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1+vJ

13 Navier's relations were finalized with the shear components by A. Cauchy in 1827-29. Navier, Poisson, Cauchy and Lame assumed that all materials were with v = 1/4 which then led to the so-called uni-constant theory. A bi-constant representation was later upheld by G. Green who showed that the number of elastic moduli goes up to 21 for crystals.

From the stress components, it is useful to derive the energy that occurs in the deformation of the volume element. The total work in the deformation of the element, or free energy, is represented per volume unit by dF = 2 = 2(1+0 (4 + T-W4), (L125)

where efk is the sum of the square of all the components of the £ik symmetric tensor, and e2 is the square of the sum of its diagonal components.

The problem of determining the displacement vector, i.e. its flexure components u, v, w, consists of finding the form of integral functions which satisfies the symmetry of the stresses or of the strains, if any, and to set up their limit values at the boundaries. In general, the three-dimensional problem is highly complex and is solved by finite element analysis and computer.

In active optics methods the mirror-plate is generally designed with rotational symmetry. The z-component w of the displacement vector is the starting feature; it represents the flexure to achieve. Hence the problem consists of determining the stress components. This leads to the determination of the resulting external forces and moments at the boundaries and to the associated determination of an appropriate thickness distribution.

1.13.5 Uniform Torsion of a Rod and Strain Components

The 3-D problem of the elastic deformation in torsion of a rod is a classical example where the analytical displacements can be easily obtained (cf. for instance Timoshenko ). Although the torsion angle t per unit length is assumed to be small, the total rotation Qm = tI of one end may reach one or several times 2n for a rod whose length-to-radius ratio i/a is large. From (1.114), the torsion angle per unit length is t = M/c where M and c are the twisting moment and the torsional rigidity. Considering the shear modulus G and a rod of radius a, from (1.115a) we obtain

In the first approximation of a small T-value, the study of the torsion of a rod leads to the conclusion that all the strain components £ik are null except for the shear strain £tz so the volume of the rod remains unchanged in a torsion since orr + att + ozz = 0.

However and as shown hereafter, when a large torsion occurs so the t2a2 terms cannot be neglected, the dimensions of a rod are modified unless opposite axial forces that increase its length are added to the opposite twisting moments generating the torsion.

• Torsion generated by twisting moments only: Let us consider a straight rod where only opposite twisting moments Mt are applied to its ends around the z-axis. A generatrix at a radial distance r = a from this axis is deformed into a helix (Fig. 1.50). At any point of this helix, the inclination angle with respect to a direction parallel to z is a constant ax whatever the z ordinate. When twisted, the two facets z = constant of a volume element remain parallel and their axial separation is dz' = dzcosra. Hence the axial strain component is

Denoting the components of the displacement vector as u = ur, v = ut and w = uz, any cross-sections perpendicular to the z-axis remain plane (dw/dO = dw/dr = 0) so the axial displacement w is a function of z only. From (1.121b),

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