3.1. Quantitative Analysis
Now, we are going to investigate the influence of the introduction of the gradual release declining time on the induced flexural stress waves and excited bending moment and thus the influence on the process of the secondary fracture during spaghetti bent break.
The physical meaning of is the time duration of the gradually declined release of initial bending moment at the free end dropping from original value to zero. The greater value of reveals a longer time duration in the boundary condition at the free end, and vice versa.
First, let us consider one of the special cases. We can see from Equations (
40) and (
41) that when the value of
approaches to zero, or to say, when the gradually declined release boundary condition of our model is simplified to a special case—the sudden release boundary condition case, the solution is accordingly simplified to
Equation (
42) is the so-called self-similar solution in terms of excited bending moment with variables of distance and time, the form of which is the same as Equation (
1). It can be easily figured out that the extremum value of the excited bending moment from Equation (
42) is twice the maximum of the Fresnel sine integral although lacking of any adjustable time parameter, that is, the excited bending moment will always get to 1.43 times initial moment somewhere in the half-infinite stick regardless of how short the propagate time will be, and then lead to a secondary fracture. Thus, the self-similar solution can only demonstrate the inevitability of a secondary fracture, but intrinsically indicates an infinite stress wave speed, which is not in line with the real situation.
Second, let us assume some typical non-zero values of to investigate the difference and improvement of the introduction of and whether varying values of affect the excited bending stress waves and their propagation and how.
Assuming
, we can get excited bending moment versus distance curves in progressive time coordinates as shown in
Figure 4. The gradually declined release of initial bending moment excites a series of flexural stress waves from the free end of the stick, and these flexural stress waves locally increase the excited bending moment to beyond a critical limit and eventually lead to a secondary fracture. The excited stress wave propagates through the stick, but will not affect the physical state far away from the wave front. The excited bending moment at the free end of the stick gradually declines with the propagation time from initial value down to zero when
, which is consistent with the boundary condition Equation (
14), and propagates through the whole stick without any external value at the free end when
. Under any time coordinate, the bending moment excited from the gradual release of the initial moment always monotonously grows to a maximum value, and this maximum value increases with time, but will not increase indefinitely. It can be seen from
Figure 4, and also can be proved mathematically from the analytical solution, that the maximum excited bending moment
has an asymptotic limit value of twice the maximum of the Fresnel sine integral, being 1.43. The difference between non-zero valued
case and approaching zero valued
case lies on the necessity of a certain propagate time when the excited moment grows to some certain value. Therefore, the introduction of gradually release boundary condition of our model brings an adjustable characteristic time parameter, and accordingly provides a possibility to predict the intrinsic fragmentation time and fragmentation length.
Then, we extend the value of
from 1 to greater numbers, and cases of two typical values of release declining time
and
are illustrated in
Figure 5 and
Figure 6, respectively.
As can be seen from
Figure 5 and
Figure 6, the linearly gradually declined release with greater-valued time duration likewise excites a series of flexural stress waves propagating from the free end of the stick to the infinite end and these flexural stress waves locally increase the excited bending moment to beyond a critical limit and eventually lead to a secondary fracture. Under any time coordinate, the excited bending moment always monotonously grows to a maximum value, and the maximum value increases with propagation time to an asymptotic extremum value of
.
Figure 4,
Figure 5 and
Figure 6 show the excited flexural waveform curves at progressive time coordinates, and we can figure out that at the very beginning of both distance and time the excited bending moment will rise to beyond its initial value, that is, at the very beginning as long as the flexural stress wave is excited, the secondary fracture is about to happen, provided that once the bending moment exceeds it initial value
a secondary fracture will start up. However, experience and experiments from us and other scholars [
11,
15] suggest that the secondary fracture will take place at a point with some certain distance from the first fracture point, which reminds that the crack criterion of the secondary fracture should be not the same as the quasi-static first fracture.
Therefore, we adopt the weakest chain principle as the crack criterion of the secondary fracture. The key to the criterion is that the secondary fracture will take place at the point of the maximum value of the excited bending moment
. However, in our model
is an asymptotic extremum which is actually unreachable through the whole bent process. Therefore, we propose three typical value of
, the contour curves of which are illustrated in
Figure 7,
Figure 8 and
Figure 9, respectively, when
with variables of distance and time.
