On the St. Venant problems for inhomogeneous circular bars

The classical St. Venant problems, i.e., simple tension, pure bending, and flexure by a transverse force, are considered for circular bars with elastic moduli that vary as a function of the radial coordinate. The problems are reduced to second-order ordinary differential equations, which are solved for a particular choice of elastic moduli. The special case of a bar with a constant shear modulus and the Poisson's ratio varying is also considered and for this situation the problems are solved completely. The solutions are then used to obtain homogeneous effective moduli for inhomogeneous cylinders. Material inhomogeneities associated with spatially variable distributions of the reinforcing phase in a composite are considered. It is demonstrated that uniform distribution of the reinforcement leads to a minimum of the Young's modulus in the class of spatial variations in the concentration considered.