Analysis of thermal stresses in FGM-Matrix medium induced by constant heat flux at the far field

Abstract

The thermal stress due to heat flux at the far field is derived for an infinitely extended elastic medium which contains a spherical inclusion made of functionally graded materials (FGM). The material properties in the inclusion phase are assumed to vary linearly while the material properties in the matrix phase are uniform. As FEM is not appropriate for this type of analysis, a semi-analytical approach is employed in which the temperature distribution for the entire medium and the thermal stress field in the matrix phase are obtained analytically while the thermal stress field in the FGM phase is obtained semi-analytically by solving a set of ordinary differential equations. The present method is preferred over FEM in that all the geometric and material parameters are kept in the formulations and the cost of computation is minimal. The von Mises stress is shown to be continuous at the FGM-matrix interface which underscores the advantage of using FGMs over conventional composites.

Keywords:

• Functionally graded material
• thermal stress
• heat flux
• spherical FGM inclusion
• von Mises stress

Notes

1 If the volume fraction of the “k₁ material” denoted by ν_f is linear inside the FGM inclusion as (2) $${\nu }_f = 1-\frac{r}{a},$$(2)

k(r) can be expressed as (3) $$\begin{array}{c}k(r)={\nu }_f{k}_1+(1-{\nu }_f)k_2\\ =(1-\frac{r}{a})k_1+\frac{r}{a}{k}_2\\ =k_1+(k_2-k_1)\frac{r}{a},\end{array}$$(3) assuming the rule of mixtures at r. This is the same as Eq. (1)(1) $$k(r)=\{\begin{array}{cc}{k}_1+(k_2-k_1)\frac{r}{a},& 0\le r\le a\\ k_2,& a<r<\infty \end{array}$$(1) .