Analytical solutions for n-phase functionally graded material cylinders under de saint venant load conditions: homogeneization and effects of poisson ratios on the overall stiffness

Object of the paper is to show the exact analytical solutions for the elastic response of a solid circular cylinder composed by the assembly of a central core and n surrounding hollow phases, all made of different homogeneous elastic materials, under de Saint Venant load conditions, finally obtaining for the whole object an equivalent one-dimensional homogenized beam model. In particular, the exact solution for the case of combined shear and bending is new for this type of heterogeneous compound materials, and hence it can turn useful for the comparison with other classical simplified solutions widely adopted in practical engineering problems. The analytical procedure, producing overall elastic relationships between generalized stresses and strains for Functionally Graded Material Cylinders (FGMCs) in cases of axial force, torque, bending and shear, leads to register possible large increases of selected homogenized stiffness coefficients as effect of mutual interactions among the constituents mediated by their different Poisson ratios. The special case of negative Poisson ratios is then considered, envisaging the possibility of exploiting their relevant effects on the overall stiffness of the heterogeneous media in the design of new materials, starting from the particular microstructures or work processes. At the end, some example applications and sensitivity analyses are shown for two-phase cylinders.