On the validity regime of the bulge equations
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The plane strain bulge test technique is a powerful and acknowledged technique for characterizing the mechanical behavior of thin films. In a bulge test analysis, the stress and strain are derived from the measured quantities using analytical approximations of the deformed geometry (bulge equations). To improve the bulge test, the systematic error introduced by these approximations is evaluated and quantified by scrutinizing the method on a finite element model of the bulge test, used as an idealized experiment.
I. INTRODUCTION
Thin films are used in many modern high-tech systems.1–3 In these systems, distinct materials with specific differentiating properties are often mixed. Therefore, profound understanding of the mechanical behavior of these thin films is required to predict failure due to thermomechanical loading.1 Many materials behave differently than their bulk counterparts when one or more dimensions are in the range of an intrinsic length scale,4–7 which is often the case for thin films. The mechanical behavior can also depend on the adjacent (adhered) films,8 the film microstructure, and the fabrication process.9 Therefore, experimental methods that can measure the mechanical performance of thin films in an accurate manner are of great value. The bulge test is such a valuable technique, acknowledged as one of the easier techniques to measure full (isothermal) stress–strain curves of freestanding thin films,10–12 even partly into the compression regime.13 In the bulge test, a membrane is deflected using a pressurized medium, while the deflection is measured [Fig. 1(a)]. Typically, the stress and strain are calculated from the pressure P and out-of-plane deflection at the membrane apex d by means of an analytical description of the membrane deformation, called the bulge equations.14,15 These bulge equations are sensitive to the membrane geometry, which has been a source of inaccuracy for bulge test measurements in the past. However, with the currently available Si micromachining sample production techniques, these problems are largely overcome.10 Nevertheless, when material parameters obtained from bulge tests are compared with parameters obtained with other smallscale testing methods, such as nanoindentation, substrate warpage, or microbeam bending, some discrepancies emerge.16,17 a)
Address all correspondence to this author. e-mail: [email protected] DOI: 10.1557/jmr.2012.69 J. Mater. Res., Vol. 27, No. 9, May 14, 2012
Bulge test experiments can be classified with respect to the shape of the tested membrane (circular, rectangular, etc.). The plane strain bulge test, which uses a rectangular membrane with aspect ratio larger than 4,5 is currently the most widely used bulge test for thin film testing, due to its favorable combination of microfabrication routes, its well-defined strain state (at least far away from the membrane boundaries), and its straightforward pressure versus deflection relation5,12 P¼2
Eh 4 Eh de0 þ d3 a2 3 ð1 tÞa4
;
ð1Þ
where P is the pressure, a is the width of the membrane, t the Pois
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