1.

Introduction

The lifetime of a brittle optical window subject to stress is limited by the rate at which microscopic cracks grow under the influence of stress until the window fractures. For many window materials, slow crack growth is promoted by atmospheric moisture, which participates in bond breaking chemical reactions at the tip of a growing crack.^1,2 It is necessary to know the upper limit for the rate of slow crack growth to establish the design life of a window.

Slow crack growth rate is commonly characterized by dynamic fatigue experiments in which test coupons are broken at a series of constant applied stress rates: ${\dot{\sigma }}_a=d{\sigma }_a/dt$, where ${\sigma }_a$ is applied stress, $t$ is time, and ${\dot{\sigma }}_a$ is the applied stress rate. The more rapidly stress is applied, the less time there is for cracks to grow and the higher the stress at which failure occurs. Coupons are tested either with their as-polished surface having a natural distribution of microscopic flaws, or they are tested with a deliberate indentation that creates a reproducible flaw size larger than natural flaws.

Dynamic fatigue experimental results are fit equally well in the measured range by two empirical crack growth rate laws: a power law and an exponential law. When extrapolated to low stress intensity factors ($K_I$) common to many window applications, the two laws diverge and can predict window lifetimes that differ by several orders of magnitude. For most materials, we do not have data at low $K_I$ to differentiate which crack growth law applies. In the absence of this knowledge, U.S. National Aeronautics and Space Administration design rules prescribe that the more conservative exponential law shall be used for window lifetime prediction.³

Power law slow crack growth parameters can be derived from dynamic fatigue measurements of as-polished coupons by using analytical equations given in ASTM C1368.⁴ Fuller et al.⁵ gave an approximate analytical fit to a numerical solution to derive power law parameters from dynamic fatigue of indented coupons. There are no analytical equations to derive exponential law parameters from dynamic fatigue of as-polished or indented coupons. Jakus et al.⁶ described a numerical method to obtain the exponential law from dynamic fatigue of as-polished (unindented) coupons. Choi et al.⁷ presented an approximate method to obtain the exponential law from dynamic fatigue of as-polished (unindented) coupons. Ritter et al.⁸ gave an approximate method to obtain the exponential law from dynamic fatigue of indented coupons. The purpose of our paper is to present a numerical method for least-squares fitting of all data points in dynamic fatigue of as-polished or indented coupons to find parameters for the exponential law and the power law.

2.

Dynamic Fatigue Measurements for BK7 Glass

Figure 1(a) shows dynamic fatigue measurements from Tables 1 and 2 for as-polished BK7 glass disks immersed in water and broken in biaxial flexure at stress rates of ${\dot{\sigma }}_a=0.39$, 7.7, 77, or $620\mathrm{MPa}/\mathrm{s}$.⁵ Figure 1(b) shows results from Tables 3 and 4 for BK7 glass disks indented by a Vickers indenter prior to measuring the dynamic fatigue strength in water.

Fig. 1

Dynamic fatigue of BK7 glass disks in distilled water at room temperature: (a) as-polished surface and (b) Vickers indented with 250 g load. Data from Fuller et al.⁵ provided by E. R. Fuller and G. D. Quinn. Inert strength in Fig. 1(a) is shifted to the left side of the graph for clarity. Data for these graphs are listed in Tables 1Table 2Table 3–4.

Table 1

Dynamic fatigue data for unindented (as-polished) BK7 glass in water.

Table 2

Inert strength of unindented (as-polished) BK7 glass in dry nitrogen.

Table 3

Dynamic fatigue data for Vickers-indented BK7 glass in water.

Table 4

Inert strength of Vickers-indented BK7 glass in dry nitrogen.

High scatter in the strength of as-polished coupons in Fig. 1(a) is typical for optically polished glass and ceramics. The standard deviation of the mean for the slope is 25% and 1.4% for the intercept. The intercept is the ordinate when $\log (\text{stress rate})=0$. Scatter is reduced by a factor of 10 for indented coupons in Fig. 1(b) because failure originates at the indentation crack, which is larger than natural flaws and has a reproducible size. Fewer indented coupons are required for testing because of the low scatter. The natural flaws in the as-polished specimens were identified by Quinn⁵ as scratch/dig flaws, invisible on the polished surfaces, created during the specimen preparation that had all surface traces eliminated by final polishing.

