Bending energy and dynamics

Bending of a rod by angle 9. Insert the elongation along a surface that is a distance y above the undeformed middle surface is y9 (and y 0 below the middle surface, in compression). 

Consider an elastic beam of length L, thickness Ly and width with Young s modulus E. It is instructive to calculate the elastic energy of bending this beam by a small angle B (see Fig. 8.12)  

The Kuhn length b determines the crossover between stiff and flexible length scales. For rods or beams with length L of the order of the Kuhn length b, the angle of thermally induced fluctuations is of the order of unity 0 1 ---------------------------------------------------------------- 

The bending energy of a bent beam [Eq. (8.100)] then can be rewritten in terms of the Kuhn length  

The last relation was obtained using Eq. (8.99) for the deformation angle 0. By writing Eq. (8,103) in terms of the Kuhn length, it becomes much more general and applies to beams with cross-sections that are not rectangular (such as the bending of a cylindrical rod). 

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