Crack divergence phenomena in tempered glass

Abstract

High speed photography using the Cranz-Schardin camera was performed to study the crack divergence and divergence angle in thermally tempered glass. A tempered 3.5 mm thick glass plate was used as a specimen. It was shown that two types of bifurcation and branching existed as the crack divergence. The divergence angle was smaller than the value calculated from the principle of optimal design and showed an acute angle.

Keywords

Crack divergence bifurcation and branching divergence angle 3.5-mm tempered glass high speed photography

1. Introduction

As tempered glass has the property of becoming fine fragments when fractured, it is socially recognized as a safety glass. Figure 1 shows an example of the fracture pattern when tempered glass is fractured from the center. The propagation of cracks, especially the divergence phenomenon, has a very important meaning, because fragmentation is caused by repeated crack propagation and divergence.

Fig. 1.

Fracture pattern in tempered glass.

In terms of repeated divergence, the following relationships between the distance ℓ from the divergence of the crack to the re-divergence and the applied stress 𝜎 from the outside have been reported, as shown in Eq. (1) below: $\begin{eqnarray}\displaystyle {\sigma}^{2}\ell =\text{const}. & & \displaystyle\end{eqnarray}$ (1) As a factor that influences crack propagation in tempered glass, the magnitude of internal stress, which is residual stress, is known. Since Eq. (1) can be applied to tempered glass, the greater the internal stress in the tempered glass, the shorter the crack length before re-divergence.

Equation (1) is in line with the general tendency that the fragment density (number of fragments/50 mm × 50 mm) increases as the internal stress of the tempered glass increases. Hara et al. [11] also reported the same results as Eq. (1). Takatsu et al. [16] experimentally determined the relationship between the fragment density and surface stress, and Akeyoshi et al. [1] determined the experimental relationship between the fragment density and tensile stress.

It is known that crack propagation in glass follows the normal stress law [9], which means that the influence of the principal stress direction in the glass is large. Takatsu et al. [15] reported the effect of local residual stress on crack propagation, and the fracture pattern differs depending on the difference in the stress pattern given in the first step when performed quenching in two stages producing tempered glass. However, general tempered glass has been produced so that the fracture pattern is uniform and the difference in the principal stress direction is also made as small as possible. Therefore, there is much uncertainty about the clear dependence of the principal stress on the fracture pattern.

In the case of tempered glass, the stress that promotes crack propagation is not applied from the outside, but is made by the internal stress existing in the tempered glass. Figure 2 shows a schematic diagram of the stress distribution in tempered and annealed glass. The view that the compressive stress-neutral point-tensile stress changes continuously from the surface layer to the inner layer and the position of the neutral point is common The crack propagating in the glass depends on its tensile stress. If the value is large, the propagation of cracks is promoted.

Fig. 2.

Stress distribution of ordinary and tempered glass.

The crack divergence was fundamentally thought to propagate in a different direction from the previous propagation direction. The crack divergence of glass could be broadly explained in two concepts. One is “the concept as a means to release elastic energy after reaching the critical velocity” [11], as shown in Fig. 4. According to this concept, the crack maintains a constant velocity after exceeding energy E_C, and crack divergence is generated when the critical energy E_B is exceeded. The other concept is “based on the union of p-c and s-cracks which were generated in a place not on the extension of the p-crack” [10], as shown in Fig. 5. As a result, it is observed as a divergence phenomenon that the s-cracks propagating toward the p-crack are connected.

Fig. 3.

Relation of crack velocity and divergence energy by the energy theory.

Fig. 4.

Generation of s- and p-cracks by the combination theory.

Currently, there is much uncertainty about crack divergence in tempered glass. In particular, there are few reports on the divergence phenomenon in tempered glass, but the safety of tempered glass is considered to depend on the fragmentation phenomenon at fracture. It is not easy to analyze the crack divergence phenomenon, because it is difficult to actually observe the crack propagation and divergence is an extremely high-speed phenomena of velocity at about 1500 m/s [8], and many new cracks called “secondary cracks” occur with a time delay in tempered glass.

Fig. 5.

Ordinary crack divergence called bifurcation.

