Strength and toughness

CHAPTER 18

Strength and toughness 18.1 Introduction Applied stress at the point of failure is termed the strength. For glass products, a measurement of strength generally consists of applying increasing magnitude of tensile stress in a defined geometry specimen until fracture occurs. Under the application of pure hydrostatic compression, glass densifies (Section 7.4). As a convention, tensile stress is denoted by +σ, or simply σ, whereas a compressive stress is denoted by
σ. Units of strength are the same as that of stress, for example, dyn/cm2 in cgs, MPa in SI, and psi in the English system. It is a common experience that glass products break readily when subjected to mechanical or thermal shock. On the other hand, bulletproof glass and glass fibers which are “stronger than steel” are not unheard of either. We could summarize our experience as follows: Strength of common glass products: about 14–70 MPa. freshly drawn glass rods: about 70–140 MPa. abraded glass rods: about 14–35 MPa. wet, scored glass rod: about 3–7 MPa armored glass: about 350–500 MPa. handled glass fibers: about 350–700 MPa. freshly drawn glass fibers: about 0.7–2.1 GPa. In contrast, most metals including steels fail around 140–350 MPa. It is no surprise that strength is not one of the more appreciated virtues of glass products. Two other common observations that contrast failure in metals to that in glass are as follows: (i) Most metals neck or yield plastically before actual failure occurs, whereas glasses do not show any ductility prior to the failure. We say that the metals show ductile failure and the glasses show brittle failure. (ii) Whereas a given group of metal specimens prepared identically shows a narrow distribution of yield stress and ultimate fracture stress, glass specimens produced apparently identically are likely to display a large variation of strength.

Fundamentals of Inorganic Glasses https://doi.org/10.1016/B978-0-12-816225-5.00018-3

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All this is despite the fact that the elastic properties and the density of glasses are generally of the same order as that of most metals! The ductile behavior of metals is easy to understand. Crystalline structures such as those of the metals contain a large number of dislocation defects which can move readily and new dislocations are created under the influence of applied stresses. An irreversible movement of dislocations appears as ductility or plastic deformation. In glasses there are no dislocations in the structure and, hence, there is no apparent ductility on the application of stress. Some plastic deformation under a microhardness indentation and irreversible compaction under large magnitude of hydrostatic stress do occur in glass (see Chapter 8). However, the amount of permanent strain involved is small compared to that due to dislocation motion in most metals. We know now that the strength of solids is also compromised by structural flaws which act as stress intensifiers. Intensification at the flaw tip makes it possible for the solid to fracture at stress magnitudes lower than that theoretically needed to pull atoms apart. Although in metals the intensified stress creates motion through dislocations and other defects such as grain boundaries, in glasses there are no grains, and hence the stress results in a propagation of the flaws creating, in principle, two new surfaces. Because flaws grow, there is fatigue over the duration of stress application, not predicted by earlier theories of the strength of brittle solids. In this chapter, our objective is to understand the marked influence of specimen size, surface condition, indentation damage due to handling, and environment on the measured strength of glass. After a brief mention of the experimental techniques of strength measurement for glasses, we will consider the theoretical strength of a “flawless” and a flawed brittle solid and explore the description of strength in terms of flaw growth kinetics and fracture statistics. After understanding why glass products are weak in practice, we will explore the current technologies and potential opportunities for glass strengthening.

18.2 Theoretical strength of a flawless brittle solid The amount of work done in pulling two atoms apart such that two new surfaces are created is the theoretical strength of a flawless solid. Consider the pulling of the atom located at a0 away from an atom located at the origin (Fig. 18.1). Because of the nature of interatomic forces, the stress reaches a maximum σ m, after which a decrease is experienced asymptotically until the separation becomes much too large for the atoms to be considered bonded, and two new surfaces having fracture surface energy γ f each are created. The instantaneous stress, σ, may be approximated by a sinusoidal form

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Fig. 18.1 Variation of stress σ with increasing interatomic spacing.




 2πx σ ¼ σ m sin : λ At small separation, sin(2πx/λ)  2πx/λ, and hence
 2πx : σ  σm λ

(18.1)

(18.2)

According to Hooke’s law σ¼

Ex , a0

(18.3)

where E is the Young’s modulus and x/a0 is the strain. Combining Eqs. (18.2) and (18.3), we obtain σm ¼

λE : 2πa0

(18.4)

The energy of two freshly fractured surfaces should equal the work of separation which is the area under the stress-distance curve approximated as a half sine wave. Thus
 ð λ=2 2πx λσ m 2γ f ¼ : (18.5) σ m sin dx ¼ 2π λ 0 Combining Eqs. (18.4) and (18.5), we obtain
1=2 γf E σm ¼ : a0

(18.6)

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Fig. 18.2 Elliptical flaw in a glass plate under tension.

Eq. (18.6) is the Orowan [1] expression for the theoretical strength of “perfect” brittle solids. If we use E ¼ 70 GPa, γ f ¼ 3.5 J/m2, and a0 ¼ 0.2 nm, we obtain σ m ¼ E/ 2 ¼ 35 GPa. (Recall: 1 Pa ¼ 1 N m
2, and 1 N m ¼ 1 J.) Strengths approaching 13–14 GPa have been reported for fine silica glass fibers freshly drawn under very careful drawing conditions. Note: The reader is to be cautioned not to confuse γ f with the interfacial surface energy γ commonly quoted for glasses as 0.35 J/m2. The normally measured interfacial energy is that of a relaxed surface having adsorbed species such as water. In a fresh fracture, the atomic bonds are separated with no adsorbed species such as water. There may also be some plastic deformation energy requirement for the bond stretching in addition to irreversible effects such as acoustic and photon emission, and conversion of some energy to heat.

18.3 Strength of a flawed brittle solid: Griffith’s analysis It was Inglis [2] who pointed out that flaws in brittle solids act as stress concentrators. On the application of an applied stress σ a, the stress σ yy at the tip

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of an assumed elliptical hole or flaw (Fig. 18.2) with minor axis 2b and major axis ¼ 2c in a plate (z-plane) is given by
 2c σ yy ¼ σ a 1 + : (18.7) b This means that when σ a is such that σ yy ¼ σ m, the specimen will fracture. Since the radius of curvature at the tip of the ellipse ρ ¼ b2/c, Eq. (18.7) yields
rffiffiffi c σ yy ¼ σ a 1 + 2 : ρ Now, since c ≫ ρ,

rffiffiffi c : σ yy ¼ 2σ a ρ

(18.8)

The curvature ρ at the crack-tip approximates atomic dimensions a0, hence, the strength of a flawed brittle solid σ f is given by rffiffiffiffiffiffiffi rffiffiffiffisffiffiffiffiffiffiffi 1 a0 γ f E 1 γ f E ¼ σ f ¼ ðσ a Þfailure ¼ (18.9) a0 2 c 2 c Eq. (18.9) implies that a flaw having c as small as about 50 μm can cause fracture at as little as 35 MPa. Note that σ f ≪ σ m. Griffith [3] suggested that having an applied stress such that the stress at the tip of the flaw exceeds the theoretical strength cannot be the sole criterion for strength. The flaw must exceed a critical length before it would propagate into a fracture. Griffith argued that the change ΔΨ in the potential energy of a plate with an elliptical flaw of length 2c is given by πc 2 σ 2a (18.10) + 4cγ f : E The first term represents the loss in strain energy, and the second the approximate gain in surface energy. The linear surface energy term causes ΔΨ to increase with c initially until c becomes c*, at which point ΔΨ begins to decrease due to the dominance of the parabolic term. c* is found by differentiating with respect to c and setting the expression equal zero. This gives rffiffiffiffiffiffiffiffiffiffi 2γ f E σ f ¼ ðσ a Þfailure ¼ : (18.11) πc ∗ ΔΨ ¼


(Note: Eq. (18.11) is valid for plane-stress condition. Replace E by E/ (1
ν2), where ν ¼ Poisson’s ratio, for a plane-strain condition.)

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Eqs. (18.9) and (18.11) hint at the effect of the environment through its effect on the surface energy. Thus, since contact with water lowers γ f, (σ a)water < (σ a)air. Also apparent is the statistical nature of the failure in brittle solids. Flaws generally occur on a free surface and have a wide distribution of lengths, which give them a wide distribution of severity in terms of their c/ρ ratio. We may argue that fracture would occur at the site of the most severe flaw much like the case of the weakest link in a chain. These flaws are called Griffith flaws. Although γ f and E depend on glass composition somewhat (range about 20%), the variation in c can be of several orders of magnitude. It is not a surprise, therefore, that the observed strength of glass is not intrinsic to its composition but may depend greatly on methods of handling. Even though the length of a typical Griffith flaw may be as little as 5–50 μm, its width is of the atomic dimensions. For this reason, Griffith flaws are not easily observable, even with the help of the highest power optical microscope. The use of high-resolution electron microscopy on thin silica films has shown some evidence [4] of an elliptical crack having about 1.5-μm tip curvature. The most important drawback in Griffith’s analysis is the lack of a time effect. Since it is known that glass fatigues over long duration of stress application, any consideration of σ f must include a time effect. The reconciliation is made through the application of fracture mechanics.

