Abstract
In the current investigation, based upon the nonlocal strain gradient theory of elasticity, an inhomogeneous size-dependent beam model is formulated within the framework of a refined hyperbolic shear deformation beam theory. Thereafter, via the constructed nonlocal strain gradient refined beam model, the nonlinear primary resonance of laminated functionally graded graphene platelet-reinforced composite (FG-GPLRC) microbeams under external harmonic excitation is studied in the presence of the both hardening-stiffness and softening-stiffness size effects. The graphene platelets are randomly dispersed in each individual layer in such a way that the weight fraction of the reinforcement varies on the basis of different patterns of FG dispersion. Based upon the Halpin–Tsai micromechanical scheme, the effective material properties of laminated FG-GPLRC microbeams are achieved. By putting the Hamilton’s principle to use, the nonlocal strain gradient equations of motion are developed. After that, a numerical solving process using the generalized differential quadrature (GDQ) method together with the Galerkin technique is employed to obtain the nonlocal strain gradient frequency response and amplitude response associated with the nonlinear primary resonance of laminated FG-GPLRC microbeams. It is found that the nonlocality size effect leads to an increase in the peak of the jump phenomenon and the associated excitation frequency, while the strain gradient size dependency results in a reduction in both of them.
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References
Yang J, Wu H, Kitipornchai S (2017) Buckling and postbuckling of functionally graded multilayer graphene platelet-reinforced composite beams. Compos Struct 161:111–118
Song M, Kitipornchai S, Yang J (2017) Free and forced vibrations of functionally graded polymer composite plates reinforced with graphene nanoplatelets. Compos Struct 159:579–588
Feng C, Kitipornchai S, Yang J (2017) Nonlinear bending of polymer nanocomposite beams reinforced with non-uniformly distributed graphene platelets (GPLs). Compos B Eng 110:132–140
Wu H, Yang J, Kitipornchai S (2017) Dynamic instability of functionally graded multilayer graphene nanocomposite beams in thermal environment. Compos Struct 162:244–254
Fu Y, Du H, Zhang S (2003) Functionally graded TiN/TiNi shape memory alloy films. Mater Lett 57:2995–2999
Fu Y, Du H, Huang W, Zhang S, Hu M (2004) TiNi-based thin films in MEMS applications: a review. Sens Actuators A 112:395–408
Kahrobaiyan MH, Asghari M, Rahaeifard M, Ahmadian MT (2010) Investigation of the size-dependent dynamic characteristics of atomic force microscope microcantilevers based on the modified couple stress theory. Int J Eng Sci 48:1985–1994
Şimşek M, Reddy JN (2013) A unified higher order beam theory for buckling of a functionally graded microbeam embedded in elastic medium using modified couple stress theory. Compos Struct 101:47–58
Lei J, He Y, Zhang B, Gan Z, Zeng P (2013) Bending and vibration of functionally graded sinusoidal microbeams based on the strain gradient elasticity theory. Int J Eng Sci 72:36–52
Reddy JN, El-Borgi S, Romanoff J (2014) Non-linear analysis of functionally graded microbeams using Eringen's non-local differential model. Int J Non Linear Mech 67:308–318
Jung W-Y, Han S-C, Park W-T (2014) A modified couple stress theory for buckling analysis of S-FGM nanoplates embedded in Pasternak elastic medium. Compos B Eng 60:746–756
Shojaeian M, Tadi Beni Y (2015) Size-dependent electromechanical buckling of functionally graded electrostatic nano-bridges. Sens Actuators A 232:49–62
Zhang B, He Y, Liu D, Shen L, Lei J (2015) Free vibration analysis of four-unknown shear deformable functionally graded cylindrical microshells based on the strain gradient elasticity theory. Compos Struct 119:578–597
Jung W-Y, Han S-C (2015) Static and eigenvalue problems of Sigmoid functionally graded materials (S-FGM) micro-scale plates using the modified couple stress theory. Appl Math Model 39:3506–3524
Sahmani S, Aghdam MM, Bahrami M (2015) On the free vibration characteristics of postbuckled third-order shear deformable FGM nanobeams including surface effects. Compos Struct 121:377–385