Fig. 6.

Experimental setup.

Aratani et al. reported that reflected waves could influence changes in fragment density (number of fragments/50 mm × 50 mm) [4], and the dynamic stress intensity factor decreases to about 1/2 with before and after crack divergence using the Cranz-Schardin camera. Also, in terms of the crack connection phenomenon [14] observed in zone-tempered glass, it could be explained [5] by introducing the concept of stress 𝜎_CR [14] generated at the propagating crack tip.

The divergence angle is very important to understand the divergence mechanism in tempered glass. However, only a few papers reported on this topic. Therefore, this study focused on the crack divergence pattern and divergence angle, as a part of investigating the divergence phenomenon in tempered glass.

2. Experimental procedures

2.1. Specimens

Tempered glass that has passed the Japanese Industrial Standard JIS R 3206 was used as a specimen. The specimen size was 600 × 400 × 3.5 (mm). An ordinary soda-lime-silicate float glass plate produced by Central Glass Co. Ltd. was used and the chemical composition is shown in Table 1.

Table 1
Chemical composition of glass specimen

SiO₂ Al₂O₃ CaO MgO Na₂O K₂O Others

72 2 8 3 13 1 1

SiO₂	Al₂O₃	CaO	MgO	Na₂O	K₂O	Others
72	2	8	3	13	1	1

(mass %)

2.2. Test equipment and method

We used a Cranz-Schardin type high speed camera [13], which was developed at the Research Institute for Applied Mechanics, Kyushu University, Japan. The layout is shown in Fig. 6. An air gun was used to fracture the glass specimen, and an extremely light impactor of 0.2 g was used to minimize the deformation of the glass sample when it fractured.

2.3. Divergence angle measurement

The angle after divergence with respect to the direction of crack propagation before divergence was defined as the divergence angle and was measured. The measurement of the divergence angle was performed by focusing on cracks that can be clearly confirmed to have diverged. Secondary cracks [7] were excluded because they are different from the divergence phenomenon. Note that these are different from secondary sub-cracks (s-cracks), which are described in the “concept based on the unification of secondary s- and p-cracks” [10].

3. Results and discussion

3.1. Divergence pattern in tempered glass

Figure 7 shows a high-speed photograph of crack propagation in general tempered glass. Bifurcation has been considered as the divergence at the time of fracture in the tempered glass, and few examples of other divergence patterns have been reported. However, it was observed that the direction of divergence is not always the same in the cracks before and after divergence. Although bifurcation (the circle surrounded by the solid line in Fig. 7) was observed as before, and different divergence patterns were also observed (the circles surrounded by dashed lines in Fig. 7). Namely, not only the pattern of bifurcation, but also another pattern of divergence with two propagating directions was observed.

Oka [12] examined the divergence of blood vessels into two patterns:

Bifurcation: Two flow directions of the blood vessel after divergence are different from the direction of the blood vessel before divergence.

Branching: One flow direction of the blood vessel after divergence is the same as the blood vessel before divergence, and another flow direction is different from that before divergence.

In this study, crack divergence in tempered glass will be divided into two divergence patterns of bifurcation and branching, according to Oka’s concept.

Fig. 7.

High-speed photography of crack propagation in tempered glass.

Fig. 8.

Two types of divergences in blood vessels: (a) bifurcation, (b) branching.

Fig. 9.

Crack divergence of branching types.

The crack divergence phenomenon in the tempered glass was analyzed by distinguishing the two types of bifurcation and branching, as below:

Bifurcation: Two crack directions after divergence are different from that before divergence.

Branching: One crack direction after divergence is the same as the crack direction before divergence, and another crack direction is different from that before divergence.

It was confirmed that not only the bifurcated divergences that have been described previously but also many branching divergences existed. In Fig. 7, the divergence surrounded by the solid circle is the bifurcation, and the divergence surrounded by the dotted line is the branching. It was thought that the two bifurcation angles were almost equal, but they did not always show the same or close values. Crack divergences with ambiguity that could be classified as either the bifurcation or branching type were also observed.

