Subcritical crack growth tests were carried out on DCDC specimens of silica by Lechenault et al. [9.7]. The water entrance into the fracture surfaces formed by the passage of the crack exposed to deuterium oxide D2O was evaluated with a neutron reflection technique to measure the penetration of the deuterium oxide into the silica glass. The authors found a satisfactory fit to the reflection data by assuming that the water concentration was constant at the surface up to a distance of L, followed by an exponential decrease in concentration for distances greater than L. Consequently, Lechenault et al. [9.7] fitted their results by the expression

$$\mathcal{C}\_{\boldsymbol{w}} \, ^{\prime}\mathcal{C}\_{\boldsymbol{w}}(\mathbf{0}) = \begin{cases} 1 & \text{for} \quad \boldsymbol{z} \le L \\ \exp[-(\boldsymbol{z} - L)/\eta] & \text{for} \quad \boldsymbol{z} > L \end{cases} \tag{9.3.1}$$

and found the parameters L=4.3 nm, η=3.5 nm for a region of low crack rates (v≈10-8 m/s) at K= 0.61 MPa∙m1/2 and L=4.6 nm, η=2.3 nm for higher crack rates (v≈4×10-6 m/s) at K= 0.77 MPa∙m1/2. These water profiles are illustrated in Fig. 9.4.

#### 9.3.2 Result by Tomozawa et al. [9.8]

The concentration of water below the crack surfaces was measured by Tomozawa et al. [9.8] using the nuclear reaction analysis (NRA). The profile of the hydrogen concentration is shown in Fig. 9.5a. In order to allow an approximate extrapolation of measured data to the surface, we fitted the data cH by a complementary error function

$$\mathbf{c}\_{\rm H} = \mathbf{c}\_{\rm H,0} \operatorname{erfc}\left[\frac{z}{2b}\right] + \underbrace{\mathbf{0}.005 \times 10^{22} / \operatorname{cm}^3}\_{\text{Background}} \tag{9.3.2}$$

taking into account the background level of 0.005×1022/cm3 . The coefficient cH,0 is the surface concentration.

The red bar in Fig. 9.5a indicates the 90%-CI of the surface concentration. The depth b at which the concentration decreased to ≅50% of the surface value in Fig. 9.5a is about 5 nm. By fitting (9.3.2) to the measurements, the best set of parameters was found in [9.9] to be

$$\alpha\_{\rm H,0} = 1.30 \,\text{[1.05S, 1.5S]} \,\text{10}^{22} \text{/cm}^3, \,\text{b} \equiv 4.95 \,\text{[3.28, 6.61]} \,\text{nm} \,(90\% \,\text{CI}).$$

with the mean squares sum of 11.65 (1022/cm3 ) 2 . The maximum hydroxyl concentration at z=0, S(0), could be computed in [9.9] with the result

$$S(0) = 16.5[14.4, 18.4] \text{ (wt\#6)}\tag{9.3.3}$$

For a second description of the water distribution let us be guided by the results of Lechenault et al. [9.7]. We fitted the measurements by Tomozawa et al. [9.8] also via eq.(9.3.1) with the parameters

$$\alpha\_{\text{H},0} = 1.12 \text{ } 10^{22}/\text{cm}^3, \, L = 3.0 \text{ nm, } \eta = 2.96 \text{ nm}$$

and a mean squares sum of 11.78×(1022/cm3 ) 2 . This result is shown in Fig. 9.5b. Having in mind the rather strong scatter of the surface values, the two representations in Figs. 9.5a and 9.5b are equal.

#### 9.4 Computation of shielding stress intensity factors

#### 9.4.1 Evaluation for undamaged material

Subtracting the background level and introducing eq.(9.3.2) into eq.(9.2.6) or (8.1.1), and assuming at the tip S/S0≅cH/cH0, yields

$$K\_{sh} = \bigcap\_{0}^{\alpha} \kappa S(\alpha')h(\alpha') \, d\alpha' = -\wp \sqrt{2/\pi} \, \frac{\kappa S(0)E}{1-\nu} \sqrt{b} \, \Gamma(\frac{3}{4}) \tag{9.4.1}$$

with the Euler Gamma function Γ that for the argument ¾ reads Γ(3/4)=1.2254, so that we can finally write with π≅=Γ 19777.0)4/3(/2

$$K\_{sh} \cong -\nu \frac{\kappa S(0)E}{1-\nu} \sqrt{b} \tag{9.4.2}$$

Comparing eqs.(9.2.2) and (9.4.2) allows to identify approximately: b≅ω. For the representation of the water profile by eq.(9.3.1), the evaluation of eq.(9.2.6) yields

$$K\_{sh} = \bigcap\_{0}^{\alpha} \kappa S(\alpha \prime) h(\alpha \prime) d\alpha \prime = -\wp \frac{\kappa S(0) E}{1 - \nu} \left( \sqrt{L} + \frac{1}{2} \sqrt{\pi \eta} \right) \exp\left( \sqrt{\frac{L}{\eta}} \right) \text{erfc} \left( \sqrt{\frac{L}{\eta}} \right) \tag{9.4.3}$$

From the surface concentration in Fig. 9.5a, given by eq.(9.3.3), the shielding stress intensity factor results as

$$K\_{sh} \equiv -0.21 \text{ MPa} \sqrt{\text{m}} \tag{9.4.4}$$

For the description by eq.(9.3.1) we obtain:

$$K\_{sh} \equiv -0.20 \text{ MPa} \sqrt{\text{m}} \tag{9.4.5}$$

where ≅73% of the total value come from the constant part of the water profile.

Fig. 9.5 Hydrogen concentration in the surface region of a growing crack via NRA-measurements by Tomozawa et al. [9.8], a) representation by eq.(9.3.2), b) expressed via eq.(9.3.1).

#### 9.5 Crack-tip shielding and threshold behaviour

When a grown crack has been unloaded and is again loaded, the crack tip will only see loading by a stress intensity factor if Kappl>-Ksh. Consequently, the shielding stress intensity factor causes a threshold value for the subcritical crack growth as is schematically illustrated in Fig 9.6b. The stress intensity at which Ktotal and crack growth rate disappear in a crack-growth test under monotonously decreasing loading may be denoted as Kappl(Ktip=0)=K0=or Kth where "th" stands for "threshold". Then it holds for a crack-arrest test under decreasing load

$$K\_{\rm anat} = K\_{\rm appl} + K\_{sh} = 0\tag{9.5.1a}$$

or
$$K\_0 = K\_{\text{sh}} = -K\_{\text{sh}}\tag{9.5.1b}$$

Fig. 9.6 a) Shielding stress intensity factors for the unloading case of Ktotal = Kappl+Ksh < 0, b) expected threshold values of the v-Kappl curve, schematic.

It should be emphasized that this threshold behaviour can only occur if the cracks have already grown subcritically (Fig. 9.1b). Cracks that are loaded for the first time can grow immediately, since the shielding zone has a heart-shaped contour (Fig. 9.1a) for which Ksh=0, [9.1]. The shielding stress intensity Ksh is plotted in Fig. 9.1 for a zone of height ω according to McMeeking and Evans [9.1]. These data can be simply described by Evans and Faber [9.10] as

$$\frac{K\_{sh}}{K\_{\circ}} = \frac{2}{\pi} \arctan(\Delta a/o) \tag{9.5.2}$$

The applied stress intensity factor for a constant applied load depends on the crack extension ∆avia

$$\frac{K\_{approx}}{K\_{approx.0}} = \sqrt{1 + \frac{\Delta a}{a\_0}} = \sqrt{1 + \frac{\Delta a / \alpha o}{a\_0 / \alpha o}}\tag{9.5.3}$$

where Kappl,0 stands for Kappl,∆a=0.

Fig. 9.7 Shielding stress intensity factor Ksh normalized on its saturation value K∞, reached for ∆a/ω→∞ obtained by McMeeking and Evans [9.1].

For a crack growing at Ktotal >0, the applied stress intensity factor increases and the shielding stress intensity factor becomes stronger negative. The total stress intensity factor must first increase and then decrease. After a minimum, the total stress intensity factor rises again.

The arrest condition is illustrated in Figs. 9.8a and 9.8b for short cracks of different initial lengths a0/ω and varying saturation values K∞ of the shielding stress intensity factor. The shielding stress intensity factor can depend in principle on the externally applied load and the actual crack rate. For reasons of transparency we assume in Figs. 9.8a and 9.8b constant values for ω and K∞. Crack arrest occurs in those cases for which the curves intersect the line Ktotal=0, indicated by the open circles. The dashed extensions are hypothetic total stress intensity factors which make no physical sense since stress singularities at a crack tip are only present for Ktotal>0. From Fig. 9.8b it becomes obvious that subcritical crack growth for small cracks can take place below that threshold stress intensity factor that occurs in the case of cracks under decreasing loading (here K0=0.2 MPa√m).

Fig. 9.8 Total stress intensity factor for K∞ = -0.2 MPa√m, open circles: arrest condition Ktotal=0; a) Ktotal vs. ∆a for a0 =3ω, b) Ktotal vs. Kappl for different initial crack lengths a0 and K0=0.2 MPa√m.

#### References

9.1 McMeeking, R.M., Evans, A.G., Mechanics of Transformation-Toughening in Brittle Materials, J. Am. Ceram. Soc. 65(1982), 242-246.

9.2 G. Rizzi, T. Fett, Defects at silica surfaces and crack-tip shielding- A Finite Element Study, Scientific Working Papers 33, 2015, ISSN: 2194-1629, Karlsruhe, KIT.

9.3 Zhao, T., Nehorai, A., Detection and localization of a moving biochemical source in a semi-infinite medium, Report, Electrical and Computer Engineering Dept. Univ. of Illinois, Chicago, USA.

9.4 Babich, V.M., On asymptotic solutions of the problems of moving sources of diffusion and oscillations, UDC 534.231.1, 517.226, 1991, pp. 3067-3071, Plenum Publishing Corp.

9.5 T. Fett, New contributions to R-curves and bridging stresses – Applications of weight functions, IAM 3, KIT Scientific Publishing, 2012, Karlsruhe.

9.6 S. M. Wiederhorn, T. Fett, G. Rizzi, M. J. Hoffmann, J.-P. Guin, The Effect of Water Penetration on Crack Growth in Silica Glass, Engng. Fract. Mech. 100(2013), 3-16.

9.7 F. Lechenault, D.L. Rountree, F. Cousin, J.-P Bouchaud, L. Ponson and E. Bouchaud, Evidence of Deep Water Penetration in Silica during Stress Corrosion Fracture, Phys. Rev. Let. 106(2011), 165504.

9.8 M. Tomozawa, W.-T. Han and W.A. Lanford, Water Entry into Silica Glass during Slow Crack Growth, J. Am. Ceram. Soc. 74 [10] (1991) 2573-2576.

9.9 G. Schell, G. Rizzi, T. Fett, Hydroxyl concentrations on crack surfaces from measurements by Tomozawa, Han and Lanford, Scientific Working Papers 82, 2018, ISSN: 2194-1629, Karlsruhe, KIT.

9.10 Evans, A.G., Faber, K.T., Crack-growth resistance of microcracking brittle materials, J. Am. Ceram. Soc. 67(1984), 255-260.

# 10 Anomalous subcritical crack growth

#### 10.1 Anomalous and normal subcritical crack growth10. 1,

Similar to other glasses, silica also shows the effect of subcritical crack growth. The fundamental effect, namely bond break processes, requires very high stresses. They are available in the singular stress field of cracks at crack-tip distances in the order of a few molecule diameters. Crack growth in silica glass has been well described by fracture mechanics techniques representing the mechanical load by the stress intensity factor. 10. 2, 10. 3, 10. 4

Subcritical crack propagation of silica is usually presented in form of v(K)-curves where v is the crack-growth rate and K the externally applied stress intensity factor. Results are reported as a function of relative humidity and temperature in [10.1-10.4]. For silica, the astonishing effect of decreasing crack-growth rate v 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 [10.5] is called anomalous subcritical crack growthbehavior. A comparison for different glasses is given in Fig. 10.1.

Whereas for silica the crack velocity decreases with temperature, it increases for all other glasses [10.5]. This conclusion is of course somewhat problematic since curves (1) and (2) are not for silicate glass and curves (3-6) obtained in vacuum cannot describe any influence that water might have. In addition, it is mentioned in [10.2] that silica in the vacuum tests failed not before toughness was reached. This implies a very steep v-K-curve and consequently subcritical crack growth at a value of K=0.8 KIc could simply not be reached.

Suratwala and Steele [10.5] assume that the negative temperature dependence will stem from a change in the slow crack growth resistance of the glass upon exposure to temperature and stress. In the following considerations we will try to explain the effect by a simpler reason. In [10.6] we discussed the apparently incompatible temperature trends qualitatively.

Fig. 10.1 Subcritical crack growth rate vs. temperature for different glasses at an individual stress intensity factor of K=0.8 KIC. Curves (1, 2, and silica): Measurements in N2 at 2700 Pa partial water vapour pressure. Curves (3-6): measurements in vacuum [10.2]. Curves (1, 2): Phosphate glasses [10.7, 10.8]. Details can be found in [10.5].

#### 10.2 Experimental crack growth results from literature

In order to discuss anomalous crack growth, two basic results from literature are compiled. In Fig. 10.2a v-K-curves for constant vapour pressure of p=2100-2200 Pa are plotted [10.5]. There is a clear decrease of crack velocity with increasing temperature visible and is the reason for the notation "anomalous subcritical crack growth". Measurements on silica in liquid water were published by Wiederhorn and Bolz [10.1] as shown in Fig. 10.2b for the temperature region 2°C≤T≤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 [10.5].

In this Section 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. 10.2a the pressure was kept constant at 2.1-2.2 kPa, the saturation pressure varies in the water tests of Fig. 10.2b strongly, namely by a factor of >100 (0.6 kPa at 2°C and 70 kPa at 90°C).

Consequently, the water concentrations at the crack tips are clearly different. This will be outlined in the following theoretical considerations.

Fig. 10.2 a) v-K-curves for constant vapour pressure of p=2.1-2.2 kPa by Suratwala and Steele [10.7], b) v-K-curves in liquid water by Wiederhorn and Bolz [10.1].

#### 10.3 Expectation from reaction rate theory

In [10.9] Wiederhorn et al. suggest for the subcritical crack growth rate v as a function of stress intensity factor K, and temperature

$$\mathcal{V} = A\_1 \, C \, k\_r(T) \exp\left[\frac{-\mathcal{Q} + b \, K}{RT}\right] \tag{10.3.1}$$

where again C is the concentration of molecular water [H2O], T the temperature in °K, R the gas constant, Q the activation energy and A1 and b are proportionality constants. In eq.(10.3.1) the quantity kr(T) is the rate coefficient for the single-step reaction showing the temperature dependency

$$k\_r(T) \propto T \exp\left[\frac{-\Delta G}{RT}\right] \tag{10.3.2}$$

(∆G=free energy of activation). From eqs.(10.3.1) and (10.3.2) we get with Q1=Q+∆G

$$\mathcal{V} = A\_2 \, CT \exp\left[\frac{-Q\_1 + b\, K}{RT}\right] \tag{10.3.3}$$

Under a water vapour pressure p, and the solubility for molecular water at p=1 kPa, C1kPa, the water concentration reads

$$C = p \, C\_{1kPa} \tag{10.3.4}$$

and equation (10.3.3) results in

$$\mathbf{V} = A\_2 \ p \ C\_{1kPa} T \exp\left[\frac{-Q\_1 + b\ K}{RT}\right] \tag{10.3.3a}$$

When evaluating the temperature dependence, it should be noted that both pressure and solubility also vary with temperature. These influencing factors will be considered in the following Sections.

Fig. 10.3 v-K-curves in liquid water by Wiederhorn and Bolz [10.1], transformed to a common saturation pressure of 3.2 kPa (saturation pressure at 25°C) according to eq.(10.3.3a).

#### 10.4 Effect of vapour pressure

The crack-growth rates from Fig. 10.2b are transformed by eq.(10.3.3a) to a constant vapour pressure of p=3.2 kPa (saturation pressure at 25°C), Fig. 10.3, exhibiting clearly the same temperature trend as found by Suratwala and Steele [10.5]. 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.

