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Nonlocal nonlinear analysis of functionally graded plates using natural neighbour Galerkin method

Abstract

In the present work, flexural response of functionally graded plates subjected to transverse loads have been investigated using the meshless natural neighbor Galerkin method (NNGM). The plate formulation has been developed based on the Reddy’s (Mechanics of laminated composite plates and shells: theory and analysis, 2nd edition, CRC Press, Boca Raton, 2014) third-order shear deformation theory (TSDT) using the von Kármán nonlinear strains. The governing equations of the TSDT have been derived accounting for the length scale/size effects considering the Eringen’s nonlocal stress-gradient model (Eringen in Microcontinuum filed theories—I: foundations and solids, Springer-Verlag, 1998). The C1 continuous shape functions have been computed using the sibson’s interpolant and generalizing a Bezier patch over the domain. The nonlocal nonlinear model of the resulting governing equations has been developed, and Newton’s iterative procedure is used for the solution of nonlinear algebraic equations. The mechanical properties of functionally graded plate are assumed to vary continuously through the thickness and obey a power-law distribution of the volume fraction of the constituents. The variation of volume fractions through the thickness have been computed using two different homogenization techniques, namely, the rule of mixtures and the Mori–Tanaka scheme. A detailed parametric study to show the effect of side-to-thickness ratio, power-law index, and nonlocal parameter on the load-deflection characteristics of plates have been presented. The central deflections obtained using (NNGM) have been compared with the results from literature based on finite element method. The results have been compared with the two homogenization schemes and also with results computed with the first-order shear deformation theory (FSDT) to show the accuracy of nonlocal nonlinear formulation based on TSDT.

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Appendices

Appendix A

The governing equations of equilibrium for the plate flexure based on Reddy’s [38] nonlinear third-order shear deformable plate theory are

$$\begin{aligned}&\dfrac{\partial N_{xx}^{nl}}{\partial x}+\dfrac{\partial N_{xy}^{nl}}{\partial y}=0\\&\dfrac{\partial N_{xy}^{nl}}{\partial x}+\dfrac{\partial N_{yy}^{nl}}{\partial y}=0\\&\dfrac{\partial \bar{Q}_x^{nl}}{\partial x}+\dfrac{\partial \bar{Q}_y^{nl}}{\partial y}\\&\quad +\dfrac{\partial }{\partial x}\left( N_{xx}^{nl}\dfrac{\partial w}{\partial x}+N_{xy}^{nl}\dfrac{\partial w}{\partial y}\right) +\dfrac{\partial }{\partial y}\left( N_{xy}^{nl}\dfrac{\partial w}{\partial x}+N_{yy}^{nl}\dfrac{\partial w}{\partial y}\right) \\&\quad +c_1\left( \dfrac{\partial ^2 P_{xx}^{nl}}{\partial x^2}+2\dfrac{\partial ^2 P_{xy}^{nl}}{\partial x \partial y}+\dfrac{\partial ^2 P_{yy}^{nl}}{\partial y^2}\right) +q=0\\&\dfrac{\partial \bar{M}_{xx}^{nl}}{\partial x}+\dfrac{\partial \bar{M}_{xy}^{nl}}{\partial y}-\bar{Q}_x^{nl}=0\\&\dfrac{\partial \bar{M}_{xy}^{nl}}{\partial x}+\dfrac{\partial \bar{M}_{yy}^{nl}}{\partial y}-\bar{Q}_y^{nl}=0 \end{aligned}$$

where

$$\begin{aligned} {\bar{M}_{\alpha \beta }^{nl}}={M_{\alpha \beta }^{nl}} -c_1{P_{\alpha \beta }^{nl}},\quad {\bar{Q}_{\alpha }^{nl}}={Q_{\alpha }^{nl}}-c_2{R_{\alpha }^{nl}} \end{aligned}$$

The local and nonlocal stress resultants are related based on Eq. (8) as.

$$\begin{aligned}&\mathcal {L}(N_{\alpha \beta }^{nl})=N_{\alpha \beta }^{nl},\quad \mathcal {L}(M_{\alpha \beta }^{nl})=M_{\alpha \beta }^{nl},\\&\mathcal {L}(P_{\alpha \beta }^{nl})=P_{\alpha \beta }^{nl},\quad \mathcal {L}(Q_{\alpha }^{nl})=Q_{\alpha }^{nl},\quad \mathcal {L}(R_{\alpha }^{nl})=R_{\alpha }^{nl} \end{aligned}$$

