Paper

Journal of the Ceramic Society of Japan 119 [2] 110-115 2011

Load dependence of densiﬁcation in glass during
Vickers indentation test
Yoshinari KATO,*,**,³ Hiroki YAMAZAKI,* Satoru ITAKURA,** Satoshi YOSHIDA** and Jun MATSUOKA***Technical Division, Nippon Electric Glass Co., Ltd., Otsu 520–8639**School of Engineering, The University of Shiga Prefecture, Hikone, Shiga 522–8533

The load dependence of densiﬁcation during a Vickers indentation test was investigated for three commercial glass compositions,soda-lime silicate glass, aluminoborosilicate glass, and lead borosilicate glass, each of which exhibits markedly diﬀerent
susceptibility to crack initiation. The contribution of densiﬁcation to the total deformation due to indentation was evaluated as
the ratio of the depths of indentation before and after heat treatment measured with an atomic force microscope (AFM). For thesoda-lime silicate and aluminoborosilicate glasses, the contribution of densiﬁcation decreases with increasing applied load, but
the rate of the decrease is less for the alumonoborosilicate glass than for the soda-lime silicate glass. For the lead borosilicate
glass, the contribution of densiﬁcation is low throughout the range of loads investigated. The residual stress can be estimatedfrom the contribution of densiﬁcation, and the variation of load dependence of the residual stress is considered to result in a large
diﬀerence in the crack initiation load among the glasses.
©2011 The Ceramic Society of Japan. All rights reserved.

Key-words : Oxide glass, Crack initiation, Indentation, Residual stress[Received October 14, 2010; Accepted November 26, 2010]

1.

Introduction

Crack initiation and crack propagation are the most importantfactors determining the strength of glass. It is well known thatglass exhibits a very high theoretical strength of close to 10 GPa.
However, once a crack initiates in the glass surface, stressconcentration occurs at the crack tip, resulting in catastrophic
fracture even under an applied stress much lower than thetheoretical strength. Thus, the evaluation of crack initiation andcrack propagation is of considerable interest for glass engineersand glass scientists.
Crack propagation is usually evaluated by measuring fracture
toughness, KIC. Glass with higher KIC has higher breaking
strength for a given ﬂaw size. Although various glass compositions have almost the same value of KIC, there is a marked
diﬀerence in the susceptibility to fracture among glasses inindustrial and practical use. It is considered that this diﬀerence isdue to a diﬀerence in crack initiation behavior. Therefore, thecharacteristics of both crack initiation and crack propagationmust be evaluated to understand glass strength.
Unlike crack propagation which can be evaluated by measuring KIC, a method of evaluating crack initiation of glass has not
yet been established, although some methods have been
proposed. Wada et al.1) proposed “crack resistance (CR)” ofglass against Vickers indentation. When a glass sample isindented with a Vickers diamond indenter, radial cracks initiate
around the indentation at a given indentation load. The value of
CR is deﬁned as the load required for the radial cracks to initiate,
indicating the diﬃculty of crack initiation. On the other hand,
Lawn and Marshall2) proposed the ratio of hardness to fracture
toughness as a simple index of brittleness, and it was reported
that this “brittleness” index has a clear relationship with crack
initiation behavior.3) In addition, Sehgal et al.4) developed aconvenient method of estimating the “brittleness” index of glass³