The near equivalence of the slopes in Figs. 1(a) and 1(b) is fortuitous because of the 25% uncertainty in the slope of Fig. 1(a). The theoretical slope in Fig. 1(a) should be $1/(n+1)$, where $n$ is the exponent in the power law Eq. (2) in the next section. The theoretical slope in Fig. 1(b) should be $1/(n^{\prime \prime }+1)$, where the relation between $n$ and $n^{\prime \prime }$ is given by Eq. (15) in the paper of Fuller et al.⁵ $n=(4n^{\prime \prime }-2)/3$. From the observed slope in Fig. 1(b) = 0.06101, we find $n^{\prime \prime }=15.391$. From this value of $n^{\prime \prime }$, we calculate $n=19.855$ and the slope of Fig. 1(a) should be $1/(n+1)=0.04795$. The observed slope in Fig. 1(a) is $0.06212\pm 25\%=0.0466$ to 0.0776, which includes the theoretical value 0.04795. The only reason why the slopes in Figs. 1(a) and 1(b) are nearly the same is because of the large scatter in the data in Fig. 1(a).

3.

Power Law and Exponential Law for Crack Growth Rate

Two types of empirical equations are commonly used to fit observed slow crack growth rates ($v$). The exponential law equation is

Equations (1) and (2) generally fit observed failure strengths equally well in dynamic fatigue experiments in the measured range of stress rates. However, Fig. 2 shows that the two crack growth laws diverge when extrapolated to low stress intensity factor ($K_I$). The exponential law predicts higher crack growth rate at low stress intensity factor than the rate predicted by the power law.

Fig. 2

Comparison of exponential and power law behavior as a function of $K_I$. Crack growth rate parameters are derived from coupons in Fig. 1 using $K_{Ic}=0.95\mathrm{MPa}\sqrt{\text{m}}$ and geometric factor $Y=2/\sqrt{\pi }$ in the fracture mechanics equation $K_I=Y{\sigma }_a\sqrt{c}$.

The low $K_I$ region is where windows with low applied stress spend most of their lives. As cracks grow under the influence of a constant stress, crack size increases and the rate of crack growth accelerates, eventually becoming catastrophic. Because most of a window’s life is spent at low $K_{\mathrm{I}}$, window lifetime predicted by the power law can be up to several orders of magnitude longer than lifetime predicted by the exponential law.

For most materials, we do not have data at low $K_I$ to know which crack growth law applies. Static fatigue experiments at a constant low stress until failure occurs can, in principle, distinguish the two growth laws, but such experiments are slow to conduct. Median times to failure for unindented material in Table 1 range from 0.2 to 210 s at decreasing stress rates. For indented material, median times range from 0.8 to 4600 s. A reviewer of this paper believes that static fatigue testing is more realistic than dynamic fatigue because trends can change at very long lifetimes. That point is well taken, but static fatigue measurements took that reviewer up to 2 years to conduct. Dynamic fatigue is a compromise between obtaining meaningful measurements and concluding the work in days instead of years. In the absence of knowing the applicable crack growth law, U.S. National Aeronautics and Space Administration design rules prescribe that the more conservative exponential law Eq. (1) shall be used for window lifetime prediction.³

Power law crack growth parameters $A^{*}$ and $n$ for Eq. (2) for unindented coupons are derived from the slope and intercept of the dynamic fatigue graph in Fig. 1(a) by analytical equations given in ASTM C1368.⁴ There are no analytical equations to derive exponential law parameters from dynamic fatigue of unindented or indented coupons because Eq. (1) must be integrated numerically. We now present a numerical method for least-squares fitting of all data points in dynamic fatigue of as-polished (unindented) or indented coupons in Fig. 1(a) or Fig. 1(b) to find parameters for the exponential law or the power law.

4.

Stress Intensity Field for Indented Test Coupons

First, we introduce equations taken from Fuller, Lawn, Cook, and coworkers^5,9–13 to analyze dynamic fatigue of indented coupons. Then we apply these equations to find the parameters $v_o$ and $\beta$ for the exponential law Eq. (1) or $A^{*}$ and $n$ for the power law Eq. (2) for indented or unindented coupons.

