Strains and stresses in multilayered elastic structures: The case of chemically vapor-deposited ZnS/ZnSe laminates

Solid structures consisting of layers of different materials created at elevated temperatures usually exhibit substantial residual stresses. These stresses are caused by intrinsic strains in addition to thermal strains, and originate from the bonding of the layers, which generates internal forces and moments that must be balanced to achieve mechanical equilibrium. It is shown that the solution proposed by Townsend [P. Townsend, D. Barnett, and T. Brunner, J. Appl. Phys. 62, 4438 (1987)] for describing elastic interactions in multilayered, elastically isotropic structures provides a powerful tool for evaluating the strains, the normal stresses off the edges, and the curvature of chemically vapor-deposited (CVD) laminates. The residual stresses acting in each layer are best expressed as follows: sigma(i)(z) = E-i'[(epsilon(i,0)-<(epsilon(0))over bar>) + (z(N)-z)K], where z measures the distance from the bottom surface, E-i' is the biaxial modulus of the layer, (epsilon(i,0)-<(epsilon(0))over bar>) characterizes the strain mismatch prior to any mechanical relaxation, and (z(N)-z)K defines the bending contribution, which depends linearly on the distance from the neutral plane and stems from the curvature K. For bilayered structures the curvature can be expressed in a relatively simple form. In conjunction with the proper expression for the average stress in the coating, it is seen that Stoney's equation holds for thickness ratios much larger than expected in the context of the thin-film approximation. The case of CVD laminates made of ZnS on ZnSe illustrates how the theory can be applied for designing structures that minimize the deformation and, thus, for obtaining optically desirable configurations. The stresses acting in bilayered ZnS/ZnSe structures are controlled by the strain differential Delta epsilon(0) = epsilon(c)(intr.) + (<(alpha(c))over bar>-<(alpha(s))over bar>)Delta T, where epsilon(c)(intr.) represents the intrinsic strain of the ZnS coating, and (<(alpha(c))over bar>-<(alpha(s))over bar>)Delta T defines the contribution originating from the thermal expansion mismatch. For ZnS deposited at 670 degrees C, measurements of the curvature yield Delta epsilon(0)similar or equal to-5.53 x 10(-4); upon evaluating available information on the thermal expansion of ZnS and ZnSe we conclude that the intrinsic strain of CVD ZnS is compressive and amounts to about 1 or 2 x 10(-4), for thicknesses ranging from 1 to 4 mm. (C) 2000 American Institute of Physics. [S0021-8979(00)05305-6].