A nonlinear formulation of piezoelectric shells with complete electro-mechanical coupling

Abstract

A nonlinear piezoelectric shell model capable of accurately expressing the direct and the converse piezoelectric effect is presented. The developed theory is meant to encompass large strains, displacements and rotations that can occur in the shell as a consequence of a complete coupling between the mechanical and the electrical fields. Based on results present in the literature according to which a linear dependence of the electric potential on the shell thickness is not adequate to represent its electrical behavior, a specific structure of the field of admissible displacements is taken into account. Warping functions characterized by an ad-hoc polynomial expansion aimed at expressing their dependence on the shell through-the-thickness coordinate are here considered to describe the shear and the extensional deformability of the shell transverse fibers. Linear constitutive relations for a transversely isotropic continuum are considered. Finally, the governing equations of motion of the shell are obtained via enforcement of the virtual work theorem.

Appendix

Different quantities are introduced in the paper in a concise form for the sake of its readability. In Eq. (73) the deformation field components are shown. As already pointed out, assuming that $w^{{\alpha }}=\zeta N^{{\alpha }}$ and $w^{3}=\zeta (N^{3}-1)$ a Kirchhoff model of shell is obtained. The complete expressions of the deformations ${\varepsilon }_{\alpha \alpha }^{k}$ read:

$$\begin{aligned} {\varepsilon }_{\alpha \alpha }^{k0}= & {} \frac{1}{2}\left( A_{\alpha \alpha }-a_{\alpha \alpha }\right) \end{aligned}$$

(103)

$$\begin{aligned} {\varepsilon }_{\alpha \alpha }^{k1}= & {} \left[ A_{\alpha }^{\gamma }\left( N_{,\alpha }^{\gamma }-N^3b_{\alpha }^{\gamma }\right) a_{\gamma \gamma }+A_{\alpha }^3\left( \frac{\partial N^3}{\partial \xi ^{\alpha }}-N^{\gamma }b_{\alpha \gamma }\right) +b_{\alpha }^{\alpha }a_{\alpha \alpha }\right] \end{aligned}$$

(104)

$$\begin{aligned} {\varepsilon }_{\alpha \alpha }^{k2}= & {} \frac{1}{2}\left[ \left( N_{,\alpha }^{\gamma }-N^3b_{\alpha }^{\gamma }\right) ^2a_{\gamma \gamma }+\left( \frac{\partial N^3}{\partial \xi ^{\alpha }}-N^{\gamma }b_{\alpha \gamma }\right) ^2 -\left( b_{\alpha }^{\alpha }\right) ^2\right] \end{aligned}$$

(105)

$$\begin{aligned} {\varepsilon }_{\alpha \beta }^{k0}= & {} \frac{1}{2}A_{\alpha \beta }\end{aligned}$$

(106)

$$\begin{aligned} {\varepsilon }_{\alpha \beta }^{k1}&= \frac{1}{2}\left[ A_{\alpha }^{\gamma }\left( N_{,\beta }^{\gamma }-N^3b_{\beta }^{\gamma }\right) a_{\gamma \gamma }\right. \\&\quad +\,A_{\beta }^{\gamma }\left( N_{,\alpha }^{\gamma }-N^3b_{\alpha }^{\gamma }\right) a_{\gamma \gamma }+A_{\alpha }^3\left( \frac{\partial N^3}{\partial \xi ^{\beta }}-N^{\gamma }b_{\beta \gamma }\right) \\&\quad \left. +\,A_{\beta }^3\left( \frac{\partial N^3}{\partial \xi ^{\alpha }}-N^{\gamma }b_{\alpha \gamma }\right) \right] \end{aligned}$$

(107)

$$\begin{aligned} {\varepsilon }_{\alpha \beta }^{k2}&= \frac{1}{2}\left[ \left( N_{,\alpha }^{\gamma }-N^3b_{\alpha }^{\gamma }\right) \left( N_{,\beta }^{\gamma }-N^3b_{\beta }^{\gamma }\right) a_{\gamma \gamma }\right. \\&\quad +\,\left. \left( \frac{\partial N^3}{\partial \xi ^{\alpha }}-N^{\gamma }b_{\alpha \gamma }\right) \left( \frac{\partial N^3}{\partial \xi ^{\beta }}-N^{\gamma }b_{\beta \gamma }\right) \right] \end{aligned}$$

(108)

where the $(\cdot )_{,}$ sign denotes covariant differentiation. It is worth noticing that different well-known formulations for the shell deformation field (based on Kirchhoff hypotheses) can be obtained through a straightforward linearization of the above expressions. If it is assumed that

$$\begin{aligned} {{\varvec{R}}}_o={{\varvec{r}}}_o+\varDelta {{\varvec{r}}}_o \end{aligned}$$

(109)

where $\varDelta {{\varvec{r}}}_o$ is an infinitesimal displacement vector, the following holds

$$\begin{aligned} {\varepsilon }_{{\alpha }{\alpha }}^{k0}= & {} \frac{\partial \varDelta {{\varvec{r}}}_o}{\partial \xi ^{{\alpha }}}\cdot {{\varvec{a}}}_{{\alpha }}\end{aligned}$$

(110)

$$\begin{aligned} {\varepsilon }_{{\alpha }{\beta }}^{k0}= & {} \frac{1}{2}\left( \frac{\partial \varDelta {{\varvec{r}}}_o}{\partial \xi ^{{\alpha }}}\cdot {{\varvec{a}}}_{{\beta }} +\frac{\partial \varDelta {{\varvec{r}}}_o}{\partial \xi ^{{\beta }}}\cdot {{\varvec{a}}}_{{\alpha }}\right) \end{aligned}$$