18.4 Elementary fracture mechanics concepts

pffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi It may be realized from Eq. (18.11) that σ f πc ∗ ¼ 2γ f E ¼ constant, depending on the material and its environment. E is determined by the material, whereas γ f is determined by both the material and the environment. pffiffiffiffiffi If p we ffiffiffiffiffifficall σ a πc ¼ K, the stress intensity factor, and call the quantity σ f πc ∗ ¼ Kc , the critical stress intensity factor, then fracture would occur when K ! Kc due to changes in σ a or c. Suitable units for K are MPa m1/2 and N mm
3/2 in SI units, and ksi in1/2 in English units. Note 1 N mm
3/2 ¼ 0.0316 MPa m1/2 and 1 ksi in1/2  1.1 MPa m1/2. According to Irwin [5], there are three possible modes of independent kinematic movements that cracked faces could have relative to each other (Fig. 18.3). In mode I (opening mode), the fracture plane (the y-plane) and the direction of propagation (x-direction) are perpendicular to the direction of the applied stress (y-direction). Mode II (sliding mode) is a failure due to shearing where the applied stress (σ yx) acts to slide fracture planes (y-plane) over each other along the direction of propagation (x). Mode III (tearing mode) is also a shear failure, except that the direction of fracture

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Fig. 18.3 The different modes of crack extension.

propagation (x) is perpendicular to that of the stress (σ yz) on the y-plane. Stress fields at a distance r and polar angle θ from the tip of a sharp slit crack can then be defined in the form of a generalized equation: σ ij ¼ K ð2πr Þ
1=2 fij ðθÞ,

(18.12)

where K represents three stress intensity factors KI (pronounced K-one), KII, and KIII corresponding to the three modes of failure. For a single crack of length 2c extending in an infinite body subjected to a uniform remotely applied stress field [σ ij]∞, the K values are given by [6]   pffiffiffiffiffi KI ¼ σ yy ∞ πc   pffiffiffiffiffi KII ¼ σ yx ∞ πc (18.13)   pffiffiffiffiffi KIII ¼ σ yz ∞ πc There are thus three critical values of K: KIc (pronounced K-one-cee), KIIc, and KIIIc. The fracture mechanics criteria for failure, then, are the approach of either of the stress intensity factors to their critical values. Of the three critical values, we are generally concerned with KIc for failure in brittle solids which represents both failure plane as well as propagation perpendicular to the principal tension. Irwin also defined1 the term strain energy release rate G equal to K2/E for the plane-stress condition, and equal to K2(1
ν2)/E for the plane-strain condition. G is the change in the strain energy of the body when a unit change occurs in crack length. [Note, as before, plane-strain calculations for mode I and mode II failure can be obtained from the plane-stress formulas by replacing E with E/(1
ν2). Henceforth, we will assume plane-stress condition.] For an elastic crack in an infinitely wide plate, 1

The letter G was used by Irwin in honor of Griffith.

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GIc ¼ KIc2 =E ¼ σ 2f πc ∗ =E: Hence, σf ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi EGIc =πc ∗ ,

(18.14)

which gives the basic definition that GIc ¼ 2γ f. Likewise GIIc ¼ K2IIc/E and GIIIc ¼ K2IIIc/2 μ; (μ ¼ shear modulus) under conditions of plane stress. Note 2 in the denominator for Mode III. We describe toughness of a solid as its ability to absorb energy prior to failure. Mathematically, it is the area under the true stress-true strain curve. In brittle solids, such a plot has little meaning on a macroscale. We must talk about the energy required to extend the flaw from some initial size to the critical size. Thus, GIc would be a measure of the fracture toughness. pffiffiffiffiffiffiffiffiffiffiBy convention, however, the critical stress intensity factor KIc ¼ EGIc itself is called the fracture toughness of the material. (Many of the classic papers such as those by Griffith and Irwin on strength and fracture mechanics are included in Ref. [7].) The inclusion of the dynamic changes in σ a and c all lumped into K makes it possible to include the time element into proper perspective while considering fracture in brittle solids.

18.5 Glass fatigue Griffith’s theory was challenged by two observations in particular: (i) Under constant load, the time-to-failure varies inversely with the load applied. (ii) Under increasing load (constant strain rate), higher stress to failure is needed for higher straining (or stressing) rate. Both these observations of delayed failure are manifestations of fatigue: the first is termed static fatigue and the second dynamic fatigue. The most cited static fatigue experiments are those carried out by Mould and Southwick [8]. These authors first obtained strength σ N of pristine glass rods immersed in liquid N2 (called the inert strength). They then plotted the strengths of rods of the same glass abraded in different manner using different grits as a fraction of the inert strength against log(t ¼ time-to-failure). All the curves were linear for the most part, approximately parallel to each other and had negative slopes (Fig. 18.4). By replotting the data of σ/σ N versus log(t/t0.5) where log t0.5 is the time corresponding to σ/σ N ¼ 0.5, it could be shown that all the data fell on a master curve (Fig. 18.5). In the linear portion, one may write

Strength and toughness

0.8

Symbol Abrasion a b c d e f

0.7 0.6 s/sN

495

0.5 0.4 0.3 0.2 10–3

10–2

10–1

1

10

102

103

Load duration (s)

Fig. 18.4 Plot of the time to fracture (x-axis) as a function of the normalized stress for various conditions of surface abrasion. (Abrasion code: a ¼ severe grit blast; b ¼ mild grit blast, emery cloth perpendicular to stress; c ¼ 150 grit; d ¼ 600 grit; e ¼ 320 grit; f ¼ 150 grit, parallel to stress.) (After R.E. Mould, R.D. Southwick, J. Am. Ceram. Soc. 42 (1959) 542; R.E. Mould, R. D. Southwick, J. Am. Ceram. Soc. 42 (1959) 582. Reproduced with permission of the American Ceramic Society.)

Fig. 18.5 “Universal static fatigue” curve of Mould and Southwick [8]. (Reproduced with permission of the American Ceramic Society.)

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Fig. 18.6 Dynamic fatigue of glass. (After J.E. Ritter, Jr., R.P. LaPorte, J. Am. Ceram. Soc. 58 (1975) 265. Reproduced with permission of the American Ceramic Society.)

σ=σ N ¼
A log ðt=t0:5 Þ + B

(18.15)

Mould and Southwick’s experiments showed that flaws apparently grow with time during the application of stress; the growth velocity must be lower at lower stresses than at higher stresses thus allowing a longer delayed failure. The dynamic fatigue experiments are conducted by applying stress at varying rates and plotting the data against log(stressing rate). Typical results (Fig. 18.6) show that higher stressing rates allow the specimens to sustain higher stresses. Apparently, higher stressing rates allow lesser time for flaws to grow, making it possible for the specimen to sustain higher stresses. Both static and dynamic fatigue experiments are now reconciled through direct measurements of what is called the subcritical crack velocity as a function of the applied stress or the stress intensity factor by Wiederhorn and his associates [9]. Wiederhorn used a microscope/cathetometer to measure the growth velocity of cracks of length a as a function of the applied load P

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on double cantilever specimens of microscope slides of soda lime silicate glass. Figure 18.7 shows the specimen geometry. A primary notch 0.4mm wide  0.5mm deep is made along the middle of the slide and is cut through at one end making an angle slightly greater than 30°. Application of a load P initiates a crack guided along the “web” of glass as shown; the distance a is measured from the center of the loading holes. The measurements were made with the specimens immersed in liquid N2, gaseous N2 at 25°C of different RH, and water. The measurements of load and crack length could be converted to stress intensity factor KI using the formula:   (18.16) KI ¼ Pa=w 1=2 b1=2 d3=2 ð3:467 + 2:315 d=aÞ where the meaning of the different terms is as shown in Fig. 18.7. A plot of the log(crack velocity) versus KI is shown in Fig. 18.8. In general, three regions of subcritical crack growth behavior are observed after which the crack has grown to the critical size required to fulfill the Griffith energy release criterion, and a spontaneous failure is imminent: (i) REGION I where the crack velocity increases exponentially with applied load as well as humidity. (ii) REGION II where the velocity is independent of the applied stress, but still depends on the humidity level. (iii) REGION III where the velocity is independent of the humidity level, but depends, once again, exponentially on the applied load. The slope in region III is much steeper than in region I. At the end of region III, KI ! KIc, and “spontaneous” failure follows. The spontaneity of the

Fig. 18.7 Double cantilever beam specimen configuration to measure crack velocity. Hatched area represents the uncracked portion of the specimen (“web”) between the guiding notches and the back surface. (Courtesy T. Kokot, Saxon Glass Technologies, Inc., after consultation with S.M. Wiederhorn).

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Fig. 18.8 Crack velocity versus applied intensity factor for soda lime silica glass in N2 gas of varying levels of humidity. (After S.M. Wiederhorn, J. Am. Ceram. Soc. 50 (1967) 407. Reproduced with permission of the American Ceramic Society.)

failure is agreed, generally, to be a velocity exceeding about 0.1 m/s. Thus, as a convention, velocities less than about 0.1 m/s are the “subcritical” crack velocities which cause the growth of a flaw up to its critical size. Wiederhorn and Bolz [9] showed that the crack velocities depend on glass composition (Fig. 18.9). There apparently existed a fourth region at very low velocities where the V-KI plot tended to become vertical. Thus, there appeared a static fatigue limit of the applied stress below which crack did not grow (to any measurable extent). In normal alcohols, the crack growth is essentially similar to that in nitrogen gas and depends on the partial pressure (¼RH) of the dissolved water [10]. In addition, Wiederhorn et al. showed that the crack velocity in region I not only depended on composition but also on temperature (Fig. 18.10) and the environment, for instance, the pH of the medium, suggesting the possibility that subcritical crack growth may be a thermally activated stress-corrosion process. Wiederhorn suggested that the behavior in region II could be explained by assuming that the crack velocity in region II was limited by the rate of water transport to the crack

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Fig. 18.9 Effect of glass composition on crack velocity-KI behavior of the various glasses tested in water at room temperature. (After S.M. Wiederhorn, C.H. Bolz, J. Am. Ceram. Soc. 53 (1970) 543. Reproduced with permission of the American Ceramic Society.)

tip. As indicated by Quakenbush and Frechette [11], the existence of region II is crack-size dependent. If the crack is sufficiently small, then water migrates to the tip as rapidly as the crack growth and there is no region II. Crack propagation studies in vacuum indicate that region III does not occur in all glasses. Fused silica, Vycor, and Pyrex show only region I prior to catastrophic failure. The dependence of the crack velocity V on KI is more popularly expressed in terms of a power law (instead of exponential functions) as V ¼ AK nI ,

(18.17)

where A and n are constants. n is called the stress-corrosion susceptibility coefficient or the crack-growth exponent, and ranges between 12 and 35 for most glasses: higher values are observed for immersion in strong HCl, and for silica glass. For a generalized case of a flaw of size a,

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Fig. 18.10 Effect of temperature on the crack velocity-KI curves for a soda lime silicate glass in water. (After S.M. Wiederhorn, C.H. Bolz, J. Am. Ceram. Soc. 53 (1970) 543. Reproduced with permission of the American Ceramic Society.)

