A third gradient-based continuum model for the mechanics of continua reinforced with extensible bidirectional fibers resistant to flexure

Abstract

The mechanics of fiber-reinforced composite materials is presented within the framework of the second strain gradient theory. As such, a continuum-based model is developed for the analysis of elastic materials reinforced with bidirectional fibers and subjected to finite plane deformations. The obtained model is subsequently applied to the cases of the unidirectional fiber composites for the purpose of model implementation. The Euler equilibrium equations and the associated boundary conditions are obtained via the variational principle and iterative integration by parts. In particular, we formulate the complete expressions of Piola-type triple stress and its coupled triple force arising in the third gradient of continuum deformations, which, in turn, yield the unique deformation maps in the presence of admissible boundary conditions of higher order. The solutions of the resulting systems of differential equations are obtained via the custom-built numerical scheme from which smooth and dilatational shear angle distributions are predicted throughout the entire domain of interest. It is also observed that the third gradient constitutive parameter is associated with the volume dilatation of third-gradient continua, which may be appeared in the form of shear band inclination angle.

Appendix: Finite element analysis of the sixth-order coupled PDE

The resulting systems of PDEs (Eqs. 87–89) are sixth-order coupled nonlinear differential equations. Demonstrating numerical analysis approaches for coupled PDE systems, especially those with higher order terms, is not trivial. For preprocessing, (Eqs. 87–89) can be recast as:

$$\begin{aligned}{} & {} \mu (Q+\chi _{1,22})-AS+BD+\frac{E_{1}}{2} (3QC^{2}+QD^{2}+2CRD-Q)-C_{1}M+A_{1}M_{,11}=0, \nonumber \\{} & {} \mu (R+\chi _{2,22})+AG-BC+\frac{E_{1}}{2} (3RD^{2}+RC^{2}+2CQD-R)-C_{1}N+A_{1}N_{,11}=0, \nonumber \\{} & {} Q-\chi _{1,11}=0, \nonumber \\{} & {} R-\chi _{2,11}=0, \nonumber \\{} & {} C-\chi _{1,1}=0, \nonumber \\{} & {} D-\chi _{2,1}=0, \nonumber \\{} & {} G-\chi _{1,2}=0, \nonumber \\{} & {} S-\chi _{2,2}=0, \nonumber \\{} & {} M-Q_{,11}=0, \nonumber \\{} & {} N-R_{,11}=0, \nonumber \\{} & {} A-\mu (Q+\chi _{1,22})-C_{1}M=0, \nonumber \\{} & {} B-\mu (R+\chi _{2,22})-C_{1}N=0, \end{aligned}$$

(104)

where $A=p_{,1},$ $B=p_{,2},$ $Q=\chi _{1,11},$ $R=\chi _{2,11},$ $C=\chi _{1,1},$ $D=\chi _{2,1},$ $G=\chi _{1,2},$ $S=\chi _{2,2},$ $M=Q_{,11}$ and $N=R_{,11}.$ As a result, we were able to reduce a sixth-order partial differential coupled system of equations to a second-order system of coupled PDEs. The above nonlinear terms (i.e., $A\chi _{2,2},B\chi _{2,1}$ etc.) can be treated via the Picard iterative procedure,

$$\begin{aligned}{} & {} -A_{\textrm{initial}}\chi _{2,2}^{\textrm{initial}}+B_{\textrm{initial}}\chi _{2,1}^{\textrm{initial}}\Rightarrow -A_{0}\chi _{2,2}^{0}+B_{0}\chi _{2,1}^{0}, \nonumber \\{} & {} \quad A_{\textrm{initial}}\chi _{1,2}^{\textrm{initial}}-B_{\textrm{initial}}\chi _{1,1}^{\textrm{initial}}\Rightarrow A_{0}\chi _{1,2}^{0}+B_{0}\chi _{1,1}^{0}, \nonumber \\{} & {} \quad -3Q_{\textrm{initial}}C_{\textrm{initial}}^{2}+Q_{\textrm{initial}}D_{\textrm{initial}}^{2}+2C_{\textrm{initial}}R_{\textrm{initial}}D_{\textrm{initial}}\Rightarrow 3Q_{0}C_{0}^{2}+Q_{0}D_{0}^{2}+2C_{0}R_{0}D_{0}, \nonumber \\{} & {} \quad -3R_{\textrm{initial}}D_{\textrm{initial}}^{2}+R_{\textrm{initial}}C^{2}+2C_{\textrm{initial}}Q_{\textrm{initial}}D_{\textrm{initial}}\Rightarrow 3R_{0}D_{0}^{2}+R_{0}C_{0}^{2}+2C_{0}Q_{0}D_{0}, \end{aligned}$$

