Paper

Journal of the Ceramic Society of Japan 118 [9] 792-798 2010

Effect of B 2 O 3 content on crack initiation under
Vickers indentation test
Yoshinari KATO,*,**,³ Hiroki YAMAZAKI,* Yoshihiro KUBO,** Satoshi YOSHIDA,**
Jun MATSUOKA** and Tomoko AKAI****Technical Division, Nippon Electric Glass Co., Ltd., Otsu, Shiga 520–8639**School of Engineering, The University of Shiga Prefecture, Hikone, Shiga 522–8533***Research institute for Innovation in Sustainable Chemistry, National Institute ofAdvanced Industrial Science and Technology (AIST), Ikeda, Osaka 562–8577

Eﬀect of B2O3 content on crack resistance was investigated by indentation tests of glass samples with various compositions of
B2O3. The ternary SiO2­B2O3­Na2O glass system (SBN series) and non-alkaline aluminoborosilicate glass system (SAB series).
When B2O3 is substituted with SiO2 in the SBN system (“SBN1” series), crack resistance has a relationship with density. In aseries of the SBN system where density did not change with B2O3 content (“SBN2” series), crack resistance decreased withincreasing B2O3 content. On the other hand, crack resistance increased with increasing B2O3 content in the SAB series, where
density did not change. According to the results of NMR measurement, boron in 4-cordination state ([4]B) increased in the SBN2
series while boron in 3-cordination state ([3]B) increased in the SAB series with increasing B2O3 content. Therefore, crack
resistance increases with increasing [3]B and decreases with increasing [4]B. The diﬀerence in structure between [3]B and [4]Bcontaining glasses leads to diﬀerent eﬀect on residual stress around the indentation, resulting in diﬀerence in crack resistance.
©2010 The Ceramic Society of Japan. All rights reserved.

Key-words : Oxide glass, Crack initiation, Indentation, B2O3
[Received March 25, 2010; Accepted June 17, 2010]

1.

Introduction

Crack initiation and crack propagation are the most importantfactors determining strength of glass. It is well known that glass
shows very high theoretical strength close to 10 GPa. However,once a crack initiates in the glass surface, stress concentration
occurs at the crack tip, resulting in catastrophic fracture byapplied stress much lower than the theoretical strength. So
evaluation of crack initiation and crack propagation is of muchinterest for glass engineers and glass scientists.
Crack propagation is usually evaluated by measuring fracture
toughness, KIC. It is expected that glass with higher KIC has
higher breaking strength. Although various glass compositions
show almost the same value of KIC, there is much diﬀerence insusceptibility of fracture in industrial and practical use among
glass compositions. It is supposed that the diﬀerence is due to adiﬀerence of crack initiation. Therefore, both characteristics of
crack initiation and crack propagation must be evaluated forunderstanding glass strength.
Some methods for evaluating crack initiation of glass were
proposed. Wada et al.1) proposed “crack resistance (CR)” of glass
against Vickers indentation. When a glass sample is indented
with a Vickers diamond indenter, radial cracks initiate around theindentation at a given indentation load. The value of CR isdeﬁned as the load for the radial cracks to initiate, indicating thediﬃculty in crack initiation. On the other hand, Lawn et al.2)
proposed the ratio of hardness to fracture toughness as a simple
index of brittleness, and Sehgal et al.3) developed a convenientmethod to estimate the “brittleness” index of glass by using theratio of characteristic crack length to the length of indentation
diagonal.
³