#### 10.5 Effect of water solubility

Since the v-K-curves should depend on the water concentrations at glass surfaces, results of water concentrations are shown in Fig. 10.4 as a function of temperature. These data represent the concentrations of the molecular and hydroxyl water species derived in Section 4 of [10.10] on the basis of measurements by Zouine et al. [10.11] under saturation pressure. According to eqs. (4.1.3) and (4.1.4a) in [10.10] it results for the molecular water C in mass units

$$C = \frac{C\_w}{1 + \frac{1}{2}k},\tag{10.5.1}$$

and for the hydroxyl concentration S

$$S = \frac{17}{18} \frac{C\_w}{\left(\frac{1}{2} + \frac{1}{k}\right)}\tag{10.5.2}$$

(Cw = total water concentration, k = equilibrium constant). Figure 10.4 shows the concentrations of the two water species S∝p and C∝p normalized on the water vapour pressure of p=1 kPa. The straight lines can be expressed by

$$\Rightarrow C\_{\text{lkPa}} \cong \exp\left[-22.79 + \frac{\text{35.84 kJ/mol}}{RT}\right] \tag{10.5.3}$$

$$S\_{\rm lkPa} \cong \exp\left[-19.41 + \frac{25.09 \,\text{kJ/mol}}{RT}\right] \tag{10.5.4}$$

103

Fig. 10.4 Solubility of water species at silica surfaces under saturation pressure derived from data by Zouine et al. [10.11], water concentrations normalized on the saturation pressure.

Fig. 10.5 Crack-growth rates normalized on the molecular water concentration and temperature T, a) data from Fig. 10.2a, b) data from Fig. 10.2b.

The crack velocities of Fig. 10.2 are plotted in Fig. 10.5 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.

#### 10.6 Effect of shielding stress intensity factor

#### 10.6.1 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 Kappl with the K-value acting at the crack tip, Ktip. It was early outlined in [10.12, 10.13] that a shielding stress intensity factor must develop as a consequence of volume increase by the hydroxyl generation (for references see e.g. [10.14]).

Computations by McMeeking and Evans [10.15] result in a shielding stress intensity factor for a grown crack which is surrounded by a zone of thickness ω undergoing a volume strain εv

$$K\_{sh} = -\nu \frac{\varepsilon\_\text{\nu} E}{1 - \nu} \sqrt{\alpha} \tag{10.6.1}$$

with ψ=0.22 [10.15]. 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 [10.16] that water-induced swelling at crack tips generates an intrinsic stress intensity factor Ksh<0 that shields a crack from an externally applied loading Kappl so that the stress intensity factor Ktip at the crack tip is reduced. The effective stress intensity factor acting at the crack tip, Ktip, represents the singular stress field at the tip. From the principle of superposition, Ktip is given by

$$K\_{\rm alp} = K\_{\rm appt} + K\_{\rm sh} \quad , \quad K\_{\rm sh} < 0 \tag{10.6.2}$$

i.e. Ktip<Kappl. The effect of the shielding stress intensity factor on the v-K-curves was outlined in [10.16]. As shown by eq.(1.1.9), the volume strain εv is related to the hydroxyl concentration S by the proportionality ε∝S. Consequently, it holds

$$K\_{sh} = -\nu \frac{\kappa SE}{1 - \nu} \sqrt{\rho o} \tag{10.6.3}$$

again with κ≅0.97. The hydroxyl concentration S in the crack-tip stress field is thermally activated as eq.(1.1.3) 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. When in a subcritical crack growth test the shielding stress intensity factor equals −Kappl, the total stress intensity factor Ktip disappears followed by abruptly decreasing crack rates, i.e. the threshold values Kth of the v-Kcurve is reached. This is equivalent to

$$K\_{\rm sh} = -K\_{\rm sh} \tag{10.6.4}$$

Threshold values for silica were evaluated by Sglavo et al. [10.17] on the basis of literature results, [10.18,10.19]. The threshold stress intensity factor as a function of temperature at ≈2 kPa partial water pressure was described by an Arrhenius temperature dependency

$$K\_{\rm \,th} \equiv K\_{\rm \,th, 0} \exp\left[\frac{-q}{RT}\right] \tag{10.6.5}$$

Kth,0=0.638 MPa√m, q=1.83 kJ/mol. The value Kth,0 in eq.(10.6.5) 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 quailitative interpretations. Equations (10.6.4) and (10.6.5) make it possible to calculate the shielding stress intensity factor Ksh and via eq.(10.6.2) the total stress intensity factor Ktip acting at the crack tip. This value is responsible for the crack-tip stresses and subcritical crack growth rates.

Finally, Fig. 10.6 gives the curves of Fig. 10.5 vs. the difference Kappl-Kth. Under the assumption that Ksh≠f(Kappl), the curves in Fig. 10.6 agree with the v-Ktip curves. The diagrams of Fig. 10.6 show that, for the same water concentration, plotting against the stress intensity factor Ktip (actually prevailing at the crack tip) leads to normal subcritical crack growth in both cases, namely increasing crack rates with increasing temperature.

Fig. 10.6 Crack-growth rates from Figs.10.5a and 10.5b plotted vs. stress intensity factor Kappl-Kth.

#### 10.6.2 Variable shielding

Deviations from the assumption of a constant shielding stress intensity factor Ksh have to be expected. With increasing crack-growth rates the time for the water entrance via diffusion decreases and, consequently, the zone height ω in eq.(10.6.3) becomes smaller. This yields in decreasing shielding stress intensity factor Ksh with increasing v or increasing load. A computation of Ksh(Kappl, T) was performed in [10.6] resulting in rather complicated equations and should not be duplicated here.

An approximate method may be suggested to estimate Ksh or, equivalently, Ktip as a function of Kappl. For this purpose, let us use the results of Section 8.2 for the case that shielding Ksh disappears at fracture toughness, Fig. 8.4c. This situation is shown in Fig. 10.7. The total or tip-stress intensity factor Ktotal=Ktip holds according to eq.(8.2.9)

$$K\_{\rm n|p} = \frac{K\_{appl} - K\_{\rm th}}{K\_{\rm lc} - K\_{\rm th}} K\_{\rm lc} \tag{10.6.6}$$

Figure 10.8 represents the v-Ktip results for the data in Figs. 10.5a and 10.5b. 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.

Finally, Fig. 10.9a shows the stress intensity factor Ktip necessary for reaching a constant normalized crack velocity v/CT. There seems to exist a discontinuity in Fig. 10.9a between 75°C and 150°C. Figure 10.9b shows the applied stress intensity factor necessary to reach a crack velocity of v=10-7 m/s at a partial water vapour pressure of pH2O=2100- 2200 as a function of temperature. In both representations, there seems a change of the crack-growth behaviour visible at about 140°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. [10.20], Grimaldi et al. [10.21], and Pallares et al [10.22]. At the high temperatures this is no longer possible. So one would expect a discontinuity between tests in "water" and tests in humid air.

Fig. 10.7 Suggestion for the estimation of the v-Ktip curve.

For the slopes of the v-K-curves one would expect from the theoretical analysis by Wiederhorn et al. [10.9] according to eq.(10.3.1)

$$
\log \nu \propto B K\_{\text{sup}} \quad , \quad B = \frac{b}{RT} \tag{10.6.7}
$$

i.e. a decrease of B with increasing reciprocal absolute temperature T. From the subcritical crack growth rates of Fig. 10.8 the slopes, B, were determined. Additionally the parameter B for a representation by log(v)∝B Kappl is introduced for the data by Wiederhorn and Bolz [10.1]. The result is illustrated in Fig. 10.10a where the parameter B from eq.(10.6.7) is plotted. The squares show the results obtained from the data in [10.5] and the circles from data in [10.1].

Fig. 10.8 Crack-growth rates from Figs. 10.5a and 10.5b plotted vs. total stress intensity factor Ktip.

The values of B increase with decreasing temperature. This applies to all results with the exception of the data by Suratwala and Steele [10.5] for temperatures below 150°C, see also the jump in Fig. 10.9a. Below this temperature, the effect seems to be slightly reversed. In Fig. 10.10b the parameter b defined in eq.(10.3.1) is shown. The parameter b is for the representation via Ktip a constant. This holds for the data from [10.1] and the data above ≈100°C from [10.5] but decreases for lower temperatures as is included in Fig. 10.10b as the dashed line. The representation by eq.(10.6.7) yields (Standard Deviations in brackets)

$$b = 0.112 \, [0.0017] \, (\text{m}^{\circ 2}/\text{mol}) \quad \text{ for } \{10.1\} \tag{10.6.8a}$$

$$b = 0.158 \,\text{[0.0073](m}^{\circ 2}\text{/mol)} \quad \text{for [10.5] and } \cong 1 \,\text{50} \,\text{°C} \tag{10.6.8b}$$

For a representation by log(v)∝B Kappl, Wiederhorn and Bolz [10.1] suggested b=0.216 m5/2/mol, introduced in Fig. 10.10b as the dash-dotted line.

Fig. 10.9 a) Stress intensity factors Ktip for constant normalized subcritical crack growth rates v/(C T) as a function of temperature, red: results in humid air by Suratwala and Steele [10.5], blue: results in liquid water by Wiederhorn and Bolz [10.1], b) applied stress intensity factors necessary for a crack-growth rate of v=10-7m/s at p=2100-2200 Pa pressure.

#### 10.6.3 Expectations for other glasses

In the previous considerations, only silica was considered. Here, we will look on other glasses, too. Density measurements on several glass compositions were reported by Scholze [23]. The materials were: (B) B2O3-glass, (K) K2O-SiO2-glass (20-80 mol-%), (Ca) Na2O-CaO-SiO2-glass (16-10-74 mol-%), (Al) Li2O-Al2O3-SiO2-glass (20-5-75 mol- %), and (Si) silica.

The related volume strains εv of all the glasses according to Section 2 in [10.10] are plotted in Fig. 10.11. Whereas the B2O3-glass shows strong volume shrinkingdue to water uptake, silica shows clear volume expansion. The effects on the other glasses are rather small. Materials Ca and Al also swell. Kshrinks slightly.

In contrast to the crack-tip shielding for silica, it has to be expected that the B2O3-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.

Fig. 10.10 a) Slopes B of subcritical crack growth curves v(Ktip) as a function of temperature, b) parameter b =B×RT, hatched area: scatter ranges of ±1 Standard Deviation, dash-dotted line: suggestion by Wiederhorn and Bolz [10.1].

Conclusions: We studied subcritical crack growth results measured by Suratwala and Steele [10.5] in water vapour and by Wiederhorn and Bolz [10.1] measured in liquid water under saturation pressure. From the results shown before, the anomalous subcritical crack growth in the experiments by Suratwala and Steele [10.5] is a consequence of the decreasing surface water solubility with increasing temperature and the plotting against the applied stress intensity factor Kappl instead of the physically acting value Ktip that governs the singular stress field at crack tips. With other words: Subcritical crack growth in silica is normal since for constant water concentration at crack tips the crack-growth rate at constant, Ktip, increases with increasing temperature.

It is also noteworthy that if the temperature dependence of the saturation pressure is eliminated exclusively in the tests in liquid water [10.1], anomalous crack growth behaviour would appear (Fig. 10.3).

This suggests that the representation of v-K-curves for constant pressure instead of constant water concentration in the glass and by plotting the crack-growth rates vs. the applied stress intensity factor (instead of the physically relevant crack-tip stress intensity factor, Ktip) is responsible for the anomalous behaviour of silica.

Finally, it can be shown that the parameter b in eq.(10.3.1) is a constant if the crack-tip stress intensity factor Ktip is used for K.

Fig. 10.11 Volume strains for several glasses from [10.10]. B: B2O3-glass, K: K2O-SiO2-glass (20-80 mol-%), Ca: Na2O-CaO-SiO2-glass (16-10-74 mol-%), Al: Li2O-Al2O3-SiO2-glass (20-5-75 mol-%), Si: Silica

### References

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10.2 S.M. Wiederhorn, H. Johnson, A.M. Diness and A.H. Heuer, Fracture of Glass in Vacuum,J. Am. Ceram. Soc.57 [188] 336-341 (1974).

10.3 S. Sakaguchi and Y. Hibino, Fatigue in low-strength silica optical fibres, J. Mater. Sci.19 3416-3420 (1984).

10.4 M. Muraoka and H. Abé, Subcritical Crack Growth in silica Optical Fibers in a Wide Range of Crack Velocities,J. Am. Ceram. Soc. 79[181] 51-57 (1996).

10.5 T.I. Suratwala and R.A. Steele, Anomalous temperature dependence of sub-critical crack growth in silica glass,J. Non-Crystalline Sol.316 174-182 (2003).

10.6 T. Fett, K.G. Schell, E.C. Bucharsky, M.J. Hoffmann, Anomalous subcritical crack growth in silica, Glass Physics and Chemistry, 47(2021), 581-589.

10.7 T.I. Suratwala, R.A. Steele, G.D. Wilke, J. Campbell, J. Non-Crystalline Sol. 263/264 (2000), 213.

10.8 S. Crichton, M. Tomozawa, J. Hayden, T. Suratwala, J. Campbell, J. Am. Ceram. Soc.82 (1999), 3097.

10.9 S.M. Wiederhorn, E.R. Fuller, R. Thomson, Micromechanisms of crack growth in ceramics and glasses in corrosive environments, Metal Science, Aug-Sept (1980), 450- 458.

10.10 T. Fett, G. Rizzi, K.G. Schell, E.C. Bucharsky, P. Hettich, S. Wagner, M. J. Hoffmann, Consequences of hydroxyl generation by the silica/water reaction Part I: Diffusion and Swelling, KIT Scientific Publishing, Karlsruhe.

10.11 A. Zouine, O. Dersch, G. Walter and F. Rauch, Diffusivity and solubility of water in silica glass in the temperature range 23-200°C, Phys. Chem. Glass: Eur. J. Glass Sci and Tech. Pt. B, 48 [2] (2007), 85-91.

10.12 S. M. Wiederhorn, T. Fett, G. Rizzi, S. Fünfschilling, M.J. Hoffmann, J.-P. Guin, Effect of Water Penetration on the Strength and Toughness of Silica Glass, J. Am. Ceram. Soc. 94 [S1] (2011), 196-S203.

10.13 S.M. Wiederhorn, T. Fett, G. Rizzi, M. Hoffmann, J.-P. Guin, Water Penetration – its Effect on the Strength and Toughness of Silica Glass, Met. Mater. Trans. A, 44(2013) [3], 1164 -1174.

10.14 Shelby, J.E., Density of vitreous silica, J. Non-Cryst. 349 (2004), 331-336.

10.15 McMeeking, R.M., Evans, A.G. Mechanics of transformation-toughening in brittle materials, J. Am. Ceram. Soc. 65(1982), 242–246.

10.16 Wiederhorn, S.M., Fett, T., Rizzi, G., Hoffmann, M.J., Guin, J.-P., The Effect of Water Penetration on Crack Growth in Silica Glass, Engng. Fract. Mech., 100 (2013), 3– 16.

10.17 V. Sglavo, T. Fett, K.G. Schell, M.J. Hoffmann, S.M. Wiederhorn, v-K-data for silica from interrupted lifetime measurements, 144, 2020, ISSN: 2194-1629, Karlsruhe, KIT.

10.18 V.M. Sglavo and D.J. Green, Fatigue limit in fused silica, J. Eur. Ceram. Soc. 21 (2001) 561-567.

10.19 J. S. Aaldenberg, P. J. Lezzi, Measurement of the silica glass fatigue limit, J. Am. Ceram. Soc., 103(2020), 3097-103.

10.20 M. Ciccotti, M. George, V. Ranieri, L. Wondraczek, and C. Marliere, Dynamic Condensation of Water at Crack Tips in Fused Silica Glass, J. Non Cryst. Solids, 354 [2– 9] 564–8 (2008).

10.21 A. Grimaldi, M. George, G. Pallares, C. Marniere, M. Ciccotti, The crack tip: A nanolab for studying confined liquids, Phys. Rev. Letters, 100, 165505 (2008).

