The giant increase of stiffness of inhomogeneous rods and beams

We demonstrate that stiffnesses of an inhomogeneous beam of coaxial structure coincide with the ones predicted by classical Bernoulli-Euler and Saint-Venant theories if and only if the indicated below conditions on the local Poisson's ratio are satisfied. If the conditions are not satisfied, the stiffnesses of the inhomogeneous beam exceed the stiffnesses predicted by classical theories. The difference in Poisson's ratios of the components of the rod/beam can result in a giant increase in stiffness when using materials possessing a negative Poisson's ratio.