Full-field optical measurement of curvatures in ultra-thin-film–substrate systems in the range of geometrically nonlinear deformations

Abstract

This article describes coherent gradient sensing ͑CGS͒ as an optical, full-field, real-time, nonintrusive, and noncontact technique for the measurement of curvatures and nonuniform curvature changes in film-substrate systems. The technique is applied to the study of curvature fields in thin Al films ͑6 m͒ deposited on thin circular silicon wafers ͑105 m͒ of ''large'' in-plane dimensions ͑50.8 mm in diameter͒ subjected to thermal loading histories. The loading and geometry is such that the system experiences deformations that are clearly within the nonlinear range. The discussion is focused on investigating the limits of the range of the linear relationship between the thermally induced mismatch strain and the substrate curvature, on the degree to which the substrate curvature becomes spatially nonuniform in the range of geometrically nonlinear deformation, and finally, on the bifurcation of deformation mode from axial symmetry to asymmetry with increasing mismatch strain. Results obtained on the basis of both simple models and more-detailed finite-element simulations are compared with the full-field CGS measurements with the purpose of validating the analytical and numerical models.

Non-uniform, axisymmetric misfit strain: in thin films bonded on plate substrates/substrate systems: the relation between non-uniform film stresses and system curvatures

“…These analyses have removed both the restrictions of an equi-biaxial curvature state as well as the assumption of infinitesimal deformations. They have allowed for the prediction of kinematically nonlinear behavior and bifurcations in curvature states which have also been observed experimentally [12,13]. These bifurcations are transformations from an initially equi-biaxial to a subsequently biaxial curvature state that may be induced by an increase in film stress beyond a critical level.…”

“…These analyses also retain the assumption of spatial curvature and stress uniformity across the system. However, they allow for deformations to evolve from an initially spherical shape to an energetically favored shape (e.g., ellipsoidal, cylindrical or saddle shapes) which features three different, still spatially constant, curvature components [12,13].…”

Untitled

“…These analyses have removed both the restrictions of an equi-biaxial curvature state as well as the assumption (ii) of infinitesimal deformations. They have allowed for the prediction of kinematically nonlinear behavior and bifurcations in curvature states which have also been observed experimentally [Lee et al 2001;Park et al 2003;Fahnline et al 1991]. These bifurcations are transformations from an initially equi-biaxial to a subsequently biaxial curvature state that may be induced by an increase in film stress beyond a critical level.…”

“…These analyses also retain the assumption (vi) of spatial curvature and stress uniformity across the system. However, they allow for deformations to evolve from an initially spherical shape to an energetically favored shape (for example, ellipsoidal, cylindrical or saddle shapes) which features three different, still spatially constant, curvature components [Lee et al 2001;Park et al 2003;Masters and Salamon 1993;Salamon and Masters 1995].…”

Transmission wavefront shearing interferometry for photoelastic materials

“…Previously, CGS in transmission has been used only for optically isotropic materials [3,4,6]. CGS in reflection has been used for opaque isotropic materials [3,4], for materials with reflective coatings [3,4,10], and for composite materials [5,11]. No previous studies have considered CGS in transmission for optically anisotropic materials.…”