To generate a peculiar flexure the elasticity problem
is to determine the variable thickness distribution associated with a
given loading configuration or the reverse, i.e. the loading
configuration associated with a constant thickness, solving
the elastical equilibrium equations, with the associated boundary conditions
(simply supported or clamped edges). In the case of small deformations applied
to a circular plate, this process
is generally reduced to solve a differential equation between the flexure,
the thickness distribution and the loading configuration, one of these
last two functions being unknown. But when the flexure become larger, the
forces and moments applied to an element generate radial and tangential
stress 
and strains 
in the mirror
structure (see Fig. 3 (click here)). Taking into account these forces we have to solve a
differential system with three 3
order, non-linear, coupled and
non-homogeneous equations.
Figure 3: Forces and moments for an element of the deformed mirror
We consider the quadratic deformation W(r)=W0(1-r2/a2), where W0
is the maximal deformation needed for the variable curvature mirror and a
the radius at the edge of the mirror.
Using a circular plate with a variable thickness distribution t(r) under
an uniform load q, the differential system of equilibrium equations can
be written as follow:
where M(r,t) are the bending moments and N(r,t) the tensiles
forces associated (Ferrari 1993).
Using the expression of the radial and tangential stress
and strains given by Timoshenko & Woinowsky-Krieger
and those of M(r,t) and ,
functions of the flexion W(r) and the thickness t(r), this system can
be transformed into another one in which the relation between W(r) and
t(r) is underlined.
where the radial shearing force, for an uniform loading configuration, is equal
to 
.
Behind this point we used a numerical method, integrating the equations, in order to determine the solutions of the differential system. But unlike usual elasticity codes which calculate the deformations of a given structure, we answer the problem backwards. We determine a structural parameter of the mirror, its thickness distribution t(r), from the desired flexure W(r). This new method is well adapted to the problem and permits to obtain solutions fast, avoiding numerous iterations between the inputed structure and the resulting deformation (Ferrari 1993).
The numerical code gives us the thickness distribution, the bending moments,
the tensile forces and the total stress to insure that we stay in the
elasticity domain of the material. From these results we first can fix the
limit of validity of the small deformations approximation. Up to a flexion
amplitude equal to 1/3 of the mirror's mean thickness (i.e. a flexion ratio
) both theories give the same thickness distributions with
only 2 or 3% deviations. After this limit, the non-linear effects introduced
by the shearing forces into the mirror become important and the differences
between the solutions can reach 50% for a flexion ratio equal to 1.5
(LOOM 1991).
The code calculates also the value of the load q to apply in order to achieve
the desired radius of curvature (or the flexion ratio 
). This relation
is displayed in Fig. 4 (click here) in a dimensionless form for stainless steel. This
material having a Poisson ratio 
equal to 0.315, against 0.20 for a glass,
the bending moments and thus the flexion achieved are larger, but the stress
are also higher. The limit of the small deflections approximation is evident
in this figure where the curves obtained with the two different theories
diverge for flexion ratios above 1/3.
Figure 4: Load/Flexion relations in the case of a simply supported
plate given by the small and large deformations theories