Corresponding author: Y. Kato; E-mail: ykato@neg.co.jp792

In our previous study,4) CR of various commercial glass
compositions are investigated by using Vickers indentations, andcompared with other mechanical properties. The value of CR
does not have a clear relationship with KIC, but it has a clear
relationship with recovery of indentation depth (RID). The valueof RID is calculated by using a diﬀerence between indentation
depth before and after heat-treatment, and it shows degree of
densiﬁcation. The value increased with increasing RID. It is
concluded that densiﬁcation should reduce residual stress aroundthe indentation to prevent cracks from initiating.
As mentioned above, a clear relationship between CR andconventional mechanical properties has not been found, but it
was found that there is diﬀerence in CR between glass with more
B2O3 content and glass with less B2O3 content. The relationshipbetween CR and KIC is shown in Fig. 1. There is a trend that CR
increases with increasing KIC among the glass containing lessthan 5% B2O3. On the other hand, glasses containing more B2O3
show higher CR compared with this trend. Therefore, it is
supposed that B2O3 content may increase CR. However, there is
no report on the eﬀect of B2O3 content on CR. In the present
study, CR of glass in the ternary system SiO2­B2O3­Na2O system(“SBN” series) and in non-alkaline aluminoborosilicate glass
system (“SAB” series) is evaluated, and eﬀect of B2O3 content
on CR is discussed.

2.
2.1

Experimental

Preparation of glass sample

Three glass compositional series were selected in this study.
Two of them are simple ternary series of SiO2­B2O3­Na2O (SBN
series). The compositions in the SBN series employed in thisstudy are shown in Fig. 2. The “SBN1” series is expressed by(80 ¹ 40x)SiO2­40xB2O3­20Na2O (x = 0, 0.125, 0.25, 0.5, 0.75,
and 1.0) and the “SBN2” series is expressed by 80SiO2­©2010 The Ceramic Society of Japan

JCS-Japan

Journal of the Ceramic Society of Japan 118 [9] 792-798 2010

Table 1. Batch compositions of the samples in the SBN series and theSAB series. The SBN1 series and the SBN2 series are expressed by theequations of (80 ¹ 40x)SiO2­40xB2O3­20Na2O and 80SiO2­10yB2O3­(20 ¹ 10y)Na2O, respectively. The SAB series is expressed approximately by the equation (70 ¹ z)SiO2­10Al2O3­(10 + z)B2O3­6CaO­
3SrO­1BaO

10000
B2O3 > 5 %B2O3 < 5 %CR / gf

1000

SrO
BaO
SiO2 B2O3 Na2O Al2O3 CaO(mol %) (mol %) (mol %) (mol %) (mol %) (mol %) (mol %)

100

10
0.5

0.6

0.7

0.8

0.9

1.0

1/2

K IC / MPa · m

Fig. 1. A relationship between fracture toughness KIC and crack
resistance CR. This ﬁgure is reproduced after Kato et al.4)

100 0
20

80

40

60

80
75
70
60
50
40

0
5
10
20
30
40

20
20
20
20
20
20

®
®
®
®
®
®

®
®
®
®
®
®

®
®
®
®
®
®

®
®
®
®
®
®

SBN2 series
y = 0.080
y = 0.580
y = 1.0
80

0
5
10

20
15
10

®
®
®

®
®
®

®
®
®

®
®
®

SAB systemz = 0.070
z = 2.5
67.5
z = 5.0
65

10
12.5
15

®
®
®

10
10
10

6
6
6

3
3
3

1
1
1

B2O3 /mol%

SiO2 /mol%

60

40

80

20

0
100

SBN1 series
x = 0.0
x = 0.125
x = 0.25
x = 0.5
x = 0.75
x = 1.0

100
80

60

40

20

0

Na 2 O/mol%
Batch compositions of samples studied in the SBN series.
Solid circles denote glass composition in the (80 ¹ 40x)SiO2­40xB2O3­
20Na2O series, “SBN1” (x = 0, 0.125, 0.25, 0.5, 0.75, and 1.0). Opensquares denote glass composition in the 80SiO2­10yB2O3­(20 ¹ 10y)Na2O series, “SBN2” (y = 0, 0.5, and 1.0).
Fig. 2.

viscosity of the melt. For the SAB series, the raw material SiO2,
Al2O3, B2O3, CaCO3, SrCO3, BaCO3, and other minor components were used. The powders were melted at 1500°C for about
22 h. After the melting mentioned above, the glass melts werepoured onto a carbon plate, and then placed in an electric furnace
to cool slowly. Glass transition temperature, Tg, was determinedby a dilatometer. The cooled glass was heated up to thetemperature of (Tg + 30°C), held for 30 min, and then cooled by
3°C/min to obtain annealed glasses. Samples of the glasses wereground, lapped with Al2O3 slurry, and then ﬁnished with cerium
oxide to get optically smooth surfaces, which were used for thefollowing indentation tests.

