Investigation of the thermal stress in a thin plate

Abstract

When the distribution of temperature in a body is non uniform, there is a state of thermal stress. The linear, quasi-static, uncoupled theory of thermoelasticity is used to investigate such a state of stress in a thin circular plate subject to purely radial heat flow. It is shown that the plane-stress hypothesis, although consistent with a two-dimensional treatment, leads to unsatisfactory results when used within the full framework of the three-dimensional theory and an alternative approach is given. The solution is obtained by the superposition of a primary stress system satisfying Saint Venant boundary conditions at the edge of the plate and a suitably chosen secondary (isothermal) stress system. The analysis of each system is executed using a method based on the asymptotic expansion technique of Reiss and Locke (1961) , the small parameter being the thickness/diameter ratio h, of the plate. It is found that a boundary layer effect occurs in the isothermal case in which the significant terms are second order (h(^2)), adding some further justification to the Saint Venant Principle. Consideration of the composite solution shows that the two-dimensional solution plays the role of the zeroth order term in the series solution, the higher order terms "being in the nature of three-dimensional corrections. These correction terms are of second order and it is concluded that for sufficiently small h the solution is plane stress except in the boundary layer. The investigation is completed by a discussion of the method in relation to a specific example. The accuracy of the series solution is considered and numerical results given.