Finite element modeling and analysis of flexoelectric plates using gradient electromechanical theory

Abstract

This work presents the development of a two-way coupled flexoelectric plate theory starting from a 3D gradient electromechanical theory. The gradient electromechanical theory considers three mechanical length scale parameters and two electric length scale parameters to account for both mechanical and electrical size effects. Variational formulation is used to derive the plate governing equations and boundary conditions considering Kirchhoff’s assumptions. A computationally efficient $C^2$ continuous non-conforming finite element is developed to solve the resulting plate equations. To assess the accuracy of the non-conforming finite element framework, the results are compared with Navier-type analytical solution for a simply supported flexoelectric plate. The finite element framework is also validated with experimental results in the existing literature for a passive micro-plate. The results show excellent agreement with both analytical and experimental results. Furthermore, computational efficiency of the non-conforming element is compared with the standard conforming element, which contains greater degrees of freedom and continuity across all elemental edges. It was observed that the non-conforming element is almost twice as fast as the conforming element without a significant loss of accuracy. The 2D finite element formulation is subsequently used to analyze the size-dependent response of flexoelectric composite plates operating in both sensor and actuator modes. Various parametric studies are performed to analyze the effect of boundary conditions, length scale parameters, size of the plate, flexoelectric layer thickness ratio, etc., on the response of flexoelectric plate-type sensors and actuators. It is found that the effective electromechanical coupling increases in a flexoelectric plate at microscale (due to the size effects), and it is higher than standard piezoelectric materials for plate thickness $h \le 8\,{{\upmu }}$m.

Appendices

Appendix A

Components of constitutive matrices: The components of $\mathbf{\overline{H}}$, $\mathbf{\overline{G}}$, $\mathbf{\overline{D}}$, $\mathbf{\overline{E}}$, $\mathbf{\overline{B}}$ and $\mathbf{\overline{K}}$ matrices defined in equations (51)–(54) are given as follows:

$$\begin{aligned} \mathbf{\overline{H}}&= \left[ \begin{array}{cccccccccccc} H_{111}&{}\quad H_{112}&{}\quad H_{113}&{}\quad H_{221}&{}\quad H_{222}&{}\quad H_{223}&{}\quad H_{121}&{}\quad H_{122}&{}\quad H_{123}\\ \end{array}\right] ^{\textrm{T}}, \end{aligned}$$

(98)

$$\begin{aligned} \mathbf{\overline{G}}&= \left[ \begin{array}{cccccccccccc} G_{111}&{}\quad G_{112}&{}\quad G_{113}&{}\quad G_{221}&{}\quad G_{222}&{}\quad G_{223}&{}\quad G_{121}&{}\quad G_{122}&{}\quad G_{123}\\ \end{array}\right] ^{\textrm{T}}, \end{aligned}$$

(99)

$$\begin{aligned} \mathbf{\overline{D}}&= \left[ \begin{array}{ccc} D_1&{}\quad D_2&{}\quad D_3\\ \end{array}\right] ^{\textrm{T}}, \quad \mathbf{\overline{E}}= \left[ \begin{array}{ccc} E_1&{}\quad E_2&{}\quad E_3\\ \end{array}\right] ^{\textrm{T}}, \end{aligned}$$

(100)

$$\begin{aligned} \mathbf{\overline{B}}&= \left[ \begin{array}{ccccccccc} B_{11}&{}\quad B_{22}&{}\quad B_{33}&{}\quad B_{12}&{}\quad B_{13}&{}\quad B_{23}\\ \end{array}\right] ^{\textrm{T}}, \end{aligned}$$

(101)

$$\begin{aligned} \mathbf{\overline{K}}&= \left[ \begin{array}{ccccccccc} K_{11}&{}\quad K_{22}&{}\quad K_{33}&{}\quad K_{12}&{}\quad K_{13}&{}\quad K_{23}\\ \end{array}\right] ^{\textrm{T}}. \end{aligned}$$

(102)

