## Abstract

In Part 2 of the two-paper series, the asymmetrically laminated piezoelectric shell subjected to distributed bias voltage as modeled in Part 1 is analytically and numerically investigated. Three out-of-plane degrees-of-freedom (DOFs) and a number of in-plane DOFs are retained to study the shell's snap-through phenomenon. A convergence study first confirms that the number of the in-plane DOFs retained affects not only the number of predicted equilibrium states when the bias voltage is absent but also the prediction of the critical bias voltage for snap-through to occur and the types of snap-through mechanisms. Equilibrium states can be symmetric or asymmetric, involving only a symmetric out-of-plane DOF, and additional asymmetric out-of-plane DOFs, respectively. For symmetric equilibrium states, the snap-through mechanism can evolve from the classical bidirectional snap-through and latching to a new type of snap-through that only allows snap-through in one direction (i.e., unidirectional snap-through), depending on the distribution of the bias voltage. For asymmetric equilibrium states, degeneration can occur to the asymmetric bifurcation points when the radii of curvature are equal. Finally, the unidirectional snap-through renders an explanation to the experimental findings in Part 1.

## Introduction

In Part 1 of the two-part paper series, a model was developed to formulate an asymmetrically laminated piezoelectric shell subjected to distributed bias voltage. The motivation is to simulate a lead-zirconate-titanate (PZT) thin-film micro-actuator and to investigate different snap-through mechanisms that can be induced via distributed bias voltage. As will be presented later, due to the converse piezoelectric effect, the piezoelectric shell laminate manifests a snap-through mechanism that is very different from elastic or electrostatic bistable structures. The unique snap-through mechanism could also be used to qualitatively explain the experimental measurement reported in Fig. 2 of Part 1 [1].

Although the discretized formulation was available, its equilibrium and stability analyses were not completed in Part 1. In order for snap-through to occur, one must show that there are multiple equilibrium states and analyze their stability. The equilibrium and stability analyses may seem straightforward. Nevertheless, there are some unique challenges.

First, the number of in-plane degrees-of-freedom (DOFs) retained in the formulation is critical. In the studies of nonlinear elastic thin-walled structures, it is well known that the number and type of in-plane DOFs retained can significantly affect the accuracy of snap-through prediction [2,3]. Therefore, it is important to find out how many in-plane DOFs are needed for an asymmetrically laminated piezoelectric shells. The second challenge is the converse piezoelectric effect. As demonstrated in Part 1, the converse piezoelectric effect can simultaneously deflect and harden the shell laminate. The interplay between deflection and hardening complicates the snap-through mechanism. Specifically, as will be presented later, it will lead to an asymptote in symmetric equilibrium states of one symmetric DOF, which can separate an equilibrium curve into two disjoint portions. Depending on the location of the asymptote, the snap-through can occur in only one direction or in both directions, leading to different snap-through mechanisms. Therefore, this feature needs to be investigated thoroughly. The third challenge is that degeneration can occur to bifurcation of asymmetric equilibrium states which involve at least one nonzero asymmetric DOF. For curved beams and shallow arches, asymmetric equilibrium states generally bifurcate from the symmetric equilibrium states. For two-dimensional shell structures, asymmetric equilibrium states for x and y directions may become degenerate or indistinguishable when the radii of curvature Rx and Ry in these two directions are the same. If so, how would the degeneration affect the bifurcation of the asymmetric equilibrium states from the symmetric ones? Finally, how does the bias voltage actually induce snap-through?

The purpose of this paper is to conduct equilibrium and stability analyses using a combined analytical and numerical methods, with the hope of qualitatively explaining the experimental findings. Through the analytical method, we are able to predict exact locations of the asymptote and how it leads to different snap-through mechanisms. The analytical method can also show how the asymmetric equilibrium states evolve when the system becomes degenerate. The numerical method complements the analytical method in many ways. For example, the numerical method predicts bifurcation points accurately and demonstrates how they evolve when a degenerate system is perturbed.

For the rest of the paper, we will first summarize the reference system and equations of motion for the numerical and analytical studies. Material and geometric properties will then be defined. Next, a convergence study is conducted to ensure that a sufficient amount of in-plane DOFs are included in the numerical study. After the convergence is secured, studies on symmetric and asymmetric equilibrium states are carried out. Moreover, bifurcation of asymmetric equilibria and its dependence on degeneracy are studied in-depth. Finally, a discussion is dedicated to qualitatively explaining the experimental findings using the analysis results.

## Reference System and Equation of Motion

This section serves as a quick summary of the reference system defined and the equation of motion derived in Part 1 of the two-paper series. The reference system is an asymmetrically laminated, doubly curved, piezoelectric shell occupying a rectangular domain $Ω≡{(x,y)|0≤x≤a,0≤y≤b}$ with a simply supported boundary conditions, where a and b are the widths of the domain. The inner electrode occupies a rectangular domain $Ω−≡{(x,y)|xle≤x≤xre,yle≤y≤yre}$, and the outer electrode is $Ω+≡Ω−Ω−$, where $xle, xre, yle$, and $yre$ are the four vertices of the domain; see Fig. 3 in Part 1.

The transverse displacement $w(x,y,t)$ is approximated via three shape functions
$w(x,y,t)≈r11(t)sin(πx/a)sin(πy/b)+r21(t)sin(2πx/a)sin(πy/b)+r12(t)sin(πx/a)sin(2πy/b)$
(1)
In Eq. (1), $r11(t)$ is the response of the symmetric transverse shape, and $r21(t)$ and $r12(t)$ are the response of antisymmetric transverse (AT) shapes. For the sake of simplicity, the subscripts 11, 21, and 12 are replaced with 1, 2, and 3 for the rest of paper, e.g., $r11=r1$, $r21=r2$, and $r12=r3$. The in-plane displacements $u(x,y,t)$ and $v(x,y,t)$ are approximated via 2N symmetric in-plane shapes, $2N2−2N$ antisymmetric in-plane shapes, and 2N electric-induced shapes (those associated with $pje$ and $qie$) as follows:
$u(x,y,t)≈∑i=1N∑j=1Npij(t)sin(iπx/a)sin(jπy/b)+∑j=1Nμ(x)sin(jπy/b)pje(t)v(x,y,t)≈∑i=1N∑j=1Nqij(t)sin(iπx/a)sin(jπy/b)+∑i=1Nθ(y)sin(iπx/a)qie(t)$
(2)

where N is the in-plane shape function number indicator, $pje(t), qie(t), μ(x)$, and $θ(y)$ have been defined in Eqs. (49), (50), (52), and (53) in Part I, respectively. Furthermore, a sufficient number of N to ensure convergence will be determined in Sec. 4.1.

