Optimum Young's Modulus of a Homogeneous Cylinder Energetically Equivalent to a Functionally Graded Cylinder

For a functionally graded (FG) circular cylinder loaded by uniform pressures on the inner and the outer surfaces and Young's modulus varying in the radial direction, we find lower and upper bounds for Young's modulus of the energetically equivalent homogeneous cylinder. That is, the strain energies of the FG and the homogeneous cylinders are equal to each other. For a typical power law variation of Young's modulus in the FG cylinder, it is shown that taking only two series terms, yields good values for bounds of the equivalent modulus. We also study two inverse problems. First, an investigation is made to find the radial variation of Young's modulus in the FG cylinder, having a constant Poisson's ratio, that gives the maximum value of the equivalent modulus. Second, the complementary problem of finding the radial variation of Poisson's ratio in the FG cylinder, having a constant stiffness, that gives the maximum value of the equivalent modulus, is considered. It is found that the spatial variation of the elastic properties, that maximizes the equivalent modulus, depends strongly upon the external loading on the cylinder.