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Size-dependent effect on biaxial and shear nonlinear buckling analysis of nonlocal isotropic and orthotropic micro-plate based on surface stress and modified couple stress theories using differential quadrature method

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Abstract

The size-dependent effect on the biaxial and shear nonlinear buckling analysis of an isotropic and orthotropic micro-plate based on the surface stress, the modified couple stress theory (MCST), and the nonlocal elasticity theories using the differential quadrature method (DQM) is presented. Main advantages of the MCST over the classical theory (CT) are the inclusion of the asymmetric couple stress tensor and the consideration of only one material length scale parameter. Based on the nonlinear von Kármán assumption, the governing equations of equilibrium for the micro-classical plate considering midplane displacements are derived based on the minimum principle of potential energy. Using the DQM, the biaxial and shear critical buckling loads of the micro-plate for various boundary conditions are obtained. Accuracy of the obtained results is validated by comparing the solutions with those reported in the literature. A parametric study is conducted to show the effects of the aspect ratio, the side-to-thickness ratio, Eringen’s nonlocal parameter, the material length scale parameter, Young’s modulus of the surface layer, the surface residual stress, the polymer matrix coefficients, and various boundary conditions on the dimensionless uniaxial, biaxial, and shear critical buckling loads. The results indicate that the critical buckling loads are strongly sensitive to Eringen’s nonlocal parameter, the material length scale parameter, and the surface residual stress effects, while the effect of Young’s modulus of the surface layer on the critical buckling load is negligible. Also, considering the size dependent effect causes the increase in the stiffness of the orthotropic micro-plate. The results show that the critical biaxial buckling load increases with an increase in G 12/E 2 and vice versa for E 1/E 2. It is shown that the nonlinear biaxial buckling ratio decreases as the aspect ratio increases and vice versa for the buckling amplitude. Because of the most lightweight micro-composite materials with high strength/weight and stiffness/weight ratios, it is anticipated that the results of the present work are useful in experimental characterization of the mechanical properties of micro-composite plates in the aircraft industry and other engineering applications.

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References

  1. Eringen, A. C. On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. Journal of Applied Physics, 54, 4703–4710 (1983)

    Article  Google Scholar 

  2. Reddy, J. N. Nonlocal theories for bending, buckling and vibration of beams. International Journal of Engineering Science, 45, 288–307 (2007)

    Article  MATH  Google Scholar 

  3. Farajpour, A., Shahidi, A. R., Mohammadi, M., and Mahzoon, M. Buckling of orthotropic micro/nanoscale plates under linearly varying in-plane load via nonlocal continuum mechanics. Composite Structures, 94, 1605–1615 (2012)

    Article  Google Scholar 

  4. Peng, X. W., Guo, X. M., Liu, L., and Wu, B. J. Scale effects on nonlocal buckling analysis of bilayer composite plates under non-uniform uniaxial loads. Applied Mathematics and Mechanics (English Edition), 36(1), 1–10 (2015) DOI 10.1007/s10483-015-1900-7

    Article  MathSciNet  MATH  Google Scholar 

  5. Akgöz, B. and Civalek, Ö. Free vibration analysis for single-layered graphene sheets in an elastic matrix via modified couple stress theory. Materials and Design, 42, 164–171 (2012)

    Article  Google Scholar 

  6. Mohammadimehr, M., Mohandes, M., and Moradi, M. Size dependent effect on the buckling and vibration analysis of double-bonded nanocomposite piezoelectric plate reinforced by boron nitride nanotube based on modified couple stress theory. Journal of Vibration and Control (2014) DOI 10.1177/1077546314544513

    Google Scholar 

  7. Thai, H. T. and Vo, T. P. A size-dependent functionally graded sinusoidal plate model based on a modified couple stress theory. Composite Structures, 96, 376–383 (2013)

    Article  Google Scholar 

  8. Chorbanpour-Arani, A., Haghparast, E., Heidari-Rarani, M., and Khoddami-Maraghi, Z. Strain gradient shell model for nonlinear vibration analysis of visco-elastically coupled Boron Nitride nano-tube reinforced composite micro-tubes conveying viscous fluid. Computational Materials Science, 96, 448–458 (2015)