Appendix B

Stiffness matrix term’s

$$\begin{aligned} K^{11}_{ij}= & {} \int _{\Omega ^e}\bigg [A_{11}\frac{\partial \psi ^{(1)}_j}{\partial x}\frac{\partial \psi ^{(1)}_i}{\partial x}+A_{66}\frac{\partial \psi ^{(1)}_j}{\partial y}\frac{\partial \psi ^{(1)}_i}{\partial y}\\&+A_{16}\bigg (\frac{\partial \psi ^{(1)}_j}{\partial y}\frac{\partial \psi ^{(1)}_i}{\partial x}+\frac{\partial \psi ^{(1)}_i}{\partial y}\frac{\partial \psi ^{(1)}_j}{\partial x}\bigg )\bigg ]dxdy \end{aligned}$$
$$\begin{aligned}K^{12}_{ij}= & {} \int _{\Omega ^e}\bigg [A_{12}\frac{\partial \psi ^{(1)}_i}{\partial x}\frac{\partial \psi ^{(1)}_j}{\partial y}+A_{16}\frac{\partial \psi ^{(1)}_i}{\partial x}\frac{\partial \psi ^{(1)}_j}{\partial x}\\&+A_{66}\frac{\partial \psi ^{(1)}_i}{\partial y}\frac{\partial \psi ^{(1)}_j}{\partial x}+A_{26}\frac{\partial \psi ^{(1)}_i}{\partial y}\frac{\partial \psi ^{(1)}_j}{\partial y}\bigg ]dxdy \end{aligned}$$
$$\begin{aligned} K^{13}_{ij}= & {} \int _{\Omega ^e}\bigg \{\frac{\partial \psi ^{(1)}_i}{\partial x}\bigg [\frac{1}{2}A_{11}\frac{\partial w}{\partial x}\frac{\partial \varphi _j}{\partial x}+\frac{1}{2}A_{12}\frac{\partial w}{\partial y}\frac{\partial \varphi _j}{\partial y}\\&+\frac{1}{2}A_{16}\bigg (\frac{\partial w}{\partial x}\frac{\partial \varphi _j}{\partial y}+\frac{\partial w}{\partial y}\frac{\partial \varphi _j}{\partial x}\bigg )\\&-c_1E_{11}\frac{\partial ^2 \varphi _j}{\partial x^2}-c_1E_{12}\frac{\partial ^2 \varphi _j}{\partial y^2}-2c_1E_{16}\frac{\partial ^2 \varphi _j}{\partial x \partial y}\bigg )\\&+\frac{\partial \psi ^{(1)}_i}{\partial y}\bigg [\frac{1}{2}A_{16}\frac{\partial w}{\partial x}\frac{\partial \varphi _j}{\partial x}+\frac{1}{2}A_{26}\frac{\partial w}{\partial y}\frac{\partial \varphi _j}{\partial y}\\&+\frac{1}{2}A_{66}\bigg (\frac{\partial w}{\partial x}\frac{\partial \varphi _j}{\partial y}+\frac{\partial w}{\partial y}\frac{\partial \varphi _j}{\partial x}\bigg )-c_1E_{16}\frac{\partial ^2 \varphi _j}{\partial x^2}\\&-c_1E_{26}\frac{\partial ^2 \varphi _j}{\partial y^2}-2c_1E_{66}\frac{\partial ^2 \varphi _j}{\partial x \partial y}\bigg ]\bigg \}dxdy \end{aligned}$$
$$\begin{aligned}K^{14}_{ij}= & {} \int _{\Omega ^e}\bigg \{\frac{\partial \psi ^{(1)}_i}{\partial x}\bigg [(B_{11}-c_1E_{11})\frac{\partial \psi ^{(2)}_j}{\partial x}\\&+(B_{16}-c_1E_{16})\frac{\partial \psi ^{(2)}_j}{\partial y}\bigg ]\\&+\frac{\partial \psi ^{(1)}_i}{\partial y}\bigg [(B_{16}-c_1E_{16})\frac{\partial \psi ^{(2)}_j}{\partial x}\\&+(B_{66}-c_1E_{66})\frac{\partial \psi ^{(2)}_j}{\partial y}\bigg ]\bigg \}dxdy \end{aligned}$$
$$\begin{aligned}K^{15}_{ij}= & {} \int _{\Omega ^e}\bigg \{\frac{\partial \psi ^{(1)}_i}{\partial x}\bigg [(B_{16}-c_1E_{16})\frac{\partial \psi ^{(2)}_j}{\partial x}\\&+(B_{12}-c_1E_{12})\frac{\partial \psi ^{(2)}_j}{\partial y}\bigg ]\\&+\frac{\partial \psi ^{(1)}_i}{\partial y}\bigg [(B_{66}-c_1E_{66})\frac{\partial \psi ^{(2)}_j}{\partial x}\\&+(B_{26}-c_1E_{26})\frac{\partial \psi ^{(2)}_j}{\partial y}\bigg ]\bigg \}dxdy \end{aligned}$$
$$\begin{aligned}K^{21}_{ij}= & {} \int _{\Omega ^e}\bigg [A_{16}\frac{\partial \psi ^{(1)}_j}{\partial x}\frac{\partial \psi ^{(1)}_i}{\partial x}+A_{66}\frac{\partial \psi ^{(1)}_i}{\partial x}\frac{\partial \psi ^{(1)}_j}{\partial y}\\&+A_{12}\frac{\partial \psi ^{(1)}_i}{\partial y}\frac{\partial \psi ^{(1)}_j}{\partial x}+A_{26}\frac{\partial \psi ^{(1)}_i}{\partial y}\frac{\partial \psi ^{(1)}_j}{\partial y}\bigg )\bigg ]dxdy \end{aligned}$$
$$\begin{aligned}K^{22}_{ij}= & {} \int _{\Omega ^e}\bigg [A_{66}\frac{\partial \psi ^{(1)}_i}{\partial x}\frac{\partial \psi ^{(1)}_j}{\partial x}+A_{22}\frac{\partial \psi ^{(1)}_i}{\partial y}\frac{\partial \psi ^{(1)}_j}{\partial y}\\&+A_{26}\bigg (\frac{\partial \psi ^{(1)}_i}{\partial y}\frac{\partial \psi ^{(1)}_j}{\partial x}+\frac{\partial \psi ^{(1)}_i}{\partial x}\frac{\partial \psi ^{(1)}_j}{\partial y}\bigg )\bigg ]dxdy \end{aligned}$$
$$\begin{aligned}K^{23}_{ij}= & {} \int _{\Omega ^e}\bigg \{\frac{\partial \psi ^{(1)}_i}{\partial x}\bigg [\frac{1}{2}A_{16}\frac{\partial w}{\partial x}\frac{\partial \varphi _j}{\partial x}+\frac{1}{2}A_{26}\frac{\partial w}{\partial y}\frac{\partial \varphi _j}{\partial y}\\&+\frac{1}{2}A_{66}\bigg (\frac{\partial w}{\partial x}\frac{\partial \varphi _j}{\partial y}+\frac{\partial w}{\partial y}\frac{\partial \varphi _j}{\partial x}\bigg )\\&-c_1E_{16}\frac{\partial ^2 \varphi _j}{\partial x^2}-c_1E_{26}\frac{\partial ^2 \varphi _j}{\partial y^2}-2c_1E_{66}\frac{\partial ^2 \varphi _j}{\partial x \partial y}\bigg )\\&+\frac{\partial \psi ^{(1)}_i}{\partial y}\bigg [\frac{1}{2}A_{12}\frac{\partial w}{\partial x}\frac{\partial \varphi _j}{\partial x}\\&+\frac{1}{2}A_{22}\frac{\partial w}{\partial y}\frac{\partial \varphi _j}{\partial y}\\&+\frac{1}{2}A_{26}\bigg (\frac{\partial w}{\partial x}\frac{\partial \varphi _j}{\partial y}+\frac{\partial w}{\partial y}\frac{\partial \varphi _j}{\partial x}\bigg )\\&-c_1E_{12}\frac{\partial ^2 \varphi _j}{\partial x^2}-c_1E_{22}\frac{\partial ^2 \varphi _j}{\partial y^2}-2c_1E_{26}\frac{\partial ^2 \varphi _j}{\partial x \partial y}\bigg ]\bigg \}dxdy \end{aligned}$$
$$\begin{aligned}K^{24}_{ij}= & {} \int _{\Omega ^e}\bigg \{\frac{\partial \psi ^{(1)}_i}{\partial x}\bigg [(B_{16}-c_1E_{16})\frac{\partial \psi ^{(2)}_j}{\partial x}\\&+(B_{66}-c_1E_{66})\frac{\partial \psi ^{(2)}_j}{\partial y}\bigg ]\\&+\frac{\partial \psi ^{(1)}_i}{\partial y}\bigg [(B_{12}-c_1E_{12})\frac{\partial \psi ^{(2)}_j}{\partial x}\\&+(B_{26}-c_1E_{26})\frac{\partial \psi ^{(2)}_j}{\partial y}\bigg ]\bigg \}dxdy \end{aligned}$$
$$\begin{aligned}K^{25}_{ij}= & {} \int _{\Omega ^e}\bigg \{\frac{\partial \psi ^{(1)}_i}{\partial x}\bigg [(B_{66}-c_1E_{66})\frac{\partial \psi ^{(2)}_j}{\partial x}\\&+(B_{26}-c_1E_{26})\frac{\partial \psi ^{(2)}_j}{\partial y}\bigg ]\\&+\frac{\partial \psi ^{(1)}_i}{\partial y}\bigg [(B_{26}-c_1E_{26})\frac{\partial \psi ^{(2)}_j}{\partial x}\\&+(B_{22}-c_1E_{22})\frac{\partial \psi ^{(2)}_j}{\partial y}\bigg ]\bigg \}dxdy \end{aligned}$$
