## Abstract

In this paper, we investigate the symmetric snap-through buckling and the asymmetric bifurcation behaviours of an initially curved functionally graded material (FGM) microbeam subject to the electrostatic force and uniform/non-uniform temperature field. The beam model is developed in the framework of Euler–Bernoulli beam theory, accounting for the through-thickness power law variation of the beam material and the physical neutral plane. Based on the Galerkin decomposition method, the beam model is simplified as a 2 d.f. reduced-order model, from which the necessary snap-through and symmetry breaking criteria are derived. The results of our work reveal the significant effects of the power law index on the snap-through and symmetry breaking criteria. Our results also reveal that the non-uniform temperature field can actuate the FGM microbeam and induce the snap-through and asymmetric bifurcation behaviours.

## 1. Introduction

The bistable micro-electro-mechanical systems (MEMS) based on initially curved microbeams have drawn considerable attention from the research community due to their potential applications as optical switches, micro-valves and non-volatile memories [1–5]. The initially curved beam (arch) under transverse forces may exhibit two main instabilities: symmetric snap-through buckling and asymmetric bifurcation. The symmetric snap-through buckling is the transition between two stable states [6]. After the snap-through, the arch shape is symmetric, as depicted in figure 1*a*. However, for the asymmetric bifurcation, the arch may exhibit one of the asymmetric deformations shown in figure 1*b*. Studies on the symmetric snap-through and asymmetric bifurcation of homogeneous macro/micro/nanobeams have been largely reported in the literature, and the influences of the initial arch rise, beam thickness, residual axial force, uniform temperature variation, size effect and surface effects have been systematically investigated [6–21].

Recently, functionally graded materials (FGMs) have been proposed to be used in MEMS [22,23] for their combined properties of improved stress distribution, enhanced thermal resistance and high fracture toughness [24,25]. FGMs are composite materials showing gradual variations of material properties (e.g. mechanical strength, thermal conductivity and thermal expansion) from one surface to another [26,27]. In the literature, various behaviours of FGM microbeams have been studied, such as bending, buckling, free vibration and pull-in instability [25,27–31]. However, to the authors’ best knowledge, studies concerned with the symmetric snap-through and asymmetric bifurcation of FGM microbeams are seldom reported.

In this paper, we extend our earlier work [19,20] to study the symmetric snap-through and asymmetric bifurcation of the initially curved FGM microbeam under thermo-electrical loadings (i.e. electrostatic force and uniform/non-uniform temperature field). The material variation through the beam thickness is accounted for by a power law model containing a power law index, and the effective material properties are estimated using the mixed rule. Owing to the variation of elastic modulus through the beam thickness, the physical neutral plane does not coincide with the geometric mid-plane and our beam model is developed by considering the physical neutral plane. Using the model, the effects of the power law index on the symmetric snap-through and asymmetric bifurcation induced by the electrostatic force are carefully investigated. Moreover, using non-uniform temperature field to induce the snap-through and asymmetric bifurcation of FGM microbeams are also proposed and examined.

## 2. Model formulation

### (a) Effective material properties of functionally graded material microbeam

The system investigated is an initially curved FGM microbeam under electrostatic force (figure 2*a*). The material on the top surface of the beam is ceramic, and that on the bottom surface is metal. It is assumed that the volume fractions of metal *V*_{m} and ceramic *V*_{c} vary gradually through the beam thickness (along *z*-coordinate in figure 2*a*), according to the following power law [27,32,33]:
*h* is the beam thickness, and *n* (≥0) is the power law index. The volume fraction of metal *V*_{m} at different levels of *n* is shown in figure 2*b*, from which it is seen that the beam becomes metal rich (*V*_{m} increases) when decreasing *n*. With the aid of equation (2.1), the effective material properties (i.e. Young's modulus *E* and coefficient of linear thermal expansion *α*) can be estimated by the mixed rule as [27,29,32,33]
*E*_{m} and *E*_{c} are the Young's moduli of metal and ceramic, respectively; *α*_{m} and *α*_{c} are the linear thermal expansion coefficients of metal and ceramic, respectively. The effective Poisson's ratio *ν* is assumed to be constant.