2.2
10yB2O3­(20 ¹ 10y)Na2O (y = 0, 0.5, and 1.0). Phase separation
was not observed with microscope in all of the samples preparedin this study. The other system is non-alkaline aluminoborosilicate system, “SAB series”. The glass compositions in thisSAB series correspond to those obtained by substituting SiO2
with B2O3 by 2.5 mol % or 5.0 mol % in composition of acommercial non-alkaline aluminoborosilicate glass. The generalcomposition of the commercial glass is as follows: 70SiO2,
10Al2O3, 10B2O3, 6CaO, 3SrO, 1BaO (by mol %), and otherminor components. Therefore, the glass composition in the SAB
series investigated in this study is expressed approximately by
the equation of (70 ¹ z)SiO2­10Al2O3­(10 + z)B2O3­6CaO­
3SrO­1BaO (z = 0.0, 2.5, and 5.0). Each glass composition isrewritten in molar percentage in Table 1.
Glass samples were prepared by melting the batch of highgrade raw materials which are used for manufacturing glass forelectrical use. For the SBN system, the powders of the raw
material SiO2, B2O3, and Na2CO3 were weighed appropriately,
and a little of reﬁning agent corresponding to a few tenth
percentage was added to prepare batches. The batches weremixed thoroughly and melted in a 500 cc Pt crucibles at 1100 to
1500°C for about 6 h in an electric furnace. The melting
temperature for each composition was chosen according to the

Measurement of glass properties

The properties, that is, density (µ), glass transition temperature(Tg), Young’s modulus (E), and bulk modulus (K), are measured.
Density was measured by Archimedes method. Young’s modulus
and shear modulus (G), were determined by cube-resonancemethod.5) Bulk modulus was calculated by using the values of E
and G. The Vickers hardness was measured with a microhardness
tester (MXT50, Matsuzawa Seiki Corp., Japan). The applied load
was 100 gf.
The value of CR was measured by Vickers indentation tests.1)
The glass sample was indented by a Vickers diamond indenter inair (25°C, 30% relative humidity), and the corners where radialcracks appeared were counted using an optical microscope.
When glass sample is indented, various types of crack initiate
around the indenter. Only radial cracks are counted for
determining crack resistance, because the cracks normal to glass
surface are critical to fracture of glass. The percentage of crack
initiation was determined as the ratio of the number of thecorners with the cracks to the total number of the corners ofindentations. The applied load was increased step-by-step andtwenty indentations were made for each applied load. The load at
which the percentage would be 50% is determined as “crack
resistance” or CR. The details of the method to measure CR aredescribed in our previous paper.4)
793

JCS-Japan

Kato et al.: Effect of B2O3 content on crack initiation under Vickers indentation test

Knoop indentation (100 gf)

Table 2. Density (µ), glass transition temperature (Tg), Young’s
modulus (E), and bulk modulus (K), Vickers hardness (Hv), and crack
resistance (CR) of the samples in the SBN series and the SAB seriesµ(g/cm3)

Tg
(°C)

E
(GPa)

K
(GPa)

Hv
(GPa)

CR
(gf)

SBN1
x = 0.0
x = 0.125
x = 0.25
x = 0.5
x = 0.75
x = 1.0

2.39
2.45
2.49
2.52
2.49
2.44

483
522
553
571
554
527

60.0
68.5
77.0
82.0
79.4
74.6

35.7
43.2
44.8
50.6
47.3
46.1

3.7
4.5
4.9
5.4
5.1
4.9

1840
270
70
30
40
70

SBN2
y = 0.0
y = 0.5
y = 1.0

2.39
2.42
2.40

483
549
598

60.0
72.3
77.0

35.7
41.9
43.1

3.7
4.2
5.1

1840
940
360

SAB systemz = 0.0z = 2.5
z = 5.0

2.49
2.49
2.48

709
692
676

71.4
69.7
68.4

46.6
46.1
46.0

5.6
5.6
5.5

1150
1420
2030

Recovery of indentationd before
Heat-treatment(0.9 x Tg - 2hr)

depth (RID):d− d after∆d
RID = before=
d befored befored after∆dFig. 3.

