Vibration and frequency analysis of edge-cracked functionally graded graphene reinforced composite beam with piezoelectric actuators

Abstract

This paper investigates vibrations of the edge-cracked functionally graded graphene reinforced composite (FG-GRC) beam with the piezoelectric actuators. The edge crack is simulated by a rotational massless spring model. The effective Young modulus of the FG-GRC beam is estimated by utilizing the modified Halpin–Tsai model. The rule of mixture is applied to calculate the mass density and Poisson ratio of the FG-GRC beam. The total energy function of the edge-cracked FG-GRC piezoelectric beam is derived through using Timoshenko beam theory and von Kármán nonlinear strain–displacement relationship. The mechanical–electrical governing equations of motion for the edge-cracked FG-GRC piezoelectric beam are obtained by applying the standard Ritz procedure and are solved by the direct iterative method. The effectiveness and accuracy of this approach are verified through comparing the present results with other research results. Both uniformly and functionally graded (FG) distributed graphene nanoplatelets (GPLs) are considered to analyze influences of the GPL weight fraction, crack depth, crack location, boundary condition, thickness of the piezoelectric layer, and applied actuator voltage on the mechanical–electrical linear and nonlinear vibrations of the edge-cracked FG-GRC beam. The numerical results can help us predict the mechanical–electrical dynamic behaviors of the FG-GRC beam with cracks and promote the development of the structural health monitoring.

Appendices

Appendix A

The trail functions in Eqs. (48)–(52) are expressed in the following forms:

$$ \begin{gathered} U_{a} \left( X \right) = \sum\limits_{j = 1}^{n} {A_{j} } H_{1j} ,\;W_{a} \left( X \right) = \sum\limits_{j = 1}^{n} {B_{j} } H_{1j} , \hfill \\ U_{b} \left( X \right) = \sum\limits_{j = 1}^{n} {\tilde{A}_{j} } H_{2j} ,\;W_{b} \left( X \right) = \sum\limits_{j = 1}^{n} {\tilde{B}_{j} } H_{2j} , \hfill \\ U_{2} \left( X \right) = \sum\limits_{j = 1}^{n} {\tilde{A}_{j} } H_{3j} ,\;W_{2} \left( X \right) = \sum\limits_{j = 1}^{n} {\tilde{B}_{j} } H_{3j} , \hfill \\ \psi_{a} \left( X \right) = \sum\limits_{j = 1}^{n} {C_{j} } H_{4j} ,\;\psi_{b} \left( X \right) = \sum\limits_{j = 1}^{n} {\tilde{C}_{j} } H_{5j} ,\;{\text{and}} \hfill \\ \psi_{c} \left( X \right) = \sum\limits_{j = 1}^{n} {\tilde{A}_{j} } H_{6j} ,\;\psi_{2} \left( X \right) = \sum\limits_{j = 1}^{n} {\tilde{C}_{j} } H_{7j} . \hfill \\ \end{gathered} $$

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Appendix B

The elements of symmetric linear stiffness matrix ${\mathbf{K}}_{{\mathbf{L}}}$ in Eq. (54) are