### (b) Governing equations of initially curved functionally graded material microbeam

Suppose that the displacements *u*_{x}, *u*_{y} and *u*_{z} (respectively, along *x*-, *y*- and *z*-coordinate) at an arbitrary point in the microbeam only depend on *x* and *z*. Further suppose that *u*_{y}=0. For a thin beam (thickness *h*≪ span *L*), the Euler–Bernoulli beam theory can be applied as
*u* and *w* are, respectively, the axial (along *x*-coordinate) and transverse (along *z*) displacements of a point on the physical neutral plane of the beam (dashed line in figure 2*a*), and *z*_{0} is the position of the physical neutral plane. With the aid of equation (2.2a), we can calculate *z*_{0} as [32,33]
*w*_{0}(*x*) by *ε*_{xx} from equation (2.5) as
*δU*_{e} of the elastic strain energy can be calculated as
*y*−*z* plane in figure 2*a*), and the stress resultants *N* and *M* are given by
*δW*_{e} of the work done by the distributed electrostatic force *f*_{e} as
*f*_{e} can be estimated as [34]
*w* depend on the coordinate system (figure 2*a*), *ε*_{0}(=8.8542×10^{−12} F m^{−1}) is the vacuum permittivity, *b* is the beam width, *V* is the voltage applied between the beam and the rigid electrode, and *g*_{0} is the gap between the beam ends and the electrode (figure 2*a*). Introducing equations (2.7) and (2.9) into the theorem of minimum potential energy: *δU*_{e}−*δW*_{e}=0, we obtain the following governing equations:
*E*, the effective linear thermal expansion coefficient *α* (given by equation (2.2)), and the constant Poisson's ratio *ν*. For an infinitely wide beam, the one-dimensional constitutive equation becomes [25]
*T* is the temperature variation. Introducing equations (2.4), (2.6) and (2.13) into equation (2.8), we obtain

and
*A*_{1} and bending stiffness *B*_{1} can be calculated with the aid of equations (2.2a) and (2.4) as
*N*_{T} and *M*_{T} are given by
*N* is constant along *x*-coordinate. Considering equation (2.14a), we can estimate *N* as the following average value
*u*(0)=*u*(*L*)=0) are used. In equation (2.17), replace *N* with equation (2.18) and we have

### (c) Reduced-order model

The Galerkin decomposition method [6,9,35] can be used to solve the governing equation (2.21) with the boundary conditions (2.22) considered. According to this method, the dimensionless deflection *ϕ*_{j} ( *j*=1,2,…,*n*_{w}) is the *j*th eigenmode of the double-clamped straight beam, and *q*_{j} is its generalized coordinate. The buckling eigenmodes have been found more suitable for the studies on the snap-through buckling and asymmetric bifurcation [16,17], so we decided to take them here
*C*_{j} is a constant satisfying *λ*_{j} is the eigenvalue satisfying

It is shown in [35] that the numerical simulations using *n*_{w}≥6 in equation (2.23) are indistinguishable from each other. It is further stated that a reasonably accurate prediction of the symmetric snap-through behaviour can be given by considering only the 1st mode, and for the asymmetric deformations, the participation of the 2nd mode is more than that of the 4th and 6th modes. So in order to simplify our studies for the analytical snap-through and symmetry breaking criteria, we decided to focus on the first two modes here (i.e. *n*_{w}=2 in equation (2.23)). Suppose that the dimensionless initial deflection *q*_{1} is the dimensionless midpoint deflection (*ϕ*_{1}(0.5)=1, *ϕ*_{2}(0.5)=0); *q*_{0} (=*r*/*g*_{0}) is the dimensionless initial arch rise, with *r* being the initial arch rise (deflection at the midpoint).

Introduce equation (2.25) into equation (2.21), multiply the result respectively by *ϕ*_{1} and *ϕ*_{2}, and then integrate over the domain [0, 1]. Further integrate by parts and consider the orthogonality of *ϕ*_{1} and *ϕ*_{2}, we obtain

where the expressions and calculated values of *b*_{11}, *b*_{22}, *s*_{11} and *s*_{22} are given in table 2, and the integrals *I*_{1}, *I*_{2}, *I*_{t1} and *I*_{t2} are given below

## 3. Functionally graded material microbeam actuated by electrical loading

### (a) Snap-through and asymmetric bifurcation behaviours

Suppose that the FGM microbeam is under a uniform temperature change Δ*T*. Then equation (2.26) can be reduced to
*P* due to temperature change can be calculated with the aid of table 1 and equations (2.2), (2.15b) and (2.19):
*F*_{1} being

*F*_{1} varies between 1 (at *n*=0, the beam only contains metal) and *α*_{c}/*α*_{m} (at *I*_{2}=0 at *q*_{2}=0, equation (3.1) always has the following solution corresponding to the symmetric deformation
*γ* and the axial force *P* with the aid of equations (2.15) and (3.2), and tables 1 and 3, and then introduce the values of *γ* and *P* into equation (3.1) and solve it numerically. The typical solutions are shown in figure 3.