Indentation of a ceramic or glass surface with a Vickers pyramidal diamond indenter leaves a residual stress field due to an elastic/plastic mismatch resulting in a localized stress intensity factor $K_r$ that depends on material and indentation load and the equation is

Here, $P$ is the indentation force, $c$ is the radius (“crack size”) of the half-penny radial or median crack produced by indentation, and ${\chi }_r$ is a proportionality constant that depends on the elastic modulus and hardness of the material. If stress ${\sigma }_a$ is applied to an indented test coupon in a dynamic fatigue experiment, the combined stress intensity factor is the sum of the residual stress intensity factor in Eq. (3) plus the applied stress intensity factor, which is given by

When an indentation crack of initial size $c_i$ is first formed, as in an equilibrium condition, the residual stress intensity factor is equal to the critical stress intensity factor, i.e., $K_r=K_{Ic}$. Substituting $c_i$ into the denominator of Eq. (3) and $K_{Ic}$ in place of $K_r$ gives the relation ${\chi }_{r}P=K_{Ic}{c}_i^{3/2}$. Substituting $K_{Ic}{c}_i^{3/2}$ for ${\chi }_{r}P$ in Eq. (5) gives

The inert strength of a material is the average strength measured when a coupon is taken to failure in an inert atmosphere (such as dry nitrogen) in which slow crack growth is negligible. We can find an expression for inert strength by replacing $K_I$ with $K_{Ic}$ at the left side of Eq. (5) and then solving for the applied stress needed to break the coupon:

The applied stress ${\sigma }_a$ in Eq. (7) passes through a maximum that we will call ${\sigma }_m$ when the crack size is $c=c_m$. The maximum stress ${\sigma }_m$ is the inert strength, which is the strength of indented coupons when there is no slow crack growth. To find an expression for ${\sigma }_m$, set the derivative $d{\sigma }_a/dc$ for Eq. (7) equal to 0 and solve for the maximum stress ${\sigma }_m$, and the equation is given by

Substitute the expression for ${\sigma }_m$ from Eq. (8) back into Eq. (7) to find an expression for $c_m$, which is given by

In summary, we have the following information from dynamic fatigue of indented coupons:

5.

Crack Propagation in Dynamic Fatigue of Indented Coupons

5.1.

Evolution of $K_I$ and Crack Size in Dynamic Fatigue

Modeling the evolution of crack size and mechanical strength during dynamic fatigue is needed to find $A^{*}$ and $n$ for the power law or $v_{\mathrm{o}}$ and $\beta$ for the exponential law. The power law has an analytical solution, but the exponential law requires numerical integration. We developed a numerical method which can be modified for use with either as-polished (unindented) or indented coupons, and for either power or exponential law crack growth.

The procedure to be described below allows us to compute curves in Fig. 3 showing the progression of stress intensity factor and crack size during dynamic fatigue. The abscissa is the fraction of time to failure. Actual times to failure decrease by four orders of magnitude as the stress rate increases by four orders of magnitude. In Fig. 3(a) the stress intensity factor decreases rapidly from its initial value $K_I=K_{Ic}$ down to a nearly constant level that persists through most of the dynamic fatigue measurement, and then climbs rapidly to failure. Crack size in Fig. 3(b) jumps from its initial value $c_i$ and then increases gradually until it rises rapidly when close to failure. It takes 90% of the measurement time for the crack to double in size from the almost level value of $\sim 20\mu \mathrm{m}$ early in the experiment up to $\sim 40\mu \mathrm{m}$ near the end of the experiment.

Fig. 3

Calculated change in (a) stress intensity factor and (b) crack size for Vickers indented BK7 glass as a function of time. Parameters $v_{\mathrm{o}}$ and $\beta$ taken from Wiederhorn et al.¹⁴: $v_{\mathrm{o}}=2.3\times {10}^{-14}\mathrm{m}/\mathrm{s}$ and $\beta =39.53{(\mathrm{MPa}\sqrt{\text{m}})}^{-1}$ for $K_I>0.4\mathrm{MPa}\sqrt{\text{m}}$ and $v_{\mathrm{o}}=3.2\times {10}^{-22}\mathrm{m}/\mathrm{s}$ and $\beta =84.96{(\mathrm{MPa}\sqrt{\text{m}})}^{-1}$ for $K_I<0.4\mathrm{MPa}\sqrt{\text{m}}$. In Wiederhorn’s paper BK7 glass is called borosilicate crown I. Other parameters used were inert $\text{strength}=117.2\mathrm{MPa}$ (Table 4), $K_{Ic}=0.95\mathrm{MPa}\sqrt{\text{m}}$ (Refs. 15 and 16), and $Y=2/\sqrt{\pi }$.