    Article  Google Scholar 

  9. Akgöz, B. and Civalek, Ö. Strain gradient elasticity and modified couple stress models for buckling analysis of axially loaded micro-scaled beams. International Journal of Engineering Science, 49, 1268–1280 (2011)

    Article  MathSciNet  Google Scholar 

  10. Xu, X. J. and Deng, Z. C. Variational principles for buckling and vibration of MWCNTs modeled by strain gradient theory. Applied Mathematics and Mechanics (English Edition), 35(9), 1115–1128 (2014) DOI 10.1007/s10483-014-1855-6

    Article  MathSciNet  MATH  Google Scholar 

  11. Mohammadi, H. and Mahzoon, M. Thermal effects on postbuckling of nonlinear microbeams based on the modified strain gradient theory. Composite Structures, 106, 764–776 (2013)

    Article  Google Scholar 

  12. Ansari, R., Mohammadi, V., Shojaei, M. F., Gholami, R., and Rouhi, H. Nonlinear vibration analysis of Timoshenko nanobeams based on surface stress elasticity theory. European Journal of Mechanics A/Solids, 45, 143–152 (2013)

    Article  MathSciNet  Google Scholar 

  13. Mohammadimehr, M., Mohammadi-Najafabadi, M. M., Nasiri, H., and Navi, B. R. Surface stress effects on the free vibration and bending analysis of the nonlocal single-layer graphene sheet embedded in an elastic medium using energy method. Proceedings of the Institution of Mechanical Engineers, Part N: Journal of Nanoengineering and Nanosystems (2014) DOI 10.1177/1740349914559042

    Google Scholar 

  14. Ke, L., Wang, Y., Yang, J., and Kitipornchai, S. Free vibration of size-dependent Mindlin microplates based on the modified couple stress theory. Journal of Sound and Vibration, 331, 94–106 (2012)

    Article  Google Scholar 

  15. Murmu, T. and Adhikari, S. Nonlocal elasticity based vibration of initially pre-stressed coupled nanobeam systems. European Journal of Mechanics A/Solids, 34, 52–62 (2012)

    Article  MathSciNet  Google Scholar 

  16. Analooei, H. R., Azhari, M., and Heidarpo, A. Elastic buckling and vibration analyses of orthotropic nanoplates using nonlocal continuum mechanics and spline finite strip method. Applied Mathematical Modelling, 37, 6703–6717 (2013)

    Article  MathSciNet  Google Scholar 

  17. Murmu, T., Sienz, J., Adhikari, S., and Arnold, C. Nonlocal buckling of double-nanoplate-systems under biaxial compression. Composite Part B: Engineering, 44, 84–94 (2013)

    Article  Google Scholar 

  18. Sahmani, S. and Ansari, R. Size-dependent buckling analysis of functionally graded third-order shear deformable microbeams including thermal environment effect. Applied Mathematical Modelling, 37, 9499–9515 (2013)

    Article  MathSciNet  Google Scholar 

  19. Xu, Y. M., Shen, H. S., and Zhang, C. L. Nonlocal plate model for nonlinear bending of bilayer graphene sheets subjected to transverse loads in thermal environments. Composite Structures, 98, 294–302 (2013)

    Article  Google Scholar 

  20. Anjomshoa, A., Shahidi, A. R., Hassani, B., and Jomehzadeh, E. Finite element buckling analysis of multi-layered graphene sheets on elastic substrate based on nonlocal elasticity theory. Applied Mathematical Modelling, 38, 5934–5955 (2014)

    Article  MathSciNet  Google Scholar 

  21. Jung, W., Han, S., and Park, W. A modified couple stress theory for buckling analysis of S-FGM nanoplates embedded in Pasternak elastic medium. Composite Part B: Engineering, 60, 746–756 (2014)

    Article  Google Scholar 

  22. Mohammadi, M., Farajpour, A., Moradi, A., and Ghayour, M. Shear buckling of orthotropic rectangular graphene sheet embedded in an elastic medium in thermal environment. Composite Part B: Engineering, 56, 629–637 (2014)