(105)

where the values of A, B, C, D, G, S, Q,  and R continue to be refreshed based on their previous estimations (i.e., $A_{0},B_{0},C_{0},D_{0},G_{0},S_{0},Q_{0}$ and $R_{0}$) as iteration progresses. As a result, the above expression can be generalized to N number of iterations as

$$\begin{aligned}{} & {} -A_{N-1}\chi _{2,2}^{N-1}+B_{N-1}\chi _{2,1}^{N-1}\Rightarrow -A_{N}\chi _{2,2}^{N}+B_{N}\chi _{2,1}^{N}, \nonumber \\{} & {} \quad A_{N-1}\chi _{1,2}^{N-1}-B_{N-1}\chi _{1,1}^{N-1}\Rightarrow A_{N}\chi _{1,2}^{N}+B_{N}\chi _{1,1}^{N}, \nonumber \\{} & {} \quad 3Q_{N-1}C_{N-1}^{2}+Q_{N-1}D_{N-1}^{2}+2C_{N-1}R_{N-1}D_{N-1}\Rightarrow 3Q_{N}C_{N}^{2}+Q_{N}D_{N}^{2}+2C_{N}R_{N}D_{N}, \nonumber \\{} & {} \quad 3R_{N-1}D^{2}+R_{N-1}C_{N-1}^{2}+2C_{N-1}Q_{N-1}D_{N-1}\Rightarrow 3R_{N}D_{N}^{2}+R_{N}C_{N}^{2}+2C_{N}Q_{N}D_{N}. \end{aligned}$$

(106)

A convergence criteria can be used to determine the number of iterations. Thus, the weak form of Eq. (104) is obtained by

$$\begin{aligned}{} & {} \int _{\Omega ^{e}}w_{1}(\mu (Q+\chi _{1,22})-A_{0}S+B_{0}D+\frac{E_{1}}{2} (3QC_{0}^{2}+QD_{0}^{2}+2RC_{0}D_{0}-Q)-C_{1}M+A_{1}M_{,11})\textrm{d}\Omega =0, \nonumber \\{} & {} \int _{\Omega ^{e}}w_{2}(\mu (R+\chi _{2,22})+A_{0}G-B_{0}C+\frac{E_{1}}{2} (3RD_{0}^{2}+RC_{0}^{2}+2QC_{0}D_{0}-R)-C_{1}N+A_{1}N_{,11})\textrm{d}\Omega =0, \nonumber \\{} & {} \int _{\Omega ^{e}}w_{3}(Q-\chi _{1,11})\textrm{d}\Omega =0, \nonumber \\{} & {} \int _{\Omega ^{e}}w_{4}(R-\chi _{2,11})\textrm{d}\Omega =0, \nonumber \\{} & {} \int _{\Omega ^{e}}w_{5}\left( C-\chi _{1,1}\right) \textrm{d}\Omega =0, \nonumber \\{} & {} \int _{\Omega ^{e}}w_{6}\left( D-\chi _{2,1}\right) \textrm{d}\Omega =0, \nonumber \\{} & {} \int _{\Omega ^{e}}w_{7}\left( G-\chi _{1,2}\right) \textrm{d}\Omega =0, \nonumber \\{} & {} \int _{\Omega ^{e}}w_{8}\left( S-\chi _{2,2}\right) \textrm{d}\Omega =0, \nonumber \\{} & {} \int _{\Omega ^{e}}w_{9}(M-Q_{,11})\textrm{d}\Omega =0, \nonumber \\{} & {} \int _{\Omega ^{e}}w_{10}(N-R_{,11})\textrm{d}\Omega =0, \nonumber \\{} & {} \int _{\Omega ^{e}}w_{11}(A-\mu (Q+\chi _{1,22})-C_{1}M)\textrm{d}\Omega =0, \nonumber \\{} & {} \int _{\Omega ^{e}}w_{12}(B-\mu (R+\chi _{2,22})-C_{1}N)\textrm{d}\Omega =0. \end{aligned}$$