It is extremely important to note that not only bifurcation divergences but also many branching divergences existed, because this denies the premise of the “concept based on the unification of secondary s- and p-cracks” [10]. Since the location of s-cracks is not premised on the extension of the p-crack, branching is out of concept, and the divergence mechanism including branching in glasses is extremely difficult to explain.

3.2. Measurement results of the divergence angle

Figure 10 shows the measurement results of the divergence angle. As mentioned above, the two divergence angles in the bifurcation did not always show the same values. The respective divergence angles (𝜃₁, 𝜃₂) and the divergence angle (𝜃₁ + 𝜃₂) between the two cracks were examined separately. Fig. 10(a) shows the divergence angle (𝜃₁, 𝜃₂) of each bifurcation, and Fig. 10(b) shows the divergence angle (𝜃₁ + 𝜃₂) between bifurcation cracks. Figure 10(c) shows the divergence angle in case of branching. All the results are shown as a Weibull plot obtained using the average rank method.

The divergence angle did not show a constant value, but varied greatly from about 5° to 60°, as shown in Fig. 10(a). As mentioned above, the divergence angles of the two cracks after divergence did not always show the same or close values, but many values of divergence angles that were far apart were also recognized. Even in the case of a branching, the divergence angle was not constant, and a variation of about 40° to 70° was observed, as shown in Fig. 10(c).

Oka [12] reported that the divergence angle (𝜃₁, 𝜃₂) in bifurcation is 51° from the principle of optimal design. Yokobori [17] stated that the rock fracture shown in Fig. 11 could be explained using Oka’s concept. The divergence angle in tempered glass was shown as an acute value compared with that of Oka’s calculation result. The divergence angle was about 28° on average, the standard deviation was 14.0°, and the Weibull coefficient was 2.3. In terms of the average value, the divergence angle was 23° sharper than the value calculated by Oka.

Fig. 10.

(a) Divergence angle of 𝜃₁ and 𝜃₂ of the bifurcation type. (b) Divergence angle of 𝜃₁ + 𝜃₂ of the bifurcation type. (c) Divergence angle of 𝜃_B of the branching type.

Fig. 11.

Fracture pattern of rock [17].

Fig. 12.

(a) Image of crack divergence of the bifurcation type in tempered glass. (b) Image of crack divergence of the branching type in tempered glass.

Fig. 13.

Two cases of crack propagation to s-crack generated in the tensile stress area (case 1) and in the compressed stress area (case 2).

This tendency was, of course, similar with the divergence angle (𝜃₁ + 𝜃₂) between the two cracks, too. Although the value of the divergence angle in this case was 102° calculated from the principle of optimal design, most of them had small values, and about 1/4 of the divergence angles smaller than 50°. The divergence angle was about 55° on average, the standard deviation was 20.2°, and the Weibull coefficient was 3.2. In terms of the average value, the divergence angle became 47° smaller.

In the case of branching, Oka stated that the divergence angle of branch cracks is 90°, based on the similar idea. However, the divergence angle in tempered glass around 90° was not observed, and the maximum was about 70°. The divergence angle was about 52° on average, the standard deviation was 9.5°, and the Weibull coefficient was 6.0. In terms of the average value, the divergence angle became 38° smaller.

An image of crack divergence in tempered glass is shown in Fig. 12. Figure 12(a) shows the case of bifurcation, and Fig. 12(b) shows the branching. The solid line shows the crack propagation direction after divergence based on the average value of divergence angles obtained in this measurement, and the dotted line shows the propagation direction based on the divergence angle calculated by Oka. The principal stress directions at the crack tip during propagation are indicated by 𝜎₁ and $\sigma_2.$ Since it propagates from the bottom to the top, the relation of principal stresses is 𝜎₁ ≧ 𝜎₂. This way, the average value of the measured results can be better understood than the result calculated from the principle of optimal design. The values of the divergence angle measured in this experiment were uneven, and most of the divergence angles in bifurcation and all of those in branching were smaller than Oka’s calculated value. This means that there is a high possibility that the stress distribution emphasized the principal stress 𝜎₁ on crack propagation.