In deriving the equations of motion, the following dimensionless quantities are introduced:
$ξ=xa, η=yb, R̂x=Rxa, R̂y=Ryb, Δx̂=Δxa, Δŷ=Δybr̂1=r1h2, r̂2=r2h2, r̂3=r3h2, Δϕ̂+/−=e312KΔϕ+/−$
(3)
where a and b are the widths of the shell, $Δx$ and $Δy$ are the widths of the inner electrode, h2 is the thickness of the piezoelectric layer, e31 is the piezoelectric constant, K is the summation of the membrane stiffness of all three layers, and $Δϕ+$ and $Δϕ−$ are electric potential difference in the outer and inner electrode domains, respectively. Following the procedure in Appendix of Part 1, one can eliminate the in-plane DOFs pij and qij and derive three transverse differential equations of motion that contain only transverse DOFs r1, r2, and r3. The three transverse differential equations of motion are
$m4r̂¨1+f1(r̂1,r̂2,r̂3;Δϕ̂−;Δϕ̂+)=F1e−Δϕ̂−+F1e+Δϕ̂+$
(4a)
$m4r̂¨2+f2(r̂1,r̂2,r̂3;Δϕ̂−;Δϕ̂+)=0$
(4b)
$m4r̂¨3+f3(r̂1,r̂2,r̂3;Δϕ̂−;Δϕ̂+)=0$
(4c)
where m is the mass of the shell laminate, and
$f1=(K1+S1e−Δϕ̂−+S1e+Δϕ̂+)r̂1+Q11r̂12+Q22r̂22+Q33r̂32+Γ111r̂13+Γ122r̂1r̂22+Γ133r̂1r̂32f2=(K2+S2e−Δϕ̂−+S2e+Δϕ̂+)r̂2+Q21r̂2r̂1+Γ222r̂23+Γ233r̂2r̂32+Γ211r̂2r̂12f3=(K3+S3e−Δϕ̂−+S3e+Δϕ̂+)r̂3+Q31r̂3r̂1+Γ333r̂33+Γ322r̂3r̂22+Γ311r̂3r̂12$
(5)

Note that the arguments $(r̂1,r̂2,r̂3;Δϕ̂−;Δϕ̂+)$ are ignored in Eq. (5) for simplicity. Definition of all coefficients in Eqs. (4) and (5) can be found in the Appendix. The equations of motion (4a)(4c) will be used to study snap-through instability of the shell laminate for the rest of the paper.

The stability of equilibria of Eq. (4) is studied by its Jacobian matrix J. The 3 × 3 Jacobian matrix is
$J(r̂1,r̂2,r̂3;Δϕ̂−;Δϕ̂+)=[∂f1∂r̂1∂f1∂r̂2∂f1∂r̂3∂f2∂r̂1∂f2∂r̂2∂f2∂r̂3∂f3∂r̂1∂f3∂r̂2∂f3∂r̂3]$
(6)
where
$∂f1∂r̂1=K1+S1e−Δϕ̂−+S1e+Δϕ̂++2Q11r̂1+3Γ111r̂12+Γ122r̂22+Γ133r̂32∂f2∂r̂2=K2+S2e−Δϕ̂−+S2e+Δϕ̂++Q21r̂1+3Γ222r̂22+Γ233r̂32+Γ211r̂12∂f3∂r̂3=K3+S3e−Δϕ̂−+S3e+Δϕ̂++Q31r̂1+3Γ333r̂32+2Γ322r̂22+Γ311r̂12∂f1∂r̂2=2Q22r̂2+Γ122r̂1r̂2, ∂f1∂r̂3=2Q33r̂3+Γ133r̂32∂f2∂r̂1=Q21r̂2+2Γ211r̂2r̂1, ∂f2∂r̂3=2Γ233r̂2r̂3∂f3∂r̂1=Q31r̂1+2Γ311r̂3r̂1, ∂f3∂r̂2=2Γ322r̂3r̂2$
(7)

For a set of equilibria $(r̂10,r̂20,r̂30)$, its stability is determined by the real part of the eigenvalues of the Jacobian matrix. If the real part of all eigenvalues is positive, the equilibria are stable; the equilibria are unstable otherwise. Note that the Jacobian matrix is a function of the bias voltages $Δϕ̂+$ and $Δϕ̂−$, i.e., it determines the stability of equilibria with arbitrary combinations of the direct current (DC) bias voltages.

For the sake of demonstration, let us assume $Δϕ̂−=ΠΔϕ̂+$, where Π indicates the relative phase and magnitude between $Δϕ̂−$ and $Δϕ̂+$. Also, for the sake of simplification, $Δϕ̂+=Δϕ̂$ for the rest of the paper.

## Geometric and Material Properties

Material and geometric properties are either measured or chosen to best mimic the micro-actuator that gives the bifurcation phenomenon reported in Fig. 2 of the Part 1 paper. Table 1 lists the properties used in the numerical study.

Table 1

Material and geometric properties of each layer

LayerMaterialThickness (nm)Young's modulus (GPa)Density (kg/m3)Poisson's ratio
Top electrodeParylene2502.411100.4
Gold20079192800.42
Chromium5027971900.21
PZTPZT100081.375000.33
Substrate and Bottom electrodePlatinum100168214500.38
Titanium50110.345000.34
Nitride20029730000.28
Oxide50074.826500.19
Parylene2502.411100.4
LayerMaterialThickness (nm)Young's modulus (GPa)Density (kg/m3)Poisson's ratio
Top electrodeParylene2502.411100.4
Gold20079192800.42
Chromium5027971900.21
PZTPZT100081.375000.33
Substrate and Bottom electrodePlatinum100168214500.38
Titanium50110.345000.34
Nitride20029730000.28
Oxide50074.826500.19
Parylene2502.411100.4

With the properties listed in Table 1, the midsurface of the piezoelectric layer is located above the modulus-weighted midsurface $S¯$ with $z¯(2)=0.35 μ$m. In other words, the substrate layer $S(1)$ is located below $S¯$, and the PZT and top electrode layers $S(2)$ and $S(3)$ are above.