(107)

Using integration by parts and Green–Stoke’s theorem (e.g., $\int _{\Omega ^{e}}w_{1}\chi _{1,11}\textrm{d}\Omega =\int _{\partial \Gamma ^{e}}\left( w_{1}\chi _{1,1}\right) N\textrm{d}\Gamma -\int _{\Omega ^{e}}w_{1,1}\chi _{1,1}\textrm{d}\Omega$). We obtain from the above that

$$\begin{aligned}{} & {} \int _{\Omega ^{e}}w_{1}\mu Q-\mu w_{1,2}\chi _{1,2}-w_{1}A_{0}S+w_{1}B_{0}D+w_{1}\frac{E_{1}}{2} (3QC_{0}^{2}+QD_{0}^{2}+2RC_{0}D_{0}-Q)-w_{1}C_{1}M \nonumber \\{} & {} \quad -\, A_{1}w_{1,1}M_{,1})\textrm{d}\Omega +\int _{\partial \Gamma ^{e}}\left( \mu w_{1}\chi _{1,2}\right) N\textrm{d}\Gamma +\int _{\partial \Gamma ^{e}}\left( A_{1}w_{1}M_{,1}\right) N\textrm{d}\Gamma =0, \nonumber \\{} & {} \int _{\Omega ^{e}}(w_{2}\mu R-\mu w_{2,2}\chi _{2,2}+w_{2}A_{0}G-w_{2}B_{0}C+w_{2}\frac{E_{1}}{2} (3RD_{0}^{2}+RC_{0}^{2}+2QC_{0}D_{0}-R)-w_{2}C_{1}N \nonumber \\{} & {} \quad -\, A_{1}w_{2,1}N_{,1})\textrm{d}\Omega +\int _{\partial \Gamma ^{e}}\left( \mu w_{2}\chi _{2,2}\right) N\textrm{d}\Gamma +\int _{\partial \Gamma ^{e}}\left( A_{1}w_{2}N_{,1}\right) N\textrm{d}\Gamma =0, \nonumber \\{} & {} \int _{\Omega ^{e}}(w_{3}Q+w_{3,1}\chi _{1,1})\textrm{d}\Omega -\int _{\partial \Gamma ^{e}}\left( w_{3}\chi _{1,1}\right) N\textrm{d}\Gamma =0, \nonumber \\{} & {} \int _{\Omega ^{e}}(w_{4}R+w_{4,1}\chi _{2,1})\textrm{d}\Omega -\int _{\partial \Gamma ^{e}}\left( w_{4}\chi _{2,1}\right) N\textrm{d}\Gamma =0, \nonumber \\{} & {} \int _{\Omega ^{e}}\left( w_{5}C-w_{5}\chi _{1,1}\right) \textrm{d}\Omega =0, \nonumber \\{} & {} \int _{\Omega ^{e}}\left( w_{6}D-w_{6}\chi _{2,1}\right) \textrm{d}\Omega =0, \nonumber \\{} & {} \int _{\Omega ^{e}}\left( w_{7}G-w_{7}\chi _{1,2}\right) \textrm{d}\Omega =0, \nonumber \\{} & {} \int _{\Omega ^{e}}\left( w_{8}S-w_{8}\chi _{2,2}\right) \textrm{d}\Omega =0, \nonumber \\{} & {} \int _{\Omega ^{e}}(w_{9}M+w_{9,1}Q_{,1})\textrm{d}\Omega -\int _{\partial \Gamma ^{e}}\left( w_{9}Q_{,1}\right) N\textrm{d}\Gamma =0, \nonumber \\{} & {} \int _{\Omega ^{e}}(w_{10}N+w_{10,1}R_{,1})\textrm{d}\Omega -\int _{\partial \Gamma ^{e}}\left( w_{10}R_{,1}\right) N\textrm{d}\Gamma =0, \nonumber \\{} & {} \int _{\Omega ^{e}}(w_{11}A-w_{11}\mu Q+\mu w_{11,2}\chi _{1,2}-w_{11}C_{1}M)\textrm{d}\Omega -\int _{\partial \Gamma ^{e}}\left( \mu w_{11}\chi _{1,2}\right) N\textrm{d}\Gamma =0, \nonumber \\{} & {} \int _{\Omega ^{e}}(w_{12}B-w_{12}\mu R+\mu w_{12,2}\chi _{2,2}-w_{12}C_{1}N)\textrm{d}\Omega -\int _{\partial \Gamma ^{e}}\left( \mu w_{12}\chi _{2,2}\right) N\textrm{d}\Gamma =0, \end{aligned}$$

