Effect of substrate deformation on the microcantilever beam-bending test
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With regard to substrate deformation, this work analyzed the microcantilever beam-bending test and provided a closed formula of deflection versus load. The substrate deformation was formulated using two coupled springs; the spring compliances were related to the elastic compliances of the substrate, the support angle between the substrate and the microcantilever beam, and the beam thickness. Finite element analysis was conducted to calculate the spring compliances and verify the analytic formula. The results showed that the proportionality factor of the load to the deflection was a third-order polynomial of the length from the loading point to the fixed beam end. Examples are also given to indicate the relative error of Young’s modulus when evaluated with the beam-bending theory without considering the substrate deformation. Characterizing mechanical properties of thin films has become a very active area of research, as illustrated by the fact that the U.S. Materials Research Society has organized eight symposia on the subject since 1988.1 The microcantilever beam-bending test is widely used to evaluate Young’s modulus of a thin film.2,3 In particular, nowadays the micromachine technique makes sample preparation convenient and a load-and-displacement-sensing nanoindenter conducts the bending test reliably.3,4 When a microcantilever beam is bent in a nanoindentation system, the resulting load–displacement behavior can be used to determine both the in-plane elastic modulus and the yield strength of the film material. For simplicity of analysis, a linear theory is generally adopted.3,5 The bending compliance of a cantilever beam, C, is experimentally determined as the proportionality factor of load (per unit width), P, to deflection, w, i.e.,
w = CP . (1) Without consideration of substrate deformation, the beam compliance is inversely proportional to the flexural modulus, D, as follows: L3 C= , (2) 3D where L is the length from the loading point to the fixed beam end. The flexural modulus is given by E E*h3 E* = D= , 12 1 − 2 for plane-strain condition along the beam width, E* = E , (3) for plane-stress condition along the beam width, a)
Address all correspondence to this author. e-mail: [email protected] b) Visiting scholar from Zhengzhou Research Institute of Mechanical Engineering. c) Visiting scholar from Beijing University of Chemical Technology. 1868
http://journals.cambridge.org
J. Mater. Res., Vol. 15, No. 9, Sep 2000 Downloaded: 13 Mar 2015
where E and are Young’s modulus and Poisson’s ratio, respectively, and h denotes the beam thickness. Young’s modulus of a thin film is usually assessed by using Eqs. (1)–(3) from experimental load–deflection data. To improve the assessment accuracy of elastic properties of thin films, Baker and Nix 4 proposed an empirical correction for deviations from the above beam-bending theory. In their correction, the real beam supported by a deformable substrate is equivalent to an ideal beam supported by a rigid substrate of somewhat greater, but unknown, length. Thus, they r
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