The components of material matrices $\mathbf{\overline{Q}^c}$, $\mathbf{\overline{Q}^g}$, $\varvec{\overline{\epsilon }}$, $\mathbf{\overline{k}}$, $\mathbf{\overline{h}}$ and $\mathbf{\overline{f}}$ defined in Eqs. (51)–(54) are given by:

$$\begin{aligned} \mathbf{\overline{Q}^c}= \left[ \begin{array}{cccc} \overline{Q}^c_{11}&{}\quad \overline{Q}^c_{12}&{}\quad 0\\ \overline{Q}^c_{21}&{}\quad \overline{Q}^c_{22}&{}\quad 0\\ 0&{}\quad 0&{}\quad \overline{Q}^c_{33}\\ \end{array}\right] , \end{aligned}$$

(103)

where $\overline{Q}_{ij}^c$, $i,j=1,2,3$ are the components of stiffness matrix and can be obtained in terms of Young’s modulus c and Poison’s ratio v using the plane stress assumptions [42].

$$\begin{aligned} \mathbf{\overline{h}}&= \left[ \begin{array}{ccccccccc} h_1+2h_2&{}\quad h_1&{}\quad h_1&{}\quad 0&{}\quad 0&{}\quad 0\\ h_1&{}\quad h_1+2h_2&{}\quad h_1&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 2h_2&{}\quad 0&{}\quad 0\\ \end{array}\right] , \end{aligned}$$

(104)

$$\begin{aligned} \mathbf{\overline{Q}^g}&= \left[ \begin{array}{cccccccccccc} \overline{Q}^g_{11}&{}\quad 0&{}\quad 0&{}\quad \overline{Q}^g_{14}&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad \overline{Q}^g_{18}&{}\quad 0\\ 0 &{}\quad \overline{Q}^g_{22}&{}\quad 0&{}\quad 0&{}\quad \overline{Q}^g_{25}&{}\quad 0&{}\quad \overline{Q}^g_{27}&{}\quad 0&{}\quad 0\\ 0 &{}\quad 0&{}\quad \overline{Q}^g_{33}&{}\quad 0&{}\quad 0&{}\quad \overline{Q}^g_{36}&{}\quad 0&{}\quad 0&{}\quad 0\\ \overline{Q}^g_{41}&{}\quad 0&{}\quad 0&{}\quad \overline{Q}^g_{44}&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad \overline{Q}^g_{48}&{}\quad 0\\ 0 &{}\quad \overline{Q}^g_{52}&{}\quad 0&{}\quad 0&{}\quad \overline{Q}^g_{55}&{}\quad 0&{}\quad \overline{Q}^g_{57}&{}\quad 0&{}\quad 0\\ 0 &{}\quad 0&{}\quad \overline{Q}^g_{63}&{}\quad 0&{}\quad 0&{}\quad \overline{Q}^g_{66}&{}\quad 0&{}\quad 0&{}\quad 0\\ 0 &{}\quad \overline{Q}^g_{72}&{}\quad 0&{}\quad 0&{}\quad \overline{Q}^g_{75}&{}\quad 0&{}\quad \overline{Q}^g_{77}&{}\quad 0&{}\quad 0\\ \overline{Q}^g_{81}&{}\quad 0&{}\quad 0&{}\quad \overline{Q}^g_{84}&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad \overline{Q}^g_{88}&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad \overline{Q}^g_{99}\\ \end{array}\right] , \end{aligned}$$

(105)