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

110

by using the ratio of the characteristic crack length to the lengthof the indentation diagonal.
In our previous study,5) CR for various commercial glass
compositions was investigated by using Vickers indentations,and CR values were compared with other mechanical properties.
It was found that the value of CR does not have a clear
relationship with general mechanical properties, such as KIC andhardness, but it has a clear relationship with the recovery ofindentation depth (RID). The value of RID is the ratio of thechange in indentation depth due to heat treatment to theindentation depth before heat treatment, and it is a measure of
the degree of densiﬁcation occurring during indentation. Densiﬁcation is an inelastic deformation that is accompanied by achange in volume. The value of CR increased with increasing
RID in our previous study.5) It was concluded that densiﬁcation
should reduce residual stress around the indentation, thus
preventing cracks from initiating.
However, the diﬀerence in crack resistance among various
glass compositions was much larger than that in the residual
stress. Figure 1 shows the relationship between crack resistanceand the estimated residual stress around an indentation at a load of
100 gf.5) The residual stress ranges over a factor of two (0.3 to
0.6 GPa) while CR ranges over a factor of about 30 (30 to1300 gf). On the other hand, it was reported that soda-lime silicate
glass exhibits a load-dependent contribution of densiﬁcation
to deformation.6) Therefore, densiﬁcation aﬀects the residual
stress, so the load dependence of the contribution of densiﬁcation
to deformation should aﬀect cracking behavior during indentation. However, the load dependences of the contribution ofdensiﬁcation for various commercial glasses have not beenclariﬁed.
In the present study, densiﬁcation during indentation tests isevaluated at various loads using a dynamic hardness tester and an
atomic force microscope, AFM, then the eﬀect of the load
dependence of the contribution of densiﬁcation on crack
resistance is discussed.
©2011 The Ceramic Society of Japan

JCS-Japan

Journal of the Ceramic Society of Japan 119 [2] 110-115 2011

10000

CR / gf

Vickers indentation

1000
d before

100

Heat−treatment(0.9 x Tg − 2h)

Recovery of indentation
depth (RID ):
RID =

V
V
d before− d afterVd before=∆dVd befored after
∆d

10
1.0

1.5

2.0

2.5

σ rs / GPa

Schematic of method used to measure recovery of indentation
depth (RID).

Fig. 2.

Fig. 1. Relationship between crack resistance, CR, and estimatedresidual stress around indentations when indented at a load of 100 gf.5)

Table 1. General compositions of glasses and their density (µ), glass transition temperature (Tg), fracture toughness (KIC),
Young’s modulus (E), bulk modulus (K), and crack resistance (CR)5)
Name

General composition(mol %)µ(g/cm3)

Tg
(°C)

KIC
(MPa·m1/2)

E
(GPa)

K
(GPa)

CR
(gf)

C
D
G

70SiO2­10Al2O3­10B2O3­10Others
70SiO2­10Na2O­10CaO­10Others
60SiO2­25PbO­5B2O3­10Others

2.48
2.49
4.44

710
540
470

0.79
0.75
0.66

70
72
63

40
40
44

1200 (900­1400)
150 (140­170)
30 (25­40)

Molar fractions of minor components in the glass composition are omitted.
Experimental uncertainties are as follows: d: «0.01 g/cm3 (Archimedes method), Tg: «2°C (dilatometer method), KIC:
«0.05 MPa·m1/2 (SEPB method), E: «1 GPa (resonance method), and K: «1 GPa (resonance method)

2.

Experimental

Among the glass compositions investigated in the previous
study,5) three glass compositions with markedly diﬀerent valuesof CR were selected for the present study. The ﬁrst is Glass C,
aluminoborosilicate glass, whose crack resistance is 1200 gf. The
second is Glass D, soda-lime silicate glass, whose crack resistanceis 150 gf. The third is Glass G, lead borosilicate glass, whosecrack resistance is 30 gf. The general compositions and propertiesof these glasses are shown in Table 1. The methods of measuring
CR and the other properties are described in the previous paper.5)
Samples of the glasses were ground, lapped with Al2O3 slurry,
and then ﬁnished with cerium oxide to obtain optically smoothsurfaces, which were used in the following indentation test.
The contribution of densiﬁcation to the total deformation due
to indentation was evaluated by measuring the recovery ofindentation depth after heat treatment.7)­9) The recovery is
attributed to annealing recovery of densiﬁed glass under a highcompressive stress.10) The procedure for measuring the recovery
of indentation depth is shown in Fig. 2. A glass sample wasindented at various loads from 5 to 200 gf for 15 s using aVickers indenter with a dynamic hardness tester (DUH-201,
Shimadzu, Japan), and the diagonal and depth of the indentation
were measured with an AFM (Nanoscope IIIa Digital Instruments, USA). The indented sample was heat-treated at atemperature of 0.9 © Tg (in °C) for 2 h, and the indentation
depth was measured again. The ratio of the change in depth to thedepth before the heat treatment was deﬁned as the recovery ofindentation depth, RID, given by the following equation:
RID ¼ ðd V before  d V after Þ=d V before ¼ d=d V before :
V

