Direct flexoelectricity appears in dielectrics as a kind of coupling effect of electric field and elastic strain gradient. Its strong size dependence makes it effective in nanostructures. In this paper, based on the flexoelectric theory and mixed finite element method (FEM), the mechanical properties of the two-dimensional functionally graded (FG) flexoelectric nanobeam are studied, and the phenomenon of the deflection platform is found. By applying various types of loads to FG nanobeams, the influence of geometric parameters, length scale, and material distribution on the phenomenon of the deflection platform is discussed. According to the simulation results, it is found that the aspect ratio and the length scale of the beam will change the shape of the deflection platform, making it change regularly among various shapes. The geometric size will greatly increase the height of the platform, but it has little effect on the dimensionless deflection. Reasonable material distribution can enhance the internal coupling effect, increase the height of the platform and reduce the required load. Our results suggest that when studying nano-scale flexoelectric models, the deflection platforms or similar phenomena should be considered.
AbdollahiAPecoCMillánD, et al. (2014) Computational evaluation of the flexoelectric effect in dielectric solids. Journal of Applied Physics116: 093502.
2.
AimmaneeSTichakornK (2018) Piezoelectrically induced snap-through buckling in a buckled beam bonded with a segmented actuator. Journal of Intelligent Material Systems and Structures29: 1862–1874.
3.
AmanatidouEAravasN (2002) Mixed finite element formulations of strain-gradient elasticity problems. Computer Methods in Applied Mechanics and Engineering191: 1723–1751.
4.
AmdouniSMansouriKRenardY, et al. (2012) Numerical convergence and stability of mixed formulation with x-fem cut-off. European Journal of Computational Mechanics/Revue Européenne de Mécanique Numérique21: 160–173.
5.
AravasN (2011) Plane-strain problems for a class of gradient elasticity models—a stress function approach. Journal of Elasticity104: 45–70.
6.
BaroudiSJemaiANajarF (2016) Modeling and parametric analysis of a piezoelectric flexoelectric nanoactuator. MATEC Web of Conferences83: 04002.
7.
BarrettaRČanadijaMFeoL, et al. (2018) Exact solutions of inflected functionally graded nano-beams in integral elasticity. Composites Part B Engineering142: 273–286.
8.
BhaskarUKBanerjeeNAbdollahiA, et al. (2016) A flexoelectric microelectromechanical system on silicon. Nature Nanotechnology11: 263–266.
9.
BirmanVByrdLW (2007) Modeling and analysis of functionally graded materials and structures. Applied Mechanics Reviews60: 195–216.
10.
BoudaoudMLuTLiangS, et al. (2018) A voltage/frequency modeling for a multi-dofs serial nanorobotic system based on piezoelectric inertial actuators. IEEE/ASME Transactions on Mechatronics23: 2814–2824.
11.
BursianEOiZMakarovK (1969) Ferroelectric plate polarization by bending. Izvestiya Akademii Nauk SSSR Seriya Fizicheskaya33: 1098.
12.
ChapelleDBatheKJ (1993) The inf-sup test. Computers & Structures47: 537–545.
13.
ChenWZhengYFengX, et al. (2015) Utilizing mechanical loads and flexoelectricity to induce and control complicated evolution of domain patterns in ferroelectric nanofilms. Journal of the Mechanics and Physics of Solids79: 108–133.
14.
ChenYYanZ (2019) Investigation of pull-in behaviors of a nanoswitch tuned by piezoelectric and flexoelectric effects. International Journal of Mechanical Sciences161–162: 105032.
15.
ChuBSalemDR (2012) Flexoelectricity in several thermoplastic and thermosetting polymers. Applied Physics Letters101: 103905.
16.
ChuLDuiGZhengY (2020a) Thermally induced nonlinear dynamic analysis of temperature-dependent functionally graded flexoelectric nanobeams based on nonlocal simplified strain gradient elasticity theory. European Journal of Mechanics-A/Solids82: 103999.
17.
ChuLLiYDuiG (2020b) Nonlinear analysis of functionally graded flexoelectric nanoscale energy harvesters. International Journal of Mechanical Sciences167: 105282.
18.
ChuLDuiGJuC (2018) Flexoelectric effect on the bending and vibration responses of functionally graded piezoelectric nanobeams based on general modified strain gradient theory. Composite Structures186: 39–49.
CummingsLJRichardsonG (2006) Bistable nematic liquid crystal device with flexoelectric switching. European Journal of Applied Mathematics17: 435–463.
21.
DavidsonAJMottramNJ (2002) Flexoelectric switching in a bistable nematic device. Physical Review E Statistical Nonlinear & Soft Matter Physics65: 051710.
22.
DengFDengQYuW, et al. (2017) Mixed finite elements for flexoelectric solids. Journal of Applied Mechanics Transactions ASME84: 081004.
23.
DengQKammounMErturkA, et al. (2014a) Nanoscale flexoelectric energy harvesting. International Journal of Solids and Structures51: 3218–3225.
24.
DengQLiuLSharmaP (2014b) Flexoelectricity in soft materials and biological membranes. Journal of the Mechanics and Physics of Solids62: 209–227.