10.22 G. Pallares, A. Grimaldi, M. George, L. Ponson, M. Ciccotti, Quantitative analysis of crack closure driven by Laplace pressure in silica glass, J. Am. Ceram. Soc. 94(2011), 2613-2618.

23 Scholze, H., Glas, Springer Verlag, Berlin, 1988.

# **Consequences of hydroxyl generation by the silica/water reaction**

# **Part III: Damage and Young's Modulus**

# 11 Damage and Young's modulus

#### 11.1 Damage by hydroxyl generation

When a hydroxyl has been formed, the initial silica ring is broken and the mechanical cohesion is weakened as is illustrated in Fig. 11.1. Such "defects" in the glass structure can be treated by using the damage variable D of continuum damage mechanics (Kachanov [11.1], Lemaitre [11.2]). This parameter is proportional to the density of micro-defects.

According to the postulate of strain equivalence by Lemaitre [11.3], the effective elastic modulus, ED, decreases with increasing damage

$$E\_D = E\_0(1 - D) \tag{11.1.1}$$

where E0 is the modulus of virgin glass. The damage variable D can be determined from module measurements via eq.(11.1.1). The occurrence of this effect is in principle known from measurements on NaO-SiO2 glass by Ito and Tomozawa [11.4]. To the authors' knowledge, so far no comparable uni-axial measurements on fused silica are available. Therefore, we temporarily consider the damage D as a certain function of the hydroxyl concentration:

$$D = f(S) \tag{11.1.2}$$

Apart from the equi-triaxial loading case with σx = σy = σz (including the case of disappearing stresses), the elastic modulus must become a tensor with components depending on the degree of loading multiaxiality.

Since this possibility would make the further treatment very difficult [11.3] and nontransparent, we assume in the following considerations that the damage remains isotropic and is considered to be of scalar nature. This is equivalent to the assumption of randomly orientated defects. Then also E remains isotropic. We assume that the defects in SiO2, caused by hydroxyl generation, might behave like normal nano-pores.

Fig. 11.1 Volume element of silica showing damage by bond breaking due to the water/silica reaction, third dimension ignored.

# 11.2 Experimental evidence for modulus reduction in silica

In literature, there is experimental evidence for modulus decrease with increasing hydroxyl content. This can be seen from measurements of sound velocity as a function of water content. Measurements on longitudinal sound velocities in silica specimens with different water content were reported by Fraser [11.5] and Le Parc et al. [11.6]. Individual least-squares fits were made resulting for the data set by Fraser [11.5]

$$\mathbf{v} \equiv \mathbf{5974} (1 - 4.18 \mathbf{S} \mathbf{S}) \quad \text{in (m/s)}\tag{11.2.1}$$

and the set by LeParc et al. [11.6]

$$\mathbf{v} \equiv \mathbf{5959(1 - \mathbf{5.34S})} \quad \text{in (m/s)}\tag{11.2.2}$$

When we normalize the results of the two test series on their individual values for S=0, we get the representation in Fig. 11.2a. A common straight-line fit of these data yields

$$\frac{\nu}{\nu\_0} = 1 - B \, S \tag{11.2.3}$$

with the parameter

$$B = \text{S.04 [4.23, 5.85]}$$

(90%-Confidence Intervals, CI, in brackets). The dependency of eq.(11.2.3) is introduced in Fig. 11.2a as the straight line. The longitudinal sound velocity depends on Young's modulus E and density ρ via *v* ∝ *E* / ρ.

Whereas the densities for the measurements by Fraser are compiled in [11.5], we used for the results by LeParc et al [11.6] S according to the measurements by Shelby [11.7] (see also Section 2.1 in [11.8]):

$$\frac{E\_D}{E\_0} = \rho / \rho\_0 \times \left(\mathbf{v} / \mathbf{v}\_0\right)^2 = 1 - \mathcal{Z}\,\mathbf{S} \tag{11.2.4}$$

with λ=10.6 [8.7, 12.5]. It has to be noted that this value holds for isotropic damage since the natural OH-content doesn't show any preference for a special direction. The Young's modulus resulting from the data by Fraser [11.5] is shown in Fig. 11.2b.

Figure 11.2b represents in addition stiffness coefficients C11 by the triangles as were reported by Kushibiki et al. [11.9].

Fig. 11.2 a) Longitudinal sound velocity in silica with different OH-content (blue circles: results by Fraser [11.5], red circles: results by LeParc et al. [11.6]), b) Young's modulus for the data of Fig. 11.2a together with stiffness results by Kushibiki et al. [11.9] as the triangles.

#### 11.3 Modelling of damage by spherical pores

Analytical computations on the reduction of Young's modulus with porosity were carried out by Wang [11. 10 ] for spherical pores of equal size and constant spacing. Unfortunately, the results were given as tabulated data. They were the basis of a number of fitting relations, mostly using exponential functions, which trivially must fail for large porosities since they could not represent the requirement of E=0 for a finite critical porosity. An overview can be obtained in [11.11, 11.12].

To overcome such problems, we fitted the analytical data by Wang [11.10] according to

$$\frac{E\_D}{E\_0} = \exp\left[-A\_1 P - \frac{A\_2}{(P\_{\text{max}} - P)^{1/8}} P^2\right] \tag{11.3.1}$$

with Pmax=0.4764 from [11.10] and the free fitting parameters A1 and A2 resulting as

$$A\_1 = \text{3.580}\left[\text{3.565}, \text{3.595}\right], A\_2 = \text{3.747}\left[\text{3.699}, \text{3.795}\right]$$

with very narrow 90% confidence intervals indicating the good agreement with the numerical data. Figure 11.3a shows good agreement between eq.(11.3.1) and the numerical data entered by the circles. The dashed line indicates the initial slope that for small porosities reads

$$\frac{E\_D}{E\_0} \equiv 1 - A\_1 P \tag{11.3.2}$$

and intersects the abscissa at a characteristic porosity Pc of

$$P\_c = 1/A\_1 = 0.278 P\_{\text{max}} \tag{11.3.3}$$

In order to give a better resolution for the agreement at low modules, Fig. 11.3b represents the same results in logarithmic ordinate scaling. For our purpose we rewrite eq.(11.3.1) by normalizing the porosities on the maximum value Pmax and then replace P by S

$$\frac{E\_D}{E\_0} = \exp\left[-B\_1(S/S\_{\text{max}}) - \frac{B\_2}{(1 - S/S\_{\text{max}})^{1/3}} \left(\frac{S}{S\_{\text{max}}}\right)^2\right] \tag{11.3.4}$$

with parameters B1=1.667, B2=0.933. From eq.(11.2.4) and the initial slope of eq.(11.3.4) we obtain via B1/Smax=λ the maximum value of hydroxyl concentration

$$S\_{\text{max}} = \frac{B\_1}{\lambda} = 0.157 \text{ [} 0.133, 0.192\text{]} \tag{11.3.5}$$

A simpler pore model was proposed by Phani and Niyogi [11.13] with the result of

$$\frac{E\_D}{E\_0} = (1 - aP)^n \tag{11.3.6}$$

This type of equation applied to the hydroxyl damage gives good agreement with the Wang solution over a large range of hydroxyl concentrations for the parameter set

$$\mathbf{n} \equiv \mathbf{2}, \; \mathbf{a} = \lambda \mathcal{I} \mathbf{2} \equiv \mathbf{5}.\mathbf{3} \tag{11.3.7}$$

The modulus ratio can be described by

$$\frac{E\_D}{E\_0} = (1 - S \, / \, S\_{\text{max}})^2 \tag{11.3.8}$$

The related dependency is shown in Fig. 11.3c by the red curve that gives disappearing of Young's modulus at Smax = 0.188 [0.16, 0.23].

Fig. 11.3 a) Effect of porosity on Young's modulus for spherical pores; analytical results from Wang [11.10] (circles) compared with suggested fitting relation eq.(11.3.1) (curve); b) logarithmic ordinate scaling, c) Young's modulus as a function of hydroxyl concentration.

Fig. 11.4 a) Results of porosity on Young's modulus for silica by Adachi and Sakka [11.14], curve: best fit according to eq.(11.3.6), b) comparison with results from longitudinal sound velocity in silica for different OH-content (red circles: results by LeParc et al. [11.6] and Fraser [11.5]).

The fact that eq.(11.3.6) is an appropriate description for silica glass in the region of low moduli EDE0→0, as can be seen from measurements by Adachi and Sakka [11.14], plotted by the circles in Fig. 11.4a. By fitting the data, we obtain the best representation for the parameter set:

$$\text{In} = 2, \text{ } \mathfrak{a} = 1.19 \tag{11.3.9}$$

introduced in Fig. 11.4a by the circles. Figure 11.4b shows all measurements from Figs. 11.2 and 11.4a in a normalized representation. The dashed line illustrates a linear extension of the small-concentration data.

#### 11.4 Effect of damage in thin surface layers

#### 11.4.1 Global stress

If the damage is concentrated on thin surface layers of a thickness d that is small to the undamaged bulk thickness W, d<<W, the strain in the damaged surface layer must equal the strain in the bulk:

$$\mathcal{E} = \frac{\sigma\_D}{E\_o} = \frac{\sigma\_0}{E\_o} \tag{11.4.1}$$

The reduced average stress in the damaged surface, σD, is therefore according to eq.(11.1.1)

$$
\sigma\_D = \sigma\_0 (1 - D) \tag{11.4.2}
$$

(σ0= stress in the undamaged bulk).

External loading in z-direction will increase the distance between atoms of the ring structure, resulting in the strain εz in loading direction. Failure of the material is reached, when the strain reaches a critical value εc. The damage in the material affects also the load-bearing capacity of any area element since it reduces its net cross section and, consequently, the strength of the damaged material σf. If σf,0 denotes the strength in the absence of damage, the strength of the damaged material is

$$
\sigma\_{f,D} = \sigma\_{f,0}(\mathrm{l} - D) \tag{11.4.3}
$$

The strength of water-damaged silica fibres is strain-controlled for d<<W. This means that the strain at failure εc is the same in the surface layer and the bulk. This condition yields

$$
\varepsilon\_c = \frac{\sigma\_{f,D}}{E\_D} = \frac{\sigma\_{f,0}(1-D)}{E\_0(1-D)} = \frac{\sigma\_{f,0}}{E\_0} \tag{11.4.4}
$$

Equation (11.4.4) tells us that the damage problem can be handled simply by using bulk material parameters (E0, σf,0) since the damage by broken bonds cancels out in (11.4.4). For some more details on fiber strength, see Section 11.7.

#### 11.4.2 Effect on crack resistance and stress intensity factors

The damage of the initial ring structure of silica, Fig. 11.1, must also affect the resistance against crack propagation. The crack growth resistance Gc represents the energy necessary to split all bonds that are broken when the crack has passed the considered volume element. Application of the damage variable yields for the crack growth resistance of the damaged material, Gc,D,

$$\mathbf{G}\_{c,D} = \mathbf{G}\_{c,0} \text{(1-D)}\tag{11.4.5}$$

(Gc,0 = crack resistance for the undamaged material, D=0) or in terms of stress intensity factors

$$K\_{c,D} = \sqrt{EG\_{c,D}} = K\_{lc}(\mathbf{l} - D) \tag{11.4.6}$$

where Kc,D is the reduced toughness of the damaged material and KIc the fracture toughness measured in fracture mechanics tests on the undamaged material.

In the special case of a crack fully embedded in a material of reduced but constant modulus, ED, the problem simplifies strongly. The applied stress intensity factor is generally given by

$$K\_{appl} = \sigma\_{appl} F \sqrt{\pi a} \tag{11.4.7}$$

(F = Fracture mechanics geometric function, a=crack length, Fig. 11.5). Using the actual stress from eq.(11.4.2), Kappl reads for the crack fully embedded in the damaged zone

$$K\_{\text{app}\ell,D} = \underbrace{\sigma\_{\text{app}\ell,0} \, F \sqrt{\pi \, a}}\_{K\_{\text{app}\ell,0}} (1 - D) = K\_{\text{app}\ell,0} (1 - D) \tag{11.4.8}$$

where Kappl,0 stands for the applied stress intensity factor formally computed with stresses σ0 as present in the bulk material

$$K\_{approx,0} = \sigma\_0 F \sqrt{\pi a} \tag{11.4.9}$$

From (11.4.6), (11.4.8) and (11.4.9) it results equivalently to eq.(11.4.4)

$$\frac{K\_{\text{app},D}}{K\_{\text{c},D}} = \frac{K\_{\text{app},0}(1-D)}{K\_{\text{fc}}(1-D)} = \frac{K\_{\text{app},0}}{K\_{\text{fc}}} \tag{11.4.10}$$

Also in terms of stress intensity factors, surface cracks fully embedded in the surface layer can be handled simply by using bulk material parameters (Kappl, KIc) instead of the unknown parameters in the layer.

#### 11.4.3 Effect on pre-existing dimples

By removing the plastic protective coating on fibers with an acid, defects will be generated that can be described as hemispherical pits [11.15]. Depending on their geometry, such defects can reduce the surface strength clearly.

At the deepest point of a pit surrounded by the hydroxyl-damaged material under tensile stress σD, the stress is increased by the stress-concentration factor β, so that the maximum stress at the deepest point (A), Fig. 11.5, is

$$
\sigma\_{\text{max},D} = \beta \,\sigma\_D \tag{11.4.11a}
$$

For a half-sphere it is for instance β≈2. Before the hydroxyl-damaged layer was generated, it did hold equivalently for the undamaged material under stress σ0

$$
\sigma\_{\text{max},0} = \beta \,\sigma\_0 \tag{11.4.11b}
$$

From (11.4.11a) and (11.4.11b) we have to conclude that in the case of failure the strengths in presence and absence of hydroxyl damage are reduced both by the common factor 1/β. When the strengths in presence of the pit are denoted by σf,pit,D and σf,pit,0, it holds

$$
\sigma\_{f, \text{plt}, 0} = \frac{1}{\mathcal{J}} \sigma\_{f, 0} \tag{11.4.12a}
$$

$$
\sigma\_{\boldsymbol{f}\_{f},\boldsymbol{\text{p}\boldsymbol{t},D}} = \frac{1}{\beta} \sigma\_{\boldsymbol{f},D} \tag{11.4.12b}
$$

Apart from the fact that the strengths are reduced, it can be seen from the combined eqs.(11.4.12a) and (11.4.12b)

$$\frac{\sigma\_{f,\text{plr},D}}{\sigma\_{f,D}} = \frac{\sigma\_{f,\text{plr},0}}{\sigma\_{f,0}} \tag{11.4.13}$$

with the consequence that also in this case the hydroxyl damage has no influence on strength.

Fig. 11.5 Cross-section through a crack and a partially embedded hemispherical pit in a hydroxyl-damaged surface layer of thickness d<<W (W= radius of the fiber) and a<<d.

#### 11.4.4 Effect of swelling stresses

If the volume swelling occurs in a constrained manner, such as at the glass surface, then the swelling causes a build-up of compressive stresses in the surface region, which can strengthen the glass significantly as has been outlined in [11.16].

The compressive swelling stresses in axial and circumferential directions, y, z, respectively, are generally proportional to the product of volume swelling strain εv and Young's modulus E

$$
\sigma\_{sw,\text{y}}, \sigma\_{sw,\text{z}} \propto \varepsilon\_{\text{v}} E \tag{11.4.14}
$$

For the most simple case of isotropic swelling, the factor of proportionality is for instance 1/(3(1-ν)), where ν is Poisson's ratio.

The related volume swelling strain, εv >0, can be expressed according to eq.(1.1.9) by

$$
\boldsymbol{\varepsilon}\_{\text{v}} = \boldsymbol{\kappa} \times \mathbf{S} \tag{11.4.15}
$$

with the coefficient κ≅0.97.

When for example σsw,z,0 denotes the axial swelling stress in the undamaged material, it holds as a consequence of eq.(11.1.1) that the swelling stress in the hydroxyl-damaged surface is

$$
\sigma\_{s\nu,\varepsilon,D} = \sigma\_{s\nu,\varepsilon,0} (1 - D) \tag{11.4.16}
$$

As an example, this dependency may be applied to the crack-like surface defects.