where $\overline{Q}^g_{11}=2(a_1+a_2+a_3+a_4+a_5)$, $\overline{Q}^g_{14}=2a_1+a_2$, $\overline{Q}^g_{18}=a_3+\frac{1}{2}a_2$, $\overline{Q}^g_{22}=2a_1+2a_4$, $\overline{Q}^g_{25}=\overline{Q}^g_{14}$, $\overline{Q}^g_{27}=\frac{1}{2}a_2+a_5$, $\overline{Q}^g_{33}=\overline{Q}^g_{22}$, $\overline{Q}^g_{36}=2a_1$, $\overline{Q}^g_{41}=\overline{Q}^g_{14}$, $\overline{Q}^g_{44}=\overline{Q}^g_{22}$, $\overline{Q}^g_{48}=\overline{Q}^g_{27}$, $\overline{Q}^g_{52}=\overline{Q}^g_{25}$, $\overline{Q}^g_{55}=\overline{Q}^g_{11}$, $\overline{Q}^g_{57}=\overline{Q}^g_{18}$, $\overline{Q}^g_{63}=\overline{Q}^g_{36}$, $\overline{Q}^g_{66}=\overline{Q}^g_{33}$, $\overline{Q}^g_{72}=\overline{Q}^g_{27}$, $\overline{Q}^g_{75}=\overline{Q}^g_{57}$, $\overline{Q}^g_{77}=a_3+2a_4+a_5$, $\overline{Q}^g_{81}=\overline{Q}^g_{18}$, $\overline{Q}^g_{84}=\overline{Q}^g_{48}$, $\overline{Q}^g_{88}=\overline{Q}^g_{77}$ and $\overline{Q}^g_{99}=2a_4$.

$$\begin{aligned} \mathbf{\overline{f}}&= \left[ \begin{array}{cccccccccccc} 2f_1+f_2&{}\quad 0&{}\quad 0\\ 0 &{}\quad f_2&{}\quad 0\\ 0 &{}\quad 0\quad f_2\\ f_2&{}\quad 0&{}\quad 0\\ 0 &{}\quad 2f_1+f_2&{}\quad 0\\ 0 &{}\quad 0&{}\quad f_2\\ 0 &{}\quad f_1&{}\quad 0\\ f_1&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 0\\ \end{array}\right] , \end{aligned}$$

(106)

$$\begin{aligned} {\varvec{\overline{\epsilon }}}&= \left[ \begin{array}{ccc} \epsilon _1&{}\quad 0&{}\quad 0\\ 0&{}\quad \epsilon _1&{}\quad 0\\ 0&{}\quad 0&{}\quad \epsilon _1\\ \end{array}\right] , \end{aligned}$$

(107)

and

$$\begin{aligned} \mathbf{\overline{k}}= \left[ \begin{array}{ccccccccc} k_1+2k_2&{}\quad k_1&{}\quad k_1&{}\quad 0&{}\quad 0&{}\quad 0\\ k_1&{}\quad k_1+2k_2&{}\quad k_1&{}\quad 0&{}\quad 0&{}\quad 0\\ k_1&{}\quad k_1&{}\quad k_1+2k_2&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 2k_2&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 2k_2&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 0&{}\quad 2k_2\\ \end{array}\right] . \end{aligned}$$

(108)

Appendix B

The matrices $\textbf{D}^\textbf{SS}$, $\textbf{D}^\textbf{SK}$, $\textbf{D}^\textbf{GG}$, $\textbf{D}^\textbf{GE}$, $\textbf{D}^\textbf{EE}$, $\textbf{D}^\textbf{EG}$, $\textbf{D}^\textbf{KK}$ and $\textbf{D}^\textbf{KS}$ defined in Eq. (84) are given by:

$$\begin{aligned} \textbf{D}^\textbf{SS}&=\int _{\alpha _3}{\mathbf{(H^S)}}^{\textrm{T}}{} \mathbf{\overline{Q}^c}{\mathbf{H^S}}\textrm{d}\alpha _3, \end{aligned}$$

(109)

$$\begin{aligned} \textbf{D}^\textbf{SK}&=\int _{\alpha _3}{\mathbf{(H^S)}}^{\textrm{T}}{} \mathbf{\overline{h}}{\mathbf{H^K}}\textrm{d}\alpha _3, \end{aligned}$$

(110)

$$\begin{aligned} \textbf{D}^\textbf{GG}&=\int _{\alpha _3}{\mathbf{(H^G)}}^{\textrm{T}}{} \mathbf{\overline{Q}^g}{\mathbf{H^G}}\textrm{d}\alpha _3, \end{aligned}$$

(111)

$$\begin{aligned} \textbf{D}^\textbf{GE}&=\int _{\alpha _3}{\mathbf{(H^G)}}^{\textrm{T}}{} \mathbf{\overline{f}}{\mathbf{H^E}}\textrm{d}\alpha _3, \end{aligned}$$

