Hooke torsional elasticity

For an ideal solid, Hooke s law holds the stress, cr, applied is proportional to the deformation, e, and the proportionality constant is the modulus of elasticity E, so a = E e. Besides E also other quantities play a role, such as the shear modulus, G, in a shearing deformation or torsion, which is related to E. For the sake of simplicity we shall mainly use as a representative quantity for the elastic stiffness in any geometry of loading. 

Let us assume that a liquid is incompressible, B oo, and discuss orientational (or torsimial) elasticity of a nematic. In a solid, the stress is caused by a change in the distance between neighbor points in a nematic the stress is caused by the curvature of the director field. Now a curvature tensor dnjdxj plays the role of the strain tensor ,y. Here, indices i,j = 1, 2, 3 and Xj correspond to the Cartesian frame axes. The linear relationship between the curvature and the torsional stress (i.e., Hooke s law) is assumed to be valid. The stress can be caused by boundary conditions, electric or magnetic field, shear, mechanical shot, etc. We are going to write the key expression for the distortion fi-ee energy density gji, related to the director field curvature . To discuss a more general case, we assume that gji t depends not only on quadratic combinations of derivatives dnjdxj, but also on their linear combinations ... 

When a solid material is subjected to a small stress (i.e., tension, compression, or torsion), the resulting strain is proportional to the applied stress and the proportional factor is called an elastic modulus, and this simple linear relationship between stress and strain is called Hooke s law. 

---