Membrane effects describe the case where the disk stretches rather than bends. Figure 5 shows the differences in the deflection and stress for the two cases. Figure 6a shows the case for a standard flexing disk where the deflection is small compared with the thickness. This is the fundamental assumption for standard thin disk stress prediction, and leads to the formulas that are currently used for designing microwave windows. The side nearest the load is under compression, whereas the side away from the loading is under tension. For an edge supported disk the maximum stress depends on load, radius and thickness as:

σ _max proportional to q(a/h)²

Figure 6b shows the case for a membrane where the deflection is much larger than the thickness, and bending moments are unimportant. In this case the entire thickness of the disk is in tension, so that the tensile stress necessary to balance the loading can be spread over the entire thickness of the disk. For an edge supported membrane the maximum stress has a quite different dependence on disk parameters:

σ _max proportional to E^1/3q^2/3(a/h)^2/3

In the membrane regime the maximum stress in a disk increases much more slowly with disk radius, and it is somewhat dependent on the Young's Modulus, which effectively measures the thickness contraction as the membrane stretches. Sapphire is much too stiff for bending moments to be totally neglected, so modeling the stress in very thin sapphire windows involves some combination of membrane and bending effects.

Large Deflection Theory. Given that the standard theory of thin flexing disks is inadequate because it does not include membrane effects, research was performed to find a tractable model that could be used to predict the response to loading of very thin sapphire disks.

The general theoretical treatment of thin circular plates is based on the following assumptions: (1) The plate is flat, of uniform thickness, and of homogeneous isotropic material, (2) The thickness of the plate is not more than about 1/4 of its least transverse dimension, (3) All forces - loads and reactions - are normal to the plane of the plate, and (4) The plate is nowhere stressed beyond its elastic limit.

For a thin plate under transverse loading, bending (flexural behavior) and stretching (membrane behavior) can both occur at the same time. Pure bending behavior is such that the mid-plane undergoes only out-of-plane deflection, which is small by definition. The relationships of stress and deformations with the transverse loading are linear and this type of behavior is governed by the small-deflection theory. The typical limit on central deflection for pure bending modeling to be valid is that the deflection be less than half the thickness of the disk.

As the disk deflection increases beyond half the disk thickness, the material in the disk begins to stretch as well as bend, and membrane effects become significant. For purely membrane behavior, the plate does not deform out of its plane. The deformations are confined to the directions parallel to the plane of the disk only, and are constant throughout the thickness of the plate for all the layers. When the normal maximum deflection of a plate reaches the order of the thickness of the plate, the membrane action becomes comparable to that of the bending. For larger deflections membrane effects dominate. [15]

The theory for disk behavior that is generalized to include membrane effects is called the large-deflection theory of plates, and it is valid even when the deflections are equal or larger than the plate thickness, although the deflections must still be small relative to the other dimensions (the diameter) of the plate. The stresses and deflections vary in a non-linear manner with the magnitude of the transverse loading.

Large deflection modeling is described in detail in a classic text by Timoshenko [15], which forms the basis for the work presented here. It describes the more complete theory of an edge-mounted circular plate where the deflection at the center of the plate under load can be on the order of the plate thickness or larger. Large deflection theory is inherently non-linear and the theory is complex enough that the work done in this program has used approximate solution techniques [15]. The general problem of the response of plates to loading is analyzed in terms of strain, displacements, forces and strain energy. Stress is a derived quantity based on equations that determine the deflection of the disk over its surface.

The transition from linear to non-linear behavior occurs over a narrow range in the load parameter, qa⁴/Eh⁴, where q is the load (Pa), a is the disk radius, E is Young's modulus, and h is the disk thickness. For a clamped disk, if the load parameter has a value of about 6 the linear theory over predicts the deflection and the stress by about 20%. In this case the deflection to thickness ratio, w₀/h, is about 1. Linear theory becomes increasingly inaccurate for load parameters larger than this, but is a good approximation when the load parameter is smaller. Note that whether a disk is in the non-linear regime depends on the load, q, as well as the radius to thickness ratio, a/h, and the properties of the material through E.

An equivalent alternative description is that each specific material that acts like a plate (vs. membrane) limited in its deflection by its specific failure strength and its stiffness. These factors are characterized together in the stress parameter, σ a²/Eh². Large deflection theory (20% error from linear theory) should be used when the stress parameter is greater than about 3. Whether the disk is thin enough that pressure loading causes a large deflection is not only determined by its thickness relative to the supporting radius but also by both its stiffness over the unsupported span and its inherent strength to withstand both stretching and bending. A thinner, stiffer disk can behave in a linear manner described by bending theory, whereas a thicker, less stiff disk can behave in a non-linear manner. For a specific material E is fixed, as is its failure stress, σ _f. Thus for a specific disk where a and h are known the stress parameter must be less than σ_fa²/Eh² for the linear theory relating load to stress to apply. The limiting value of the stress parameter based on failure strength determines whether the membrane forces important in large deflection theory can be achieved under sufficient load, and how far into the large deflection regime the disk is operating when it fails.

Figure 5. Load/deflection diagram for membrane effects.

Figure 6a. Deflection vs. load factor for linear vs. large deflection theory.

Figure 6b. Stress vs. deflection for linear vs. large deflection theory.

The divergence between the predictions of linear theory and those of large deflection theory are shown in Figs. 6a and 6b. [15]. Figure 6a shows how as the load factor increases, the actual deflection increases much more gradually with load than as predicted by linear theory. Figure 6b allows the difference in theory to be defined in terms of stress. Figure 6b shows that for small deflection membrane effects are negligible and the linear theory is accurate. At deflections (w₀/h) of 0.5 membrane stresses become significant and the linear theory breaks down. Note that the graph in Fig. 6b is deceptive in indicating that the non-linear, large deflection theory predicts larger stresses than that predicted by the linear theory. Figure 6b shows that for the same deflection the sum of membrane plus bending stresses are greater than that predicted by linear theory. Figure 6a, however, shows that for large deflections the large deflection theory predicts much lower deflections for a given load, and thus lower stresses.

As an example of the behavior of a disk made of a common strong material, the degree to which steel is in the large deflection regime can be calculated as follows. AISI-SAE 1020 hardened plain carbon steel has a tensile strength of 620 MPa, and a Young's Modulus of 200 GPa. The parameters applicable to the present microwave window case are that the disk cover a 100 mm aperture and support a minimum of 2 atm (1 atm plus a safety factor of 100%). Based on linear flexing disk theory (Eq. 1), setting σ _max = σ _f, and a = 50 mm, a steel disk with h = 0.9 mm will fail at 2 atm pressure. The radius to thickness ratio, a/h, is 55, and the stress parameter is 9. This means that under these conditions near failure steel is in the large deflection regime, but not by a large margin. Since the steel disk is operating in the large deflection regime, the linear theory implies that a larger thickness is needed than is actually necessary. The true minimum design thickness leads to a higher stress factor and behavior that is farther into the large deflection regime after each iteration. The immediate conclusion is that most optical materials will never operate near the large deflection regime, since they are weaker by orders of magnitude than steel and thus they will require a much greater thickness to support the pressure and they will have a much smaller stress factor.