The failure data is shown for the 12.5 mm diameter disks is shown in Fig. 12. The average failure pressure for the standard disks is 12 MPa, whereas the average failure pressure for the epi disks is 23 MPa.
Figure 12. Failure pressure of a set of 12.5 mm diameter, 051 mm thick sapphire windows with a standard 80/50 polish (Tests # 1-5) and a nominal epitaxial finish (Tests # 7-11).
For these tests the variation in strength was approximately a factor of 4. This is a small sample, and confusion is added to the standard sample by not placing any constraint on either the quality or the orientation of the standard disks. The disks had no observable defects, either by eye or through crossed polarizers. The epi samples were best quality sapphire, but they were cut from a larger disk, which may have introduced flaws. Furthermore, it is known that the large disks have more defects at the edge than at the center - sample 7 was taken from the edge.
The apparent disagreement between the theoretical prediction and the average of the standard disk experimental data is explained by postulating that some of these disks have been strengthened, and some of the epi polished disks weakened, both by the chance inclusion or exclusion of weakening flaws. If this were the case the lowest failure pressure values for the standard polish would be expected to agree with the theory for the unstrengthened sapphire, as well as the highest failure pressure values matching theory for the epi polished disks. The data indicates that this is the case. The failure pressure varies by a factor of 4 for all of the disks, which is a very large variation in strength. The strength of the 80/50 polish disks varies by a factor of 2.7; the epi disk strength varies by a factor of 1.7.
The degree of variation in the failure strength data does not allow the prediction of standard failure strength, unless the minimum strength results are used. Given the large variation, quite a few more tests would have to be done to verify that the minimums are indeed typical minimums. For the purposes of engineering, the minimum strength would have to be used, which, for sapphire, is usually given as approximately 420 MPa in tension at room temperature. The variation in these results makes it very difficult to use for supporting any modeling calculations.
Strain/Deflection Testing. Other properties than strength, such as Poisson's ratio, ν , and Young's modulus, E, are well known and accurately specified for sapphire. This means that although the failure strength cannot be readily predicted, the strain (stress) and deflection of the disk in response to load are much more accurately known. As a result, stress/strain and central deflection measurements provide a much more reliable test and support of large deflection modeling.
A complication in correlating experimental stress/strain and deflection vs. load with large deflection theory was found to be the O-ring sealing. O-rings require compression to seal, and most of the window pressure testing was done without any significant static clamping on the window. High-force clamping would seat the O-ring, but it would also lead to high stresses at the aperture rather than the distributed stresses sought to optimize window performance. Experimentally, the windows begin pressurization seated and sealed against the O-ring, but not in contact with the flange face or aperture. As the loading built up the disk would both deflect at the center and the outer edge would move toward the flange face. At some load the disk would come in contact with the aperture. For lower loads the disk would act as a simply supported disk, supported at the O-ring radius. For higher loads, the disk would act in a cantilevered manner, closer to the edge immovable and clamped boundary condition based on a disk with a radius equal to the aperture radius. This effect will be discussed with respect to the detail test results discussed below.
Deflection measurements were performed on 2.5 cm, 5.0 cm, 7.5 cm, and 10.0 cm diameter disks. Strain gauge measurements were made on 5.0 cm, and 10.0 cm diameter disks. Because of the progressive seating of the O-ring, the disks are described as 1) initially edge mounted and simply supported at the O-ring diameter until it deflects enough to come in contact with the flange at the aperture, at which point it becomes 2) an edge mounted disk supported at the aperture diameter with an immovable edge (constrained by the material beyond the aperture) that is progressively more clamped as the pressure increases and the force on the part of the disk beyond the aperture increases.
Experimental results best agree with large deflection modeling in the case where the disk is purposely clamped into the flange by adding a spacer ring between the flanges. Clamping in not perfect, because the primary clamp between the flange must be the conflat knife edge that forms the pressure seal. Figures 13a and b show the results of strain and displacement testing on a 100 cm diameter clamped disk. The aperture radius is 44.6 mm. Figure 13a shows the central displacement vs. load. The experimental results agree closely with the large deflection model for the clamped, immovable edge case. The prediction of the linear model and some other boundary conditions for the large deflection model are also shown for comparison. Central deflection for the clamped, edge immovable case is approximately 1/3 of what it would be for the simply supported case. Figure 13b shows the central stress for the clamped, edge immovable case, which also agrees well with the model. As expected the agreement improves at higher loads. At a load of 1 atm. the central stress is reduced by a over a factor of 3 compared with linear theory, but it must be remembered that the stress is higher at the aperture. The strain and derived stress experimental results appeared to be less reliable than the central deflection measurements, which were a simpler measurement.