KI ¼ σ a Yc 1=2

(18.18)

1/2

where Y is a geometry parameter (¼π , e.g., for a slit crack in an infinite plate as in Eq. (18.13)). Now, since V ¼ dc/dt, Eqs. (18.17) and (18.18) yield dc ¼ 2(KI/Y2σ 2a )dKI, which when integrated between the limits of time t ¼ 0 and t ¼ tf gives   2 KIi2
n
KIc2
n tf ¼ : AY 2 σ 2a ðn
2Þ ≫ K2
n Since K2
n Ii Ic for large values of n, the above expression reduces to tf ¼

2KIi2
n : AY 2 σ 2a ðn
2Þ

(18.19)

Now, KIi ¼ ðσ a =σ Ic ÞKIc ,

(18.20)

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where σ Ic is the critical fracture stress in an inert environment. It follows that

n 2
n 2 σa KIc =σ Ic : (18.21) tf ¼ 2 σ Ic AY 2 ðn
2Þ The time-to-failure t0.5 at half the inert strength (as discussed in the Mould and Southwick experiments earlier) is given by

n 2
n 2 1 KIc =σ Ic t0:5 ¼ 2 (18.22) 2 AY 2 ðn
2Þ Dividing Eq. (18.21) by Eq. (18.22), we obtain tf =t0:5 ¼ ð2σ a =σ Ic Þ
n : After rearranging and taking logarithms of both sides, we obtain
 tf 1
log ð2Þ: (18.23) log ðσ a =σ Ic Þ ¼
log t0:5 n Eq. (18.23) yields a “universal fatigue curve” similar to the experimental observations of Mould and Southwick (Eq. 18.15) as long as the value of n remains unchanged in a given group of specimens with a given type of flaws. Now, dσ/dc ¼ [dσ/dt]/V. Hence, substituting Eqs. (18.17) and (18.18),

dσ a dc n σ a dσ ¼ : dt AY n c n=2 Integrating both sides over the extent of the crack growth (from ci to c*), and assuming that the stressing rate dσ/dt is constant,

ð ð σf dσ a 1 c∗
n=2 n σ a dσ ¼ c dc: dt AY n ci 0 This yields " # σ nf + 1



h i dσ a 1 ð2
nÞ=2 : ¼2   c ∗ð2
nÞ=2
ci n n+1 dt AY ð2
nÞ

and c1/2 ¼ KIc/σ Ic, we obtain Since c∗(2
n)/2≪ c(2
n)/2 i i



 dσ a σ n
2 dσ a n+1 Ic : σ f ¼ 2ðn + 1Þ 2 n
2 ¼ B dt dt ðn
2ÞAY KIc

(18.24)

where B is a constant. This shows that a plot of log(σ f) versus log(dσ a/dt) has a slope of 1/(n + 1). The gentle upward slope is as observed in dynamic fatigue experiments.

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18.6 Mechanism of strength based on slow crack growth The experimental observations of static and dynamic fatigue under different environment support the notion that the strength of glass is determined by a thermally activated, stress-assisted crack-growth. The most successful explanation along this argument has been put forward by Charles and Hillig [12]. These authors suggested that flaws in glass grow by stress corrosion. The resulting subcritical crack velocity V is given by


ΔE + ðσΔυ=3Þ
υm γ f =ρ V ¼ V0 exp , (18.25) RT where ΔE is the activation energy of the chemical reaction in the stress-free state, Δυ is the activation volume, and υm is the molar volume of the glass. The third term within the brackets accounts for changes in the surface curvature. The stress varies along the sides of the flaw according to Inglis criterion (Eq. 18.8), being the highest at the tip and the lowest along the wall. In an annealed glass, the corrosion may be uniform (Fig. 18.11A). Corrosion of the tip causes c to increase, whereas corrosion of the sides causes ρ to increase. At high values of applied stress, the differential corrosion causes c/ρ of the flaw to increase, that is, the flaw sharpens (Fig. 18.11B). This, in turn, causes increased stress at the flaw tip, which leads to more rapid propagation of the crack and ultimately, to failure. On the other hand, in the presence of low applied stress, c/ρ does not increase much, and may even decrease, leading, ultimately, to flaw blunting (Fig. 18.11C). Because of thermal activation, there is very little corrosion at very low temperatures, such as the liquid N2 temperature. One would tend to get essentially no subcritical crack growth, or in other words, the “inert” strength would approach the theoretical strength. As the temperature rises, more and more subcritical crack growth occurs leading to lower observed

Fig. 18.11 Effect of stress corrosion on the geometry of the crack tip. (After W.B. Hillig, R.J. Charles, in: V.F. Zackay (Ed.), High Strength Materials, John Wiley and Sons, New York, 1965, pp. 682–705.)

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strengths. The possibility of a fatigue limit is indicated by the situation when changes in σΔυ/3 are compensated by changes in υmγ f/ρ. One may question as to whether crack healing would occur if the applied stress were removed. From Griffith’s energy criterion, it is a natural expectation that if the crack did not reach its critical length then the removal of the applied stress should lead to crack closure. Crack healing in water or air is observed [13] experimentally in soda lime glasses; however, it is not readily obvious how stress-corrosion chemistry would reverse its course at the tip of an advancing crack front. An atomistic model to explain corrosion and crack healing has been suggested by Michalske and Freiman [14].

18.7 Elementary fractographic analysis A fracture in glass leaves its “fingerprints” behind, which can be recognized to gain useful information regarding the cause of the fracture. Fractographic analysis is basically a postmortem analysis where the confidence level is high when all the pieces are available. The best sources for learning the details of fractographic analysis in brittle solids are books by Frechette [15] and Quinn [16]. Examination of the fracture is easier with very careful lighting conditions, a low-power pocket magnifier or an adjustable power stereo microscope with a portable reflected light and camera accessory. A typical fracture in glass is shown in Fig. 18.12. Fracture itself is a kinetic phenomenon: it

Fig. 18.12 Fracture surface of a soda lime silicate glass specimen broken at room temperature. (Modified from an image provided by George Quinn.)

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originates from a critical-sized flaw and propagates across the body. Surrounding the origin is a shiny region called the mirror which ends into fine radial fibrous or misty texture, called the mist. Surrounding the mist are wider and deeper radial ridges with slivers of glass lifted out, called the hackle, leading ultimately to crack branching. In the mirror region, the fracture front travels with increasing velocity. The mist begins to form as the fracture velocity approaches its terminal velocity generally equal to about 0.6 times the velocity Vs of transverse acoustic (shear) waves in the medium (typically Vs  3500 m/s). In rare cases of extraordinarily fast loading, the terminal velocity has been reported to be as much as 95% of the transverse acoustic wave velocity. The hackle region represents the motion of the fracture front at terminal velocity along parallel but slightly different coplanar steps. Much of the elastic energy is liberated in the hackle region, and hence, the width of the hackle reflects the degree of violence associated with the fracture. After the hackle region, the fracture decelerates. In its onward journey, it may encounter a defect or a surface. Interaction of the fracture front with stress waves originating from the defect or the surface produces Wallner lines (Fig. 18.12). Fracture travels faster in the tensile region, and hence, the Wallner lines are advanced in the direction of increasing tension. Twist hackle (Fig. 18.13), also called river pattern, is formed when the tension at the head

Fig. 18.13 Twist hackle in a fractured glass specimen. Crack was propagating in the direction of the arrow. (Courtesy of James Varner, Alfred University, Alfred, New York.)

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Fig. 18.14 Hertzian cone caused by high-velocity impact from a small, hard body against the center. (A) Top view. (B) Cross-sectional. (Photograph courtesy of J.R. Varner, Alfred University, Alfred, New York.)

of the crack front is twisted. The crack front breaks up into a group of closely spaced, nearly parallel but noncoplanar surfaces. The fracture surface is completed by the segments curving over each other. Long, needle-shaped slivers of glass are often lifted out at the curve, making the marks look like tributaries joining into a river. In an end view, the twist hackle would look like a series of steps. The twist hackle is observed often when the trailing edge of a

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fracture front extends into compression region. When fracture occurs in the presence of a corrosive medium such as water, scarps may be produced. These scarps represent smooth boundaries between the liquid-filled and the dry regions of the fracture front. When crack branching occurs, the maximum angle of forking θ depends [17] on the ratio of the principal stress magnitudes by: θ ¼ 180° when σ y ¼ σ x (uniform biaxial stress), and 90° when σ y ¼ σ x/2 (cylindrical vessel bursting due to internal pressure), 45° when σ y ¼ 0 (simple tension), and θ ¼ 15° when σ y ¼
σ x (simple twist). If a hard spherical ball is dropped on a plate of glass, a thin plate may flex such that tension is created on the opposite side. Cracking, if any, would then originate on the tension side. For a given force of impact, the flexural stress decreases as d
2, where d is the thickness of the plate. When the plate is thick and the impacting object is relatively smooth or blunt, the impacting face develops [18] Hertzian or bearing stresses which increase roughly as d3/2. Fracture may start as a circular crack perpendicular to the tensile radial stress around the circle of contact, moving vertically downwards at first, followed by flaring out to form a conoidal fracture reaching the back surface. This type of fracture is termed Hertzian cone fracture (Figs. 18.14 and 18.18), typical of bullet holes in glass. Radial cracks may also form if the impact is violent. Hertzian stress analysis due to spherical ball loading is discussed further. Pointed projectiles produce more complicated fracture. Of particular interest, the cracks produced by a Vickers microhardness indenter, shown schematically in Fig. 18.15, are discussed further. A review on this subject is provided by Cook and Pharr [19]. Elastic-plastic analysis of stresses in the glass by a pointed indenter has been provided by Yoffe [20] and is discussed in Section 18.8. At low loads, nothing may happen during the loading cycle. However, during the unloading cycle the compressive elastic field in the surface region diminishes, leaving a tensile residual field. The tensile residual may cause the nucleation of radial cracks (also called Palmqvist cracks (see Shetty et al. [21]). These cracks propagate radially along the median planes close to the surface (Figs. 18.15B and 18.16). At higher loads, growth of penny-shaped median cracks propagating downward along the median planes occurs during loading because of the subsurface tensile elastic stress field surrounding the plastic deformation region (Fig. 18.15A). In addition to the median/radial crack system, the gradual retreat of the compression zone due to elastic recovery brings tensile region closer to the shear fault lines. Thus upon unloading after a high loading, circular lateral cracks develop [22] originating at the base of the plastic zone, in addition to the radial cracks closer to the surface (Figs. 18.15B and 18.16D). With

