A most general strain gradient plate formulation for size-dependent geometrically nonlinear free vibration analysis of functionally graded shear deformable rectangular microplates

Abstract

In this work, the size-dependent geometrically nonlinear free vibration of functionally graded (FG) microplates is investigated. For this purpose, with the aid of Hamilton’s principle, a nonclassical rectangular microplate model is developed based on Mindlin’s strain gradient theory, Mindlin’s plate theory and the von Kármán geometric nonlinearity. For some specific values of the length scale material parameters, the simple form of size-dependent mathematical formulation based on the modified strain gradient theory (MSGT) and modified couple stress theory is obtained. The generalized differential quadrature method, numerical Galerkin scheme, periodic time differential operators and pseudo arc-length continuation method are utilized to determine the geometrically nonlinear free vibration characteristics of FG microplates with different boundary conditions. The parametric effects of thickness-to-material length scale ratio, material gradient index, length-to-thickness ratio, length-to-width ratio and boundary conditions on the nonlinear free vibration characteristics of FG microplates are studied through various numerical examples presented. It is found that a considerable difference exists between the results of various elasticity theories at small values of length scale parameter. A more precise prediction can be provided by using the size-dependent plate model based on the MSGT.

Appendices

Appendix 1

Considering Eqs. (9), (10), (12) as well as the following constants

$$\begin{aligned} \beta _1= & {} 2\left( {a_1 +a_2 +a_3 +a_4 +a_5 } \right) ,\beta _2 =a_1 +2a_2 ,\nonumber \\ \beta _3= & {} a_1 +2a_3,\beta _4 =a_1 +2a_5,\beta _5 =2\left( {a_2 +a_4 } \right) ,\nonumber \\ \beta _6= & {} a_3 +2a_4+a_5 \end{aligned}$$

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and introducing the following parameters

$$\begin{aligned} \left\{ {A_{11} ,B_{11} ,D_{11} } \right\}= & {} \int _{-h/2}^{h/2} \left( {\lambda +2\mu } \right) \left\{ {1,x_3 ,x_3^2 } \right\} \hbox {d}x_3 ,\nonumber \\ \left\{ {A_{12} ,B_{12} ,D_{12} } \right\}= & {} \int _{-h/2}^{h/2} {\lambda }\left\{ {1,x_3 ,x_3^2 } \right\} \hbox {d}x_3 ,\nonumber \\ \left\{ {A_{55} ,B_{55} ,D_{55} } \right\}= & {} \int _{-h/2}^{h/2} {\upmu }\left\{ {1,x_3 ,x_3^2 } \right\} \hbox {d}x_3 ,\nonumber \\ A_i= & {} \int _{-h/2}^{h/2} a_i \hbox {d}x_3 , \left\{ {E_i ,F_i ,G_i } \right\} \nonumber \\= & {} \int _{-h/2}^{h/2} \beta _i \left\{ {1,x_3 ,x_3^2 } \right\} \hbox {d}x_3 \end{aligned}$$

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the classical and nonclassical resultant forces and moments $\left( {N_{ij} ,Q_i ,M_{ij} ,T_{ijk} ,M_{ijk} } \right)$ can be written as follows

$$\begin{aligned} N_{11}= & {} A_{11} \kappa _{11}^0 +B_{11} \kappa _{11}^1 +A_{12} \kappa _{22}^0\nonumber \\&+B_{12} \kappa _{22}^1 , N_{12} =2A_{55} \kappa _{12}^0 +2B_{55} \kappa _{12}^1 ,\nonumber \\ N_{22}= & {} A_{11} \kappa _{22}^0 +B_{11} \kappa _{22}^1 +A_{12} \kappa _{11}^0\nonumber \\&+B_{12} \kappa _{11}^1 , \left( {Q_1 ,Q_2 } \right) =k_s A_{55} \left\{ {\gamma _1 ,\gamma _2 } \right\} , \end{aligned}$$