It is seen from figure 3 that the deformation behaviour of the beam depends on the dimensionless initial arch rise *q*_{0}. At very small *q*_{0} (figure 3*a*), the midpoint deflection of the beam decreases gradually with the increase of the applied voltage (①→*q*_{p}) until reaching the extreme point *q*_{p}, where a slight increase in voltage leads to a sudden collapse of the beam onto the rigid electrode (*q*_{p}→②, referred to as pull-in instability).

When *q*_{0} becomes larger (figure 3*b*), two more extreme points *q*_{s} and *q*_{r} appear, which correspond, respectively, to the snap-through buckling and the release (snap-back). During the electrical loading (*β*_{v} increases), the midpoint deflection decreases gradually (①→*q*_{s}) until reaching *q*_{s} where two stable states (*q*_{s} and ②) coexist. A slight increase in *β*_{v} makes the state at *q*_{s} unstable, which results in a sudden transition from *q*_{s} to ② (referred to as snap-through buckling). After that, the midpoint deflection continues to decrease gradually with the increase of *β*_{v} until reaching *q*_{p} where the beam collapses onto the electrode.

The beam deformations in figure 3*a*,*b* are symmetric (*q*_{2}=0). When *q*_{0} is large, the solutions of the asymmetric deformations (*q*_{2}≠0) exist, then the beam may deform asymmetrically. In figure 3*c* where *q*_{0} is large enough, the bifurcation points *q*_{b1} and *q*_{b2} are located on the stable parts of the symmetric equilibrium curve. So during the quasi-static loading and unloading, *q*_{b1} and *q*_{b2} can be reached before the snap-through points *q*_{s} and *q*_{r}. In this case, the microbeam exhibits asymmetric bifurcation instead of snap-through buckling. By gradually decreasing the midpoint deflection *q*_{1}, the beam deforms symmetrically (①→*q*_{b1} in figure 3*c*) until reaching the first bifurcation point *q*_{b1} where there are two stable asymmetric deformations. Following one of the asymmetric deformations (A or A^{′}), the beam deforms asymmetrically (*q*_{b1}→*q*_{b2}) until reaching the second bifurcation point *q*_{b2}. After *q*_{b2}, the beam returns to deform symmetrically with the decrease of the midpoint deflection.

In summary, to exhibit snap-through buckling or asymmetric bifurcation, the microbeam should have a large dimensionless initial arch rise *q*_{0}. Moreover, the minimum allowable *q*_{0} for asymmetric bifurcation is larger than that for snap-through buckling.

It is noted that in figure 3*b*,*c*, the arch rise at the beginning of loading (marked as ①) is slightly larger than the initial arch rise *q*_{0}. This is due to the fact that the axial compression from thermal loading (temperature increase of 1 K) increases the arch rise at the beginning.

### (b) Effect of power law index

The power law index *n* is related to the distributions of metal and ceramic in the FGM microbeam (equation (2.1)). The effects of *n* on the snap-through and asymmetric bifurcation behaviours are shown in figure 4 by plotting the numerical solutions to equation (3.1) at different levels of *n* (0∼10). The other parameters are taken or calculated from table 3.

It is seen from figure 4*a* that the critical voltage (normalized as *β*_{v}) at snap-through (point *q*_{s}) increases with the decrease of *n*, while that at the release (*q*_{r}) decreases. By decreasing *n*, the beam becomes metal rich (figure 2*b*), so the axial compression *P* (calculated by equation (3.2)) due to the temperature increase becomes larger (*α*_{m}>*α*_{c} in table 3). Since *P* stabilizes the buckling configurations, the transitions between the configurations (snap-through and release) become more difficult. As a result, more voltage (i.e. larger *β*_{v}) is required to induce the forward transition (snap-through) during loading, and more voltage decrease (smaller *β*_{v}) is required to enable the reverse transition (release) during unloading.

It should be noted that the axial compression also depends on Young's modulus. For the studied FGM microbeam (table 3), Young's moduli of the ceramic and the metal are close (*E*_{c}/*E*_{m}=310 GPa/204 GPa=1.5), when compared with their thermal expansion coefficients (*α*_{m}/*α*_{c}=(13.2×10^{−6} K^{−1})/(3.4×10^{−6} K^{−1})=3.9). This explains why the effect of the thermal expansion coefficient is dominant.