Strength and toughness

507

Fig. 18.15 Idealized crack morphologies at the site of a Vickers indentation contact. (A) Median cracks originating at the edge of the plastic deformation zone and propagating deeper into the glass along median planes, formed during loading. The cracks are circular, or parts of a circle truncated by the deformation zone. (B) Radial (or Palmqvist) cracks emanating at the ends of the diagonal near the surface in the form of near-elliptical shape propagating radially outward along median planes containing the direction of loading, formed during unloading. (C) Half-penny crack formed when the median and radial cracks join during unloading. (D) Lateral cracks upon unloading. These originate at the base of the deformation zone, propagating perpendicular to the direction of loading, and curving toward the surface.

continued unloading, all the cracks grow in the residual field. Median cracks grow mostly toward the surface so as to join with the radially growing radial cracks, making a half-penny crack (Fig. 18.15C). Occasionally, the circular fronts of the lateral cracks may grow up to the surface, in which case, chipping occurs. The basic morphological features summarized above are often sufficient to trace back the origin of the fracture, the sign of stresses at the surfaces (the origin invariably would be on the tensile surface). A judgment of the violence associated with the fracture and counting the number of fragments produced can often give at least a semiquantitative idea of the stresses that lead to the fracture.

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Fundamentals of Inorganic Glasses

Fig. 18.16 Top view of the cracking in Vickers indentation. (A) Partial load increasing ¼ 47 N. (B) Median cracks at maximum load ¼ 90 N. (C) Partial unloading, load ¼ 30 N. (D) Upon full unloading. Circular lateral cracks may be seen as shadows. Width of field ¼ 720 μm. (After D.B. Marshall, B.R. Lawn, J. Mater. Sci. 14 (1979) 2001. Reproduced with permission of Chapman & Hall. Photographs courtesy of B.R. Lawn, National Institute of Standards and Technology, Gaithersburg, Maryland.)

18.8 Damage introduction in glass Here we address the question as to whether surface damage in glass products has any relationship to the glass network yield strengths introduced in Section 18.4. It is worth asking the questions: (1) Can flaws be generated during glass forming process when glass might be molten to semisolid? (2) Would seemingly benign contact of solid glass with some foreign body have the potential to create new flaws or intensify the existing flaws? To answer the first question, we invoke our discussion on non-Newtonian viscosity (Section 9.8). It is shown that when shearing strain rate is high even liquids can fracture. Fig. 9.18 shows that the cohesive strength of the network decreases with decrease in Newtonian viscosity. Fig. 18.17 from Ito and Taniguchi [23] shows that the application of 5 GPa uniaxial tension on a silica liquid generates a cavitation within the network. The cavitation can be considered a flaw embryo. We may conclude that

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Cavity Formation

Fig. 18.17 Generation of a flaw embryo by application of 5 GPa extensional stress. (After Ref. S. Ito, T. Taniguchi, J. Non-Cryst. Solids 349 (2004) 173.)

Fig. 18.18 Ring/cone crack in Hertzian impact.

flaw embryos generate due to the application of tensile stresses that exceed the hydrostatic yield strength for dilation during glass product forming stage in molten to semisolid form. The second question is answered by looking at Hertz analysis [24] of a spherical ball contact with a solid glass surface (a hard elastic half-space, see Fig. 18.18), modified by Yoffe [20] for a sharp indenter contact to include plastic deformation. All components of the stress field are compressive under the indenter, hence, there is no tendency to develop any cracking. Hertz [24] showed that the radial stress σr at a distance r from the center of the hard sphere elastic contact outside the contact circle is given by  σ r ¼ ð1=2Þ P=r 2 ð1
2ν1 Þ (18.26) where P is the load, and ν1 is the Poisson’s ratio of the perfectly elastic target half-space (glass substrate). The hoop stress σ ϕ is negative but around the same magnitude as σ r. [Notation is based on spherical coordinates; origin is the contact point, θ is the angle to the loading (symmetry) axis, and ϕ is the hoop angle about the loading axis.] The hoop stress is tensile immediately outside the contact circle and drops off rapidly with r.

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Fundamentals of Inorganic Glasses

Yoffe [20] used superposition of a Boussinesq field from a point loading of the surface plus a localized “blister” field owing to the plastically deformed (densified) zone. The localized blister field drops off as 1/r3. For a conicalshaped indenter contact during the loading cycle and assuming Poisson’s ratio ¼ 0.25 for the substrate, stresses on the surface (θ ¼ π/2) are given by

 σ r , π=2 ¼ P=4πr 2
7B=r 3 σ θ, π=2 ¼ τrθ ¼ 0, and

 σ ϕ, π=2 ¼
P=4πr 2 + 2B=r 3

(18.27)

Depending on the value of P and B, σ r and σ ϕ can be tensile or compressive. A tensile σ r may lead to the Hertzian ring crack around the contact. Along the loading axis, below the indenter θ ¼ 0, it may be shown that

 σ θ ¼ P=8πr 2
B=r 3 (18.28) which is tensile at high P and causes median cracks to develop. The deformation zone under the indenter and the cracking during loading are shown in Fig. 18.19.

Fig. 18.19 Crack system associated with sharp indenter during loading. Median crack initially generates from the plastic contact zone (hatched) as a full penny; stretching upward with time. (After C. Atkinson, J.M. Martinez-Esnaola, M.R. Elizalde, Mat. Sci. Technol. 28 (2012) 1079.)

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Fig. 18.20 Crack system associated with sharp indenter during unloading. Saucershaped lateral cracks generate from the base; radial cracks combine with preexisting median cracks (dashed lines) to make half-penny cracks. (After C. Atkinson, J.M. Martinez-Esnaola, M.R. Elizalde, Mat. Sci. Technol. 28 (2012) 1079.)

Upon unloading (Fig. 18.20), P ¼ 0 but B remains unchanged. Hence, on the surface (θ ¼ π/2): σ r , π=2 ¼
7B=r 3 ; σ θ, π=2 ¼ 0 and σ ϕ, π=2 ¼ 2B=r 3

(18.29)

The tensile hoop σ ϕ,π/2 leads to radial cracks. Also, at the base of the plastic zone below indenter, θ ¼ 0. Yoffe’s analysis [20] shows that: σ r , 0 ¼ 12B=r 3 and σ θ, 0 ¼ σ ϕ, 0 ¼
B=r 3

(18.30)

The tensile radial stress at the base creates lateral cracks. Also, σ ϕ becomes tensile at θ  36° from the loading axis and has a maximum on the surface (θ ¼ π/2). The tensile σ ϕ causes existing median cracks to spread upward to combine with radial cracks to form half-penny cracks. During unloading, the unspent elastic energy is recovered by a “spring-back.” Yoffe suggested that B is small for porous or solids that can be densified to a significant amount such as silica glass. B would be large for less compactible solids such as a soda lime silicate glass. In summary, we have shown that median cracks tend to develop during loading and particularly so in glasses such as silica which have a high yield strength (lower B). When yield strength is lower (larger B), radial and lateral cracks grow on unloading. No matter what the glass composition, one system of cracks or the other will develop during loading or unloading cycle of a contact with sharp solid. Thus, damage of glass surface is inevitable due to it being handled subsequent to its forming.

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Fundamentals of Inorganic Glasses

18.9 Fracture statistics Fracture statistics are based on the weakest link criterion according to which the body will fail when the stress at any defect is sufficient to cause unstable crack propagation of that defect. Examples of weakest link distributions are the Weibull distribution, flaw density distributions, and extreme value distributions. Of these, the Weibull distribution is the most popular because of its mathematical simplicity and closer agreement with experimental data. The Weibull theory states that the cumulative probability of failure P for a body is given by P ¼ 1
exp ð
RÞ where R is the risk of rupture given by  ð
σ
σu m dυ, for σ > σ u R¼ σ0 υ

(18.31)

(18.32)

¼ 0, for σ < σ u where υ is the volume, σ is the stress over some volume element dυ, σ u is the minimum stress level for expectation of failure (the fatigue limit), σ 0 is a normalizing stress parameter, and m is called the Weibull modulus. Often σ u is set at 0 without introducing much error. It may be recognized that Y and υ are material and loading constants which can be lumped back into a redefined σ 0 (with units of stress) such that a simplified form of the Weibull distribution is
m

σ : P ¼ 1
exp
σ0

(18.33)

The derivative of the Weibull distribution dP/dσ gives the probability density function p which is the probability that the specimen will fail within σ and σ + dσ, that is, the strength distribution

m
1
m

dP m σ σ p¼ ¼ : exp
dσ σ0 σ0 σ0

(18.34)

Plots of Eqs. (18.33) and (18.34) are shown in Fig. 18.21. Note the skewness of the strength distribution. Taking logarithms of the survival probability (1
P) twice, we obtain from Eq. (18.33):

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513

Fig. 18.21 Weibull distribution. Top figure is the cumulative fracture probability, P, plotted against the applied stress σ for a normalizing stress σ 0 ¼ 50 MPa. Lower figure is the probability density, p (¼ dP/dσ), that a failure will occur between stress σ and σ + dσ. Strength distribution is narrower for a larger value of the Weibull modulus, m.