Schematics of measuring recovery of indentation depth or RID.

Densiﬁcation was evaluated by shrinkage of indentation
through heat-treatment.6)­9) The shrinkage is attributed toannealing recovery of densiﬁed glass under high compressive
stress.10) The procedure to measure the recovery of indentation
depth is shown in Fig. 3. A glass sample was indented with a
Knoop indenter at an applied load of 100 gf, and the depth of theindentation was measured by an atomic force microscope, AFM(Nanoscope IIIa Degial Instrument, USA). A Knoop indenter wasused because of its tendency to inhibit crack formation. The
indented sample was heat-treated at temperature of 0.9 © Tg (in°C) for 2 hours, and the indentation depth was measured again.
The ratio of the depth change to the depth before the heattreatment was deﬁned as recovery of indentation depth or RID as
the following equation,
RID ¼ ðdbefore  dafter Þ=dbefore ¼ d=dbefore :ð1ÞHere, dbefore and dafter are Knoop indentation depths before andafter the heat-treatment, respectively, and ¦d is diﬀerence
between the depths. The details of the method are also describedin our previous paper.4)

2.3

NMR measurement

By using NMR, coordination state of boron was determined. It
is well known that there is two types of coordination state ofboron in glass: 3-coordination state ([3]B) and 4-coordination
state ([4]B). 11B NMR spectra were collected with a Chemmagnetics CMX-200 spectrometer (4.7 T) at a resonance frequency of
64.25 MHz. The 90° pulse length is 4.5 ¯s. All spectra wererecorded using 0.5 ¯s pulse excitation. Delay time of 3 s wasused. The fractions of 4-copordinated boron, N4, and 3coordinated boron, N3, were obtained by simulating the NMR
line shape with a home-built software developed by National
Institute of Advanced Industrial Science and Technology (AIST),
Japan.11)

3.

Results

Results of µ, Tg, E, K, and Hv are shown in Table 2. In the
SBN1 series, density increases with increasing x, shows amaximum at x = 0.5, and then decreases. The values of Tg, E, K,
and Hv show maxima at x = 0.5. In the SBN2 series, there is nochange in density as a function of y. The values of Tg, E, K, and
Hv increases with increasing y (increasing B2O3). In the SAB
series, density does not change with increasing z, similar to the
SBN2 series. However, unlike the SBN2 series, the values of Tg,
E, and K decrease with increasing z.
The relationship between indentation load and percentage ofcrack initiation in the SBN1, the SBN2, and the SAB series areshown in Figs. 4(a), (b), and (c), respectively. The percentage ofcrack initiation increases with increasing the applied load. From
794

Experimental uncertainties are as follows. d: «0.01 g/cm3, Tg: «2°C,
E: «1 GPa, K: «1 GPa, Hv: «0.3 GPa.

these ﬁgures, the load at which the percentage would be 50% is
determined as crack resistance or CR. The results of CR are also
shown in Table 2. Relationships between CR and each compositional parameter (x, y, and z) are shown in Fig. 5. Results of µ are
also plotted in this ﬁgure for comparison. In the SBN1 series, CR
decreases with increasing x, until x = 0.5. On the other hand, CR
increases with increasing x in the region of x > 0.5. Therefore,
CR shows a minimum at x = 0.5. Note that the trend of CR is
opposite to that of µ. According to Sehgal et al.,12) “brittleness”
index, or susceptibility of cracking, has strong relationship withdensity. Brittleness of normal glass with density of more than
2.4 g/cm3 increases with density. The trend agrees with thatof the SBN1 series because higher CR corresponds to low
brittleness index. On the other hand, there is little change indensity in both the SBN2 series and the SAB series. However,
the trend of CR with increase of B2O3 content is diﬀerent
between these two series. The value of CR decreases withincreasing y in the SBN2 series while CR increases withincreasing z.
The diagonal length of Vickers indentation, 2a, the depth of
Knoop indenter before and after the heat-treatment, dbefore anddafter, and RID are shown in Table 3. In the SBN1 series, RID
decreases with increasing x until x = 0.5, shows a minimum at
x = 0.5, and then increased. This trend is similar to that of CR,
and opposite to that of µ. In the SBN2 or the SAB series, RID
does not change much with increasing y or z, respectively,
although we found the changes in CR in both series.
Examples of NMR spectra in the SBN1, the SBN2, andthe SAB series are shown in Fig. 6. These spectra of glass
containing B2O3 consist of two signals. One is a sharp signal(signal A in the ﬁgure) and the other is the broad doublet signal(signal B ³ B¤). The former is originated from the four-coordinated boron, [4]B, and the latter from the three-coordinated boron,[3]
B. Large amount of boron is found as [4]B in the SBN1 and
SBN2 series while large amount of boron is found as [3]B in the
SAB series. The ratio of the amount of [4]B or [3]B to the total
amount of B, N4 or N3, respectively, is measured by deconvoluting the spectra and integrating each area. The values of N4 and N3

