From the stress-strain relation one shows that the radial and tangential stresses orr and ott are equal. Their value are maximum at the surfaces of the plate, i.e. for z = ± t/2, and expressed by

The basic solution for a plain plate providing a curvature mode Cv 1 is displayed by Fig. 2.1. Two designs using an outer ring built-in at the perimeter plate are equivalent and allow generating the bending moment by mean of axial forces (Fig. 2.1). The intensity of these forces can be derived from the vase form study in Chap. 7.

Fig. 2.1 Variable Curvature Mirrors derived from the CTD class. Up: Basic solution - uniform bending moment applied along the perimeter, Down: Axial forces on a vase form providing equivalent bending moments

2.1.2 Plates of Variable Thickness Distribution - VTD -Cycloid-Like form - Tulip-Like Form

Considering variable thickness distributions (VTDs), it is possible to find several configurations which actively generate the first-order curvature mode Cv 1 (Lemaitre [35, 36]). We will see below, from the thin plate theory, that the possible VTD

geometries depend on the load distributions and associated reactions at the substrate boundaries. The radial and tangential bending moments Mr and Mt are represented by

where D(r) = Et3(r)/[12(1 - v2)] is the variable rigidity. The static equilibrium between the components Mr, Mt of the bending moments and the shearing force Qr acting in a plate element is derived about a local tangential axis. This equilibrium writes dMr

After substitution of Mr, dMr/dr, Mt and division by rD, this differential equation becomes d3z / 1 dD 1 \ d2z f v dD 1 \ dz Qr that is,

Notation (2.1) of a flexural curvature mode leads to V2z = 4A20. After substitutions, the first derivative of the rigidity is dD Q' - ' Q,, (2.14)

dr 2(1 + v)A20 1 + v thus a direct function of the shearing force.

Three loading configurations and boundary reactions on the substrate are of interest for practical applications. Each of them is associated with a particular shearing force.

• VTD Type 1 - Uniform loading and reaction at edge: A uniform load q is applied all over the surface of the substrate in reaction at the edge r = a. At a current radius r of the substrate, the shearing force is defined by the equilibrium nr2q + 2nrQr = 0 of the inner element to r, that is

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