The so-called shielding stress intensity factor Ksh<0 that shields the crack tip partially from the applied load is in the damaged material

$$K\_{sh,D} = \sigma\_{sw, \varepsilon, D} F \sqrt{\pi a} \ a \quad = \sigma\_{sw, \varepsilon, 0} F \sqrt{\pi a} \ (1 - D) = K\_{sh, 0} (1 - D) \tag{11.4.17}$$

where Ksh,0 stands for the shielding stress intensity factor in the undamaged material.

In presence of crack-tip shielding, failure occurs, when the total stress intensity factor Ktip reaches fracture toughness. In the damaged material the principle of superposition results in

$$K\_{\rm tip,D} = K\_{\rm appl,D} + K\_{\rm sh,D} = K\_{\rm fc,D} \quad , \quad K\_{\rm sh,D} < 0 \tag{11.4.18}$$

Using eqs.(11.4.6), (11.4.8) and (11.4.17) gives the same result in terms of the stress intensity factors in the undamaged material

$$K\_{\rm 4p,D} = (K\_{\rm appl,0} + K\_{\rm sh,0})(1 - D) = K\_{\rm 4c,0}(1 - D) \tag{11.4.19}$$

The equal sign on the right of (11.4.19) makes clear, that the shielding effect can be described by stress intensity factors in the undamaged material.

For the case of the surface additionally damaged by pit-like surface defects, the tensile stress σD, has to be replaced by the total stress σD+σsw,z,D resulting, equivalently to eq.(11.4.19), in

$$
\sigma\_{\max,D} = \beta \left( \sigma\_D + \sigma\_{sw,\varepsilon,D} \right) = \beta (\sigma\_0 + \sigma\_{sw,\varepsilon,0})(1 - D) = \sigma\_{f,0}(1 - D) \qquad (11.4.20)
$$

Again from the equal sign on the right, it becomes obvious that treatment with damaged quantities gives the same results as are obtained when hydroxyl damage is completely ignored.

# 11.5 Example of application: Uniaxial stress-strain curve

#### 11.5.1 Swelling excluded

As has been outlined in [11.17], the hydroxyl concentration is stress-dependent. In order to allow a transparent derivation, swelling stresses may be neglected first. The hydroxyl concentration as a function of stress can be written

$$S = S\_0 \exp\left[\sigma \frac{\Delta V\_{\text{eff}}}{RT}\right] \tag{11.5.1}$$

At temperatures >500°C, the parameter ∆Veff reads for uniaxial loading [11.17]

$$
\Delta V\_{\text{eff}} = 14.4 \text{ cm}^3/\text{mol} \tag{11.5.2}
$$

The total strain ε under the applied stress is

$$\mathcal{E} = \frac{\sigma\_{appl}}{E\_D(S)}\tag{11.5.3}$$

where for ED(S) the equations (11.3.4) or (11.3.8) have to be introduced. The maximum stress, asymptotically reached for ε→∞, is

$$
\sigma\_{\text{max}} = -\frac{RT}{\Delta V\_{\text{eff}}} \text{Log} \left[ \frac{S\_0}{S\_{\text{max}}} \right] \tag{11.5.4}
$$

In Fig. 11.6 the σ-ε-curves are shown for 650°C (355 Torr water vapour pressure) and for 500°C with parameters S0 (6 Torr pressure) taken from [11.17]. The dashed lines indicate the two asymptotes for ε→0 and ε→∞ of the solid curves.

Fig. 11.6 Stress-strain curves for 650°C, 355 Torr and 500°C 6 Torr, hydroxyl concentrations S0 from [11.17]; Approximate description of the stress-strain curves by eq.(11.5.5) (dashed lines).

For simpler mechanics computations, the stress-strain curves can be approximated by the two asymptotes (dashed lines in Fig. 11.6) where the rising part represents Young's modulus and the horizontal line the saturation stress given by eq.(11.5.4), i.e. a uniaxial "yield stress" σY = σmax. Then it holds

Fig. 11.7 Stress-strain curves, curve for 650°C and 500°C, black: without swelling, red: swelling included, a) total strain as the abscissa, b) strain by external loading as abscissa.

#### 11.5.2 Swelling included

When swelling strains are taken into consideration, eq.(11.5.3) has to be replaced by

$$
\varepsilon = \frac{\sigma}{E\_p(S)} + \varepsilon\_{\text{sw}, \varepsilon} \tag{11.5.6}
$$

where εsw,z is the swelling strain in axial direction. According to [11.17], the eq.(11.5.1) yields

$$
\sigma\_{sw,\varepsilon} = \alpha \,\,\kappa \,\, S = \alpha \,\,\kappa \,\, S\_0 \exp[\sigma\_{opt} \frac{\Delta V\_{off}}{RT}] \tag{11.5.7}
$$

with κ=0.97, a=1.92 [11.17]. The stress-strain curve obtained via eqs. (11.5.6) and (11.5.7) is shown in Fig. 11.7a as the red curves together with the curves by eq.(11.5.3) as the black ones. The offset of the red curves comes from the fact that even in the absence of stresses a swelling strain occurs that is caused by the initial hydroxyl concentration S0. When εsw,z,0 is the swelling strain caused by the initial concentration S0, the strain increase under load is ∆ε=ε−εsw,z,0. This is shown in Fig. 11.7b.

#### 11.6 Deviating behaviour under compressive loading

In the previous considerations, we applied the model of pore-like defects. This model describes symmetrical material response under tension and compression loading. For special cases of applications, the description by crack-like defects may be of advantage. This model allows introducing non-symmetry of deformation in tension and compression. Due to the fact that the crack surfaces cannot interpenetrate each other under compressive normal stress, the reduction of Young's modulus is less strong than in tension, represented by

$$E = \begin{cases} E\_0 \ (\text{l} - D) & \text{for} \qquad \sigma \ge 0 \\ E\_0 \text{(l} - hD) & \text{for} \qquad \sigma < 0 \end{cases} \tag{11.6.1}$$

with the crack closure parameter 0≤h≤1, often found to be h=0.2 [11.18].

#### 11.6.1 Shielding-stress intensity factor for damaged material

A consequences of damage by hydroxyl generation is the reduction of Young's modulus E. When ED is the modulus in the damaged state and E0 the value for undamaged silica, we could derive the relation [11.19, 11.20] (see Section 11.3)

$$\frac{E\_D}{E\_0} = (1 - \frac{1}{2}\mathcal{A}\,S)^2 = (1 - S\,/S\_{\text{max}})^2\tag{11.6.2}$$

with λ=10.6 [8.7, 12.5] (90%-CI in brackets). The hydroxyl concentration at which the Young's modulus disappears is Smax =2/λ= 0.188 [0.16, 0.23].

Including the varying modulus in the computation of the shielding stress intensity factor requires a modification of the equations in Section 9.2.2. Now the stress intensity factor results from

$$K\_{sh} = \frac{1}{S(0)E\_0} \int\_0^\circ (S \times E) \left(\frac{\partial K\_{sh}}{\partial \phi'}\right) d\phi' \tag{11.6.3}$$

The concentration profiles by Lechenault et al. [11.21], Fig. 9.4, clearly show the characteristic shape with a plateau as would be expected from the Irwin and Dugdale crack-tip zone models for "plastic flow" behaviour.

#### 11.6.2 Shielding stress intensity factor for asymmetric damage

In the case of disappearing total stress intensity factor at the crack tip, Ktotal =Kappl +Ktip ≤0, all volume elements in the water-affected zone are under compression due to volume swelling by hydroxyl generation [11.22]. In the previous considerations, we applied the model of pore-like defects. This model describes symmetrical material response under tension and compression loading. The Young's modulus in compression is not necessarily identical in tension and compression.

Figure 11.8b again shows the shielding stress intensity factor for the hydroxyl concentration according to eq.(11.6.3) for different values of the non-symmetry coefficient or crack closure parameter 0≤h≤1. The asymmetry gives reason to define an upper limit for the shielding stress intensity factor. The zone in front of the crack tip is under tension for a load of Ktip > 0, Fig. 11.8a. The crack flanks, on the other hand, are stress-free with regard to the applied load and are under pressure due to the swelling stresses. In this region the lower expression in (11.6.1) holds. As the limit approximation, the reduced modulus in the zone ahead of the tip is assumed to be free of any stress (ED→0) and the crack-face region is assumed to exhibit the initial modulus, i.e. h→0. This approximation is described by using the value ψ = 0.37 according to McMeeking and Evans [11.23]. Finally, Fig. 11.8b represents results from [11.19] for an erfc-shaped hydroxyl distribution with a surface value S(0) and a parameter bof 4.95 nm (solid curves), according to

$$S = S(0)\text{erfc}\left[\frac{z}{2b}\right] \tag{11.6.4}$$

The dashed line in Fig. 11.8b shows the upper limit solution for the same S-distribution addressed before.

Fig. 11.8 a) Negative and positive contributions to the total shielding stress intensity factor, b) shielding stress intensity factors for an erfc-shaped hydroxyl distribution, eq.(11.6.4), as a function of the surface concentration S(0). Solid curves: from [11.21], dashed curve: via eq.(9.2.2) with ψ=0.37, E=E0.

#### 11.7 Global strength of a fiber

The hydroxyl-damage in the glass also affects the load-bearing capacity of any area element, since it reduces its net cross-section, AD

$$A\_D = A\_0(1 - D) \tag{11.7.1}$$

where A0 denotes the total geometrical cross section subsuming damaged and undamaged regions.

When again σf,0 denotes the strength in the absence of hydroxyl damage, and σf,D the strength in the hydroxyl-damaged state, the strength results as given in eq.(11.4.3).

Let us now consider the strength of silica fibers. First, the water distribution may be replaced by C=C0=const. over the thickness b and C=0 in the bulk indicated in the inset of Fig. 11.9. The total cross section area A of the fiber with radius Ris

$$A = R^2 \pi \tag{11.7.2a}$$

that of the inner circle of radius Rb

$$A\_{\rm inv} = (R - b)^2 \pi \tag{11.7.2b}$$

and the ring-shaped area is

$$A\_b = R^2 \pi - (R - b)^2 \pi = (2R - b)\pi \, b \tag{11.7.2c}$$

Failure in the undamaged fiber occurs at the surface, when the stress reaches the strength σf,0. The force P0 at failure is then

$$P\_0 = \sigma\_{f,0} \mathbb{R}^2 \pi \tag{11.7.3}$$

When the surface is damaged by an amount of D, the strength in the layer b decreases to σf,D according to eq.(11.4.3). Since the Young's modulus decreases in the same way, eq.(11.1.1), it is clear that the critical strain εc remains unaffected as is visible from eq.(11.4.4).

Fig. 11.9 Failure load of a fiber for differently damaged surface regions as a function of layer thickness b.

The Bernoulli Theorem tells us that plane cross sections must remain plane. It implies that the strain in the damaged and undamaged parts of the fiber must be the same.

This again implies that at failure σ=σf,0 is in the inner area Ainn and σ=σf,D=σf,0(1-D) in the surface layer. The total load PD of the damaged fiber at failure is the sum of the load in the inner (undamaged) area and the load in the damaged ring-shapedzone, i.e.

$$P\_D = \sigma\_{f,0} (R - b)^2 \pi + \sigma\_{f,0} (1 - D)(2R - b) b \,\pi \tag{11.7.4}$$

The reduction of the fracture load follows from eqs.(11.7.3) and (11.7.4)

$$\frac{P\_D}{P\_0} = \frac{(R-b)^2 + (1-D)b(2R-b)}{R^2} = 1 - \frac{2bD}{R} + \frac{b^2D}{R^2} \tag{11.7.5}$$

and is plotted in Fig. 11.9 for several damage values D as a function of the layer thickness. It becomes clear, that for small ratios of b/R<0.02, the strength (computed from the load PD) is hardly affected.

In the special case of thin layers, b<<R, it results approximately

$$\frac{P\_D}{P\_0} \cong 1 - \frac{2b}{R}D\tag{11.7.6}$$

Figuratively speaking, it can be said that even in the absence of any load-carrying contribution from the stresses in the surface layer, D=1, the strength of the entire fiber is hardly affected.

#### References

11.1 L.M. Kachanov, Time of the rupture process under creep conditions, TVZ Akad Nauk S.S.R. Otd Tech. Nauk 8(1958)

11.2 J. Lemaitre, Evaluation of dissipation and damage in metals, Proc. I.C.M. 1, Kyoto Japan (1971).

11.3 J. Lemaitre, How to use damage mechanics, Nuclear Engng. Design 80(1984), 233- 245.

11.4 S. Ito and M. Tomozawa, J. de Phys. C9 (1982), 68.

11.5 D.B. Fraser, Factors Influencing the Acoustic Properties of Vitreous Silica, Citation: Journal of Applied Physics 39(1968), 5868-5878.

11.6 R Le Parc, C Levelut, J Pelous, V Martinez, B Champagnon, Influence of fictive temperature and composition of silica glass on anomalous elastic behavior J. Phys.: Condens.Matter 18(2006), 7507-27.

11.7 Shelby, J.E., Density of vitreous silica, J. Non-Cryst. 349 (2004), 331-336.

11.8 T. Fett, G. Rizzi, K.G. Schell, E.C. Bucharsky, P. Hettich, S. Wagner, M. J. Hoffmann, Consequences of hydroxyl generation by the silica/water reaction Part I: Diffusion and Swelling, KIT Scientific Publishing, Karlsruhe.

11.9 J. Kushibiki, M. Arakawa, Y. Yohashi, K. Suzuki, T. Maruyama, A super-precise CTE evaluation method for ultra-low-expansion-glasses using the LFB ultrasonic material characterization system, Jap. J. of Appl. Phys. 44(2005), 4374-4380.

11.10 J.C. Wang, Young's modulus of porous materials, J. Mater. Sci. 19(1984), 801-808.

11.11 K.K. Phani, S.K., Niyogi, Elastic-modulus–porosity relationship for Si3N4, J. Mater. Sci. Letters 6(1987), 511–515

11.12 K.K. Phani, S.K. Niyogi, A.K. De, Porosity dependence of fracture mechanical properties of reaction sintered Si3N4, J. Mater. Sci. Letters 7(1988), 1253–1256.

11.13 K.K. Phani, S.K. Niyogi, A.K. De, Young's modulus of porous brittle solids, J. Mater. Sci. 22(1987), 257–263.

11.14 Adachi, T, Sakka, S., Dependence of the elastic moduli of porous silica gel prepared by the sol-gel method on heat-treatment, J. Mater. Sci. 25(1990), 4732-37.

11.15 D. Inniss, Q. Zhong, C.R. Kurkjian, Chemically corroded pristine silica fibers: Blunt or sharp flaws?, J. Am. Ceram. Soc. 76(1993), 3173-77.

11.16 S.M. Wiederhorn, T. Fett, G. Rizzi, M. Hoffmann, J.-P. Guin, Water Penetration – its Effect on the Strength and Toughness of Silica Glass, Met. Mater. Trans. A, 44 [3], 1164 -1174 (2013).

11.17 S. M. Wiederhorn, G. Rizzi, S. Wagner, M. J. Hoffmann, and T. Fett, Stress-Enhanced Swelling of Silica: Effect on Strength, J. Am. Ceram. Soc. 99(2016), 2956-63.

11.18 J. Lemaitre, J.P. Sermage, One damage law for different mechanisms, Comp. Mech., 20(1997), 84-88.

11.19 T. Fett, G. Schell, C. Bucharsky, Hydroxyl Damage in Silica: Full-range description including large damages 126, 2019, ISSN: 2194-1629, Karlsruhe, KIT.

11.20 K.G. Schell, T. Fett, C. Bucharsky, M.J. Hoffmann, S.M. Wiederhorn, Damage by Hydroxyl Generation in Silica, Glass Physics and Chemistry, 46(2020), 424-428.

11.21 F. Lechenault, D.L. Rountree, F. Cousin, J.-P Bouchaud, L. Ponson and E. Bouchaud, "Evidence of Deep Water Penetration in Silica during Stress Corrosion Fracture," Phys. Rev. Let. 106(2011), 165504.