$$ \begin{gathered} {\mathbf{K}}_{{{\mathbf{L}}\left( {j,m} \right)}} = \int_{0}^{{X_{0} }} {\left( {a_{{11}} \frac{{\partial H_{{1j}} }}{{\partial X}}\frac{{\partial H_{{1m}} }}{{\partial X}}} \right)} dX,\;{\mathbf{K}}_{{{\mathbf{L}}\left( {j,n + m} \right)}} = \int_{0}^{{X_{0} }} {\left( {a_{{11}} \frac{{\partial H_{{1j}} }}{{\partial X}}\frac{{\partial H_{{2m}} }}{{\partial X}} + b_{{11}} \frac{{\partial H_{{1j}} }}{{\partial X}}\frac{{\partial H_{{6m}} }}{{\partial X}}} \right)} dX, \hfill \\ {\mathbf{K}}_{{{\mathbf{L}}\left( {j,2n + m} \right)}} = {\mathbf{K}}_{{{\mathbf{L}}\left( {j,3n + m} \right)}} = 0,\;{\mathbf{K}}_{{{\mathbf{L}}\left( {j,4n + m} \right)}} = \int_{0}^{{X_{0} }} {\left( {b_{{11}} \frac{{\partial H_{{1j}} }}{{\partial X}}\frac{{\partial H_{{4m}} }}{{\partial X}}} \right)} dX, \hfill \\ {\mathbf{K}}_{{{\mathbf{L}}\left( {j,5n + m} \right)}} = \int_{0}^{{X_{0} }} {\left( {b_{{11}} \frac{{\partial H_{{1j}} }}{{\partial X}}\frac{{\partial H_{{5m}} }}{{\partial X}}} \right)} dX, \hfill \\ {\mathbf{K}}_{{{\mathbf{L}}\left( {n + j,n + m} \right)}} = \int_{0}^{{X_{0} }} {\left[ {a_{{11}} \frac{{\partial H_{{2j}} }}{{\partial X}}\frac{{\partial H_{{2m}} }}{{\partial X}} + b_{{11}} \left( {\frac{{\partial H_{{2j}} }}{{\partial X}}\frac{{\partial H_{{6m}} }}{{\partial X}} + \frac{{\partial H_{{2m}} }}{{\partial X}}\frac{{\partial H_{{6j}} }}{{\partial X}}} \right)} \right.} + d_{{11}} \frac{{\partial H_{{6j}} }}{{\partial X}}\frac{{\partial H_{{6m}} }}{{\partial X}} \hfill \\ \left. { + a_{{55}} \eta ^{2} H_{{6j}} H_{{6m}} } \right]dX + \int_{{X_{0} }}^{1} {\left( {a_{{11}} \frac{{\partial H_{{3j}} }}{{\partial X}}\frac{{\partial H_{{3m}} }}{{\partial X}}} \right)dX + K_{T}^{ * } \left( {H_{{6j}} H_{{6m}} } \right)_{{X = X_{0} }} } , \hfill \\ {\mathbf{K}}_{{{\mathbf{L}}\left( {n + j,2n + m} \right)}} = \int_{0}^{{X_{0} }} {\left( {a_{{55}} \eta H_{{6j}} \frac{{\partial H_{{1m}} }}{{\partial X}}} \right)} dX,\;{\mathbf{K}}_{{{\mathbf{L}}\left( {n + j,3n + m} \right)}} = \int_{0}^{{X_{0} }} {\left( {a_{{55}} \eta H_{{6j}} \frac{{\partial H_{{2m}} }}{{\partial X}}} \right)} dX, \hfill \\ {\mathbf{K}}_{{{\mathbf{L}}\left( {n + j,4n + m} \right)}} = \int_{0}^{{X_{0} }} {\left( {b_{{11}} \frac{{\partial H_{{2j}} }}{{\partial X}}\frac{{\partial H_{{4m}} }}{{\partial X}} + d_{{11}} \frac{{\partial H_{{6j}} }}{{\partial X}}\frac{{\partial H_{{4m}} }}{{\partial X}} + a_{{55}} \eta ^{2} H_{{6j}} H_{{4m}} } \right)\;} dX + K_{T}^{ * } \left( {H_{{6j}} H_{{4m}} } \right)_{{X = X_{0} }} , \hfill \\ {\mathbf{K}}_{{{\mathbf{L}}\left( {n + j,5n + m} \right)}} = \int_{0}^{{X_{0} }} {\left( {b_{{11}} \frac{{\partial H_{{2j}} }}{{\partial X}}\frac{{\partial H_{{5m}} }}{{\partial X}} + d_{{11}} \frac{{\partial H_{{6j}} }}{{\partial X}}\frac{{\partial H_{{5m}} }}{{\partial X}} + a_{{55}} \eta ^{2} H_{{6j}} H_{{5m}} } \right)\;} dX \hfill \\ + \int_{{X_{0} }}^{1} {\left( {b_{{11}} \frac{{\partial H_{{3j}} }}{{\partial X}}\frac{{\partial H_{{7m}} }}{{\partial X}}} \right)dX + K_{T}^{ * } \left( {H_{{6j}} H_{{5m}} - H_{{6j}} H_{{7m}} } \right)_{{X = X_{0} }} } , \hfill \\ {\mathbf{K}}_{{{\mathbf{L}}\left( {2n + j,2n + m} \right)}} = \int_{0}^{{X_{0} }} {\left( {a_{{55}} \frac{{\partial H_{{1j}} }}{{\partial X}}\frac{{\partial H_{{1m}} }}{{\partial X}} - \frac{1}{2}\tilde{N}^{P} \frac{{\partial H_{{1j}} }}{{\partial X}}\frac{{\partial H_{{1m}} }}{{\partial X}}} \right)} dX, \hfill \\ {\mathbf{K}}_{{{\mathbf{L}}\left( {2n + j,3n + m} \right)}} = \int_{0}^{{X_{0} }} {\left( {a_{{55}} \frac{{\partial H_{{1j}} }}{{\partial X}}\frac{{\partial H_{{2m}} }}{{\partial X}} - \frac{1}{2}\tilde{N}^{P} \frac{{\partial H_{{1j}} }}{{\partial X}}\frac{{\partial H_{{2m}} }}{{\partial X}}} \right)} dX, \hfill \\ {\mathbf{K}}_{{{\mathbf{L}}\left( {2n + j,4n + m} \right)}} = \int_{0}^{{X_{0} }} {\left( {a_{{55}} \eta \frac{{\partial H_{{1j}} }}{{\partial X}}H_{{4m}} } \right)} dX,\;{\mathbf{K}}_{{{\mathbf{L}}\left( {2n + j,5n + m} \right)}} = \int_{0}^{{X_{0} }} {\left( {a_{{55}} \eta \frac{{\partial H_{{1j}} }}{{\partial X}}H_{{5m}} } \right)} dX, \hfill \\ {\mathbf{K}}_{{{\mathbf{L}}\left( {3n + j,3n + m} \right)}} = \int_{0}^{{X_{0} }} {\left( {a_{{55}} \frac{{\partial H_{{2j}} }}{{\partial X}}\frac{{\partial H_{{2m}} }}{{\partial X}} - \frac{1}{2}\tilde{N}^{P} \frac{{\partial H_{{2j}} }}{{\partial X}}\frac{{\partial H_{{2m}} }}{{\partial X}}} \right)} dX \hfill \\ + \int_{{X_{0} }}^{1} {\left( {a_{{55}} \frac{{\partial H_{{3j}} }}{{\partial X}}\frac{{\partial H_{{3m}} }}{{\partial X}} - \frac{1}{2}\tilde{N}^{P} \frac{{\partial H_{{3j}} }}{{\partial X}}\frac{{\partial H_{{3m}} }}{{\partial X}}} \right)} dX, \hfill \\ {\mathbf{K}}_{{{\mathbf{L}}\left( {3n + j,4n + m} \right)}} = \int_{0}^{{X_{0} }} {\left( {a_{{55}} \frac{{\partial H_{{2j}} }}{{\partial X}}H_{{4m}} } \right)} dX, \hfill \\ {\mathbf{K}}_{{{\mathbf{L}}\left( {3n + j,5n + m} \right)}} = \int_{0}^{{X_{0} }} {\left( {a_{{55}} \frac{{\partial H_{{2j}} }}{{\partial X}}H_{{5m}} + a_{{55}} \frac{{\partial H_{{3j}} }}{{\partial X}}H_{{7m}} } \right)} dX, \hfill \\ {\mathbf{K}}_{{{\mathbf{L}}\left( {4n + j,4n + m} \right)}} = \int_{0}^{{X_{0} }} {\left( {d_{{11}} \frac{{\partial H_{{4j}} }}{{\partial X}}\frac{{\partial H_{{4m}} }}{{\partial X}} + a_{{55}} \eta ^{2} H_{{4j}} H_{{4m}} } \right)} dX + K_{T}^{ * } \left( {H_{{4j}} H_{{4m}} } \right)_{{X = X_{0} }} , \hfill \\ {\mathbf{K}}_{{{\mathbf{L}}\left( {4n + j,5n + m} \right)}} = \int_{0}^{{X_{0} }} {\left( {d_{{11}} \frac{{\partial H_{{4j}} }}{{\partial X}}\frac{{\partial H_{{5m}} }}{{\partial X}} + a_{{55}} \eta ^{2} H_{{4j}} H_{{5m}} } \right)} dX + K_{T}^{ * } \left( {H_{{4j}} H_{{5m}} - H_{{4j}} H_{{7m}} } \right)_{{X = X_{0} }} ,\;{\text{and}} \hfill \\ {\mathbf{K}}_{{{\mathbf{L}}\left( {5n + j,5n + m} \right)}} = \int_{0}^{{X_{0} }} {\left( {d_{{11}} \frac{{\partial H_{{5j}} }}{{\partial X}}\frac{{\partial H_{{5m}} }}{{\partial X}} + a_{{55}} \eta ^{2} H_{{5j}} H_{{5m}} } \right)} dX + \int_{{X_{0} }}^{1} {\left( {d_{{11}} \frac{{\partial H_{{7j}} }}{{\partial X}}\frac{{\partial H_{{7m}} }}{{\partial X}} + a_{{55}} \eta ^{2} H_{{7j}} H_{{7m}} } \right)} dX \hfill \\ + K_{T}^{ * } \left( {H_{{7j}} H_{{7m}} + H_{{5j}} H_{{5m}} - H_{{5j}} H_{{7m}} - H_{{7j}} H_{{5m}} } \right)_{{X = X_{0} }} . \hfill \\ \end{gathered} $$

