Anomalous Subcritical Crack Growth in Silica

Silica shows the effect of subcritical crack growth in humid environments. Measurements in liquid water show increasing subcritical crack growth velocities when the temperature is increased as was shown by Wiederhorn and Bolz. Since this has been generally found for glasses, this effect is called normal subcritical crack growth. For measurements on silica in water vapour environment the astonishing effect of decreasing crack-growth rate ν at an increased temperature was observed for constant partial water pressure in the humid environment. This surprising result observed in v–K experiments by Suratwala and Steele is called anomalous subcritical crack growth behavior. In the present report we consider the effects of reduced water solubility at silica surfaces and crack-tip shielding as the reasons for anomalous subcritical crack growth. Three influences were found, namely the water vapour pressure, the temperature-dependent surface solubility of water, and crack-tip shielding. From our computation, we can conclude that silica shows normal subcritical crack growth, when it is taken into account that the real physical stress intensity factor KI is used that describes the stresses in the singular crack-tip field, i.e. when crack growth velocity v is plotted vs. Ktip.


INTRODUCTION
Subcritical crack growth behavior of silica is usually presented in form of v(K)-curves where ν is the crackgrowth rate and K the externally applied stress intensity factor. Results are reported as a function of relative humidity and temperature in [1][2][3]. They show an increase of subcritical crack growth rates at constant K with increasing temperatures. For silica the astonishing effect of decreasing crack-growth rate ν at an increased temperature was observed for constant partial water pressure in the humid environment. This surprising result observed in v(K) experiments by Suratwala and Steele [4] is called anomalous subcritical crack growth behavior. In [5] we discussed the apparently incompatible temperature trends qualitatively. In this paper quantitative computations will be carried out.

EXPERIMENTAL CRACK GROWTH RESULTS
FROM LITERATURE In order to discuss anomalous crack growth, some basic results from literature are compiled. In Fig. 1a v(K)-curves for constant vapour pressure of p = 2100-2200 Pa are plotted [4]. There is a clear decrease of crack velocity with increasing temperature θ visible that is the reason for the notation "anomalous" subcritical crack growth.
Measurements on silica in liquid water were published by Wiederhorn and Bolz [1] as shown in Fig. 1b for the temperature region 2°C ≤ θ ≤ 90°C. From this figure it is evident that the crack-growth rates increase with increasing temperature. At first glance, this seems to be in contrast to the results of Suratwala and Steele [4].
In this report it will be shown that the main difference between these contrary results is the fact of strongly different water vapour pressure in the two tests. Whereas in the tests of Fig. 1a the pressure was kept constant at 2.1-2.2 kPa, the saturation pressure varies in the water tests of Fig. 1b strongly, namely a factor of >100 (0.6 kPa at 2°C and 70 kPa at 90°C).
Consequently, the water concentrations at the glass surface and at the crack tip are clearly different. This will be outlined in the following theoretical considerations.
RT v constants. In Eq. (1) the quantity k r (T) is the rate coefficient for the single-step reaction showing the temperature dependency with ΔG as the free energy of activation. From Eqs. (1) and (2) we get with Q 1 = Q + ΔG Under a water vapour pressure p, and the solubility for molecular water at the surface, C 1kPa , the water concentration reads When evaluating the temperature dependence, it should be noted that both pressure and solubility also vary with temperature. These influencing factors will be considered here.

EFFECT OF VAPOUR PRESSURE
The crack-growth rates from Fig. 1b are transformed by Eq. (3a) to a constant vapour pressure of p = 3.2 kPa (saturation pressure at 25°C), Fig. 2, exhibiting clearly the same temperature trend as found by Suratwala and Steele [4]. It should be noted that in the plot only the ordinate has been reduced to constant pressure and the abscissa values remain unaffected. From this plot one could claim that even in liquid water anomalous subcritical crack growth is present.
EFFECT OF WATER SOLUBILITY Since the v-K-curves should depend on the water concentrations at the glass surface, Eq. (1), results of water concentrations are shown in Fig. 3a as a function of temperature. These data represent the concentra-  tions of the molecular and hydroxyl water species obtained from measurements by Zouine et al. [7] under saturation pressure. Silica reacts with the silica network according to (5) with the concentration of the hydroxyl S = [≡SiOH] and that of the molecular water C = [H 2 O]. In molar units, the total water concentration C w is given by (6) where k is the equilibrium constant describing the ratio of k = S/C.
A relation for the ratio S/C of the two water species, where S stands for the hydroxyl concentration and C for the concentration of molecular water, was given by Wiederhorn et al. [8]. For the temperature range of 90°C ≤ θ ≤ 350°C (T = 273°C + θ), the data are fitted by the straight line (7) with A = 32.3 and Q = 10.75 kJ/mol. Equations (6) and (7) result in (8) (9) and in mass units Note: the ratio 17/18 reflects the different mole masses of water and hydroxyl. Figure 3a shows the concentrations of the two water species under saturation pressure. In Fig. 3b the water concentrations S ∝ p and C ∝ p are normalized on the water vapour pressure of p = 1 kPa. The straight lines can be expressed by The crack velocities of Fig. 1 are plotted in Fig. 4 in the normalized representation as v/(C × T) vs. applied K for the partial water vapour pressure of p = 3.2 kPa (i.e. the saturation pressure at 25°C). It can be seen that the temperature tendency of Fig. 4b is now again as normal crack-growth behaviour. This makes clear that even for constant vapour pressure the anomalous crack-growth behaviour is at least partially caused by the reduced surface water concentration with increasing temperature.

