Due to constant heat flux, the thermal distribution around an insulated barrier at the interface of substrate and functionally graded material (FGM) which are essentially two-phase particulate composites is examined in such a way that the volume fractions of the constituents vary continuously in the thickness direction. Using integral transform method, two-dimensional steady-state diffusion equation with variable conductivity is turned into constant coefficient differential equation. Reducing that equation to a singular integral equation with Cauchy type, the temperature distribution around the barrier is obtained by defining an unknown function, which is called density function, as a series expansion of orthogonal polynomials. Results are shown for different thickness and nonhomogeneity parameters of FGM.

There are many engineering applications under severe thermal loading that require high temperature resistant materials in various forms of composites and bonded materials such as power generation, transportation, aerospace, and thermal barrier coatings. New developments in science and technology rely on the developments of new materials. Composites appear to provide the necessary flexibility in the design of these new materials, which are essentials that every part of the material in use exhibits uniform properties. In the mid-1980s, a new composite material, which was initially designed as a kind of thermal barrier coating used in aerospace structural applications and fusion reactors, was found by a group of scientists in Japan. Because of the material’s structure, it is named as functionally graded material (FGM). FGMs were used in modern technologies as advanced structures where the composition or the microstructure is locally varied so that a certain variation of the local material properties is achieved [

The aircraft and aerospace industry and the computer circuit industry are very interested in the possibility of materials that can withstand very high thermal gradients. This is normally achieved by using a ceramic layer connected with a metallic layer. The composition profile which varies from 0% ceramic at the interface to 100% ceramic near the surface, in turn, is selected in such a way that the resulting nonhomogeneous material exhibits the desired thermomechanical properties. The concept of FGMs could provide great flexibility in material design by controlling both composition profile and microstructure [

A number of reviews dealing with various aspects of FGMs have been published in the past few decades. They show that most of early research studies in FGMs had focused more on thermal stress analysis and fracture mechanics. Fracture mechanics of FGMs have been studied analytically by Erdogan and coworkers [

Due to the material mismatch at the interface of substrate and coating, the thermal distributions and thermal stresses on the crack or insulated barrier at the interface are examined by researchers. A general analysis of one-dimensional steady-state thermal stresses in a hollow thick cylinder made of functionally graded material is developed by Jabbari et al. [

Jin and Noda [

In this study, the Hankel integral transform method will be used to solve the heat equation in axisymmetric coordinate system. Problem will be examined as a one-dimensional and a two-dimensional heat conduction problem that is a mixed boundary value problem over the real line. Using mixed boundary conditions a Fredholm integral equation will be obtained with Cauchy type singularity and then it will be solved by using some known numerical techniques [

The thermal distribution around a penny-shaped barrier at the interface of graded composite coating and a substrate is given by the following steady-state heat equation in axisymmetric coordinate system:

Geometry of the heat conduction problem.

The solution can be obtained using superposition method which is an addition of one- and two-dimensional heat conduction problems,

(a) One-dimensional heat conduction without an insulated barrier, (b) Two-dimensional heat conduction with an insulated barrier.

Rewriting (

The unknown value

The integral equation in (

Because of the nature of the problem it is necessary to increase the density of the collocation points near the ends

The function

Due to the logarithmic singularity as

This work is supported by the Scientific Research Fund of Fatih University under Project number P50040801-2.