For the displacement-controlled asymmetric bifurcation behaviour, figure 4*b* shows that the critical midpoint deflection at the first bifurcation point *q*_{b1} increases with the decrease of the power law index *n*, while that at the second bifurcation point *q*_{b2} decreases, so the domain (*q*_{b2}∼*q*_{b1}) of the asymmetric deformation becomes larger. Reducing *n* leads to an increase of the axial compression *P*. Since the 2nd buckling mode becomes more preferable than the 1st mode when increasing the compression, the domain where the 2nd buckling mode exists (i.e. asymmetric deformation) is broadened.

The considered thermal loading in figure 4 is uniform temperature increase. If a temperature decrease is applied, we can predict that the effects of the power law index *n* are opposite to those observed in figure 4. This is because that the temperature decrease will result in an axial tension, whose effects on snap-through and asymmetric bifurcation are opposite to those of compression.

### (c) Snap-through and symmetry breaking criteria

#### (i) Snap-through criterion

The symmetric snap-through buckling takes place at the extreme points on the *β*_{v}−*q*_{1} curve (see *q*_{s} and *q*_{r} in figure 3*b*). Considering the symmetric deformation described by equation (3.4), we can obtain the extreme points by solving the following equation:
*I*_{3} being
*γ* and *P* with the aid of equations (2.15) and (3.2), and tables 1 and 3, introduce the values of *γ* and *P* into equation (3.5) and solve it numerically. The solutions are shown in figure 5*a*. It is seen that *q*_{0} should be larger than a critical value *q*_{s} and *q*_{r}. At *q*_{s} and *q*_{r} are close to 0. As a first approximation of *q*_{1}=0 in equation (3.5) and find
*f*_{1} and *m*_{11} are given in table 2. In equation (3.7), replace *q*_{0}, *γ* and *P* with the expressions in table 1 and equations (2.15) and (3.2), and we obtain the following requirement (i.e. minimum allowable ratio between the initial arch rise *r* and the beam thickness *h*) for the existence of snap-through points:
*F*_{2} is given by
*g*_{0}/*h*) must be large enough (>2 for the double-clamped beam, which can be found in [16,20]). When the distance between the electrode and the beam is small (i.e. small (*g*_{0}/*h*)), the second stable configuration cannot be realized, and the beam is observed to directly collapse onto the electrode [16]. At *n*=0 or *T*=0, equation (3.8) can be reduced to the symmetric snap-through criterion for the homogeneous microbeam in the literature [6,19].

By taking the parameters in table 3 (except the power law index *n* and the gap *g*_{0}), equation (3.8) is plotted in figure 5*b* at different levels of *n*. It is seen that (*r*/*h*)_{min−1} decreases with the decrease of *n*. By decreasing *n*, the beam becomes metal rich (figure 2*b*), which raises the axial compression *P* from the temperature increase (calculated by equation (3.2)). Since *P* has the effect of increasing the arch rise and larger *P* results in larger increase, the required initial arch rise (normalized as (*r*/*h*)_{min−1}) for snap-through buckling is reduced. If a temperature decrease is applied, the resulted axial force is tension, which has the effect of decreasing the arch rise. In this case, (*r*/*h*)_{min−1} will increase when decreasing *n*.

It is noted that if (*r*/*h*)_{min−1} calculated from equation (3.8) is negative or an imaginary number, the snap-through may take place in an initially straight beam. In this case, (*r*/*h*)_{min−1} is taken to be 0 in figure 5*b*.