(108)

where $\Omega$, $\partial \Gamma$ and $\textbf{N}$ are the domain of interest, the associated boundary, and the rightward unit normal to the boundary in the sense of the Green–Stoke’s theorem respectively. The unknowns $\chi _{1},$ $\chi _{2},Q,$ R, C, D, G, S, M, N, A and B can be expressed in the form of Lagrangian polynomial as

$$\begin{aligned} \left( *\right) =\sum _{j=1}^{n=4}\left[ \left( *\right) _{j}\Psi _{j}(x,y)\right] , \end{aligned}$$

(109)

where $\left( *\right)$ represents any of the twelve unknowns. Therefore, the test function w is obtained as

$$\begin{aligned} \left( w_{k}\right) =\sum _{i=1}^{n=4}\left[ w_{k}^{i}\Psi _{i}(x,y)\right] ; \text { }k=1,2,3,4,\ldots 12, \end{aligned}$$

(110)

where $w_{i}$ is the weight of the test function and $\Psi _{i}(x,y)$ are the corresponding shape function for the four-node rectangular elements such that

$$\begin{aligned} \Psi _{1}=\frac{(x-2)(y-1)}{2},\Psi _{2}=\frac{x(y-1)}{-2},\Psi _{3}=\frac{xy }{2}\text { and }\Psi _{4}=\frac{y(x-2)}{-2}. \end{aligned}$$

(111)

