The primary objective of this study is to examine the bending behavior of graphene nanoplates under asymmetric multi-part conditions. Efforts focus on distinguishing the commonly used Modified Couple Stress Theory (MCST) and the newly proposed Nonlocal Modified Couple Stress Theory (NMCST) in capturing size dependency, including a detailed investigation of the influences of material length scale parameters (MLSPs). Constitutive equations for nonlinear bending analysis of orthotropic rectangular micro/nanoplates are derived using three-dimensional elasticity theory and NMCST. Unlike MCST, which uses a single MLSP and suits isotropic materials, NMCST incorporates three MLSPs, better suited to the non-isotropic nature of graphene nanoplates. This enables studying distinct effects of each MLSP, particularly under asymmetric conditions. Furthermore, NMCST offers deeper insight into the influence and material dependency of MLSP values. Employing 3D elasticity theory, the study avoids typical approximations in displacement fields defined by conventional plate theories. Consequently, governing equations based on 3D elasticity theory and NMCST are developed in Cartesian coordinates for the first time. These equations are numerically solved using a newly introduced Semi-Analytical Polynomial Method (SAPM), tailored to graphene’s orthotropic properties. The method addresses complex asymmetric conditions such as multi-part boundary scenarios, loading distributions, and elastic foundations—by segmenting the nanoplate and applying distinct conditions to each part. SAPM is presented as a robust technique for solving various partial differential equations under symmetric and asymmetric configurations. The numerical solution explores behavior and interaction of MLSPs across diverse conditions, emphasizing their dependency on the base material. Due to scarce data on bending behavior under multi-part conditions, validation is performed using Molecular Mechanics Method (MMM) simulations of graphene nanoplates. Results obtained via MM, FE, and SAPM approaches are compared and verified for reliability and consistency.
RafieeMARafieeJWangZ, et al. Enhanced mechanical properties of nanocomposites at low graphene content. ACS Nano2009; 3: 3884–3890.
2.
GolkarianARJabbarzadehM.The attitude of variation of elastic modules in single wall carbon nanotubes: nonlinear mass-spring model. J Solid Mech2012; 4: 106–113.
3.
GolkarianARJabbarzadehM.The density effect of van der Waals forces on the elastic modules in graphite layers. Comput Mater Sci2013; 74: 138–142.
4.
ShariatiAAGolkarianARJabbarzadehM.Investigation of vibrational behavior of perfect and defective carbon nanotubes using Non–Linear mass–spring model. Journal of Solid Mechanics2014; 6: 255–264.
5.
GolkarianARJabbarzadehM.The influence of interlayer interactions on the mechanical properties of polymeric nanocomposites. JSCS2015; 80(11): 1449–1459.
6.
BaoWXZhuCC.Simulation of young's modulus of single-walled carbon nanotubes by molecular dynamics. Physica B2044; 352: 156–163.
7.
ZhaoHMinKAluruNR.Size and chirality dependent elastic properties of graphene nanoribbons under uniaxial tension. Nano Lett2009; 9: 3012–3015.
8.
FleckNAMullerGMAshbyMF, et al. Strain gradient plasticity: theory and experiment. Acta Metall Mater1994; 42: 475–487.
9.
LamDCCYangFChongACM, et al. Experiments and theory in strain gradient elasticity. J Mech Phys Solids2003; 51: 1477–1508.
10.
LamDCCChongACM. Indentation model and strain gradient plasticity law for glassy polymers. J Mater Res1999; 14: 3784–3788.
11.
MaQClarkeDR.Size dependent hardness of silver single crystals. J Mater Res1995; 10: 853–863.
12.
StelmashenkoNAWallsMGBrownLM, et al. Microindentations on W and Mo oriented single crystals: an STM study. Acta Metall Mater1993; 41: 2855–2865.
13.
MindlinRDEshelNN.On first strain-gradient theories in linear elasticity. Int J Solids Struct1968; 4: 109–124.
14.
ToupinRA.Elastic materials with couple-stresses. Arch Ration Mech Anal1962; 11: 385–414.
15.