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The elements of the symmetric mass matrix ${\mathbf{M}}$ in Eq. (54) are

$$ \begin{gathered} {\mathbf{M}}_{{\left( {j,m} \right)}} = \int_{0}^{{X_{0} }} {\tilde{I}_{1} H_{1j} H_{1m} } dX,\;{\mathbf{M}}_{{\left( {j,n + m} \right)}} = \int_{0}^{{X_{0} }} {\left( {\tilde{I}_{1} H_{1j} H_{2m} + \tilde{I}_{2} H_{1j} H_{6m} } \right)} dX, \hfill \\ {\mathbf{M}}_{{\left( {j,2n + m} \right)}} = {\mathbf{M}}_{{\left( {j,3n + m} \right)}} = 0,\;{\mathbf{M}}_{{\left( {j,4n + m} \right)}} = \int_{0}^{{X_{0} }} {\tilde{I}_{2} H_{1j} H_{4m} } dX,\;{\mathbf{M}}_{{\left( {j,5n + m} \right)}} = \int_{0}^{{X_{0} }} {\tilde{I}_{2} H_{1j} H_{5m} } dX, \hfill \\ {\mathbf{M}}_{{\left( {n + j,n + m} \right)}} = \int_{0}^{{X_{0} }} {\left( {\tilde{I}_{1} H_{2j} H_{2m} + \tilde{I}_{2} H_{2j} H_{6m} + \tilde{I}_{2} H_{2m} H_{6j} + \tilde{I}_{3} H_{6j} H_{6m} } \right)} dX + \int_{{X_{0} }}^{1} {\tilde{I}_{1} H_{3j} H_{3m} } dX, \hfill \\ {\mathbf{M}}_{{\left( {n + j,2n + m} \right)}} = {\mathbf{M}}_{{\left( {n + j,3n + m} \right)}} = 0,\;{\mathbf{M}}_{{\left( {n + j,4n + m} \right)}} = \int_{0}^{{X_{0} }} {\left( {\tilde{I}_{2} H_{2j} H_{4m} + \tilde{I}_{3} H_{6j} H_{4m} } \right)} dX, \hfill \\ {\mathbf{M}}_{{\left( {n + j,5n + m} \right)}} = \int_{0}^{{X_{0} }} {\left( {\tilde{I}_{2} H_{2j} H_{5m} + \tilde{I}_{3} H_{6j} H_{5m} } \right)} dX + \int_{{X_{0} }}^{1} {\tilde{I}_{2} H_{3j} H_{7m} } dX, \hfill \\ {\mathbf{M}}_{{\left( {2n + j,2n + m} \right)}} = \int_{0}^{{X_{0} }} {\tilde{I}_{1} H_{1j} H_{1m} } dX,\;{\mathbf{M}}_{{\left( {2n + j,3n + m} \right)}} = \int_{0}^{{X_{0} }} {\tilde{I}_{1} H_{1j} H_{2m} } dX, \hfill \\ {\mathbf{M}}_{{\left( {2n + j,4n + m} \right)}} = {\mathbf{M}}_{{\left( {2n + j,5n + m} \right)}} = 0,\;{\mathbf{M}}_{{\left( {3n + j,3n + m} \right)}} = \int_{0}^{{X_{0} }} {\tilde{I}_{1} H_{2j} H_{2m} } dX + \int_{{X_{0} }}^{1} {\tilde{I}_{1} H_{3j} H_{3m} } dX, \hfill \\ {\mathbf{M}}_{{\left( {3n + j,4n + m} \right)}} = {\mathbf{M}}_{{\left( {3n + j,5n + m} \right)}} = 0,\;{\mathbf{M}}_{{\left( {4n + j,4n + m} \right)}} = \int_{0}^{{X_{0} }} {\tilde{I}_{3} H_{4j} H_{4m} } dX,\;{\text{and}} \hfill \\ {\mathbf{M}}_{{\left( {4n + j,5n + m} \right)}} = \int_{0}^{{X_{0} }} {\tilde{I}_{3} H_{4j} H_{5m} } dX,\;{\mathbf{M}}_{{\left( {5n + j,5n + m} \right)}} = \int_{0}^{{X_{0} }} {\tilde{I}_{3} H_{5j} H_{5m} } dX + \int_{{X_{0} }}^{1} {\tilde{I}_{3} H_{7j} H_{7m} } dX. \hfill \\ \end{gathered} $$

