A size-dependent functionally graded nanocomposite Mindlin plate model based on consistent generalized continuum theory

• Autorzy: Roshanbakhsh M. Z. , Tavakkoli S. M.

Identyfikatory

DOI

10.24423/aom.4459

• Języki publikacji: EN

Abstrakty

EN

This paper presents an effective consistent-continuum model to analyse the behaviour of functionally graded nanocomposite (FG-NC) Mindlin plates based on the consistent couple stress theory (CCST) and the non-classical finite element method. A novel unified form is presented based on the Halpin–Tsai model to capture the small-scale heterogeneity, which can simultaneously consider the grading effects of the matrix and reinforcement phases along with the dispersion distribution through the plate thickness. To meet the C1 continuity requirements of the couple stress theory, a four-node rectangular element is adopted by using the Hermitian approach and in the way of a sub-parametric manner. The element has 20 degrees of freedom (DOF) at each node, which is reduced to 12 DOF in a bending mode without stretching deformation. FG-NC plates’ bending, free vibration, and buckling behaviour are investigated. Graphene oxide (GO), reduced graphene oxide (rGO), and silver-reduced graphene oxide (Ag-rGO) are considered for the dispersed phase. Size-dependent optimal values for the material and geometrical properties of the FG-NC plate model are presented, which minimize its mass with the frequency constraint. The effects of various parameters such as grading index, weight fraction, dispersion pattern, filler aspect/thickness ratio, and length scale parameter are examined, and benchmark examples are provided.

Słowa kluczowe

EN

C-CST; size-dependent finite element; FG-nanocomposite Mindlin plate; graphene-based reinforcement

• Wydawca: Instytut Podstawowych Problemów Techniki PAN
• Czasopismo: Archives of Mechanics
• Rocznik: 2024
• Tom: Vol. 76, nr 1-2
• Strony: 93--141
• Opis fizyczny: Bibliogr. 74 poz., rys., tab., wykr.

Twórcy

autor

Roshanbakhsh M. Z.

• Department of Civil Engineering, Shahrood University of Technology, Shahrood, Iran

autor

Tavakkoli S. M.

• Department of Civil Engineering, Shahrood University of Technology, Shahrood, Iran