JCS-Japan

2.50

x
0.0
0.125
0.25
0.5
0.75
1.0

50

100

2.40

10

2.35

0
1000

10000

0.5

0.75

x

(a) SBN1 series
2.55yρ / g · cm-3

50

0.0
0.5
1.0
0
1

10

100
Load /gf

1000

2.50

1000

2.45

100

2.40

10

2.35

1
0

10000

0.5

1

y

(b) SBN2 series(b) SBN2 series
SAB :(70-z )SiO2-10Al2O3-(10+z )B2O3-6CaO-3SrO-1BaO

ρ / g · cm-3

100

50

2.55

10000

2.50

1000

2.45

100

z

2.40

0.0
2.5
5.0

2.35

10

ρ
CR

1
0

0
500

10000

ρ
CR

SBN2 : 80SiO2 - 10y B2O3 - (20-10y )Na2O

100

1CR / gf100

0.25

CR / gf

1010(a) SBN1 series
Percentage of crack initiation(%)

1000

2.45

Load / gf

Percentage of crack initiation(%)

10000

ρ
CR

CR / gf

2.55

SBN1 : (80-40x )SiO2-40x B2O3-20Na2O

100

ρ / g · cm-3

Percentage of crack initiation(%)

Journal of the Ceramic Society of Japan 118 [9] 792-798 2010

2.5

5

z
1000

1500

2000

Load / gf(c) SAB series
Fig. 4. A relationship between the applied load and the percentage ofcrack initiation. (a) SBN1 series, (b) SBN2 series, and (c) SAB series.

are shown in Table 4. Dell et al.13) showed that the ratio of [4]B
can be predicted from the combination of simple structuralgroups. For rNa2O­B2O3­kSiO2 glasses (k ¼ 8), N4 is r, whenr ¼ 1/2 + K/16, and N4 is 1/2 + K/16, when 1/2 + K/16 ¼
r ¼ 1/2 + K/4. The results calculated by this equation are also
shown in Table 4. The results in this study agree well withcalculated values. The amount of the [4]B unit or the [3]B unit isdetermined as the molar content of B2O3 multiplied by N4 or N3,
and denoted as B[4] or B[3], respectively. The amounts of B[3]and B[4] are also shown in Table 4, and plotted against eachcompositional parameter in Fig. 7. In the SBN1 series, B[4]increases with increasing x until x < 0.5, while B[3] does notincrease. On the other hand, in the region of x > 0.5, B[3]increases with increasing x more than B[4]. It is well known that(c) SAB series
Change of density and crack resistance as a function of glass
compositional parameter (x, y, and z). (a) SBN1 series, (b) SBN2 series,and (c) SAB series.
Fig. 5.

the coordination state of boron changes by an addition of alkalineoxide as below:½BO3  þ 1=2Na2 O ) ½BO4  þ Naþ :¹