The experimental results shows that the divergence angle in tempered glass tends to be sharper than the calculated value by Oka, which was inferred to be caused by the difference in the divergence conditions. The formation and divergence phenomena of blood vessels is thought to be in a very static field based on the optimal design in nature. With regards to the occurrence of rock fissures and divergence [17], cracks did not propagate with a critical velocity, but grew in a static condition in the natural world. On the other hand, crack propagation in tempered glass is a propagating phenomenon at the terminal velocity of about 1500 m/s, and is presumed to be a phenomenon similar to a dynamic field. Since the propagation energy 𝜎_CR [3] generated at the propagating crack tip also affects the propagation of the crack, it is presumed that the terminal velocity of the propagating crack is influenced on the propagation energy and divergence angle.

It was reported that the divergence pattern of blood vessels was caused by their diameters. Namely, bifurcation occurs when the diameter of blood vessels after divergence is almost the same, and branching occurs when the diameter of blood vessels after divergence is almost different. If the diameter of the blood vessel before and after divergence does not change, the blood vessel will be divided into two directions. When there is a large difference in the diameter of blood vessels, thin blood vessels diverge from the trunk blood vessels. Expanding this concept into crack divergence in glass, the bifurcation occurs when the propagation energy of the cracks immediately after the divergence has similar values, and the branching occurs when the propagation energy of the cracks immediately after the divergence has different values.

Considering the propagation energy before and after divergence in bifurcation, the following equations can be presumed: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \text{E}_{\text{A}}=\text{E}_{0}\cos {\theta}_{\text{A}}-\text{E}_{0}\sin {\theta}_{\text{A}}+\text{E}_{\text{SA}}\end{eqnarray}$ (2) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \text{E}_{\text{B}}=\text{E}_{0}\cos {\theta}_{\text{B}}-\text{E}_{0}\sin {\theta}_{\text{B}}+\text{E}_{\text{SB}}.\end{eqnarray}$ (3) Here, E_A and E_B are the propagation energy in the two cracks immediately after divergence, E₀ is the crack propagation energy before divergence, and 𝜃_A and 𝜃_B are the divergence angles, respectively. E_SA and E_SB are correction terms, which is energy that increases or decreases based on changes in the stress field in tempered glass, and can be positive or negative. The first term in Eqs (2) and (3) is the energy spent for crack propagation after divergence, and the second term is the energy that depends on the repulsive force for separating the two cracks. When the energy of crack propagation before divergence and of two cracks after divergence is equal, Eq. (4) holds: $\begin{eqnarray}\displaystyle \text{E}_{0}=\text{E}_{\text{A}}+\text{E}_{\text{B}}. & & \displaystyle\end{eqnarray}$ (4) From Eqs (2)–(4): $\begin{eqnarray}\displaystyle \text{E}_{0}\text{} & =\text{} & \displaystyle \text{E}_{0}\cos {\theta}_{\text{A}}-\text{E}_{0}\sin {\theta}_{\text{A}}+E_{\text{SA}}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼\text{E}_{0}\cos {\theta}_{\text{B}}-\text{E}_{0}\sin {\theta}_{\text{B}}+E_{\text{SB}}.\end{eqnarray}$ (5) To simplify the crack propagation after divergence, it is assumed that the principal stresses 𝜎₁ and 𝜎₂ of the tempered glass at the divergence are almost equal. $\begin{eqnarray}\displaystyle E_{\text{SA}}=E_{\text{SB}}=0. & & \displaystyle\end{eqnarray}$ (6) Furthermore, when the divergence angle of the two cracks are assumed to be equal, Eq. (5) changes: $\begin{eqnarray}\displaystyle \text{E}_{0}\text{} & =\text{} & \displaystyle \text{E}_{0}\cos {\theta}_{\text{A}}-\text{E}_{0}\sin {\theta}_{\text{A}}+\text{E}_{0}\cos {\theta}_{\text{B}}-\text{E}_{0}\sin {\theta}_{\text{B}}\nonumber\\ \displaystyle \text{} & =\text{} & \displaystyle 2\text{E}_{0}(\cos {\theta}_{\text{A}}-\sin {\theta}_{\text{A}}).\end{eqnarray}$ (7) Since E₀ has a finite value: $\begin{eqnarray}\displaystyle 1/2=(\cos {\theta}_{\text{A}}-\sin {\theta}_{\text{A}}). & & \displaystyle\end{eqnarray}$ (8) Square Eq. (8): $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle 0.25=\cos ^{2}{\theta}_{\text{A}}-2\cos {\theta}_{\text{A}}\cdot \sin {\theta}_{\text{A}}+\sin ^{2}{\theta}_{\text{A}}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \sin 2{\theta}_{\text{A}}=0.75\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle 2{\theta}_{\text{A}}=48.6^{\circ }\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle {\theta}_{\text{A}}=24.3^{\circ }.\nonumber\end{eqnarray}$

