Elastic calculation of the thermal strains and stresses of the multilayered plate

Abstract

An elastic calculation has been carried out on the thermal deformation strains, and stresses of a multilayered plate (MLP) which is composed of a circular substrate and thin films on it. The MLP is assumed to be constructed with no initial stresses at a temperature T0, and to be given a temperature change ΔT. In the calculation, the variation method has been utilized in such a manner as to minimize the total Helmholtz-free energy of the MLP by varying its shape of deformation. Without using the assumption of the plane stress state, a closed form solution has been obtained which determines the radius of curvature of the warpage, strains εij (i, j=x,y,z) and stresses τij in each layer by ΔT, αm the thermal expansion coefficient, Em the Young’s modulus, σm the Poisson’s ratio, and tm the thickness of each layer. The solution obtained has reproduced the plane stress state as an approximate description of the deformation. A solution in the case of the large radius MLP, where the effect of the shear strains has been taken into account, has also been shown. The small radius solution without the shear strain correction is in agreement with the earlier ones in the thin-film case, but the shear strain correction in the large radius solution exceeds, in magnitude, the film thickness correction included in the earlier solutions.