(112)

$$\begin{aligned} \textbf{D}^\textbf{EE}&=\int _{\alpha _3}{\mathbf{(H^E)}}^{\textrm{T}}{\varvec{\overline{\epsilon }} }{\mathbf{H^E}}\textrm{d}\alpha _3, \end{aligned}$$

(113)

$$\begin{aligned} \textbf{D}^\textbf{EG}&=\int _{\alpha _3}{\mathbf{(H^E)}}^{\textrm{T}}{} \mathbf{\overline{f} }^{\textrm{T}}{\mathbf{H^G}}\textrm{d}\alpha _3, \end{aligned}$$

(114)

$$\begin{aligned} \textbf{D}^\textbf{KK}&=\int _{\alpha _3}{\mathbf{(H^K)}}^{\textrm{T}}{} \mathbf{\overline{k} }{\mathbf{H^K}}\textrm{d}\alpha _3. \end{aligned}$$

(115)

$$\begin{aligned} \textbf{D}^\textbf{KS}&=\int _{\alpha _3}{\mathbf{(H^K)}}^{\textrm{T}}{} \mathbf{\overline{h} }^{\textrm{T}}{\mathbf{H^S}}\textrm{d}\alpha _3, \end{aligned}$$

(116)

The finite element matrices $\textbf{K}^\textbf{SS}$, $\textbf{K}^\textbf{SK}$, $\textbf{K}^\textbf{GG}$, $\textbf{K}^\textbf{GE}$, $\textbf{K}^\textbf{EE}$, $\textbf{K}^\textbf{EG}$, $\textbf{K}^\textbf{KK}$ and $\textbf{K}^\textbf{KS}$ defined in Eq. (85) are given by:

$$\begin{aligned} \textbf{K}^\textbf{SS}&=\int _{\alpha ^e_1}\int _{\alpha ^e_2}{\mathbf{(B^S)}}^{\textrm{T}}{} \mathbf{D^{SS}}{\mathbf{B^S}}\textrm{d}\alpha ^e_2\textrm{d}\alpha ^e_1, \end{aligned}$$

(117)

$$\begin{aligned} \textbf{K}^\textbf{SK}&=\int _{\alpha ^e_1}\int _{\alpha ^e_2}{\mathbf{(B^S)}}^{\textrm{T}}{} \mathbf{D^{SK}}{\mathbf{B^K}}\textrm{d}\alpha ^e_2\textrm{d}\alpha ^e_1, \end{aligned}$$

(118)

$$\begin{aligned} \textbf{K}^\textbf{GG}&=\int _{\alpha ^e_1}\int _{\alpha ^e_2}{\mathbf{(B^G)}}^{\textrm{T}}{} \mathbf{D^{GG}}{\mathbf{B^G}}\textrm{d}\alpha ^e_2\textrm{d}\alpha ^e_1, \end{aligned}$$

(119)

$$\begin{aligned} \textbf{K}^\textbf{GE}&=\int _{\alpha ^e_1}\int _{\alpha ^e_2}{\mathbf{(B^G)}}^{\textrm{T}}{} \mathbf{D^{GE}}{\mathbf{B^E}}\textrm{d}\alpha ^e_2\textrm{d}\alpha ^e_1, \end{aligned}$$

(120)

$$\begin{aligned} \textbf{K}^\textbf{EE}&=\int _{\alpha ^e_1}\int _{\alpha ^e_2}{\mathbf{(B^E)}}^{\textrm{T}}{} \mathbf{D^{EE}}{\mathbf{B^E}}\textrm{d}\alpha ^e_2\textrm{d}\alpha ^e_1, \end{aligned}$$

(121)

$$\begin{aligned} \textbf{K}^\textbf{EG}&=\int _{\alpha ^e_1}\int _{\alpha ^e_2}{\mathbf{(B^E)}}^{\textrm{T}}{} \mathbf{D^{EG}}{\mathbf{B^G}}\textrm{d}\alpha ^e_2\textrm{d}\alpha ^e_1, \end{aligned}$$

