Practical Application of a Geometrically Nonlinear Stress-Curvature Relation
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PRACTICAL APPLICATION OF A GEOMETRICALLY NONLINEAR STRESS-CURVATURE RELATION CHRISTINE B. MASTERS AND N.J. SALAMON The Pennsylvania State University, Department of Engineering Science and
Mechanics, University Park, PA 16802 ABSTRACT A recently developed geometrically nonlinear stress-curvature relation based on a minimization of the total strain energy, which predicts a bifurcation in shape as the magnitude of intrinsic film stress increases, is discussed in this paper. It is compared with the linear theories of Stoney and Brenner & Senderoff for a thin molybdenum film on silicon substrates with various thicknesses. Although the ratio of film to substrate elastic modulus is only 2, Stoney's equation generates significant error for this film/substrate system and the Brenner & Senderoff relation should be used for calculating initial film stress when plate deflections are small. When deflections exceed approximately half the substrate thickness the Brenner & Senderoff equation produces over 10% error and consequently, the nonlinear stress-deflection relation should be used to relate plate curvatures to initial film stress. INTRODUCTION The stresses which are developed in a thin film as it is deposited onto a substrate are a major point of interest in much thin film research [1,2]. And one of the most commonly used methods for measuring intrinsic thin film stress is the beam or plate bending method. In this stress measuring technique, the initial film stress causes the flexible film/substrate system to deflect or curve. This curvature is then measured and used to calculate the initial film stress produced during deposition. Typically the linear relation of Stoney [3] is used to predict initial film stresses from measured beam curvatures (deflections) 2 a* = Ea8a
s
tf
Here a* is the initial film stress causing the deflection, Es is the substrate elastic modulus, ts and tf are the substrate and film thicknesses respectively, and a is the beam curvature. This equation assumes that the film is very much thinner than the substrate (tf/ts
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