$$\begin{aligned}K^{31}_{ij}= & {} \int _{\Omega ^e}\bigg [\frac{\partial \varphi _i}{\partial x}\bigg (A_{11}\frac{\partial w}{\partial x}\frac{\partial \psi _j^{(1)}}{\partial x}+A_{16}\frac{\partial w}{\partial x}\frac{\partial \psi _j^{(1)}}{\partial y}\\&+A_{16}\frac{\partial w}{\partial y}\frac{\partial \psi _j^{(1)}}{\partial x}+A_{66}\frac{\partial w}{\partial y}\frac{\partial \psi _j^{(1)}}{\partial y}\bigg )\\&+\frac{\partial \varphi _i}{\partial y}\bigg (A_{16}\frac{\partial w}{\partial x}\frac{\partial \psi _j^{(1)}}{\partial x}+A_{66}\frac{\partial w}{\partial x}\frac{\partial \psi _j^{(1)}}{\partial y}\\&+A_{12}\frac{\partial w}{\partial y}\frac{\partial \psi _j^{(1)}}{\partial x}+A_{26}\frac{\partial w}{\partial y}\frac{\partial \psi _j^{(1)}}{\partial y}\bigg )\\&+\bigg (-c_1E_{11}\frac{\partial \psi ^{(1)}_j}{\partial x}\frac{\partial ^2\varphi _i}{\partial x^2}-c_1E_{16}\frac{\partial \psi ^{(1)}_j}{\partial y}\frac{\partial ^2\varphi _i}{\partial x^2}\\&-c_1E_{12}\frac{\partial \psi ^{(1)}_j}{\partial x}\frac{\partial ^2\varphi _i}{\partial y^2}-c_1E_{26}\frac{\partial \psi ^{(1)}_j}{\partial y}\frac{\partial ^2\varphi _i}{\partial y^2}\\&-2c_1E_{16}\frac{\partial \psi ^{(1)}_j}{\partial x}\frac{\partial ^2\varphi _i}{\partial x\partial y}-2c_1E_{66}\frac{\partial \psi ^{(1)}_j}{\partial y}\frac{\partial ^2\varphi _i}{\partial x\partial y}\bigg )\bigg ]dxdy\\ K^{32}_{ij}= & {} \int _{\Omega ^e}\bigg [\frac{\partial \varphi _i}{\partial x}\bigg (A_{12}\frac{\partial w}{\partial x}\frac{\partial \psi _j^{(1)}}{\partial y}+A_{16}\frac{\partial w}{\partial x}\frac{\partial \psi _j^{(1)}}{\partial x}\\&+A_{26}\frac{\partial w}{\partial y}\frac{\partial \psi _j^{(1)}}{\partial y}+A_{66}\frac{\partial w}{\partial y}\frac{\partial \psi _j^{(1)}}{\partial x}\bigg )\\&+\frac{\partial \varphi _i}{\partial y}\bigg (A_{26}\frac{\partial w}{\partial x}\frac{\partial \psi _j^{(1)}}{\partial y}+A_{66}\frac{\partial w}{\partial x}\frac{\partial \psi _j^{(1)}}{\partial x}\\&+A_{22}\frac{\partial w}{\partial y}\frac{\partial \psi _j^{(1)}}{\partial y}+A_{26}\frac{\partial w}{\partial y}\frac{\partial \psi _j^{(1)}}{\partial x}\bigg )\\&+\bigg (-c_1E_{12}\frac{\partial \psi ^{(1)}_j}{\partial y}\frac{\partial ^2\varphi _i}{\partial x^2}-c_1E_{16}\frac{\partial \psi ^{(1)}_j}{\partial x}\frac{\partial ^2\varphi _i}{\partial x^2}\\&-c_1E_{22}\frac{\partial \psi ^{(1)}_j}{\partial y}\frac{\partial ^2\varphi _i}{\partial y^2}-c_1E_{26}\frac{\partial \psi ^{(1)}_j}{\partial x}\frac{\partial ^2\varphi _i}{\partial y^2}\\&-2c_1E_{26}\frac{\partial \psi ^{(1)}_j}{\partial y}\frac{\partial ^2\varphi _i}{\partial x\partial y}-2c_1E_{66}\frac{\partial \psi ^{(1)}_j}{\partial x}\frac{\partial ^2\varphi _i}{\partial x\partial y}\bigg )\bigg ]dxdy \end{aligned}$$
$$\begin{aligned}K^{33}_{ij}= & {} \int _{\Omega ^e}\bigg [\frac{\partial \varphi _i}{\partial x}\bigg (A_{55}\frac{\partial \varphi _j}{\partial x}-2c_2D_{55}\frac{\partial \varphi _j}{\partial x}+c_2^2F_{55}\frac{\partial \varphi _j}{\partial x}\\&+A_{45}\frac{\partial \varphi _j}{\partial y}-2c_2D_{45}\frac{\partial \varphi _j}{\partial y}+c_2^2F_{45}\frac{\partial \varphi _j}{\partial y}\bigg )\\&+\frac{\partial \varphi _i}{\partial y}\bigg (A_{44}\frac{\partial \varphi _j}{\partial y}-2c_2D_{44}\frac{\partial \varphi _j}{\partial y}+c_2^2F_{44}\frac{\partial \varphi _j}{\partial y}\\&+A_{45}\frac{\partial \varphi _j}{\partial x}-2c_2D_{45}\frac{\partial \varphi _j}{\partial x}+c_2^2F_{45}\frac{\partial \varphi _j}{\partial x}\bigg )\\&+\bigg (\frac{1}{2}A_{11}\bigg (\frac{\partial w}{\partial x}\bigg )^2+\frac{1}{2}A_{12}\bigg (\frac{\partial w}{\partial y}\bigg )^2\\&+A_{16}\frac{\partial w}{\partial x}\frac{\partial w}{\partial y}\bigg )\frac{\partial \varphi _i}{\partial x}\frac{\partial \varphi _j}{\partial x}\\&+\bigg (\frac{1}{2}A_{16}\bigg (\frac{\partial w}{\partial x}\bigg )^2+\frac{1}{2}A_{26}\bigg (\frac{\partial w}{\partial y}\bigg )^2+A_{66}\frac{\partial w}{\partial x}\frac{\partial w}{\partial y}\bigg )\\&\bigg (\frac{\partial \varphi _i}{\partial x}\frac{\partial \varphi _j}{\partial y}+\frac{\partial \varphi _i}{\partial y}\frac{\partial \varphi _j}{\partial x}\bigg )\\&+\bigg (\frac{1}{2}A_{12}\bigg (\frac{\partial w}{\partial x}\bigg )^2+\frac{1}{2}A_{22}\bigg (\frac{\partial w}{\partial y}\bigg )^2\\&+A_{26}\frac{\partial w}{\partial x}\frac{\partial w}{\partial y}\bigg )\frac{\partial \varphi _i}{\partial y}\frac{\partial \varphi _j}{\partial y}\\&+\frac{\partial \varphi _i}{\partial x}\bigg (-c_1E_{11}\frac{\partial w}{\partial x}\frac{\partial ^2\varphi _j}{\partial x^2}-c_1E_{12}\frac{\partial w}{\partial x}\frac{\partial ^2\varphi _j}{\partial y^2}\\&-2c_1E_{16}\frac{\partial w}{\partial x}\frac{\partial \varphi _j}{\partial x\partial y}\\&-c_1E_{16}\frac{\partial w}{\partial y}\frac{\partial ^2\varphi _j}{\partial x^2}-c_1E_{26}\frac{\partial w}{\partial y}\frac{\partial ^2\varphi _j}{\partial y^2}\\&-2c_1E_{66}\frac{\partial w}{\partial y}\frac{\partial \varphi _j}{\partial x\partial y}\bigg )\\&+\frac{\partial \varphi _i}{\partial y}\bigg (-c_1E_{16}\frac{\partial w}{\partial x}\frac{\partial ^2\varphi _j}{\partial x^2}-c_1E_{26}\frac{\partial w}{\partial x}\frac{\partial ^2\varphi _j}{\partial y^2}\\&-2c_1E_{66}\frac{\partial w}{\partial x}\frac{\partial \varphi _j}{\partial x\partial y}-c_1E_{12}\frac{\partial w}{\partial y}\frac{\partial ^2\varphi _j}{\partial x^2}\\&-c_1E_{22}\frac{\partial w}{\partial y}\frac{\partial ^2\varphi _j}{\partial y^2}\\&-2c_1E_{26}\frac{\partial w}{\partial y}\frac{\partial \varphi _j}{\partial x\partial y}\bigg )\\&+\frac{\partial ^2\varphi _i}{\partial x^2}\bigg (-\frac{1}{2}c_1E_{11}\frac{\partial w}{\partial x}\frac{\partial \varphi _j}{\partial x}-\frac{1}{2}c_1E_{12}\frac{\partial w}{\partial y}\frac{\partial \varphi _j}{\partial y}\\&+\frac{1}{2}c_1E_{16}\bigg (\frac{\partial w}{\partial x}\frac{\partial \varphi _j}{\partial y}+\frac{\partial w}{\partial y}\frac{\partial \varphi _j}{\partial x}\bigg )\\&+c_1^2H_{11}\frac{\partial ^2 \varphi _j}{\partial x^2}+c_1^2H_{12}\frac{\partial ^2 \varphi _j}{\partial y^2}+2c_1^2H_{16}\frac{\partial ^2 w}{\partial x \partial y}\bigg )\\&+\frac{\partial ^2\varphi _i}{\partial y^2}\bigg (-\frac{1}{2}c_1E_{12}\frac{\partial w}{\partial x}\frac{\partial \varphi _j}{\partial x}\\&-\frac{1}{2}c_1E_{22}\frac{\partial w}{\partial y}\frac{\partial \varphi _j}{\partial