Tension, bending, and flexure of functionally graded cylinders

The classical St. Venant problems (tension, bending and ¯exure) for isotropic elastic prismatic bars with the elastic moduli varying across the cross-section are examined. Inequalities relating the appropriate e€ective overall Youngʹs modulus to averages of the actual moduli are derived. The strain energy density for a composite with N elastic phases is examined, and it is found that the strain energy density and thus the elastic moduli are convex functions of the volume fractions. This result is then used to show that, in simple tension, the e€ective Youngʹs modulus is a minimum for the homogeneous distribution of the phases. It is also shown that, in bending and ¯exure, the e€ective Youngʹs modulus can be increased by concentrating the elastic components with the greater Youngʹs modulus further from the axis of bending.