ð1Þ

V
before and d after are the Vickers indentation depths before

Here, dand after heat treatment, respectively, and ¦d is the diﬀerence
between the depths. In our previous study,5) a Knoop indenter

was used to measure RID because it induces less crack formation
than a Vickers indenter. In the present study, however, a Vickers
indenter was used to evaluate the load dependence of RID andcrack initiation in the case of Vickers indentation. The relationship between the recovery after Knoop and Vickers indentation
was described in our previous study.5)

3.

Results and discussion

Photographs of indentations on the three glasses at a load of
100 gf are shown in Fig. 3. There are some diﬀerences in the
shape and position of cracks among the glasses, as well as in the
shape of indentation. In Glass C, which has the highest value ofCR, no radials crack initiate. In Glass D, radial cracks initiate
near some of the corners of the indentation, and the edge of theindentation are concave. In Glass G, radial cracks initiate at all
corners of the indentation. Thus, even at the same indentationload, the number of initiating cracks and the shape of theindentation diﬀer among the glass compositions.
Examples of AFM images of indentations in the glass samples
before and after heat treatment are shown in Fig. 4. The recoveryof the indentation diagonal during the heat treatment is very
limited for all the glass samples. However, a remarked recovery of
the opposite side (face-to-face) length of the indentation during
the heat treatment was found in Glasses C and D, while therecovery of the opposite side length is small in Glass G. On theother hand, the annealing recovery of indentation depth is alsosigniﬁcant in Glasses C and D. Examples of cross-section proﬁlesmeasured with an AFM are shown in Fig. 5. In this ﬁgure, thedepth recovery of Glass G is not distinct as compared with theother glasses. In all the glasses, the recovery of depth is the mostremarked than those of diagonal length and opposite-side length.
As for the soda-lime glass, it was also reported that the depth
recovery was the largest among the annealing recoveries of
diagonal length, opposite-side length, and depth.11) Therefore,111

JCS-Japan(a) Glass C

Kato et al.: Load dependence of densiﬁcation in glass during Vickers indentation test

RID would be representative of the annealing recovery ofindentation, and be one of the measures of the contribution of
the densiﬁcation under a Vickers indenter. Eﬀects of indenter
geometry of indentation recovery will be discussed later.
The indentation diagonal and RID for various applied loads areshown in Table 2, and the relationship between RID and theapplied load is shown in Fig. 6. For Glasses C and D, the RID
decreases with increasing applied load, similar to the results of
Yoshida et al.6) Note that Glasses C and D have almost the samevalue of RID at loads of up to 20 gf. However, RID for Glass D
decreases more rapidly with increasing load than that for Glass
C. Glass C has the largest value of RID at 200 gf. On the other
hand, RID for Glass G is smaller than that for the other glasses at
every indentation load.
Densiﬁcation decreases the residual stress beneath the indentation.12) Lawn et al.13) proposed a method of estimating theresidual stress using a simple cavity model. According to thismodel, the residual stress, ·rs, can be expressed as(b) Glass D(c) Glass G
Microscopic images of indentations at a load of 100 gf: (a) GlassC, (b) Glass D, (c) Glass G. The width of the images is 45 ¯m.
Fig. 3.

· rs ¼ K  Vpf =Vpz :Here, ·rs is the estimated residual stress, K is the bulk modulus,¦Vpf is the plastic ﬂow volume, and Vpz is the volume of the plastic
zone around the indentation. By modifying Eq. (2), the residual(a) Glass C (Before)(b) Glass C (After)(c) Glass D (Before)(d) Glass D (After)(e) Glass G (Before)(f) Glass G (After)(Color online) Examples of AFM images of Vickers indentations before and after heat treatment: Glass C [(a), (b)],Glass D [(c), (d)], and Glass G [(e), (f)]. The indentation load is 100 gf. The heat treatment temperature and holding time are
0.9 © Tg (°C) and 2 h, respectively.
Fig. 4.