Bibliografia

• 1. B.P. Mishra, D. Mishra, P. Panda, A. Maharana, An experimental investigation of the effects of reinforcement of graphene fillers on mechanical properties of bidirectional glass/epoxy composite, Materials Today: Proceedings, 33, 5429–5441, 2020, doi: 10.1016/J.MATPR.2020.03.154.
• 2. S. Najafishad, H.D. Manesh, S.M. Zebarjad, N. Hataf, Y. Mazaheri, Production and investigation of mechanical properties and electrical resistivity of cementmatrix nanocomposites with graphene oxide and carbon nanotube reinforcements, Archives of Civil and Mechanical Engineering, 20, 2, 57, 2020, doi: 10.1007/S43452-020-00059-5/METRICS.
• 3. Z. Hu, G. Tong, D. Lin, C. Chen, H. Guo, J. Xu, L. Zhou, Graphene-reinforced metal matrix nanocomposites – a review, Materials Science and Technology, 32, 9, 930–953, 2016, doi: 10.1080/02670836.2015.1104018.
• 4. G. Mittal, V. Dhand, K.Y. Rhee, S.J. Park, W.R. Lee, A review on carbon nanotubes and graphene as fillers in reinforced polymer nanocomposites, Journal of Industrial and Engineering Chemistry, 21, 11–25, 2015, doi: 10.1016/J.JIEC.2014.03.022.
• 5. G.A. Maugin, A.V. Metrikine, Mechanics of generalized continua: one hundred years after the Cosserats, Springer Science & Business Media, New York, 2010, doi: 10.1007/978-1-4419-5695-8.
• 6. A.C. Eringen, Nonlocal continuum field theories, Springer Science & Business Media, New York, 2002, doi: 10.1007/b97697.
• 7. A.C. Eringen, Microcontinuum Field Theories, Springer Science & Business Media, New York, 1999, doi: 10.1007/978-1-4612-0555-5.
• 8. R.D. Mindlin, H.F. Tiersten, Effects of couple-stresses in linear elasticity, Archive for Rational Mechanics and Analysis, 11, 1, 415–448, 1962, doi: 10.1007/BF00253946.
• 9. R.D. Mindlin, Influence of couple-stresses on stress concentrations, Experimental Mechanics, 3, 1, 1–7, 1963, doi: 10.1007/bf02327219.
• 10. F. Yang, A.C.M. Chong, D.C.C. Lam, P. Tong, Couple stress based strain gradient theory for elasticity, International Journal of Solids and Structures, 39, 10, 2731–2743, 2002, doi: 10.1016/S0020-7683(02)00152-X.
• 11. H. Wang, Z. Li, S. Zheng, Size-dependent deflection of cross-ply composite laminated plate induced by piezoelectric actuators based on a re-modified couple stress theory, Archives of Mechanics, 71, 3, 177–205, 2019, doi: 10.24423/AOM.3047.
• 12. I. Eshraghi, S. Dag, Transient dynamic analysis of functionally graded micro-beams considering small-scale effects, Archives of Mechanics, 73, 4, 303–337, 2021, doi: 10.24423/AOM.3786.
• 13. A.R. Hadjesfandiari, G.F. Dargush, Couple stress theory for solids, International Journal of Solids and Structures, 48, 18, 2496–2510, 2011, doi: 10.1016/J.IJSOLSTR.2011.05.002
• 14. A.R. Hadjesfandiari, G.F. Dargush, Fundamental solutions for isotropic size-dependent couple stress elasticity, International Journal of Solids and Structures, 50, 9, 1253–1265, 2013, doi: 10.1016/J.IJSOLSTR.2012.12.021.
• 15. A.R. Hadjesfandiari, Character of couple and couple-stress in continuum mechanics, 2022, doi: 10.48550/arXiv.2202.02152.
• 16. A. Hajesfandiari, A.R. Hadjesfandiari, G.F. Dargush, Boundary element formulation for plane problems in size-dependent piezoelectricity, International Journal for Numerical Methods in Engineering, 108, 7, 667–694, 2016, doi: 10.1002/nme.5227.
• 17. B.T. Darrall, A.R. Hadjesfandiari, G.F. Dargush, Size-dependent piezoelec-tricity: A 2D finite element formulation for electric field-mean curvature coupling in dielectrics, European Journal of Mechanics - A/Solids, 49, 308–320, 2015, doi: 10.1016/j.euromechsol.2014.07.013.
• 18. A.R. Hadjesfandiari, Size-dependent piezoelectricity, International Journal of Solids and Structures, 50, 18, 2781–2791, 2013, doi: 10.1016/J.IJSOLSTR.2013.04.020.
• 19. A.R. Hadjesfandiari, Size-dependent thermoelasticity, Latin American Journal of Solids and Structures, 11, 9, 1679–1708, 2014, doi: 10.1590/S1679-78252014000900010.
• 20. H. Vaghefpour, H. Arvin, Nonlinear free vibration analysis of pre-actuated isotropic piezoelectric cantilever Nano-beams, Microsystem Technologies, 25, 11, 4097–4110, 2019, doi: 10.1007/S00542-019-04351-0/METRICS.
