A numerical scheme for geometrically exact flexoelectric microbeams using the weak form quadrature element method

Abstract

In this article, a geometrically exact flexoelectric beam model and its weak form quadrature element formulation are established. The coupling of strain gradient and electric field in flexoelectricity involving size effects is taken into account and the resulting C1 continuity requirements are satisfied. The formulation owns the advantage of high-order approximation to circumvent locking problems after discretization. Several numerical examples originating from benchmark nonlinear beam problems are studied to validate the feasibility of the formulation. Compared to existing flexoelectric beam models, the use of geometrically exact beam theory in the present model makes an innovative contribution to analyzing flexoelectric microbeams undergoing large displacements and rotations. This study is a step forward to better understandings of flexoelectric effects on microbeams under large deflections, while also providing a viable approach to predict their nonlinear behaviors in such conditions.

Appendices

Appendix A Constitutive and element tangent stiffness matrix

1.1 Constitutive matrix

By denoting the initial shear-axial strain vector and bending-torsional strain vector as $\varvec{\gamma }_0$ and $\varvec{\kappa }_0$, the stress resultant ${\textbf{n}}$, ${\textbf{m}}$ and the double stress resultant ${\tilde{\textbf{n}}}$, ${\tilde{\textbf{m}}}$ in Eq. (31) are given by

$$\begin{aligned} \begin{array}{l} {{\textbf{n}}} = \displaystyle \int _A {{\textbf{s}}}{\text { d}}A = {{\textbf{D}}_1}\left( {\varvec{\gamma }}-{\varvec{\gamma }_0}\right) ,\\ {{\textbf{m}}} = \displaystyle \int _A {\left( {\hat{\varvec{\lambda }}{{\textbf{s}}} +\hat{{\textbf{e}}}_\alpha {{\tilde{{\textbf{s}}}}_\alpha }} \right) }{\text { d}}A = {{\textbf{D}}_2}\left( {\varvec{\kappa }} -{\varvec{\kappa }_0}\right) + {{\textbf{D}}_3}\left( {\varvec{\gamma }'}-{\varvec{\gamma }'_0}\right) , \\ {\tilde{{\textbf{n}}}} = \displaystyle \int _A {{{{\tilde{{\textbf{s}}}}}_3}{\text { d}}A} ={{\textbf{D}}_3^T}\left( {\varvec{\kappa }}-{\varvec{\kappa }_0} \right) + {{\textbf{D}}_4}\left( {\varvec{\gamma }'}-{\varvec{\gamma }'_0}\right) , \\ {\tilde{{\textbf{m}}}} = \displaystyle \int _A {\hat{\varvec{\lambda }}{{\tilde{{\textbf{s}}}}_3}{\text { d}}A} = {{\textbf{D}}_5}\left( {\varvec{\kappa }'}-{\varvec{\kappa }'_0}\right) , \end{array} \end{aligned}$$

(A1)

with

$$\begin{aligned} {\textbf{D}}_1= & {} \text { diag}\begin{bmatrix} k_{s1}GA&\,\, k_{s2}GA&\,\, EA \end{bmatrix} \end{aligned}$$

(A2)

$$\begin{aligned} {\textbf{D}}_2= & {} \text { diag}\begin{bmatrix} EI_1+l^2EA+\dfrac{f_1^2A}{\epsilon }&\,\, EI_2+l^2EA +\dfrac{f_1^2A}{\epsilon }&\,\, GJ_{sv}+2l^2GA \end{bmatrix} \end{aligned}$$

(A3)

$$\begin{aligned} {\textbf{D}}_3= & {} \frac{1}{\epsilon }\left[ \begin{array}{ccc} 0&{}\quad f_1 f_2A &{}\quad 0 \\ -f_1 f_2A&{}\quad 0&{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 \\ \end{array} \right] \end{aligned}$$

(A4)

$$\begin{aligned} {\textbf{D}}_4= & {} \text { diag}\begin{bmatrix} l^2k_{s1}GA+\dfrac{f_2^2A}{\epsilon }&\,\, l^2k_{s2}GA +\dfrac{f_2^2A}{\epsilon }&\,\, l^2EA \end{bmatrix} \end{aligned}$$

(A5)

$$\begin{aligned} {\textbf{D}}_5= & {} \text { diag}\begin{bmatrix} l^2EI_1&\,\, l^2EI_2&\,\, l^2GJ_{sv}+\dfrac{f_2^2 J_{sv}}{\epsilon } \end{bmatrix} \end{aligned}$$

(A6)

where $k_{s1}$ and $k_{s2}$ are the shear correction factors; $I_1$ and $I_2$ are the corresponding principal bending inertial moments of the cross section; and $J_{sv}$ is the torsional inertial moment.

