A two-step load-deflection procedure applicable to extract the Young's modulus and the residual tensile stress of circularly shaped thin-film diaphragms

Abstract

We report on a novel two-step load-deflection (LD) formula and technique that enables an accurate extraction of the Young's modulus and the residual tensile stress from LD measurements on circularly shaped thin-film diaphragms. This LD relationship is derived from an adaptation of Timoshenko's plate bending theory, where the in-plane and out-of-plane deflections are approximated by series expansions. Utilizing the minimum total potential energy principle yields an infinitedimensional system of equations which is solved analytically resulting in a compact closed-form solution. In the appendant measurement procedure, the whole transverse bending characteristic of the diaphragm as a function of the radial coordinate is recorded for different pressure loads and introduced into the novel LD equation in order to determine the elastomechanical parameters of interest. The flexibility of this approach is demonstrated by ascertaining the Young's modulus and the residual tensile stress of two disparate diaphragm materials made of either micromachined silicon or microfiltered buckypapers composed of carbon nanotube compounds. V C 2014 AIP Publishing LLC.

Thickness dependence of Young's modulus and residual stress of sputtered aluminum nitride thin films

“…The same behavior is observed in silicon diaphragms. 8 This shift is clearly visible when the pressure dependence of the coefficients C 1 and C 2 is analyzed, as yielded in Fig. 5.…”

“…It is worth noting that the characteristics for pure plate wðrÞ=w 0 ¼ ð1 À q 2 Þ 2 and pure membrane behaviour wðrÞ=w 0 ¼ 1 À q 2 correspond to a ¼ ð1; À1; 0; …Þ and a ¼ ð1; 0; 0; …Þ, respectively. 8 The expansion is stopped at K ¼ 3, thus a 2k ¼ 0 for k > 3. This proves to be sufficient to accurately describe all measured bending curves.…”

“…9 It should be noted that w 0 is typically increased for larger values of R. The radius R ¼ 400 lm is chosen in order to maximize w 0 and in regard to the upper limit defined by the white light interferometer. The expressions for C 1 , C 2 , and C 3 are determined by applying the minimum total energy principle, as demonstrated by Beigelbeck et al 8 The evaluation procedure consists of four parts:…”

“…The experimental setup is described in more detail by Schalko et al 7 In order to accurately determine Y and r f , an advanced theoretical approach is utilized, which does not rely on a fixed bending behavior (i.e., plate or membrane-type). 8 The normalized bending characteristics are described using the functional series expansion…”

Elastic–plastic characterization of microstructures through pull-in 4 point bending test

“…Using the bulge test, the mechanical properties are generally obtained by fitting the analytical load-deflection expression to the experimental values. This requires an accurate analytical model of the test structure which is mostly dependent on the shape of the micro-plate and its supporting boundary conditions [13]. The quasi-static pull-in instability is an another very important bending test to measure the mechanical properties of the metal thin films.…”