11.22 S. M. Wiederhorn, T. Fett, G. Rizzi, M. J. Hoffmann, J.-P. Guin, The Effect of Water Penetration on Crack Growth in Silica Glass, Engng. Fract. Mech. 100(2013), 3-16.

11.23 McMeeking, R.M., Evans, A.G., Mechanics of Transformation-Toughening in Brittle Materials, J. Am. Ceram. Soc. 65(1982), 242-246.

# 12 Shielding affected by damage

#### 12.1 Effect of local swelling at surface cracks

The reaction of water and silica in the surface diffusion zone, eq. (1.1.1), affects the fracture mechanics stress intensity factor K at the tips of cracks. As a consequence of the high near-tip stresses, very high hydroxyl concentrations must occur in the crack-tip region. This allows the assumption that near a crack tip the water is present predominantly in the form of hydroxyl water.

Due to restrictions in free expansion, the swelling strains εv result in swelling stresses which give rise for the "shielding" stress intensity factor Ksh<0. At first loading, the swelling zone at the crack tip r(σh,ϕ) is heart-shaped as illustrated in Fig. 12.1a.

Fig. 12.1 a) Swelling zone at the tip of an arrested crack under mechanical loading, caused by stressenhanced diffusion, b) zone for a crack grown by ∆a.

The shielding stress intensity factor caused by a swelling zone of height ω is [12.1, 12.2]

$$K\_{\omega\_h} = -\nu \frac{\varepsilon\_\nu E}{1 - \nu} \sqrt{\alpha} \tag{12.1.1}$$

where Eis Young's modulus and ν Poisson's ratio.

# 12.2 Shielding stress intensity factors for a damaged material

A consequence of damage by hydroxyl generation is the reduction of Young's modulus E. When ED is the modulus in the damaged state and E0 the value for undamaged silica, we could derive the relation (11.3.8), [12.2]

$$\frac{E\_D}{E\_0} = (1 - S \, / \, S\_{\text{max}})^2 \tag{12.2.1}$$

with Smax =0.188 [0.16, 0.23].

Fig. 12.2 a) Water profile on DCDC specimen on the faces of a growing crack fractured in liquid water by Tomozawa et al. [12.3].

Including the varying modulus into the computation of the shielding stress intensity factor requires a modification of eq.(9.2.6). Now the stress intensity factor results from

$$K\_{sh} = \frac{1}{S(0)E\_0} \prod\_{0}^{\alpha} (S \times E) \left(\frac{\partial K\_{sh}}{\partial \phi}\right)\_{\phi = \phi^{\circ}} d\phi^{\circ} \tag{12.2.2}$$

In order to explain the evaluation of eq.(12.2.2), again the measurements by Tomozawa et al. [12.3], plotted already in Fig. 9.5, may be used here again, Fig. 12.2.

# 12.3 Shielding stress intensity factor for asymmetric damage

In the case of disappearing total stress intensity factor at the crack tip, Ktotal = Kappl +Ktip ≤0, all volume elements in the water-affected zone are under compression due to volume swelling by hydroxyl generation [12.4]. In the previous considerations, we applied the model of pore-like defects. This model describes symmetrical material response under tension and compression loading. The Young's modulus in compression is not necessarily identical in tension and compression, see Section 11.6.

Figure 12.3b again shows the shielding stress intensity factor for the hydroxyl concentration according to eqs.(12.2.1) and (11.6.1) for different values of the non-symmetry coefficient or crack closure parameter 0≤h≤1.

Fig. 12.3 Distribution of the product S×E relevant for eq.(11.2.2), computation for h =1 in eq.(11.6.1), b) for the representations by Fig. 12.2a (blue curve) and Fig. 12.2b (red curve).

The product S×E that governs the shielding stress intensity factor according to eq.(12.2.2) is shown in Fig. 12.3. The blue curve shows the result for the representation by Fig.12.2a and the red curve by Fig. 12.2b. The region at a depth of z≈5-6 nm has the strongest influence on shielding.

Fig. 12.4 Shielding stress intensity factors for an erfc-shaped hydroxyl distribution, as a function of the surface concentration S(0) from Fig. 11.8b. Square: Result for S(0)= 16.5% and h≅0.05.

Figure 12.4 again represents the results of Fig. 11.8b for an erfc-shaped hydroxyl distribution with a surface value S(0) and a parameter bof 4.95 nm (solid curves).

An effective value of h can be estimated from the experimental result of S(0)=16.5%, a crack-growth rate of v ≅10-9m/s, and a related applied stress intensity factor Kappl≅0.43 MPa√m (see Fig. 8.4a). According to eq.(8.2.8), we obtain for KIc=0.75 MPa√m and Kth=0.3 MPa√m [12.5]: Ksh=-0.213 MPa√m as introduced as the square in Fig. 12.4. This value is equivalent to an asymetry parameter of

$$h \equiv 0.05\tag{12.3.1}$$

### **12.4 Mechanical interpretation of the limit case under compression**

The question remains how to explain such an asymmetry in the modulus of elasticity. A possible interpretation of the asymmetry coefficient h is illustrated in Figure 12.5a, where a water molecule may attack a load-carrying bond of the silica net structure.

A SiO2-ring of the silica network may be described by a spherical (nano)-void in a continuous material. This allows a fracture mechanics treatment by an FE-study. According to the finite crack-tip curvature for cracks, the radius of such a void has to be expected in the order of ρ = 0.5 nm [12.6], Fig. 12.5a.

Under a remote tensile load in z-direction, σ∞, the red Si-O-bond may be cracked by reaction with molecular water (blue/red). In the tensile case the opened bond cannot transfer any significant tensile forces, Fig. 12.5b. Consequently, it would be h→1 in eq.(11.6.1).

Fig. 12.5 a) Bond breaking under uni-axial tension by reaction with molecular water (attacked Si-O-bond in red).

Due to the initially load-carrying but now fractured bond, the stress concentrations increase at the ends of the now tablet-shaped void as indicated by the red circles. Due to the exponential increase of the hydroxyl concentration with stress, newly reacting water molecules will predominantly react at these locations.

Fig. 12.6 a) Additionally cracked bond at a location of high stress concentration, b) limit case of a wedgelike contact of the hydroxyls under compression.

### References

12.1 McMeeking, R.M., Evans, A.G., Mechanics of Transformation-Toughening in Brittle Materials, J. Am. Ceram. Soc. 65(1982), 242-246.

12.2 T. Fett, G. Schell, C. Bucharsky, Hydroxyl Damage in Silica: Full-range description including large damages 126, 2019, ISSN: 2194-1629, Karlsruhe, KIT.

12.3 M. Tomozawa, W.-T. Han and W.A. Lanford, Water Entry into Silica Glass during Slow Crack Growth, J. Am. Ceram. Soc. 74 [10] (1991) 2573-2576.

12.4 S. M. Wiederhorn, T. Fett, G. Rizzi, M. J. Hoffmann, J.-P. Guin, The Effect of Water Penetration on Crack Growth in Silica Glass, Engng. Fract. Mech. 100(2013), 3-16.

12.5 V. Sglavo, T. Fett, K.G. Schell, M. J. Hoffmann, S. M. Wiederhorn, v-K-data for silica from interrupted lifetime measurements 137, 2020, ISSN: 2194-1629, Karlsruhe, KIT.

12.6 S.M. Wiederhorn, E.R. Fuller, Jr. and R. Thomson, Micromechanisms of crack growth in ceramics and glasses in corrosive environments, Metal Science, 14(1980), 450-8.

# **APPENDIX**

# A1 Modifications of shielding by additional effects

#### A1.1 Deformed steady-state swelling zones

The computations of the swelling zones are mostly carried out for diffusion in a motionless coordinate system with a fixed crack tip, i.e. for an arrested crack. Whereas for a growing crack the transversal diffusion normal to the crack plane is hardly affected by the moving crack tip, the diffusion and the crack propagation compete in crack direction [A1.1, A1.2]. The consequence for strongly stress enhanced diffusion zones with cordial shape is a reduced ratio of the zone size in length direction r0 to the zone height ω (Fig. A1.1a), which for an arrested crack is [A1.3]

$$\frac{r\_0(0)}{\rho o(0)} = \frac{8}{\sqrt{27}}\tag{A1.1.1}$$

In this relation r0(0), ω(0) stands for r0(v=0) and ω(v=0), respectively.

Fig. A1.1 a) Swelling zone ahead the tip of an arrested crack, b) zone with reduced length r0(v) for a crack growing at a constant crack-growth rate v(schematic).

For the stress intensity factor of the deformed zones one can use some single results from literature. Depending on the subcritical crack growth rate, a growing crack can show a crack-tip diffusion zone with a reduced value of r0/ω as studied in [A1.4]. Such a zone is illustrated in Fig. A1.1b. The stress intensity factors for three different ratios of r0/ω are available from [A1.3]. These cases are illustrated in Fig. A1.2a. Figure..A1.2b shows the related coefficients ψ for the general representation

$$K\_{\rm sh} = -\nu \frac{\varepsilon\_0 E}{1 - \nu} \sqrt{\alpha} \tag{A1.1.2}$$

by the circles. The interpolating curve through the data points in Fig. A1.2b. Figure A1.2can be expressed simply by

$$\forall \nu \equiv 0.37 \left( 1 - \frac{1}{2} \tanh \left( \frac{7}{9} \frac{r\_0}{\alpha} \right) \right) \tag{A1.1.3}$$

This interpolation relation enables the computation of stress intensity factors for deformed diffusion zones.

Fig. A1.2 a) Swelling zones with different shape at the zone end, b) shielding stress intensity factors by eq.(A1.1.2) for the zones given in a), expressed by the coefficient ψ.

#### A1.2 Crack leaving a swelling zone

At higher temperatures and after long time, also unloaded cracks may develop a swelling zone of thickness b by water diffusion into the crack faces, Fig. A1.3a. In this case, the zone end is modeled by a semicircle. If the swelling strain is constant in the swelling zone, εv=ε0=const., and εv=0 outside the zone, the shielding stress intensity factor is

$$K\_{sh} = -\nu \frac{\varepsilon\_0 E}{1 - \nu} \sqrt{b} \tag{A1.2.1}$$

with the coefficient ψ depending on the zone shape in the crack-tip region. If such a crack grows by an amount of ∆a, it will escape from the initial swelling zone, as illustrated in Fig. A1.3a-Fig. A1.3d for different zone ends. The shielding stress intensity factors for two limit cases of the zone shape near the crack tip are plotted in Fig. A1.3e. For the circular zone end (Fig. A1.3b), the shielding stress intensity factor is given by the squares. A straight zone end with r0/b=0 yields the dash-dotted curve. The two dependencies can be described by eq.(A1.2.1) where now ψ is a function of ∆a/b. In order to minimize the uncertainty in the true shape of the zone ends it was recommended in [A1.5] to use for computations r0/b= 0.5 with coefficient ψ= 0.3.

The stress intensity factor solution for the case r0/b=½ (Fig. A1.3d) can be obtained by linear interpolation of the two limit cases resulting in the dashed curve of Fig. A1.3e. The coefficient ψ may be approximated by

$$\forall \nu \equiv \begin{cases} 0.305 + \frac{1}{12} \Delta a / b & \text{for } \quad \Delta a / b \le \frac{1}{2} \\ 0.34 \exp[-2\sqrt{\Delta a / b - \frac{1}{2}}] & \Delta a / b > \frac{1}{2} \end{cases} \tag{A1.2.2}$$

It should be mentioned that the curves in Fig. A1.3e change abruptly caused by the assumption of a sharp layer boundary. For an arbitrarily varying strain distribution, the weight function or Green's function method from [A1.6] can be used. The stress intensity factor caused by a swelling layer of thickness db' in the surface distance z=b' (a) undergoing the strain ε(b') results as

$$d(K\_{\prec h}) = K\_{\prec h}(b' + db') - K\_{\prec h}(b') \tag{A1.2.3}$$

This gives rise for a Green's- or "weight function" hdefined as

$$h = \frac{d(K\_{sh})}{db'} \tag{A1.2.4}$$

For any swelling distribution ε(z)=ε(b'), the related shielding stress intensity factor is then obtained by summing over all the zone increments db' multiplied with the local swelling strain

$$K\_{sh} = \frac{1}{\mathcal{E}\_0} \int\_0^\circ \varepsilon(b')h(b')db', \quad \varepsilon\_0 = \varepsilon(b'=0) \tag{A1.2.5}$$

Fig. A1.3 a) Crack with a thick circular swelling layer of constant swelling strain, b) crack after additional extension ∆a for a circular zone, c) for a rectangular zone, d) for a zone with r0/b=1/2, e) variation of the shielding stress intensity factor for a crack growing in and outside the initial layer according to [A1.3] (dash-dotted curve) and [A1.5] (squares); dashed curve: interpolated from the continuous and dash-dotted curves for r0/b=1/2.

In the absence of stress-enhanced diffusion, the swelling profiles are represented by the complementary error function. The volumetric strain at the location z=b' is in this case

$$
\varepsilon = \varepsilon\_0 \operatorname{erfc}\left(\frac{b'}{2\sqrt{D\_o t}}\right) \tag{A1.2.6}
$$

where ε0 is the strain at the free surface, D0 the diffusion coefficient and t the time after the first water contact.

Equation (A1.2.5) has been evaluated for the case of r0/b=½. The resulting shielding stress intensity factors related to the erfc-shaped swelling profiles are shown in Fig. A1.4b. The numerical result for r0/b=½ can approximated by equation (A1.2.1) with

$$\nu \approx \frac{0.305}{1 + \left(\frac{\Delta a}{b}\right)^2} \tag{A1.2.7}$$

Fig. A1.4 a) Green's function as the shielding stress intensity factor of a thin swelling layer of thickness b', b) stress intensity factors according to the dashed curve in (d) for an erfc-shaped strain distribution (insert); curves according to Fig. A1.3e.

#### A1.3 Blunt crack with finite notch root radius

Two crack-tip models are commonly used in fracture mechanics for the description of stresses ahead of a crack. Whereas continuum mechanics assumes the existence of a mathematically sharp crack, in a micro-structurally motivated approach the slender notch with a finite radius at the crack end is considered. Stresses at a sharp crack tip are described by the crack-tip stress intensity factor Ktip. They become singular if the tip is approached, σ→∞ for r→0. However, the K-description via continuum mechanics must fail for very short distances, when the tip distance competes with the microstructure of the material. In the micro-structurally motivated approach, the crack tip region can be considered as a slender notch with root radius ρ in the order of the average radius of the SiO2 rings. Wiederhorn et al. [A1.7] suggest a crack-tip radius ρ = 0.5 nm for the silica network.

Whereas shielding stress intensity factors are well known for sharp cracks [A1.6], there is a lack in multiaxiality behaviour of swelling stresses for the slender notch model. The individual swelling stress components and also the hydrostatic stress terms are necessary for computations on stress-enhanced equilibrium of the silica/water reaction. In contrast to internal swelling, the stress components by externally applied loading are known from Creager and Paris [A1.8].

#### A1.3.1 Finite element model

Finite Element computations in order to determine swelling stresses for the slender notch model as a function of the distance from the notch root and differently high swelling zones were carried out in [A1.9]. A single crack embedded in a swelling zone of height h and constant volume swelling is shown in Fig. A1.5. The crack length, a, is small compared to the specimen dimensions, realizing the edge-cracked semi-infinite body. On the other hand, the crack length is at least a factor of 10 larger than the notch radius ρ so that a "slender" notch is sufficiently guaranteed. Zones of different heights were modelled in a finite element (FE) study. Solid continuum elements (8-node bi-quadratic) were chosen and the computations were carried out with ABAQUS Version 6.8. The volume strain was replaced by the equivalent thermal problem by heating the swelling zone by ∆T keeping T=0 in the rest of the structure. This results in the strain

$$
\varepsilon\_\circ = \mathfrak{Z}\alpha\Delta T \tag{A1.3.1}
$$

(α=thermal expansion coefficient) with isotropic expansion in x-, y-, and z-directions

$$
\mathfrak{E}\_{\flat} = \mathfrak{E}\_{\times} + \mathfrak{E}\_{\flat} + \mathfrak{E}\_{\natural} \tag{A1.3.2}
$$

Requirements for the z-coordinate were chosen as "plane strain" as is fulfilled at crack tips.