The shell widths $a=560 μ$m and $b=560 μ$m used in Table 1 are different from the actual micro-actuator diaphragm widths of $800 μ$m for the following reasons. Due to fabrication limitation, unetched silicon residue builds up around the outer perimeter of the micro-actuator [4]. Finite element analyses were conducted to estimate an effective diaphragm area accordingly [5]. The predicted diaphragm deflection mimics the deflection of a simply supported shell with widths $a=560 μ$m and $b=560 μ$m approximately. Finally, $Δx=392 μ$m and $Δy=392 μ$m are the actual widths of the inner electrode of the micro-actuator.

## Single-Degree-of-Freedom Analysis

As a first attempt, a single DOF $r̂1$ is retained in Eq. (4) and the bias voltage is set to be zero. This is to determine possible equilibrium states of the piezoelectric micro-actuator in its natural and no-load condition. The simplicity of the single-DOF analysis not only provides a platform to examine convergence of numerical solutions but also an analytical expression to study stability of the equilibrium states. They are described in detail as follows.

### Equilibrium Positions and Convergence Study.

There are two goals in the equilibrium study. The first goal is to derive equilibrium states. The second goal is to conduct a convergence study to determine a sufficient number of N, the in-plane shape function number indicator introduced in Eq. (2), to predict the equilibrium states and snap-through instability. With $r̂2$ and $r̂3$ ignored, the equation of motion (4a) leads to an equilibrium condition $r̂1=r̂10$, where $r̂10$ is a constant satisfying
$[K1+(ΠS1e−+S1e+)Δϕ̂]r̂10+Q11r̂102+Γ111r̂103=(ΠF1e−+F1e+)Δϕ̂$
(8)
According to classical theories of 1DOF systems, a necessary condition for the shell laminate to leave the trivial equilibrium state and remain at a new equilibrium state after the external load is removed is existence of multiple equilibrium states while no external load is present. By imposing $Δϕ̂=0$ in Eq. (8), equilibrium states of $r̂10$ satisfy the following algebraic equation:
$r̂10(Γ111r̂102+Q11r̂10+K1)=0$
(9)
It is clear that Eq. (9) has either one or three equilibrium states. To have three equilibrium states, it is necessary and sufficient that the quadratic expression in Eq. (9) has a positive determinant, i.e.,
$Δ(R̂x,R̂y)=Q112−4Γ111K1>0$
(10)

Note that after substituting the geometric and material properties listed in Table 1, Q11, $Γ111$, and K1 are functions of unknown radii of curvature $R̂x$ and $R̂y$, and so is Eq. (10). The contour $Δ(R̂x,R̂y)=0$ defines a boundary on the $R̂x$$R̂y$ plane within which $R̂x$ and $R̂y$ are admissible to produce three equilibrium states. Every set of radii of curvature $R̂x$ and $R̂y$ that falls outside the contour leads to only one equilibrium. The contour $Δ(R̂x,R̂y)=0$, however, depends on the number of in-plane shape functions used. By plotting the contour $Δ(R̂x,R̂y)=0$ on the $R̂x$$R̂y$ plane, one can examine how the number of in-plane shape functions affects the number of equilibrium states. To this end, the inertial terms of Eq. (4) are set to be zero and the remaining equations are symbolically solved to obtain Eq. (10) in terms of radii of curvature $R̂x$ and $R̂y$. Then, the contour is plotted for three cases of N: (a) N = 2, (b) N = 3, and (c) N = 4, as shown in Fig. 1(a). The material and geometric properties are kept identical for all cases.

Fig. 1
Fig. 1
Close modal

There are several things worth noting in Fig. 1(a). First, as the number of in-plane shape functions increases, the contour gradually enlarges and encloses more admissible radii of curvature. In other words, prediction of multiple equilibria is not possible if insufficient in-plane shape functions are retained. Second, Fig. 1(a) is only meaningful when the radii of curvature are sufficiently large to not violate the shallow shell assumptions. Third, the contours of negative and positive radii of curvature are not identical because the shell laminate is asymmetrically laminated in the thickness direction.

The number of in-plane shape functions not only affects the number of multiple equilibria but also the prediction of the snap-through voltage (critical voltage for snap-through to occur). To demonstrate this, let us first define a sequence of relative error of the snap-through voltage as
$εN=Δϕ̂N*−Δϕ̂N−1*Δϕ̂N−1*, N=3,4,5,6$
(11)

where $Δϕ̂N*$ is the snap-through voltage obtained for $N=3,4,5,6$, which can be obtained by solving Eq. (8) for the limit points. Next, two different sets of radii of curvature ($R̂x=R̂y=10$ and $R̂x=R̂y=16$) and two values of Π are chosen for the demonstration. Finally, εN of each combination of $R̂x, R̂y$, and Π is plotted in Fig. 1(b). As shown, the relative error quickly decreases as the number of N increases. Specifically, it reduced to be less than 0.5% after $N≥5$, indicating convergence.