• 21. G. Deng, G.F. Dargush, Mixed variational principle and finite element formulation for couple stress elastostatics, International Journal of Mechanical Sciences, 202, 106497, 2021, doi: 10.1016/j.ijmecsci.2021.106497.
• 22. F. Abbaspour, H. Arvin, Vibration and thermal buckling analyses of three-layered centrosymmetric piezoelectric microplates based on the modified consistent couple stress theory, Journal of Vibration and Control, 26, 15–16, 1253–1265, 2020, doi: 10.1177/1077546320924273.
• 23. B.N. Patel, D. Pandit, S.M. Srinivasan, A simplified moment-curvature based approach for large deflection analysis of micro-beams using the consistent couple stress theory, European Journal of Mechanics – A/Solids, 66, 45–54, 2017, doi: 10.1016/J.EUROMECHSOL.2017.06.002.
• 24. A.R. Hadjesfandiari, G.F. Dargush, Comparison of theoretical elastic couple stress predictions with physical experiments for pure torsion, 2016, doi: 10.48550/arXiv.1605.02556.
• 25. R. Soroush, A. Koochi, M. Keivani, M. Abadyan, A bilayer model for incorporating the coupled effects of surface energy and microstructure on the electromechanical stability of NEMS, International Journal of Structural Stability and Dynamics, 17, 1771005, 2017, doi: 10.1142/S0219455417710055.
• 26. A.K. Geim, K.S. Novoselov, The rise of graphene, Nature Materials, 6, 3, 183–191, 2007, doi: 10.1038/nmat1849.
• 27. P. Avouris, C. Dimitrakopoulos, Graphene: synthesis and applications, Materials Today, 15, 3, 86–97, 2012, doi: 10.1016/S1369-7021(12)70044-5.
• 28. A.M. Dimiev, S. Eigler, Graphene Oxide: Fundamentals and Applications, John Wiley & Sons, 2016, doi: 10.1002/9781119069447.
• 29. Y. Guo, X. Yang, K. Ruan, J. Kong, M. Dong, J. Zhang, J. Gu, Z. Guo, Reduced graphene oxide heterostructured silver nanoparticles significantly enhanced thermal conductivities in hot-pressed electrospun polyimide nanocomposites, ACS Applied Materials & Interfaces, 11, 28, 25465–25473, 2019, doi: 10.1021/acsami.9b10161.
• 30. S.K. Krishnan, E. Singh, P. Singh, M. Meyyappan, H.S. Nalwa, A review on graphene-based nanocomposites for electrochemical and fluorescent biosensors, RSC Advances, 9, 16, 8778–8881, 2019, doi: 10.1039/C8RA09577A.
• 31. T. Yang, W. Chen, F. Yan, H. Lu, Y.Q. Fu, Effect of reduced graphene oxides decorated by Ag and Ce on mechanical properties and electrical conductivity of copper matrix composites, Vacuum, 183, 109861, 2021, doi: 10.1016/J.VACUUM.2020.109861.
• 32. A. Kumar Srivastava, D. Kumar, Postbuckling behaviour of graphene-reinforced plate with interfacial effect, Archives of Mechanics, 70, 1, 3–36, 2018, doi: 10.24423/AOM.2796.
• 33. T. Farsadi, D. Asadi, H. Kurtaran, Frequency study of functionally graded multilayer graphene platelet-reinforced polymer cylindrical panels, Archives of Mechanics, 73, 5–6, 471–498, 2021, doi: 10.24423/AOM.3761.
• 34. Q. Qian, Y. Wang, F. Zhu, C. Feng, J. Yang, S. Wang, Primary nonlinear damped natural frequency of dielectric composite beam reinforced with graphene platelets (GPLs), Archives of Civil and Mechanical Engineering, 22, 1, 53, 2022, doi: 10.1007/S43452-021-00369-2/METRICS.
• 35. N. Guarín-Zapata, J. Gomez, A.R. Hadjesfandiari, G.F. Dargush, Variational principles and finite element Bloch analysis in couple stress elastodynamics, Wave Motion, 106, 102809, 2021, doi: 10.1016/J.WAVEMOTI.2021.102809.
• 36. G.F. Dargush, G. Apostolakis, A.R. Hadjesfandiari, Two- and three-dimensional size-dependent couple stress response using a displacement-based variational method, European Journal of Mechanics - A/Solids, 88, 104268, 2021, doi: 10.1016/J.EUROMECHSOL.2021.104268.
• 37. G. Apostolakis, G.F. Dargush, Size-dependent couple stress natural frequency analysis via a displacement-based variational method for two- and three-dimensional problems, Acta Mechanica, 234, 3, 891–910, 2023, doi: 10.1007/S00707-022-03421-1/METRICS.
