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

Figure 2 shows the half schematic view of the microstructure with respect to the center of the test specimen. In the figure, L_p is the distance of the top suspended plate from the axis of rotation to the edge, L_s is the length of the central test specimen and L_e is the distance of the center of the bottom electrode to the axis of rotation.

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With an applied actuation voltage between the top suspended plate and bottom fixed electrode, an electrostatic force acts on the plate given as;

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{F}_{e}}=\frac{{{\varepsilon }_{o}}A{{V}^{2}}}{2{{\left(g-{{h}_{1}} \right)}^{2}}}\nonumber \end{align} \tag{ 1 }$$

where ${{\varepsilon }_{o}}$ is the permitivity, A is the overlap area, V is the actuation voltage, g is the air gap thickness and ${{h}_{1}}$ is the deflection in the top plate at the center. Under this electrostatic force, the top plate moves downwards with an angle α. From figure 2, the resultant elongation in the central specimen can be obtained as;

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \Delta L={{h}_{3}}-{{h}_{2}}\nonumber \end{align} \tag{ 2 }$$

where ${{h}_{2}}$ is the upwards deflection in the suspended plate at the edge and ${{h}_{3}}$ is the upwards defletion in the central specimen. For the test microstructure, since the stiffness of the suspended plate is much higher than the stiffness of the central of specimen, it can be assumed that the specimen undergoes a pure bending. Thus, the equilibrium equation for the central specimen can be written as;

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{F}_{e}}\cdot ({{L}_{e}}+{{L}_{p}})-{{F}_{r}}\cdot ({{L}_{p}})=M\nonumber \end{align} \tag{ 3 }$$

where M is the bending moment applied to the central specimen about its thickness at a distance ${{L}_{p}}$ from the anchor point and ${{F}_{e}}$ is the electrostatic force applied at a distance $({{L}_{e}}+{{L}_{p}})$ from the central specimen. The microstructure is symmetric and the idealized anchors have a reaction force ${{F}_{r}}$ equal to ${{F}_{e}}$. The axial and bending stress in the central specimen can be evaluated by considering the deflection in the specimen under an applied electrostatic force ${{F}_{e~}}$. For the central specimen undergoing pure bending, the relation between the bending moment and radius of curvature can be obtained as;

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \frac{1}{R}=\frac{M}{EI~}\nonumber \end{align} \tag{ 4 }$$

where R is the radius of curvature of the central specimen, E is the Young's modulus and I is the bending moment of inertia. From figure 2, the elongation in the central specimen can be related to the radius of curvature of the central specimen under pure bending as;

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \Delta L={{h}_{3}}-{{h}_{2}}=R-\sqrt{{{R}^{2}}-\frac{L_{s}^{2}}{4}}.\nonumber \end{align} \tag{ 5 }$$

The above equation shows that the radius of curvature of the central specimen with length ${{L}_{s}}$ can be calculated by using the experimentally measured maximum deflection in the central specimen. The above equation shows that the radius of curvature of the central specimen with length ${{L}_{s}}$ can be calculated by using the experimentally measured maximum deflection in the central specimen. The maximum bending stress at the extreme fibers at the top and bottom of the central specimen can be obtained as;

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{\sigma }_{b\max }}=\frac{Mt}{2I ~ }=\frac{4Et}{{{L}_{s}}}\alpha =\frac{4Et}{{{L}_{s}}}{{\sin }^{-1}}\frac{{{h}_{2}}}{{{L}_{p}}}.\nonumber \end{align} \tag{ 6 }$$

where $M=\frac{4EI}{{{L}_{s}}}\alpha$ is the bending moment, $I=\frac{w{{t}^{3}}}{12}$ is bending moment of inertia, $\alpha ={{\sin }^{-1}}\frac{{{h}_{2}}}{{{L}_{p}}}$ is the deflection angle, w and t are the width and thickness of the central specimen respectively. From figure 2, the reaction force acting on the central specimen corresponding to the electrostatic force $({{F}_{e}})$ can be calculated as;

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle P=EA\frac{\Delta L}{{{L}_{s}}}=\frac{2EA}{{{L}_{s}}}\left({{L}_{p}}-L_{p}^{\prime } \right)=\frac{2EA}{{{L}_{s}}}\left({{L}_{p}}-\sqrt{L_{p}^{2}-h_{2}^{2}} \right).\nonumber \end{align} \tag{ 7 }$$

Using above equation the axial force and hence the axial stress (${{\sigma }_{a}}=P/A$) acting on the central specimen can be calculated by measuring the upwards defelction in th suspended plate (${{h}_{2}}$). The equations (2), (6) and (7) shows that the total stress (${{\sigma }_{a}}+{{\sigma }_{b}}$) and strain in the specimen can be evaluated by the experimental measure of the upwards deflection in the central specimen corresponding to an applied actuation voltage.