Kiani K (2016) Free dynamic analysis of functionally graded tapered nanorods via a newly developed nonlocal surface energy-based integro-differential model. Compos Struct 139:151–166
Akbarzadeh Khorshidi M, Shariati M, Emam SA (2016) Postbuckling of functionally graded nanobeams based on modified couple stress theory under general beam theory. Int J Mech Sci 110:160–169
Safaei B, Naseradinmousavi P, Rahmani A (2016) Development of an accurate molecular mechanics model for buckling behavior of multi-walled carbon nanotubes under axial compression. J Mol Graph Model 65:43–60
Sahmani S, Aghdam MM (2017) Imperfection sensitivity of the nonlinear axial buckling behavior of FGM nanoshells in thermal environments based on surface elasticity theory. Int J Comput Mater Sci Eng 6:1750003
Ziaee S (2017) The steady-state response of size-dependent functionally graded nanobeams to subharmonic excitation. J Eng Math 104:19–39
Tao C, Fu Y (2017) Thermal buckling and postbuckling analysis of size-dependent composite laminated microbeams based on a new modified couple stress theory. Acta Mech 228:1711–1724
Guo J, Chen J, Pan E (2017) Free vibration of three-dimensional anisotropic layered composite nanoplates based on modified couple-stress theory. Phys E 87:98–106
Sahmani S, Aghdam MM, Akbarzadeh A (2018) Surface stress effect on nonlinear instability of imperfect piezoelectric nanoshells under combination of hydrostatic pressure and lateral electric field. AUT J Mech Eng 2:177–190
Lu L, Guo X, Zhao J (2017) Size-dependent vibration analysis of nanobeams based on the nonlocal strain gradient theory. Int J Eng Sci 116:12–24
Sohi AN, Nieva PM (2018) Size-dependent effects of surface stress on resonance behavior of microcantilever-based sensors. Sens Actuators A 269:505–514
Attia MA, Abdel Rahman AA (2018) On vibrations of functionally graded viscoelastic nanobeams with surface effects. Int J Eng Sci 127:1–32
Sahmani S, Aghdam MM, Rabczuk T (2018) A unified nonlocal strain gradient plate model for nonlinear axial instability of functionally graded porous micro/nano-plates reinforced with graphene platelets. Mater Res Express 5:045048
Ghanati P, Safaei B (2019) Elastic buckling analysis of polygonal thin sheets under compression. Indian J Phys 93:47–52
Sahmani S, Fotouhi M, Aghdam MM (2019) Size-dependent nonlinear secondary resonance of micro-/nano-beams made of nano-porous biomaterials including truncated cube cells. Acta Mech 230:1077–1103
Safaei B, Moradi-Dastjerdi R, Chu F (2019) Effect of thermal gradient load on thermo-elastic vibrational behavior of sandwich plates reinforced by carbon nanotube agglomerations. Compos Struct 192:28–37
Lu L, Guo X, Zhao J (2019) A unified size-dependent plate model based on nonlocal strain gradient theory including surface effects. Appl Math Model 68:583–602
Safaei B, Moradi-Dastjerdi R, Qin Z, Chu F (2019) Frequency-dependent forced vibration analysis of nanocomposite sandwich plate under thermo-mechanical loads. Compos B Eng 161:44–54
Sahmani S, Safaei B (2019) Nonlinear free vibrations of bi-directional functionally graded micro/nano-beams including nonlocal stress and microstructural strain gradient size effects. Thin Walled Struct 140:342–356
Safaei B, Moradi-Dastjerdi R, Behdinan K, Chu F (2019) Critical buckling temperature and force in porous sandwich plates with CNT-reinforced nanocomposite layers. Aerosp Sci Technol 91:175–185
Safaei B, Moradi-Dastjerdi R, Qin Z, Behdinan K, Chu F (2019) Determination of thermoelastic stress wave propagation in nanocomposite sandwich plates reinforced by clusters of carbon nanotubes. J Sandwich Struct Mater. https://doi.org/10.1177/1099636219848282
Lim CW, Zhang G, Reddy JN (2015) A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation. J Mech Phys Solids 78:298–313
Li L, Hu Y (2015) Buckling analysis of size-dependent nonlinear beams based on a nonlocal strain gradient theory. Int J Eng Sci 97:84–94
Li L, Hu Y (2016) Wave propagation in fluid-conveying viscoelastic carbon nanotubes based on nonlocal strain gradient theory. Comput Mater Sci 112:282–288