1 ln ln 1
P






 1 ¼ m ln σ + ln m : σ0

(18.35)

This means that a plot of ln[ln(1/(1
P))] versus ln σ is a straight line whose slope is the Weibull modulus m (Fig. 18.22). Typical values of m are

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Fundamentals of Inorganic Glasses

Fig. 18.22 Weibull plot of the survival probability (1
P) versus the applied stress σ.

between 5 and 15, with 10 as a typical average. Occasionally, one may get more than one straight-line sections indicating the presence of a multimodal distribution of flaws. One could rewrite the Weibull distribution to include the effects of time under stress (which represents static fatigue) and the specimen size (which reflects volume or surface effects as above) as
m
r
q

σ t L P ¼ 1
exp
, (18.36) σ0 t0 L0 where r and q are Weibull exponents for time and spatial dimensions, respectively. Eqs. (18.33) and (18.36) are instructive in the sense that they suggest relatively large increases in fractional failure when any of the processing stresses, time under stress, or dimensions over which stresses are distributed, are increased even a small amount. A typical illustration could be by showing that a glass plant operating at 2% glassware cracks due to processing thermal stresses would experience a 60% increase when the expansion coefficient of the glass rises only 5% from the usual 35  10
7/°C. Note that conventional tests in an R&D laboratory on the glass will not detect any statistically significant

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515

change in strength due to slight expansion increase, but the number of scrap glass barrels will surely rise on the factory floor. Likewise, examining stressed glassware for breakage after a specific time of holding in the warehouse is quite useful for quality control. Because of the varying values of Y, groups of specimens with identical flaw distributions are likely to yield different strength distributions when tested in varying modes.

18.10 Life prediction If a crack velocity versus KI curve is available, then life prediction at constant stress may be made using stepwise integration of the curve of small time steps Δt. The subcritical crack velocity V at some applied stress intensity factor causes the crack to grow Δc, where Δc/Δt ¼ V. A new KI corresponding to length c + Δc is computed which, then, determines new V, and so on. The accumulated time when c equals specimen dimensions gives the life expectancy. For a given material, the quantity within the parentheses is a constant, and hence, we may write log tf ¼
n log σ a + C,

(18.37)

which may be used for life prediction at some other unknown constant applied stress. One other method of estimating life at constant stress is to perform proof testing. Proof testing often involves subjecting a group of specimens to a stress higher than the known service stress. The survival of the specimens at this proof stress guarantees that KI at the tip of the most serious flaw is less than KIc, that is, KIi =σ a ¼ ½KI proof =σ p < KIc =σ Ic : Hence,

  KIi < σ a =σ p KIc :

(18.38)

Substituting Eq. (18.38) into Eq. (18.19) we get a minimum time-tofailure
n
2 .

n
2 2 2  σp KIc σ a AY ðn
2Þ : (18.39) tmin ¼ 2 σa

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Fundamentals of Inorganic Glasses

In Eq. (18.39), tmin is independently a function of (σ p/σ a) ratio and σ a. A plot of log (tmin) versus log(σ a) yields [25] a family of shallow lines as shown in Fig. 18.23, which can be used for life prediction after the specimens have survived a proof stress. Clearly, the larger the ratio (σ p/σ a) the longer is the survival time at any given applied stress. If Weibull probability data of fast fracture are available, then one does not need proof testing for life prediction. Instead, the procedure recommended by Evans and Wiederhorn [25] may be followed to yield a tmin for a given fracture probability. One question often asked is, “Would the specimens weaken after a proof test?” The question is based on the argument that subcritical growth of flaws is likely to have occurred during proof testing which could lead to weakening.

Fig. 18.23 Proof test diagram for soda lime silicate glass in water. The numbers below each line refer to the proof stress/applied stress ratio. (After A.G. Evans, S.M. Wiederhorn, Int. J. Fract. 10 (1974) 379. Reproduced with permission of Kluwer Academic Publishers.)

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517

The counter-argument is that the testing would have weeded out the low strength specimens; hence, the average failure strength σ ∗f of the remainder is likely to be higher than the average failure strength σ fi of the initial population. Evans and Wiederhorn have shown that σ ∗f > σ fi, so long as the Weibull exponent m < (n
2), where n is as defined in Eq. (18.17).

18.11 Experimental measurement of glass strength and fracture toughness Conveniently shaped specimens (generally circular or rectangular crosssection rods/beams) are subjected to increasing loads using a universal mechanical tester, for instance, an Instron machine. Strength is given by the load divided by the area at the point of fracture. The simple experiment, unfortunately, is loaded with problems. If a uniaxial pull is carried out, one must make sure, by using universal ball joint motion, that the pull is indeed along the length of the specimen and that there is no shear component of the pull acting particularly at the grips. Often it helps to use self-aligning stranded wire and specimen which has larger cross section at the ends to be gripped than in the middle. A more convenient way to test the strength is to perform centrally loaded three-point, or preferably, four-point beam bending. The measured strength in such testing is also called the modulus of rupture (MOR) (see ASTM C158-84). In a three-point loading scheme (Fig. 8.3, Chapter 8) a rod of glass is supported on two pegs (generally steel pins placed in a grooved steel plate) and loaded with a force F in the middle (again, using a steel pin). The stress σ on the mid-vertical plane at a point y away from the neutral axis is determined by the well-known formula: σ¼

My , I

(18.40)

where M ¼ maximum bending moment ¼ (F/2)(L/2). L ¼ length of the beam between the pegs. I ¼ geometric moment of the beam cross section. ¼bh3/12 for a rectangular beam (b ¼ width, h ¼ height). ¼πa4/4 for a circular rod of radius a. Maximum tension is on the convex surface. Thus, y ¼ h/2 for a rectangular beam, or a for a circular rod. In the four-point bending experiment (Fig. 8.4, Chapter 8), the maximum tensile stress is uniform along the outer convex surface between the inner pegs. The value is obtained by substituting M ¼ (F/2)[(L1
L2)/2] in Eq. (18.40).

518

Fundamentals of Inorganic Glasses

Another popular flexure test is the ball-on-ring test (ASTM F-394), where a load F is concentrically applied using a ball on a thin plate of thickness t supported on a ring of radius a. When flexure is taken to the point of rupture, the concentric ball behaves as if it was a contact ring of radius b over which the stress is uniformly constant. The radial and the hoop stress σ at the center are equal and given by

3F ð1 + νÞ σ¼ 4πt 2





b2 ð1
νÞa2 1 + 2 ln , + 1
2 2a ð1 + νÞR2 b a




(18.41)

where R is the radius of the plate (> a). It has been shown [26] using strain gages that the formula works best if we assume b ¼ t/3. The ring-on-ring flexure test (ASTM C-1499-15) allows sampling of a larger inner volume for strength measurement, much like a 4-point bending test. In the MOR tests, the biggest problems are the friction at the pegs and the applicability of the linear beam theory. Depending on the beam length to height ratio, the center deflection should not exceed 2–3 times the beam height. Biaxial ring tests have similar problems; however, one major advantage of the biaxial tests is the absence of edges in the region of the maximum stress. Glass containers are often tested using one of two methods: thermal shock stress test (ASTM C-149) and the burst-pressure test (ASTM C-147). Thermal shock testing was discussed in Section 10.7. In a burst-pressure test, containers are pressurized using fluids or gas until failure occurs. The measurement setup can be quite complex for concerns over operator safety. The advantage, however, is that the entire glass product volume exposed to internal pressure is tested for strength. Strength measurements may also be performed by impact testing, generally by impact of a steel ball by a pendulum swing (DIN 52295-2010), or dropping a stainless steel ball from a known height (ASTM F-3007 for laminated glass). For instance, mobile devices with chemically strengthened cover glass are expected to survive a series of 1.6 m drops onto rough asphalt. To ensure that the failure does not originate from an experimental artifact, it is strongly advised that strength tests are carried out, in general, in conjunction with a determination of the fracture origin. To keep the pieces of the specimen together after failure, one may use an adhesive tape, at least on the nontensile surfaces. Because failure generally originates at the surface from the site of a severe flaw, most as-received products show wide variation in measured strengths

Strength and toughness

519

because of a wide distribution of surface flaws. A 20%–50% variation is not uncommon. Strength measurements, therefore, involve breaking a number of specimens. Recommended minimum number of samples is generally 30 for the measurements to give statistically meaningful results. If the intention is to examine the effects of a certain processing step not related to the surface, then it is an accepted technique to abrade the specimen with a known distribution of flaws and obtain what is called the abraded MOR. Dry SiC grit (of a specific mesh size) may be dropped from a fixed height on the specimen surface to be tested (see ASTM C158-84). Fracture toughness and the stress-corrosion susceptibility coefficient n of glass are measured by subjecting specimens of a defined initial crack geometry to a variety of loading techniques and measuring the time-to-failure under constant stress (static fatigue), failure stress as a function of stressing rate (dynamic fatigue), or crack velocity. Appropriate equations to calculate KIc and n are discussed earlier. Popular standard specimen configurations and loading conditions are the double cantilever, double torsion, and the notched bend methods outlined by Freiman [27]. Fracture parameters may be determined from fractographic features. This branch has been of particular interest in recent years. Fig. 18.24 shows redrawn schematics of Fig. 18.12. Each of the boundaries is characterized by their radii. The initial flaw of size ai (semiminor axis on the fracture plane)

Fig. 18.24 Features observed on a glass fracture surface: ri, mist boundary; ro, hackle boundary; rcb, macroscopic crack branching; ai, initial flaw size; acr, size at which flaw extends catastrophically; bi, the initial semimajor axis of the flaw at the open surface. (The dimensions are often, asymmetrical.) (After J.J. Mecholsky, A.C. Gonzalez, S.W. Freiman, J. Am. Ceram. Soc. 62 (1979) 577. Reproduced with permission of the American Ceramic Society.)