#### (ii) Symmetry breaking criterion

Near the asymmetric bifurcation points *q*_{b1} and *q*_{b2}, *q*_{2} is around 0 (figure 3*c*). So to obtain *q*_{b1} and *q*_{b2}, we can linearize the governing equations (equation (3.1)) around *q*_{2}=0 as [16]
*I*_{4} being
*q*_{2}≠0), equation (3.10b) can be further reduced to
*β*_{v}, respectively, from equations (3.10a) and (3.12), equate these expressions, and we obtain the following equation for *q*_{1} corresponding to the asymmetric bifurcation points *q*_{b1} and *q*_{b2}:
*γ* and *P*, introduce them into equation (3.13) and solve it numerically. The solutions are shown in figure 6*a*, from which it is found that when the dimensionless initial arch rise *q*_{0} becomes larger than a critical value *q*_{b1} and *q*_{b2} exist. It is further found that at *q*_{b2} are close to 0. So to estimate *q*_{1}=0 in equation (3.13) and obtain
*m*_{22} is given in table 2. In equation (3.14), replace *q*_{0}, *γ* and *P* with the expressions in table 1 and equations (2.15) and (3.2), and we obtain the following requirement for the existence of the asymmetric bifurcation points:
*F*_{1} and *F*_{2} are, respectively, given in equations (3.3) and (3.9). Equation (3.15) is the necessary criterion for symmetry breaking. To guarantee that the asymmetric bifurcation can take place, the bifurcation points should be located on the stable parts of the symmetric equilibrium curve [16]. It is noted that when *n*=0 or *T*=0, we can reduce equation (3.15) to the necessary symmetry breaking criterion for the homogeneous microbeam [6].

With the parameters in table 3 (except *n* and *g*_{0}), equation (3.15) is plotted in figure 6*b*, from which similar conclusions to those in §3c(i) can be drawn: under the uniform temperature increase, (*r*/*h*)_{min−2} decreases with the decrease of *n*; while under the temperature decrease, it increases when decreasing *n*. These results are due to the fact that the power law index *n* influences the temperature-induced axial force, which in turn influences (*r*/*h*)_{min−2}. Detailed explanations are given in §3c(i).

## 4. Functionally graded material microbeam actuated by thermal loading

### (a) Snap-through and asymmetric bifurcation behaviours

Thermal loadings can also induce the snap-through and asymmetric bifurcation behaviours of the FGM microbeam. Suppose that we can generate a uniform heat source *Q*(W m^{−3}) in the beam, and apply a temperature change of Δ*T*_{1} to the beam ends. The heat can be generated by passing the electric current through the beam. Then the stabilized temperature change Δ*T*(*x*) along beam length (*x*-coordinate in figure 2*a*) can be obtained by solving the following one-dimensional steady-state heat equation:
*k*_{m} and *k*_{c} are the thermal conductivity of metal and ceramic, respectively. In equation (4.1), the temperature variation along beam thickness (*z*-coordinate in figure 2*a*) is neglected. Since the beam thickness *h* is much smaller than the beam span *L*, the temperature variation along beam length is dominant. Further neglect the thermal convection and take into account the boundary conditions Δ*T*(0)=Δ*T*(*L*)=Δ*T*_{1}, we can solve equation (4.1) as
*P* due to temperature change and the transverse force *I*_{t1} due to thermal moment are calculated with the aid of table 1, equations (2.2), (2.4), (2.15b), (2.16b), (2.19), (2.27c) and (4.3):
*F*_{1} and *f*_{1} are given respectively in equation (3.3) and table 2, and *F*_{3} is given below:
*I*_{t1} can actuate the initially curved beam (i.e. bends it towards the electrode) and induce the snap-through and asymmetric bifurcation. However, for the homogeneous microbeams (i.e. power law index *n*=0 or *I*_{t1} is always null (*F*_{3}=0 at *n*=0 and

The axial force *P* due to temperature variation can be tension or compression. The axial compression works against the transverse force *I*_{t1} to move the beam away from the electrode (i.e. raises the arch rise). The axial tension reduces the arch rise, but this results in the increase of the minimum allowable initial arch rise for snap-through and asymmetric bifurcation [16,20]. So both forces hinder the snap-through and asymmetric bifurcation behaviours. To reduce *P*, we can control the temperature change Δ*T*_{1} applied to the beam ends. It is found from equation (4.5a) that *P* is null at
*Q* in the beam, and apply a temperature change of *γ* and the voltage parameter *β*_{v} (at *V* =50 V) with the aid of equation (2.15) and tables 1 and 3, and then introduce *γ* and *β*_{v} into equation (4.8) and solve it numerically. The typical solutions are shown in figure 8, from which it is seen that the proposed thermal loading can induce the snap-through and asymmetric bifurcation of FGM microbeam. Careful comparisons between figures 3 and 8 also reveal that the pull-in deflection (around −0.7 at *q*_{p} in figure 3) of the electrically actuated microbeam is larger than that (around −0.9 in figure 8) of the thermally actuated microbeam. The proposed thermal loading (with a small constant voltage<50 V) can help increase the working distance (from the initial deflection to the pull-in deflection) of the microbeam.