The elastic–plastic analysis in beams is generally based on the development of the 'plastic hinges'. These plastic hinges start to form when the stress at the extreme fibers of a critical section of the beam reach the yield stress value in the uni-axial tension. This results in the triangular distribution of the elastic stresses to be changed to the fully plastic rectangular stress distribution as shown in the figure 3. In the elastic region of the beam there is a linear distribution of stresses over the cross section. The maximum stress occurs at the extreme fibers and this value cannot exceed the yield stress. With an increase in the beam bending the extreme fibers yield and become plastic. This yielding continues with the applied bending unless the elastic region completely diminishes and the beam becomes fully plastic. Assuming a microbeam of symmetric cross-section A, thickness of t, having a distance of  ±d from the neutral axis to the plastic region and a ratio of $\beta =2d/t$; the bending for the microbeam in the figure 3 is given as;

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle M\left(x \right)=2\int_{0}^{t/2}{\sigma \left(x,y \right)ydA}.\nonumber \end{align} \tag{ 8 }$$

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The stress in the elastic region i.e. $-d\leqslant y\leqslant d$ is given by $\sigma =\pm {{\sigma }_{y}}\frac{y}{d}$ while the stress in the plastic region i.e. $d\leqslant y\leqslant t/2$ and $-t/2\leqslant y\leqslant -d$ is equal to the yield stress ${{\sigma }_{y}}$. Thus, for a rectangular cross-section of width b(y)  =  b and thickness t the equation of bending can be written as;

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle M\left(x \right)=\frac{1}{4}{{\sigma }_{y}}b{{t}^{2}}(1-\frac{1}{3}\left({{\left(\frac{2d}{t} \right)}^{2}} \right)={{M}_{p}}\left(1-\frac{1}{3}{{\beta }^{2}} \right)\nonumber \end{align} \tag{ 9 }$$

or

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \beta \left(x \right)=\sqrt{3\left(1-\frac{M\left(x \right)}{{{M}_{p}}} \right)};\quad 0\leqslant \beta \leqslant 1.\nonumber \end{align} \tag{ 10 }$$

The above equation shows that since at $x={{x}_{0}}$, $\beta \left(x \right)=0$ thus $M\left({{x}_{0}} \right)={{M}_{p}}=\frac{1}{4}{{\sigma }_{y}}b{{t}^{2}}$. This shows that the microbeam is in fully plastic state at $x={{x}_{0}}$. For $x~\leqslant {{x}_{1}}$ and $x\geqslant {{x}_{2}}$; the value of $\beta \left(x \right)=1$ and hence $\left| M\left(x \right) \right|\leqslant \frac{2}{3}{{M}_{p}}$. This depicts that microbeam is fully elastic for $x\leqslant {{x}_{1}}$ and $x\geqslant {{x}_{2}}$ with a maximum stress of ${{\sigma }_{y}}$ at the points ${{x}_{1}}$ and ${{x}_{2}}$. Similarly, for the region ${{x}_{1}}\leqslant x\leqslant {{x}_{2}}$, the microbeam is in elastic–plastic region and the width of this region is determined by the value of the bending moment $M\left(x \right)$. This analysis shows that β is a function of x and hence the location of the plastic hinge varies along the length of the microbeam. For the test microstructure design presented in this work, the formation of plastic hinge can be assumed to be exactly in the center of the central specimen due to symmetric electrostatic loading and geometric shape of the central specimen.