112

ð2Þ

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Journal of the Ceramic Society of Japan 119 [2] 110-115 2011

2

2
Glass C

Glass C

Diagonal
direction1
Depth / µm

Depth / µm

1

0
−1

−10

0
−1

Before
After

−2
−20

0
Distance / µm

10

20

−10

0
Distance / µm1020(b) Glass C (Opposite−side direction)

2

2
Diagonal
direction

Glass D

Opposite−sidedirection

Glass D

1

1
Depth / µm

Depth / µm

Before
After

−2
−20

(a) Glass C (Diagonal direction)

0

−1

0

−1

BeforeAfter

Before
After

−2
−20

−10

0
Distance / µm

10

20

−2
−20

−10

0
Distance / µm1020(d) Glass D (Opposite−side direction)(c) Glass D (Diagonal direction)
2

2
Diagonal
direction

Glass G

Opposite−sidedirection

Glass G

1

1
Depth / µm

Depth / µm

Opposite−sidedirection

0

−1

0

−1

Before
After

−2
−20

Before
After

−10

0
Distance / µm1020(e) Glass G (Diagonal direction)

−2
−20

−10

0
Distance / µm

1020(f) Glass G (Opposite−side direction)

Examples of cross-section proﬁles in diagonal direction and in opposite-side deirection before and after heat
treatment. Glass C [(a), (b)], Glass D [(c), (d)], and Glass G [(e), (f)]. The indentation load is 100 gf. The heat-treatment
temperature and holding time is 0.9 © Tg (°C) and 2 h, respectively.
Fig. 5.

stresses around the indentations of various commercial glasses
were estimated in our previous study.5) Yoﬀe14) assumed that theplastic zone has a hemispherical shape with a diameter equal tothe diagonal length of the Vickers indentation. If the indentation
diagonal is 2a, Vpz can be expressed by the following equation:
V pz ¼ ð1=2Þ  ð4=3Þ  ³  a3 ¼ 2:09  a3 :ð3ÞThe plastic ﬂow volume, ¦Vpf, is assumed to equal the volume of
the indentation after the heat treatment. Lawn et al.13) assumed
that ¦Vpf was equal to the volume of the indentation without any
heat treatment. However, the part of the indentation volume
attributed to densiﬁcation contributes a little to stress generation
because the densiﬁed part of the volume does not expand theplastic zone. Therefore, assuming that the diagonal length, 2a,does not change during the heat treatment and that the projectedarea of the Vickers indentation has a square shape, ¦Vpf can beapproximated by
Vpf ¼ ð1=3Þ  S  d V after ¼ ð2=3Þ  a2  d V after :ð4ÞHere, S is the projected area of the Vickers indentation and dVafteris the Vickers indentation depth after the heat treatment.
By substituting Eqs. (3) and (4) into Eq. (2), the residual stressis expressed as· rs ¼ 0:319  K  d V after =a:ð5Þ

It is assumed that the estimated residual stress is constant over thesemi-sphere surface plastic zone. Although this cavity model is
too simple to estimate the stress distribution around theindentation, it would be possible to evaluate the load andcompositional dependences of the estimated residual stress113

JCS-Japan

Kato et al.: Load dependence of densiﬁcation in glass during Vickers indentation test

Table 2. Indentation depth before and after heat treatment (dVbefore anddVafter), recovery of indentation depth (RID), and indentation diagonallength of Vickers indentation (2a) for each type of glass
Load/gf

dVbefore/¯mdVafter/¯m

RID

2a/¯m

C

5
10
20
50
100
200

0.21
0.35
0.54
0.97
1.40
2.21

0.11
0.20
0.34
0.63
0.93
1.60

0.48
0.43
0.37
0.35
0.34
0.28

3.6
4.9
7.3
12.6
18.2
26.7

5
10
20
50
100
200

0.21
0.33
0.56
0.95
1.38
2.32

0.11
0.19
0.34
0.67
1.05
1.83

0.48
0.42
0.39
0.29
0.24
0.21

3.4
5.1
7.6
12.7
18.5
26.4

10
100
200

0.54
2.20
3.13

0.48
2.05
2.75

0.11
0.07
0.12

5.9
19.7
28.2

D

G

Experimental uncertainties are as follows. 2a: «0.5 ¯m, dbefore and dafter:«0.03 ¯m, RID: «3%.