6.

Method Validation

To validate our results for unindented (as-polished) coupons, Table 5 compares the power law derived by our numerical method with the power law derived from analytical equations of ASTM C1368. Parameters derived by both methods for three different materials are almost identical.

Table 5

Dynamic fatigue power law validation data for unindented materials in water.

To validate our method for indented coupons, we can compare our power law parameters with those from the approximate equations given by Fuller et al.^5,11 Table 6 shows good agreement between the two methods for three different glass materials. Raw data for indented BK7 glass and soda-lime glass have relatively low scatter (slope uncertainties of 2% and 6%). Higher scatter of 18% in the raw data for the specialty glass did not lead to much discrepancy in the parameters derived by our method and Fuller’s method. Fuller et al.¹¹ state that their analytical equations are within 1% of those from numerical integration for the exponent $n$ and within 10% for the multiplier ($A^{*}$). We observe the same level of agreement (1% and 10%) between our numerical integration and Fuller’s equations.

Table 6

Dynamic fatigue power law validation data for indented materials in water.

We also verified that our numerical method is predicting close to the same failure stress calculated by the equations of ASTM C1368 for unindented coupons or by Fuller’s approximate equations for indented coupons over the wide range of input parameters in Table 7. Over the parameter space with 6 million combinations in Table 7, 99% of results are within 0.2% of the ASTM method for unindented coupons, and within 0.84% of Fuller’s approximate method for indented coupons. Tables 5 and 6 demonstrate that our numerical method finds the same global minimum expected from the ASTM or Fuller equations.

Table 7

Parameter space used in model testing.

7.

Uncertainty in Crack Growth Parameters

To estimate uncertainties in Eq. (1) crack growth parameters $\beta$ and $\log v_{\mathrm{o}}$, we employ the jackknife procedure.^21,22 In this procedure, we delete the first point (${\dot{\sigma }}_a=77.17\mathrm{MPa}/\mathrm{s}$, ${\sigma }_f=63.97\mathrm{MPa}$) in Table 3 and compute the best values of $\beta$ and $\log v_{\mathrm{o}}$ from the remaining 33 points. The first row of Table 8 shows that the best values are $\beta =42.74$ and $\log v_{\mathrm{o}}=-14.08$. For the second row of Table 8, we replace the first point back into the set and delete the second point. We continue deleting one point at a time from the original data to generate 34 new values of $\beta$ and $\log v_{\mathrm{o}}$ in columns 2 and 3 of Table 8.

Table 8

Jackknife error analysis for data from Table 1 data.

The standard error ($u=\text{standard deviation}$ of the mean) of a parameter is an estimate of the uncertainty of that parameter. At the bottom of Table 8, we compute the mean and standard error of the 34 values of $\beta$ and $\log v_{\mathrm{o}}$. The jackknife estimates of the standard error $(u_{\beta })$ and variance $({u_{\beta }}^2)$ for $\beta$ are

Covariance is a measure of the magnitude and direction in which a change in one variable affects another variable. Columns 4 and 5 of Table 8 compute the jackknife estimates of the covariance of $\beta$ and $\log v_{\mathrm{o}}$ as follows:

Let us estimate the uncertainty in crack velocity $u_v$ from the exponential crack growth law $v=v_o{e}^{\beta K_I}$ for best fit parameters $\beta =42.39$ and $\log v_o=-14.02$. For a chosen stress intensity factor $K_I=0.40\mathrm{MPa}\sqrt{\text{m}}$, the crack velocity is $v=v_o{e}^{\beta K_I}=2.238\times {10}^{-7}\mathrm{m}/\mathrm{s}$. To find the uncertainty in crack velocity, we will use standard errors at the bottom of Table 8: $u_a=0.2374$, $u_{\beta }=1.1264$, and covariance $u_{a\beta }=-0.2644$. The uncertainty in crack velocity $u_v$ is found from the equation