In the elastic region, the pull-in voltage of microstructures including fixed-free microcantiliver and fixed–fixed microbeams is dependent on the structural parameters, air gap height and Young's modulus of the material along with a constant shape factor. The constant shape factor is a function of boundary conditions or geometric configuration. For example, the pull-in voltage for the fixed-free microcantiliver and fixed–fixed microbeams is given as [42]

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \left\{\begin{array}{@{}cccccccccccccccccccc@{}} {{V}_{PE\left({\rm microcantiliver} \right)}}=\frac{\sqrt{2}}{3}\left(\sqrt{\frac{E{{t}^{3}}{{g}^{3}}}{{{\varepsilon }_{0}}{{L}^{4}}}} \right) \nonumber \\ {{V}_{PE\left({\rm fixed}-{\rm fixed}\,{\rm microbeam} \right)}}=\sqrt{\frac{128}{27}}\left(\sqrt{\frac{E{{t}^{3}}{{g}^{3}}}{{{\varepsilon }_{0}}{{L}^{4}}}} \right) \nonumber \\ \end{array} \right.\nonumber \end{align} \tag{ 11 }$$

where E, t, g, L and ${{\varepsilon }_{0}}$ are the Young's modulus, thickness, air gap height, length and air permittivity respectively for the microstructures. The equation (11) shows that the pull-in voltage is a function of same geometric parameters for the fixed-free microcantiliver and fixed–fixed microbeams with a difference of a constant shape factor value i.e. 0.47 and 2.18 for fixed-free microcantiliver and fixed–fixed microbeam, respectively. These values are verified through FEM simulations and are obtained as 0.42 for fixed-free microcantiliver and 2.2 for fixed–fixed microbeam. These results show that the constant shape factor for the central specimen in the test microstructure, presented in this work, can be estimated by considering the pull-in voltage value obtained through FEM simulations by using the following relation;

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{V}_{PE\left({\rm central}\,{\rm specimen} \right)}}={{Y}_{s}}\left(\sqrt{\frac{Et_{s}^{3}{{g}^{3}}}{{{\varepsilon }_{o}}{{\left({{L}_{e}}+{{L}_{p}} \right)}^{4}}}} \right).\nonumber \end{align} \tag{ 12 }$$

Where ${{Y}_{s}}$ is the shape factor constant, ${{t}_{s}}$ is the thickness of the central specimen, (${{L}_{e}}+{{L}_{p}}$) is the length of the plate from the central specimen to the center of the bottom electrode. The value of ${{Y}_{s}}$ calculated for the central specimen using equation (12) for a pull-in voltage of 118 V obtained through FEM simulations is 1.13. This value lies in between the values of shape factor constant obtained for the fixed-free microcantiliver and fixed–fixed microbeams.

In the elasto-plastic region, the equation (12) for the pull-in voltage can be modified by introducing a β factor which incorporates the effect of plasticity. In the elasto-plastic region, the stress–strain curve can be evaluated either without considering the strain hardening effect i.e. elastic–perfect plastic behavior or with a constant strain hardening effect, as shown in figure 4. Thus, the pull-in voltage in the elasto-plastic region can be expressed as;

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{V}_{PE\left({\rm central}\,{\rm specimen} \right)}}={{Y}_{s}}\left(\sqrt{\frac{\left[ E{{\beta }^{3}}t_{s}^{3}+{{E}_{t}}\left(1-{{\beta }^{3}} \right)t_{s}^{3} \right]{{g}^{3}}}{{{\varepsilon }_{o}}{{\left({{L}_{e}}+{{L}_{p}} \right)}^{4}}}} \right).\nonumber \end{align} \tag{ 13 }$$

Where $\beta =2d/{{t}_{s}}$ is a ratio between the distance of the plastic hinge with respect to the central specimen neutral axis and total thickness of the central specimen. The ${{E}_{t}}$ is the tangent modulus that evaluates the strain hardening effect after the yield limit. As discussed in section 3.2, with β  =  1 and β  =  0, the central specimen is in fully elastic and plastic state respectively. However, for 0  <  β  <  1 the central specimen is in the elasto-plastic region.

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The stress distribution and vertical deflection in the central specimen of the test microstructures is obtained using multiphysics non-linear FEM simulations in ANSYS. The electrostatic-mechanical coupling is modeled using reduced order transducer elements (TRANS126). For the parallel plates, the fringe field capacitance is a function of (a) air gap thickness to bottom electrode width and (b) top plate thickness to bottom electrode width. The fringe field capacitance effect is more pronounced for the increasing values of these two ratios [43]. For the proposed test microstructure, the air gap thickness to bottom electrode width ratio is 0.056 and top plate thickness to bottom electrode width ratio is 0.16. These low ratio values suggest that the effect of fringe field capacitance in the FEM analysis can be ignored. Figures 5(a) and (b) shows the vertical deflection and normal stress distribution profile in the central specimen respectively, for an applied actuation voltage. The stress is mostly concentrated in the central specimen with upper and bottom surface experiencing tensile and compressive stress respectively. The stress varies linearly from bottom to the top surface with nearly zero stress in the centroid of the specimen cross-section. Moreover, the stress value in the torsion springs is negligible in comparison to the central specimen, thus minimizing the local plasticity effect in the torsion springs. Figures 6(a) and (b) shows the vertical deflection and normal stress curves for the central specimen with respect to actuation voltage. The deflection and normal stress in the specimen varies non-linearly with increasing voltage with a maximum deflection of 1.2 µm and normal stress of 177 MPa just before the occurrence of the pull-in phenomenon at 118 V. This high value stress in the specimen as compared to the yield stress of gold thin film (100–120 MPa), before pull-in, proves the suitability of the proposed design to study the plastic and elastic–plastic behavior in gold thin film.