To demonstrate how the in-plane shape function numbers influence the equilibrium states of Eq. (8), let us consider a particular set of radii of curvature $R̂x=16$ and $R̂y=16$. As seen in Fig. 1, this set falls outside the contours (I) and (II), but lies within the contour (III). In other words, expansions (I) and (II) will predict only one equilibrium state. In contrast, expansion (III) will lead to three equilibrium states. With $R̂x=16$ and $R̂y=16$, we can solve Eq. (8) for equilibrium positions $r̂10$ given a bias voltage $Δϕ̂$; see Fig. 2 for four cases of N of interest. As seen in Figs. 2(a) and 2(b), there is only one stable equilibrium state A for cases (I) and (II) with $Δϕ̂=0$; therefore, even though snap-through does occur, there is no latching, which means that the voltage corresponding to the release point (snap-back) is positive and not negative (the interested reader is referred to Ref. [6] for the definition of latching.). In other words, when the bias voltage is removed, the shell laminate will resume its original state at A. In Fig. 2(c), there are three equilibrium states A, B, and C for case (III) with $Δϕ̂=0$. If both equilibrium states A and C are stable, the system could start from equilibrium A to reach equilibrium C by increasing the bias voltage $Δϕ̂$. In addition, Fig. 2(d) shows an almost identical situation when N = 6, indicating that $R̂x=16$ and $R̂y=16$ indeed lead to three equilibria.

Fig. 2
Fig. 2
Close modal

For the rest of the paper, we will retain N = 6 to conduct the snap-through study. With the selection, the numerical results in Fig. 1(b) demonstrate reasonable convergence while the computational efforts remain manageable.

### Characteristics of Voltage-Equilibrium Curve.

The voltage-equilibrium curves shown in Figs. 2(c) and 2(d) have two unique features worth noting, because they provide important clues to the stability of all equilibrium states. The first feature is the slope at equilibrium states A, B, and C. To obtain the slopes, one can differentiate Eq. (8) with respect to $r̂10$ and impose $Δϕ̂=0$ (note that $Δϕ̂=0$ at A, B, and C) to obtain
$3Γ111r̂102+2Q11r̂10+K1=[(ΠF1e−+F1e+)−(ΠS1e−+S1e+)r̂10]d(Δϕ̂)dr̂1|r̂1=r̂10$
(12)
For the trivial equilibrium state A with $r̂10=0$, one can further obtain
$K1=(ΠF1e−+F1e+)d(Δϕ)dr̂1|r̂1=r̂10$
(13)
Note that $K1>0$ because it is the linear stiffness of $r̂1(t)$ in Eq. (4a) when $Δϕ̂=0$. Therefore, the trivial equilibrium state A is stable. Equation (13) is indeed an important feature of the dual electrode design. By choosing a proper magnitude and phase indicator Π, the slope of the voltage-equilibrium curve at A can be positive or negative. The second feature is the existence of an asymptote, which can be found from Eq. (12) by setting $[(ΠF1e−+F1e+)−(ΠS1e−+S1e+)r̂1]=0$. As a result, the asymptote is described as
$r̂1=ΠF1e−+F1e+ΠS1e−+S1e+≡D$
(14)

Also, by choosing a proper Π, one can control the location of the asymptote. The physical meaning of the asymptote can be briefly explained as follows. From Eq. (8), it can be observed that the coefficient $S1e$ manifests a softening or hardening effect due to the converse piezoelectric effect, whereas the coefficient F1 has a forcing effect that deflects the piezoelectric micro-actuator. When S1 and F1 differ by a sign, which is the case for the reference system considered herein, $Δϕ̂$ deflects, and at the same time, hardens the shell laminate, making $r̂1$ approach D. By the time $r̂1$ reaches D, the forcing effect will be completely negated by the hardening effect, resulting in zero displacement despite a finite bias voltage. In other words, the location of the asymptote indicates the largest deflection the shell laminate can undergo given a bias voltage if the equilibrium curve is continuous.

Since there are two nontrivial equilibrium states B and C, there are three possibilities: (a) C < D < B meaning that the asymptote is between the equilibrium states B and C, (b) C < B < D meaning that the equilibrium states B and C are on the left side of the asymptote, and (c) D < C < B meaning that the equilibrium states B and C are on the right side of the asymptote. Where the asymptote appears depends on the setup of the asymmetric laminated shell and the value of Π. For the reference system and $Π=−0.48$, B and C appear on both sides of the asymptote; see Figs. 2(c) and 2(d).

To demonstrate how the location of the asymptote changes the snap-through mechanism, the voltage-response of Eq. (8) is numerically computed with $Π=−0.45, Π=−0.48$, and $Π=−0.58$. The results are plotted in Fig. 3. There are several things worth noting in Fig. 3. First, when $Π=−0.45$, Fig. 3(a) shows C < B < D. As a result, the snap-through occurs with no latching. As such, the shell laminate resumes its original state A after the bias voltage is removed. Second, when $Π=−0.48$, Fig. 3(b) shows C < D < B, and snap-through is observed. However, there is no snap-back, indicating that the shell laminate will remain at C after snap-through, and is not likely to resume A even if a reversed bias voltage is applied. Finally, when $Π=−0.58$, Fig. 3(c) shows D < C < B, and both snap-through and latching are observed.

Fig. 3
Fig. 3
Close modal

Of the three cases, Fig. 3(b) is the most special and is worthy of further investigation. Specifically, it resembles Fig. 7 in the work of Varelis and Saravanos [7], where the voltage-displacement curve of a bimorph curved beam of partial surface piezoelectric coverage subjected to bias voltage was also observed to separate into two disjoint portions before and after snap-through. Yet, their work was based on finite element simulations, and the cause of two disjoint portions was left unexplained. In this work, on the other hand, it is clearly explained that the asymptote is the cause.

Finally, the shape of the shell laminate's cross section along the x axis with different bias voltages is shown in Fig. 3(d). The circled numbers indicate the sequence of applying a bias voltage. As seen in Fig. 3(d), from points ① to ②, the deflection increases as the bias voltage increases. From points ② to ③, a light increase of the voltage causes the shell laminate to snap-through. After snap-through, the slope of the voltage-response curve changes its sign. Consequently, from points ③ to ④, an increase in the voltage instead reduces the deflection. Furthermore, the reduction is marginal because the deflection is reaching the asymptote. Finally, from points ④ to ⑤, a decrease in the voltage causes the shell laminate to deflect further downward, resulting in no snap-back.

