In vacuum, the strength of glass is time-independent.1 In the presence of humidity, how-
ever, stress corrosion causes flaws to grow slowly when they are exposed to a positive crack opening stress. This means that a glass element which is stressed below its momentary strength will still fail after the time necessary for the most critical flaw to grow to its critical size at that particular stress level. The momentary strength of a loaded glass element therefore decreases with time, even if it is exposed to static loads only. This phenomenon, which is fundamental for the structural use of glass, was already discovered in 1899 by Grenet[178]. The growth of a surface flaw depends on the properties of the flaw and the glass, the stress history that the flaw is exposed to, and the relationship between crack velocity and stress intensity.
In the present document, the term ‘stress corrosion’ is used to refer to the chemical phenomenon. The term ‘subcritical crack growth’ is used to refer to the consequence of stress corrosion, i. e. the growth of surface flaws.2
3.2.1 Relationship between crack velocity and stress intensity
First systematic investigation of stress corrosion was conducted by Levengood[237]. An explanation for the chemical process behind the phenomenon was put forward by Charles and Hilling[64] and further developed by Michalske and Freiman [252]. This theory, also known as the ‘classical stress corrosion theory’, involves the chemical reaction of a water molecule with silica at the crack tip (Figure 3.1):3
Si-O-Si+H2O → Si-OH+HO-Si (3.1)
According to this theory, the crack velocity scales with the kinetics of this chemical reaction. Its activation energy depends on the local stress and on the radius of curvature at the
1Even in vacuum, the resistance of many glasses is in fact slightly time-dependent. This effect, called ‘inert
fatigue’, is however of no practical relevance for structural engineering applications.
2In academic publications, this distinction is not always made and other terms, such as ‘slow crack growth’,
‘static fatigue’, and ‘environmental fatigue’ are in use.
3The classical interpretation involving a chemical reaction at the very tip of a crack is questioned by
Tomozawa[321]. As the diffusion of molecular water into the glass is activated by stress, he suggests that this diffusion process and the modification of the glass properties in the crack tip area that it causes might explain subcritical crack growth. A more in-depth discussion is beyond the scope of this document. The interested reader should refer to specialized texts on this subject, e. g.[56, 170, 184, 252, 321, 338].
Glass Water 1 Si O Si Si O H H O O H H O Si Si Si 2 3 H H O Figure 3.1:
Stress corrosion, chemical reaction at the crack tip: (1) adsorption of water to Si–O bond, (2) concerted reaction involving simultaneous proton and electron transfer, and (3) formation of surface hydroxyl groups[252].
crack tip. The theory involves a first order chemical process, which is consistent with the observed linear correlation between the logarithm of the crack velocity v and the logarithm of the humidity ratio H (except for very low H or v)[338].
Figure 3.2 shows the simplified, schematic relationship between crack velocity v and
stress intensity factor KIthat is commonly used for glass lifetime prediction. For values of
KIclose to the fracture toughness KIc(definition→ p. 57) or even above, v is independent
of the environment and approaches a characteristic crack propagation speed (about 1 500 m/s for soda lime silica glasses) very rapidly. In a narrow region below KIc(region III), the curve is very steep, v lying between 0.001 m/s and 1 m/s. In inert environments (cf. Section 3.3.3), this curve would extrapolate linearly to lower crack velocity. In normal environments, the behaviour strongly depends on the environmental conditions. The empirical relationship
v= S · KIn, (3.2)
which was originally proposed by Evans and Wiederhorn[163], provides a good approxi- mation for region I.4The parameters S and n need to be determined from experiments. The unit of S depends on the value of n. This can be avoided with the following equivalent formulation (S= v0· K−n
Ic ):
v= v0 KI/KIcn
(3.3) The crack velocity parameter v0has the units of speed (length/time), n is dimensionless.
When the v-KI-curve is plotted on logarithmic scales, v0represents its position and n its
slope. KIcis a material constant that is known with a high level of precision and confidence
(cf. Section 3.3.1). Regions I and III are connected by region II. In this region, the kinetics of the chemical reaction at the crack tip are no longer controlled by the activation of the chemical process, but by the supply rate of water. It takes time for a water molecule to be transported to the crack tip, such that a shortage in the supply of water occurs as the crack velocity increases[338]. The crack velocity v is, therefore, essentially independent of KI but depends on the amount of humidity in the environment. Below a certain threshold
stress intensity Kth(see Section 3.2.2), no crack growth occurs.