520

Fundamentals of Inorganic Glasses

and semimajor axis bi grow to critical size acr and bcr at which point KIc is reached and the flow propagates rapidly. (Unfortunately, there are no markings on the surface at acr.) Experimentally it is observed that
1=2

σ f ¼ σ a + σ r ¼ Mj rj

,

(18.42)

where j ¼ 0, 1, 2, 3 refers to critical flaw, mirror-mist boundary, mist-hackle boundary, and crack branching, respectively, σ f is the true stress at failure, σ a is the applied stress, and σ r is any residual stress, if present. The terms Mj are called mirror constants. The validity of Eq. (18.42) is demonstrated in Fig. 18.25 where the log(σ f) has been plotted against log r1 for a variety of glasses with solid lines drawn at slope ¼
1/2. Also, ðacr bcr Þ1=2 ¼ c ∗ ¼ constant

(18.43)

In effect, c* is the equivalent radius of a semicircular crack which would lead to spontaneous crack propagation under the applied stress [28].

Fig. 18.25 Plot of the log(fracture stress) versus outer mirror radius for various glasses. Solid lines are drawn with a slope
1/2. As2S3 (); vitreous carbon (), borosilicate (▪); soda lime silicate (●); aluminosilicate (5); lead silicate ( ); leached Vycor (□). (After J.J. Mecholsky, R.W. Rice, S.W. Freiman, J. Am. Ceram. Soc. 57 (1974) 440. Reproduced with permission of the American Ceramic Society.)

Strength and toughness

521

It has been found that: (1) r1/c* ¼ 5–7 in the absence of residual stress. (2) Mmirror-mist ¼ 1.7–1.9 MPa m1/2 and Mmist-hackle ¼ 2.0–2.1 MPa m1/2. Now, for the elliptical flaw with equivalent semicircular radius c*: pffiffiffiffiffiffiffiffiffiffiffiffi σ f 1:2πc ∗ KIc ¼ , (18.44) ϕ where ϕ is an elliptical integral of the second kind:
 ð π=2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2
a2 2 ϕ¼ 1
2 sin θ dθ b 0 π ¼ for b ¼ a ðhalf -penny shapeÞ , 2 we obtain ϕKIc Mj ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1:2πc ∗ =rj

(18.45)

(18.46)

When M is plotted against KIc, data on most brittle solids fall in a band around hackle/flaw size ratio equal to about 13. Vickers indentation may be used to determine KIc. For the Vickers indentation, Anstis et al. [29] have developed the following expression for KIc:
1=2 E
3=2 KIc ¼ ΩP c0 , (18.47) H where 2c0 is the post-indentation equilibrium length of the radial crack on the surface from end to end (>3 times the indentation diagonal), E is the Young’s modulus, H is the hardness, and Ω is a material-dependent constant for Vickers-produced cracks ¼ 0.016 for glasses. Because of the problems associated with a complex crack system, the values of KIc so obtained are not very reliable and are at best within about 20% of those measured by more conventional techniques. It is often recommended to call this measurement the indentation toughness KS. Some research [30] has also been performed to employ the indentation itself as a precracked specimen with a defined KI and allow these cracks to propagate under different conditions of loading.

18.12 Data on strength and fatigue parameters As indicated in Section 18.3, strength is ordinarily not an intrinsic property to quote; strength of a glass product varies with the extent of its handling.

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Fundamentals of Inorganic Glasses

The majority of research interest has been on the issue as to how close one could get to the theoretical strength predicted by Eq. (18.6). In all such studies, specimens have been glass fibers, the interest motivated by the potential application of fibers in long distance telecommunications (see Chapter 19). Some of the highest values reported are 13 GPa for fused silica, 8.5 GPa for S-glass and 6.3 GPa for E-glass [31]. The reported high strengths undoubtedly reflect the corrosion-free and abrasion-free environment processing in addition to the low total surface area to decrease the absolute number of surface defects. Fracture toughness KIc and the stress-corrosion susceptibility coefficient n have been studied for several glasses. Typical values for fused silica, conventional soda lime silicate, Pyrex borosilicate, aluminosilicate (57SiO2 15Al2O3  5B2O3  7MgO  10CaO 6BaO; wt%), and a lead silicate (35SiO2  4Na2O 61MgO; wt%) are 0.75, 0.7, 0.77, 0.85, and 0.63 MPam1/2, respectively. Values of n generally range between 12 and 35, although occasionally as high as 70 has been reported. The rather low values of KIc relative to those found typically for metals (15–50 MPa m1/2 for aluminum alloys, 50–200 MPa m1/2 for steels, and 75–125 MPa m1/2 for titanium alloys) show why metals may be preferred in applications requiring high strength under load over long periods. Even polycarbonates have KIc of the order of 3 MPa m1/2 despite the low fracture strengths. Chalcogenide glasses in the Ge
Sb
As
Se
Te systems have been reported [32, 33] to have even lower KIc values. These range between 0.15 and 0.3 MPa m1/2. Fracture surface energies γ f for a number of glasses are shown [27] in Table 18.1. Variation of γ f versus Young’s modulus, Al2O3 and B2O3 content in a compilation of roughly 300 separate records of various glasses is shown by Freiman et al. [34] There are some interesting trends, perhaps with E, but much of the trends are inconclusive.

18.13 Stronger glass products: Strengthening, hardening, and toughening By now, we should be able to distinguish between the terms strengthening, hardening, and toughening. Strengthening implies increasing stress-tofracture σ f. Hardening implies reducing the plastic deformation around an indent. Vickers hardness Hv is a measure of this. Toughening implies increasing the area under the stress-strain curve prior to fracture and is measured by KIc. Strengthening would normally lead to toughening, but the reverse is

Strength and toughness

Table 18.1 Fracture surface energies of some glasses Glass Environment

Pyrex

Soda-lime (or float glass)

Fused silica

Aluminosilicate

Air, 20°C, 20% RH Air, 22°C, 40% RH N2 (gas), 27°C, <0.1% RH N2 (l), 77 K Water, 20°C Air, 20°C, 20% RH N2 (l), 77 K N2 (l), 77 K N2 (gas), 27°C, <1% RH Air, 22°C, 40% RH Vacuum, 10
1 Torr N2 (gas), 27°C, <1% RH N2 (l), 77 K Air, 22°C, 40% RH N2 (gas), 27°C, <1% RH N2 (l) 77 K Air, 22°C, 40% RH

523

γ f (J/m2)

4.7 4.0 4.5–4.8 4.7 2.5 3.9 4.1 4.5–4.6 3.8–3.9 3.5 5.0 4.3–4.4 4.6 3.7 4.6–4.7 5.2 3.7

After S. W. Freiman, T.L. Baker, J.B. Wachtman, in: C.R. Kurkjian (Ed.), Strength of Inorganic Glasses, Plenum Press, New York, 1985, pp. 597–608.

not true. Toughening is important when the applied force is momentary, such as in an impact particularly in a ballistic situation. The applied energy is absorbed by tough products, leaving little energy to cause crack propagation. A hardened surface is important mostly to reduce the creation of surface flaws due to abrasion with contacting bodies. One must also argue that there is no such thing as an “unbreakable glass,” a “bullet-proof glass,” a “fire-proof glass,” or similar terms used in (occasionally deceptive) marketing of glass products. Just like hearts, given enough applied stress, a glass product will break. Essentially everything, including metals, ceramics, glass, dust particles, and even human hands, coming into contact with glass has the potential of causing surface damage resulting in a lowered practical strength. Strengthening, hardening, and toughening of glass products, therefore, is of considerable importance to glassware manufacturers. At stake are the issues involving product safety in the hands of consumer including legal costs involved in personal injury and property damage, process yields during manufacture, and competition from alternate materials. The last item involves issues such as lightweighting where glass containers should be made lighter for a higher strength/weight ratio to retain their market share.

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Fundamentals of Inorganic Glasses

The underlying concepts to strengthen glass, that is, increase σ f are essentially six: (1) reduce the severity of the flaws (reduce KI); (2) control of environment in the immediate vicinity of the crack tip (increase KIc); (3) polymeric coatings to reduce risk of surface damage (reduce c); (4) lamination to increase mechanical stiffness and to arrest crack propagation, if any; (5) introduce compression in the surface (increase σ a to reach σ f level) by thermal tempering, chemical strengthening, mechanical tempering, overglazing, or by surface crystallization (6) crack pinning, deflection or crack-tip shielding (increase KIc). Firepolishing glassware during glass forming is the simplest way to reduce the severity (c/ρ) of the flaws. The flow of glass rounds out or even closes the flaws such that either c decreases or ρ increases or both. Likewise, etching, generally using 1%–6% HF or 1%–6% ammonium bifluoride [NH4HF2 (“ABF”)], can evenly dissolve glass such that flaws are reduced. (Note: higher strength HF can be quite corrosive and is likely to have a weakening effect on the glass.) In many instances, autoclaving glassware under pressurized steam or water can also remove the flawed surface. Because subsequent glass handling by consumer will always produce flaws on exterior surfaces, these methods are helpful only during glass processing or for strengthening unexposed glass surfaces. The control of environment in the immediate vicinity of the crack tip and reduced risk of surface damage can both be achieved by coating with nonpolar polymers such as lacquers, Teflon, Lexan, UV-curable polyacrylates, silanes, etc. The polymers act as barriers to reduce the thermodynamic activity of water and provide an inert environment around the crack tip such that not only the chemical durability is improved but also the fracture surface energy is increased, thus causing increased strength. Additionally, the risk of surface damage to glass products is reduced by “hot-end” and “coldend” coatings. The hot-end coatings essentially consist of SnO2 or TiO2 deposited by passing a mixture of SnCl4/TiCl4 vapors and steam near the entrance to the annealing lehr. Both the oxides are believed to increase the abrasion resistance of the glass surface, thus preventing them from being damaged by contacts with metal surfaces (particularly the conveyor belt). They do not, however, protect glass from getting abraded by rubbing against each other. The cold-end coatings are stearates, oleates (soaps and lubricants), and polyurethanes applied by spraying at the exit end of the annealing