MindlinRDTierstenHF.Effects of couple-stresses in linear elasticity. Arch Ration Mech Anal1962; 11: 415–448.
16.
KoiterW.Couple stresses in the theory of elasticity, I and II. Nederl. Akad. Wetensch. Proc. Ser. B1964; 17–29.
17.
YangFChongACMLamDCC, et al. Couple stress based strain gradient theory for elasticity. Int J Solids Struct2002; 39: 2731–2743.
18.
EringenAC.Nonlocal polar elastic continua. Int J Eng Sci1972; 10(1): 1–16.
19.
GurtinMEWeissmullerJLarcheF.A general theory of curved deformable interfaces in solids at equilibrium. Philos Mag1998; 78: 1093–1109.
20.
AshooriAMahmoodiMJ.The modified version of strain gradient and couple stress theories in general curvilinear coordinates. Eur J Mech A Sol2015; 49: 441–454.
21.
ChenWLiX.A new modified couple stress theory for anisotropic elasticity and microscale laminated Kirchhoff plate model. Arch Appl Mech2014; 84(3): 323–341.
22.
MiandoabEMPishkenariHNYousefi-KomaA, et al. Polysilicon nano-beam model based on modified couple stress and Eringen’s nonlocal elasticity theories. Physica E Low Dimens Syst Nanostruct2014; 63: 223–228.
23.
LiZHeYLeiJ, et al. A standard experimental method for determining the material length scale based on modified couple stress theory. Int J Mech Sci2018; 141: 198–205.
24.
ParkSKGaoXL.Bernoulli–Euler beam model based on a modified couple stress theory. J Micromech Microeng2006; 16: 2355–2359.
25.
LeiJHeYGuoS, et al. Size-dependent vibration of nickel cantilever microbeams: experiment and gradient elasticity. AIP Adv2016; 6: 105202.
26.
Akbarzadeh KhorshidiM. The material length scale parameter used in couple stress theories is not a material constant. Int J Eng Sci2018; 133: 15–25.
27.
KaramiBGhayeshMHFantuzziN.Quasi-3D free and forced vibrations of poroelastic microplates in the framework of modified couple stress theory. Compos Struct2024; 330: 117840.
28.
BuiXBNguyenTKKaramanliA, et al. Size-dependent behaviours of functionally graded sandwich thin-walled beams based on the modified couple stress theory. Aerosp Sci Technol2023; 142: 108664.
29.
YeXMaHLiuX, et al. Size-dependent thermal bending of bilayer microbeam based on modified couple stress theory and Timoshenko beam theory. Eur J Mech A Sol2023; 100: 105029.
30.
ZhengYFZhouYWangF, et al. Nonlinear deformation analysis of magneto-electro-elastic nanobeams resting on elastic foundation by using nonlocal modified couple stress theory. Eur J Mech A Sol2024; 103: 105158.
31.
UzunBYaylıMÖ.Porosity dependent torsional vibrations of restrained FG nanotubes using modified couple stress theory. Mater Today Commun2022; 32: 103969.
32.
Al-ShewailiahDMRAl-ShujairiMA. Static bending of functionally graded single-walled carbon nanotube conjunction with modified couple stress theory. Mater Today Proc2022; 61(3): 1023–1037.
33.
SalehipourHNahviHShahidiH, et al. 3D elasticity analytical solution for bending of FG micro/nanoplates resting on elastic foundation using modified couple stress theory. Appl Math Model2017; 47: 174–188.
34.
KarimipourITadi BeniYArvinH, et al. Dynamic wave propagation in micro-torus structures: implementing a 3D physically realistic theory. Thin-Walled Struct2021; 165: 107995.
35.
KarimipourITadi BeniYAkbarzadehAH.Modified couple stress theory for three-dimensional elasticity in curvilinear coordinate system: application to micro torus panels. Meccanica2020; 55: 2033–2073.
36.
AziziBShariatiMSouqSSMN, et al. Bending and stretching behavior of graphene structures using continuum models calibrated with modal analysis. Appl Math Model2023; 114: 466–487.
37.