• 38. H.X. Nguyen, T.N. Nguyen, M. Abdel-Wahab, S.P.A. Bordas, H. Nguyen-Xuan, T.P. Vo, A refined quasi-3D isogeometric analysis for functionally graded microplates based on the modified couple stress theory, Computer Methods in Applied Mechanics and Engineering, 313, 904–940, 2017, doi: 10.1016/J.CMA.2016.10.002.
• 39. A. Farzam, B. Hassani, Isogeometric analysis of in-plane functionally graded porous microplates using modified couple stress theory, Aerospace Science and Technology, 91, 508–524, 2019, doi: 10.1016/J.AST.2019.05.012.
• 40. H.M. Ma, X.L. Gao, J.N. Reddy, A non-classical Mindlin plate model based on a modified couple stress theory, Acta Mechanica, 220, 1–4, 217–235, 2011, doi: 10.1007/S00707-011-0480-4/METRICS.
• 41. B. Zhang, Y. He, D. Liu, Z. Gan, L. Shen, A non-classical Mindlin plate finite element based on a modified couple stress theory, European Journal of Mechanics - A/Solids, 42, 63–80, 2013, doi: 10.1016/J.EUROMECHSOL.2013.04.005.
• 42. C.H. Thai, A.J.M. Ferreira, T.D. Tran, P. Phung-Van, A size-dependent quasi-3D isogeometric model for functionally graded graphene platelet-reinforced composite microplates based on the modified couple stress theory, Composite Structures, 234, 111695, 2020, doi: 10.1016/J.COMPSTRUCT.2019.111695.
• 43. C.P. Wu, H.X. Hu, A unified size-dependent plate theory for static bending and free vibration analyses of micro- and nano-scale plates based on the consistent couple stress theory, Mechanics of Materials, 162, 104085, 2021, doi: 10.1016/J.MECHMAT.2021.104085.
• 44. C.P. Wu, E.-L. Lin, Free vibration analysis of porous functionally graded piezoelectric microplates resting on an elastic medium subjected to electric voltages, Archives of Mechanics, 74, 6, 463–511, 2022, doi: 10.24423/AOM.4150.
• 45. C.P. Wu, Y.A. Lu, A hermite-family c1 finite layer method for the three-dimensional free vibration analysis of exponentially graded piezoelectric microplates based on the consistent couple stress theory, International Journal of Structural Stability and Dynamics, 23, 4, 2350044, 2023, doi: 10.1142/S021945542350044X.
• 46. C.P. Wu, Y.S. Lyu, An asymptotic consistent couple stress theory for the three-dimensional free vibration analysis of functionally graded microplates resting on an elastic medium, Mathematical Methods in the Applied Sciences, 46, 4, 4891–4919, 2023, doi: 10.1002/MMA.8810.
• 47. C.P. Wu, C.H. Hsu, A three-dimensional weak formulation for stress, deformation, and free vibration analyses of functionally graded microscale plates based on the consistent couple stress theory, Composite Structures, 296, 115829, 2022, doi: 10.1016/J.COMPSTRUCT.2022.115829.
• 48. C.P. Wu, Y.A. Lu, 3D static bending analysis of functionally graded piezoelectric microplates resting on an elastic medium subjected to electro-mechanical loads using a size-dependent Hermitian C 2 finite layer method based on the consistent couple stress theory, Mechanics Based Design of Structures and Machines, 1–43, 2023, doi: 10.1080/15397734.2023.2209633.
• 49. H.P. Wu, Y. Shang, S. Cen, C.F. Li, Penalty C0 8-node quadrilateral and 20-node hexahedral elements for consistent couple stress elasticity based on the unsymmetric finite element method, Engineering Analysis with Boundary Elements, 147, 302–319, 2023, doi: 10.1016/J.ENGANABOUND.2022.12.008.
• 50. Y.H. Mao, Y. Shang, Y.D. Wang, Non-conforming Trefftz finite element implementation of orthotropic Kirchhoff plate model based on consistent couple stress theory, Acta Mechanica, 234, 5, 1857–1887, 2023, doi: 10.1007/S00707-023-03479-5/TABLES/13.
• 51. A. Farzam, B. Hassani, Isogeometric analysis of FG polymer nanocomposite plates reinforced with reduced graphene oxide using MCST, Advances in Aircraft and Spacecraft Science, 9, 1, 69–93, 2022, doi: 10.12989/aas.2022.9.1.069.
• 52. M.A. Roudbari, T.D. Jorshari, C. Lu, R. Ansari, A.Z. Kouzani, M. Amabili, A review of size-dependent continuum mechanics models for micro- and nano-structures, Thin-Walled Structures, 170, 108562, 2022, doi: 10.1016/J.TWS.2021.108562.
• 53. J. Reddy, Theory and analysis of elastic plates and shells, CRC press, 2006, doi: 10.1201/9780849384165.
