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Fig. 2.15 Plasticity and hysteresis of quenched Fe87Cr13 alloys. Left: Stress-strain diagram, Ewing-Muir linearization by prestressing. Right: Hysteresis loop in extended elastic domain after prestressing respect to reference lens calibers that are concave spheres of discrete curvature. These are mounted onto a wheel facing the VCM in a Fizeau mounting.

In a first approximation, the difference between the initial and final interfero-grams shows that the plastic deformation is of quadratic form

From Type 1 VCMs built with the design parameters in Fig. 2.4, the maximum tested elastic sag was zElas = - 381 ¡im for the last caliber R = 84 mm; the mean value of the measured plastic sags was zPlas = -14 ¡m. Thus, the typical plasto-elastic deformation ratio is

ZPlas/ZElas = (CQ - C0)/(Cmax - CQ) = 3.67 ± 0.15% . (2.52)

These results can be used for analysis with a plasticity theory (Lubliner [40]) for model investigations. Given the stress distribution of ar (Fig. 2.4), that is also of cycloid-like form, the plastic deformation appears near the substrate faces from mirror axis to a radius about half the aperture, r « a/2.

In order to compensate for the effect induced by the plastic deformation, the following conditions apply (Lemaitre et al. [38]):

Plasticity Compensation: Assuming a VCM figured quasi-flat with curvature C0 at rest, which becomes CQ at rest after prestressing, and denoting APlas = CQ -C0,

1 ^ if the optical figurings are always executed at same curvature before and after pre-stressing,

2 ^ and if the rear side of the substrate ZRS(r) is defined by the co-addition of the thickness distribution (elasticity term) and of a lens shape (plasticity term) following the sign of loading q as

Fig. 2.16 VCM shapes during prestressing cycle - VTD Type 1. He-Ne interferograms with respect to discrete curvature calibers. The plastic deformation is derived from the two patterns on the left (Loom)

Fig. 2.16 VCM shapes during prestressing cycle - VTD Type 1. He-Ne interferograms with respect to discrete curvature calibers. The plastic deformation is derived from the two patterns on the left (Loom)

ZRS = t(r) - ACPlas (a2 - r2)/2 for q > 0 i.e. VC < 0, or (2.53a) ZRS = t(r) + ACPlas r2/2 for q < 0 i.e. VC > 0, (2.53b)

then the optimal properties of the VCM design with the large deformation theory are recovered.

These conditions can be realized by the construction and prestressing tests of preliminary prototypes allowing the determination of the lens shaped correction. Hence, the plasticity correction was set up in the final design of Type 1 VCMs, following (2.53a), which generates a zoom range with all negative curvature (VC < 0). This correction is included into the thicknesses t*(r) of Table 2.1 with C0 = 0, C* = - 0.4410-3 mm-1 and APlas = - C0 = C$.

### 2.7.2 Hysteresis Compensation and Curvature Control

In Sect. 2.2, we have shown from the large deformation theory that the representation of the loading as a function of the flexure ratio is not linear (2.36) but expressed by an odd series in z/t0. For Type 1 VCMs of zooming range [ f/^-f/2.5 ] deformed by air pressure, this representation provides a convenient accuracy by limiting the power series development up to i = 5. Considering the curvature C instead of the flexure ratio, the load-curvature relationship derived from (2.36) can be represented by (Ferrari et al. [22])

q = ft (C - Co) + fa (C - Co)3 + ft (C - Co)5 , (2.54)

where fai are coefficients, C0 the mirror curvature at rest and VC < 0. For large deformations, metal substrates show a flexural hysteresis:

1 ^ During the de-loading, the same curvatures as during loading are obtained by lower applied loads.

2 ^ After the loading and subsequent de-loading sequence, the initial and final curvatures are identical.

The largest of the hysteresis loops is the path AIWJA (Fig. 2.14 Right), where the extremal working point W of maximum load qmax and curvature Cmax is reached. Considering a loading sequence up to qseq, and provided qseq < qmax < qp.s such as defined from prestressing, the above load-curvature relationship is only valid for increasing pressures; when decreasing the load from a qseq loading, the fai coefficients become slightly different. Let fa;|seq be those coefficients. Given a curvature C, the load difference Aq between the increasing and decreasing pressures is a function of the maximum pressure qseq or of the associated curvature Cseq reached in the going up sequence. The hysteresis amplitudes Aq, increasing with higher deformation sequences, are also represented by fifth-order odd polynomials

Aq |seq = fa* (C - Co) + fa* (C - Co)3 + (C - Co)5 , (2.55)

Remaining under the maximum working stress defined by the prestressing (Sect. 2.6.1), measurements have been carried out on 12 cycloid-type VCMs having a zoom-range [f/^-f/2.5]. Hysteresis amplitudes were determined by Shack-Hartmann optical tests for loading sequences qseq < qmax. Considering a representation of the hysteresis Aq in function of the load q instead of the curvature, we obtain the form

Aq |seq = q + 8 q3 + 85 q5, q < qSeq < qmax, (2.56)

where Si coefficients are deduced from series (2.54) and (2.55).

From the (2.56) form, we can model all the hysteresis loops from simple properties. By definition, given a sequence, the hysteresis is null at the maximum load of the sequence, i.e. at q = qseq. In addition, the results from Shack-Hartmann tests show that the slopes of the hysteresis loops at q = 0 and q = qseq are opposite (Ferrari [22]). Then, we have the two conditions

dq seq q = o which entails

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