1.2 Element tangent stiffness matrix

According to the definition in Eq. (58), the element tangent stiffness matrix is derived as

$$\begin{aligned} {{\textbf{K}}}^{(e)} = \sum \limits _{i = 1}^n {{\mu _i}} {w_i}\left( \begin{aligned}{} & {} {{\textbf{A}}}_i^T{\varvec{\Gamma }}_i^T{{{\textbf{D}}}_{\text { I}}}{{\varvec{\Gamma }}_i}{{{\textbf{A}}}_i} + {{\textbf{A}}}_i^T{\varvec{\Gamma }}_i^T{{{\textbf{D}}}_{{\text { II}}}}{\overline{\varvec{\Gamma }} _i}{\overline{{\textbf{A}}} _i} + \overline{{\textbf{A}}} _i^T\overline{\varvec{\Gamma }} _i^T{{{\textbf{D}}}_{\text { II}}^T}{{\varvec{\Gamma }}_i}{{{\textbf{A}}}_i} \\ {}{} & {} \quad + \overline{{\textbf{A}}} _i^T\overline{\varvec{\Gamma }} _i^T{{{\textbf{D}}}_{{\text { III}}}}{\overline{\varvec{\Gamma }} _i}{\overline{{\textbf{A}}} _i} + {{\textbf{A}}}_i^T{{\mathbf {\Xi }}_i}{{{\textbf{A}}}_i} + \overline{{\textbf{A}}} _i^T{\overline{\mathbf {\Xi }} _i}{\overline{{\textbf{A}}} _i} \\ \end{aligned} \right) . \end{aligned}$$

(A7)

where ${{\textbf{D}}}_{\text { I}}$, ${{\textbf{D}}}_{\text { II}}$ and ${{\textbf{D}}}_{\text { III}}$ are defined as

$$\begin{aligned} \begin{aligned} {\textbf{D}}_{\text { I}} = \left[ {\begin{array}{*{20}{c}} {\textbf{D}}_1&{}\quad {\textbf{0}} \\ {\textbf{0}}&{}\quad {\textbf{D}}_2 \end{array}} \right] , \\ {{{\textbf{D}}}_{{\text { II}}}} = \left[ {\begin{array}{*{20}{c}} {\textbf{0}}&{}\quad {\textbf{0}} \\ {\textbf{D}}_3&{}\quad {\textbf{0}} \end{array}} \right] , \\ {{{\textbf{D}}}_{{\text { III}}}} = \left[ {\begin{array}{*{20}{c}} {\textbf{D}}_4&{}\quad {\textbf{0}} \\ {\textbf{0}}&{}\quad {\textbf{D}}_5 \end{array}} \right] . \\ \end{aligned} \end{aligned}$$

(A8)

${{\mathbf {\Xi }}_i}$ and ${\overline{\mathbf {\Xi }} _i}$ are given by

$$\begin{aligned} {{\mathbf {\Xi }}_i} = \left[ {\begin{array}{*{20}{c}} {\textbf{0}}&{}\quad {\textbf{0}}&{}\quad { - {{\mathbf {\Lambda }}_i}{{{\hat{{\textbf{n}}}}}_i}{\mathbf {\Lambda }}_i^T}, \\ {\textbf{0}}&{}\quad {\textbf{0}}&{}\quad { - {{\mathbf {\Lambda }}_i}{{{\hat{{\textbf{m}}}}}_i}{\mathbf {\Lambda }}_i^T}, \\ {{{\mathbf {\Lambda }}_i}{{{\hat{{\textbf{n}}}}}_i}{\mathbf {\Lambda }}_i^T}&{}\quad {\textbf{0}}&{}\quad {{\hat{{\textbf{r}}}}_i^\prime {{\mathbf {\Lambda }}_i}{{{\hat{{\textbf{n}}}}}_i}{\mathbf {\Lambda }}_i^T} \end{array}} \right] . \end{aligned}$$

(A9)

$$\begin{aligned} {{{\bar{\mathbf {\Xi }} }}_i} = \left[ {\begin{array}{*{20}{c}} {\textbf{0}}&{}\quad {\textbf{0}}&{}\quad {\textbf{0}}&{}\quad {\textbf{0}}&{}\quad { - {{\mathbf {\Lambda }}_i}{{{\hat{{\tilde{\textbf{n}}}}}}_i}{\mathbf {\Lambda }}_i^T} \\ {\textbf{0}}&{}\quad {\textbf{0}}&{}\quad {\textbf{0}}&{}\quad {\textbf{0}}&{}\quad { - {{\mathbf {\Lambda }}_i}{{{\hat{{\tilde{\textbf{m}}}}}}_i}{\mathbf {\Lambda }}_i^T} \\ {\textbf{0}}&{}\quad {\textbf{0}}&{}\quad {\textbf{0}}&{}\quad { - {{\mathbf {\Lambda }}_i}{{{\hat{{\tilde{\textbf{n}}}}}}_i}{\mathbf {\Lambda }}_i^T}&{}\quad { - \widehat{{\mathbf \Lambda ^\prime _i}{{{\tilde{{\textbf{n}}}}}_i}}}, \\ {\textbf{0}}&{}\quad {\textbf{0}}&{}\quad {{{\mathbf {\Lambda }}_i}{{{\hat{{\tilde{\textbf{n}}}}}}_i}{\mathbf {\Lambda }}_i^T}&{}\quad { - {{\mathbf {\Lambda }}_i}{{{\hat{{\tilde{\textbf{m}}}}}}_i}{\mathbf {\Lambda }}_i^T}&{}\quad {\hat{{\textbf{r}}}_i^\prime {{\mathbf {\Lambda }}_i}{{{\hat{{\tilde{\textbf{n}}}}}}_i}{\mathbf {\Lambda }}_i^T - \widehat{{\mathbf \Lambda ^\prime _i}{{{\tilde{{\textbf{m}}}}}_i}}} \\ {{{\mathbf {\Lambda }}_i}{{{\hat{{\tilde{\textbf{n}}}}}}_i}{\mathbf {\Lambda }}_i^T}&{}\quad {\textbf{0}}&{}\quad {\widehat{{\mathbf \Lambda ^\prime _i}{{{\tilde{{\textbf{n}}}}}_i}}}&{}\quad {\hat{{\textbf{r}}}_i^\prime {{\mathbf {\Lambda }}_i}{{{\hat{{\tilde{\textbf{n}}}}}}_i}{\mathbf {\Lambda }}_i^T}&{}\quad {\hat{{\textbf{r}}}_i^\prime \widehat{{\mathbf \Lambda ^\prime _i}{{{\tilde{{\textbf{n}}}}}_i}} + \hat{{\textbf{r}}}_i^{\prime \prime }{{\mathbf {\Lambda }}_i}{{{\hat{{\tilde{\textbf{n}}}}}}_i}{\mathbf {\Lambda }}_i^T} \end{array}} \right] . \end{aligned}$$

