A bulge test based methodology for characterizing ultra-thin buckled membranes
Introduction
The bulge test methodology has become the standard technique for mechanical characterization of thin films [1] especially for freestanding membranes. This is due to (1) the possibility of precise sample processing facilitated by recent developments in micro-fabrication technology; (2) the need for minimal sample handling, which is especially challenging at small scales; (3) the relatively simple data processing for determining the membrane stress and strain values, needed to extract the mechanical properties. The method essentially involves fixing a freestanding membrane over a small window opening and applying a known pressure to it, while measuring the resulting membrane deflection (or curvature). Various models have been developed, based on the sample geometry, to convert the pressure-deflection data to the a (elasto-plastic) stress-strain curve, which is used to determine mechanical properties such as Young's modulus, Poisson's ratio, residual stresses, and plasticity parameters [[1], [2], [3]]. During the last 30 years ample research has been devoted to improve the accuracy of the bulge test by studying the underlying assumptions such as the influence of bending stiffness [4] and initial conditions e.g. initial film thickness and residual stress [1,5,6]. Among the different varieties of the bulge test method, the plane-strain bulge test is most popular [7], where it was shown that for rectangular membranes with in-plane aspect ratio larger than 4, the stress state in the center of the membrane reduces to a plane strain condition. This means that the stress and the strain are given by [4]:where κtt is the curvature in the transverse direction, a is half of the width of the membrane, δ is the deflection of the apex of the membrane, P is the applied pressure and h is the membrane thickness. σtt and εtt denote the normal stress and strain in the transverse direction, respectively. As a consequence of the plane strain condition in the center of the rectangular membrane, the transverse stress and the transverse strain can be related by the following constitutive equation:where is the plane strain modulus. To extract the Young's modulus (E) and the Poisson's ratio (ν) separately, an additional test needs to be performed, e.g., a bulge test on a circular or square membrane for which a different stress-strain equation holds, resulting in the biaxial modulus [2,8].
However, such freestanding membranes often buckle as a result of processing induced (compressive) residual stresses in combination with their small out-of-plane bending stiffness, particularly for ultra-thin membranes. In some cases, the buckling is exploited as a functional part in devices, e.g., in bi-stable micro actuators [[9], [10], [11]]. Moreover, the buckling phenomenon in freestanding thin membranes has gained a lot of attention in Micro Solid Oxide Fuel Cells (μSOFC), where stacks of freestanding membranes serving as electrodes or solid electrolytes, are often buckled as a result of the processing. While initially buckling was considered an issue [12], recent literature suggests that it can actually be beneficial to have these membranes in a buckled state to enhance their functional properties. It has been shown that buckled membranes are mechanically more stable at elevated temperatures, i.e. lower thermomechanical tensile stresses develop compared to a ‘flat’ membrane, often having significant tensile stresses already at room temperature [13,14]. Mechanical models have been developed to exploit the behavior of such μSOFC membranes in the post-buckling regime and consequently expand the design space into the low-stress post-buckling regime [12,13]. Recently, controlled buckling patterns in μSOFC solid electrolyte membranes (Fig. 1a) using ‘strain engineering’ have been employed to demonstrate local tuning of ionic conductivity of the electrolyte as an alternative of solid solution doping [15]. Furthermore, in the exciting field of graphene, where buckles and ripples are intrinsically present in the suspended configuration (Fig. 1b), these phenomena are receiving considerable attention [16] to be exploited in various applications [17], such as improved hydrogen absorption on a rippled graphene surface (due to local curvature), for future efficient hydrogen-based fuel cells [18].
The presence of such buckling patterns in test samples, as shown in Fig. 2, prevents the application of the conventional bulge test. All available literature to date confirms that the conventional bulge test methodology cannot be used to characterize buckled membranes since even at high pressures, buckling patterns typically do not disappear in the membranes and some stress components near the edge of the membrane stay compressive [2,19]. Only in very few cases, the samples can be pressurized, beyond the point where the buckling pattern completely disappears, where the bulge Eqs. (1), (2), (3), (4) might apply [2,19]. Alternatively, the sample may deform plastically before entering the cylindrical regime. Therefore, such buckled samples are typically discarded and the processing needs to be modified to prevent the buckles to occur, in order to mechanically characterize the membranes accurately. Such processing modifications can be time consuming, costly and sometimes even infeasible or undesired. Moreover, any processing change could influence the actual properties to be determined.
Clearly there is a need for a convenient characterization methodology to determine the material properties of the buckled samples in their original (buckled) state.
This paper introduces a characterization methodology for testing buckled samples, which builds on the bulge test theory and thus exploits its aforementioned advantages. The approach adopted here is to numerically model the bulge test-like pressure loading of the buckled membrane in the rippled regime, to which the meandering pattern (see Fig. 2 bottom) starts transitioning as soon as even minute pressure is applied [20], to understand the mechanics and provide relations for the relevant membrane stress and strain components. To relate the stresses and strains using simple constitutive equations for extracting the material properties, regions of interest (ROI) with simplified stress states are identified and explored. Furthermore, to accurately measure the complex non-uniform three-dimensional displacement field of the buckled membranes, recent advances in bulge test methodology involving integration of Global Digital Image Correlation (GDIC) with conventional bulge test theory to [8] are exploited.
Section snippets
Digital height correlation based bulge test
In conventional bulge test theory, the stresses and the strains obtained using Eq. (2), (3) are based on the assumption that they are homogeneous over the membrane, i.e. an infinitely long cylinder or full sphere is assumed. This assumption does not hold anymore when the bending effects at the boundaries play a significant role for films with a relatively large thickness [8]. Inhomogeneous fields are also expected in the case of buckled membranes, even for a very small thickness, and in the
Simulation results
Using the three step loading procedure, explained in the previous section, the numerical model adequately captures the three different regimes seen in the experiments, i.e. the rippled, meandering regime, and the cylindrical regime. Moreover, the evolution of the rippled regime, with increasing pressure as well as the transitions between the different regimes seems are well captured. Comparison with experimentally observed regimes shown in Fig. 6 provides a qualitative validation of the model.
Proof of principle experiment
Here, the results of a successful test serving as a proof of principle experiment to show the feasibility of application of the method in a real experiment are shown. Pressure increments of 2.5 kPa were applied to the sample and kept constant while the topographical images were acquired with the profilometer.
An ROI shown in Fig. 4 from the peak to valley of a ripple is selected for DHC analysis. As discussed in Ref. [8], a limited number of degrees of freedom (dofs) has to be used to capture
Conclusions and recommendations
It is well known from literature that conventional bulge testing, which is frequently used to characterize freestanding membranes, does not apply to the particular class of buckled membranes. In this paper, the conventional bulge test methodology has been extended to characterize the elastic properties of buckled membranes. This has been achieved by developing a validated FE based numerical model, which captures the complex mechanics of the (pressure-loaded) buckled membrane. A recently
Acknowledgments
This work was supported by the Vidi funding of J.H. (project number 12966) within the Netherlands Organization for Scientific Research (NWO). The authors would also like to greatly thank Johan Klootwijk from Philips Research for providing the samples, Jan Neggers for sharing his digital image correlation code, Roel Donders for upgrading the bulge test setup and help with experiments and Marc van Maris for technical support in the laboratory.
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