### Stability of Equilibrium States.

To determine the stability of each equilibrium A, B, and C in Fig. 3(b), we use the Jacobian matrix in Eq. (6). Specifically, we ignore $r̂2$ and $r̂3$, set $Δϕ̂=0$, and assume the equilibria of Eq. (8) as $r̂10$. Consequently, the stability criterion becomes
$K1+2Q11r̂10+3Γ111r̂102>0$
(15)
With Eqs. (12) and (14), we can rewrite Eq. (15) as
$[(ΠS1e−+S1e+)(D−r̂10)d(Δϕ̂)dr̂1]>0$
(16)

If $[(ΠS1e−+S1e+)(D−r̂10)(d(Δϕ̂)/dr̂1)]<0$, the equilibrium state is unstable. For the reference system considered herein, $(ΠS1e−+S1e+)>0$. Therefore, Eq. (16) implies that we can use the relative position of the equilibrium state and the asymptote (i.e., $D−r̂10$) and the slope at the equilibrium state (i.e., $d(Δϕ̂)/dr̂1$) to determine the stability of A, B and C as follows. In Fig. 3(b), equilibrium B is on the right side of the asymptote and $D−r̂10<0$, but the slope at B is positive. Therefore, $[(ΠS1e−+S1e+)(D−r̂10)(d(Δϕ̂)/dr̂1)]<0$ and equilibrium state B is unstable. In contrast, equilibrium C is on the left side of the asymptote and $D−r̂10>0$. Also, the slope at C is positive; therefore, $[(ΠS1e−+S1e+)(D−r̂10)(d(Δϕ̂)/dr̂1)]>0$. Equilibrium state C is stable. The same procedure can be applied to Figs. 3(a) and 3(c) to show that B is an unstable equilibrium and C is a stable equilibrium.

## Multiple-Degrees-of-Freedom Analysis

The foregoing analysis addresses equilibrium and stability of the PZT thin-film micro-actuator when the voltage is absent. Nevertheless, snap-through from one stable equilibrium to another is achieved via the applied voltage $Δϕ̂$. Therefore, one must specify the type of applied voltage $Δϕ̂$ in studying the snap-through. Also, additional DOFs $r̂2$ and $r̂3$, which involve asymmetric shape functions (cf. Eq. (1)), must be included to study asymmetric snap-through instability. In this section, we study snap-through when $Δϕ̂$ is quasi-static, which means the change in $Δϕ̂$ is very slow and thus the inertia effect is negligible.

When the applied bias voltage is quasi-static, the inertia of the system is neglected. As a result, the equations of motion (4) become a set of algebraic equation whose solutions predict all possible equilibrium states. These equilibrium states bifurcate as $Δϕ̂$ is varied. When a voltage $Δϕ̂$ is quasi-statically incremented, the system migrates to new equilibrium positions and eventually reaches the snap-through state.

Based on Eq. (4), there are three types of equilibrium states. For type (a) equilibrium, $r̂1≠0$ and $r̂2=r̂3=0$. Since the displacement involves only symmetric transverse shape, it is denoted as symmetric equilibrium. For type (b) equilibrium, $r̂1, r̂2≠0$ and $r̂3=0$ or $r̂1, r̂3≠0$ and $r̂2=0$. Since one AT shape is involved, it is denoted as asymmetric equilibrium of the first kind. For type (c) equilibrium, $r̂1, r̂2, r̂3≠0$. Since two AT shapes are involved, it is denoted as asymmetric equilibrium of the second kind.

### Symmetric Equilibrium States.

Since $r̂2=r̂3=0$, the symmetric equilibrium state is the same as predicted by the single-DOF analysis shown in Figs. 2(d) and 3(b). There are, however, several things worth noting and they are explained as follows via Fig. 3(b).

First, equilibrium state A is stable; therefore, the branch originating from A must be stable as well. Similarly, equilibrium state B is unstable, and the branch originating from B must be unstable. These two branches meet at the limiting point S. For equilibrium state C, it is stable and the branch originating from C is also stable. In this case, snap-through appears in the following manner. Initially, the micro-actuator is at rest at stable equilibrium state A with no voltage applied. When $Δϕ̂$ is increased beyond point S, snap-through occurs and the system migrates to the C-branch. When the voltage $Δϕ̂$ is removed, the micro-actuator stays at stable equilibrium state C. Note that the system will not be able to return to the stable equilibrium state A. In other words, the snap-through is unidirectional.

For the system shown in Fig. 3(a), the snap-through phenomenon is entirely different. When $Δϕ̂$ is increased, the micro-actuator may experience a jump phenomenon; the equilibrium position remains in the A branch. As a result, when the voltage $Δϕ̂$ is removed, the micro-actuator will return to its initial equilibrium state A. A snap-through from state A to C is not possible via a quasi-statically applied $Δϕ̂$.

Yet, Fig. 3(c) shows another possible snap-through scenario. When $Δϕ̂$ is increased beyond point S, snap-through occurs and the system migrates to the C-branch. When $Δϕ̂$ is removed, the micro-actuator stays at state C and snap-through occurs. However, if $Δϕ̂$ is reversed significantly enough, snap-through occurs again and the system migrates to the A-branch. When $Δϕ̂$ is removed, the micro-actuator stays in its original state A. In other words, the snap-through is bidirectional.

### Asymmetric Equilibrium States.

Asymmetric equilibrium states of types (b) and (c) can be obtained numerically by solving Eq. (4) (with inertial effect ignored) through a procedure suggested by Krylov and coworkers [8]. The voltage-response diagrams and equilibrium states of the reference system with equal radii of curvature $R̂x=R̂y=16$ are plotted in Figs. 4(a), 5(a), and 5(c) (i.e., the left column of Figs. 4 and 5). In Figs. 4 and 5, superscripts a, b, and c refer to equilibrium states of types (a), (b), and (c), respectively. These three subplots are explained in detail as follows.