### (b) Snap-through and symmetry breaking criteria

Considering equation (4.9), we can obtain the critical snap-through points (*q*_{s}, *q*_{r}) by solving the following equation:
*I*_{3} is given in equation (3.6). In equation (4.10), *γ* and *β*_{v} (at *V* =50 V) are calculated with the aid of equation (2.15) and tables 1 and 3. The numerical solutions to equation (4.10) are shown in figure 9*a*, from which it is seen that *q*_{0} should be larger than a critical value *q*_{s} and *q*_{r}. By taking *q*_{1}=0 in equation (4.10), we find
*q*_{0} and *γ* with the expressions in table 1, and take into account equation (2.15), we obtain the following necessary snap-through criterion:
*F*_{2} is given in equation (3.9).

To obtain the asymmetric bifurcation points (*q*_{b1}, *q*_{b2}), we linearize equation (4.8) around *q*_{2}=0:
*I*_{4} is given in equation (3.11). For the asymmetric deformations (*q*_{2}≠0), equation (4.13b) can be further reduced to
*γ* and *β*_{v} (at *V* =50 V) using equation (2.15) and tables 1 and 3, introduce them into equation (4.14) and solve it numerically. The solutions are shown in figure 9*b*, from which it is seen that the solutions of asymmetric bifurcation points *q*_{b1} and *q*_{b2} exist when the dimensionless initial arch rise *q*_{0} becomes larger than a critical value *q*_{1}=0 in equation (4.14) and we obtain
*q*_{0} and *γ* with the expressions in table 1 and equation (2.15), and we obtain the necessary symmetry breaking criterion as follows:
*F*_{2} is given in equation (3.9).

Equations (4.12) and (4.16) show that the applied voltage (normalized as *β*_{v}) has the effect of decreasing the minimum allowable ratios between the initial arch rise *r* and beam thickness *h* for the symmetric snap-through ((*r*/*h*)_{min−1}) and asymmetric bifurcation ((*r*/*h*)_{min−2}). The electrostatic force (due to the applied voltage) helps the thermal transverse force *I*_{t1} to induce snap-through and asymmetric bifurcation, so both (*r*/*h*)_{min−1} and (*r*/*h*)_{min−2} are reduced when the voltage is applied. It is noted at last that by taking *V* =0 and *n*=0 or

## 5. Conclusion

This paper is concerned with a unified study on the symmetric snap-through and asymmetric bifurcation behaviours of an initially curved FGM microbeam subject to an electrostatic force and uniform/non-uniform temperature field. The two-constituent material variation through the beam thickness is taken into account using a power law model. The governing equations, which consider the physical neutral plane, are developed and solved by the Galerkin decomposition method. Our results, which are based on the 2 d.f. reduced-order model, show that the power law index plays a significant role in the snap-through and symmetry breaking criteria. Our results also show that the FGM microbeam can be actuated by the non-uniform temperature field, and the symmetric snap-through and asymmetric bifurcation behaviours can be induced.

The intermolecular forces (van der Waals and Casimir forces) are ignored in the paper, since they are negligible with respect to the electrostatic force for cases involving the snap-through buckling of microbeams [19]. The fringing field effect due to the finite size of the microbeam width is also neglected. When the fringing fields at the edges of the microbeam are taken into account, the critical loadings at snap-through and asymmetric bifurcation will decrease, since the fringing fields increase the total voltage between the beam and the rigid electrode, and as a result, smaller loadings are required to induce the snap-through [19] and asymmetric bifurcation. Moreover, this paper only studies the double-clamped FGM microbeam. The beam can also be supported by other boundary conditions such as ‘simply-supported’. Compared with the double-clamped microbeam, the simply-supported microbeam will have smaller critical loadings. The reduced rotation at the two ends of the double-clamped microbeam makes it more difficult to bend, and as a result, larger loadings are needed for the snap-through [20] and asymmetric bifurcation.

## Authors' contributions

X.C. carried out the research work and drafted the manuscript. S.A.M. helped in correcting and improving the manuscript. Both authors gave final approval for publication.

## Competing interests

We have no competing interests.

## Funding

The research is funded by the Natural Sciences and Engineering Research Council of Canada (NSERC), the Discovery Accelerator Supplements (DAS) and the Qatar National Research Foundation under the National Priority Research Program.

## Acknowledgements

The financial support provided by the Natural Sciences and Engineering Research Council of Canada, the Discovery Accelerator Supplements, and the Qatar National Research Foundation under the National Priority Research Program is gratefully acknowledged.

- Received August 24, 2015.
- Accepted January 7, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.