y}+\frac{1}{2}c_1E_{26}\bigg (\frac{\partial w}{\partial x}\frac{\partial \varphi _j}{\partial y}+\frac{\partial w}{\partial y}\frac{\partial \varphi _j}{\partial x}\bigg )\\&+c_1^2H_{12}\frac{\partial ^2 \varphi _j}{\partial x^2}+c_1^2H_{22}\frac{\partial ^2 \varphi _j}{\partial y^2}\\&+2c_1^2H_{26}\frac{\partial ^2 \varphi _j}{\partial x \partial y}\bigg ) +\frac{\partial ^2\varphi _i}{\partial x\partial y}\bigg (-c_1E_{16}\frac{\partial w}{\partial x}\frac{\partial \varphi _j}{\partial x}\\&-c_1E_{26}\frac{\partial w}{\partial y}\frac{\partial \varphi _j}{\partial y}\\&-c_1E_{66}\bigg (\frac{\partial w}{\partial x}\frac{\partial \varphi _j}{\partial y}+\frac{\partial w}{\partial y}\frac{\partial \varphi _j}{\partial x}\bigg )+2c_1H_{16}\frac{\partial ^2\varphi _j}{\partial x^2}\\&+2c_1H_{26}\frac{\partial ^2\varphi _j}{\partial y^2}+4c_1^2H_{66}\frac{\partial ^2 \varphi _j}{\partial x \partial y}\bigg )\bigg ]dxdy \end{aligned}$$
$$\begin{aligned}K^{34}_{ij}= & {} \int _{\Omega ^e}\bigg \{\frac{\partial \varphi _i}{\partial x}\bigg [(A_{55}-2c_2D_{55}+c_2^2F_{55})\psi ^{(2)}_j\\&+(B_{16}-c_1E_{16})\frac{\partial \psi _j^{(2)}}{\partial x}\frac{\partial w}{\partial y}\\&+(B_{66}-c_1E_{66})\frac{\partial \psi _j^{(2)}}{\partial y}\frac{\partial w}{\partial y}\\&+(B_{11}-c_1E_{11})\frac{\partial \psi _j^{(2)}}{\partial x}\frac{\partial w}{\partial x}\\&+(B_{16}-c_1E_{16})\frac{\partial \psi _j^{(2)}}{\partial y}\frac{\partial w}{\partial x}\bigg ]\\&+\frac{\partial \varphi _i}{\partial y}\bigg [(A_{45}-2c_2D_{45}+c_2^2F_{45})\psi ^{(2)}_j\\&+(B_{66}-c_1E_{66})\frac{\partial \psi _j^{(2)}}{\partial y}\frac{\partial w}{\partial x}\\&+(B_{16}-c_1E_{16})\frac{\partial \psi _j^{(2)}}{\partial x}\frac{\partial w}{\partial x}\\&+(B_{12}-c_1E_{12})\frac{\partial \psi _j^{(2)}}{\partial x}\frac{\partial w}{\partial y}\\&+(B_{26}-c_1E_{26})\frac{\partial \psi _j^{(2)}}{\partial y}\frac{\partial w}{\partial y}\bigg ]\\&+\frac{\partial ^2\varphi _i}{\partial x^2}\bigg [-c_1F_{11}\frac{\partial \psi ^{(2)}_j}{\partial x}\\&-c_1F_{16}\frac{\partial \psi ^{(2)}_j}{\partial y}+c_1^2H_{11}\frac{\partial \psi ^{(2)}_j}{\partial x}+c_1^2H_{16}\frac{\partial \psi ^{(2)}_j}{\partial y}\bigg ]\\&+\frac{\partial ^2\varphi _i}{\partial y^2}\bigg [-c_1F_{12}\frac{\partial \psi ^{(2)}_j}{\partial x}-c_1F_{26}\frac{\partial \psi ^{(2)}_j}{\partial y}\\&+c_1^2H_{12}\frac{\partial \psi ^{(2)}_j}{\partial x}+c_1^2H_{26}\frac{\partial \psi ^{(2)}_j}{\partial y}\bigg ]\\&+\frac{\partial ^2\varphi _i}{\partial x \partial y}\bigg [-2c_1F_{16}\frac{\partial \psi ^{(2)}_j}{\partial x}-2c_1F_{66}\frac{\partial \psi ^{(2)}_j}{\partial y}\\&+2c_1^2H_{16}\frac{\partial \psi ^{(2)}_j}{\partial x}+2c_1^2H_{66}\frac{\partial \psi ^{(2)}_j}{\partial y}\bigg ]\bigg \}dxdy \end{aligned}$$
$$\begin{aligned}K^{35}_{ij}= & {} \int _{\Omega ^e}\bigg \{\frac{\partial \varphi _i}{\partial x}\bigg [(A_{45}-2c_2D_{45}+c_2^2F_{45})\psi ^{(2)}_j\\&+(B_{12}-c_1E_{12})\frac{\partial \psi _j^{(2)}}{\partial y}\frac{\partial w}{\partial x}\\&+(B_{16}-c_1E_{16})\frac{\partial \psi _j^{(2)}}{\partial x}\frac{\partial w}{\partial x}\\&+(B_{26}-c_1E_{26})\frac{\partial \psi _j^{(2)}}{\partial y}\frac{\partial w}{\partial y}\\&+(B_{66}-c_1E_{66})\frac{\partial \psi _j^{(2)}}{\partial x}\frac{\partial w}{\partial y}\bigg ]\\&+\frac{\partial \varphi _i}{\partial y}\bigg [(A_{44}-2c_2D_{44}+c_2^2F_{44})\psi ^{(2)}_j\\&+(B_{26}-c_1E_{26})\frac{\partial \psi _j^{(2)}}{\partial y}\frac{\partial w}{\partial x}\\&+(B_{66}-c_1E_{66})\frac{\partial \psi _j^{(2)}}{\partial x}\frac{\partial w}{\partial x}\\&+(B_{22}-c_1E_{22})\frac{\partial \psi _j^{(2)}}{\partial y}\frac{\partial w}{\partial y}\\&+(B_{26}-c_1E_{26})\frac{\partial \psi _j^{(2)}}{\partial x}\frac{\partial w}{\partial y}\bigg ]\\&+\frac{\partial ^2\varphi _i}{\partial x^2}\bigg [-c_1F_{12}\frac{\partial \psi ^{(2)}_j}{\partial y}\\&-c_1F_{16}\frac{\partial \psi ^{(2)}_j}{\partial x}+c_1^2H_{12}\frac{\partial \psi ^{(2)}_j}{\partial y}+c_1^2H_{16}\frac{\partial \psi ^{(2)}_j}{\partial x}\bigg ]\\&+\frac{\partial ^2\varphi _i}{\partial y^2}\bigg [-c_1F_{22}\frac{\partial \psi ^{(2)}_j}{\partial y}-c_1F_{26}\frac{\partial \psi ^{(2)}_j}{\partial x}\\&+c_1^2H_{22}\frac{\partial \psi ^{(2)}_j}{\partial y}\\&+c_1^2H_{26}\frac{\partial \psi ^{(2)}_j}{\partial x}\bigg ] +\frac{\partial ^2\varphi _i}{\partial x \partial y}\bigg [-2c_1F_{26}\frac{\partial \psi ^{(2)}_j}{\partial y}\\&-2c_1F_{66}\frac{\partial \psi ^{(2)}_j}{\partial x}+2c_1^2H_{26}\frac{\partial \psi ^{(2)}_j}{\partial y}\\&+2c_1^2H_{66}\frac{\partial \psi ^{(2)}_j}{\partial x}\bigg ]\bigg \}dxdy \end{aligned}$$
$$\begin{aligned}K^{41}_{ij}= & {} \int _{\Omega ^e}\bigg \{\frac{\partial \psi ^{(2)}_i}{\partial x}\bigg [(B_{11}-c_1E_{11})\frac{\partial \psi ^{(1)}_j}{\partial x}\\&+(B_{16}-c_1E_{16})\frac{\partial \psi ^{(1)}_j}{\partial y}\bigg ]\\&+\frac{\partial \psi ^{(2)}_i}{\partial y}\bigg [(B_{16}-c_1E_{16})\frac{\partial \psi ^{(1)}_j}{\partial x}\\&+(B_{66}-c_1E_{66})\frac{\partial \psi ^{(1)}_j}{\partial y}\bigg ]\bigg \}dxdy\\ K^{42}_{ij}= & {} \int _{\Omega ^e}\bigg \{\frac{\partial \psi ^{(2)}_i}{\partial x}\bigg [(B_{12}-c_1E_{12})\frac{\partial \psi ^{(1)}_j}{\partial y}\\&+(B_{16}-c_1E_{16})\frac{\partial \psi ^{(1)}_j}{\partial x}\bigg ]\\&+\frac{\partial \psi ^{(1)}_i}{\partial y}\bigg [(B_{26}-c_1E_{26})\frac{\partial \psi ^{(1)}_j}{\partial y}\\&+(B_{66}-c_1E_{66})\frac{\partial \psi ^{(1)}_j}{\partial x}\bigg ]\bigg \}dxdy \end{aligned}$$