    Article  Google Scholar 

  23. Golmakani, M. E. and Rezatalab, J. Nonuniform biaxial buckling of orthotropic nanoplates embedded in an elastic medium based on nonlocal Mindlin plate theory. Composite Structures, 119, 238–250 (2015)

    Article  Google Scholar 

  24. Daneshmehr, A., Rajabpoor, A., and Pourdavood, M., Stability of size dependent functionally graded nanoplate based on nonlocal elasticity and higher order plate theories and different boundary conditions. International Journal of Engineering Science, 82, 84–100 (2014)

    Article  Google Scholar 

  25. Yao, G. and Li, F. M. Nonlinear vibration of a two-dimensional composite laminated plate in subsonic air flow. Journal of Vibration and Control, 21, 662–669 (2015)

    Article  Google Scholar 

  26. Yao, G. and Li, F. M. Chaotic motion of a composite laminated plate with geometric nonlinearity in subsonic flow. International Journal of Non-Linear Mechanics, 50, 81–90 (2013)

    Article  Google Scholar 

  27. Mohammadimehr, M., Rousta-Navi, B., and Ghorbanpour-Arani, A. Free vibration of viscoelastic double-bonded polymeric nanocomposite plates reinforced by FG-SWCNTs using MSGT, sinusoidal shear deformation theory and meshless method. Composite Structures, 131, 654–671 (2015)

    Article  Google Scholar 

  28. Reddy, J. N. Mechanics of Laminated Composite Plates and Shells-Theory and Analysis, 2nd ed., CRC Press, Boca Raton (2004)

    MATH  Google Scholar 

  29. Shu, C. and Richards, B. E. Application of generalized differential quadrature to solve twodimensional incompressible Navier-Stokes equations. International Journal of Numerical Methods in Fluids, 15, 791–798 (1992)

    Article  MATH  Google Scholar 

  30. Shu, C. Differential Quadrature and Its Application in Engineering, Springer, London (2000)

    Book  MATH  Google Scholar 

  31. Ventsel, E. and Krauthammer, T. Thin Plates and Shells: Theory, Analysis, and Application, Marcel Dekker, Inc., New York (2001)

    Book  Google Scholar 

  32. Pradhan, S. C. and Murmu, T. Small scale effect on the buckling analysis of single-layered graphene sheet embedded in an elastic medium based on nonlocal plate theory. Physica E, 42, 1293–1301 (2010)

    Article  Google Scholar 

  33. Malekzadeh, P., Setoodeh, A. R., and Beni, A. A. Small scale effect on the thermal buckling of orthotropic arbitrary straight-sided quadrilateral nanoplates embedded in an elastic medium. Composite Structures, 93, 2083–2089 (2011)

    Article  Google Scholar 

  34. Cook, I. T. and Rockey, K. C. Shear buckling of rectangular plates with mixed boundary conditions. Aeronautical Quarterly, 14, 349–356 (1963)

    Google Scholar 

  35. Bassily, S. F. and Dickinson, S. M. Buckling and lateral vibration of rectangular plates subject to in-plane loads a Ritz approach. Journal of Sound Vibration, 29, 219–239 (1972)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Solid Mechanics, Faculty of Mechanical Engineering, University of Kashan, Kashan, 87317-51167, Iran

    M. Mohammadimehr, M. A. Mohammadimehr & P. Dashti

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  1. M. Mohammadimehr
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  2. M. A. Mohammadimehr
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  3. P. Dashti
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Correspondence to M. Mohammadimehr.

Additional information

Project supported by the Iranian Nanotechnology Development Committee and the University of Kashan (No. 363452/10)

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Mohammadimehr, M., Mohammadimehr, M.A. & Dashti, P. Size-dependent effect on biaxial and shear nonlinear buckling analysis of nonlocal isotropic and orthotropic micro-plate based on surface stress and modified couple stress theories using differential quadrature method. Appl. Math. Mech.-Engl. Ed. 37, 529–554 (2016). https://doi.org/10.1007/s10483-016-2045-9

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  • DOI: https://doi.org/10.1007/s10483-016-2045-9

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Chinese Library Classification

2010 Mathematics Subject Classification

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