Fig. 4
Fig. 4
Close modal
Fig. 5
Fig. 5
Close modal

Figure 4(a) shows the voltage-equilibrium curve, i.e., $Δϕ̂$ versus $r̂1$. This curve is particularly meaningful for two reasons. First, all three types of equilibrium states involve a nonzero $r̂1$; therefore, all equilibrium branches will appear in this plot. Second, comparison of Fig. 4(a) with Fig. 3(b) reveals the difference of 1DOF and 3DOF analyses. In Fig. 4(a), there are three sets of voltage-equilibrium curves. The thin thickness curve refers to type (a) equilibrium (i.e., symmetric equilibrium states) and is the same as the 1DOF results shown in Fig. 3(b). When the applied voltage $Δϕ̂$ is increased, type (a) equilibrium bifurcates into type (b) and type (c) equilibria (i.e., asymmetric equilibrium states of the first and the second kind) at the same time. The bifurcation occurs at bifurcation points $AS1b$ for type (b) equilibrium (in medium thickness curves) and $AS1c$ for type (c) equilibrium (in large thickness curves). Moreover, $AS1b$ and $AS1c$ coincide because the radii of curvature $R̂x$ and $R̂y$ are identical. As the micro-actuator further deflects, the types (b) and (c) equilibrium curves “bypass” the asymptote (in a dynamic snap-through process) and bifurcate back into the type (a) equilibrium curve at $AS2b=AS2c$ (not labeled in Fig. 4(a)). When the micro-actuator deflects furthermore, the type (a) equilibrium bifurcates into type (b) and type (c) equilibria again at $AS3b=AS3c$.

One important thing to note is that Eqs. (4b) and (4c) are symmetric when $R̂x=R̂y$. If one swaps $r̂2$ and $r̂3$ in Eq. (4b), one will recover Eq. (4c). It has several profound implications. First, all type (b) equilibrium states will have equal $r̂2b$ and $r̂3b$. As explained earlier, type (b) equilibrium states include $(r̂1b,r̂2b,0)$ and $(r̂1b,0,r̂3b)$. When $(r̂1b,r̂2b,0)$ is substituted into Eq. (4b) and $(r̂1b,0,r̂3b)$ is substituted in Eq. (4c), the two resulting equations are identical. It implies that $r̂2b=r̂3b$ for a given $r̂1b$. Therefore, only type (b) equilibrium states with $(r̂1b,r̂2b,0)$ are plotted in Fig. 4(a) to avoid repetition. The second implication is that type (c) equilibrium states are degenerate; they are linear combinations of the type (b) equilibrium states within one arbitrary parameter. Let us consider a type (c) equilibrium state as
$(r̂1c,r̂2c,r̂3c)=(r̂1c,ρ̂c cos θ,ρ̂c sin θ)$
(17)

where $ρ̂c$ and θ are constant. When Eq. (17) is substituted into Eqs. (4b) and (4c), one will find that the two equations are linearly dependent with θ being arbitrary. Subsequent substitution of either of the resultant equations into Eq. (4a) to eliminate $Δϕ̂+$ and $Δϕ̂−$ will lead to an identical equation that describes a surface of revolution about the $r̂1$ axis in the $r̂1$$r̂2$$r̂3$ space. In other words, the type (b) equilibrium states are merely of principal curves on the surface and thus special cases of the type (c) equilibrium states shown in Eq. (17). Therefore, the type (b) and type (c) equilibrium curves in Fig. 4(a) are identical.

To better depict the bifurcation points, Fig. 5(a) depicts type (a) and type (b) equilibrium curves in the $r̂1$$r̂2$$r̂3$ space. The straight line is type (a) equilibrium with $r̂2=r̂3=0$. The curves indicated by $r̂2b$ are type (b) equilibrium with $r̂2≠0$ and $r̂3=0$, where the curves indicated by $r̂3b$ are type (b) equilibrium with $r̂2=0$ and $r̂3≠0$. As the micro-actuator deflects, these equilibrium states bifurcate into five branches at $AS1b$, coalesce at $AS2b$, and bifurcate again into five branches at $AS3b$.

Similarly, Fig. 5(c) depicts type (a) and type (c) equilibrium curves in the $r̂1$$r̂2$$r̂3$ space. The straight line is type (a) equilibrium with $r̂2=r̂3=0$. The curves indicated by $(r̂2c,r̂3c)$ are four representative curves of type (c) equilibrium with $r̂2≠0$ and $r̂3≠0$. As explained in Eq. (17), type (c) equilibrium is degenerate, implying that type (c) equilibrium forms a surface of revolution about the $r̂1$ axis in the $r̂1$$r̂2$$r̂3$ space. Therefore, if one rotates the four representative curves in Fig. 5(c) about the $r̂1$ axis, one will generate the entire type (c) equilibrium surface. Moreover, intersection of the type (c) equilibrium surface with $r̂2=0$ and $r̂3=0$ planes, respectively, will generate type (b) equilibrium.

For the equilibrium states described earlier, there are several issues worth discussing. First, the bifurcation behavior described earlier at the bifurcation points ASb and ASc (and thus in Figs. 4 and 5) is similar to that of a curved beam as studied by Krylov and coworkers [8]. Second, the location of bifurcation points can be determined by analyzing the behaviors near the small neighborhood around the parent branch where the bifurcation occurs. Table 2 lists the numerical values of the bifurcation points. Note that all three bifurcation points $AS1b, AS2b$, and $AS3b$ of type (b) equilibrium match with $AS1c, AS2c$, and $AS3c$ of type (c) equilibrium, respectively, as a result of equal radii of curvature. Finally, the first set of bifurcation points $AS1b$ and $AS1c$ does not coincide with the limit point S. Instead, they occur at a more negative value of $r̂1$ and at a lower value of $Δϕ̂$ than what S has. Therefore, snap-through from stable equilibrium A to C in Fig. 2(c) cannot be achieved quasi-statically via the asymmetric equilibrium states by increasing the voltage $Δϕ̂$. This type of asymmetric bifurcation is termed as” subcritical pitchfork bifurcation” in Ref. [9], where the limit point of the symmetric equilibrium branch precedes the branching on the asymmetric equilibria. The switching between the limit point and the asymmetric bifurcation point, corresponding to a supercritical pitchfork bifurcation, depends on the radii of curvature, which is similar to the shallow arches' dependence on the arch height. For the reference system, the critical radii of curvature (when they are equal) for the switching occur around $R̂x=R̂y=10$, which corresponds to a too high initial rise of the shell laminate (around $7 μ$m or seven times the PZT layer thickness); thus, it is of less interest for the current work. Moreover, a stability test using the Jacobian matrix defined in Eq. (6) shows that the asymmetric equilibrium states between $AS1b$ and $AS2b$ are unstable for both subcritical and supercritical pitchfork bifurcation, which matches the stability analysis reported in Ref. [9].