Equation (108) can be reacst using Eqs. (109) and (110) as

$$\begin{aligned}{} & {} \sum _{i,j=1}^{n}\left\{ \int _{\Omega ^{e}}\left( \mu \Psi _{i}\Psi _{j}+3 \frac{E_{1}}{2}\Psi _{i}\Psi _{j}C_{0}^{2}+\frac{E_{1}}{2}\Psi _{i}\Psi _{j}D_{0}^{2}-\frac{E_{1}}{2}\Psi _{i}\Psi _{j}\right) \textrm{d}\Omega \right\} Q_{j}\\{} & {} \quad -\sum _{i,j=1}^{n}\left\{ \int _{\Omega ^{e}}\left( \mu \Psi _{i,2}\Psi _{j,2}\right) \textrm{d}\Omega \right\} \chi _{1j}\\{} & {} \quad -\sum _{i,j=1}^{n}\left\{ \int _{\Omega ^{e}}\left( \Psi _{i}\Psi _{j}A_{0}\right) \textrm{d}\Omega \right\} S_{j}+\sum _{i,j=1}^{n}\left\{ \int _{\Omega ^{e}}\left( \Psi _{i}\Psi _{j}B_{0}\right) \textrm{d}\Omega \right\} D_{j}\\{} & {} \quad -\sum _{i,j=1}^{n}\left\{ \int _{\Omega ^{e}}\left( \Psi _{i}\Psi _{j}C_{1}+\Psi _{i,1}\Psi _{j,1}A_{1}\right) \textrm{d}\Omega \right\} M_{j}\\{} & {} \quad +\sum _{i,j=1}^{n}\left\{ \int _{\Omega ^{e}}\left( \frac{E_{1}}{2}\Psi _{i}2\Psi _{j}C_{0}D_{0}\right) \textrm{d}\Omega \right\} R_{j}+\int _{\partial \Gamma ^{e}}\left( \mu \Psi _{i}\chi _{1,2}\right) N\textrm{d}\Gamma +\left( \int _{\partial \Gamma ^{e}}A_{1}\Psi _{i}M_{,1}\right) N\textrm{d}\Gamma =0, \\{} & {} \quad \times \, \sum _{i,j=1}^{n}\left\{ \int _{\Omega ^{e}}\left( \mu \Psi _{i}\Psi _{j}+3 \frac{E_{1}}{2}\Psi _{i}\Psi _{j}D_{0}^{2}+\frac{E_{1}}{2}\Psi _{i}\Psi _{j}C_{0}^{2}-\frac{E_{1}}{2}\Psi _{i}\Psi _{j}\right) \textrm{d}\Omega \right\} R_{j}\\{} & {} \quad -\sum _{i,j=1}^{n}\left\{ \int _{\Omega ^{e}}\left( \mu \Psi _{i,2}\Psi _{j,2}\right) \textrm{d}\Omega \right\} \chi _{2j}\\{} & {} \quad +\sum _{i,j=1}^{n}\left\{ \int _{\Omega ^{e}}\left( \Psi _{i}\Psi _{j}A_{0}\right) \textrm{d}\Omega \right\} G_{j}-\sum _{i,j=1}^{n}\left\{ \int _{\Omega ^{e}}\left( \Psi _{i}\Psi _{j}B_{0}\right) \textrm{d}\Omega \right\} C_{j}\\{} & {} \quad -\sum _{i,j=1}^{n}\left\{ \int _{\Omega ^{e}}\left( \Psi _{i}\Psi _{j}C_{1}+\Psi _{i,1}\Psi _{j,1}A_{1}\right) \textrm{d}\Omega \right\} N_{j}\\{} & {} \quad +\sum _{i,j=1}^{n}\left\{ \int _{\Omega ^{e}}\left( \frac{E_{1}}{2}\Psi _{i}2\Psi _{j}C_{0}D_{0}\right) \textrm{d}\Omega \right\} Q_{j}+\int _{\partial \Gamma ^{e}}\left( \mu \Psi _{i}\chi _{2,2}\right) N\textrm{d}\Gamma +\int _{\partial \Gamma ^{e}}\left( A_{1}\Psi _{i}N_{,1}\right) N\textrm{d}\Gamma =0,\\{} & {} \quad \times \, \sum _{i,j=1}^{n}\left\{ \int _{\Omega ^{e}}\left( \Psi _{i}\Psi _{j}\right) \textrm{d}\Omega \right\} Q_{j}+\sum _{i,j=1}^{n}\left\{ \int _{\Omega ^{e}}\left( \Psi _{i,1}\Psi _{j,1}\right) \textrm{d}\Omega \right\} \chi _{1j}-\int _{\partial \Gamma ^{e}}\left( \Psi _{i}\chi _{1,1}\right) N\textrm{d}\Gamma =0, \\{} & {} \quad \times \, \sum _{i,j=1}^{n}\left\{ \int _{\Omega ^{e}}\left( \Psi _{i}\Psi _{j}\right) \textrm{d}\Omega \right\} R_{j}+\sum _{i,j=1}^{n}\left\{ \int _{\Omega ^{e}}\left( \Psi _{i,1}\Psi _{j,1}\right) \textrm{d}\Omega \right\} \chi _{2j}-\int _{\partial \Gamma ^{e}}\left( \Psi _{i}\chi _{2,1}\right) N\textrm{d}\Gamma =0, \end{aligned}$$

