b 2 ,c 2 can be obtained by the dimensionless condition dim ( and with 1+a-3b-c=0, b+c=0, and -2c=0, one can obtain a=-1, b=0, and c=0. Bringing in the determined a, b, and c, the first dimensionless quantity can be obtained as follows: where Π 1 has been rewritten into Π 1,modif ied , which is the diameter-to-length ratio.
In the same way, one can obtain a 1 = 0, b 1 = 0, and c 1 = −1, and the second dimensionless quantity is where Π 2 has also been rewritten into Π 2,modif ied , which means the ultimate strain, i.e., the ratio of ultimate stress to the elastic modulus. Similarly, one obtains a 2 = 1, b 2 = 0, and c 2 = 0, so the third dimensionless quantity is According to dimensional analysis theory, one can obtain Π 3 = f (Π 1,modif ied , Π 2,modif ied ), which could be written as Assuming that the ultimate strain of spaghetti is fixed, then c is a constant and Eq. (6) can be rewritten as According to Barenblat's suggestion [31] , f ( D L ) can be changed to an expansion similar to a power series: where C and α are constants. Inserting Eq. (8) into Eq. (7), Since L = 240mm, the constants C and α can be determined by data fitting. Through FE data fitting, the relationship between the limit curvature and diameter of spaghetti at a humidity of 21% − 34%, temperature of 21% − 26%, and quench speed of 300mm/s can be obtained; in addition, C = 0.091 and α = −0.366, as shown in Fig. 10: The relationship between the radius and limit curvature when the length of the spaghetti is L = 240mm is thus obtained.

DYNAMIC MATHEMATICAL MODEL AFTER FRACTURE
Audoly believes that the multi-segment fracture of spaghetti is due to the cascading effect caused by the transmission of bending waves after the first fracture. Taking Fig. 6(c) 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, and section b denotes the spaghetti segments with two free ends. Only the cantilever situation is discussed here, as shown in Figs. 10(a) and 10(c). Section a is taken as an example. The Euler-Bernoulli beam equation [30] can be applied to the spaghetti after breaking: where y is the deflection, s the arc length, The self-similarity solution of Eq. (13) is where C 1 , C 2 , C 3 , and C 4 are undetermined coefficients. The place s = 0 is the free end, s = L is the fixed end, and the boundary conditions are brought in as κ(0, t) = 0, κ(s, 0) = κ 0 , ∂κ ∂s (0, t) = 0, and ∂ 2 κ ∂s 2 (0, t) = 0 (κ 0 is the initial curve), after which one obtains C 1 = C 3 = C 4 = 0, C 2 = 2κ 0 .
FIG. 14: Numerical solution of Eq. (15) 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. Figure 13 shows that, when ξ reached the maximum value; at this time, the curvature κ is 1.428 times the initial curvature κ 0 . Audoly believes that the coefficient 1.428 is universal [12] , which is double the maximum value of the Fresnel sine integral. The coefficient of κ κ0 = 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 10mm long for plug-in articulation in this paper. The inserted part can be regarded as a fixed end relative to the other parts of the spaghetti, so it meets the boundary and initial conditions; this process does not involving any material properties. The curvature κ a of the breaking point a at t = 0 relaxed in a short period of time. This sudden relaxation produces an explosive stress wave, strong enough to break the remaining part at this time, κ a = 7.968m −1 . When t = 0.0017s, κ b = 11.571m −1 at point b, and point b breaks when t=0.0018 s. In this process, κ b is 1.452 times κ a . The difference between 1.452 and 1.428 is 1.7%. This also supports Audoly's κ κ0 = 1.428 coefficient.

DIMENSIONAL ANALYSIS AND FITTING OF FRACTURED SEGMENT NUMBER
Audoly's work [12] 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 diameter D is in the range of 1 to 3 mm. The relationship between the fractured 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 converges to a value with increasing diameterto-length ratio. To study the relationship between them, polynomial fitting is performed on the FE data, and the formulation is Drawing the fitting curve together with the FE data, the plot shown in Fig. 18 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.

CONCLUSIONS
In terms of fracture limit curvature, the relationship between fracture limit curvature and diameter is obtained through dimensional analysis as exemplified by Eq. (9). After fitting the FE simulation data, it was found that when the spaghetti length L is constant, the limit curvature is inversely proportional to the diameter of the elastic rod, and the relationship between them is expressed by Eq. (10).
In terms of the dynamic mathematical model after fracture, the Euler-Bernoulli beam equation and FE method are used in this paper to confirm the conclusion of Audoly that κ κ0 = 1.428. This conclusion 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 fracture segment numbers when the spaghetti lengths are L=200, 220, and 240 mm and the diameters D are in the range of 1 to 3 mm was simulated. The FE simulation results show that the fracture segment number of segments gradually decreases from eight with increasing diameterto-length ratio, and converges to 3.6. The relationship between them is expressed by Eq. (16). The decreasing trend of their fracture segments is in line with the actual law.
In this paper, the length of the spaghetti selected and value range of the diameter does not exert much influence. If more simulations under different sizes and more data can be obtained, it is our belief that the limitcurvature-to-diameter-length ratio can be obtained. The relationship between the number of fractured segments and diameter-to-length ratio is more accurate.