Table 2

Location of bifurcation points of different types of equilibrium branches

Radii of curvatureEquilibrium branchesBifurcation pointsLocation ($r̂1, r̂2, r̂3$)
$R̂x=R̂y=16$($r̂2b,0)$, (0, $r̂3b)$, ($r̂2c, r̂3c$)$AS1b$ and $AS1c$(−6.12, 0., 0.)
$AS2b$ and $AS2c$(−9.28, 0., 0.)
$AS3b$ and $AS3c$(−13.26, 0., 0.)
$R̂x=16$ and $R̂y=16.1$$(r̂2b,0)$$AS11b$(−6.12, 0., 0.)
$AS21b$(−9.17, 0., 0.)
$AS31b$(−13.07, 0., 0.)
$(0,r̂3b)$$AS12b$(−6.17, 0., 0.)
$AS22b$(−9.29, 0., 0.)
$AS32b$(−13.37, 0., 0.)
($r̂2c, r̂3c$)$AS11c$(−16.34, 0., −1.57)
$AS12c$(−16.34, 0., 1.57)
Radii of curvatureEquilibrium branchesBifurcation pointsLocation ($r̂1, r̂2, r̂3$)
$R̂x=R̂y=16$($r̂2b,0)$, (0, $r̂3b)$, ($r̂2c, r̂3c$)$AS1b$ and $AS1c$(−6.12, 0., 0.)
$AS2b$ and $AS2c$(−9.28, 0., 0.)
$AS3b$ and $AS3c$(−13.26, 0., 0.)
$R̂x=16$ and $R̂y=16.1$$(r̂2b,0)$$AS11b$(−6.12, 0., 0.)
$AS21b$(−9.17, 0., 0.)
$AS31b$(−13.07, 0., 0.)
$(0,r̂3b)$$AS12b$(−6.17, 0., 0.)
$AS22b$(−9.29, 0., 0.)
$AS32b$(−13.37, 0., 0.)
($r̂2c, r̂3c$)$AS11c$(−16.34, 0., −1.57)
$AS12c$(−16.34, 0., 1.57)

### Effects of Unequal Radii of Curvature.

Now let us consider the case of unequal radii of curvature. Figures 4(b), 5(b), and 5(d) show the equilibrium states for the case of $R̂x=16$ and $R̂y=16.1$. First, let us focus on Fig. 5(b), which shows type (a) and type (b) equilibrium states. Compared with their counterpart in Fig. 5(a), these equilibrium states do not change their topology significantly and seem to be a small perturbation from the case of equal radii of curvature. Nevertheless, the bifurcation points for type (b) equilibrium states start to split. As listed in Table 2, the bifurcation point for $(r̂2b,0)$ and $(0,r̂3b)$ equilibrium states is identical when $R̂x=R̂y$. They become different when $R̂x≠R̂y$. For example, the first bifurcation point $AS11b$ of $(r̂2b,0)$ equilibrium states occurs at $r̂1b=−6.12$, while the first bifurcation point $AS12b$ of $(0,r̂3b)$ equilibrium states occurs at $r̂1b=−6.17$. Similarly, the second and the third sets of bifurcation points are also split as shown in Table 2.

The split of bifurcation points is a natural consequence of unequal radii of curvature. When $R̂x≠R̂y$, the symmetry in Eqs. (4b) and (4c) is destroyed. As a result, equilibrium states with an arbitrary parameter θ shown in Eq. (17) are no longer possible, and type (c) equilibrium states in the form of an axisymmetric surface of revolution about the $r̂1$ axis disappear. This can be easily seen in Fig. 5(d), where type (c) equilibrium states are shown (in curves indicated by $(r̂2c,r̂3c)$). There are no type (c) equilibrium states between bifurcation points $AS12b$ and $AS22b$ (not shown). Type (c) equilibrium states recede and do not appear until $r̂1<−16.34$.

Figure 4(b) summarizes how the equilibrium position $r̂1$ evolves and its corresponding applied voltage $Δϕ̂$. Again, type (a) equilibrium curves do not change their topology, e.g., there is an asymptote. It seems that type (b) equilibrium curves also provide a passage for the equilibrium states to bypass the asymptote. However, since this branch of equilibrium states is unstable according to the stability analysis stated earlier, the shell laminate cannot follow this branch and bypass the asymptote under quasi-statically applied bias voltage. Finally, type (c) equilibrium states recede significantly and bifurcate from type (b) equilibrium.

## Discussion

Thus far, the preceding analyses have revealed three different snap-through mechanisms that can occur to the piezoelectric shell laminate. The first is the unidirectional snap-through, illustrated in Fig. 3(b); the second is the bidirectional snap-through, illustrated in Fig. 3(c); and the third is asymmetric snap-through, illustrated in Figs. 4 and 5. Next, the possibility of each type of snap-through that can lead to the experimental findings in Fig. 2 of Part 1 is discussed. To this end, the natural frequency bifurcation from $1$ to $5$ is reviewed herein. First, when the DC bias voltage exceeded 2.5 V, the natural frequency of the micro-actuator bifurcated into two branches, indicated by $3$. Second, as the bias voltage was reversed to be lower than −3.5 V, the natural frequency bifurcated into a new branch, indicated by $4$, and remained at this branch. The natural frequency bifurcation from $1$ to $3$ can be understood as limit-point snap-through, where the micro-actuator migrated from its trivial equilibrium state to a new one, e.g., from A to C in Fig. 3(b) or Fig. 3(c). The natural frequency bifurcation from $3$ to $4$, however, cannot be explained by the bidirectional snap-through shown in Fig. 3(c). For the bidirectional snap-through, the equilibrium state will resume its trivial state after the reversed voltage exceeds a threshold, i.e., from C back to A, via snap-back. However, that is not what happened in Fig. 2 of Part 1.