$$\begin{aligned}{} & {} \quad \times \, \sum _{i,j=1}^{n}\left\{ \int _{\Omega ^{e}}\left( \Psi _{i}\Psi _{j}\right) \textrm{d}\Omega \right\} C_{j}-\sum _{i,j=1}^{n}\left\{ \int _{\Omega ^{e}}\left( \Psi _{i}\Psi _{j,1}\right) \textrm{d}\Omega \right\} \chi _{1j}=0, \nonumber \\{} & {} \quad \times \, \sum _{i,j=1}^{n}\left\{ \int _{\Omega ^{e}}\left( \Psi _{i}\Psi _{j}\right) \textrm{d}\Omega \right\} D_{j}-\sum _{i,j=1}^{n}\left\{ \int _{\Omega ^{e}}\left( \Psi _{i}\Psi _{j,1}\right) \textrm{d}\Omega \right\} \chi _{2j}=0, \nonumber \\{} & {} \quad \times \, \sum _{i,j=1}^{n}\left\{ \int _{\Omega ^{e}}\left( \Psi _{i}\Psi _{j}\right) \textrm{d}\Omega \right\} G_{j}-\sum _{i,j=1}^{n}\left\{ \int _{\Omega ^{e}}\left( \Psi _{i}\Psi _{j,2}\right) \textrm{d}\Omega \right\} \chi _{1j}=0, \nonumber \\{} & {} \quad \times \, \sum _{i,j=1}^{n}\left\{ \int _{\Omega ^{e}}\left( \Psi _{i}\Psi _{j}\right) \textrm{d}\Omega \right\} S_{j}-\sum _{i,j=1}^{n}\left\{ \int _{\Omega ^{e}}\left( \Psi _{i}\Psi _{j,2}\right) \textrm{d}\Omega \right\} \chi _{2j}=0, \nonumber \\{} & {} \quad \times \, \sum _{i,j=1}^{n}\left\{ \int _{\Omega ^{e}}\left( \Psi _{i}\Psi _{j}\right) \textrm{d}\Omega \right\} M_{j}+\sum _{i,j=1}^{n}\left\{ \int _{\Omega ^{e}}\left( \Psi _{i,1}\Psi _{j,1}\right) \textrm{d}\Omega \right\} Q_{j}-\int _{\partial \Gamma ^{e}}\left( \Psi _{i}Q_{,1}\right) N\textrm{d}\Gamma =0, \nonumber \\{} & {} \quad \times \, \sum _{i,j=1}^{n}\left\{ \int _{\Omega ^{e}}\left( \Psi _{i}\Psi _{j}\right) \textrm{d}\Omega \right\} N_{j}+\sum _{i,j=1}^{n}\left\{ \int _{\Omega ^{e}}\left( \Psi _{i,1}\Psi _{j,1}\right) \textrm{d}\Omega \right\} R_{j}-\int _{\partial \Gamma ^{e}}\left( \Psi _{i}R_{,1}\right) N\textrm{d}\Gamma =0, \nonumber \\{} & {} \quad \times \, \sum _{i,j=1}^{n}\left\{ \int _{\Omega ^{e}}\left( \Psi _{i}\Psi _{j}\right) \textrm{d}\Omega \right\} A_{j}-\sum _{i,j=1}^{n}\left\{ \int _{\Omega ^{e}}\left( \mu \Psi _{i}\Psi _{j}\right) \textrm{d}\Omega \right\} Q_{j}+\sum _{i,j=1}^{n}\left\{ \int _{\Omega ^{e}}\left( \mu \Psi _{i,2}\Psi _{j,2}\right) \textrm{d}\Omega \right\} \chi _{1j} \nonumber \\{} & {} \quad -\sum _{i,j=1}^{n}\left\{ \int _{\Omega ^{e}}\left( \Psi _{i}\Psi _{j}C_{1}\right) \textrm{d}\Omega \right\} M_{j}-\int _{\partial \Gamma ^{e}}\left( \mu \Psi _{i}\chi _{1,2}\right) N\textrm{d}\Gamma =0, \nonumber \\{} & {} \quad \times \, \sum _{i,j=1}^{n}\left\{ \int _{\Omega ^{e}}\left( \Psi _{i}\Psi _{j}\right) \textrm{d}\Omega \right\} B_{j}-\sum _{i,j=1}^{n}\left\{ \int _{\Omega ^{e}}\left( \mu \Psi _{i}\Psi _{j}\right) \textrm{d}\Omega \right\} R_{j}+\sum _{i,j=1}^{n}\left\{ \int _{\Omega ^{e}}\left( \mu \Psi _{i,2}\Psi _{j,2}\right) \textrm{d}\Omega \right\} \chi _{2j} \nonumber \\{} & {} \quad -\sum _{i,j=1}^{n}\left\{ \int _{\Omega ^{e}}\left( \Psi _{i}\Psi _{j}C_{1}\right) \textrm{d}\Omega \right\} N_{j}-\int _{\partial \Gamma ^{e}}\left( \mu \Psi _{i}\chi _{2,2}\right) N\textrm{d}\Gamma =0. \end{aligned}$$

