Analytical Solutions of Stresses in a Cylindrical Plate Due to Polynomial Radial Temperature Distributions

Thermal stress is a significant contributor to a component’s failure during manufacturing processes. In the electronic industry, for example, it is very common that components experience an extensive number of nonuniform, local heating cycles throughout its life time. In order to promote reliability, components are put through burn-in which creates a very nonuniform temperature distribution. In order to reduce cost, reworkability is considered to be a necessary option for the manufacturing processes to achieve high yield; these rework processes usually require localized heating. In addition, due to certain functionality requirements, materials with different coefficients of thermal expansion are cast together. The thermal mismatch caused by nonuniform temperature and/or different coefficients of thermal expansion will create thermal stress which could result in the cracking of the components. The fracture often initiates on the interface between the different materials or at the free edge of the surface. To make the problems mathematically tractable, the problems here are simplified as linear thermo-elastic and axisymmetric. It is concluded that the displacement distribution is one order higher than the temperature distribution if the temperature is a polynominal function of the radial distance from the center of a disk. The solutions also show the effect of the edge boundary conditions on the stress level; namely a certain degree of constrained edge support will reduce the tensile stress around the edge of the plate. This will reduce the failure rate of the plate, particularly for a brittle material. Finally, a numerical finite element solution for a square plate with a localized heating source is given to demonstrate the applications of the analytical solutions to fixture design during the process development.