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 elasti c moduli that vary as a function of the the radial coordinate. The problems are reduced to second-order ordinary differential equations, which are sol ved for a particular choice of elastic moduli. The special case of a bar wi th a constant shear modulus and the Poisson's ratio varying is also conside red and for this situation the problems are solved completely. The solution s are then used to obtain homogeneous effective moduli for inhomogeneous cy linders. Material inhomogeneities associated with spatially variable distri butions of the the reinforcing phase in a composite are considered. It is d emonstrated that uniform distribution of the reinforcement leads to a minim um of the Young's modulus in the class of spatial variations in the concent ration considered.