4The exponential functions v= v
i· eβKI and v= vi· eβ(KI−KIc) were also proposed to model the v-KI
relationship. In practice, the difference between a power law with a high exponent and an exponential function is very small. An exponential function has the main advantage of being consistent with the kinematics of the above-mentioned chemical reaction. Equation (3.2), however, allows for much simpler calculations, which explains its predominant use.
Figure 3.2: Idealized v-KI-relationship. Kth log KI Cr ac k v elocit y, v KIB KIc log v
Stress intensity factor, KI
I II III threshold vacuum environment 0( /I Ic)n v=v K K I 0 Ic
log( )v = ⋅n log( ) log(K + v K⋅ −n)
In view of the order of magnitude of glass elements in buildings (mm to m), the typical depth of surface flaws (µm to mm) and the service life generally required, only the range of extremely slow subcritical crack growth, region I, is relevant for determining the design life of a glass element. The contribution of regions II and III to an element’s lifetime is negligible.
3.2.2 Crack healing, crack growth threshold and hysteresis effect
In 1958, Levengood[237] found that aging has an effect on glass surface flaws. Further experimental work in laboratory conditions showed that the strength of flawed specimens increases during stress-free phases[317, 336]. Looking at it in more detail, this effect, generally called crack healing, is a consequence of two phenomena, the crack growth threshold and the hysteresis effect.
At stress intensities below the crack growth threshold5K
th, no significant crack growth
occurs. For typical soda lime silica glass at a moderate pH value, Kth is about 0.2 to
0.3 MPa m0.5(see Haldimann[187] for an overview of available data).
The crack growth threshold was originally explained by a rounding of the crack tip (‘crack tip blunting’) at slow crack velocities[27, 64]. More recent investigations, however, strongly support the hypothesis that alkali are leached out of the glass and that this change in the chemical composition at the tip of the crack is responsible for the crack growth threshold rather than a geometrical change (blunting). Observations of aged indentation cracks by atomic force microscopy did not give any evidence of blunting. Sodium containing crystallites were actually found on the surface of glass close to the tip of the indentation crack. This is more consistent with alkali ions’ migration under the high stress at the crack tip and their exchange with protons or hydronium ions6from the
environment[171, 183, 256].
In alkali containing glasses, there is also a hysteresis effect: an aged crack will not repropagate immediately on reloading. The hysteresis effect is convincingly explained by renucleation of the aged crack in a plane different from the original one, as if the path of
5Also known as ‘stress corrosion limit’, ‘crack growth limit’, ‘threshold stress intensity’, or ‘fatigue limit’. 6A hydronium ion is the cation H
the crack has to turn around the area just in front of the former crack tip. This non-coplanar re-propagation was directly observed by atomic force microscopy[195, 335, 339].
A comprehensive probabilistic crack propagation model, which accounts for the above- mentioned effects, was proposed by Charles et al.[65].
Although its favourable influence can be considerable, crack healing has not been taken into account (at least not explicitly) by design proposals to this day. Because of the strong dependence on the environmental conditions, crack healing is difficult to quantify. The threshold appears to depend strongly on the environmental conditions and on the glass’s chemical composition. It is, for instance, more easily evidenced with alkali containing glasses and in neutral or acidic environments, while there is no evidence of a threshold in alkaline environments[177]. In static long-term outdoor tests, in contrast to tests in the climatic chamber, no evidence of any substantial crack healing or of a crack growth threshold was found[167]. For structural applications, in which safety is a
major concern, it therefore remains advisable not to take any threshold or healing effects into account.
3.2.3 Influences on the relationship between stress intensity and crack growth It is important to bear in mind that the relationship between stress intensity and crack velocity is very sensitive to a number of aspects. A short overview is given in the following. For more details, see[187].
u Humidity. As mentioned before, the water content of the surrounding medium7
strongly influences subcritical crack growth. The effect of an increasing water content is essentially a parallel shift of regions I and II of the v-KI relationship towards higher crack velocities[337].
u Temperature. An increasing temperature causes mainly a parallel shift of the curve
towards higher crack velocities. Furthermore, the slope decreases slightly[338].
