An accurate solution for thermo-mechanical stresses in tri-material assembly in electronic packages

The closed-form solution of thermo-mechanical analysis of tri-material assembly is initially provided by Schmidt [1] in 1999 and Suhir (2] in 2003. However, there exist some contradictions and inconsistencies in both the solutions of Schmidt and Suhir. The contradiction arises in consideration of the exponent parameter kappa in the characteristic equation. Both Schmidt and Suhir showed that the exponent parameter kappa in the shearing stresses contains two roots. But for no particular reason both of them considered only one root for kappa and as a consequence it leads to an ambiguous and incorrect solution. In the present paper a complete model for shearing stress for uniform temperature change is presented using both the roots for kappa. Subsequently a model for peeling stress is proposed from the consideration of moment equilibrium combined with the above mentioned shearing stress model. The contradictions in Schmidt's and Suhir's solutions are clearly highlighted in this paper. Analytical and FEM solutions are presented for the same tri-material package as used by Suhir [2]. The comparison of the analytical results and the numerical simulation is in a good matching agreement.