Modeling of relaxation of viscoelastic stresses in multi-layered thin films/substrate systems due to thermal mismatch

Assume the ratio of the total axial rigidity of thin films to that of the substrate is smaller than 0.02, an approximate closed-form solution for viscoelastic stresses in multi-layered thin films/substrate systems due to thermal mismatch is derived. This is achieved by utilizing the analogy between the governing field equation of elasticity and the Laplace transform with respect to time of the viscoelastic field equation. Based on two solutions, simplified solutions for relaxation of residual stresses distributed in multiple layers of thin films deposited on a thick substrate are obtained. The effect of the thickness, thermal expansion coefficient, Young's modulus, and viscosity coefficient of the substrate and thin films on the relaxation of residual stresses is considered. This simplified solution can be applied to some special cases such as one layered or periodic multi-layered thin films on a thick substrate.