• 54. M. Asghari, Geometrically nonlinear micro-plate formulation based on the modified couple stress theory, International Journal of Engineering Science, 51, 292–309, 2012, doi: 10.1016/J.IJENGSCI.2011.08.013.
• 55. J.N. Reddy, J. Berry, Nonlinear theories of axisymmetric bending of functionally graded circular plates with modified couple stress, Composite Structures, 94, 12, 3664–3668, 2012, doi: 10.1016/J.COMPSTRUCT.2012.04.019.
• 56. K.F. Wang, T. Kitamura, B. Wang, Nonlinear pull-in instability and free vibration of micro/nanoscale plates with surface energy – A modified couple stress theory model, International Journal of Mechanical Sciences, 99, 288–296, 2015, doi: 10.1016/J.IJMECSCI.2015.05.006.
• 57. M.Z. Roshanbakhsh, S.M. Tavakkoli, B. Navayi Neya, Free vibration of functionally graded thick circular plates: An exact and three-dimensional solution, International Journal of Mechanical Sciences, 188, 105967, 2020, doi: 10.1016/J.IJMECSCI.2020.105967.
• 58. S. Zhao, Z. Zhao, Z. Yang, L.L. Ke, S. Kitipornchai, J. Yang, Functionally graded graphene reinforced composite structures: A review, Engineering Structures, 210, 110339, 2020, doi: 10.1016/J.ENGSTRUCT.2020.110339.
• 59. M.A. Rafiee, J. Rafiee, Z. Wang, H. Song, Z.Z. Yu, N. Koratkar, Enhanced mechanical properties of nanocomposites at low graphene content, ACS Nano, 3, 12, 3884–3890, 2009, doi: 10.1021/NN9010472.
• 60. J.C. Haplin, J.L. Kardos, The Halpin–Tsai equations: A review, Polymer Engineering & Science, 16, 5, 344–352, 1976, doi: 10.1002/pen.760160512.
• 61. J.C. Halpin, R.L. Thomas, Ribbon reinforcement of composites, Journal of Composite Materials, 2, 4, 488–497, 1968, doi: 10.1177/002199836800200409.
• 62. F. Ebrahimi, A. Dabbagh, Mechanics of nanocomposites?: homogenization and analysis, CRC Press, 2020, doi: 10.1201/9780429316791.
• 63. J.N. Reddy, Introduction to the Finite Element Method, McGraw-Hill Education, 2005.
• 64. F.K. Bogner, R.L. Fox, L.A. Schmit, The generation of interelement compatible stiffness and mass matrices by the use of interpolation formulae, Proceedings of the First Conference on Matrix Methods in Structural Mechanics AFFDL-TR-66-80, 1966.
• 65. A. Adini, R.W. Clough, Analysis of plate Bending by the Finite Element Method, University of California, California, 1960.
• 66. D.C.C. Lam, F. Yang, A.C.M. Chong, J. Wang, P. Tong, Experiments and theory in strain gradient elasticity, Journal of the Mechanics and Physics of Solids, 51, 8, 1477–1508, 2003, doi: 10.1016/S0022-5096(03)00053-X.
• 67. X. Liu, Q. Yang, Molecular dynamic simulation of mechanical behaviour of RGO produced by thermal reduction method, Micro & Nano Letters, 12, 9, 638–642, 2017, doi: 10.1049/MNL.2017.0072.
• 68. J. Wan, J.W. Jiang, H.S. Park, Negative Poisson’s ratio in graphene oxide, Nanoscale, 9, 11, 4007–4012, 2017, doi: 10.1039/C6NR08657H.
• 69. M.R. Safaei, H.R. Goshayeshi, I. Chaer, Solar still efficiency enhancement by using graphene oxide/paraffin nano-PCM, Energies (Basel), 12, 10, 2002, 2019, doi: 10.3390/EN12102002.
• 70. M.F. Ashby, D.R.H. Jones, Engineering Materials 1: An Introduction to Properties, Applications, and Design, Elsevier, 2012, doi: 10.1016/C2009-0-64288-4.
• 71. G.C. Tsiatas, A new Kirchhoff plate model based on a modified couple stress theory, International Journal of Solids and Structures, 46, 13, 2757–2764, 2009, doi: 10.1016/J.IJSOLSTR.2009.03.004.
• 72. H.T. Thai, S.E. Kim, A size-dependent functionally graded Reddy plate model based on a modified couple stress theory, Composites Part B: Engineering, 45, 1, 1636–1645, 2013, doi: 10.1016/J.COMPOSITESB.2012.09.065.
• 73. H. Salehipour, H. Nahvi, A.R. Shahidi, H.R. Mirdamadi, 3D elasticity analytical solution for bending of FG micro/nanoplates resting on elastic foundation using modified couple stress theory, Applied Mathematical Modelling, 47, 174–188, 2017, doi: 10.1016/J.APM.2017.03.007.
• 74. H.T. Thai, D.H. Choi, Size-dependent functionally graded Kirchhoff and Mindlin plate models based on a modified couple stress theory, Composite Structures, 95, 142–153, 2013, doi: 10.1016/J.COMPSTRUCT.2012.08.023.