(122)

$$\begin{aligned} \textbf{K}^\textbf{KK}&=\int _{\alpha ^e_1}\int _{\alpha ^e_2}{\mathbf{(B^K)}}^{\textrm{T}}{} \mathbf{D^{KK}}{\mathbf{B^K}}\textrm{d}\alpha ^e_2\textrm{d}\alpha ^e_1, \end{aligned}$$

(123)

$$\begin{aligned} \textbf{K}^\textbf{KS}&=\int _{\alpha ^e_1}\int _{\alpha ^e_2}{\mathbf{(B^K)}}^{\textrm{T}}{} \mathbf{D^{KS}}{\mathbf{B^S}}\textrm{d}\alpha ^e_2\textrm{d}\alpha ^e_1, \end{aligned}$$

(124)

Appendix C

Analytical solution for simply supported flexoelectric composite plates: The derived governing equations for the classical flexoelectric plate are solved analytically for simply supported boundary conditions. The governing differential equations (62)–(66) are obtained in the explicit form using Eqs. (58)–(61). The Navier solution is used to express the primary variables in terms of series solution as follows:

$$\begin{aligned} u_0&= \sum _{m=1}^\infty \sum _{n=1}^\infty U_{mn}\textrm{cos}(p\alpha _1) \textrm{sin}(l\alpha _2) , \end{aligned}$$

(125)

$$\begin{aligned} v_0&= \sum _{m=1}^\infty \sum _{n=1}^\infty V_{mn}\textrm{sin}(p\alpha _1) \textrm{cos}(l\alpha _2), \end{aligned}$$

(126)

$$\begin{aligned} w_0&= \sum _{m=1}^\infty \sum _{n=1}^\infty W_{mn}\textrm{sin} (p\alpha _1) \textrm{sin}(l\alpha _2), \end{aligned}$$

(127)

$$\begin{aligned} \phi _1&=\sum _{m=1}^\infty \sum _{n=1}^\infty { \Phi }_{mn}\textrm{sin}(p\alpha _1) \textrm{sin}(l\alpha _2), \end{aligned}$$

(128)

$$\begin{aligned} \phi _0&=\sum _{m=1}^\infty \sum _{n=1}^\infty { \gamma }_{mn}\textrm{sin}(p\alpha _1) \textrm{sin}(l\alpha _2), \end{aligned}$$

(129)

where $p=\pi m /a$ and $l=\pi n /b$. The coefficients $U_{mn},\, V_{mn},\, W_{mn},$ $\Phi _{mn}$ and $\gamma _{mn}$ are unknowns to be solved using the Navier solution technique.

In addition, transverse load $q(\alpha _1,\alpha _2)$ is expressed in terms of Fourier series as

$$\begin{aligned} q=\sum _{m=1}^\infty \sum _{n=1}^\infty q_{mn}\,\textrm{sin}(p\alpha _1)\textrm{sin}(l\alpha _2), \end{aligned}$$

(130)

where

$$\begin{aligned} q_{mn} = \frac{4}{ab}\int _{0}^b\int _{0}^a q(\alpha _1,\alpha _2) \,\textrm{sin}(p\alpha _1)\,\textrm{sin}(l\alpha _2) \, \textrm{d}\alpha _1\textrm{d}\alpha _2, \end{aligned}$$

(131)

The series solutions (125)–(130) are substituted into the explicit form of governing equations, resulting in algebraic equations of the form:

$$\begin{aligned} \mathbf{R^g}{} \mathbf{U^g} \, = \, \mathbf{F^g}, \end{aligned}$$

(132)

where $\mathbf{U^g}$ is the vector of unknowns to be solved, which is given by

$$\begin{aligned} \mathbf{U^g} = \begin{bmatrix} U_{mn}&V_{mn}&W_{mn}&\Phi _{mn}&\gamma _{mn} \end{bmatrix}^\textrm{T}, \end{aligned}$$

$\mathbf{F^g}$ is the resultant force matrix, and $\mathbf{R^g}$ is the $5 \times 5$ resultant stiffness matrix. Equation (132) is solved to obtain the displacements and electrostatic potential in terms of Navier coefficients. This general formulation is used to analyze the flexoelectric plate, both in actuator and sensor modes.