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 E-1 and v(1) power law functions of r, and for E-1 an exponential but v(1) an affine function of r, we derive explicit expressions for stresses and displacements. Here E-1 and v(1) 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 E-1 and v(1) are equal then stresses in the cylinder are independent of v(1); however, displacements depend upon v(1). 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 v(1) 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 v(1). Conversely, we find how E-1 and v(1) 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, E-1 and v(1) must be affine functions of r. For the in-plane shear stress to be uniform through the cylinder thickness, E-1 and v(1) must be functions of r(2). Exact solutions and optimal design parameters presented herein should serve as benchmarks for comparing approximate solutions derived through numerical algorithms.