0.5

Glass C
Glass D
Glass G

RID

0.4
0.3
0.2
0.1
0.0
1
Fig. 6.

10

P / gf

100

1000

Relationship between applied load (P) and RID.

qualitatively. The relationship between the calculated residual
stress and the applied load is shown in Fig. 7. The residual stressincreases with increasing load, not in a proportional manner but
in a logarithmic manner.
A diﬀerence between the susceptibility to densiﬁcation andthat to plastic ﬂow is considered to result in the load dependenceof the residual stress. Densiﬁcation is inelastic deformation withvolume change while plastic ﬂow is inelastic deformation withvolume conservation. Densiﬁcation proceeds through a process
in which bond angles and the dihedral angle of the glass network
are changed by external stress, while plastic ﬂow proceeds
through a process in which the bonds break and recombine
sequentially. Since changing the bond angle requires less energy
than bond breakage, the densiﬁcation should occur at a lower
stress than plastic ﬂow. In previous papers,15),16) it was reported
that the activation energy of the densiﬁcation recovery process
was much lower than that of viscous ﬂow, which indicates that
irreversible densiﬁcation occurs more easily than plastic ﬂow. On
the other hand, a Vickers indenter has a ﬁnite tip radius, such as
several hundred nanometers. This means that the maximum
pressure under the indenter tip is not inﬁnity but a ﬁnite valuethat increases with increasing load. The increased pressure results
in a larger volume of plastic ﬂow and a decrease in the114

σ rs / GPa

Glass

3.0

2.0

Glass G

1.0

Glass D
Glass C

0.0
1

10

100

1000

10000

P / gfFig. 7. Relationship between applied load (P) and estimated residual
stress (·rs).

contribution of densiﬁcation to the total deformation. With afurther increase in the load, however, the eﬀect of the tip radius
should decrease and the contribution of densiﬁcation should
approach a constant value, which is determined by the shape of
the indenter.
The packing density of ions in glass aﬀects densiﬁcation
character among glasses. The packing density, the ratio of thetotal volume of constitutive ion spheres to the volume of theglass, can be calculated from the ionic radius estimated by themethod of Pauling.17) The packing densities of Glasses C, D, andG were calculated to be 0.542, 0.544, and 0.561, respectively.
Since a glass with a higher packing density has less free volume,
Glass G exhibits the least densiﬁcation. Since the packing densityof Glass C is almost the same as that of Glass D, Glass C exhibits
similar densiﬁcation to Glass D.
The diﬀerence in the load dependence of the contribution ofdensiﬁcation among glasses is considered to be related to thecompositional variation of the relationship between pressure andthe increase in density of glass. Rouxel et al.18) investigated therelationship between hydrostatic pressure and the increase indensity of glass. With increasing applied hydrostatic pressure, thedensity of soda-lime glass does not increase until the hydrostaticpressure reaches a threshold pressure of about 10 GPa, above
which it increases steeply. On the other hand, the density of B2O3
glass increases gradually above the threshold pressure of about
5 GPa. It is considered that such diﬀerences in the pressure
dependence of hydrostatic densiﬁcation result in diﬀerences inthe distribution of densiﬁed region around an indentation, even if
two glasses exhibit the same contribution of densiﬁcation to
deformation. In the case of a glass with a low threshold pressure,the wide distribution of the densiﬁed region is thought to form.
Therefore, it is thought that the contribution of densiﬁcation ofglasses with a low threshold pressure does not decreasesigniﬁcantly with increasing load. Our previous study19) showedthat RID increases with increasing amount of 3-coordinatedboron and that almost all the boron in Glass C is 3-coordinatedboron. The 3-coordinated boron in Glass C may be the origin of
its low threshold pressure and the small slope of its load
dependence of densiﬁcation.
The large diﬀerence in CR among the glasses can be explainedby the diﬀerence in the load dependence of densiﬁcation orresidual stress. It is assumed that cracks initiate at the load at
which the residual stress around the indentation exceeds thecritical stress required for crack initiation. The small diﬀerence inthe slope of the load dependence of the residual stress between
the glasses is thought to result in a large diﬀerence in the applied