A very different test case of the displacement and stress theory is the data for a 100 mm window supported and sealed near its edge, with an aperture radius of 44.6 mm and a seal radius of 46.5 mm. Figure 14a shows the displacement data, together with the simply supported and clamped immovable edge cases predicted by the large deflection theory. The experimental data measures the displacement of the window that includes the compression of the O-ring. The window is approximately 0.35 mm above the flange surface at zero pressure. At 2 atm loading the disk is cantilevered at the aperture, so that the true central deflection is 1.55 mm - 0.35 mm, or 1.2 mm, in close agreement again with the simply supported, edge immovable data. The experimental data and theory for stress vs. load is shown in Fig. 14b. The measured stress is what would be expected of a disk that began to be loaded at the O-ring diameter and approached large deflection theory for a disk the same diameter as the aperture at higher loading.
Extensive displacement data was taken for 75 mm, 0.43 mm thick sapphire windows that were soldered onto copper cups that formed one half of a standard double-disk microwave window fixture. The copper cups had a nominal 64 mm ID. The soldering process bowed the disks toward the high pressure side by approximately 0.15 mm at the center due to the larger contraction of the copper and solder relative to the sapphire. After the joint had hardened the copper contracted more than the sapphire, bowing the sapphire away from the copper cup at the center. This added an effective cantilevering preload to the sapphire. Under 1 atm vacuum, the deflection at the center of the disk was 0.21 mm, giving a total of 0.36 mm central deflection in response to 1 atm pressure. Large deflection theory predicts a central deflection of 0.42 mm for a clamped, immovable edge 64 mm diameter disk. The soldered disk must bend the copper at its edge, as well, presumably accounting for the reduced total deflection.
Figure 13. Large deflection modeling compared with experimental results for a 100 mm diameter thin clamped sapphire window: a) central deflection vs. load, and b) central stress vs. load.
Stress and deflection measurements were also made on a 50 mm diameter, 33 mm thick disk. The displacement and stress theory and data for a 50 cm window with an aperture radius of 20.9 mm are shown in Fig. 15 together with the stress and displacement data. Figure 15a shows the displacement data together with the simply supported, simply supported edge immovable, and clamped immovable edge cases theory predictions. The experimental data measures the displacement of the window that includes the compression of the O-ring. The window is approximately 0.5 mm above the flange surface at zero pressure. At 2 atm loading the disk is cantilevered at the aperture, so that the true central deflection is 8.4 mm - 0.5 mm, or 3.4 mm, in close agreement with the simply supported, edge immovable data. The experimental data and theory for stress vs. load is shown in Fig. 15b. The measured stress at high loading approaches the simply supported, edge immovable data as expected. At lower pressures the disk behaves as if it were simply support, edge free with a diameter equal to the O-ring diameter. Since this is a significantly larger diameter than the aperture diameter, the stresses are higher. The 50 mm disk measurements are thus fully consistent with large deflection theory.
Deflection data was also taken for 25 mm diameter, 0.51 mm thick disks. The disks had an aperture diameter of 19 mm and a seal diameter of approximately 24.6 mm. Deflection data was again very close to the simply supported, edge immovable case.
All of the data together strongly support the use of large deflection theory to predict the behavior of the O-ring sealed windows. The edge immovable, simply supported boundary condition based on the aperture diameter seems to be the appropriate input parameters for the theory for the testing procedures and disk parameters used in this program. This boundary condition does not imply high edge stresses, so it should give good predictions of the maximum stress in the disk. It is not clear whether the edge immovable boundary condition represents the true physical condition of the disk, or whether the combination of partial clamping and extended material makes this condition hold as an average condition. It does seem appropriate to use the edge immovable, simply supported boundary condition of large deflection theory based on the flange aperture to predict failure stress from failure pressure measurements. This will allow some conclusions to be reached about polish strengthening and the statistics of the sapphire failure.
Figure 14. Large deflection modeling compared with experimental results for a thin unclamped 100 mm diameter sapphire window: a) central deflection vs. load, and b) central stress vs. load.
