Functionally graded cylinders : Asymptotically exact analytical formulations

This paper uses Variational Asymptotic Method (VAM) to obtain asymptotically exact analytical solutions for cylinders made up of Functionally Graded Materials (FGM). VAM splits a three-dimensional elasticity problem into a two-dimensional linear cross-sectional problem and a one-dimensional beam problem. It ensures the asymptotic correctness since the method makes no ad hoc assumptions. This is accomplished by taking advantage of certain small parameters inherent to beam-like structures. This technique has been successfully used for a variety of problems but it has never been implemented on cylinders made up of FGM. Starting with the variable geometry of the cylindrical cross-section made up of radially non-homogeneous material properties controlled by a volume fraction variable, the 3-D problem has been formulated and solved analytically despite the presence of geometrical non-linearities. Closed form analytical solutions are obtained to predict the exact response of functionally graded cylinders. The influence of inner and outer material properties, radius values and the variation trend of material composition on the mechanical behavior is highlighted. Results obtained from the present theory are successfully validated using a commercially available 3-D FEM solver ABAQUS. Analytical solutions obtained can analyze the behavior of any cylinder with FGM quickly and accurately.