Any point in a contour curve reaches the set value with the corresponding spacial coordinate and time coordinate, and the point in the contour curve who owns the minimum value of time is believed to be the case when and where a secondary fracture is about to happen, and is marked. It can be seen from
Figure 7 that when
, the fracture distances are
,
, and
times nondimensional length or radii of gyration respectively with progressive set critical moment from
to
, which indicates that greater set critical value of bending moment leads to greater value of fracture distance, and vice versa, and
Figure 8 and
Figure 9 share the same fact with greater values of
. Meanwhile, the fracture time are
,
, and
times nondimensional time, respectively, when
, which indicates that greater set critical value of bending moment as well leads to greater value of fracture time, and vice versa, and cases with greater values of
share the same fact.
It also can be seen from the three contour curves that when the set value is the same, for instance , the fracture distance are 8, 25, and 35 times nondimensional length or radii of gyration and the fracture time are times nondimensional time at , respectively, indicating that greater value of leads to greater value of fracture distance and fracture time, and vice versa.
As discussed above, we investigate the effect and influence of release declining time on the secondary fracture process from one perspective of the contour curves with variables of distance and time, and subsequently we are going to continue to investigate from another perspective of the envelope curves of the maximum excited bending moment from the excited flexural waveform curves such as
Figure 4,
Figure 5 and
Figure 6. In fact, we provide
Figure 10 of envelope curves extended to seven different values of
along spatial distance. We can see from the curves that whatever value
takes, as long as it is non-zero, the maximum bending moment will always progressively increase and then to the asymptotic extremum value of 1.43 times initial value. However, greater value of
, or longer release declining time duration, leads to farther distance when the excited moments reach to some certain set value marked in our schematic, and vice versa.
Meanwhile,
Figure 10 reveals improvement and advantage between solutions to our gradual release model and the sudden release model, that by introducing a non-zero valued release declining time
, the excited maximum bending moment will not directly increase to its asymptotic extremum, but will undergo a certain time duration and distance before get to some set critical value. Therefore, a characteristic time parameter is brought into solution when the release declining time
is brought to the boundary condition, which is an adjustable time parameter to indicate further understanding towards secondary fracture during spaghetti bent break. As a result, as long as we give a crack criterion of the secondary fracture in terms of bending moment, we are able to get the fracture distance and fracture time from our solution, which are also the characteristic fragmentation length and the characteristic fragmentation time of the secondary fracture.
3.2. Estimated Value of
Subsequently, let us discuss what value of should be taken physically and theoretically.
According to the linear elastic theory, the ultimate crack speed is Rayleigh wave speed (
). However, Schardin and Struth [
16] found that the maximum velocity reached by the fast impact crack is material characteristic speed and less than the Rayleigh wave speed. This result was confirmed in other materials using different measurement methods, such as for noncrystalline materials with crack speed range of 0.4 to 0.7
[
17], and for crystal materials ranging from 0.63 to 0.90
[
18]. These studies showed that the ultimate crack speed is constant for each material and occupies a specific proportion in the elastic wave speed.
For linear elastic material of positive Poisson’s ratio, the Rayleigh wave speed equals 0.862–0.955 times of the shear wave speed (). Moreover, the shear wave speed is about 0.577–0.707 times the elastic wave speed (). Therefore, the Rayleigh wave speed equals to 0.497–0.675 times the elastic wave speed .
This means that the crack speed is about 0.199–0.473 of the elastic wave speed for noncrystalline material such as spaghetti sticks. As a result, the time duration of the first fracture during spaghetti bent break is the stick thickness
h divided by crack speed,
The result in Equation (
43) is just the release declining time in our model, or the estimated value of
. Therefore, under linear elastic theory, the value of
should be taken as 8–20 times nondimensional time.
Moreover, in fact, the research of the dynamic fracture process of a long rod subjected to pure bending, mainly the first fracture, was originated from Freund and Herrmann’s work [
19]. They reported that the crack tip rapidly accelerates to near the characteristic terminal speed, maintains this speed to travel through most of the stick thickness
h, and then decelerates quickly. They also presented that the bending moment on the fracture section decreases monotonically down to zero with a time duration of
, which is
or 20 times the nondimensional time in our model. Later works by Adeli [
20] and Levy [
21] further investigated the dynamics of this process and shared the same results of the crack tip transmission time and the moment declining time. In addition, numerical simulation results by the authors suggests the time duration of the first fracture of brittle elastic material subjected to four points bending is 17 times nondimensional time.
As a result, that the value of
is estimated as 8–20 is reasonable, and we can figure out from
Figure 10 that the fracture length between the first and second crack is 22–34 times nondimensional length or 6–9 times stick thickness accordingly, when
is chosen as the set critical value.