The divergence angle of the bifurcation was calculated at about 24° assuming that the total energy before and after divergence is the same. The divergence angle of the two cracks after divergence are the same, and the effect of the principal stress in the tempered glass is extremely small. This value is much smaller than 51° calculated by Oka from the principle of optimal design. However, this divergence angle is very close to the average value 28°, which is obtained in this measurement and 50% value of Weibull probability as shown in Fig. 10(a). It may also be considered as the median value of the divergence angle, which greatly varied from about 5° to 60°. On the other hand, the result including this variation is greatly influenced by the principal stress in tempered glass, and means that the correction terms E_SA and E_SB cannot be ignored. There are differences in the stress field of tempered glass at the divergence point, and it is considered to be a major cause of the variation in the divergence angle at the bifurcation.

On the other hand, it is extremely difficult to use the above assumptions for branching. The crack before divergence and the trunk crack propagate in the same direction, and only the branch crack propagates in the other direction. Energy is required to be applied in a direction different from the propagating direction of the p-crack and the trunk crack, only for branch cracks propagating in the other direction. The stress field of tempered glass can be considered for this energy, and the existence of a principal stress that is different from the propagation direction of the p-crack and the trunk crack is required. It is necessary to confirm whether there is an energy difference as described by Oka between the trunk crack and the branch crack, including the occurrence of repulsive force.

The above results have something in common with the “concept as a means to release elastic energy after reaching the critical velocity” [11]. However, bifurcation and branching phenomena difference and deviation of divergence angles cannot be explained by the above concept only. It is necessary to conduct a more detailed study based on this concept.

3.3. Validity of two crack divergence theories

Currently, there is much uncertainty about the crack divergence phenomenon of glass, and the divergence of glass is largely explained in two ways as mentioned above. One is “the concept as a means to release elastic energy after reaching the critical velocity” [11], and the other is “the concept based on the union of p- and s-cracks which were not generated on the extension of the p-crack” [10]. Although the latter concept is supported by many researchers, it is extremely difficult to explain the divergence phenomenon especially in tempered glass.

To explain the concept in a bit more detail, it is said that the crack will diverge through the following steps:

Fracture velocity reaches the maximum speed.

Tension appears in places other than the extension of the p-crack tip.

Secondary cracks (s₁, s₂) occur independently.

The initial crack propagates by connecting the secondary cracks.

There are many questions about the generation of s-cracks. It is extremely difficult to assume a mechanism that appears at a position other than the tip of the p-crack. As mentioned above, crack propagation in tempered glass does not occur due to external load, but is caused by residual stress in glass. Generation of tension requires additional loading, for example from stress generated by propagating cracks due to internal stress, and the position of the new tension generation does not have to deviate from the pcrack extension. On the contrary, it is more natural to occur on the extension of the propagating p-crack.

There is also a problem with secondary cracks (s₁, s₂) that occur independently. In order to generate s-cracks independently, large tensile stress must be generated. In the tensile stress that newly generates a crack and the tensile stress that propagates a crack, it is clear that the former is larger. It is extremely unlikely that a new crack will be generated under the condition of propagating the crack.

There is also the problem of connecting the initial and secondary cracks. In order to connect with p- and s-cracks, the principal stress directions at the tips of both cracks must be completely similar. Takahashi et al. reported a crack connection phenomenon in a specific region of zone-tempered glass [14], however, the principal stress direction is almost fixed in this specific region. In tempered glass with different principal stress directions, it is extremely unlikely that the principal stress directions at the two crack tips will coincide.

