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9.2.3 Expansion Representation of the Flexure

From the previous analysis, we consider now a representation of the flexure of a lens by an expansion. Assuming that the inequality (9.43) is satisfied, the last term of the flexure in (9.41) can be expanded as p2 p2 p4 p6

so the first three terms in the expansion of the flexure are ap

9 Singlet Lenses and Elasticity Theory of Thin Plates 1

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To summarize, the first three terms of the flexure of a single lens can be represented by the following equation set z = aq(A2p2 + A4P4 + A6p6 + •••), 0 < p < 1 A2, A4, A6 obtained from (9.44) with |8| > 1 A given by (9.40)

p = 6(1 - v2)q/Em3 m = a(cx - C2)/2 = aK/2(n - 1) T = m(8 -p2) with To = m 8 > 0.

Given a central reduced thickness T0 = t0/a and elasticity constants E, v of the glass, the variation of A2, A4, and A6 coefficients of the reduced flexure per load unit, z/aq, with respect to the pseudo-power m are displayed by Fig. 9.8 and Table 9.1.

9.2.4 Maximum Stresses at the Lens Surfaces

In order to avoid the risk of breakage of the glass during the figuring, one must select a convenient maximal value for the maximum stresses which appear at the surface of the lens. Given a material and a loading delay, this value can be established from Table 5.2 by introducing a reduction factor, say ~ 1/2 or 1/3, from the ultimate tensile strength (cf. Sect. 5.2.5). The other parameters available to avoid the rupture are the central thickness to of the lens and the intensity q of the load. The radial and tangential maximum stresses at the surface are

where the sign depends on the considered surface of the lens. From (9.22a), we obtain for the radial component

After substitution of the slope a as given by (9.38), the term in parentheses becomes

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