Geometrically Nonlinear Stress-Deflection Relations for Thin Film/Substrate Systems With a Finite Element Comparison

Abstract

A new higher order geometrically nonlinear relation is developed to relate the deflection of a thin film /substrate system to the intrinsic film stress when these deflections are larger than the thickness of the substrate. Using the Rayleigh-Ritz method, these nonlinear relations are developed by approximating the out-of-plane deflections by a second-order polynomial and midplane normal strains by sixthorder polynomials. Several plate deflection configurations arise in an isotropic system: at very low intrinsic film stresses, a single, stable, spherical plate configuration is predicted; as the intrinsic film stress increases, the solution bifurcates into one unstable spherical shape and two stable ellipsoidal shapes; in the limit as the intrinsic film stress approaches infinity, the ellipsoidal configurations develop into cylindrical plate curvatures about either one of the two axes. Curvatures predicted by this new relation are significantly more accurate than previous theories when compared to curvatures calculated from three-dimensional nonlinear finite element deflection results. Furthermore, the finite element results display significant transverse stresses in a small boundary region near the free edge.

The role of the sample shape and size on the internal stress induced curvature of thin-film substrate systems

“…1). These values are compared in Table 1 with the analytical results of Masters and Salamon (1994) and Harper and Wu (1990). They are very close to those obtained by Masters and Salamon, in particular when computed at the quarter of the plate.…”

“…2), but leads to the perfectly symmetrical unstable one (curve BE). Consequently, a slight imperfection must be introduced to allow the system to converge towards one of the two stable solutions (Freund, 2000;Lee et al, 2001;Masters and Salamon, 1994). It is applied in the form of an anisotropic mismatch stress so that:…”

“…Thus, the results of our computations are compared with those obtained analytically and numerically by these authors. To facilitate the comparison, our computations are carried out with the parameter values they have chosen (Masters and Salamon, 1994), namely:…”

Stoney Formula: Investigation of Curvature Measurements by Optical Profilometer

“…For instance, Masters and Salamon [7] and Harper and Wu [8] used punctual curvatures measured in the center and at a quarter of the sample. Ngo et al [8] showed that the film stress depends on curvatures of the entire substrate and non-locally on the substrate curvature.…”

Untitled

“…This crucial assumption is often violated in practice since film stresses and the associated system curvatures are nonuniformly distributed over the plate area [Masters and Salamon 1994]. Recently and relaxed the assumption (vi) (and also (iv) and (v)) to study the thin film/substrate system subject to nonuniform, Figure 1.…”