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lehr at a temperature not too high to impart thermal shock, but high enough to dry them. The increased lubricity reduces risk of damage during packing in the cartons. In recent years, laminated glass has become extremely popular with its use as a structural component for architectural applications (ASTM C-1172). Flat glass plates are bonded to each other using commercially available sheets of tear-resistant and ultraviolet (UV)-resistant polymeric interlayer (such as polyvinyl butyral “PVB”, ethylene vinyl acetate “EVA,” or thermoplastic polyurethane “TPU”). The purpose of the polymeric interlayer is threefold: (a) It provides an effectively “seamless” thicker glass which can readily take a desired load. (b) It keeps any broken glass shards together. Risk of personal injury is lower if the thickness of the inboard ply is 1.5 mm or less. (c) The interlayer is able to absorb some impact energy. Thus, it becomes a barrier for continuing crack propagation. It is possible for only one layer to break while keeping the other layer intact. Use of laminated glass as automotive windshields, storefront windows, display cases, floors, staircases, walls, and columns is now a commonplace for esthetic reasons. Outstanding example is the Skywalk on Grand Canyon and the glass floor in the upper deck of the Tower Bridge in London. Aircraft cockpit windshields are actually laminates of two or three plies of chemically strengthened glass bonded to each other using thick layers of PVB. Glasses for ballistic applications invariably are laminated plies with the inboard ply being a polycarbonate. Compressive stresses in glass surfaces are helpful in strengthening because any applied tension must overcome the compression before subcritical crack growth can occur. Surface compression is readily introduced by thermal, mechanical, and chemical tempering, glazing, and surface crystallization. In Section 13.8 we saw that individual layers of glass cooler on the outside pass through the glass transition range with varying cooling rates and at different instants of time develop compression in the surface and a balancing tension in the interior. The processing of glass to develop
70 to
200 MPa (
10,000 to
30,000 psi) surface compression is called thermal tempering. The magnitude of surface compression increases with increased cooling rate, thermal expansion coefficient of the glass, and the difference between the expansion coefficient of the glass and the supercooled liquid. Thus, thermal tempering normally involves cooling or quenching a glass rapidly from about 50°C to 100°C above its transition range using jets of cool air (Section 13.9). Occasionally, contact with liquid medium, for instance, molten tin, molten salts

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Fundamentals of Inorganic Glasses

and oils, or contact with solid pads may also be used to effect rapid quenching. Commercially available fully tempered flat glass is air jet cooled and generally has
100 MPa surface compression. Because the stress profile is more or less parabolic (Section 13.8) for the case of air tempering, higher surface compression would lead to the development of higher interior tension which, in turn, poses increased risk of tension-driven glass fracture from an internal or edge flaw during the tempering process. Hence, thermal tempering is rarely carried out to develop surface compression higher than about
200 MPa. Thermal tempering is not very effective for thin-walled glass product, such as a fiber, because it is not easy to establish a meaningful temperature gradient or a cooling rate gradient across the fiber diameter by normal means. Because of a high level of stored elastic energy, thermally tempered products dice into a large number of pieces when a sharp object penetrates through the compressive layer. The pieces so produced are dull-edged and pose little body injury in terms of cuts. The number of pieces and their size is determined by the degree of temper; the higher the temper, the larger the number and the smaller the size. The use of tempered glass instead of normal product in places, such as in doors in building access areas, bathroom sliding doors, and automobile windows (except the front windshield) is, hence, government regulated. (For a suggested reading, see Ref. [35].) Mechanical tempering can be useful for composite glass fiber-type products. Core-clad glass fibers where the cladding has lower glass transition temperature are allowed to cool under simple tensile-loaded condition. The temperature is such that the cladding relaxes; the entire load is borne by the stiffer core. Upon cooling, the redistribution of the “frozen tensile stress” in the core develops surface compression in the clad; the fiber is strengthened [36]. A similar process can be applied to strengthen glass-coated tungsten corona wires used in xerography [37]. Chemical strengthening by ion-exchange stuffing process involves immersing the glass in an electrically heated steel bath of molten alkali salt for typically several hours at temperatures close to but less than the strain point of the glass (Section 14.8.1). For effective strengthening, two features must be considered: the magnitude of surface compression (“CS”), and the depth of this compression layer below the surface, otherwise called the case depth or depth of layer (DOL). A high magnitude of surface compression (of the order of
500 to
700 MPa), but a shallow case depth (of the order of 30–40 μm) may be sufficient for ordinary reusable beverage containers. However, for aircraft windshield applications where bird hits are a major concern, the

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presence of a substantially increased case depth (>300 μm) is vital. For compression to develop on the outside surface, the size of the bath alkali ions should be greater than that in the host glass. Hence, KNO3 bath is typically used for a sodium containing glass, and a NaNO3 bath is employed to strengthen a lithium containing glass. The stress builds up as a result of the dominance of stress production by ion stuffing over stress relaxation due to glass fluidity. The stress production is directly related to the concentration of alkali ions in the glass and their interdiffusion coefficient with the salt ions which typically has an activation energy generally of the order of 120–160 kJ mol
1. Hence, for an overall successful chemical tempering, firstly, the glass composition should have a high concentration of mobile ions, and secondly, since the relaxation is viscosity controlled which has a relatively high activation energy (300–600 kJ mol
1), the lower the exchange temperature the greater the net residual stress. At temperatures above the annealing point, stresses produced are essentially relaxed continuously. Products of fused silica or a Pyrex-type borosilicate glass having only about 4% alkali are not suitable for chemical tempering for failure to meet the first requirement. Common soda lime silicate glass, unfortunately, is generally not very suitable because any efforts to increase the case depth lead to the relaxation of surface compression. Glasses which meet both the requirements reasonably for a commercial operation are mixed alkali glasses, in particular of the Li2O
Na2O
Al2O3
SiO2 system. Apparently, additions of ZnO and P2O5 in the glass composition are generally helpful. Because a typical stress profile in the diffusion zone is quite steep (Fig. 14.13), the interior tension is usually small, and hence, glass products can be tempered to high levels of surface compression, as high as 1 GPa, without the risk of tension-driven failure from an internal or edge flaw. For the same reason, dicing on penetration of the compressive layer does not occur, and even fine glass fibers may be chemically tempered. The compressive stress profile in a glass can be measured using polarized light microscopy assisted with a calibrated strain compensator. Unstressed glass is optically symmetric; however, birefringence readily generates along stress axes (see Section 19.2.6) which is directly related to the difference of principal stresses. One therefore obtains a thin polished section and measures the birefringence as a function of depth below surface using compensators such as a quartz wedge, Babinet-Soleil, Berek or Senarmont [38]. The classification of chemically strengthened glass is done according to the magnitude of surface compression and the DOL (case-depth) as per ASTM C-1422. A more convenient method applicable for flat glasses only is to

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Fundamentals of Inorganic Glasses

use an ellipsometer in which parallel and perpendicularly polarized light rays impinge on the stressed surface and, after punching in to the case-depth level, are internally reflected back out. Example of such an instrument is the Orihara “FSM.” There are some interesting variations applied on the chemical tempering process. These are (i) combination of thermal and chemical tempering, (ii) two-step chemical tempering where the glass may be immersed in a bath containing smaller ions at above the strain point, and subsequently exchanged in a bath containing larger ions at below strain point, (iii) three-step process where ion size for exchange and air reheat steps are carefully separated, (iv) contact pastes, (v) vapor-phase exchange, (vi) electrical field assistance, (vii) microwave assistance, and finally (viii) sonic assistance. Some of these techniques are reviewed by Bartholomew and Garfinkel [39], Varshneya and LaCourse [40], and Karlsson et al. [41]. Smaller ions at high temperatures can penetrate deeply into the glass: a subsequent exchange with larger ions can result in significantly deeper compressive layer relative to that obtained in a single step process. Contact pastes made with exchange salt mixed with an inert carrier such as clay have found much use in stained glass products, particularly for Cu and Ag staining. Because of the long-time processing requirement, most large-scale commercial glass products are not very amenable to the ion-exchange strengthening technique. To increase the kinetics, electric field assistance has been studied. The setup requires electrical separation of the molten salt by the wall of the product, electrodes are inserted on each side, and the current is passed for a certain length of time in a forward direction and reversed for half the time. (See Abou-el-leil and Cooper [42] for the mathematics of the fieldassisted exchange.) The electrical separation itself requires that at least some portion of the glass product stay above the molten salt line. Unfortunately, the portion above the salt line then develops high tensile stresses on the surface after exchange (by stress reversal) and acts as the Achilles’ heel. For this reason, field-assisted ion-exchange process is not a commercial process even for containers where the geometry is natural for salt separation. A sequential short-term traditional or paste-based exchange strengthening of the edges followed by electric field-assisted strengthening can offer some protection at the expense of total process time [43]. Microwave-assisted ion exchange has been shown [44] to yield greatly increased depth of ion exchange. However, beneficial strengthening is yet to be reported. It is possible that increased depth of penetration results from unrecorded increases in the ion exchange temperature, which would