The partial derivative $\partial v/\partial a=\partial v/\partial (\log v_o)$ is obtained by expressing the exponential law in the form $v=v_o{e}^{\beta K_{\mathrm{I}}}=e^{\ln v_o}{e}^{\beta K_I}=e^{(\log v_o)(\ln 10)}{e}^{\beta K_I}=e^{a\ln 10}{e}^{\beta K_I}$, giving $\partial v/\partial a=(\ln 10)e^{a\ln 10}{e}^{\beta K_I}=5.154\times {10}^{-7}$. Similarly, $\partial v/\partial \beta =K_I{v}_o{e}^{\beta K_I}=8.953\times {10}^{-8}$. With these values, the uncertainty in crack velocity from Eq. (23) is $u_v=2.740\times {10}^{-8}\mathrm{m}/\mathrm{s}$. Combining the velocity with its uncertainty, we can say $v=(2.238\pm 0.274)\times {10}^{-7}=(1.96$ to $2.51)\times {10}^{-7}\mathrm{m}/\mathrm{s}$. In this example, the covariance term in Eq. (23) is almost as large as the sum of the two variance terms, but opposite in sign. If we had neglected covariance, the relative uncertainty in velocity would have been 71% instead of 12%, which emphasizes the importance of including covariance.

To assess the jackknife procedure applied to finding standard errors $u_{\text{slope}}$, $u_{\text{intercept}}$, and covariance $u_{\text{slope},\text{intercept}}$ for fitting a straight line, we calculated the standard errors and covariance for the slope and intercept of the straight line in Fig. 1(b) using analytical equations 11, 13, and X1.11 in ASTM C1368.⁴ Jackknife standard errors differed from those of the analytical equations by $-1\%$, $+6\%$, and $-27\%$ for $u_a$, $u_{\beta }$, and $u_{a\beta }$, respectively.

8.

Discussion and Conclusion

Equations (5(6)(7)(8)(9)(10)(11))–(12) from indentation fracture mechanics^5,9–13 predict that the initial size of the indentation flaw is $c_i=(0.3968)c_m$, where $c_m$ is the crack size at failure in an inert atmosphere. Our calculations are predicated on this calculated value of $c_i$. It was observed at NIST that the size of the unstable crack grows in air before the dynamic fatigue experiment begins. It is also assumed that residual stress from indentation continues to exist throughout the mechanical test, but diminishes according to Eq. (3) as crack size grows during the mechanical test. We could question what is the “right” initial crack size to use in simulating dynamic fatigue and does the residual stress relax by mechanisms other than just crack extension during the test?

Notice in Fig. 3 that the stress intensity factor drops and the crack size increases in a fraction of a second to values that remain almost flat for much of the dynamic fatigue experiment. We find from simulations in Table 9 that the starting value of crack size [$c_{t-1}$ in Eq. (16) for the first step of the simulation] of dynamic fatigue can be varied over a factor of 2.5 with little effect on the derived values of the slow crack growth parameters. We still use the same theoretical value of the indentation crack size $c_i=(0.3968)c_m$, but we allow the crack to propagate after indentation and prior to testing in these simulations. For the data in Tables 3 and 4, we conclude that if the crack grows to less than $2.5*c_i$ between indentation and testing, then pre-test crack growth has little effect on the derived slow crack growth parameters.

Table 9

Effect of crack growth after indentation, but before testing.

The other question raised above is does residual stress relax over time by mechanisms other than crack extension? We cannot provide a universal answer to this question, but we can cite one study²³ of soda-lime glass in which there was no significant decline in residual stress in air or water even up to 40 days. Spontaneous, post-indentation crack growth was observed for more than a day, but the residual stress driving the crack growth appeared to be constant.

In conclusion, we presented a step-by-step recipe for numerical integration of the slow crack growth exponential rate law Eq. (1) and power rate law Eq. (2) to derive the least-squares best-fit parameters $v_{\mathrm{o}}$ and $\beta$ or $A^{*}$ and $n$ from dynamic fatigue experiments such as those in Figs. 1(a) and 1(b) with as-polished or indented test coupons. A link to our MATLAB code is provided. This code has been optimized to handle data sets with hundreds of points in a period of minutes. Tables 5 and 6 validate the method by demonstrating that the code gives the same crack growth parameters as found with the analytical equations of ASTM C1368 or close to the approximate equations of Fuller et al.^5,11 Section 7 describes how to use the jackknife method to estimate the variance and covariance of the crack growth parameters and to estimate the uncertainty in crack growth rate from those parameters. Table 9 demonstrates that limited crack growth up to 2.5 times the initial crack size after indentation and before dynamic fatigue testing has little effect on the derived crack growth parameters.