Fig. A1.5 Slender notch at a surface with swelling zone of thickness h.

#### A1.3.2 Results

Swelling stresses in x-, y-, and z-directions are plotted in Figs. A1.6a-d for zone heights of h/ρ=0.2, 0.5 and 1.0. The black data represent the hydrostatic stress, defined by

$$
\sigma\_h = \frac{1}{3} (\sigma\_\times + \sigma\_\times + \sigma\_\times) \tag{A1.3.3}
$$

Whereas all stresses are compressive within the swelling zone, they roughly disappear outside for the z-direction and for the hydrostatic stress. The x-component remains in compression and the y-component becomes tension as a consequence of the equilibrium condition.

Fig. A1.6 a)-c) Swelling stresses, normalized on the volume expansion strain for 3 different zone heights h, d) Effect of zone height h on swelling stresses at the notch root, x=0.

The surface stresses (x→0) are represented in Fig. A1.6d as a function of the relative height h/ρ of the swelling zone. The squares indicate theoretically known values for an infinitely thin surface layer: At a free surface, the stress state is plane stress and, consequently, also stresses caused by swelling are equi-biaxial (σx=0)

$$
\sigma\_{\text{y}} = \sigma\_{\text{z}} = -\frac{\varepsilon\_{\text{v}} E}{\Im(1 - \nu)}\tag{A1.3.4}
$$

where E is Young's modulus and ν is Poisson's ratio. For silica it is ν=0.17, consequently: σ/εvE = −0.4016 for the y- and z-directions. The hydrostatic stress reads

$$
\sigma\_h = \frac{1}{3} (\sigma\_\wp + \sigma\_\varepsilon) = -\frac{2\,\,\varepsilon\_\wp \, E}{9(1 - \nu)}\tag{A1.3.5}
$$

i.e. σh/(εvE)=0.2677.

In order to characterize the "multiaxiality" of the swelling stresses, we plot in Fig. A1.7a the hydrostatic stress normalized on the z-component. This parameter is 2/3≈0.67 for thin swelling layers and decreases slightly for σh/σy and increases for σh/σz. The variation over the swelling zones is negligible as can be concluded from Fig. A1.7b.

Fig. A1.7 a) Effect of zone height h on multiaxiality at the notch root, x=0, b) variation of the multiaxiality ratio σh/σz over the swelling zones.

#### A1.3.3 Effect of finite notch length

From Fig. A1.6a-d it is visible that the hydrostatic swelling stress shows a small variation over the swelling zone. For an understanding of this small effect, we increased for b/ρ=1 the notch length to a/ρ=30. Figure A1.8a shows the effect for σh. In this plot the data for a/ρ=10 are represented by the black symbols, those for a/ρ=30 by the red ones. It is obvious that the variation over the region 0≤x≤b clearly decreases. In Fig. A1.8b the effect on the normal swelling stress σy is shown. Here, the variation remains unaffected over the zone size. Only the absolute stress values σ/εvEshift slightly.

Fig. A1.8 Effect of the notch length on the hydrostatic and the normal swelling stress.

#### A1.3.4 Including shielding stress intensity factors

Stresses at slender notches were given by Creager and Paris [A1.8]. The stress component normal to the crack plane (the stress σy) is in distance xfrom the notch root

$$
\sigma\_y = \frac{2K}{\sqrt{\pi \left(\rho + 2x\right)} \,\rho + 2x} \tag{A1.3.6}
$$

The other stress components are

$$
\sigma\_x = \frac{2K}{\sqrt{\pi(\rho + 2x)}\ \rho + 2x} \tag{A1.3.7}
$$

$$
\sigma\_{\varepsilon} = \frac{2\nu K}{\sqrt{\pi \left(\rho + 2\varepsilon\right)}}\tag{A1.3.8}
$$

Consequently, the hydrostatic stress term from the applied stresses results as

$$
\sigma\_{appl,h} = \frac{2(1+\nu)\,K}{3\sqrt{\pi(\rho+2x)}}\tag{A1.3.9}
$$

Equations (A1.3.6-A1.3.9) are valid for applied stress intensity factors Kappl, shielding stress intensity factors Ksh, and total stress intensity factor Ktip, which are defined as

$$K\_{\rm nlp} = K\_{\rm aqpl} + K\_h \tag{A1.3.10}$$

The shielding stress intensity factor for an edge crack of finite length a, embedded in a swelling zone of height b, was already computed via FE in [A1.10]. The results can be expressed by the relation

$$K\_{sh}(b/a) \equiv K\_{sh}(0) \exp\left[-1.18 \frac{b}{a}\right] \tag{A1.3.11}$$

with

$$K\_{sh}(0) = -\wp(0)\frac{\varepsilon\_v E}{1-\nu}\sqrt{b} \tag{A1.3.12}$$

and different zone shape [A1.3]

$$\boldsymbol{\nu}(0) = \begin{cases} 0.22 & \text{for heart shape} \\ 0.25 & \text{for semi circle} \end{cases} \tag{A1.3.13}$$

Now we can use this solution for the computation of the shielding stress intensity factor in eqs.(A1.3.6-A1.3.9), resulting in swelling stresses contributed by Ksh. The resulting stresses are then subtracted from the results in Fig. A1.6d. The difference is the contribution by local swelling exclusively, i.e. for Ksh=0.

Two different procedures are now possible for the computation of swelling stresses:

Since the resolved data of Fig. A1.9 vary clearly stronger with the distance x from the notch root it is recommended to use results of Fig. A1.6d. For the hydrostatic stress term, it is suggested for 0<x≤b:

$$
\sigma\_h \left( \frac{x}{\rho}, \frac{b}{\rho} \right) = -\frac{2 \,\, \varepsilon\_v \, E}{9(1 - \nu)} \left( 1 + 0.078 \frac{b}{\rho} \right) \tag{A1.3.14}
$$

Fig. A1.9 Swelling stresses for a disappearing shielding stress intensity factor, Ksh=0.

### References

A1.1 Zhao, T., Nehorai, A., Detection and localization of a moving biochemical source in a semi-infinite medium, Report, Electrical and Computer Engineering Dept. Univ. of Illinois, Chicago, USA.

A1.2 Babich, V.M., On asymptotic solutions of the problems of moving sources of diffusion and oscillations, UDC 534.231.1, 517.226, 1991, pp. 3067-3071, Plenum Publishing Corp.

A1.3 McMeeking, R.M., Evans, A.G., Mechanics of Transformation-Toughening in Brittle Materials, J. Am. Ceram. Soc. 65(1982), 242-246.

A1.4 T. Fett, New contributions to R-curves and bridging stresses – Applications of weight functions, IAM 3, KIT Scientific Publishing, 2012, Karlsruhe.

A1.5 Fett, T., Guin, J.P., Wiederhorn, S.M., Interpretation of effects at the static fatigue limit of soda-lime-silicate glass, Engng. Fract. Mech. 72(2005), 2774-279.

A1.6 Wiederhorn, S.M., Fett, T., Rizzi, G., Hoffmann, M.J., Guin, J.-P., The Effect of Water Penetration on Crack Growth in Silica Glass, Engng. Fract. Mech. 100(2013), 3- 16.

A1.7 S.M. Wiederhorn, E.R. Fuller, Jr. and R. Thomson, Micromechanisms of crack growth in ceramics and glasses in corrosive environments, Metal Science, 14(1980), 450-8.

A1.8 Creager, M., Paris, P.C., Elastic field equations for blunt cracks with reference to stress corrosion cracking, Int. J. Fract. 3(1967), 247-252.

A1.9 G. Rizzi, G. Schell, T. Fett, Swelling stresses ahead of slender notches, 52, 2016, ISSN: 2194-1629, Karlsruhe, KIT.

A1.10 G. Rizzi, T. Fett, Defects at Silica Surfaces - A Finite Element Study, Scientific Working Papers 33, 2015, KIT Scientific Publishing, Karlsruhe.

# A2 Finite Element study on heartshaped zones

#### A2.1 Zone shape

The singular hydrostatic near-tip stresses are given as

$$
\sigma\_h = \frac{2}{3} (1+\nu) \frac{K}{\sqrt{2\pi r}} \cos(\varphi/2) \tag{A2.1.1}
$$

where r and ϕ are the polar coordinates with the origin at the crack tip, see Fig. A2.1a. Equations (9.1.1) and (A2.1.1) imply that in the high crack-tip stress field nearly all water is present in form of hydroxyl S.

The shape r(ϕ) of the zone contour for constant sh in plane strain results from eq.(A2.1.1) as

$$r = \frac{8}{3\sqrt{3}}\rho\cos^2(\theta/2) \tag{A2.1.2}$$

with the height ω of the zone for a prescribed hydrostatic stress

$$\rho = \frac{(1+\nu)^2}{4\sqrt{3}\pi} \left(\frac{K\_1}{\sigma\_{\text{h}}}\right)^2 \tag{A2.1.3}$$

Fig. A2.1 a) Contour for a constant hydrostatic stress ahead of a loaded crack, b) variation of Young's modulus in a frontal zone E(x) (dashed curve), replaced by an average (effective) modulus E.

The contour of constant hydrostatic stress according to eqs.(A2.1.2, A2.1.3) is shown in Fig. A2.1a as the heart-shaped curve. This contour also describes the damage zone in front of the tip of a crack that has not yet grown by subcritical or stable crack growth. The hydroxyl generation by the reaction (1.1.1) causes damage since the originally intact silica ring structure is cracked by the water attack. One of the consequences of such damage is the reduction of Young's modulus E(see Section 11).

As Fig. A2.1a shows, the tip of the crack is not embedded in the heart-shaped zone but terminates at r=0. This point in principle belongs to both the frontal zone and the undamaged surrounding material. A description by eq.(11.3.8) leads to a variable module distribution E(x), which is replaced here by an average constant value for reasons of simplicity, Fig. A2.1b. The module E0 prevails outside the crack-tip zone. Another reason for the decrease in the elasticity module in the "frontal process zone" is microcracking in polycrystalline ceramics as was studied by Evans and Faber [A2.1].

# A2.2 Crack terminating at ϕ=π of the heart-shaped zone

#### A2.2.1 Constant modulus over the zone size

First we studied the case that a constant modulus is prescribed over the whole zone. In the heart-shaped zone the Young's modulus is E ≤E0 and outside the zone E0. FE calculations were performed with ABAQUS 6.9. In order to obtain normalized stress intensity factors K' according to (A2.2.2a), ω=1, sappl=1, and E0=1 were chosen. Table A2.1 shows the J-integrals according to Rice [A2.2] and the stress intensity factors obtained from the first contours for the lowest Young's modulus of E=0.01 E0. Apart from the first contour, the results are almost constant. Due to symmetry, no mode-II stress intensity factor could appear. The average mode-I stress intensity factor over contour 2-8 is 1.974 and the Standard Deviation 0.00125 (0.06%). The interrelation between crack opening displacement v and stress intensity factor KI is for the near-tip field along the symmetry line (here: −x=r<<ω) in plain strain with E'=E0/(1-ν2 )

$$K\_I = \sqrt{\frac{\pi}{8}} \nu \frac{E^\circ}{\sqrt{r}} \tag{A2.2.1}$$

Normalized displacements v' represented in Fig. A2.2 are defined as

$$\mathbf{V} = \mathbf{V} \frac{E\_0}{\sigma\_{appl} \, o(1 - \nu^2)} \tag{A2.2.1a}$$

The stress intensity factors obtained from the least-squares fit of the displacements in Fig. A2.2 are compiled in Table A2.2, Column 3. In this context, it has to be noted that crack opening displacements COD are defined as the distance from the symmetry line to one of the surfaces. Unfortunately, the (total) crack opening, i.e. the distance between the crack flanks, is sometimes also referred to as COD.

Table A2.1 J-integral and stress intensity factors for the first contours (E=0.01, E0 =1, ν=0.17, ω=1) in normalized form.

An independent procedure for the stress intensity factor determination is the evaluation of the near-tip stresses ahead of the crack tip, i.e. the stresses along the prospective crack propagation plane. For the normal component on this plane it holds for the first terms

$$
\sigma\_{yy} = \frac{K}{\sqrt{2\pi x}} + A\sqrt{x} \tag{A2.2.2}
$$

A regression analysis of the data in Fig. A2.3 provides the stress intensity factors. The results obtained from the data fit of Fig. A2.3 are compiled in Table A2.2, Column 4 in the dimensionless form of

$$K = \frac{K}{\sigma\_{app}\sqrt{\alpha}}\tag{A2.2.2a}$$

Fig. A2.2 FE-results of near-tip CODs fitted according to eq.(A2.2.1).

Fig. A2.3 Normalized stress syy /sappl along the prospective crack propagation line (ϕ=0), lines: fitting curves according to eq.(A2.2.2), yielding the stress intensity factor for the data points in the range of 0.0005<x/ω<0.004.

#### A2.2.2 Step-shaped modulus distribution

So far only cases with a constant modulus were studied, Fig. A2.4a. An additional FEevaluation was made for the case of a variable modulus in the zone ahead the tip. As Fig. A2.4b shows, the modulus was varied in 4 steps. In the inner zone of Fig. A2.4b, the modulus E was E=0.25E0, then E=0.57E0, E=0.72E0, and outside E=E0. The result from the J-Integral was (for E0 =1, ν=0.17, sappl=1 and ω=1): K'=14.83, hardly different from the case of constant modulus E=0.25 E0. Also this value is introduced in the second Column of Table A2.2.

Table A2.2 Normalized stress intensity factors K' from J-Integral, COD, and near-tip stress syy in the form of eq.(A2.2.2a).

The K-values for the single zones of Fig. A2.4a, compiled in Table A2.2, are plotted in Fig. A2.5a by the symbols (for the geometric data see the insert). In this plot the stress intensity factors were normalized on the J-based K value at E/E0=1. The red triangle corresponds to the varying Young's modulus shown in Fig. A2.4b. Figure A2.5b represents the ratio of the data for COD- and syy-based stress intensity factors normalized on the stress intensity factors from the J-Integral. The hatched area indicates the region of data scatter with an average value and "scatter" of about

$$K(E)/K(E) \cong 0.993 \pm 0.015$$

Fig. A2.4 a) Crack-tip zone with a Young's modulus E deviating from the bulk modulus E0, E<E0, b) stepwise decreasing modulus in zones of height 1/3, 2/3, and 3/3ω.

Fig. A2.5 a) Stress intensity factors from J-Integral, COD, and near-tip stress syy, (K-values normalized on the result from J-Integral for E/E0=1), red triangle for step-shaped E-distribution, b) comparison of Kfrom COD and syy with the individual results from J-Integral

#### A2.2.3 Conclusions from the FE-analysis

From the results of Fig. A2.5 it becomes evident that

• the straight-line behaviour of Fig. A2.5a suggests the representation by

$$K \cong K\_{optl} \sqrt{\frac{E}{E\_0}}\tag{A2.2.3}$$

The FE-results confirm eq.(A2.2.3) as had to be expected from the theoretical analysis by Merkle [A2.3] on slender notches embedded in the reduced modulus at the notch-root.

From the agreement between the theoretical solution by Merkle [A2.3] and the FE-results for 0.1≤E/E0≤1, it can be concluded that our crack-zone modelling and the accuracy of our FE-mesh is sufficient at least for this range. For the lowest value of E/E0≤0.01, the FE-program indicated convergence problems. With regard to the data accuracy it should be noted that the theoretical analysis was carried out for an infinite body (ω/a, ω/H, ω/W)→∞, while the FE calculation was carried out on a finite one.

For conditions comparable with "small-scale yielding" in elastic-plastic fracture mechanics, i.e. ω << (a,Wa), the effect of the zone with reduced E on Kappl becomes a "nulleffect" [A2.2]. Then it holds

$$K\_{appl} = \sigma\_0 F(a/W, H/W) \sqrt{\pi a} \tag{A2.2.4}$$

with the geometric function F (a/W, H/W) and the remote normal tractions s0 at the plate ends.