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also result in increased stress relaxation. Likewise, although sonic-assisted ion exchange appears to improve strength somewhat [45], large increases in strength are yet to be reported. Again, it is possible that local atomic relaxation is also assisted by the same sonic wave resulting in no significant strengthening. Glazing is based on developing a surface layer whose thermal contraction coefficient is lower than the matrix. Upon cooling, the higher contracting matrix forces the surface layers to be in compression with the matrix developing the balancing tension. The compression is limited to the lower expansion layer which can be as thin as desired. Glazing can be accomplished in many ways. The simplest one is to fuse a lower softening layer on all outer surfaces (encasement) without significantly distorting the main body of the product. Glass pastes may be applied by a screen printing process, or by flame/plasma spraying. Away from the edges, the body acts like a sandwich or composite cylinder “bead” seal for which the stress formulas are given by Varshneya [46]. For an engineering computation of stresses, one needs to consider the total contraction of the glaze from its set point down to the room temperature relative to that of the substrate glass over the same range. The set point may be thought of as an effective temperature below which the glass acts essentially as a solid and above which essentially a liquid. An example of a commercial product is the Corning Corelle ware, which is a threelayer sandwich product. A selective area glaze-spray method to strengthen glass containers has also been described [47]. Likewise, an effort was also made to develop a “cased-gob” to make a composite thin-walled container by Owens-Illinois [48, 49]. Unfortunately, technological difficulties presumably associated with strengthening the bearing surface and the heel of the container, which generated a low-strength tail, resulted in the abandonment of many such glazing projects. Low-expansion surface layers may also be produced by dealkalization of the surface with steam, or SO2 treatments used in improving chemical durability which produce silica-rich surface layers. In addition, if a small ion after exchange causes the outer layers of glass to have a lower thermal contraction coefficient, then just the one-step process of above strain point exchange with smaller ions can result in the development of surface compression. Likewise, vapor phase treatments at above strain point with salts which have smaller ions, for instance, CuCl, may also produce surface compression. Strengthening by surface crystallization is based on actually two principles. Firstly, the relative contraction of the surface from the set point of the glass to room temperature after the onset of surface crystallization is lesser for the

530

Fundamentals of Inorganic Glasses

surface layers than for the glass substrate. Secondly, second-phase inclusions act to increase the effective KIc, which leads to strengthening for an assumed initial flaw distribution. Surface reduction of heavy metal ions such as Pb by H2, which causes PbO to reduce to metallic Pb clusters [50], and inclusion of fine ductile metal whiskers in the glass also increase glass strength by increasing KIc. In fiber-reinforced plastics, the strength of the composite is actually lower than that of pristine bundle of fibers, but fracture does not occur till much higher strains, in other words, the toughness is considerably higher than that of either of the components. Effective increase of KIc of composites by management of second phase materials such as ductile and brittle precipitates, whiskers, and continuous fibers is a lengthy and complex topic by itself. Advanced readers may find it useful to review Evans [51]. The key idea is that the fracture surface energy of the composite is not a weighted sum of the γ f values of the individual components. There are many possible mechanisms for rather large increases in the composite γ f. The detailed mechanisms depend on the interaction of the advancing crack front with the second phase. Strengthening/surface hardening methods by ion implantation followed by nucleation and crystallization techniques are newer. Laboratory experiments for a Li2O
Al2O3
SiO2 glass using Au+ ion implantation have been described by Arnold [52]. Surface crystallization methods clearly suffer from the technological difficulty of raising the glass product surface temperature to softening which would likely result in some degradation of the product geometry.

18.14 Topological considerations of glass strength Glass network topology is known to control yield strengths [53]. Topology is determined by the glass composition and the various forming treatments applied to it. Example of a compositional approach to improving the strength of glass, otherwise called the “less brittle glass” by Sehgal and Ito [54], is shown in Fig. 18.26. The authors plotted “brittleness” defined by hardness/toughness as measured by measuring the radial cracks at the corners of a Vickers indentation under varying loads against density of a variety of glasses. They showed that silicates displayed a distinct minimum in brittleness around 2.35 g/cm3 density. Apparently, this “least brittle” soda lime silicate glass could withstand a load of 1 kg without initiating radial cracking. In B2O3-containing glasses, on the other hand, brittleness increased monotonically with density. As suggested by Varshneya recently [55], the stress fields around an indentation can be “managed” by optimizing the strength of the blister field versus that of the elastic field as in Yoffe’s [20] description.

Strength and toughness

531

10 Anomalous Glasses (BC)

C A

Brittleness (mm-1/2)

8

6

Normal Glasses (ABDE) B

4 D

SiO2-based Glasses

2

B2O3-based Glasses E

Error in Brittleness Measurements is ±0.1 mm-1/2

0 1.8

2.0

2.2

2.4

2.6

2.8

Density (g/cc)

Fig. 18.26 Brittleness of glass versus density. (After Ref. J. Sehgal, S. Ito, J. Non-Cryst. Solids 253 (1999) 126.)

Specifically, shear yield strength should be maximized [56] whereas the compressive yield strength should be low. As Yoffe’s analysis shows, a glass which deforms fully elastically is likely to develop median and ring/cone cracks during loading an indenter (Fig. 18.18). On the other hand, a glass with high values of the blister field leading to plasticity is likely to develop radial and lateral cracks upon unloading. A useful guidance is shown in Fig. 8.17 where Sehgal and Ito’s glasses fall around ratio of elastic recovery to the total is 70% (corresponding to a Poisson’s ratio of 0.22). Generation of flaw embryo during forming of molten glass can be reduced by having a high yield strength for hydrostatic dilation; network bonds do not stretch to cause interstitial cavitation. We are thus led to conclude that the propensity for flaw nucleation and generation in glass can be reduced by actively managing glass network topology and that glass strength is entirely not dependent on extrinsic factors such as product handling and the surrounding environment as has been taught traditionally. The chemical composition can play a very significant role (the intrinsic factor) in generating and reducing the damage. In years to come, we hope that this structural approach will lead to yet more and more strengthened glass products.

Summary Strength is the applied stress at failure. Glass fails in tension only and densifies under hydrostatic compression. The theoretical strength of a flawless brittle

532

Fundamentals of Inorganic Glasses

solid silicate glass is 35 GPa. Although fibers having strengths 13 GPa have been produced, most glass products fail at 20–70 MPa. The reduction in strength is ascribed to the presence of Griffith flaws on the surface which can be thought of as oblate ellipsoidal hole of length c into glass having a tip radius of atomic dimensions. Fracture occurs when the applied stress σ a at the tip approaches the theoretical strength. According to the static fatigue argument, flaws in glass grow due to stress-assisted chemical corrosion by surrounding fluid such as humidity. The growth of flaws in the subcritical region eventually leads to a high enough velocity when the failure becomes imminent. Glass can withstand low loads for longer times or at higher loading rates. There may exist a minimum value, called fatigue limit, of tensile stress below which crack growth is nonobservable. Glass can fail in one of the three possible modes; mode I is the most common where both the fracture plane and the direction of crack propagation are perpendicular to a normal stress. Mode II and mode III are failures due to shear stresses. In each case, the stress intensity factor, K, defined as pffiffiffiffiffi σ a πc , approaches a critical value KIc. Strength is usually measured in a three-point or a four-point beam bending mode, or a ball-on-ring or ring-on-ring arrangement for flat plates. The most common technique of measuring KIc is to measure the length of the cracks that grow at the corners of a Vickers indentation under a large enough load. The examination of a fractured surface in glass usually reveals a shiny mirror-like region (called “mirror”) surrounding the origin, ending into a fibrous misty texture region (called “mist”), and finally into more jagged region called “hackle.” Further away, one should be able to note Wallner lines which result from the interaction of the fracture front with a reflected stress wave, and Wallner lines are curved away from the origin, and are receded in the region of lower tension. Because of the probabilistic nature of the flaws, values of strength can vary widely. The strength distribution is often described in terms of a Weibull distribution which is based on the weakest link argument. Weibull slopes for common glasses are generally on the order of 5–15. Common commercial ways of improving glass strength are (i) removing the surface flaws, (ii) introducing surface compression, and (iii) pinning the advancing crack fronts, otherwise called toughening. Surface flaws can be removed by fire polishing the glass or by etching with 4%–6% HF. Surface compression can be introduced by thermal tempering (Section 13.9), by

Strength and toughness

533

chemical tempering (Section 14.8.1), or by coating with a low thermal expansion coefficient glaze. Addition of second-phase particles, generally by partial crystallization in the surface, helps to pin the fracture fronts. The newer concepts to improving the strength of glass products could be based on managing the glass network topology which reduces the generation of flaw embryos in semi-molten glass during the forming stage and their growth during solid state of a product.

Exercises (1) Groups of glass specimens having an identical cross-sectional area, square and circular, are being tested for strength under (i) uniaxial stress, (ii) three-point bend, and (iii) four-point bend modes. Will there be any difference between the observed strength? Why? (2) Why does a concentric ring test give a more reliable measure of the strength of glass plates than a four-point method on beam-shaped specimens of the same plate? (3) The difference between the breaking strengths of specimens before and after ion exchange strengthening is usually significantly less than the surface compression. Why? What if the surface flaws from the glass were removed (by etching with dilute HF) prior to ion exchanging? [Hint: See D.H. Roach, A.R. Cooper, J. Am. Ceram. Soc. 71 (1988) C192.] (4) A 15Na2O10CaO75SiO2 (mol%) glass plate 10 cm  10 cm  5 mm is ion exchanged by immersing in KNO3 salt bath at 475°C for 11.11 h. Assume that the interdiffusion coefficient is constant and is given by 10
10 cm2 s
1. (a) Calculate the diffusion profile of K+ (ignore diffusion at the ends). (b) Compute a network dilation coefficient from database densities of the termini compositions. (c) Calculate the stress profile from the diffusion profile. (5) Several years ago, the baking dishes were made using a low-expansion sodium borosilicate glass. The vendor then switched the glass to a thermally tempered soda lime silicate glass. Consumer reports of glass breakage have increased. Discuss potential causes of breakage and why the breakage may be acceptable to the manufacturer. (6) Why is a low strength tail undesirable? Why does one prefer to have a high Weibull modulus?

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Fundamentals of Inorganic Glasses

(7) Internal burst-pressure test is being conducted on a group of bottles at a manufacturing plant. Discuss why one might expect to see multimodal distribution of strengths. (8) What are the primary obstacles in “lightweighting” a glass bottle? (9) What are the primary obstacles in “lightweighting” a glass window?

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