(111)

representing the linear part of the Green–Lagrange strain tensor. The ${{\varvec{N}}}$ vector normal to the deformed shell midsurface can be written in terms of the infinitesimal displacement vector $\varDelta {{\varvec{r}}}_o$ as

$$\begin{aligned} {{\varvec{N}}}= & {} \frac{({{\varvec{a}}}_1+\frac{\partial \varDelta {{\varvec{r}}}_o}{\partial \xi ^1}) \times ({{\varvec{a}}}_2+\frac{\partial \varDelta {{\varvec{r}}}_o}{\partial \xi ^2})}{\sqrt{\Vert ({{\varvec{a}}}_1+\frac{\partial \varDelta {{\varvec{r}}}_o}{\partial \xi ^1}) \Vert ^2\Vert ({{\varvec{a}}}_2+\frac{\partial \varDelta {{\varvec{r}}}_o}{\partial \xi ^2})\Vert ^2}} \\&\quad \frac{}{\overline{-[({{\varvec{a}}}_1+\frac{\partial \varDelta {{\varvec{r}}}_o}{\partial \xi ^1}) \cdot ({{\varvec{a}}}_2+\frac{\partial \varDelta {{\varvec{r}}}_o}{\partial \xi ^2})]^2}} \end{aligned}$$

(112)

and, consequently

$$\begin{aligned} {{\varvec{N}}}\sim {{\varvec{n}}}+\varDelta {{\varvec{n}}}\end{aligned}$$

(113)

with

$$\begin{aligned} \varDelta {{\varvec{n}}}&= \frac{1}{\sqrt{a}}\left( {{\varvec{a}}}_1\times \frac{\partial \varDelta {{\varvec{r}}}_o}{\partial \xi ^2} +\frac{\partial \varDelta {{\varvec{r}}}_o}{\partial \xi ^1}\times {{\varvec{a}}}_2\right) \\&\quad -\,{{\varvec{n}}}\left( \frac{{{\varvec{a}}}_1}{a_{11}}\cdot \frac{\partial \varDelta {{\varvec{r}}}_o}{\partial \xi ^1} +\frac{{{\varvec{a}}}_2}{a_{22}}\cdot \frac{\partial \varDelta {{\varvec{r}}}_o}{\partial \xi ^2}\right) \end{aligned}$$

(114)

The above expressions for the terms ${\varepsilon }_{{\alpha }{\alpha }}^{k1}$ and ${\varepsilon }_{{\alpha }{\beta }}^{k1}$ can be thus rewritten as

$$\begin{aligned} {\varepsilon }_{{\alpha }{\alpha }}^{k1}= & {} \frac{\partial \varDelta {{\varvec{n}}}}{\partial \xi ^{{\alpha }}} \cdot {{\varvec{a}}}_{{\alpha }}-\frac{1}{R_{{\alpha }}}{\varepsilon }_{{\alpha }{\alpha }}^{k0}\end{aligned}$$

(115)

$$\begin{aligned} {\varepsilon }_{{\alpha }{\beta }}^{k1}&= \frac{1}{2}\left[ \left( \frac{\partial \varDelta {{\varvec{n}}}}{\partial \xi ^{{\beta }}}\cdot {{\varvec{a}}}_{{\alpha }}+\frac{\partial \varDelta {{\varvec{n}}}}{\partial \xi ^{{\alpha }}}\cdot {{\varvec{a}}}_{{\beta }}\right) \right. \\&\quad \left. -\,\left( \frac{1}{R_{{\alpha }}}\frac{\partial \varDelta {{\varvec{r}}}_o}{\partial \xi ^{{\beta }}}\cdot {{\varvec{a}}}_{{\alpha }}+\frac{1}{R_{{\beta }}} \frac{\partial \varDelta {{\varvec{r}}}_o}{\partial \xi ^{{\alpha }}}\cdot {{\varvec{a}}}_{{\beta }}\right) \right] \end{aligned}$$

(116)

The choice of retaining different parts in the expression of $\varDelta {{\varvec{n}}}$ can bring to various formulations [28]. If the whole term $\varDelta {{\varvec{n}}}$ is neglected, the above expressions simplify into

$$\begin{aligned} {\varepsilon }_{{\alpha }{\alpha }}^{k1}= & {} -\frac{1}{R_{{\alpha }}}{\varepsilon }_{{\alpha }{\alpha }}^{k0}\end{aligned}$$

(117)

$$\begin{aligned} {\varepsilon }_{{\alpha }{\beta }}^{k1}= & {} -\frac{1}{2}\left( \frac{1}{R_{{\alpha }}}\frac{\partial \varDelta {{\varvec{r}}}_o}{\partial \xi ^{{\beta }}}\cdot {{\varvec{a}}}_{{\alpha }}+\frac{1}{R_{{\beta }}}\frac{\partial \varDelta {{\varvec{r}}}_o}{\partial \xi ^{{\alpha }}}\cdot {{\varvec{a}}}_{{\beta }}\right) \end{aligned}$$

(118)

As final remark, it can be demonstrated that higher-order terms (${\varepsilon }_{{\alpha }{\alpha }}^{k2}$ and ${\varepsilon }_{{\alpha }{\beta }}^{k2}$) become trivial as consequence of the thin-shell hypothesis ($\frac{\zeta }{R}\ll 1$).