Yang WD, Yang FP, Wang X (2016) Coupling influences of nonlocal stress and strain gradients on dynamic pull-in of functionally graded nanotubes reinforced nano-actuator with damping effects. Sens Actuators A 248:10–21
Simsek M (2016) Nonlinear free vibration of a functionally graded nanobeam using nonlocal strain gradient theory and a novel Hamiltonian approach. Int J Eng Sci 105:10–21
Sahmani S, Aghdam MM (2017) Size-dependent axial instability of microtubules surrounded by cytoplasm of a living cell based on nonlocal strain gradient elasticity theory. J Theor Biol 422:59–71
Sahmani S, Aghdam MM (2017) Nonlinear vibrations of pre-and post-buckled lipid supramolecular micro/nano-tubules via nonlocal strain gradient elasticity theory. J Biomech 65:49–60
Sahmani S, Aghdam MM (2018) Nonlinear instability of hydrostatic pressurized microtubules surrounded by cytoplasm of a living cell including nonlocality and strain gradient microsize dependency. Acta Mech 229:403–420
Sahmani S, Aghdam MM (2018) Nonlocal strain gradient beam model for postbuckling and associated vibrational response of lipid supramolecular protein micro/nano-tubules. Math Biosci 295:24–35
Li L, Tang H, Hu Y (2018) Size-dependent nonlinear vibration of beam-type porous materials with an initial geometrical curvature. Compos Struct 184:1177–1188
Radwan AF, Sobhy M (2018) A nonlocal strain gradient model for dynamic deformation of orthotropic viscoelastic graphene sheets under time harmonic thermal load. Phys B 538:74–84
Wang J, Shen H, Zhang B, Liu J, Zhang Y (2018) Complex modal analysis of transverse free vibrations for axially moving nanobeams based on the nonlocal strain gradient theory. Phys E 101:85–93
Sahmani S, Aghdam MM (2017) A nonlocal strain gradient hyperbolic shear deformable shell model for radial postbuckling analysis of functionally graded multilayer GPLRC nanoshells. Compos Struct 178:97–109
Sahmani S, Aghdam MM (2017) Nonlinear instability of axially loaded functionally graded multilayer graphene platelet-reinforced nanoshells based on nonlocal strain gradient elasticity theory. Int J Mech Sci 131:95–106
Sahmani S, Aghdam MM (2017) Nonlocal strain gradient beam model for nonlinear vibration of prebuckled and postbuckled multilayer functionally graded GPLRC nanobeams. Compos Struct 179:77–88
Sahmani S, Aghdam MM (2017) Axial postbuckling analysis of multilayer functionally graded composite nanoplates reinforced with GPLs based on nonlocal strain gradient theory. Eur Phys J Plus 132:490
Zeighampour H, Tadi Beni Y, Dekhordi MB (2018) Wave propagation in viscoelastic thin cylindrical nanoshell resting on a visco-Pasternak foundation based on nonlocal strain gradient theory. Thin Walled Struct 122:378–386
Halpin JC, Kardos JL (1976) The Halpin-Tsai equations: a review. Polym Eng Sci 16:344–352
Hejazi SM, Abtahi SM, Safaie F (2016) Investigation of thermal stress distribution in fiber reinforced roller compacted concrete pavements. J Ind Text 45:869–914
Faghih Shojaei M, Ansari R, Mohammadi V, Rouhi H (2014) Nonlinear forced vibration analysis of postbuckled beams. Arch Appl Mech 84:421–440
Liu F, Ming P, Li J (2007) Ab initio calculation of ideal strength and phonon instability of graphene under tension. Phys Rev B 76:064120
Rafiee MA, Rafiee J, Wang Z, Song H, Yu Z-Z, Koratkar N (2009) Enhanced mechanical properties of nanocomposites at low graphene content. ASC Nano 3:3884–3890
Sahmani S, Ansari R (2012) Small scale effect on vibrational response of single-walled carbon nanotubes with different boundary conditions based on nonlocal beam models. Commun Nonlinear Sci Numer Simul 17:1965–1979
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This work was supported by the National Natural Science Foundation of China (NSFC) (No. 61705200).
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Wu, Q., Chen, H. & Gao, W. Nonlocal strain gradient forced vibrations of FG-GPLRC nanocomposite microbeams. Engineering with Computers 36, 1739–1750 (2020). https://doi.org/10.1007/s00366-019-00794-1
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DOI: https://doi.org/10.1007/s00366-019-00794-1