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To validate the proposed kinematic model for the test microstructures, discussed in section 3, the vertical deflection in the central specimen obtained through coupled electrostatic-mechanical analysis is used as input to the model and corresponding axial stress, bending stress and equivalent stress values are obtained. Figure 7 shows the comparison of the stress values in the central specimen obtained directly through FEM analysis with that of obtained using FEM based vertical deflection in the developed kinematic model. The results show a good correspondence between the stress values, thus validating the accuracy of the proposed kinematic model for stress evaluation in the test microstructure. Moreover, the results show that the axial stress values is negligible as compared to the bending stress in the central specimen.

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Initially, one of the test microstructure is tested in the elastic region, by measuring the vertical deflection in the central specimen with increasing actuation voltage up to 100 V. As shown in figure 6(b), the actuation voltage of 100 V results in normal stress of 90 MPa in the central specimen by FEM analysis, which is fairly lower than the yield stress of the gold thin film. Thus, specimen is in the elastic region for actuation voltage up to 100 V. The test microstructure is tested for three consecutive loading cycles and corresponding vertical deflection in the central specimen is measured, as shown in figure 8. The results show that the test microstructure fully recovers to its initial position after each loading cycle and deflection for each cycle is also nearly same. Thus, with an applied actuation voltage of 100 V, with stress value of 90 MPa in the central specimen, the specimen remains in the elastic region and there is no onset of plasticity or formation of plastic hinges.

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For the second test microstructure, the actuation voltage is increased to 105 V to achieve higher stresses in the central specimen as compared to the first test microstructure. The normal stress value at 105 V is 110 MPa. The test microstructure is loaded for multiple cycles. Figure 9 shows that for the first loading cycle, the deflection in the central specimen with increasing voltage follows a non-linear curve with a maximum deflection of 0.7 µm at 105 V. However, for the next three loading cycles, a pull-in phenomenon occurs with a progressive decrease in the pull-in voltage value for each loading cycle. These results show that there is onset of plasticity in the specimen with an actuation voltage of 105 V and progressive decrease in the pull-in voltage value with each loading cycle can be attributed to the formation of the plastic hinge and elastic–plastic phenomenon with progressive increase in the plastic region with each loading cycle.

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The extension in the plastic hinge and corresponding decrease in the pull-in voltage for each loading cycle, with respect to the β factor can be obtained using equation (13). Assuming that the initial stiffness of the central specimen is 98.5 GPa and strain hardening behavior is given by ${{E}_{t}}$  =  0.6 E, the analytically calculated values of the β factor are shown in table 2. The results show a decrease in the β factor and hence decrease in the elastic stiffness for pull-in voltage values of second to fourth cycle. For the first loading cycle up to 105 V, the behavior of the specimen is fully elastic with β factor value of 1. For the second and third cycle, the β factor value progressively decreases which shows formation and extension of plastic hinge with each cycle. The pull-in voltage of 95 V for the fourth cycle, can be explained with a β factor value of 0 which shows that the specimen is in fully plastic state with extension of plastic hinge throughout the specimen.

To verify the elastic–plastic phenomenon for the test microstructures at an actuation voltage of 105 V, another specimen is tested for six consecutive loading cycles and deflection in the central specimen is measured. Figure 10 shows the results for the six loading cycles. At first loading cycle the deflection in the central specimen at 105 V is 0.8 µm which is same as that of test microstructure 2 for the first loading cycle. In the second and third loading cycle the specimen does not snap down at 105 V but there is a higher deflection in the central specimen as compared to the first loading cycle. For the first three loading cycle, the pull-in does not occur and the specimen seems to be in the elastic region. This behavior can be attributed to the variation in the yield point for the specimen in comparison to the second microstructure and the induced stress in the central specimen is near the yield point. However, for the fourth to sixth loading cycle the pull-in phenomenon occurs and there is also a progressive decrease in the pull-in voltage value for each loading cycle and the behavior is exactly same like the second microstructure. These results show accumulation of plastic strain in the specimen with each loading cycle with central specimen being in the elastic–plastic stress state.