u Corrosive media and pH value. The crack velocity generally increases as the pH
value of the surrounding medium increases. Furthermore, the pH value has a certain effect on the slope of the v-KI relationship and a particularly strong influence on
the crack growth threshold Kth[170].
u Chemical composition of the glass. All parameters of subcritical crack growth are
influenced by the chemical composition of the glass[338].
u Loading rate. According to Haldimann[187], the v-KIrelationship does not only
depend on environmental conditions, but is also strongly loading rate dependent. As mentioned before, stress corrosion requires humidity. If an element is loaded rapidly, the diffusion process is not fast enough, so that a shortage in the supply of water to the crack tip slows down stress corrosion and therefore the subcritical growth of flaws. Consequently, the v-KIrelationship of an element is shifted towards
lower crack velocities when loaded rapidly.
Figure 3.3 gives an overview of published v-KI-data8[43, 90, 184, 195, 284, 285, 298,
302, 324, 338]. The following can be concluded:
7It is actually the ratio of the actual partial pressure to the partial pressure at saturation. In air, this
corresponds to the relative humidity.
8When modelling subcritical crack growth, the v-K
I-relationship is generally assumed to be valid over the
full KI-range. This is why the curves that represent design models extend to the entire range of the figures’
0.2 0.3 0.4 0.5 0.6 0.7 10-12 10-10 10-8 10-6 10-4 10-2 100 Cr ac k ve lo cit y, v (m /s )
Stress intensity factor, KI (MPa m0.5)
Models proposed based on experimental data: Richter (1974) - 50% RH - DCB (through crack) Ullner (1993) - Air
Dwivedi & Green (1995) - 65% RH - 4PB dyn. fat. V Dwivedi & Green (1995) - 65% RH - direct optical V Design models:
Blank (1993) - 'summer' (16/4.5) Blank 1993 - 'winter' (16/8.2) n = 16, v0 = 6 mm/s
Experimental data:
Hénaux & Creuzet (1997) - 50% RH - V - AFM
0.2 0.3 0.4 0.5 0.6 0.7 10-12 10-10 10-8 10-6 10-4 10-2 100 Cr ac k ve lo cit y, v (m /s )
Stress intensity factor, KI (MPa m0.5)
Models proposed based on experimental data: Richter (1974) - DCB (through crack) Ritter et al. (1985) - dyn. fatigue (cross lab) Sglavo et al. (1997) - cycl. fat. - V (ai+a) Sglavo & Green (1999) - dyn. fat. - V (ai) Sglavo & Green (1999) - dyn. fat. - V (a) Ullner (1993)
Design models: n = 16, v0 = 6 mm/s
Experimental data:
Gy (2003) - as float (rcs = 0.75 MPa) - DT Gy (2003) - special annealing (rcs = 0.25 MPa) - DT Wiederhorn and Bolz (1970) - Water - 90°C - DCB Wiederhorn and Bolz (1970) - Water - 25°C - DCB Wiederhorn and Bolz (1970) - Water - 2°C - DCB
Figure 3.3: Crack growth data overview in air (above) and in water (below).
(V= Vickers indentation, ai = as indented, a = annealed, DT = double torsion test, DCB = double cantilever beam test, dyn. fat.= dynamic fatigue, rcs = residual core stress.
u General. Crack velocity parameters vary widely and depend on several influences,
including environmental conditions and the loading rate. Fracture strength predic- tions for service lives of many years are, therefore, of limited accuracy.
u Structural design. For structural design, a constant value of n= 16 is a reasonable
assumption. For general applications, v0= 6 mm/s should be conservative9. For
glass elements that are permanently immersed in water, a higher value of e. g.
v0= 30 mm/s is more appropriate.
u Interpretation of experiments. Strength data from tests at ambient conditions are
inevitably dependent on the surface condition and on crack growth behaviour. The large variability of the crack velocity parameters makes it very difficult to obtain accurate surface condition information from tests at ambient conditions [187]. Inaccurate estimation of the crack velocity during testing can yield unsafe design parameters. Testing at inert conditions is, therefore, preferable (Section 6.4).
9Further differentiation of environmental conditions, e. g. considering summer and winter conditions, is
not recommended for modelling purposes. The potential difference between the two cases is very small compared to the scatter of the data. The definition of two parameter sets would therefore be rather arbitrary and would not necessarily increase the accuracy of the model. The complexity of the calculation process, on the other hand, would be increased considerably.