Furthermore, the location generated by the assumption of s-cracks will be considered. As previously mentioned, cracks in tempered glass propagate based on the tensile stress as shown in Fig. 2. Figure 13 shows a schematic diagram of crack divergence in tempered glass based on “the concept based on the union of secondary s- and p-cracks” [10].

Case 1 at the top of Fig. 13 shows that an s-crack occurs between the tensile stress and the neutral point, and case 2 at the bottom of Fig. 13 shows that an s-crack occurs between the compressive stress and the neutral point. In case 1, the s-crack occurs within the tensile stress, that is, it exists between the two neutral points. Cracks can propagate from both ends of s- and p-cracks. In this case, many combined cracks, which consist of p-cracks, crack which combines p- and s-cracks, and s-cracks and cracks that started from the opposite side of the s-crack, will exist in the tempered glass. In this state, the crack propagating velocity will become faster compared with non-divergence. However, the experimental result [8] shows that the crack propagation velocity did not increase when the crack diverged.

If there is a crack propagation as described above, it is characterized by the glass surface of the cross section side, which becomes clear by performing fractographic analysis. It is known that fractographic changes, called mirror-mist-hackle, occur near the fracture origin. A smooth region like a mirror is shown near the fracture origin. After that, the crack propagates while changing to the area called the mist and the area called the hackle area. It has been reported [2] that the crack propagation velocity decreases after divergence and that hackle marks occur at divergence points in underwater experiments. Namely, the crack reaches the divergence after a fractographic change in the mirror-mist-hackle, then propagates at a certain distance, and re-diverges while making a fractographic change of the mirror-mist-hackle again. The propagation of cracks can be inferred by observing the glass surface of the cross section at or near the divergence point. As far as the author is aware, there are no results that indicate that s-cracks were present at or near the divergence points in tempered glass. The assumption that s-cracks are generated between the tensile stress and neutral point seems to be unreasonable in reality.

In case 2, the s-crack occurs within the compressive stress, that is, it exists between the neutral point and glass surface. In this case, since cracks cannot propagate under compressive stress, s-cracks do not propagate and remain stopped. In addition, cracks cannot propagate into the tensile stress. For this reason, it is extremely difficult to connect with the p-crack. Even if it can be connected, it will have a special mark at the connection point with the p-cracks propagate under tensile stress. However, there are no reports that such a mark was observed by fractographic analysis. In other words, it is extremely difficult to explain the divergence mechanism even if it is assumed that cracks are generated under compressive stress. In other words, it is extremely difficult to explain the divergence mechanism even if it is assumed that s-cracks are generated under compressive stress and the tips are under compressive stress. Although the authors have performed many high-speed photographs of crack propagation phenomena in tempered glass, the occurrence of s-cracks has never been observed.

From the above results, the occurrence of s-cracks which are stated in “the concept based on the union of p- and s-cracks which were not generated on the extension of the p-crack” [10] must be denied in crack divergence of tempered glass. On the other hand, there is a high possibility that “the concept as a means to release elastic energy after reaching the critical velocity” [11] can be supported. However, as mentioned in Section 3.2, the bifurcation and branching phenomena difference and deviation of divergence angle cannot be explained by the above concept only. It is necessary to conduct a more detailed study based on this concept.

4. Conclusions

As a part of investigating the crack divergence phenomenon in tempered glass, the divergence pattern and angle were mainly considered. As a result, the following findings were obtained:

As for the divergence pattern of tempered glass, it was observed that branching exists in addition to the bifurcation described previously.

The divergence angle of tempered glass was smaller than the value calculated from the principle of optimal design and showed an acute angle.

The divergence angle of the bifurcation was calculated at about 24° assuming that the total energy before and after divergence and the divergence angle of the two cracks after divergence are the same, and the effect of the principal stress in the tempered glass is extremely small.

The occurrence of s-cracks which is stated in “the concept based on the union of p- and s-cracks which were not generated on the extension of the p-crack” [10] must be denied in crack divergence of tempered glass.

