The exact solution of viscoelastic stresses in the bilayer system due to thermal and/or lattice mismatch is derived if both layers are Maxwell materials. When the thickness of one layer is much smaller than that of the other layer, the viscoelastic stresses in the bilayer system can be reduced to that of the thin film/substrate system. The relative film thickness and the position in the thin film/substrate systems are included in this solution. The average film stress decreases with increasing the normalized time and finally approaches zero in a long time. As the relative film thickness is equal to or less than 0.001, the average film stresses of the zeroth-order approximation, first-order approximation, and Hsueh and Lee model [J. Appl. Phys.91, 2760 (2002)] are close to that of exact solution. Nevertheless, as the relative film thickness is larger than 0.001, the accuracies of the zeroth-order approximation, first-order approximation, and Hsueh and Lee model are dependent on the normalized time and relative film thickness.
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This work was supported by National Science Council, Taiwan.
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Hsueh, HC., Chiang, D. & Lee, S. Modeling of exact viscoelastic stresses in bilayer systems due to thermal and/or lattice mismatch: Maxwell model. Journal of Materials Research 26, 1392–1398 (2011). https://doi.org/10.1557/jmr.2011.100