(A10)

Appendix B Analytical solutions for the simply-supported beam

We consider a plane simply-supported beam with the governing equation Eq. (37) reduced as

$$\begin{aligned}&k_{s1} GA\left( {u_1^{{\prime } {\prime } } - \varphi ^{\prime } } \right) + \frac{{f_1 f_2 A}}{\varepsilon }\varphi ^{{\prime } {\prime } {\prime } } - \left( {k_{s1} l^2 GA + \frac{{f_2^2 A}}{\varepsilon }} \right) \left( {u_1^{{\prime } {\prime } {\prime } {\prime } } - \varphi ^{{\prime } {\prime } {\prime } } } \right) = - p, \nonumber \\&k_{s1} GA\left( {u_1^{\prime } - \varphi } \right) + \left( {EI + l^2 EA + \frac{{f_1^2 + f_1 f_2 }}{\varepsilon }} \right) \varphi ^{{\prime } {\prime } } \nonumber \\&\qquad - \left( {k_{s1} l^2 GA + \frac{{f_2^2 + f_1 f_2 }}{\varepsilon }A} \right) \left( {u_1^{{\prime } {\prime } {\prime } } - \varphi ^{{\prime } {\prime } } } \right) - l^2 EI\varphi ^{{\prime } {\prime } {\prime } {\prime } } = - m, \end{aligned}$$

(B1)

where $u_1$ is the deflection and $\varphi$ denotes the rotation. p and m are the external distributed force and moment, respectively. Referring to previous works [40, 49] for strain gradient beams, the following boundary conditions

$$\begin{aligned} \begin{array}{cc} u_1=u''_1=\varphi '=0,&\text { at } s=0, L \end{array} \end{aligned}$$

(B2)

are employed. The deflection $u_1$ and rotation $\varphi$ are assumed to take the following Fourier series

$$\begin{aligned} \begin{array}{cc} u_1(s)=\sum \limits _{n=1}^{\infty }U_n \sin \dfrac{n\pi s}{L},&\varphi (s)=\sum \limits _{n=1}^{\infty }\Phi _n \cos \dfrac{n\pi s}{L}, \end{array} \end{aligned}$$

(B3)

where $U_n$ and $\Phi _n$ are Fourier coefficients to be determined for each n. External load is taken as

$$\begin{aligned} \begin{array}{cc} p(s)=p_0 \sin \dfrac{\pi s}{L},&m=0, \end{array} \end{aligned}$$

(B4)

with a constant $p_0$. According to Eqs. (B1), (B3) and (B4), after somewhat straightforward calculations, the final expression of $u_1$ and $\varphi$ is given as

$$\begin{aligned} \begin{array}{c} u_1(s)=\dfrac{a_3}{a_2^2-a_1a_3}p_0\sin \beta s,\\ \varphi (s)=\dfrac{a_2}{a_1a_3-a_2^2}p_0\cos \beta s. \end{array} \end{aligned}$$

(B5)

with

$$\begin{aligned} \begin{array}{l} a_1=-k_{s1}GA\beta ^2-k_{s1}l^2GA\beta ^4-\frac{f_2^2A}{\epsilon }\beta ^4,\\ a_2=k_{s1}GA\beta +\left( {k_{s1}}{l^2}GA + \frac{f_2^2+{f_1}{f_2}}{\epsilon }A\right) \beta ^3,\\ a_3=-k_{s1}GA-\left( EI+l^2EA+k_{s1}l^2GA+\frac{\left( f_1+f_2\right) ^2}{\epsilon }A\right) \beta ^2-l^2EI\beta ^4, \end{array} \end{aligned}$$

(B6)

where $\beta =\pi /L$.