Footnotes

Acknowledgements

The author is grateful to Professor Emeritus A. Toshimitsu Yokobori, Jr of Tohoku University for making suggestions, Professor Emeritus Kiyoshi Takahashi, the late Mr. Haruo Komatsu, Mr. Toshio Mada of Kyushu University for making suggestions and performing experiments, Mr. Masashi Kikuta for providing Weibull analysis soft, and Central Glass Company Ltd. for providing glass specimens.

Conflict of interest

None to report.

References

Akeyoshi

Kanai

Yamamoto

and Shima

, Study on the physical tempering of glass plates (Part 1) - Effects of thermal conditions on permanent stress and fragment density of tempered glass plates, Rept. Res. Lab., Asahi Glass Co., Ltd. 17 (1967), 23–36.

Aoki

Shimokawa

Hirano

and Sakata

, Study on the dynamic crack propagation in brittle materials, JSME 41 (1975), 1942–1948, (in Japanese).

Aratani

, Estimation of the stress σ_CR generated at propagating crack front in zone-tempered glass, J. Ceram. Soc. Jpn. 126(4) (2018), 246–252.

Aratani

Oginoh

and Takatsu

, Crack propagation in tempered glass, J. Mater. Sci. Jpn. 40(448) (1991), 46–50 (in Japanese).

Aratani

Tahara

Ohmi

and Nishi

, Crack propagation and stress field change in zone-tempered glass, J. Jpn. Soc. Strength Frac. Materi. 48(1) (2014), 1–9 (in Japanese).

Aratani

Yamauchi

Kusumoto

and Takahashi

, Dynamic fracture process of thermally tempered glass studied by the method of caustics, J. Ceram. Soc. Jpn., Int. Edition 101 (1993), 783–786.

Aratani

Yamauchi

Oginoh

and Takahashi

, Generation of secondary cracks in tempered soda-lime-silica glass, J. Ceram. Soc. Jpn. 105(9) (1993), 506–507.

Aratani

Yamauchi

and Takahashi

, Crack velocity in thermally tempered glass, J. Ceram. Soc. Jpn. 105(9) (1997), 789–794 (in Japanese).

Hara

, Fracture phenomena of glass, J. Mater. Sci. Soc. Jpn. (1) (1964), 21–26 (in Japanese).

10.

Hara

, Understanding of mechanical properties and the estimation methods, in: Mechanical Properties of Ceramics, Yogyo-Kyokai, Japan, 1979, pp. 49–57 (in Japanese).

11.

Hara

and Shinkai

, Fracture of glass, J. Mater. Sci. Soc. Jpn. 10 (1973), 34–40 (in Japanese).

12.

Oka

, Rheology, Shohkabo (1974), 424–431 (in Japanese).

13.

Takahashi

, Trial of multi-spark camera by pulse control, J. Jpn. Soc. Aeronautical Space Sci. 9(331) (1981), 447–452 (in Japanese).

14.

Takahashi

Aratani

and Yamauchi

, Dynamic fracture in zone-tempered glasses observed by high-speed photoelastic colour photography, JMS Letters 11 (1992), 15–17.

15.

Takatsu

Aratani

Hibino

and Mishima

, Effect of local residual stress on crack propagation in thermal tempered glass, J. Mater. Sci. Jpn. 33(375) (1984), 1540–1544 (in Japanese).

16.

Takatsu

and Ikeda

, An experimental observation of tempering of flat glass, Yogyo-kyokai-shi 84(1) (1976), 19–23 (in Japanese).

17.

Toshimitsu Yokobori

, Fracture pattern of rock, Private communication.

Crack divergence phenomena in tempered glass

Abstract

Keywords

1. Introduction

2.1. Specimens

Table 1 Chemical composition of glass specimen SiO2 Al2O3 CaO MgO Na2O K2O Others 72 2 8 3 13 1 1

2.3. Divergence angle measurement

3. Results and discussion

3.1. Divergence pattern in tempered glass

4. Conclusions

Footnotes

Acknowledgements

Conflict of interest

References

Table 1
Chemical composition of glass specimen

SiO₂ Al₂O₃ CaO MgO Na₂O K₂O Others

72 2 8 3 13 1 1