We used in our study a ratio of ω/a=1/30 and expected that the zone would be sufficiently small to fulfill geometry conditions for small-scale yielding. Despite this relatively small zone, influences from this zone can still be seen in Fig. A2.5a. This may at least partially be caused by the influence of an effective zone length aeff=a+λω, λ≥0 with λ=0 for the case of homogeneous material.

In the case of λ>0, eq.(A2.2.4) reads

$$K\_{appl} = \sigma\_0 F(a\_{\text{eff}} / W, H / W) \sqrt{\pi a\_{\text{eff}}} \tag{A2.2.5}$$

The stress intensity factors for E/E0=0.25 and ω/a=0, 1/30, and 1/90 are plotted in Fig. A2.6. A clear trend with ω/a is visible, showing an increase of the stress intensity factor with increasing relative zone sizes.

Fig. A2.6 Stress intensity factor from J-Integral for (E/E0=0.25).

# A2.3 Derivation of J-Integral for slender notches by Merkle

Merkle [A2.3] showed that the relation (A2.2.3) is exact in the case of a slender notch ending in a region of Young's modulus E different from the bulk material E0. In his analytical derivation he assumes a half-infinite crack in an infinite body loaded by remote tension in y-direction resulting in the applied stress intensity factor Kappl. Our FEcomputations were carried out on a finite rectangular plate (inset in Fig. A2.5a). The stress intensity factors were evaluated via the J-integral defined by

$$J = \int\_{\Gamma} W \, d\mathbf{y} - \mathbf{T} \frac{\partial u}{\partial \mathbf{x}} ds \,\tag{A2.3.1}$$

where W is the stain energy density, Γ the path around the crack tip, T the tractions on the path, u the displacement in y-direction, and dsa length increment of Γ.

For reasons of simplicity we used the usual path along the surfaces represented in Fig. A2.7 with the line segments (1)-(8). Along the contours (3), (5) and (7) no tractions act and on (4) and (6) constant tensile stresses are prescribed.

The J-integral along the surface segment (1) is

$$J\_1 = \int\_{\Gamma\_1} W \, d\mathbf{y} \tag{A2.3.2}$$

In order to simplify the path-integration, the slender notch was approximated by a narrow elliptic notch with parabolic end shape (Fig. A2.8).

Fig. A2.7 Integration path for J-evaluation (red arrows)

An arbitrary point on the curved surface of Fig. A2.8c is described by the geometrical interrelations

$$\mathbf{v} = r \sin \phi \mathbf{\hat{z}} \,, \quad r = \frac{\rho}{1 + \cos \phi} \Longrightarrow \mathbf{dy} = \frac{\rho \, d\phi}{1 + \cos \phi} \, \tag{A2.3.3}$$

First it has to be noted that in the J-Integral only the modulus at the surface, Es, enters and the variation with the distance from the surface doesn't play any role. In a subcritically grown crack the reduced value Es is constant along the whole notch surface.

Merkle [A2.3] showed that the elastic strain energy along the crack surfaces (1, 2, 8) is

$$W = \frac{\sigma\_{\prime}^{2}}{2E\_{s}^{\prime}} = \frac{K^{2}}{E\_{s}^{\prime}} \frac{\cos^{2}(\wp/2)}{\pi r} \tag{A2.3.4}$$

with the tangential stresses st. From the geometrical relations in eq.(A2.3.3) it follows after a simple mathematical manipulation

$$W = \frac{K^2}{E\_s'} \frac{\left(1 + \cos \varphi\right)^2}{2\pi \,\rho} \tag{A2.3.5}$$

Consequently, the J-integral along path parts (1, 2, 8) reads

$$J\_{\downarrow} = \frac{K^2}{E\_{\,\,s}^{\prime}} \frac{1}{2\pi} \int\_{-\pi}^{\pi} (1 + \cos\varphi) d\varphi \Longrightarrow J\_{\downarrow} = \frac{K^2}{E\_{\,\,s}^{\prime}} \tag{A2.3.6}$$

This of course holds for large zones, ω>ρ to ω>>ρ, for which ϕ→π can be sufficiently fulfilled. Since the results of eq.(A2.3.6) and (A2.3.7) no longer depend on the notch radius, they should also apply to the sharp crack by taking ρ→0. The remote (or "applied") stress sappl at lines (4) and (6) cause the applied stress intensity factor Kappl and the related "applied J-integral"

$$J\_{4,6} = \frac{K\_{appl}^2}{E\_{\circ}^{\prime}}\tag{A2.3.7}$$

Fig. A2.8 a) Contour at the tip of a subcritically grown crack, b) detail of a), c) crack surface approximated by an elliptical notch showing the same curvature radius ρ, schematic.

The J-integral along a closed path must disappear as shown by Rice [A2.2], i.e.

$$J\_{4,6} = J\_1 \Longrightarrow \frac{K\_{appl}^2}{E\_{\phantom{\alpha}}'} = \frac{K^2}{E\_{\phantom{\alpha}}'} \Longrightarrow K = K\_{appl} \sqrt{\frac{E}{E\_0}}\tag{A2.3.8}$$

It should be mentioned, that the derivation by Merkle [A2.3] deals with the case of a zone of different Young's modulus ahead of a crack tip with finite notch root radius ρ>0.

When the radius ρ becomes smaller in the limit-case consideration, the geometry of Figs. A2.8b and A2.8c remains self-similar and not necessarily represents the sharp crack.

#### A2.4 Results for extended zones

While in Section A2.2 only zones in front of the crack tip were considered, in this Section, we additionally compile results for zones extending over the entire crack length. In this case, the crack tip is fully embedded in the zone of reduced module. The plate dimensions and the crack length were increased by a factor of about 3 in order to reduce any influence of ω/afurthermore. This is shown in the plots of Fig. A2.9.

Fig. A2.9 a) Stress intensity factors from J-Integral, KJ (normalized on the value K0 obtained for E=E0) for embedded crack tips (squares) compared with the KJ-results of Fig. A2.5a (plotted by circles and triangle), solid curve: Approximation via eq.(A2.4.1), b) Stress intensity factors from J-Integral, dashed line: asymptote with slope=1/2, dash-dotted line: asymptote with slope=1.

The stress intensity factors exclusively on the basis of the J-Integral are given in Fig. A2.9a. The squares for E/E0 ≤0.25 were fitted by

$$\frac{K(E)}{K(E\_0)} \cong \frac{2E}{E + E\_0} \tag{A2.4.1}$$

This expression is introduced by the solid curve.

We also evaluated COD-results as done in Section A2.2 and obtained the near-tip profiles given in Fig. A2.10. Stress intensity factors KCOD were computed from the Irwin-parabola

$$\mathbf{V} = \sqrt{\frac{8}{\pi}} \frac{K\_{\text{CO}}}{E/(1-\nu^2)} \sqrt{r} \tag{A2.4.2}$$

where v is the opening displacement and ν Poisson's ratio (0.17 for silica). The displacements in the crack wake are given for the two geometries ω/a=30 and 100. In both cases a/W=a/H=0.5 was kept constant. The plot v vs. √rshowed very straight lines.

Fig. A2.10 Crack-opening profiles close to the tip versus (r/ω) 1/2 for a crack of a/ω=100.

The stress intensity factors from eq.(A2.2.1) are plotted in Fig. A2.11a as the solid circles together with the results from the J-Integral in Fig. A2.9a as the squares. In this representation no differences of the two K-values can be seen. Figure A2.11b shows the ratio of the two stress intensity factors. Note the significantly higher resolution of the ordinate. The mean value and the standard deviation SD (in brackets) are

$$\frac{K\_{\text{COD}}}{K\_{\text{J}}} = 0.999987 \text{ [} 0.001134 \text{]}$$

Although in principle no theoretical facts can be proven by numerical calculations, this good agreement of the K factors speaks for the validity of KJ using different methods.

Conclusions: From the FE results and the theoretical solution by Merkle [A2.3] it was obtained

$$K = K\_{eqpl} \sqrt{\frac{E}{E\_0}}$$

When comparing the results for the extended areas of reduced module (here: length L≈a) with the solution according to Merkle [A2.3], no agreement can be expected a priori. The zones on which the FE calculations are based do no longer meet the basic requirements of the analytical calculation, namely (ω, L)<<(a,W,H).

Fig. A2.11 a) KJ vs. E/E0 as squares, results from Fig. A2.5 as open circles, KCOD from Fig. A2.10 via eq.(A2.2.1) as solid circles, b) ratio KCOD/KJ.

#### A2.5 Crack-Opening Displacements (COD)

We evaluated COD-results in Section A2.2 in order to compute the stress intensity factors for different Young's moduli and obtained the near-tip profiles. Stress intensity factors KCOD were computed from the Irwin-parabola, eq.(A2.4.2). Again v is the opening displacement in the distance r from the crack tip and ν=0.17 is Poisson's ratio. Normalized displacements v' represented in Fig. A2.12 are defined by

$$\mathbf{V}' = \mathbf{V} \frac{E\_0}{K(E\_0)\sqrt{\alpha o}(1-\nu^2)}\tag{A2.5.1}$$

where K(E0) denotes the stress intensity factor for the case of homogeneous material. In the following COD in larger distance are considered.

#### A2.5.1 COD for heart-shaped zones

Crack-opening displacements for the crack with a heart-shaped zone ahead of the tip are shown in Fig. A2.12a in the form v'=f(r/ω) and in Fig. A2.12b in the form of (v')2 =f(r/ω). From the diagram of (v')2 =f(r/ω) it becomes obvious that the squares of CODs tend to straight lines for about r/ω>1.5. The straight lines for E/E0<1 show fictive crack origins as indicated by the intersections of the dashed extensions of the crack profiles with the abscissa. In the case of very short crack-tip distance the COD is similar to crack-tip blunting in elastic-plastic fracture mechanics although the material behaviour is purely linear-elastic. In larger distances of about r/ω>1.5, the straight line for E/E0=0.01 shows nearly the same slope as obtained for E/E0=1 and seems to be shifted by the constant amount (v'(0))2 (see the inset in Fig. A2.13).

Fig. A2.12 Influence of reduced Young's modulus in the heart-shaped crack-tip zone, a) linear representation of the normalized COD according to eq.(A2.5.1), b) square of the COD as the ordinate.

The values of the straight lines v' at r=0 are called in elastic-ideal plastic fracture mechanics the "Crack-Tip-Opening Displacements (CTOD)" and used as material property for yielding. Figure A2.13 again shows the COD-profiles for the extremely different Young's moduli ratios E/E0 =1 and 1/100. The apparent crack-tip blunting is illustrated by the inset in Fig. A2.13, when the COD is simplified by the thick red contours, resulting in an apparent crack-tip opening and an apparent increase ∆a of the fictive crack length.

Table A2.3 Parameters of straight lines in Fig. A2.12b: Slopes of straight lines, opening displacements (v'(r=0)) 2 at the crack tip, fictive crack-length increase ∆a as a function of Young's modulus ratio E/E0 for ν=0.17.

Fig. A2.13 Representation of near-tip COD by the crack-tip opening displacement.

#### A2.5.2 COD for zones extended in the crack wake

For the zone extending over the whole crack wake, we performed the same evaluations as done for the heart-shaped zone. Figure A2.14 and Table A2.4 show the results.

Table A2.4 Parameters characterizing the dashed straight lines in Fig. A2.14.

Fig. A2.14 Influence of reduced Young's modulus in the extended zone, square of the COD as the ordinate.

# A2.6 Comparison of parameters for the two zone shapes

Figure A2.15 shows again the parameters from Tables A2.3 and A2.4. In Fig. A2.15a the slopes are given and in Fig. A2.15b the offsets of the straight line asymptotes at r/ω=0. It is obvious that the slopes hardly vary noticeably with the modulus of elasticity. Only very slight changes with increasing modulus can be stated.

Summary on COD: We studied the COD-behaviour of cracks in a bi-material. The following points may be emphasized:

Fig. A2.15 a) Comparison of slopes and b) crack-tip displacements from Tables A2.3 and A2.4.

### References

A2.1 Evans, A.G., Faber, K.T., Crack-growth resistance of microcracking brittle materials, J. Am. Ceram. Soc. 67(1984), 255-260.

A2.2 Rice, J.R., A path independent integral and the approximate analysis of strain concentration by notches and cracks, Trans. ASME, J. Appl. Mech. (1986), 379-386.

A2.3 J. G. Merkle, An application of the J-integral to an incremental analysis of blunt crack behavior, Mechanical Engineering Publications, London, 1991, 319-332.

# A3 Hydrostatic stress at crack tips

#### A3.1 Hydrostatic stress under hydroxyl damaging

The hydroxyl generation by the reaction eq.(1.1.1) causes damage since the originally intact silica ring structure is cracked by the water attack. One of the consequences of such damage is the reduction of Young's modulus E. In order to describe this E-decrease, we used in [A3.1] the rather simple damage model proposed by Phany and Niyogi [A3.2].

When ED is the modulus in the damaged state and E0 the value for undamaged silica, we could derive the relation (see Section 11 and [A3.1])

$$\frac{E\_D}{E\_0} = (1 - S \, / \, S\_{\text{max}})^2 \tag{A3.1.1}$$

with Smax =0.188. Under the condition that the damage is isotropic, the multiaxiality in damaged states remains unchanged. In a volume element that is hydrostatically loaded by tensile stresses σx=σy =σz, the externally applied hydrostatic stress is

$$
\sigma\_{h,appl} = \frac{1}{3} (\sigma\_x + \sigma\_y + \sigma\_z) \tag{A3.1.2}
$$

The hydroxyl concentration as a function of stress, reads by using eq.(A3.1.1)

$$S = S\_0 \exp\left[\sigma\_{h,D} \frac{\Delta \overline{V}}{RT}\right] = S\_0 \exp\left[\sigma\_{h,appl} (1 - S / S\_{\text{max}})^2 \frac{\Delta \overline{V}}{RT}\right] \tag{A3.1.3}$$

In Fig. A3.1a, the hydroxyl concentration is plotted vs. the applied hydrostatic stress for S0=0.1 wt-%, and θ=90°C. The dash-dotted line indicates the maximum hydroxyl concentration of Smax=18.8 wt-%. Figure A3.1b shows the related hydrostatic stress in the damaged state, µσh,D, vs. the applied hydrostatic stress µσh,appl. In the notation of the coordinate axis the abbreviation µ is defined as

$$
\mu = \frac{\Delta \overline{V}}{RT} \tag{A3.1.4}
$$

The maximum hydrostatic stress, asymptotically reached for σh.appl→∞, is

$$
\mu \sigma\_{h,D,\max} = -\text{Log}\left[\frac{S\_0}{S\_{\text{max}}}\right] \tag{A3.1.5}
$$

The curves σ h,D(σh,appl) of Fig. A3.1b may be simplified by a bi-linear description as introduced by the dash-dotted lines in Fig. A3.1b.

In this case, the maximum possible stress in the damaged state is depending on the temperature. In the sense of an upper limit solution, the stresses can be approximated as

$$
\sigma\_{h,D} = \begin{cases}
\sigma\_{h,appl} & \text{for} \quad \sigma\_{h,appl} < \sigma\_{D,h,\text{max}} \\
\sigma\_{h,D,\text{max}} & \text{else}
\end{cases}
\tag{A3.1.6}
$$

#### A3.2 Irwin and Dugdale model

Due to the damage via hydroxyl generation, the stress-strain behavior at a crack tip is no longer linear. By application of the bi-linearized dependency eq.(A3.1.6), fracture mechanics problems can approximately be solved applying for instance the well-known Irwin and Dugdale models, originally developed for elastic/plastic material behaviour.

The Irwin [A3.3] and the Dugdale model [A3.4] describe plastic deformations in a zone ahead of the crack. Whereas in the Irwin model the size of the "plastic zone" was computed from "yield"-condition σappl,y=σY and by force equilibrium along the prospective crack plane, the size of the Dugdale zone results from the condition that the stress intensity factor for the fictive crack in Fig. A3.2a disappears. So the shape of the zones is not fixed, especially not for the Dugdale zone. The latter is mostly concentrated along the x-axis.