EFFECT OF SHIELDING STRESS INTENSITY FACTOR Constant Shielding
In our opinion, at least a part of the observed trend is caused by crack-tip shielding. In the preceding considerations we implicitly identified the externally applied stress intensity factor K appl with the K-value acting at the crack tip, K tip . It was early outlined in [9,10] that a shielding stress intensity factor must develop as a consequence of volume increase by the hydroxyl generation (for references see e.g. [11]). Computations by McMeeking and Evans [12] result in a shielding stress intensity factor K sh for a grown crack which is surrounded by a zone of thickness ω undergoing a volume strain ε v (12a) with ψ = 0.22 [12]. E is Young's modulus and ν Poisson's ratio. In case of glass, ω is the thickness of the water diffusion zone around the crack tip. It has been shown in [13] that water-induced swelling at crack tips generates an intrinsic stress intensity factor K sh < 0 that shields a crack from an externally applied loading K appl so that the stress intensity factor K tip at the crack tip is reduced. The effective stress intensity factor acting at the crack tip, K tip , represents the singular stress field at the tip. From the principle of superposition, K tip is given by (13) The effect of the shielding stress intensity factor on the v(K)-curves was outlined in [13]. The volume strain ε v is related to the hydroxyl concentration S by the proportionality (14) with κ ≅ 0.97 at least at low hydroxyl concentrations S as could be concluded from the density decrease reported by Shelby [11]. Consequently, it holds

SE K
The hydroxyl concentration S in the crack-tip stress field is thermally activated as Eq. (7) implies. Since also the diffusivity is thermally activated, the water diffusion zone, ω = f(√D t), increases with temperature due to the increase of the diffusivity. Consequently, the shielding stress intensity factor K sh = f(S, ω) must become stronger with increasing temperature via an Arrhenius equation. When in a subcritical crack growth test the shielding stress intensity factor equals −K appl , the total stress intensity factor K tip disappears followed by abruptly decreasing crack rates, i.e. the threshold values K th of the ν(K)-curve is reached. This is equivalent to (15) Threshold values for silica were evaluated by Sglavo et al. [14] on the basis of literature results, [15,16]. The threshold stress intensity factor as a function of temperature at ≈2 kPa partial water pressure was described by an Arrhenius temperature dependency (16) with K th,0 = 0.638 MPa√m and q = 1.83 kJ/mol.
The value K th,0 in Eq. (16) depends on the water partial pressure, for which unfortunately no measurements are currently available. For this reason, the procedure described here is not suitable for real predictions, but may be suitable for qualitative interpretations.
Equations (15) and (16) make it possible to calculate the shielding stress intensity factor K sh and via Eq. (13) the total stress intensity factor K tip acting at the crack tip. This value is responsible for the crack-tip stresses and subcritical crack growth rates. Fig. 4. Crack-growth rates normalized on the molecular water concentration and temperature T. Data from Fig. 1a (a), data from Basically, the steepness of the ν(K appl )-curve is influenced by two opposite effects. On the one hand, an increase of K appl causes an increase of the shieldingstress intensity factor [12]. On the other hand, the increase in K appl and thus in the crack velocity ν greatly shortens the time available for diffusion in the crack tip region and reduces the size of the shielding zone.
Since there are no model calculations on the steepness of the v(K appl )-curve in the literature, it remains unclear which influence dominates. An influence of the applied stress intensity factor on the shielding term is illustrated here by two arbitrarily assumed behaviours, namely constant and decreasing shielding with increasing stress intensity factor. Finally, Fig. 5 gives the curves of Fig. 4 νs. the difference K appl -K th . Under the assumption that K sh ≠ f(K appl ), the curves in Fig. 5 agree with the v(K tip )curves.
The diagrams of Fig. 5 show that, for the same water concentration, plotting against the stress intensity factor K tip (actually prevailing at the crack tip) leads to normal subcritical crack growth in both cases, namely increasing crack rates with increasing temperature.