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$$\begin{aligned} M_{11}= & {} B_{11} \kappa _{11}^0 +D_{11} \kappa _{11}^1 +B_{12} \kappa _{22}^0\nonumber \\&+D_{12} \kappa _{22}^1 , M_{12} =2B_{55} \kappa _{12}^0 +2D_{55} \kappa _{12}^1 ,\nonumber \\ M_{22}= & {} B_{11} \kappa _{22}^0 +D_{11} \kappa _{22}^1\nonumber \\&+B_{12} \kappa _{11}^0 +D_{12} \kappa _{11}^1 ,\nonumber \\ T_{111}= & {} E_1 \kappa _{11,1}^0 +E_2 \kappa _{22,1}^0 +E_3 \kappa _{12,2}^0\nonumber \\&+F_1 \kappa _{11,1}^1 +F_2 \kappa _{22,1}^1 +F_3 \kappa _{12,2}^1 ,\nonumber \\ T_{211}= & {} E_4 \kappa _{12,1}^0 +E_5 \kappa _{11,2}^0 +E_2 \kappa _{22,2}^0 +F_4 \kappa _{12,1}^1\nonumber \\&+F_5 \kappa _{11,2}^1 +F_2 \kappa _{22,2}^1 ,\nonumber \\ T_{122}= & {} E_2 \kappa _{11,1}^0 +E_5 \kappa _{22,1}^0 +E_4 \kappa _{12,2}^0\nonumber \\&+F_2 \kappa _{11,1}^1 +F_5 \kappa _{22,1}^1 +F_4 \kappa _{12,2}^1 ,\nonumber \\ T_{222}= & {} E_1 \kappa _{22,2}^0 +E_3 \kappa _{12,1}^0 +E_2 \kappa _{11,2}^0\nonumber \\&+F_1 \kappa _{22,2}^1 +F_3 \kappa _{12,1}^1 +F_2 \kappa _{11,2}^1 ,\nonumber \\ T_{112}= & {} ( 2E_6 \kappa _{12,1}^0 +E_4 \kappa _{11,2}^0 +E_3 \kappa _{22,2}^0\nonumber \\&+2F_6 \kappa _{12,1}^1 +F_4 \kappa _{11,2}^1 +F_3 \kappa _{22,2}^1 )/2,\nonumber \\ T_{221}= & {} ( 2E_6 \kappa _{12,2}^0 +E_3 \kappa _{11,1}^0 +E_4 \kappa _{22,1}^0 +2F_6 \kappa _{12,2}^1\nonumber \\&+F_3 \kappa _{11,1}^1+F_4 \kappa _{22,1}^1 )/2, \end{aligned}$$

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$$\begin{aligned} T_{311}= & {} \frac{1}{2}\left( {E_4 \gamma _{1,1} +2E_5 \kappa _{11}^1 +A_1 \gamma _{2,2} +4A_2 \kappa _{22}^1 } \right) ,T_{322}\nonumber \\&=\frac{1}{2}\left( {E_4 \gamma _{2,2} +2E_5 \kappa _{22}^1 +A_1 \gamma _{1,1} +4A_2 \kappa _{11}^1 } \right) ,\nonumber \\ T_{xxz}= & {} \left( {E_6 \gamma _{1,1} +E_4 \kappa _{11}^1 +A_1 \kappa _{22}^1 +A_3 \gamma _{2,2} } \right) /2,\nonumber \\&T_{yyz} =\left( {E_6 \gamma _{2,2} +E_4 \kappa _{22}^1 +A_1 \kappa _{11}^1 +A_3 \gamma _{1,1} } \right) /2,\nonumber \\ T_{213}= & {} \left( {2A_4 \gamma _{1,2} +A_5 \gamma _{2,1} +2A_5 \kappa _{12}^1 } \right) /2, T_{123}\nonumber \\&=\left( {2A_4 \gamma _{2,1} +A_5 \gamma _{1,2} +2A_5 \kappa _{12}^1 } \right) /2,\nonumber \\ T_{312}= & {} \left( {4A_4 \kappa _{12}^1 +A_5 \gamma _{2,1} +A_5 \gamma _{1,2} } \right) /2,\nonumber \\ M_{xxx}= & {} F_1 \kappa _{11,1}^0 +F_2 \kappa _{22,1}^0 +F_3 \kappa _{12,2}^0 +G_1 \kappa _{11,1}^1\nonumber \\&+\,G_2 \kappa _{22,1}^1 +G_3 \kappa _{12,2}^1 ,\nonumber \\ M_{yxx}= & {} F_4 \kappa _{12,1}^0 +F_5 \kappa _{11,2}^0 +F_2 \kappa _{22,2}^0 +G_4 \kappa _{12,1}^1\nonumber \\&+\,G_5 \kappa _{11,2}^1 +G_2 \kappa _{22,2}^1 ,\nonumber \\ M_{xyy}= & {} F_2 \kappa _{11,1}^0 +F_5 \kappa _{22,1}^0 +F_4 \kappa _{12,2}^0 +G_2 \kappa _{11,1}^1\nonumber \\&+\,G_5 \kappa _{22,1}^1 +G_4 \kappa _{12,2}^1 ,\nonumber \\ M_{yyy}= & {} F_1 \kappa _{22,2}^0 +F_3 \kappa _{12,1}^0 +F_2 \kappa _{11,2}^0 +G_1 \kappa _{22,2}^1\nonumber \\&+\,G_3 \kappa _{12,1}^1 +G_2 \kappa _{11,2}^1 ,\nonumber \\ M_{112}= & {} ( 2F_6 \kappa _{12,1}^0 +F_4 \kappa _{11,2}^0 +F_3 \kappa _{22,2}^0 +2G_6 \kappa _{12,1}^1\nonumber \\&+\,G_4 \kappa _{11,2}^1 +G_3 \kappa _{22,2}^1 )/2,\nonumber \\ M_{221}= & {} ( 2F_6 \kappa _{12,2}^0 +F_3 \kappa _{11,1}^0 +F_4 \kappa _{22,1}^0 +2G_6 \kappa _{12,2}^1\nonumber \\&+\,G_3 \kappa _{11,1}^1 +G_4 \kappa _{22,1}^1 )/2. \end{aligned}$$

