## Date of Award

2010

## Degree Type

Thesis

## Degree Name

Doctor of Philosophy

## Program

Mechanical and Materials Engineering

## Supervisor

John R. Dryden

## Abstract

In this thesis, the elastic field in circular beams and pipes made of functionally graded materials is considered. The following aspects are presented

First, the thermoelastic stress field in a functionally graded curved beam, where the elastic stiffness varies in the radial direction, is considered. An analytical solution is obtained where the radial variation of the stiffness is represented by a fairly general form. The stress fields corresponding to two different cases for the elastic properties are examined. The flexural stress in the curved beam is then compared with that of a ring. A relatively simple ap proximate solution is then developed and this is shown to be in good agreement with the analytical results.

Secondly, the effect of a nonconstant Poisson’s ratio upon the elastic field in functionally graded axisymmetric solids is analyzed. Both of the elastic coefficients, i.e. Young’s modulus and Poisson’s ratio, are permitted to vary in the radial direction. These elastic coefficients are considered to be func tions of composition and are related on this basis. This allows a closed form solution for the stress function to be obtained. Two cases are discussed in this investigation: a) both Young’s modulus and Poisson’s ratio are allowed to vary across the radius and the effect of spatial variation of Poisson’s ratio upon the maximum radial displacement is investigated; b) Young’s modulus is taken as constant and the change in the maximum hoop stress resulting from a variable Poisson’s ratio is calculated.

Thirdly, the stress concentration factor around a circular hole in an infinite plate subjected to uniform biaxial tension and pure shear is considered. The plate is made of a functionally graded material where both Young’s modulus and Poisson’s ratio vary in the radial direction. For plane stress conditions, the governing differential equation for the stress function is derived and solved. A general form for the stress concentration factor in case of biaxial tension is presented. Using a Frobenius series solution, the stress concentration factor is calculated for pure shear case. The stress concentration factor for uniax ial tension is then obtained by superposition of these two modes. The effect of nonhomogeneous stiffness and varying Poisson’s ratio upon the stress con centration factors are analyzed. A reasonable approximation in the practical range of Young’s modulus is obtained for the stress concentration factor in pure shear loading.

## Recommended Citation

Mohammadi, Mohsen, "Analysis of the Elastic Field in Functionally Graded Materials" (2010). *Digitized Theses*. 3718.

https://ir.lib.uwo.ca/digitizedtheses/3718