Exact solutions are derived for three-dimensional, anisotropic, linearly elastic, and functionally graded rectangular composite laminates under simply supported edge conditions. The solutions are expressed in terms of an elegant formalism that resembles the Stroh formalism, and the composite laminates can be made of multilayered functionally graded materials with their properties varying exponentially in the thickness direction. The present solution extends Pagano's solution to the functionally graded material, and can serve as a benchmark to the modeling of functionally graded composite laminates based on various numerical methods. Typical results of the present solution are discussed for a single functionally graded plate and a bi-layer plate with a functionally graded layer overlying a homogeneous layer. For both plates, a simple load is applied on their top surfaces. It is shown that with a suitable functionally graded layer, the tensile stress on the top surface (or the compressive stress on the bott...

https://blogs.uakron.edu/ernianpan/files/2014/09/070_2003JCompositeMatPan.pdf

Exact Solution for Functionally Graded Anisotropic Elastic Composite Laminates

An approximate solution for the static analysis of three-dimensional, anisotropic, elastic plates composed of functionally graded materials (FGM) is presented. The solution is obtained by using a discrete layer theory in combination with the Ritz method in which the plate is divided into an arbitrary number of homogeneous and/or FGM layers. Two types of functionally graded materials are considered: an exponential variation of the mechanical properties through the thickness of the plate, and mechanical properties as a function of the fiber orientation, which varies quadratically through the laminate thickness. The present approach is not dependent on a specific transition function, and any continuous function representing the variation of the material properties in the thickness direction may be incorporated in the model. The method is validated by solving the problem of a single simply supported FGM plate, for which excellent agreement with the exact solution is obtained. Two more examples with different boundary conditions and different material configurations are presented in order to demonstrate the applicability of this solution. Homogeneous, graded, and bi-layer plates are examined in order to study potential advantages of using FGM.

https://blogs.uakron.edu/ernianpan/files/2014/09/091_2006CompositeBRamirezHeyligerPan.pdf

Static analysis of functionally graded elastic anisotropic plates using a discrete layer approach

Fracture analysis of functionally graded materials by a BEM

Free space Green s functions are derived for graded materials in which the thermal conductivity varies exponentially in one coordinate. Closed-form expressions are obtained for the steady-state diffusion equation, in two and three dimensions. The corresponding boundary integral equation formulations for these problems are derived, and the three-dimensional case is solved numerically using a Galerkin approximation. The results of test calculations are in excellent agreement with exact solutions and finite element simulations.

Green's Functions and Boundary Integral Analysis for Exponentially Graded Materials: Heat Conduction

3D crack analysis in functionally graded materials

The problem of a point force acting in an unbounded, three–dimensional, isotropic elastic solid is considered. Kelvin solved this problem for homogeneous materials. Here, the material is inhomogeneous; it is ‘functionally graded’. Specifically, the solid is ‘exponentially graded’, which means that the Lame moduli vary exponentially in a given fixed direction. The solution for the Green9s function is obtained by Fourier transforms, and consists of a singular part, given by the Kelvin solution, plus a non–singular remainder. This grading term is not obtained in simple closed form, but as the sum of single integrals over finite intervals of modified Bessel functions, and double integrals over finite regions of elementary functions. Knowledge of this new fundamental solution for graded materials permits the development of boundary–integral methods for these technologically important inhomogeneous solids.

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J. D. Richardson

Papers

On Green's function for a three-dimensional exponentially graded elastic solid