ReddyJNBerryJ.Nonlinear theories of axisymmetric bending of functionally graded circular plates with modified couple stress. Compos Struct2012; 94(12): 3664–3668.
38.
ReddyJNKimJ.A nonlinear modified couple stress-based third-order theory of functionally graded plates. Compos Struct2012; 94: 1128–1143.
39.
DastjerdiSJabbarzadehM.Nonlinear bending analysis of bilayer orthotropic graphene sheets resting on Winkler–Pasternak elastic foundation based on non-local continuum mechanics. Compos Part B2016; 87: 161–175.
40.
DastjerdiSJabbarzadehMAliabadiS.Nonlinear static analysis of single layer annular/circular graphene sheets embedded in Winkler–Pasternak elastic matrix based on non-local theory of Eringen. Ain Shams Eng J2016; 7: 873–884.
41.
DastjerdiSLotfiMJabbarzadehM.The effect of vacant defect on bending analysis of graphene sheets based on the Mindlin nonlocal elasticity theory. Compos Part B2016; 98: 78–87.
42.
DastjerdiSJabbarzadehM.Nonlocal bending analysis of bilayer annular/circular nano plates based on first order shear deformation theory. J of Solid Mech2016; 8: 645–661.
43.
DastjerdiSJabbarzadehM.Non-Local thermo-elastic buckling analysis of multi- layer annular/circular nano-plates based on first and Third Order Shear deformation theories using DQ method. J of Solid Mech2016; 8: 859–874.
44.
ThaiHTChoiDH.Size-dependent functionally graded Kirchhoff and Mindlin plate models based on a modified couple stress theory. Compos Struct2013; 95: 142–153.
45.
ThaiHTKimSE.A size-dependent functionally graded reddy plate model based on a modified couple stress theory. Compos Part B2013; 45: 1636–1645.
46.
ThaiHTVoTP.A size-dependent functionally graded sinusoidal plate model based on a modified couple stress theory. Compos Struct2013; 96: 376–383.
47.
SahmaniSAnsariR.On the free vibration response of functionally graded higher-order shear deformable microplates based on the strain gradient elasticity theory. Compos Struct2013; 95: 430–442.
48.
DaneshmehrARajabpoorApourdavoodM.Stability of size dependent functionally graded nanoplate based on nonlocal elasticity and higher order plate theories and different boundary conditions. Int J Eng Sci2014; 82: 84–100.
49.
TielkingJT.Asymmetric bending of annular plates. Int J Solids Struct1980; 16: 361–373.
50.
PardoenGC.Asymmetric bending of circular plates using the finite element method. Comput Struct1975; 5: 197–202.
51.
AliAJ.Mechanical modelling for the behaviour of the metallic plate under the effect of bending loads. Energy Proc2015; 74: 1119–1132.
52.
GolkarianAJabbarzadehMDastjerdiSH.A Novel Method for Numerical Analysis of 3D Nonlinear Thermo-Mechanical Bending of Annular and Circular Plates with Asymmetric Boundary Conditions Using SAPM. Journal of Solid Mechanics2019; 11(3): 498–512.
53.
YangZHeD.Vibration and buckling of orthotropic functionally graded micro-plates on the basis of a re-modified couple stress theory. Results Phys2017; 7: 3778–3787.
54.
Hosseini KordkheiliSAMoshrefzadeh-SaniH. Mechanical properties of double-layered graphene sheets. Comput Mater Sci2013; 69: 335–343.
55.
LiCChouT-W.A structural mechanics approach for the analysis of carbon nanotubes. Int J Solids Struct2003; 40: 2487–2499.
56.
TserpesKIPapanikosP.Finite element modeling of single-walled carbon nanotubes. Composites: Part B2005; 36: 468–477.
57.
AghababaeiRReddyJN.Nonlocal third-order shear deformation plate theory with application to bending and vibration of plates. J Sound Vib2009; 326: 277–289.
58.
SobhyM.Levy-type solution for bending of single-layered graphene sheets in thermal environment using the two-variable plate theory. Int J Mech Sci2015; 90: 171–178.