JCS-Japan

Journal of the Ceramic Society of Japan 119 [2] 110-115 2011

σ rs / σ CR
Glass D
Glass C1
σ 100 gf (D )σ 100 gf (C )an indentation. Therefore, direct measurement of the residual
stress distribution is necessary to explain the diﬀerence in the
stress distribution among the glass compositions. Moreover,stress distribution around an indentation varies dynamicallyduring the process of indentation, which aﬀects crack initiation
behavior such as timing and position of initiating cracks.
Therefore, the in situ observation of crack initiation during
indentation and direct measurement of the residual stress
distribution are future works that are expected to quantitatively
clarify the diﬀerences in crack resistance.

4.
100 gf

CR (D)CR (C)

log P

Fig. 8. Schematic diagram showing the relationship between the crack
resistance and the critical stress for crack initiation. The horizontal axis is
the logarithm of applied load (P), and the vertical axis is the normalized
residual stress. The normalized residual stress is derived by dividing theresidual stress (·rs) by the residual stress at the load corresponding to thecrack resistance (·CR). CR (C) and CR (D) are the crack resistance ofGlasses C and D, respectively. ·100 gf (C) and ·100 gf (D) are thenormalized residual stresses at a load of 100 gf for Glasses C and D,
respectively.

load at which the critical stress is generated. The reason for this is
explained in Fig. 8. For simpliﬁcation, only approximate linesfor Glasses C and D are shown. In this ﬁgure, the horizontal axis
is the logarithm of the applied load and the vertical axis is theresidual stress normalized by the stress at a load corresponding toCR (·CR). [The approximate line in Fig. 5 for each glass isderived by the least-squares method, from which ·CR for eachglass can be estimated. The estimated values of ·CR are 1.7 GPa
for Glass D (CR = 150 gf) and 1.9 GPa for Glass C (CR =1200 gf).] The normalized residual stresses at a load of 100 gf for
Glasses C and D are also shown in Fig. 8 as ·100 g (C) and·100 g (D), respectively. The value of CR, that is, CR (C) or
CR (D) in Fig. 8, corresponds to the load at which the residual
stress equals the critical stress (·rs/·CR = 1). Because theresidual stress increases logarithmically with increasing appliedload, the diﬀerence in crack resistance, CR (C) ¹ CR (D), ismuch larger than the diﬀerence in residual stress, ·100 gf (C) ¹·100 gf (D). Therefore, the diﬀerence in the load dependence ofdensiﬁcation among the glass samples may result in the large
diﬀerence in CR, which cannot be explained by the diﬀerence inother properties, such as KIC, E, and Hv.
The diﬀerence in CR among the three glasses including Glass
G cannot yet be explained quantitatively. The residual stressdiscussed in this paper is a highly simpliﬁed model. However,
the actual stress distribution around indentations is more
complex. As shown in Figs. 4 and 5, there is a large recoveryof the opposite side length of the indentation, while there is little
recovery of the indentation diagonal. This probably results fromthe variation of microscopic density of the glass under theindentation. The contribution of densiﬁcation should be larger inthe face of Vickers indentation than at the edge. As a result, a
residual stress distribution should form in which the residual
stress at the indentation corner is larger than that at theindentation edges. However, no theory has been proposed thatcan quantitatively describe the residual stress distribution around

Conclusion

The load dependence of densiﬁcation was investigated for threecommercial glass compositions. For the aluminoborosilicate glass(Glass C) and the soda-lime silicate glass (Glass D), the recovery
of indentation depth (RID), which indicates the contribution ofdensiﬁcation to the total deformation, decreases with increasing
applied load. For the lead borosilicate glass (Glass G), RID is low
throughout the range of applied loads. The estimated residual
stress increases logarithmically with increasing applied load, andthe slope of the load dependence of the stress is diﬀerent among
the glasses. This diﬀerence is thought to result in the large
diﬀerence in crack resistance among the glasses.
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