Exact solutions for functionally graded micro-cylinders in first gradient elasticity

An exact solution is obtained for a functionally graded (FG) micro-cylinder subjected to internal and external pressures. And in the current model, its material properties are assumed to be isotropic and exponentially-varying elastic modulus in radial direction and a material length scale parameter is incorporated to capture the size effect. To this end, a theoretical formulation including the effect of size dependency is derived in the framework of the first gradient elasticity and through Hamilton's principle. Then, a fourth-order governing ordinary differential equation (ODE) with variable coefficients is developed and the corresponding solution is rather difficult to be determined for inhomogeneous problem. Next, the efficient tensor algorithm operator is utilized to reduce the fourth-order homogeneous ODE to a second-order non-homogeneous one. Finally, by using method of variation of constant, the exact solution for FG micro-cylinder problem containing the material length scale parameter and power index is constructed perfectly, which is qualitatively different from existing Lame's solution in classical elasticity. When ignoring the inhomogeneity of material, the newly obtained exact solution reduces to the ordinary one. The numerical results reveal that increasing characteristic length parameter leads to the decrease of the maximum radial and tangential stresses, and the power index has also a considerable effect on the stress distribution of FG micro-cylinders. A key physical insight that emerges from our analysis is that the newly obtained solution form can be applied directly to practical engineering structures.