$$\begin{aligned}K^{43}_{ij}= & {} \int _{\Omega ^e}\bigg \{\frac{\partial \psi ^{(2)}_i}{\partial x}\bigg [\frac{1}{2}(B_{11}-c_1E_{11})\frac{\partial w}{\partial x}\frac{\partial \varphi _j}{\partial x}\\&+\frac{1}{2}(B_{12}-c_1E_{12})\frac{\partial w}{\partial y}\frac{\partial \varphi _j}{\partial y}\\&+\frac{1}{2}(B_{16}-c_1E_{16})\bigg (\frac{\partial w}{\partial x}\frac{\partial \varphi _j}{\partial y}+\frac{\partial w}{\partial y}\frac{\partial \varphi _j}{\partial x}\bigg )\\&+c_1(c_1H_{11}-F_{11})\frac{\partial ^2 \varphi _j}{\partial x^2}+c_1(c_1H_{12}-F_{12})\frac{\partial ^2 \varphi _j}{\partial y^2}\\&+2c_1(c_1H_{16}-F_{16})\frac{\partial ^2 \varphi _j}{\partial x\partial y}\bigg ]+\frac{\partial \psi ^{(2)}_i}{\partial y}\\&\bigg [\frac{1}{2}(B_{16}-c_1E_{16})\frac{\partial w}{\partial x}\frac{\partial \varphi _j}{\partial x}\\&+\frac{1}{2}(B_{26}-c_1E_{26})\frac{\partial w}{\partial y}\frac{\partial \varphi _j}{\partial y}\\&+\frac{1}{2}(B_{66}-c_1E_{66})\bigg (\frac{\partial w}{\partial x}\frac{\partial \varphi _j}{\partial y}+\frac{\partial w}{\partial y}\frac{\partial \varphi _j}{\partial x}\bigg )\\&+c_1(c_1H_{16}-F_{16})\frac{\partial ^2 \varphi _j}{\partial x^2}+c_1(c_1H_{26}-F_{26})\frac{\partial ^2 \varphi _j}{\partial y^2}\\&+2c_1(H_{66}-c_1F_{66})\frac{\partial ^2 \varphi _j}{\partial x \partial y}\bigg ] +\psi ^{(2)}_i\\&\bigg [(A_{55}-2c_2D_{55}+c_2^2F_{55})\frac{\partial \varphi _j}{\partial x}\\&+(A_{45}-2c_2D_{45}+c_2^2F_{45})\frac{\partial \varphi _j}{\partial y}\bigg ]\bigg \}dxdy \end{aligned}$$
$$\begin{aligned}K^{44}_{ij}= & {} \int _{\Omega ^e}\bigg \{\frac{\partial \psi ^{(2)}_i}{\partial x}\bigg [(D_{11}-2c_1F_{11}+c_1^2H_{11})\frac{\partial \psi ^{(2)}_j}{\partial x}\\&+(D_{16}-2c_1F_{16}+c_1^2H_{16})\frac{\partial \psi ^{(2)}_j}{\partial y}\bigg ]\\&+\frac{\partial \psi ^{(2)}_i}{\partial y}\bigg [(D_{16}-2c_1F_{16}+c_1^2H_{16})\frac{\partial \psi ^{(2)}_j}{\partial x}\\&+(D_{66}-2c_1F_{66}+c_1^2H_{66})\frac{\partial \psi ^{(2)}_j}{\partial y}\bigg ]\\&+\psi ^{(2)}_i\bigg [(A_{55}-2c_2D_{55}+c_2^2F_{55})\psi ^{(2)}_j\bigg ]\bigg \}dxdy\\ K^{45}_{ij}= & {} \int _{\Omega ^e}\bigg \{\frac{\partial \psi ^{(2)}_i}{\partial x}\bigg [(D_{12}-2c_1F_{12}+c_1^2H_{12})\frac{\partial \psi ^{(2)}_j}{\partial y}\\&+(D_{16}-2c_1F_{16}+c_1^2H_{16})\frac{\partial \psi ^{(2)}_j}{\partial x}\bigg ]\\&+\frac{\partial \psi ^{(2)}_i}{\partial y}\bigg [(D_{26}-2c_1F_{26}+c_1^2H_{26})\frac{\partial \psi ^{(2)}_j}{\partial y}\\&+(D_{66}-2c_1F_{66}+c_1^2H_{66})\frac{\partial \psi ^{(2)}_j}{\partial x}\bigg ]\\&+\psi ^{(2)}_i\bigg [(A_{45}-2c_2D_{45}+c_2^2F_{45})\psi ^{(2)}_j\bigg ]\bigg \}dxdy \end{aligned}$$
$$\begin{aligned}K^{51}_{ij}= & {} \int _{\Omega ^e}\bigg \{\frac{\partial \psi ^{(2)}_i}{\partial x}\bigg [(B_{16}-c_1E_{16})\frac{\partial \psi ^{(1)}_j}{\partial x}\\&+(B_{66}-c_1E_{66})\frac{\partial \psi ^{(1)}_j}{\partial y}\bigg ]\\&+\frac{\partial \psi ^{(1)}_i}{\partial y}\bigg [(B_{12}-c_1E_{12})\frac{\partial \psi ^{(1)}_j}{\partial x}\\&+(B_{26}-c_1E_{26})\frac{\partial \psi ^{(1)}_j}{\partial y}\bigg ]\bigg \}dxdy\\ K^{52}_{ij}= & {} \int _{\Omega ^e}\bigg \{\frac{\partial \psi ^{(2)}_i}{\partial x}\bigg [(B_{26}-c_1E_{26})\frac{\partial \psi ^{(1)}_j}{\partial y}\\&+(B_{66}-c_1E_{66})\frac{\partial \psi ^{(1)}_j}{\partial x}\bigg ]\\&+\frac{\partial \psi ^{(1)}_i}{\partial y}\bigg [(B_{22}-c_1E_{22})\frac{\partial \psi ^{(1)}_j}{\partial y}\\&+(B_{26}-c_1E_{26})\frac{\partial \psi ^{(1)}_j}{\partial x}\bigg ]\bigg \}dxdy \end{aligned}$$
$$\begin{aligned}K^{53}_{ij}= & {} \int _{\Omega ^e}\bigg \{\frac{\partial \psi ^{(2)}_i}{\partial x}\bigg [\frac{1}{2}(B_{16}-c_1E_{16})\frac{\partial w}{\partial x}\frac{\partial \varphi _j}{\partial x}\\&+\frac{1}{2}(B_{26}-c_1E_{26})\frac{\partial w}{\partial y}\frac{\partial \varphi _j}{\partial y}\\&+\frac{1}{2}(B_{66}-c_1E_{66})\bigg (\frac{\partial w}{\partial x}\frac{\partial \varphi _j}{\partial y}+\frac{\partial w}{\partial y}\frac{\partial \varphi _j}{\partial x}\bigg )\\&+c_1(c_1H_{16}-F_{16})\frac{\partial ^2 \varphi _j}{\partial x^2}+c_1(c_1H_{26}-F_{26})\frac{\partial ^2 \varphi _j}{\partial y^2}\\&+2c_1(c_1H_{66}-F_{66})\frac{\partial ^2 \varphi _j}{\partial x\partial y}\bigg ]\\&+\frac{\partial \psi ^{(2)}_i}{\partial y}\bigg [\frac{1}{2}(B_{12}-c_1E_{12})\frac{\partial w}{\partial x}\frac{\partial \varphi _j}{\partial x}\\&+\frac{1}{2}(B_{22}-c_1E_{22})\frac{\partial w}{\partial y}\frac{\partial \varphi _j}{\partial y}\\&+\frac{1}{2}(B_{26}-c_1E_{26})\bigg (\frac{\partial w}{\partial x}\frac{\partial \varphi _j}{\partial y}+\frac{\partial w}{\partial y}\frac{\partial \varphi _j}{\partial x}\bigg )\\&+c_1(c_1H_{12}-F_{12})\frac{\partial ^2 \varphi _j}{\partial x^2}+c_1(c_1H_{22}-F_{22})\frac{\partial ^2 \varphi _j}{\partial y^2}\\&+2c_1(H_{26}-c_1F_{26})\frac{\partial ^2 \varphi _j}{\partial x \partial y}\bigg ] +\psi ^{(2)}_i\\&\bigg [(A_{45}-2c_2D_{45}+c_2^2F_{45})\frac{\partial \varphi _j}{\partial x}\\&+(A_{44}-2c_2D_{44}+c_2^2F_{44})\frac{\partial \varphi _j}{\partial y}\bigg ]\bigg \}dxdy \end{aligned}$$
$$\begin{aligned}K^{54}_{ij}= & {} \int _{\Omega ^e}\bigg \{\frac{\partial \psi ^{(2)}_i}{\partial x}\bigg [(D_{16}-2c_1F_{16}+c_1^2H_{16})\frac{\partial \psi ^{(2)}_j}{\partial x}\\&+(D_{66}-2c_1F_{66}+c_1^2H_{66})\frac{\partial \psi ^{(2)}_j}{\partial y}\bigg ]\\&+\frac{\partial \psi ^{(2)}_i}{\partial y}\bigg [(D_{12}-2c_1F_{12}+c_1^2H_{12})\frac{\partial \psi ^{(2)}_j}{\partial x}\\&+(D_{26}-2c_1F_{26}+c_1^2H_{26})\frac{\partial \psi ^{(2)}_j}{\partial y}\bigg ]\\&+\psi ^{(2)}_i\bigg [(A_{45}-2c_2D_{45}+c_2^2F_{45})\psi ^{(2)}_j\bigg ]\bigg \}dxdy \end{aligned}$$
$$\begin{aligned}K^{55}_{ij}= & {} \int _{\Omega ^e}\bigg \{\frac{\partial \psi ^{(2)}_i}{\partial x}\bigg [(D_{26}-2c_1F_{26}+c_1^2H_{26})\frac{\partial \psi ^{(2)}_j}{\partial y}\\&+(D_{66}-2c_1F_{66}+c_1^2H_{66})\frac{\partial \psi ^{(2)}_j}{\partial x}\bigg ]\\&+\frac{\partial \psi ^{(2)}_i}{\partial y}\bigg [(D_{22}-2c_1F_{22}+c_1^2H_{22})\frac{\partial \psi ^{(2)}_j}{\partial y}\\&+(D_{26}-2c_1F_{26}+c_1^2H_{26})\frac{\partial \psi ^{(2)}_j}{\partial x}\bigg ]\\&+\psi ^{(2)}_i\bigg [(A_{44}-2c_2D_{44}+c_2^2F_{44})\psi ^{(2)}_j\bigg ]\bigg \}dxdy \end{aligned}$$
$$\begin{aligned}F_i^1= & {} \oint _{\Gamma ^e}(N_{xx} \hat{n}_x+N_{xy}\hat{n}_y)\psi ^{(1)}_ids,\\ F_i^2= & {} \oint _{\Gamma ^e}(N_{xy} \hat{n}_x+N_{yy}\hat{n}_y)\psi ^{(1)}_ids\\ 0= & {} \int _{\Omega ^e}q(1-\mu \nabla ^2)\varphi _idxdy +\oint _{\Gamma ^e}\\&\biggr \{(\bar{Q}_{x}\hat{n}_x+\bar{Q}_{y}\hat{n}_y)+( N_{xx}w_{0,x}+ N_{xy}w_{0,y})\hat{n}_x\\&+(N_{xy}w_{0,x}-N_{yy}w_{0,y})\hat{n}_y+c_1\\&\bigg [P_{xx,x} \hat{n}_x+P_{yy,y}\hat{n}_y+(P_{xy,x}\hat{n}_y+P_{xy,y}\hat{n}_x)\bigg ]\\&-c_1\bigg [P_{xx}\hat{n}_x+P_{yy}\hat{n}_y+(P_{xy}\hat{n}_y+P_{xy} \hat{n}_x)\bigg ]\biggr \}ds\\ F_i^4= & {} \oint _{\Gamma ^e}(M_{xx} \hat{n}_x+M_{xy}\hat{n}_y)\psi ^{(2)}_ids,\\ F_i^5= & {} \oint _{\Gamma ^e}(M_{xy} \hat{n}_x+M_{yy}\hat{n}_y)\psi ^{(2)}_ids \end{aligned}$$