ð2Þ

Here, [BO3] and [BO4] indicate the B triangle unit and the[4]
B tetrahedral unit, respectively. Since the molar amount of
Na2O is more than that of B2O3 in the region of x from 0.0 to
0.5, the added amount of B2O3 changes to [4]B unit. At thecomposition of x = 0.5 (that is, 60SiO2­20B2O3­20Na2O), theamount of B2O3 equals to that of Na2O, so the [3]B unit increases
more in the range of x > 0.5 because the amount of Na2O is notenough to convert all the B[3] into B[4] through Eq. (2). Since
the amount of Na2O is much more than that of B2O3 in the SBN2
series, the [4]B unit increases with increasing the B2O3 content.
[3]

795

JCS-Japan

Kato et al.: Effect of B2O3 content on crack initiation under Vickers indentation test

Vickers indentation
2a
(¯m)
SBN1
x = 0.00
x = 0.125
x = 0.25
x = 0.5
x = 0.75
x = 1.00

Knoop indentation
dafterRID
dbefore
(¯m)
(¯m)
(%)

22.1
20.0
19.2
18.4
18.8
19.2

1.20
1.01
0.99
0.91
0.91
0.92

®
0.73
0.74
0.80
0.72
0.68

®
0.28
0.25
0.13
0.21
0.27

25
B [3] or B [4] / mol%Table 3. The diagonal of Vickers indentation (2a), the depth of Knoop
indentation before and after the heat-treatment at 0.9 © Tg (in °C) for 2 h,
(dbefore and dafter), and recovery of indentation depth (RID). Both of theindentations are made at the load of 100 gf

B[3]
B[4]

20
15
10
5
0
0.0

0.5

1.0

x

(a) the SBN1 series
22.1
20.7
18.9

1.20
1.02
0.94

®
0.697
0.625

®
0.32
0.34

SAB systemz = 0.0z = 2.5
z = 5.0

18.0
18.0
18.3

0.99
1.03
0.98

0.61
0.59
0.56

0.39
0.42
0.42

Experimental uncertainties are as follows. 2a: «0.3 ¯m, dbefore or dafter:«0.03 ¯m, RID: «3%.

10B[3]B [3] or B [4] / mol%SBN2
y = 0 (x = 0.0)
y = 0.5
y = 1.0

B[4]

5

0
0.0

5 kHz

0.5

1.0

y

A

(b) the SBN2 series
Intensity (a.u.)(a) SBN1 ( x =0.5 )

B[3]

B

A

B'

B [3] or B [4] / mol%(b) SBN2 ( y =1.0 )(c) SAB ( z = 5.0 )

B[4]
10

5

0

Freaquency
Examples of NMR spectra in the SBN1 series, the SBN2 series,
and the SAB series. (a) SBN1 series (x = 0.5), (b) the SBN2 series(y = 1.0), and (c) the SAB series (z = 5.0).
Fig. 6.

Table 4. The ratio of 4-coordinated boron and 3-coordinated boron,
N4 and N3, and the amount of the [4]B unit or the [3]B unit. The valuesof the column “N4 (Dell)” are calculated according to the estimation byDell et al.12)
B2O3(mol %)

N4

N3

B[4]
(mol %)

B[3](mol %)

N4(Dell)

x = 0.125
x = 0.25
x = 0.5
x = 0.75
x = 1.00

5
10
20
30
40

1.00
1.00
0.80
0.61
0.49

0.00
0.00
0.20
0.39
0.51

5.0
10.0
16.0
18.3
19.6

0.0
0.0
4.0
11.7
20.4

0.94
0.69
0.60
0.50

y = 0.5
y = 1.0

5
10

1.00
0.82

0.00
0.18

5.0
8.2

0.0
1.8

®
®

SAB 0.0
SAB 2.5
SAB 5.0

10
12.5
15.0

0.05
0.07
0.06

0.93
0.93
0.94

0.5
0.9
0.9

9.5
11.6
14.1

®
®
®

796

15

A

0.0

2.5

5.0

z

(c) the SAB series
Fig. 7. A relationship between each compositional parameter and theamount of B[3] or B[4]. (a) the SBN1 series, (b) SBN2 series, and (c) the
SAB series. B[3] and B[4] denote the molar fractions of [3]B unit and [4]B
unit, respectively.