The natural frequency bifurcation from $3$ to $4$ cannot be understood as asymmetric snap-through either. According to Ref. [9] and the stability analysis conducted earlier, the asymmetric equilibrium states associated with the bifurcation points AS1 and AS2 (regardless of type (b) or (c)) in Fig. 5 are unstable. In other words, they do not provide any new stable equilibrium states. Instead, they merely participate in a dynamic snap-through process where the shell laminate is in transition from A to C. After the bias voltage is removed, the shell laminate stays at C, the only stable equilibrium state nearby. Furthermore, the transition is also bidirectional, i.e., the shell laminate can migrate from C back to A when the bias voltage is reversed to exceed a threshold. Yet, that is not what happened in Fig. 2 of Part 1 either.

By elimination, the remaining possibility is the unidirectional snap-through. The unidirectional snap-through is a tenable explanation for two reasons. First, it does not involve any latching or snap-back, which can be used to explain why the natural frequency bifurcation from $3$ did not go back to $1$ after the bias voltage was reversed. Second, with a reversed bias voltage, it causes the shell laminate to deflect further (cf. ⑤ in Fig. 3) and may lead to a new equilibrium state, which can be used to explain why the natural frequency bifurcated into $4$, a different equilibrium state from $1$.

Although the unidirectional snap-through provides a possible experimental explanation, it requires more thorough experimental and numerical verification. For example, experimental measurement on the slope of the voltage-deflection response at $3$ will provide direct evidence. Furthermore, numerical analyses that examine dynamics effects, such as inertia effects, initial conditions, and damping, e.g., see Ref. [10], will also be helpful. Nonetheless, each of the above can constitute an independent topic and thus is out of the scope of this work.

## Conclusions

In this paper, we reported simulation findings of a PZT thin-film micro-actuator, which was modeled as an asymmetrically laminated piezoelectric shell. Moreover, the model has three out-of-plane DOFs: one symmetric DOF $r̂1$ and two asymmetric DOFs $r̂2$ and $r̂3$. The model also has a sufficient number of in-plane DOFs to ensure convergence. With the presentation above, we reach the following conclusions:

1. (1)

A convergence study shows that the number of in-plane shape functions retained is of primary importance for three reasons.

• (a)

It will affect the number of equilibrium states for a given set of radii of curvature.

• (b)

It will affect the prediction of the snap-through voltage.

• (c)

It will affect the prediction of the snap-through mechanisms.

2. (2)

When the micro-actuator is subjected to quasi-static bias voltage, it presents three types of equilibrium states: symmetric equilibrium, asymmetric equilibrium of the first kind, and asymmetric equilibrium of the second kind. The symmetric equilibrium of the first kind only involves $r̂1$. The asymmetric equilibrium of the first kind involves $r̂1$ and an asymmetric DOF $r̂2$ or $r̂3$, while the asymmetric equilibrium of the second kind involves all three DOFs $r̂1, r̂2$, and $r̂3$. All these equilibrium states are functions of the bias voltage.

• (a)

For the symmetric equilibrium, its voltage-equilibrium curve is influenced by the distribution of bias voltages, and consequently, can have two different snap-through mechanisms. The first is the unidirectional snap-through, where the voltage-equilibrium curve is discontinuous due to the presence of an asymptote. The unidirectional snap-through has only one limit point that allows for snap-through. The second is the bidirectional snap-through, which has two limit points that allow for snap-through and snap-back. Depending on the bias voltage, the micro-actuator may have one or three symmetric equilibrium states. In the case of one equilibrium state, it is always stable. In the case of three equilibrium states, two are stable and one is unstable. When the bias voltage is increased to reach the limit point, the micro-actuator will experience snap-through. When the bias voltage is reduced, the micro-actuator, however, may or may not stay in the snapped position depending on the location of the asymptote.

• (b)

Voltage-equilibrium curves of asymmetric equilibrium of the first and second kind bifurcate from those of symmetric equilibrium. Bifurcation of asymmetric equilibrium of the first kind is very similar to those obtained in one-dimensional bistable structures (e.g., curved beams and shallow arches).

• (c)

Asymmetric equilibrium states highly depend on the symmetry of radii of curvature. When the radii of curvature are equal, asymmetric equilibrium of the second type becomes degenerate, and it does not have a voltage-equilibrium curve. Instead, it has a voltage-equilibrium surface, which can be generated by revolving the voltage-equilibrium curve of the asymmetric equilibrium of the first kind.

## Acknowledgment

Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

### Appendix

The coefficients in Eqs. (4) and (5) can be found as follows. Linear stiffness
$K1=K1111; K2=K2121; K3=K1212$
(A1)
$Q11=Q111111nlh2; Q22=Q112121nlh2; Q33=Q111212nlh2Q21=Q212111nlh2; Q31=Q121211nlh2$
(A2)
Cubic stiffness
$Γ111=Γ11111111nlh22; Γ122=Γ11112121nlh22; Γ133=Γ11111212nlh22Γ222=Γ21212121nlh22; Γ233=Γ21211212nlh22; Γ211=Γ21211111nlh22Γ333=Γ12121212nlh22; Γ322=Γ12122121nlh22; Γ311=Γ12121111nlh22$
(A3)
Electric hardening or softening stiffness (depending on the signs of $Δϕ̂−$ and $Δϕ̂+$)
$S1e−=(S1111nle−+∑m,n=16S1111mnnle)2K/e31S2e−=(S2121nle−+∑m,n=16S2121mnnle)2K/e31S3e−=(S1212nle−+∑m,n=16S1212mnnle)2K/e31$
(A4)

To obtain $S1e+, S2e+$, and $S3e+$, one simply needs to substitute the superscripts – for +.

Electric forcing coefficients
$F1e+=(−F11e++∑k,l=16F11kle−B11e+B11e+)2K/(h2e31);F1e−=(−F11e−−∑k,l=16F11kle+B11e)2K/(h2e31)$
(A5)

Definition of the coefficients on the right-hand side of Eqs. (A3)(A5) is provided in the Appendix of Part 1.

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## Funding Data

• National Science Foundation (Grant No. CBET-1159623).