(112)

The local stiffness matrix and forcing vector for each element can be found as

$$\begin{aligned} \begin{bmatrix} K_{11}^{11} &{} K_{12}^{11} &{} K_{13}^{11} &{} K_{14}^{11} \\ K_{21}^{11} &{} K_{22}^{11} &{} K_{23}^{11} &{} K_{24}^{11} \\ K_{31}^{11} &{} K_{32}^{11} &{} K_{33}^{11} &{} K_{34}^{11} \\ K_{41}^{11} &{} K_{42}^{11} &{} K_{43}^{11} &{} K_{44}^{11} \end{bmatrix} _{\text {Local}} \begin{bmatrix} \chi _{1}^{1} \\ \chi _{1}^{2} \\ \chi _{1}^{3} \\ \chi _{1}^{4} \end{bmatrix} _{\text {Local}}= \begin{bmatrix} F_{1}^{1} \\ F_{2}^{1} \\ F_{3}^{1} \\ F_{4}^{1} \end{bmatrix} _{\text {Local}}, \end{aligned}$$

(113)

or alternatively, in a compact form,

$$\begin{aligned} \begin{bmatrix} K_{ij}^{11} \end{bmatrix} \begin{bmatrix} \chi _{1}^{i} \end{bmatrix} = \begin{bmatrix} F_{i}^{1} \end{bmatrix} \text { for }i,j=1,2,3,4, \end{aligned}$$

(114)

where

$$\begin{aligned} \begin{bmatrix} K_{ij}^{11} \end{bmatrix} =-\int _{\Omega ^{e}}\left( \mu \Psi _{i,2}\Psi _{j,2}\right) \textrm{d}\Omega , \end{aligned}$$

(115)

and

$$\begin{aligned} \begin{bmatrix} F_{i}^{1} \end{bmatrix} =-\int _{\partial \Gamma ^{e}}\left( \mu \Psi _{i}\chi _{1,2}\right) N\textrm{d}\Gamma -\left( \int _{\partial \Gamma ^{e}}A_{1}\Psi _{i}M_{,1}\right) N\textrm{d}\Gamma . \end{aligned}$$

(116)

Finally, we obtain the following global systems of equations for each individual elements as

