Elastic Green’s Functions for a Specific Graded Material With a Quadratic Variation of Elasticity

Abstract

In this work, Green’s functions for unbounded elastic domain in a functionally graded material with a quadratic variation of elastic moduli and constant Poisson’s ratio of 0.25 are derived for both two-dimensional (2D) and three-dimensional (3D) cases. The displacement fields caused by a point force are derived using the logarithmic potential and the Kelvin solution for 2D and 3D cases, respectively. For a circular (2D) or spherical (3D) bounded domain, analytical solutions are provided by superposing the above solutions and corresponding elastic general solutions. This closed form solution is valuable for elastic analysis with material stiffness variations caused by temperature, moisture, aging effect, or material composition, and it can be used to perform early stage verification of more complex models of functionally graded materials. Comparison of theoretical solution and finite element method results demonstrates the application and accuracy of this solution.