In the SAB series, which contains no alkaline ion, B[3]increased with increasing B2O3. The [3]B­[4]B conversion reaction
would also occur in non-alkaline glass due to the reaction withalkaline-earth ion (R¤2+) according to the following equation,
2½BO3  þ R0 O ) 2½BO4  þ R02þ :ð3Þ

Yamashita et al. showed that alkaline-earth atom reacts more
preferably with Al2O3, not with B2O3 in non-alkaline aluminoborosilicate glass.14) In the SAB series, the amount of Al2O3 is
15% and the total amount of R¤O is about 10%, so it is estimatedthat all of R¤O reacts with Al2O3. Therefore, the reaction shown in
Eq. (3) does not occur and then B[3] increases with increasing
B2O3.

JCS-Japan

Journal of the Ceramic Society of Japan 118 [9] 792-798 2010

Table 5. Regression coeﬃcient, ¢(NWF), of logarithm of CR with four
variable, e.g., the molar fraction of [3]B2O3, [3]B2O3, SiO2, and Al2O3, andintercept, C, obtained by multivariable linear regression analysis

10000
SBN1
SBN2
SAB system
Commertial glass [3]CR / gf

1000

100

Regression coeﬃcient, ¢(NWF)
[3]
B2O3
[4]
B2O3
SiO2
Al2O3

+0.098¹0.092+0.038¹0.067

Intercept, C+0.17910

1.5
2.0
Estimated residual stress, σ / GPa

2.5

Fig. 8. A relationship between CR and the estimated residual stress, ·,around the indentation at a load of 100 gf. The results of commercial
glasses are the results in the previous study.4)

As mentioned above, CR has strong relationship with density.
In the case of the glass series without density change, eﬀect of
B2O3 on CR depends on the glass series. Comparing Fig. 5 with
Fig. 7, it is supposed that CR increases with increasing B[3] orwith decreasing B[4].

4.

4.0
SBN1
log 10 (CR / gf)

1
1.0

3.0

SAB
2.0
1.0
0.0
0

In the previous study, we found that CR has a clear relationshipwith densiﬁcation, or RID.4) This should be due to the fact thatdensiﬁcation decreases the residual stress beneath the indentation,15) resulting in the increase in crack resistance. The residual
stress can be derived from a simple cavity model.16) According tothis model, the residual stress, ·, can be expressed as follows;
ð4Þ

Here, · is the estimated residual stress, K is the bulk modulus,
and "V the indentation volume and V the volume of plastic zone
around the indentation. By modifying Eq. (4), the residual
stresses around the indentation of various commercial glasses
were estimated in the previous study.4) In the previous study, thefollowings are assumed for the modiﬁcation; (1) the diameterof hemispherical plastic zone is equal to that of the Vickers
indentation diagonal, 2a,17) (2) "V equals to the volume of theindentation after the heat-treatment, and (3) dafter, which is
depth of Knoop indentation after the heat-treatment at the RID
measurement, is proportional to the value of dVafter, which is
depth of Vickers indentation after the heat-treatment for all of theglass compositions (dVafter = 1.34 · dafter). Finally, the residual
stress around the indentation was estimated to be as follows:· ¼ 0:427  K  dafter =a:

1

2

3

4

∑ β (NWF) · R (NWF) + C

Discussion· ¼ K  V =V :

SBN2

ð5Þ

Details of the assumption were described in our previous paper.4)
The calculated residual stress is shown in Fig. 8. The results ofthe commercial glass in the previous study4) are also shown inthis ﬁgure for comparison. The value of CR decreases withincreasing the residual stress in the SBN series and the SAB
series, similar to the results of commercial glasses.
As shown in Eq. (5), the residual stress is aﬀected by bulk
modulus and the value of dafter/a. The value of dafter/a correlateswith the ratio of the cavity radius to the radius of plastic zone.
The smaller value of 1/a, or the smaller hardness, indicates themore plastic deformation, while the smaller dafter indicates larger
annealing recovery, or less shear ﬂow. Therefore, the value ofdafter/a is one measure of contribution of densiﬁcation. The larger
contribution of densiﬁcation corresponds to the smaller value ofComparison between the value of log10 CR and the value of­¢(NWF)·R(NWF). Here, ¢(NWF) and R(NWF) are regression coefﬁcient and molar fraction of network component NWF, respectively, andC is an intercept. The regression coeﬃcients of ¢(NWF) are obtained bymultivariable linear analysis.
Fig. 9.