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Appendix 2

1.1 Hadamard and Kronecker products

Definition 1

Considering the matrices $\mathbf{A}=\left[ {A_{ij} } \right] _{N\times M}$ and $\mathbf{B}=\left[ {B_{ij} } \right] _{N\times M}$, the Hadamard product of these matrices can be expressed as $\mathbf{A}\circ \mathbf{B}=\left[ {A_{ij} B_{ij} } \right] _{N\times M}$.

Definition 2

If $\mathbf{A}$ and $\mathbf{B}$ are m-by-n and $p-$by-q matrices, then the Kronecker product $\mathbf{A}\otimes \mathbf{B}$ is an mp-by-nq block matrix and expressed as

$$\begin{aligned} \mathbf{A}\otimes \mathbf{B}= \left[ {{\begin{array}{ccccc} {a_{11} \mathbf{B}}&{} \cdots &{} {a_{1n} \mathbf{B}} \\ \vdots &{} \ddots &{} \vdots \\ {a_{m1} \mathbf{B}}&{} \cdots &{} {a_{mn} \mathbf{B}} \\ \end{array} }} \right] _{mp\times nq} \end{aligned}$$

1.2 Integral matrix operators

$$\begin{aligned} \int _{x_1 }^{x_N } f\left( x \right) \hbox {d}x= & {} \left( {\mathop \sum \limits _{r=0}^{N-1} \tilde{\mathbf{X}} ^{\left( r \right) }{} \mathbf{D}_x^{\left( r \right) } } \right) \mathbf{F}=\mathbf{S}_x\\ \mathbf{F}, \mathbf{S}_x= & {} \left[ {S_x } \right] _{1\times N} \end{aligned}$$

where $\mathbf{D}_x^{\left( r \right) }$ is the GDQ differential operator, and

$$\begin{aligned} \tilde{\mathbf{X}} ^{\left( r \right) }= & {} \left[ \frac{\left( {x_{2} -x_{1} } \right) ^{r+1}}{2^{r+1}\left( {r+1} \right) !},\ldots ,\right. \nonumber \\&\quad \frac{\left( {x_{i+1} -x_i } \right) ^{r+1}-\left( {x_{i-1} -x_i } \right) ^{r+1}}{2^{r+1}\left( {r+1} \right) !},\ldots ,\nonumber \\&\quad \left. -\,\frac{\left( {x_{N-1} -x_N } \right) ^{r+1}}{2^{r+1}\left( {r+1} \right) } \right] ,\\&\qquad i=2,3,\ldots ,N-1 \end{aligned}$$

Appendix 3

Considering the constants defined Eqs. (19), (21) based on the modified strain gradient theory can be expressed as

Compared to the size-dependent microplate model based on the modified couple stress theory given in [13], the colored terms are added to the governing equation.