Buckling of piezoelectric sandwich microplates with arbitrary in-plane BCs rested on foundation: effect of hygro-thermo-electro-elastic field

Abstract

Buckling and post-buckling behaviors of simply supported microplates in complex environment are studied, where elastic foundation and hygro-thermal-electro-mechanical loads are considered. The first-order shear deformation theory is used to establish basic equations of the microplate considering the von Kármán’s nonlinearity. The size-dependent effect is characterized by the modified couple stress theory. A unified boundary condition model is introduced to discuss various in-plane boundary conditions (BCs). Analytical solutions for critical mechanical/hygrothermal buckling loads and post-buckling paths of the microplate under different in-plane BCs are obtained by using the perturbation method and the Galerkin method, respectively. Results reveal that size-dependent effect and elastic foundation enhance the stiffness of the microplate. Transverse displacement of the microplate in the post-buckling stage increases with the external compressive load, temperature and moisture concentration, expressing a nonlinear curve. When the displacement constraint in the normal direction is applied on the microplate edge, the critical mechanical/hygrothermal buckling load decreases. These results can be utilized in the optimization design of the micro-electro-mechanical systems.

Graphical abstract

Appendices

Appendix A

The expressions of $A_{ij}$, $D_{ij}$, $A_{n}$, $D_{n}$, $N_{\mathrm{T}}$, $N_{\mathrm{C}}$, $N_{\mathrm{P}}$, $M_{T}$, $M_{C}$ and $M_{P}$ are given as

$$\begin{aligned}&A_{ij}=\int _{-\frac{h_{e}}{2}}^{\frac{h_{e}}{2}}Q_{eij}\mathrm{d}z +2\int _{\frac{h_{e}}{2}}^{\frac{h}{2}}Q_{pij}\mathrm{d}z\quad \left( ij=11,12,66\right) \nonumber \\&A_{44}=k_{s}\left( \int _{-\frac{h_{e}}{2}}^{\frac{h_{e}}{2}}Q_{e44}\mathrm{d}z +2\int _{\frac{h_{e}}{2}}^{\frac{h}{2}}Q_{p44}\mathrm{d}z\right) \nonumber \\&D_{ij}=\int _{-\frac{h_{e}}{2}}^{\frac{h_{e}}{2}}Q_{eij}z^{2}\mathrm{d}z +2\int _{\frac{h_{e}}{2}}^{\frac{h}{2}}Q_{pij}z^{2}\mathrm{d}z\quad \left( ij=11,12,66\right) \nonumber \\&[A_{n},D_{n}]=\int _{-\frac{h_{e}}{2}}^{\frac{h_{e}}{2}}\frac{E_{e}l^{2}}{2\left( 1+\nu _{e}\right) } [1,z^{2}]\mathrm{d}z\nonumber \\&N_{\mathrm{T}}=\int _{-\frac{h_{e}}{2}}^{\frac{h_{e}}{2}}\left( Q_{e11}+Q_{e12}\right) \alpha _{e}T\mathrm{d}z +2\int _{\frac{h_{e}}{2}}^{\frac{h}{2}}\left( Q_{p11}+Q_{p12}\right) \alpha _{p}T\mathrm{d}z\nonumber \\&M_{T}=\int _{-\frac{h_{e}}{2}}^{\frac{h_{e}}{2}}\left( Q_{e11}+Q_{e12}\right) \alpha _{e}Tz\mathrm{d}z +\int _{\frac{h_{e}}{2}}^{\frac{h}{2}}\left( Q_{p11}+Q_{p12}\right) \alpha _{p}Tz\mathrm{d}z\nonumber \\&\quad +\,\int _{-\frac{h}{2}}^{-\frac{h_{e}}{2}}\left( Q_{p11}+Q_{p12}\right) \alpha _{p}Tz\mathrm{d}z\nonumber \\&N_{\mathrm{P}}=\int _{\frac{h_{e}}{2}}^{\frac{h}{2}}{\overline{e}}_{31}E_{z}\mathrm{d}z +\int _{-\frac{h}{2}}^{-\frac{h_{e}}{2}}{\overline{e}}_{31}E_{z}\mathrm{d}z\nonumber \\&M_{P}=\int _{\frac{h_{e}}{2}}^{\frac{h}{2}}{\overline{e}}_{31}E_{z}z\mathrm{d}z +\int _{-\frac{h}{2}}^{-\frac{h_{e}}{2}}{\overline{e}}_{31}E_{z}z\mathrm{d}z\nonumber \\&N_{\mathrm{C}}=\int _{-\frac{h_{e}}{2}}^{\frac{h_{e}}{2}}\left( Q_{e11}+Q_{e12}\right) \beta _{e}C\mathrm{d}z +2\int _{\frac{h_{e}}{2}}^{\frac{h}{2}}\left( Q_{p11}+Q_{p12}\right) \beta _{p}C\mathrm{d}z\nonumber \\&M_{C}=\int _{-\frac{h_{e}}{2}}^{\frac{h_{e}}{2}}\left( Q_{e11}+Q_{e12}\right) \beta _{e}Cz\mathrm{d}z +\int _{\frac{h_{e}}{2}}^{\frac{h}{2}}\left( Q_{p11}+Q_{p12}\right) \beta _{p}Cz\mathrm{d}z\nonumber \\&\ \ \ \ \ \ \ \ +\int _{-\frac{h}{2}}^{-\frac{h_{e}}{2}}\left( Q_{p11}+Q_{p12}\right) \beta _{p}Cz\mathrm{d}z \end{aligned}$$

