## V

Corrected wavefront

Fig. 1.41 Principle of adaptive optics closed-loop control

Complementary DMs with ~ 104 actuators under development for the ELT projects are: large and very thin mirrors of 2-3 m aperture directly usable as a component of the telescope main optical train; smaller mirrors of 10-20 cm aperture located after a telescope focus; micro-opto-electro-mechanical mirrors (Moems) of 20-30 mm aperture also implemented after a telescope focus.

Whatever the principle of the acting forces - magnetostriction induced strain or piezoelectric bimorph strain - the wavefront correction is actually generated onto a thin and continuous reflective plate. Although in most DM systems the reflective plate is totally of uniform thickness, it is clear that the intermediate regions where no force is applied should be designed with a smaller thickness than that in the regions where the forces must act. A mirror plate made of continuous tulip-like thickness distributions, similar to those found in Sect. 2.1.2, could improve the performance in terms of dynamic range and influence function when the tulip profile element has a biaxial or a threefold symmetry.

### 1.13 Elasticity Theory

The elaboration of elasticity theory is based on infinitesimal formalism and is a more recent science than dioptrics. However, one may notice that for particular cases such as typically with the development of the bow over the millenniums, the mean to store a maximum energy was empirically found by giving the bow cross-sections an appropriate thickness distribution. This problem leads to equal constraint cantilever bars of particular thickness distributions whose area of the cross-section may be null at the unclamped end (cf. Galileo [63], Clebsch [34], Saint-Venant [138]), but may also remain of finite value (Lemaitre [95]).

Noticeable accounts on the development of elasticity theory are given by Todhunter and Pearson [161], Love [97] and Timoshenko [156, 158, 159], Timoshenko and Woinowsky-Krieger [155], and Landau and Lifshitz [92]. One briefly summarizes hereafter the first milestones in the early development of elasticity theory.

### 1.13.1 Historical Introduction

It was Galileo Galilei who introduced the first problem of elasticity in considering the strength of solids to rupture a cantilever beam in Discorsi e Dimostrazioni Matematiche (1638). This, known as Galileo's problem, consisted of determining the cross-section thickness distribution along the horizontal axis of a beam whose one end is built into a wall whilst the tendency to break it arises from its own weight or a load applied to its other end (Fig. 1.42).

Although not knowing the law linking the elastic displacements to the forces producing them, Galileo nevertheless clearly enounced and emphasized this important question of strength. This led to the fundamental discovery of the proportionality

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