Modeling the spaghetti in FEDEM
The spaghetti is represented by a string of beam elements, with the following physical properties, as found for Barilla spaghetti no. 1:
The elastic modulus and bending capacity is not printed on the package, but was found by means of a ruler, an electronic scale and some table-top experimenting. The shear modulus was calculating by assuming a Poisson’s ratio of 0.3.
There were a total of 25 beams, each with a length of 1cm, i.e. a total spaghetti length of 25cm. The string of beam was created by using the beam string file import. In the definition file, it was specified that each beam was to be connected with its neighbor with a joint. A Rayleigh stiffness proportional damping factor of 1E-5 was applied to the beams.
After importing the beam string each of the 6 joint DOFs is given stiffness as follows:
This stiffness values is high enough to ensure that the connections are stiff compared to the elasticity of the beam elements.
A key feature is to introduce the breaking moment. The bending in this model is around the z-axis. Hence the Rz spring must be given a maximum allowed moment. This is defined in a feature called “Advanced spring characteristics” which lets you assign an upper and lower limit for the force in a spring. For the Rz spring the breaking force (here the force is a moment) was set to +/- 0.012Nm. After exceeding these limits the spring constant will drop to zero. The #FailAll toggle will cause all remaining DOF’s of the joint to fail if one fails. This is required to ensure that the beam string is fully decoupled after a failure occurs.
Figure 1. Property panel for the Advanced spring characteristics. When this used for a rotation DOF the force is here a moment, which is set to 0.012Nm.
Defining the event
As pointed out in the paper by Audoly and Neukirch, the spaghetti does not need to be bent till it breaks. It is sufficient to bend it close to the breaking limit and release it. It is the sudden release that causes the multiple fractures. We did the same thing in this experiment by fixing the spring in one end, applying an increasing moment at the other end, which is then released abruptly just before the breaking moment is reached.
Figure 2. Pricipal layout of the model.
The moment is a linear ramp that reaches 0.01Nm at t=0.4 seconds. Thereafter it drops to zero at t=0.401s. The timestep can be varying in FEDEM, which is a useful feature for this case: In the bending phase before t=0.4s we used a timestep of Δt=0.005s. After t=0.4s the timestep is reduced to Δt=0.0002s – because after the release things happens quite fast!
Figure 3. Functions used for Moment ramp (left) and time step (right).
Burst of moments
The two first figure below shows the moment in each joint vs. time. From t=0 to 0.4s one observes a linear increase with some overlaid oscillations with a period about 0.19s, i.e 5.3 Hz. The oscillations are is the first eigenfrequency of the cantilever beam and can be verified by the following equation:
After t=0.4 one observes 3 sequential burst of moments where capacity limit of 0.012 is reached with breaking of the spaghetti. The first burst follows from the release of the external moment and leads to the first break. The break leads to the second burst, giving a second break shich again causes the third burst and break. The 3 breaks leave us with 4 pieces of spaghetti; three of them spinning in the air while the last one still stuck in the wall. The last piece forms a cantilever, oscillating at a frequency about 100Hz, which agrees with the frequency scaling of length (black curves). The flying pieces are having internal oscillations with higher frequencies.
The last figure shows moment along the spaghetti string at different time instants after the moment release. One can clearly see how a moment wave sets off from the end giving the first break at L=17cm, measured from the fixed end. This triggers a new inward wave giving the next break at L=12cm.
Figure 4. Moment in all joints as function of time.
Figure 5. Moment in all joint as function of time. Close-up at breaking time.
Figure 6. Moment in the spaghetti vs. length - shown at 3 different time instances shortly after the break. At firs a travelling wave sets off from the end giving a break at L=17mm 0.4ms after the release of moment. A new wave sets off, and give a break at L=12 after the release.
Limitations of the current work
There is no doubt that interesting core physics of multiple fracturing is captured by the simulation model. The current work was a straight-forward approach. A more in-depth study one should investigate effects of beam lengths, time step, structural damping, etc., in order to check the convergence of these parameters settings. It is also possible that a deeper investigation by the this method can spin off new scientific results. However, the scope here has been mainly to demonstrate the capabilities of structural dynamics simulation in FEDEM.