Material tailoring and universal relations for axisymmetric deformations of functionally graded rubberlike cylinders and spheres

We find the spatial variation of material parameters for pressurized cylinders and spheres composed of either an incompressible Hookean, neo-Hookean, or Mooney–Rivlin material so that during their axisymmetric deformations either the in-plane shear stress or the hoop stress has a desired spatial variation. It is shown that for a cylinder and a sphere made of an incompressible Hookean material, the shear modulus must be a linear function of the radius r for the hoop stress to be uniform through the thickness. For the in-plane shear stress to be constant through the cylinder (sphere) thickness, the shear modulus must be proportional to r2 (r3). For finite deformations of cylinders and spheres composed of either neo-Hookean or Mooney–Rivlin materials, the through-the-thickness variation of the material parameters is also determined, for either the in-plane shear stress or the hoop stress, to have a prespecified variation. We note that a constant hoop stress eliminates stress concentration near the innermost surface of a thick cylinder and a thick sphere. A universal relation holds for a general class of materials irrespective of values of material parameters. Here, for axisymmetric deformations, we have derived expressions for the average hoop stress and the average in-plane shear stress, in terms of external tractions and the inner and the outer radii of a cylinder and a sphere, that hold for their elastic and inelastic deformations and for all (compressible and incompressible) materials.