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.

Boundary integral based graded element for elastic analysis of 2D functionally graded plates

“…In this section, basic equations and the corresponding fundamental solutions for FGMs presented in (Yuan and Yin, 2011) are briefly reviewed to provide notations and references for the subsequent sections.…”

“…To obtain the fundamental displacement solution for the equilibrium Eq (13) the following transformation is established for the displacement vector (Yuan and Yin, 2011)…”

“…If the concentrated force acts at the origin, using the logarithmic potential, Yuan and Yin (2011) obtain…”

“…Based on the displacement formulation (19) (Yuan and Yin, 2011), we obtain the strain components by differentiating the solution ( 19) with respect to the spatial variable x j…”

A boundary element approach for solving plane elastostatic equations of anisotropic functionally graded materials

“…In many papers, the grading function g is chosen to be given by specific elementary functions. For example, g is an exponential function of x 1 and x 2 in Kuo and Chen [1], Petrova and Sadowski [2], and Sladek et al [3], while it is a quadratic function in Wang and Qin [4] and Yuan and Yin [5]. Of interest here is the numerical solution of the system (1) together with (4) in a two-dimensional region R on the Ox 1 x 2 plane subject to suitably prescribed conditions on the boundary of R which is denoted by C. More specifically, if p k are the Cartesian tractions on C as defined by…”