Exact Solutions and Material Tailoring for Functionally Graded Hollow Circular Cylinders

We employ the Airy stress function to derive analytical solutions for plane strain static deformations of a functionally graded (FG) hollow circular cylinder with Young’s modulus E and Poisson’s ratio v taken to be functions of the radius r . For E1 and v1 power law functions of r , and for E1 an exponential but v1 an affine function of r , we derive explicit expressions for stresses and displacements. Here E1 and v1 are effective Young’s modulus and Poisson’s ratio appearing in the stress-strain relations. It is found that when exponents of the power law variations of E1 and v1 are equal then stresses in the cylinder are independent of v1; however, displacements depend upon v1. We have investigated deformations of a FG hollow cylinder with the outer surface loaded by pressure that varies with the angular position of a point, of a thin cylinder with pressure on the inner surface varying with the angular position, and of a cut circular cylinder with equal and opposite tangential tractions applied at the cut surfaces. When v1 varies logarithmically through-the-thickness of a hollow cylinder, then the maximum radial stress, the maximum hoop stress and the maximum radial displacements are noticeably affected by values of v1. Conversely, we find how E1 and v1 ought to vary with r in order to achieve desired distributions of a linear combination of the radial and the hoop stresses. It is found that for the hoop stress to be constant in the cylinder, E1 and v1 must be affine functions of r . For the in-plane shear stress to be uniform through the cylinder thickness, E1 and v1 must be functions of r2. Exact solutions and optimal design parameters presented herein should serve as benchmarks for comparing approximate solutions derived through numerical algorithms.