(57)

The elements of the nonlinear stiffness matrices matrix ${\mathbf{K}}_{{{\mathbf{NL}}1}}$ and ${\mathbf{K}}_{{{\mathbf{NL}}2}}$ in Eq. (54) are given as

$$ \begin{gathered} {\mathbf{K}}_{{{\mathbf{NL}}1\left( {j,m} \right)}} = {\mathbf{K}}_{{{\mathbf{NL}}1\left( {j,n + m} \right)}} = 0,\;{\mathbf{K}}_{{{\mathbf{NL}}1\left( {j,2n + m} \right)}} = \int_{0}^{{X_{0} }} {\left( {\frac{{a_{{11}} }}{\eta }\frac{{\partial W_{1} }}{{\partial X}}\frac{{\partial H_{{1j}} }}{{\partial X}}\frac{{\partial H_{{1m}} }}{{\partial X}}} \right)} dX, \hfill \\ {\mathbf{K}}_{{{\mathbf{NL}}1\left( {j,3n + m} \right)}} = \int_{0}^{{X_{0} }} {\left( {\frac{{a_{{11}} }}{\eta }\frac{{\partial W_{1} }}{{\partial X}}\frac{{\partial H_{{1j}} }}{{\partial X}}\frac{{\partial H_{{2m}} }}{{\partial X}}} \right)} dX,\;{\mathbf{K}}_{{{\mathbf{NL}}1\left( {j,4n + m} \right)}} = {\mathbf{K}}_{{{\mathbf{NL}}1\left( {j,5n + m} \right)}} = {\mathbf{K}}_{{{\mathbf{NL}}1\left( {n + j,n + m} \right)}} = 0, \hfill \\ {\mathbf{K}}_{{{\mathbf{NL}}1\left( {n + j,2n + m} \right)}} = \int_{0}^{{X_{0} }} {\left( {\frac{{a_{{11}} }}{\eta }\frac{{\partial W_{1} }}{{\partial X}}\frac{{\partial H_{{2j}} }}{{\partial X}}\frac{{\partial H_{{1m}} }}{{\partial X}} + \frac{{b_{{11}} }}{\eta }\frac{{\partial W_{1} }}{{\partial X}}\frac{{\partial H_{{6j}} }}{{\partial X}}\frac{{\partial H_{{1m}} }}{{\partial X}}} \right)} dX, \hfill \\ {\mathbf{K}}_{{{\mathbf{NL}}1\left( {n + j,3n + m} \right)}} = \int_{0}^{{X_{0} }} {\left( {\frac{{a_{{11}} }}{\eta }\frac{{\partial W_{1} }}{{\partial X}}\frac{{\partial H_{{2j}} }}{{\partial X}}\frac{{\partial H_{{2m}} }}{{\partial X}} + \frac{{b_{{11}} }}{\eta }\frac{{\partial W_{1} }}{{\partial X}}\frac{{\partial H_{{6j}} }}{{\partial X}}\frac{{\partial H_{{2m}} }}{{\partial X}}} \right)} dX \hfill \\ + \int_{{X_{0} }}^{1} {\left( {\frac{{a_{{11}} }}{\eta }\frac{{\partial W_{2} }}{{\partial X}}\frac{{\partial H_{{3j}} }}{{\partial X}}\frac{{\partial H_{{3m}} }}{{\partial X}}} \right)} dX, \hfill \\ {\mathbf{K}}_{{{\mathbf{NL}}1\left( {n + j,4n + m} \right)}} = {\mathbf{K}}_{{{\mathbf{NL}}1\left( {n + j,5n + m} \right)}} = 0,\;{\mathbf{K}}_{{{\mathbf{NL}}1\left( {2n + j,2n + m} \right)}} = \int_{0}^{{X_{0} }} {\frac{{a_{{11}} }}{\eta }\left( {\frac{{\partial U_{1} }}{{\partial X}} + \frac{{\partial \psi _{1} }}{{\partial X}}} \right)} \frac{{\partial H_{{1j}} }}{{\partial X}}\frac{{\partial H_{{1m}} }}{{\partial X}}dX, \hfill \\ {\mathbf{K}}_{{{\mathbf{NL}}1\left( {2n + j,3n + m} \right)}} = \int_{0}^{{X_{0} }} {\frac{{a_{{11}} }}{\eta }\left( {\frac{{\partial U_{1} }}{{\partial X}} + \frac{{\partial \psi _{1} }}{{\partial X}}} \right)} \frac{{\partial H_{{1j}} }}{{\partial X}}\frac{{\partial H_{{2m}} }}{{\partial X}}dX, \hfill \\ {\mathbf{K}}_{{{\mathbf{NL}}1\left( {2n + j,4n + m} \right)}} = \int_{0}^{{X_{0} }} {\left( {\frac{{b_{{11}} }}{\eta }\frac{{\partial W_{1} }}{{\partial X}}\frac{{\partial H_{{1j}} }}{{\partial X}}\frac{{\partial H_{{4m}} }}{{\partial X}}} \right)} dX, \hfill \\ {\mathbf{K}}_{{{\mathbf{NL}}1\left( {2n + j,5n + m} \right)}} = \int_{0}^{{X_{0} }} {\left( {\frac{{b_{{11}} }}{\eta }\frac{{\partial W_{1} }}{{\partial X}}\frac{{\partial H_{{1j}} }}{{\partial X}}\frac{{\partial H_{{5m}} }}{{\partial X}}} \right)} dX,\;{\mathbf{K}}_{{{\mathbf{NL}}1\left( {3n + j,3n + m} \right)}} = 0, \hfill \\ {\mathbf{K}}_{{{\mathbf{NL}}1\left( {3n + j,4n + m} \right)}} = \int_{0}^{{X_{0} }} {\left( {\frac{{b_{{11}} }}{\eta }\frac{{\partial W_{1} }}{{\partial X}}\frac{{\partial H_{{2j}} }}{{\partial X}}\frac{{\partial H_{{4m}} }}{{\partial X}}} \right)} dX, \hfill \\ {\mathbf{K}}_{{{\mathbf{NL}}1\left( {3n + j,5n + m} \right)}} = \int_{0}^{{X_{0} }} {\left( {\frac{{b_{{11}} }}{\eta }\frac{{\partial W_{1} }}{{\partial X}}\frac{{\partial H_{{2j}} }}{{\partial X}}\frac{{\partial H_{{5m}} }}{{\partial X}}} \right)} dX + \int_{{X_{0} }}^{1} {\left( {\frac{{b_{{11}} }}{\eta }\frac{{\partial W_{2} }}{{\partial X}}\frac{{\partial H_{{3j}} }}{{\partial X}}\frac{{\partial H_{{7m}} }}{{\partial X}}} \right)} dX, \hfill \\ {\mathbf{K}}_{{{\mathbf{NL}}1\left( {4n + j,4n + m} \right)}} = {\mathbf{K}}_{{{\mathbf{NL}}1\left( {4n + j,5n + m} \right)}} = {\mathbf{K}}_{{{\mathbf{NL}}1\left( {5n + j,5n + m} \right)}} = 0, \hfill \\ {\mathbf{K}}_{{{\mathbf{NL}}2\left( {j,m} \right)}} = {\mathbf{K}}_{{{\mathbf{NL}}2\left( {j,n + m} \right)}} = {\mathbf{K}}_{{{\mathbf{NL}}2\left( {j,2n + m} \right)}} = {\mathbf{K}}_{{{\mathbf{NL}}2\left( {j,3n + m} \right)}} = {\mathbf{K}}_{{{\mathbf{NL}}2\left( {j,4n + m} \right)}} = {\mathbf{K}}_{{{\mathbf{NL}}2\left( {j,5n + m} \right)}} = 0, \hfill \\ {\mathbf{K}}_{{{\mathbf{NL}}2\left( {n + j,m} \right)}} = {\mathbf{K}}_{{{\mathbf{NL}}2\left( {n + j,n + m} \right)}} = {\mathbf{K}}_{{{\mathbf{NL}}2\left( {n + j,2n + m} \right)}} = {\mathbf{K}}_{{{\mathbf{NL}}2\left( {n + j,3n + m} \right)}} = 0, \hfill \\ {\mathbf{K}}_{{{\mathbf{NL}}2\left( {n + j,4n + m} \right)}} = {\mathbf{K}}_{{{\mathbf{NL}}2\left( {n + j,5n + m} \right)}} = 0, \hfill \\ {\mathbf{K}}_{{{\mathbf{NL}}2\left( {2n + j,2n + m} \right)}} = \frac{{3a_{{11}} }}{{2\eta ^{2} }}\int_{0}^{{X_{0} }} {\left( {\frac{{\partial W_{1} }}{{\partial X}}} \right)} ^{2} \frac{{\partial H_{{1j}} }}{{\partial X}}\frac{{\partial H_{{1m}} }}{{\partial X}}dX, \hfill \\ {\mathbf{K}}_{{{\mathbf{NL}}2\left( {2n + j,3n + m} \right)}} = \frac{{3a_{{11}} }}{{2\eta ^{2} }}\int_{0}^{{X_{0} }} {\left( {\frac{{\partial W_{1} }}{{\partial X}}} \right)} ^{2} \frac{{\partial H_{{1j}} }}{{\partial X}}\frac{{\partial H_{{2m}} }}{{\partial X}}dX, \hfill \\ {\mathbf{K}}_{{{\mathbf{NL}}2\left( {2n + j,4n + m} \right)}} = {\mathbf{K}}_{{{\mathbf{NL}}2\left( {2n + j,5n + m} \right)}} = 0, \hfill \\ {\mathbf{K}}_{{{\mathbf{NL}}2\left( {3n + j,3n + m} \right)}} = \frac{{3a_{{11}} }}{{2\eta ^{2} }}\int_{0}^{{X_{0} }} {\left( {\frac{{\partial W_{1} }}{{\partial X}}} \right)} ^{2} \frac{{\partial H_{{2j}} }}{{\partial X}}\frac{{\partial H_{{2m}} }}{{\partial X}}dX + \frac{{3a_{{11}} }}{{2\eta ^{2} }}\int_{{X_{0} }}^{1} {\left( {\frac{{\partial W_{2} }}{{\partial X}}} \right)} ^{2} \frac{{\partial H_{{3j}} }}{{\partial X}}\frac{{\partial H_{{3m}} }}{{\partial X}}dX,\;{\text{and}} \hfill \\ {\mathbf{K}}_{{{\mathbf{NL}}2\left( {3n + j,4n + m} \right)}} = {\mathbf{K}}_{{{\mathbf{NL}}2\left( {3n + j,5n + m} \right)}} = {\mathbf{K}}_{{{\mathbf{NL}}2\left( {4n + j,4n + m} \right)}} = {\mathbf{K}}_{{{\mathbf{NL}}2\left( {4n + j,5n + m} \right)}} = {\mathbf{K}}_{{{\mathbf{NL}}2\left( {5n + j,5n + m} \right)}} = 0. \hfill \\ \end{gathered} $$

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where j, m = 1, 2, …, n.