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Table 3 shows the increase in the plasticity in the central specimen in terms of β factor for the six loading cycles. For the first three cycles the pull-in does not occur for loading up to 105 V. This means that the specimen is in elastic state and β factor is 1. However, for the fourth cycle the pull-in occurs at 105 V, showing initialization of the plastic hinge formation process. The corresponding β factor value is 0.75. For the fifth and sixth cycle there is a progressive decrease in the pull-in voltage value with β factor values of 0.65 and 0 respectively. This shows that the plastic hinge develops with a small localized plasticization of the specimen in the fourth cycle, expands in the next loading cycle and eventually leads to fully plasticized region in the sixth cycle with β factor value equal to 0.

For the test microstructure 4, the static deflection in the central specimen is measured for both loading and unloading cycles. In the first loading–unloading cycle the actuation voltage is initially increased to 95 V in step by step and then decreased down to zero. The difference in the deflection values for both the loading and unloading curves is negligible and after the cycle the specimen restores to its original position as shown in figure 11. Thus showing that up to 95 V the stresses in the central specimen are in the elastic region. For the second loading–unloading cycle the actuation voltage is first increased to 105 V and maximum deflection of 0.75 µm is measured with higher deflection values for each voltage value as compared to the first loading cycle. Upon unloading to 0 V, a permanent deflection of 0.16 µm is measured for the central specimen. This shows an accumulation of plastic strain in the second loading–unloading cycle. The test microstructure is tested for the third loading–unloading cycle by increasing the actuation voltage to 110 V at which the pull-in phenomenon occurred. The actuation voltage is decreased in steps from 110 V to 0 V and pull-out voltage is observed to be 40 V. The deflection in the central specimen after third loading–unloading cycle is measured to be 0.56 µm which is much higher than the second loading–unloading cycle. This higher deflection in the central specimen can be explained with the fact that the with an actuation voltage of 110 V, the central specimen is loaded above the yield point instantaneously with the higher value of induced plasticity in the specimen. For the fourth loading–unloading cycle, the specimen snapped down at 100 V with much higher deflection at each voltage value during loading as compared to the first three cycles. The decrease in pull-in voltage value of 10 V from third to fourth cycle can be attributed to the high value of plasticity induced in the central specimen with an actuation voltage of 110 V in the preceding third cycle. During the fourth unloading, the pull-out voltage is 40 V and permanent deflection in the central specimen is 0.66 µm. Finally, the test microstructure is tested for the fifth loading–unloading cycle and a pull-in voltage of 100 V during loading and pull-out voltage of 40 V during unloading is obtained. The permanent deflection in the central specimen is measured to be 0.72 µm. The results for the second to fifth loading–unloading cycles clearly depict the onset of elastic–plastic phenomenon in the central specimen with progressive increase in the plastic strain with each loading cycle.

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Table 4 shows the extension of plasticity in the specimen for test microstructure 4 in terms of β factor. For the first and second loading cycle where the specimen is loaded to 95 V and 105 V respectively, the pull-in does not occur. This shows that the specimen is in a fully elastic state with β factor of 1. In the third cycle, the specimen is loaded to above 105 V and pull-in occurs at 110 V. This shows the onset of plastic hinge formation with a localized plasticity with a β factor value of 0.85. The loading of the specimen for the fourth and fifth cycle shows a large decrease in the pull-in voltage i.e. 110 V for the third cycle and 100 V for the fourth and fifth cycle. This large decrease in the pull-in voltage for two consecutive cycles shows that the loading of the specimen for a voltage of 110 V in the third cycles results in fast extension of the plastic hinge and hence the pull-in occurs at a much lower value of 100 V in the next fourth and fifth cycle. Moreover, the variation in the β factor value for the central specimen in the test microstructure 4 is exactly the same as that of the specimen in test microstructure 2 and 3 i.e. with increasing the loading above the yield point the β factor value decreases progressively with a fully plasticized value of zero and corresponding pull-in of 94 V for each specimen.