A finite element formulation using four-unknown incorporating nonlocal theory for bending and free vibration analysis of functionally graded nanoplates resting on elastic medium foundations

Abstract

A finite element model using four-unknown shear deformation theory integrated with the nonlocal theory is proposed for the bending and free vibration analysis of functionally graded (FG) nanoplates resting on elastic foundations. The present study developed the four-node quadrilateral element using Lagrangian and Hermitian interpolation functions for analysis of the membrane and bending displacement fields of FG nanoplates. Such a finite element formulation is suitable to investigate for the FG nanoplates resting on the elastic medium foundation with the stiffness matrices, the mass matrices and the load vectors using the second derivatives. The material properties of FG nanoplates are assumed to vary through the thickness direction by a power rule distribution of volume-fractions of the constituents. The equation of motion for FG nanoplates resting on the elastic foundation is obtained through Hamilton’s principle. Several numerical results are presented to demonstrate the accuracy and reliability of the present approach in comparison with other existing methods. In addition, the effects of geometrical parameters, material parameters, nonlocal parameters on the static bending and the free vibration responses of the nanoplates is also investigated in detail.

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Acknowledgements

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant number 107.02–2019.330.

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Correspondence to Quoc-Hoa Pham.

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Appendices

Appendix A

$$\begin{gathered} \lambdabar _{1} = \frac{1}{4}\left( {1 - \chi } \right)\left( {1 - \zeta } \right);\lambdabar _{2} = \frac{1}{4}\left( {1 + \chi } \right)\left( {1 - \zeta } \right); \hfill \\ \lambdabar _{3} = \frac{1}{4}\left( {1 + \chi } \right)\left( {1 + \zeta } \right);\lambdabar _{4} = \frac{1}{4}\left( {1 - \chi } \right)\left( {1 + \zeta } \right); \hfill \\ \hbar _{1} = \frac{1}{8}\left( {1 - \chi } \right)\left( {1 - \zeta } \right)\left( {2 - \chi - \zeta - \chi ^{2} - \zeta ^{2} } \right); \hfill \\ \hbar _{2} = \frac{1}{8}\left( {1 - \chi } \right)\left( {1 - \zeta } \right)\left( {1 - \chi ^{2} } \right)~ \hfill \\ \hbar _{3} = \frac{1}{8}\left( {1 - \chi } \right)\left( {1 - \zeta } \right)\left( {1 - \zeta ^{2} } \right) \hfill \\ \hbar _{4} = \frac{1}{8}\left( {1 + \chi } \right)\left( {1 - \zeta } \right)\left( {2 + \chi - \zeta - \chi ^{2} - \zeta ^{2} } \right); \hfill \\ \hbar _{5} = - \frac{1}{8}\left( {1 + \chi } \right)\left( {1 - \zeta } \right)\left( {1 - \chi ^{2} } \right) \hfill \\ \hbar _{6} = \frac{1}{8}\left( {1 + \chi } \right)\left( {1 - \zeta } \right)\left( {1 - \zeta ^{2} } \right) \hfill \\ \hbar _{7} = \frac{1}{8}\left( {1 + \chi } \right)\left( {1 + \zeta } \right)\left( {2 + \chi + \zeta - \chi ^{2} - \zeta ^{2} } \right); \hfill \\ \hbar _{8} = - \frac{1}{8}\left( {1 + \chi } \right)\left( {1 + \zeta } \right)\left( {1 - \chi ^{2} } \right); \hfill \\ \hbar _{9} = - \frac{1}{8}\left( {1 + \chi } \right)\left( {1 + \zeta } \right)\left( {1 - \zeta ^{2} } \right); \hfill \\ \hbar _{{10}} = \frac{1}{8}\left( {1 - \chi } \right)\left( {1 + \zeta } \right)\left( {2 - \chi + \zeta - \chi ^{2} - \zeta ^{2} } \right); \hfill \\ \hbar _{{11}} = \frac{1}{8}\left( {1 - \chi } \right)\left( {1 + \zeta } \right)\left( {1 - \chi ^{2} } \right); \hfill \\ \hbar _{{12}} = - \frac{1}{8}\left( {1 - \chi } \right)\left( {1 + \zeta } \right)\left( {1 - \zeta ^{2} } \right). \hfill \\ \end{gathered}$$

(\(\chi ;\text{\hspace{0.17em}}\zeta\): are natural coordinates)

Appendix B

$$\begin{gathered} B_{1} = \left[ {\begin{array}{*{20}l} {N_{u,x} } \\ {N_{v,y} } \\ {N_{u,y} + N_{v,x} } \\ \end{array} } \right];\;B_{2} = - \left[ {\begin{array}{*{20}l} {N_{wb,xx} } \\ {N_{wb,yy} } \\ {2N_{wb,xy} } \\ \end{array} } \right];\;B_{3} = - \left[ {\begin{array}{*{20}l} {N_{ws,xx} } \\ {N_{ws,yy} } \\ {2N_{ws,xy} } \\ \end{array} } \right];\;B_{1}^{b} = \left[ {\begin{array}{*{20}l} {N_{ws,x} } \\ {N_{ws,y} } \\ \end{array} } \right]; \hfill \\ B_{wx2} = \left[ {N_{wb,xx} + N_{ws,xx} } \right];B_{wy2} = \left[ {N_{wb,yy} + N_{ws,yy} } \right]; \hfill \\ B_{wx2} = \left[ {N_{wb,xx} + N_{ws,xx} } \right];\,B_{wy2} = \left[ {N_{wb,yy} + N_{ws,yy} } \right] \hfill \\ {\rm N} = \left[ {\begin{array}{*{20}l} {N_{u}^{T} } & {N_{v}^{T} } & {N_{wb}^{T} } & {N_{ws}^{T} } & {N_{wb,x}^{T} } & {N_{ws,x}^{T} } & {N_{wb,y}^{T} } & {N_{ws,y}^{T} } \\ \end{array} } \right]^{T} ;\,{\rm N}_{x} = \frac{\partial N}{{\partial x}};{\rm N}_{y} = \frac{\partial N}{{\partial y}} \hfill \\ D_{m} = \left[ {\begin{array}{*{20}l} {I_{0} } & 0 & 0 & 0 & {I_{1} } & {J_{1} } & 0 & 0 \\ {} & {I_{0} } & 0 & 0 & 0 & 0 & {I_{1} } & {J_{0} } \\ {} & {} & {I_{0} } & 0 & 0 & 0 & 0 & 0 \\ {} & {} & {} & {I_{0} } & 0 & 0 & 0 & 0 \\ {} & {} & {} & {} & {I_{2} } & {J_{2} } & 0 & 0 \\ {} & {} & {} & {} & {} & {K_{2} } & 0 & 0 \\ {} & {} & {} & {} & {} & {} & {I_{2} } & {J_{2} } \\ {sym} & {} & {} & {} & {} & {} & {} & {K_{2} } \\ \end{array} } \right] \hfill \\ \end{gathered}$$

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Tran, VK., Pham, QH. & Nguyen-Thoi, T. A finite element formulation using four-unknown incorporating nonlocal theory for bending and free vibration analysis of functionally graded nanoplates resting on elastic medium foundations. Engineering with Computers (2020). https://doi.org/10.1007/s00366-020-01107-7

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Keywords

  • Nonlocal elasticity
  • FG material
  • Elastic foundation
  • Four-unknown shear deformation theory
  • FEM