Thermo-mechanical constitutive equations for glass and its numerical formulation for warpage analysis of silicon-glass multilayered structure

• We developed thermo-mechanical constitutive equations for glass.
• We derived numerical formulations of the developed model.
• We validated the model for the warpage of the bonded silicon-glass wafer.
• The structural and stress relaxation affects the warpage significantly.

Thin multilayered structures with glass substrates are widely used in electronic devices. When layers having different thermal or mechanical properties are subjected to processes with temperature variations, warpage and residual stresses may develop incurred by the mismatch of volume change, possibly leading to the failure of the device. In order to analyze the thermo-mechanical behavior of glass including the warpage in multilayered structures with glass substrates, the thermo-mechanical constitutive equations of glass are developed here by coupling the stress relaxation (for mechanical behavior) and structural relaxation models (for thermal behavior). The constitutive equations for structural relaxation are generalized for the case with temperature dependent thermal expansion coefficients, while considering both formulations based on the specific volume and the fictive temperature. Besides, the numerical formulations of the constitutive equations for finite element analysis are derived for three-dimensional and plane stress conditions. The developed constitutive model is validated with experiments for the warpage of silicon-glass multilayered structures undergoing thermal cycles.