Appendix C

Tangent Stiffness matrix term’s

$$\begin{aligned} T_{ij}^{11}= & {} K_{ij}^{11}+\sum _{\gamma =1}^{5}\sum _{k=1}^{n}\frac{\partial K_{ik}^{1\gamma }}{\partial u_j}\Delta ^\gamma _k=K_{ij}^{11},\\ T_{ij}^{12}= & {} K_{ij}^{12}+\sum _{\gamma =1}^{5}\sum _{k=1}^{n}\frac{\partial K_{ik}^{1\gamma }}{\partial v_j}\Delta ^\gamma _k=K_{ij}^{12}\\ T_{ij}^{13}= & {} K_{ij}^{13}+\sum _{\gamma =1}^{5}\sum _{k=1}^{n}\frac{\partial K_{ik}^{1\gamma }}{\partial w_j}\Delta ^\gamma _k=K_{ij}^{13}+\sum _{k=1}^{n}\frac{\partial K_{ik}^{13}}{\partial w_j}w_k\\= & {} K_{ij}^{13}+\int _{\Omega ^e}\bigg \{\frac{\partial \psi ^{(1)}_i}{\partial x}\bigg [\frac{1}{2}A_{11}\frac{\partial w}{\partial x}\frac{\partial \varphi _j}{\partial x}\\&+\frac{1}{2}A_{12}\frac{\partial w}{\partial y}\frac{\partial \varphi _j}{\partial y}+\frac{1}{2}A_{16}\bigg (\frac{\partial w}{\partial x}\frac{\partial \varphi _j}{\partial y}+\frac{\partial w}{\partial y}\frac{\partial \varphi _j}{\partial x}\bigg )\\&+\frac{\partial \psi ^{(1)}_i}{\partial y}\bigg [\frac{1}{2}A_{16}\frac{\partial w}{\partial x}\frac{\partial \varphi _j}{\partial x}+\frac{1}{2}A_{26}\frac{\partial w}{\partial y}\frac{\partial \varphi _j}{\partial y}\\&+\frac{1}{2}A_{66}\bigg (\frac{\partial w}{\partial x}\frac{\partial \varphi _j}{\partial y}+\frac{\partial w}{\partial y}\frac{\partial \varphi _j}{\partial x}\bigg )\bigg ]\bigg \}dxdy= T_{ji}^{31} \end{aligned}$$
$$\begin{aligned}T_{ij}^{14}= & {} K_{ij}^{14}+\sum _{\gamma =1}^{5}\sum _{k=1}^{n}\frac{\partial K_{ik}^{1\gamma }}{\partial X_j}\Delta ^\gamma _k=K_{ij}^{14},\end{aligned}$$
$$\begin{aligned}T_{ij}^{15}= & {} K_{ij}^{15}+\sum _{\gamma =1}^{5}\sum _{k=1}^{n}\frac{\partial K_{ik}^{1\gamma }}{\partial Y_j}\Delta ^\gamma _k=K_{ij}^{15} \end{aligned}$$
$$\begin{aligned}T_{ij}^{21}= & {} K_{ij}^{21}+\sum _{\gamma =1}^{5}\sum _{k=1}^{n}\frac{\partial K_{ik}^{2\gamma }}{\partial u_j}\Delta ^\gamma _k=K_{ij}^{21}, \end{aligned}$$
$$\begin{aligned}T_{ij}^{22}= & {} K_{ij}^{22}+\sum _{\gamma =1}^{5}\sum _{k=1}^{n}\frac{\partial K_{ik}^{2\gamma }}{\partial v_j}\Delta ^\gamma _k=K_{ij}^{22} \end{aligned}$$
$$\begin{aligned}T_{ij}^{23}= & {} K_{ij}^{23}+\sum _{\gamma =1}^{5}\sum _{k=1}^{n}\frac{\partial K_{ik}^{2\gamma }}{\partial w_j}\Delta ^\gamma _k=K_{ij}^{23}+\sum _{k=1}^{n}\frac{\partial K_{ik}^{23}}{\partial w_j}w_k\\= & {} K_{ij}^{23}+\int _{\Omega ^e}\bigg \{\frac{\partial \psi ^{(1)}_i}{\partial x}\bigg [\frac{1}{2}A_{16}\frac{\partial w}{\partial x}\frac{\partial \varphi _j}{\partial x}\\&+\frac{1}{2}A_{26}\frac{\partial w}{\partial y}\frac{\partial \varphi _j}{\partial y}+\frac{1}{2}A_{66}\bigg (\frac{\partial w}{\partial x}\frac{\partial \varphi _j}{\partial y}+\frac{\partial w}{\partial y}\frac{\partial \varphi _j}{\partial x}\bigg )\\&+\frac{\partial \psi ^{(1)}_i}{\partial y}\bigg [\frac{1}{2}A_{12}\frac{\partial w}{\partial x}\frac{\partial \varphi _j}{\partial x}+\frac{1}{2}A_{22}\frac{\partial w}{\partial y}\frac{\partial \varphi _j}{\partial y}\\&+\frac{1}{2}A_{26}\bigg (\frac{\partial w}{\partial x}\frac{\partial \varphi _j}{\partial y}+\frac{\partial w}{\partial y}\frac{\partial \varphi _j}{\partial x}\bigg )\bigg ]\bigg \}dxdy= T_{ji}^{32} \end{aligned}$$
$$\begin{aligned}T_{ij}^{24}= & {} K_{ij}^{24}+\sum _{\gamma =1}^{5}\sum _{k=1}^{n}\frac{\partial K_{ik}^{2\gamma }}{\partial X_j}\Delta ^\gamma _k=K_{ij}^{24}, \end{aligned}$$
$$\begin{aligned}T_{ij}^{25}= & {} K_{ij}^{25}+\sum _{\gamma =1}^{5}\sum _{k=1}^{n}\frac{\partial K_{ik}^{2\gamma }}{\partial Y_j}\Delta ^\gamma _k=K_{ij}^{25} \end{aligned}$$
$$\begin{aligned}T_{ij}^{31}= & {} K_{ij}^{31}+\sum _{\gamma =1}^{5}\sum _{k=1}^{n}\frac{\partial K_{ik}^{3\gamma }}{\partial u_j}\Delta ^\gamma _k=K_{ij}^{31}, \end{aligned}$$
$$\begin{aligned}T_{ij}^{32}= & {} K_{ij}^{32}+\sum _{\gamma =1}^{5}\sum _{k=1}^{n}\frac{\partial K_{ik}^{3\gamma }}{\partial v_j}\Delta ^\gamma _k=K_{ij}^{32} \end{aligned}$$
$$\begin{aligned} T_{ij}^{33}= & {} K_{ij}^{33}+\sum _{\gamma =1}^{5}\sum _{k=1}^{n}\frac{\partial K_{ik}^{3\gamma }}{\partial w_j}\Delta ^\gamma _k\\= & {} K_{ij}^{33}+\sum _{k=1}^{n}\bigg (\frac{\partial K_{ik}^{31}}{\partial w_j}u_k+\frac{\partial K_{ik}^{32}}{\partial w_j}v_k+\frac{\partial K_{ik}^{33}}{\partial w_j}w_k\\&+\frac{\partial K_{ik}^{34}}{\partial w_j}X_k+\frac{\partial K_{ik}^{35}}{\partial w_j}Y_k\bigg )-\frac{\partial F_i^{3T}}{\partial w_j}\\= & {} K_{ij}^{33}+\int _{\Omega ^e}\bigg [\frac{\partial \varphi _i}{\partial x}\bigg (A_{11}\frac{\partial u}{\partial x}\frac{\partial \psi _j^{(1)}}{\partial x}+A_{16}\frac{\partial u}{\partial x}\frac{\partial \psi _j^{(1)}}{\partial y}\\&+A_{16}\frac{\partial u}{\partial y}\frac{\partial \psi _j^{(1)}}{\partial x}+A_{66}\frac{\partial u}{\partial y}\frac{\partial \psi _j^{(1)}}{\partial y}\bigg )\\&+\frac{\partial \varphi _i}{\partial y}\bigg (A_{16}\frac{\partial u}{\partial x}\frac{\partial \psi _j^{(1)}}{\partial x}+A_{66}\frac{\partial u}{\partial x}\frac{\partial \psi _j^{(1)}}{\partial y}+A_{12}\frac{\partial u}{\partial y}\frac{\partial \psi _j^{(1)}}{\partial x}\\&+A_{26}\frac{\partial u}{\partial y}\frac{\partial \psi _j^{(1)}}{\partial y}\bigg ) \bigg ]dxdy\\&+\int _{\Omega ^e}\bigg [\frac{\partial \varphi _i}{\partial x}\bigg (A_{12}\frac{\partial v}{\partial x}\frac{\partial \psi _j^{(1)}}{\partial y}+A_{16}\frac{\partial v}{\partial x}\frac{\partial \psi _j^{(1)}}{\partial x}\\&+A_{26}\frac{\partial v}{\partial y}\frac{\partial \psi _j^{(1)}}{\partial y}+A_{66}\frac{\partial v}{\partial y}\frac{\partial \psi _j^{(1)}}{\partial x}\bigg )\\&+\frac{\partial \varphi _i}{\partial y}\bigg (A_{26}\frac{\partial v}{\partial x}\frac{\partial \psi _j^{(1)}}{\partial y}+A_{66}\frac{\partial v}{\partial x}\frac{\partial \psi _j^{(1)}}{\partial x}\\&+A_{22}\frac{\partial v}{\partial y}\frac{\partial \psi _j^{(1)}}{\partial y}\\&+A_{26}\frac{\partial v}{\partial y}\frac{\partial \psi _j^{(1)}}{\partial x}\bigg )\bigg ]dxdy\\&+\int _{\Omega ^e}\bigg [\bigg (A_{11}\bigg (\frac{\partial w}{\partial x}\bigg )^2+A_{12}\bigg (\frac{\partial w}{\partial y}\bigg )^2\\&+2A_{16}\frac{\partial w}{\partial x}\frac{\partial w}{\partial y}\bigg )\frac{\partial \varphi _i}{\partial x}\frac{\partial \varphi _j}{\partial x}\\&+\bigg (A_{16}\bigg (\frac{\partial w}{\partial x}\bigg )^2+A_{26}\bigg (\frac{\partial w}{\partial y}\bigg )^2\\&+2A_{66}\frac{\partial w}{\partial x}\frac{\partial w}{\partial y}\bigg )\bigg (\frac{\partial \varphi _i}{\partial x}\frac{\partial \varphi _j}{\partial y}+\frac{\partial \varphi _i}{\partial y}\frac{\partial \varphi _j}{\partial x}\bigg )\\&+\bigg (A_{12}\bigg (\frac{\partial w}{\partial x}\bigg )^2\\&+A_{22}\bigg (\frac{\partial w}{\partial y}\bigg )^2+2A_{26}\frac{\partial w}{\partial x}\frac{\partial w}{\partial y}\bigg )\frac{\partial \varphi _i}{\partial y}\frac{\partial \varphi _j}{\partial y}\\&+\frac{\partial \varphi _i}{\partial x}\bigg (-c_1E_{11}\frac{\partial w}{\partial x}\frac{\partial ^2\varphi _j}{\partial x^2}-c_1E_{12}\frac{\partial w}{\partial x}\frac{\partial ^2\varphi _j}{\partial y^2}\\&-2E_{16}\frac{\partial w}{\partial x}\frac{\partial \varphi _j}{\partial x\partial y}\\&-c_1E_{16}\frac{\partial w}{\partial y}\frac{\partial ^2\varphi _j}{\partial x^2}-c_1E_{26}\frac{\partial w}{\partial y}\frac{\partial ^2\varphi _j}{\partial y^2}\\&-2E_{66}\frac{\partial w}{\partial y}\frac{\partial \varphi _j}{\partial x\partial y}\bigg )\\&+\frac{\partial \varphi _i}{\partial y}\bigg (-c_1E_{16}\frac{\partial w}{\partial x}\frac{\partial ^2\varphi _j}{\partial x^2}-c_1E_{26}\frac{\partial w}{\partial x}\frac{\partial ^2\varphi _j}{\partial y^2}\\&-2E_{66}\frac{\partial w}{\partial x}\frac{\partial \varphi _j}{\partial x\partial y}\\&-c_1E_{12}\frac{\partial w}{\partial y}\frac{\partial ^2\varphi _j}{\partial x^2}-c_1E_{22}\frac{\partial w}{\partial y}\frac{\partial ^2\varphi _j}{\partial y^2}\\&-2E_{26}\frac{\partial w}{\partial y}\frac{\partial \varphi _j}{\partial x\partial y}\bigg )\\&+\frac{\partial ^2\varphi _i}{\partial x^2}\bigg (-\frac{1}{2}c_1E_{11}\frac{\partial w}{\partial x}\frac{\partial \varphi _j}{\partial x}-\frac{1}{2}c_1E_{12}\frac{\partial w}{\partial y}\frac{\partial \varphi _j}{\partial y}\\&+\frac{1}{2}E_{16}\bigg (\frac{\partial w}{\partial x}\frac{\partial \varphi _j}{\partial y}+\frac{\partial w}{\partial y}\frac{\partial \varphi _j}{\partial x}\bigg )\bigg )\\&+\frac{\partial ^2\varphi _i}{\partial y^2}\bigg (-\frac{1}{2}c_1E_{12}\frac{\partial w}{\partial x}\frac{\partial \varphi _j}{\partial x}-\frac{1}{2}c_1E_{22}\frac{\partial w}{\partial y}\frac{\partial \varphi _j}{\partial y}\\&+\frac{1}{2}E_{26}\bigg (\frac{\partial w}{\partial x}\frac{\partial \varphi _j}{\partial y}+\frac{\partial w}{\partial y}\frac{\partial \varphi _j}{\partial x}\bigg )\bigg )\\&+\frac{\partial ^2\varphi _i}{\partial x\partial y}\bigg (-c_1E_{16}\frac{\partial w}{\partial x}\frac{\partial \varphi _j}{\partial x}-c_1E_{26}\frac{\partial w}{\partial x}\frac{\partial \varphi _j}{\partial x}\\&-c_1E_{66}\bigg (\frac{\partial w}{\partial x}\frac{\partial \varphi _j}{\partial y}+\frac{\partial w}{\partial y}\frac{\partial \varphi _j}{\partial x}\bigg )\bigg )\bigg ]dxdy\\&+\int _{\Omega ^e}\bigg \{\frac{\partial \varphi _i}{\partial x}\bigg [(B_{16}-c_1E_{16})\frac{\partial \psi _j^{(2)}}{\partial x}\frac{\partial \phi _x}{\partial y}\\&+(B_{66}-c_1E_{66})\frac{\partial \psi _j^{(2)}}{\partial y}\frac{\partial \phi _x}{\partial y}\\&+(B_{11}-c_1E_{11})\frac{\partial \psi _j^{(2)}}{\partial x}\frac{\partial \phi _x}{\partial x}\\&+(B_{16}-c_1E_{16})\frac{\partial \psi _j^{(2)}}{\partial y}\frac{\partial \phi _x}{\partial x}\bigg ]\\&+\frac{\partial \varphi _i}{\partial y}\bigg [(B_{66}-c_1E_{66})\frac{\partial \psi _j^{(2)}}{\partial y}\frac{\partial \phi _x}{\partial x}\\&+(B_{16}-c_1E_{16})\frac{\partial \psi _j^{(2)}}{\partial x}\frac{\partial \phi _x}{\partial x}\\&+(B_{12}-c_1E_{12})\frac{\partial \psi _j^{(2)}}{\partial x}\frac{\partial \phi _x}{\partial y}\\&+(B_{26}-c_1E_{26})\frac{\partial \psi _j^{(2)}}{\partial y}\frac{\partial \phi _x}{\partial y}\bigg ] \bigg \}dxdy\\&+\int _{\Omega ^e}\bigg \{\frac{\partial \varphi _i}{\partial x}\bigg [(B_{12}-c_1E_{12})\frac{\partial \psi _j^{(2)}}{\partial y}\frac{\partial \phi _y}{\partial x}\\&+(B_{16}-c_1E_{16})\frac{\partial \psi _j^{(2)}}{\partial x}\frac{\partial \phi _y}{\partial x}\\&+(B_{26}-c_1E_{26})\frac{\partial \psi _j^{(2)}}{\partial y}\frac{\partial \phi _y}{\partial y}\\&+(B_{66}-c_1E_{66})\frac{\partial \psi _j^{(2)}}{\partial x}\frac{\partial \phi _y}{\partial y}\bigg ]\\&+\frac{\partial \varphi _i}{\partial y}\bigg [(B_{26}-c_1E_{26})\frac{\partial \psi _j^{(2)}}{\partial y}\frac{\partial \phi _y}{\partial x}\\&+(B_{66}-c_1E_{66})\frac{\partial \psi _j^{(2)}}{\partial x}\frac{\partial \phi _y}{\partial x}\\&+(B_{22}-c_1E_{22})\frac{\partial \psi _j^{(2)}}{\partial y}\frac{\partial \phi _y}{\partial y}\\&+(B_{26}-c_1E_{26})\frac{\partial \psi _j^{(2)}}{\partial x}\frac{\partial \phi _y}{\partial y}\bigg ]\bigg \}dxdy\end{aligned}$$