According to these models the "length" L of the "plastic" zone, is under plane stress conditions

$$L\_{\rm Jup} = \frac{1}{\pi} \left(\frac{K\_{\rm appl}}{\sigma\_Y}\right)^2, \ L\_{\rm Doppler} = \frac{\pi}{8} \left(\frac{K\_{\rm appl}}{\sigma\_Y}\right)^2 \tag{A3.2.1}$$

where σY is the "yield stress" for the case of a linear-elastic/ideal-plastic material. The opening at the tip of the physical crack, called Crack Tip Opening Displacement (CTOD), δt, results from [A3.3, A3.4] as

$$\mathcal{S}\_{\mathfrak{t},\text{Irwin}} = \frac{4}{\pi} \frac{K\_{appl}^2}{E\sigma\_Y}, \; \mathcal{S}\_{\mathfrak{t},\text{Doughle}} = \frac{K\_{appl}^2}{E\sigma\_Y} \tag{A3.2.2}$$

The CTOD, δt, is illustrated in Figs. A3.2b and A3.2c. Whereas the Irwin model [A3.3] is preferred for circular "plastic zones", the Dugdale model [A3.4] describes also plastic deformations in a narrow strip in front of the crack. In the derivation of the Dugdale formula, it was assumed that there exists an internal through-the-thickness crack in an infinite plate and that the yield stress is constant over the zone size L regardless of the locally varying strains. This provided the analytical solutions (A3.2.1) and (A3.2.2) for the zone size and the CTOD.

Remark: It should be emphasized here once again that it is not claimed that the glass behaves in an elastic/ideally plastic manner. This approach only leads to a simple fracture mechanical treatment of the crack tip problem for the "deformation" curves represented by the limit case eq.(A3.1.6).

Fig. A3.2 a) Stress distribution curve near a crack tip (black curve) approximated by an elastic-"plastic" material behaviour (red curve) in the crack-tip region (zone shape is not necessarily a circle), b) crack tip opening displacement (CTOD) δt, c) observable crack profile.

### References

A3.2 Phani, K.H. Niyogi, K. De, Young's modulus of porous brittle solids, J. Mater. Sci. 22(1987), 257–263.

A3.3 G. R. Irwin, Plastic zone near a crack and fracture toughness, Proc. 7th Sagamore Conf. 1960, p. IV-63 (1960).

A3.4 D.S. Dugdale, Yielding of steel sheets containing slits. J. Mech. Phys. Solids 8 (1960), 100-8.

A3.1 T. Fett, G. Schell, C. Bucharsky, Hydroxyl Damage in Silica: Full-range description including large damages 126, 2019, ISSN: 2194-1629, Karlsruhe, KIT.

# A4 Swelling stresses from crackterminating angles

#### A4.1 Estimation of the magnitude of residual stresses

#### A4.1.1 Shielding stress intensity factors and residual stresses

Volume expansions occur by swelling of the glass volume due to hydroxyl generation [A4.1, A4.2, A4.3]. The results from [A4.1] can be expressed by eq.(1.1.9) in terms of the hydroxyl concentration Sin mass units.

Due to the mechanical boundary conditions, the swelling strains result in swelling stresses proportional to the amount of hydroxyl concentration as were described by eq.(1.1.11). A volume element in a thick plate that undergoes swelling cannot freely expand. If the diffusion zone is small compared to the component dimensions, expansion is completely prevented in the plane of the surface and can only take place normal to the surface plane.

In Section 2 we determined the shielding stress intensity factor from the externally applied stress intensity factor Kappl and the total stress intensity factor Ktip. In the case of silica, Ktip is for spontaneous and stable crack extension identical with the toughness KIc and was in our rather fast subcritical crack growth tests on silica about Ktip=0.55-0.6 MPa√m. Therefore, we used an average of Ktip=0.575 MPa√m.

The crack-shielding stress intensity factor must be proportional to the swelling stresses ssw acting normal on the crack plane and the square-root of the layer thickness

$$K\_{sh} = F(\varphi, b) \, \sigma\_{sw} \sqrt{\pi \, b} \tag{A4.1.1}$$

The fracture mechanics geometric function for this loading over a small region, F(ϕ,b), is so far unknown, but may be estimated by not too large effort. Very often the stresses caused by diffusion processes (here: water diffusion into silica) and the stress distribution can be described by

$$
\sigma\_{sv} = \sigma\_{sv,0} \operatorname{erfc}\left[\frac{z}{2\,b}\right] \tag{A4.1.2}
$$

where ssw,0 is the surface value, z=0, of the stresses and b is the diffusion depth at which the water concentration and consequently the swelling stresses are reduced to ≅50 % of the surface value as is illustrated in Fig. A4.1a.

Fig. A4.1 a) Stresses in the surface region according to eq.(A4.1.2), b) semi-circular surface crack loaded by the erfc-shaped stresses.

#### A4.1.2 Stress intensity factor solutions for residual stresses

In order to give an estimation for the maximum swelling stresses in eq.(A4.1.2) on the basis of the experimentally obtained shielding stress intensity factor, the geometric function F(ϕ,b) must be known. Following an approximate procedure proposed by Underwood [A4.4], the interrelation between shielding and applied stress intensity factors reads

$$K\_{sh}(\wp, b) = \lambda \frac{K\_{appl}(\wp)}{K\_{appl}(90^\circ)} K\_{sh}(90^\circ, b) \tag{A4.1.3}$$

However, this procedure can yield larger errors if the shapes of the two cracks to be compared are significantly different. In the original paper by Underwood [A4.4] it was shown that the stress intensity factors for complicated cracks (semi-elliptical surface cracks in thick-walled tubes) under complicated stresses (tension and bending superimposed) were always smaller than those for the reference crack (continuous crack in a plate under pure tension), compare e.g. Fig. 6 in [A4.4]. This fact may be indicated by the parameter l≤1 in the following eq.(A4.1.3). The equal sign, l=1, trivially holds when the crack to be predicted is identical with the reference case. From the special result of [A4.4], we can conclude for the coefficient l, introduced in eq.(A4.1.3), l≈2/3.

The shielding stress intensity factor is available for the special case of a crack terminating with ϕ=90°. For this purpose let us replace the straight crack by a semi-circular surface crack of depth a that might be large compared to the layer thickness, a>>b. The stress intensity factor at the surface point (B), Fig. A4.1b, obtained from [A4.3], is

$$K\_{sh,B} \cong 1.29 \,\sigma\_{\approx 0} \sqrt{a} \tanh\left[1.327 \sqrt{\frac{b}{a}} + 0.064 \frac{b}{a}\right] \tag{A4.1.4}$$

A series expansion of the FE-results in [A4.3] for b/a→0 results in the simple expression

$$K\_{\rm sh,B} \cong 1.72 \sigma\_{\rm sw,0} \sqrt{b} \approx \sigma\_{\rm sw,0} \sqrt{\pi b} \tag{A4.1.5}$$

When calculating K-factors, it does not matter whether the applied stresses are externally applied or internal ones. Therefore, both must show the same type of angle dependencies. In our first derivation in [A4.5] it was implicitly assumed that the coefficient l can be regarded as a constant value. From the comparison with experimental results in Fig. 4 of [A4.5], however, a dependence on the terminating angle emerged, l=l(ϕ). This finding should be taken into account in the current evaluation. The shielding stress intensity factor Ksh(ϕ,b) can now be obtained as

$$K\_{sh}(\varphi, b) = \mathcal{\lambda}(\varphi) \frac{1}{0.00658 \,\varphi + 7.66 \times 10^{-13} \,\varphi^6} \sigma\_{s \le 0} \sqrt{\pi b} \tag{A4.1.6}$$

from which the swelling stress ssw,0 can be estimated.

The results for Ksh in Table 2.1 shall be used here to estimate at least the order of magnitude of the residual stresses.

From the shielding stress intensity factors we estimated the residual stresses [A4.5]. For the tests by Schell et al. [A4.6] and Haranoh et al. [A4.7], crack extension is governed by Ktip=KIc. Fracture toughness of soda-lime glass is according to Wiederhorn [A4.8]: KIc=0.75 MPa√m and Poisson's ratio ν=0.225.

From [A4.5] it follows for the applied stress intensity factor of the crack-terminating angle ϕ (soda-lime glass ν=0.225)

$$K\_{sh}(\rho, b) = \lambda(\rho) \frac{1}{0.00606 \,\rho + 8.54 \times 10^{-13} \,\rho^6} \sigma\_{s\nu, 0} \sqrt{\pi b} \tag{A4.1.7}$$

#### A4.2 Discussion of stresses

All residual stresses in Table A4.1 were found to be negative indicating that the water/silica reaction, the ion exchange, and chemical toughening are accompanied by volume expansion. The absolute values of the compressive stresses resulting by the extension of the Underwood procedure depend on the coefficient l that is not yet known. As can be concluded from Underwood [A4.4], the value of l decreases with increasing deviation from the reference crack. A rough estimation of l as a function of ϕ may be suggested based on soda-lime glass undergoing ion-exchange.

In [A4.6] the stresses by ion exchange were concluded from strength measurements as sres = −2425 MPa. A theoretical value of −2300 MPa was determined in [A4.9, A4.10] for the volume change by the ion-exchange fully transformed in stresses. As an average value, sres = −2.4 GPa may be used to compute the value of l. From the data of −lsres in Table A4.1 and the related terminating angles ϕ, ϕ0 in Table 2.1, the red circles in Fig. A4.2a were obtained. In this plot, the terminating angles ϕ were normalized on ϕ=90° representing the reference case (black circle).

The solid curve represents a quadratic fit based on the results for the soda-lime glass with the "weight" for the 1-week result (in parentheses) described by

$$\mathcal{A}(\varphi) \equiv \frac{1}{2} \frac{\varphi}{90^{\circ}} \left( 1 + \frac{\varphi}{90^{\circ}} \right) \tag{A4.2.1}$$

In the case of swelling stresses in silica, the surface stresses at θ=250°C can be computed from the hydroxyl concentration at the free surface as shown in [A4.11]. The swelling stress components in the surface plane are

$$
\sigma\_{\text{sw},\chi} = \sigma\_{\text{sw},\chi} = \frac{\frac{3}{2} \frac{13.75 \,\text{MPa}}{\text{exp}(0.00868 \,\theta)} \exp(0.00868 \,\theta)}{\frac{1}{2} + \frac{1}{A} \exp(Q/RT)}\tag{A4.2.2}
$$

with A=32.3 and Q=10.75 kJ/mol, T=θ+273°.

Fig. A4.2 Coefficient l of eq.(A4.1.3) as a function of the crack-terminating angle ϕ; red circles: results for soda-lime glass, red line: related fitting curve for reduced weight of the data point at ϕ=71° (in parentheses); blue circles: results for silica, b) swelling stress under saturation pressure from crack-terminating angles compared with results from disk experiments, and water concentrations by Zouine et al [A4.12], full symbols: liquid water, open symbols: water vavour.

Equation (A4.2.2) yields for θ=250°C: ssw,x,y = −209 MPa. The results for silica predicted with eq.(A4.2.1), are introduced in Fig. A4.2a by the blue circles, showing compressive stresses in the range of -130 MPa to -170 MPa.

Applying eq.(A4.2.1) on the chemically toughened soda-lime glass gives a residual stress of sres=−1980 MPa. This result is in agreement with a surface stress of "more than −1000 MPa" as had been reported by Haranoh et al. [A4.7].

Table A4.1 Estimates of residual stresses, sres=ssw, in soda-lime glass and silica, predicted residual stresses for silica and chemically toughened soda-lime glass by use of eq.(A4.2.1).

Figure A4.2b gives a comparison between the swelling stresses from crack terminating angles with data from disk bending experiments, [A4.2], and hydroxyl concentrations via eqs.(1.1.11) and (3.1.4). The results for liquid water and 192 h soaking time are about 15% below the curve predicted by eq.(A4.2.2). For the shorter soaking time, the deviation is somewhat larger. Vapour soaking results in clearly deviations from the curve and show the same effect as found for the disk tests, in both cases an indication for the fact, that saturation was not reached.

The silica/water reaction at silica surfaces exposed to water causes volume expansion in the water diffusion layer and as the consequence of mechanical boundary conditions compressive stresses. Evaluation of crack-terminating angles resulted in strongly negative shielding stress intensity factors of about Ksh=−2.5 MPa√m (see Table 2.1), a clear evidence for compressive stresses in the water-affected surface layer. Consequently, the existence of swelling stresses due to the silica/water reaction is proven. For silica, heattreated in humid environments, compressive stresses in the order of about -130 MPa to - 170 MPa are obtained for 250°C.

### References

A4.1 Shelby, J.E., Density of vitreous silica, J. Non-Cryst. 349 (2004), 331-336

A4.2 S.M. Wiederhorn, T. Fett, G. Rizzi, S. Fünfschilling, M.J. Hoffmann and J.-P. Guin, Effect of Water Penetration on the Strength and Toughness of Silica Glass, J. Am. Ceram. Soc. 94 (2011) [S1] S196-S203.

A4.3 T. Fett, G. Rizzi, M. Hoffmann, S. Wagner, and S.M. Wiederhorn, Effect of Water on the inert Strength of Silica Glass: Role of Water Penetration, J. Am. Ceram. Soc. 95 (2012) [12], 3847-3853.

A4.4 H.J. Underwood, Stress intensity factors for internally pressurized thick-wall cylinders, ASTM STP 53(1972), 59.

A4.5 P. Hettich, K. G. Schell, G. Rizzi, S. Wagner, T. Fett, Estimation of swelling stresses from crack-terminating angles, Scientific Working Papers, SWP 119, 2019, ISSN: 2194-1629, Karlsruhe, KIT.

A4.6 K.G. Schell, S. Wagner, P. Hettich, T. Fett, G. Rizzi, M.J. Hoffmann, Identification of residual stress layers at glass surfaces via crack terminating angles, J. Am. Ceram. Soc. 100(2017), 4173-4179.

A4.7 T. Haranoh, H. Ishikawa, N. Shinkai, M. Mizuhashi, Crack evolution in Vickers indentation for soda-lime-silica glass, J. of Mater. Science 17(1982) 1493-1500.

A4.8 S.M. Wiederhorn, Fracture Surface Energy of Glass, J. Am. Ceram. Soc. 52 [2] 99- 105 (1969).

A4.9 T. Fett, J.P. Guin, S.M. Wiederhorn, Stresses in ion-exchange layers of soda-limesilicate glass, Fatigue Fract. Engng. Mater. Struct 28(2005), 507-514.

A4.10 J.P. Guin, S.M. Wiederhorn, T. Fett, Crack-tip structure in soda-lime-silicate glass J. Am. Ceram. Soc. 88(2005), 652-59.

A4.11 T. Fett, K.G. Schell, Elimination of swelling stresses from measurements of the equilibrium constant in silica, Scientific Working Papers, SWP 117, 2019, ISSN: 2194- 1629, Karlsruhe, KIT.

A4.12 A. Zouine, O. Dersch, G. Walter and F. Rauch, Diffusivity and solubility of water in silica glass in the temperature range 23-200°C, Phys. Chem. Glass: Eur. J. Glass Sci and Tech. Pt. B, 48 [2] (2007), 85-91.

### **Schriftenreihe des Instituts für Angewandte Materialien**

#### **ISSN 2192-9963**

Die Bände sind unter www.ksp.kit.edu als PDF frei verfügbar oder als Druckausgabe bestellbar.

Die Bände sind unter www.ksp.kit.edu als PDF frei verfügbar oder als Druckausgabe bestellbar.

Die Bände sind unter www.ksp.kit.edu als PDF frei verfügbar oder als Druckausgabe bestellbar.

### KARLSRUHER INSTITUT FÜR TECHNOLOGIE (KIT) SCHRIFTENREIHE DES INSTITUTS FÜR ANGEWANDTE MATERIALIEN

Water diffusing into silica surfaces gives rise for several effects on diffusion behaviour and mechanical properties. In a preceding booklet, we focused on diffusion and fiber strengths and deformations which were obtained by water soaking under external loading. In the present booklet we deal with results and interpretations of strength increase in the absence of applied stresses. In detail the following topics are discussed:

**T. FETT & G. SCHELL**

Consequences of Hydroxyls in Silica – Part II & III

ISSN 2192-9963 ISBN 978-3-7315-1159-5 102