$$\begin{aligned} \begin{bmatrix} \left[ K^{11}\right] &{} \left[ K^{12}\right] &{} \left[ K^{13}\right] &{} \left[ K^{14}\right] &{}\ldots &{} \left[ K^{19}\right] &{} \left[ K^{110}\right] &{} \left[ K^{111}\right] &{} \left[ K^{112}\right] \\ \left[ K^{21}\right] &{} \left[ K^{22}\right] &{} \left[ K^{23}\right] &{} \left[ K^{24}\right] &{}\ldots &{} \left[ K^{29}\right] &{} \left[ K^{210}\right] &{} \left[ K^{211}\right] &{} \left[ K^{212}\right] \\ \left[ K^{31}\right] &{} \left[ K^{32}\right] &{} \left[ K^{33}\right] &{} \left[ K^{34}\right] &{}\ldots &{} \left[ K^{39}\right] &{} \left[ K^{310}\right] &{} \left[ K^{311}\right] &{} \left[ K^{312}\right] \\ \left[ K^{41}\right] &{} \left[ K^{42}\right] &{} \left[ K^{43}\right] &{} \left[ K^{44}\right] &{}\ldots &{} \left[ K^{49}\right] &{} \left[ K^{410}\right] &{} \left[ K^{411}\right] &{} \left[ K^{412}\right] \\ \left[ K^{51}\right] &{} \left[ K^{52}\right] &{} \left[ K^{53}\right] &{} \left[ K^{54}\right] &{}\ldots &{} \left[ K^{59}\right] &{} \left[ K^{510}\right] &{} \left[ K^{511}\right] &{} \left[ K^{512}\right] \\ \left[ K^{61}\right] &{} \left[ K^{62}\right] &{} \left[ K^{63}\right] &{} \left[ K^{64}\right] &{}\ldots &{} \left[ K^{69}\right] &{} \left[ K^{610}\right] &{} \left[ K^{611}\right] &{} \left[ K^{612}\right] \\ \left[ K^{71}\right] &{} \left[ K^{72}\right] &{} \left[ K^{73}\right] &{} \left[ K^{74}\right] &{}\ldots &{} \left[ K^{79}\right] &{} \left[ K^{710}\right] &{} \left[ K^{711}\right] &{} \left[ K^{712}\right] \\ \left[ K^{81}\right] &{} \left[ K^{82}\right] &{} \left[ K^{83}\right] &{} \left[ K^{84}\right] &{}\ldots &{} \left[ K^{89}\right] &{} \left[ K^{810}\right] &{} \left[ K^{811}\right] &{} \left[ K^{812}\right] \\ \left[ K^{91}\right] &{} \left[ K^{92}\right] &{} \left[ K^{93}\right] &{} \left[ K^{94}\right] &{}\ldots &{} \left[ K^{99}\right] &{} \left[ K^{910}\right] &{} \left[ K^{911}\right] &{} \left[ K^{912}\right] \\ \left[ K^{101}\right] &{} \left[ K^{102}\right] &{} \left[ K^{103}\right] &{} \left[ K^{104}\right] &{}\ldots &{} \left[ K^{109}\right] &{} \left[ K^{1010}\right] &{} \left[ K^{1011}\right] &{} \left[ K^{1012}\right] \\ \left[ K^{111}\right] &{} \left[ K^{112}\right] &{} \left[ K^{113}\right] &{} \left[ K^{114}\right] &{}\ldots &{} \left[ K^{119}\right] &{} \left[ K^{1110}\right] &{} \left[ K^{1111}\right] &{} \left[ K^{1112}\right] \\ \left[ K^{121}\right] &{} \left[ K^{122}\right] &{} \left[ K^{123}\right] &{} \left[ K^{124}\right] &{}\ldots &{} \left[ K^{129}\right] &{} \left[ K^{1210}\right] &{} \left[ K^{1211}\right] &{} \left[ K^{1212}\right] \end{bmatrix} \begin{bmatrix} \chi _{1}^{i} \\ \chi _{2}^{i} \\ Q_{i} \\ R_{i} \\ C_{i} \\ D_{i} \\ G_{i} \\ S_{i} \\ M_{i} \\ N_{i} \\ A_{i} \\ B_{i} \end{bmatrix} {\tiny =} \begin{bmatrix} \left\{ F^{1}\right\} \\ \left\{ F^{2}\right\} \\ \left\{ F^{3}\right\} \\ \left\{ F^{4}\right\} \\ \left\{ F^{5}\right\} \\ \left\{ F^{6}\right\} \\ \left\{ F^{7}\right\} \\ \left\{ F^{8}\right\} \\ \left\{ F^{9}\right\} \\ \left\{ F^{10}\right\} \\ \left\{ F^{11}\right\} \\ \left\{ F^{12}\right\} \end{bmatrix}. \end{aligned}$$

(117)