$$\begin{aligned} T_{ij}^{34}= & {} K_{ij}^{34}+\sum _{\gamma =1}^{5}\sum _{k=1}^{n}\frac{\partial K_{ik}^{3\gamma }}{\partial X_j}\Delta ^\gamma _k=K_{ij}^{34},\\ T_{ij}^{35}= & {} K_{ij}^{35}+\sum _{\gamma =1}^{5}\sum _{k=1}^{n}\frac{\partial K_{ik}^{3\gamma }}{\partial Y_j}\Delta ^\gamma _k=K_{ij}^{35}\\ T_{ij}^{41}= & {} K_{ij}^{41}+\sum _{\gamma =1}^{5}\sum _{k=1}^{n}\frac{\partial K_{ik}^{4\gamma }}{\partial u_j}\Delta ^\gamma _k=K_{ij}^{41},\\ T_{ij}^{42}= & {} K_{ij}^{42}+\sum _{\gamma =1}^{5}\sum _{k=1}^{n}\frac{\partial K_{ik}^{4\gamma }}{\partial v_j}\Delta ^\gamma _k=K_{ij}^{42}\end{aligned}$$
$$\begin{aligned}T_{ij}^{43}= & {} K_{ij}^{43}+\sum _{\gamma =1}^{5}\sum _{k=1}^{n}\frac{\partial K_{ik}^{4\gamma }}{\partial w_j}\Delta ^\gamma _k=K_{ij}^{43}+\sum _{k=1}^{n}\frac{\partial K_{ik}^{43}}{\partial w_j}w_k\\= & {} K_{ij}^{43}+\int _{\Omega ^e}\bigg \{\frac{\partial \psi ^{(2)}_i}{\partial x}\bigg [\frac{1}{2}(B_{11}-c_1E_{11})\frac{\partial w}{\partial x}\frac{\partial \varphi _j}{\partial x}\\&+\frac{1}{2}(B_{12}-c_1E_{12})\frac{\partial w}{\partial y}\frac{\partial \varphi _j}{\partial y}\\&+\frac{1}{2}(B_{16}-c_1E_{16})\bigg (\frac{\partial w}{\partial x}\frac{\partial \varphi _j}{\partial y}+\frac{\partial w}{\partial y}\frac{\partial \varphi _j}{\partial x}\bigg ) \bigg ]\\&+\frac{\partial \psi ^{(2)}_i}{\partial y}\bigg [\frac{1}{2}(B_{16}-c_1E_{16})\frac{\partial w}{\partial x}\frac{\partial \varphi _j}{\partial x}\\&+\frac{1}{2}(B_{26}-c_1E_{26})\frac{\partial w}{\partial y}\frac{\partial \varphi _j}{\partial y}+\frac{1}{2}(B_{66}-c_1E_{66})\\&\bigg (\frac{\partial w}{\partial x}\frac{\partial \varphi _j}{\partial y}+\frac{\partial w}{\partial y}\frac{\partial \varphi _j}{\partial x}\bigg )\bigg ]\bigg \}dxdy= T_{ji}^{34}\\ T_{ij}^{44}= & {} K_{ij}^{44}+\sum _{\gamma =1}^{5}\sum _{k=1}^{n}\frac{\partial K_{ik}^{4\gamma }}{\partial X_j}\Delta ^\gamma _k=K_{ij}^{44},\\ T_{ij}^{45}= & {} K_{ij}^{45}+\sum _{\gamma =1}^{5}\sum _{k=1}^{n}\frac{\partial K_{ik}^{4\gamma }}{\partial Y_j}\Delta ^\gamma _k=K_{ij}^{45}\\ T_{ij}^{51}= & {} K_{ij}^{51}+\sum _{\gamma =1}^{5}\sum _{k=1}^{n}\frac{\partial K_{ik}^{5\gamma }}{\partial u_j}\Delta ^\gamma _k=K_{ij}^{51},\\ T_{ij}^{52}= & {} K_{ij}^{52}+\sum _{\gamma =1}^{5}\sum _{k=1}^{n}\frac{\partial K_{ik}^{5\gamma }}{\partial v_j}\Delta ^\gamma _k=K_{ij}^{52}\end{aligned}$$
$$\begin{aligned}T_{ij}^{53}= & {} K_{ij}^{53}+\sum _{\gamma =1}^{5}\sum _{k=1}^{n}\frac{\partial K_{ik}^{5\gamma }}{\partial w_j}\Delta ^\gamma _k=K_{ij}^{53}+\sum _{k=1}^{n}\frac{\partial K_{ik}^{53}}{\partial w_j}w_k\\= & {} K_{ij}^{53}+\int _{\Omega ^e}\bigg \{\frac{\partial \psi ^{(2)}_i}{\partial x}\bigg [\frac{1}{2}(B_{16}-c_1E_{16})\frac{\partial w}{\partial x}\frac{\partial \varphi _j}{\partial x}\\&+\frac{1}{2}(B_{26}-c_1E_{26})\frac{\partial w}{\partial y}\frac{\partial \varphi _j}{\partial y}\\&+\frac{1}{2}(B_{66}-c_1E_{66})\bigg (\frac{\partial w}{\partial x}\frac{\partial \varphi _j}{\partial y}+\frac{\partial w}{\partial y}\frac{\partial \varphi _j}{\partial x}\bigg ) \bigg ]\\&+\frac{\partial \psi ^{(2)}_i}{\partial y}\bigg [\frac{1}{2}(B_{12}-c_1E_{12})\frac{\partial w}{\partial x}\frac{\partial \varphi _j}{\partial x}\\&+\frac{1}{2}(B_{22}-c_1E_{22})\frac{\partial w}{\partial y}\frac{\partial \varphi _j}{\partial y}\\&+\frac{1}{2}(B_{26}-c_1E_{26})\bigg (\frac{\partial w}{\partial x}\frac{\partial \varphi _j}{\partial y}+\frac{\partial w}{\partial y}\frac{\partial \varphi _j}{\partial x}\bigg )\bigg ] \bigg \}dxdy\\= & {} T_{ji}^{35} \end{aligned}$$
$$\begin{aligned}T_{ij}^{54}= & {} K_{ij}^{54}+\sum _{\gamma =1}^{5}\sum _{k=1}^{n}\frac{\partial K_{ik}^{5\gamma }}{\partial X_j}\Delta ^\gamma _k=K_{ij}^{54},\\ T_{ij}^{55}= & {} K_{ij}^{55}+\sum _{\gamma =1}^{5}\sum _{k=1}^{n}\frac{\partial K_{ik}^{5\gamma }}{\partial Y_j}\Delta ^\gamma _k=K_{ij}^{55}\end{aligned}$$

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Basant, K., Srividya, S., Gupta, R.K. et al. Nonlocal nonlinear analysis of functionally graded plates using natural neighbour Galerkin method. Ann. Solid Struct. Mech. 12, 97–122 (2020). https://doi.org/10.1007/s12356-020-00067-3

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Keywords

  • Functionally graded materials
  • Third-order plate theory
  • Eringen’s stress-gradient model
  • The von Kármán nonlinearity
  • Nonlocal effects