dafter/a, resulting in the smaller residual stress which would be
driving force of crack initiation. The RID values measured in thisstudy were also another measure of densiﬁcation, which is
proportional to the value of dafter/a.
As mentioned above, it seems that eﬀect of B2O3 on CR
depends on the coordination state of boron. However, since
network structure should aﬀect CR, eﬀects of SiO2 and Al2O3 also
have to be taken into account. Then, by multivariable linear
regression analysis for logarithm of CR, regression coeﬃcients ofmolar fraction of [3]B2O3, [4]B2O3, SiO2, and Al2O3 were obtained,
which indicate eﬀect of each component on log10 CR. That is, thevalue of log10 CR is approximated as the following equation;
log10 CR ¼ ¢ðNWFÞ  RðNWFÞ þ C;

ð6Þ

where ¢(NWF) and R(NWF) are regression coeﬃcient and molar
fraction of the network component NWF, respectively, and C is
an intercept. The regression coeﬃcients and the intercept arelisted in Table 5. The value of log10 CR is plotted against thevalue of ­¢(NWF) · R(NWF) + C, in Fig. 9. The correlationcoeﬃcient for the analysis of 0.99 is obtained. The eﬀects of[3]
B2O3 and SiO2 are positive while the eﬀects of [4]B2O3 andAl2O3 are negative. It indicates that [3]B2O3 and SiO2 increase CR
and that [4]B2O3 and Al2O3 decrease.
It is considered that the large eﬀect of [3]B2O3 is due to thefollowing three factors: single bond strength, openness, andplanar structure. Since crack initiation corresponds to break of
network bond, it is reasonable that high single bond strength
increase CR. The order of single bond strength with oxygen:18)[3]
B­O (499 kJ/mol) > Si­O (444 kJ/mol) > Al­O (293­423
kJ/mol) > [4]B­O (373 kJ/mol) agree with that of the eﬀect on
log10 CR. However, the single bond strength is not much
diﬀerent among them. Therefore, the openness and the planar
797

JCS-Japan

Kato et al.: Effect of B2O3 content on crack initiation under Vickers indentation test

structure of [3]B2O3 unit have more eﬀects on CR than the bondstrength. The open structure leads to an easy densiﬁcation byindentation, resulting in a small residual stress. The tetrahedra of[4]
BO4 and AlO4 are accompanied with modiﬁer cations in theglass network for charge compensation, to make the network
more packed. On the other hand, the SiO4 tetrahedron and the[3]BO3 planar triangle do not need such charge compensation, so
they make open network structure. The speciﬁc planar structureof [3]B unit would also contribute to a large densiﬁcation. The
introduction of the [3]B planar triangle unit into main SiO2
network makes the network less rigid, which allows densiﬁcation
under a stress. It is deduced that the open network and the planar
structure cause an easier densiﬁcation. For further discussion, it
will be needed to get an insight into the intermediate range orderof [3]B2O3 containing glass under a stress.

5.

2)
3)
4)
5)
6)
7)
8)
9)

Conclusion

By using glasses in the SiO2­B2O3­Na2O ternary (SBN) series
and non alkaline aluminoborosilicate (SAB) series, eﬀect of
B2O3 content on crack resistance (CR) was investigated. The
value of CR has a relationship with density in the SBN system. In
the SBN2 series where density does not change with B2O3
content, CR decreases with increasing B2O3. On the other hand,
CR increases with increasing B2O3 content in the SAB series
where density does not change, neither. From the results of NMR
spectra, it is found that the reason of the diﬀerence is due to thediﬀerence in the coordination state of boron. The value of CR
increases with increasing 3-coordinated boron [3]B and decreasing 4-coordinated boron [4]B. The value of CR has a relationshipwith the residual stress around the indentation. The open networkand the planar structure of [3]B unit should result in an increase incontribution of the densiﬁcation and a decrease in the residual
stress.

798

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