Variable Shielding
Deviations from the assumption of a constant shielding stress intensity factor K sh have to be expexted. With increasing crack-growth rates the time for the water entrance via diffusion decreases and, consequently, the zone height ω in Eq. (12b) becomes smaller. This causes a decreasing shielding stress intensity factor K sh with increasing v or increasing load. A computation of K sh (K appl ,T) was performed in [5] resulting in rather complicated equations and should not be duplicated here. An approximate method may be suggested to estimate K sh or, equivalently, K tip as a function of K appl . For this purpose some basic results may be used.
(1) According to the derivation by Wiederhorn, the stress intensity factor K in Eq. (1) is the physically active total stress intensity factor K tip governing the singular crack-tip stress field. In a log(v)-K tip -plot this must result in a straight line: (17a) (2) As the subcritical crack growth curve for silica in liquid water at 25°C (Wiederhorn and Bolz [1]) shows, the crack velocity as a function of the applied stress intensity factor is also very straight over a large region of crack rates, Fig. 6a. This can be expressed as (17b) Equation (13) requires that even the shielding stress intensity factor must fulfil a similar relation, namely, If K sh depends on v there must occur an intersection of the straigt lines according to (17a) and (17b).
(3) When K tip reaches fracture toughness K Ic , the crack extension becomes very fast. Velocities in the order of sound velocity occur. Consequently, diffusion effects of surface water into the crack-tip region will disappear as well as the related shielding term, i.e. K sh (K Ic ) → 0. This situation is indicated by the circle in Fig. 6b.
From the points (1)-(3) we obtain in terms of K tip (18) and in terms of K appl or (20) Figure 7 represents the ν(K tip ) results for the data in Figs. 4a, 4b. Even from these plots normal subcritical crack growth can be stated. With increasing temperature, the crack growth rates for identical water concentrations at the glass surfaces increase, too. This holds for tests in liquid water as well as tests in humid air. Figure 8 shows the stress intensity factor K tip necessary for reaching a constant normalized crack velocity.
There seems to exist a discontinuity in Fig. 8 between 75 and 150°C. A possible reason might be capillary condensation at crack tips. A zone of liquid water can be identified at a crack tip at low temperatures in humid air as had been shown by Ciccotti et al. [17], Grimaldi et al. [18], and Pallares et al. [19]. At the high temperatures this is no longer possible. So, one would expect a discontinuity for tests in "water" to tests in humid air.
From the subcritical crack growth rates of Fig. 7 the slopes, B, were determined. Additionally the parameter B for a representation by logv ∝ B K appl is introduced for the data by Wiederhorn and Bolz [1]. For the slopes B one would expect from the theoretical analysis by Wiederhorn et al. [6] according to Eq. (1) to all results with the exception of the data by Suratwala and Steele [4] for temperatures below 150°C, see also the jump in Fig. 8. Below this temperature, the effect seems to be slightly reversed. In Fig. 9b the parameter b defined in Eq. (1) is shown. The parameter b is for the representation via K tip a constant. This holds for the data from [1] and the data above ≈100°C from [4] but decreases for lower temperatures as is included in Fig. 9b as the dashed line. The representation by (21) yields (Standard Deviations in brackets) (22a) for the data given in [1] and (22b) for the data given in [4] and ≥150°C.

Expectations for Other Glasses
In the previous considerations, only silica was considered. In this Section, we will look on other glasses, too. Density measurements on several glass compositions were reported by Scholze [20]. The relative density changes due to water content C w are given in Fig. 10a. Whereas for silica a non-linear behaviour Δρ(C w ) is reported in [20], for all other glasses straight-line behaviour was found. It has to be taken into account that the data scatter for silica is rather large as the data reported by Shelby [11] show which are introduced in red. The straight line represents the average of data for as-  received water in silica measured by several authors and quoted by Shelby [11]. The circles are results by Brückner [21] measured by water removal (also quoted in [11]). Whereas silica shows a density decrease with increasing water content, the densities of the other glasses increased with increasing C w . The relative density changes Δρ/ρ 0 were transformed into volume strains ε v via (23) with the results plotted in Fig. 10b and compiled in Table 1.
Whereas the B 2 O 3 -glass shows strong volume shrinking due to water uptake, silica shows clear volume expansion. The effects on the other glasses are rather small. Materials Ca and Al also swell. K shrinks slightly.
In contrast to the crack-tip shielding for silica, it has to be expected that the B 2 O 3 -glass B should exhibit an increase of total stress intensity factors. Slight shielding effects may occur in the soda-lime glass as well as in the Al-glass.

CONCLUSION
We studied subcritical crack growth results on silica measured in both, water vapour and in liquid water under saturation pressure. From the results shown before, the anomalous subcritical crack growth in the experiments by Suratwala and Steele is a consequence of the decreasing surface water solubility with increasing temperature and the plotting against the applied stress intensity factor K appl instead of the physically acting value K tip that governs the singular stress field at crack tips. With other words: Subcritical crack growth in silica is normal since for constant water concentra-  Table 1. Swelling and shrinking strains derived from density curves given by Scholze [20] Glass Composition (in mol %) (in wt %) ρ 0 , g/cm 3 ε v /C w It is also noteworthy that if the temperature dependence of the saturation pressure is eliminated exclusively in the tests in liquid water, anomalous crack growth behaviour would appear (Fig. 2).
This suggests that the representation of v(K)-curves for constant pressure instead of constant water concentration in the glass and by plotting the crackgrowth rates vs. the applied stress intensity factor (instead of the physically relevant crack-tip stress intensity factor, K tip ) is responsible for the apparently anomalous behaviour of silica.