(59)

Appendix B

The present equilibrium equations are validated by the degeneration analyses as follows:

1.1 Ignoring size-dependent effect

The size-dependent effect is captured by the MCST. The additional stiffness $A_{n}$ and $D_{n}$ appear in equations. By setting $l=0$, several coefficients in Eqs. (15), (16) and (59) become $A_{n}=D_{n}=0$, $X_{i}=R_{j}=0$, $(i=x,y,z,yz,xz,xy,$$j=yz,xz)$. The governing equations degenerate to the state for the classical first-order shear deformation theory expressed as

$$\begin{aligned} \begin{aligned}&N_{x,x}+N_{xy,y}=0\\&N_{xy,x}+N_{y,y}=0\\&Q_{xz,x}+Q_{yz,y}+{\tilde{N}}-k_{w}w+k_{p}\nabla ^{2}w=0\\&M_{x,x}+M_{xy,y}-Q_{xz}=0\\&M_{xy,x}+M_{y,y}-Q_{yz}=0\\ \end{aligned} \end{aligned}$$

(60)

Equation (60) is the same as the equilibrium equations expressed by Eq. (15) in paper [40] without considering the transverse load and kinetic energy.

1.2 Ignoring Pasternak foundation

The effect of Pasternak foundation on the microplate is studied by introducing two parameters $k_{w}, k_{p}$. Setting $k_{w}=k_{p}=0$ yields

$$\begin{aligned} \begin{aligned}&N_{x,x}+N_{xy,y}+\frac{1}{2}(X_{xz,xy}+X_{yz,yy})=0\\&N_{xy,x}+N_{y,y}-\frac{1}{2}(X_{xz,xx}+X_{yz,xy})=0\\&Q_{xz,x}+Q_{yz,y}+\frac{1}{2}(X_{xy,xx}+X_{y,xy}-X_{xy,yy}-X_{x,xy})+{\tilde{N}}=0\\&M_{x,x}+M_{xy,y}-Q_{xz}+\frac{1}{2}(X_{xy,x}+X_{y,y}-X_{z,y}+R_{xz,xy}+R_{yz,yy})=0\\&M_{xy,x}+M_{y,y}-Q_{yz}-\frac{1}{2}(X_{x,x}+X_{xy,y}-X_{z,x}+R_{xz,xx}+R_{yz,xy})=0\\ \end{aligned} \end{aligned}$$

(61)

Equation (61) is identical to the Eq. (27) proposed in paper [28] when the body force and transverse loads vanish.

1.3 Ignoring geometric nonlinearity

Without the aid of the von Kármán’s nonlinear strains, the strain field of the microplate becomes

$$\begin{aligned} \begin{aligned}&\varepsilon _{xx}=u_{,x}+z\varphi _{x,x},\quad \varepsilon _{yy}=v_{,y}+z\varphi _{y,y},\quad \varepsilon _{zz}=0\\&\gamma _{yz}=\varphi _y+w_{,y},\quad \gamma _{xz}=\varphi _x+w_{,x},\quad \gamma _{xy}=u_{,y}+v_{,x}+z(\varphi _{x,y}+\varphi _{y,x})\\ \end{aligned} \end{aligned}$$

The equilibrium Eqs. (6)–(10) are rewritten as

$$\begin{aligned} \begin{aligned}&N_{x,x}+N_{xy,y}+\frac{1}{2}\left( X_{xz,xy}+X_{yz,yy}\right) =0\\&N_{xy,x}+N_{y,y}-\frac{1}{2}\left( X_{xz,xx}+X_{yz,xy}\right) =0\\&Q_{xz,x}+Q_{yz,y}+\frac{1}{2}\left( X_{xy,xx}+X_{y,xy}-X_{xy,yy}-X_{x,xy}\right) -k_{w}w+k_{p}\nabla ^{2}w=0\\&M_{x,x}+M_{xy,y}-Q_{xz}+\frac{1}{2}\left( X_{xy,x}+X_{y,y}-X_{z,y}+R_{xz,xy}+R_{yz,yy}\right) =0\\&M_{xy,x}+M_{y,y}-Q_{yz}-\frac{1}{2}\left( X_{x,x}+X_{xy,y}-X_{z,x}+R_{xz,xx}+R_{yz,xy}\right) =0\\ \end{aligned} \end{aligned}$$

(62)

Equation (62) is the same as the equilibrium Eq. (6) provided in paper [41] when the transverse loads are ignored.