25.
GarzaHGhatkesarMBasakS, et al. (2015) Nano-workbench: a combined hollow afm cantilever and robotic manipulator. Micromachines6: 600–610.
26.
GhasemiHParkHSRabczukT (2017) A level-set based iga formulation for topology optimization of flexoelectric materials. Computer Methods in Applied Mechanics and Engineering313: 239–258.
27.
GhasemiHParkHSRabczukT (2018) A multi-material level set-based topology optimization of flexoelectric composites. Computer Methods in Applied Mechanics and Engineering332: 47–62.
28.
GuoHZhuangXRabczukT (2019) A deep collocation method for the bending analysis of kirchhoff plate. Cmc-Computers Materials & Continua59: 433–456.
29.
HamdiaKMGhasemiHZhuangX, et al. (2018) Sensitivity and uncertainty analysis for flexoelectric nanostructures. Computer Methods in Applied Mechanics and Engineering337: 95–109.
30.
HanJKJeon DoHChoSY, et al. (2016) Nanogenerators consisting of direct-grown piezoelectrics on multi-walled carbon nanotubes using flexoelectric effects. Scientific Reports6: 29562.
31.
HuSShenS (2010) Variational principles and governing equations in nano-dielectrics with the flexoelectric effect. Science China Physics Mechanics and Astronomy53: 1497–1504.
32.
KeLLYangJKitipornchaiS, et al. (2014) Axisymmetric postbuckling analysis of size-dependent functionally graded annular microplates using the physical neutral plane. International Journal of Engineering Science81: 66–81.
33.
KoizumiM (1993) The concept of FGM. Ceramic transactions34: 3–10.
34.
KrichenSSharmaP (2016) Flexoelectricity: a perspective on an unusual electromechanical coupling. Journal of Applied Mechanics83: 030801.
35.
LazopoulosKALazopoulosAK (2010) Bending and buckling of thin strain gradient elastic beams. European Journal of Mechanics A-Solids29: 837–843.
36.
LiangXHuSShenS (2013) Bernoulli–Euler dielectric beam model based on strain-gradient effect. Journal of Applied Mechanics-Transactions of the ASME80: 369–384.
37.
LiangXHuSShenS (2014) Effects of surface and flexoelectricity on a piezoelectric nanobeam. Smart Materials and Structures23: 035020.
38.
LiangXHuSShenS (2015) Size-dependent buckling and vibration behaviors of piezoelectric nanostructures due to flexoelectricity. Smart Materials and Structures24: 105012.
39.
LoghmanAParsaH (2014) Exact solution for magneto-thermo-elastic behaviour of double-walled cylinder made of an inner fgm and an outer homogeneous layer. International Journal of Mechanical Sciences88: 93–99.
40.
MajdoubMSSharmaPCaginT (2008) Enhanced size-dependent piezoelectricity and elasticity in nanostructures due to the flexoelectric effect. Physical Review B77: 125424.
41.
MaoSPurohitPK (2014) Insights into flexoelectric solids from strain-gradient elasticity. Journal of Applied Mechanics81: 081004.
42.
MaoSPurohitPKAravasN (2016) Mixed finite-element formulations in piezoelectricity and flexoelectricity. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences472: 20150879.
43.
MarangantiRSharmaNDSharmaP (2006) Electromechanical coupling in nonpiezoelectric materials due to nanoscale nonlocal size effects: Green’s function solutions and embedded inclusions. Physical Review B74: 014110.
44.
MashkevichVTolpygoK (1957) Electrical, optical and elastic properties of diamond type crystals. Soviet Physics JETP5: 435–439.
45.
MehralianFBeniYT (2016) Size-dependent torsional buckling analysis of functionally graded cylindrical shell. Composites Part B Engineering94: 11–25.
46.
MiyamotoYKaysserWRabinB, et al. (1999) Functionally Graded Materials:Design, Processing and Applications, Hague. Boston / Dordrecht / London: Kluwer Academic Publishers.
47.
NanthakumarSSZhuangXParkHS, et al. (2017) Topology optimization of flexoelectric structures. Journal of the Mechanics and Physics of Solids105: 217–234.
48.
PetrovAG (2002) Flexoelectricity of model and living membranes. Biochimica et Biophysica Acta (BBA)-Biomembranes1561: 1–25.
49.
RakotondrabeMIvanIA (2011) Development and force/position control of a new hybrid thermo-piezoelectric microgripper dedicated to micromanipulation tasks. IEEE Transactions on Automation Science and Engineering8: 824–834.
50.
RayMC (2014) Exact solutions for flexoelectric response in nanostructures. Journal of Applied Mechanics81: 091002.
51.
RequichaAA (2003) Nanorobots, NEMS, and nanoassembly. Proceedings of the IEEE9: 1922–1933.
52.
SadeghiHBaghaniMNaghdabadiR (2012) Strain gradient elasticity solution for functionally graded micro-cylinders. International Journal of Engineering Science50: 22–30.
53.
SamaniegoEAnitescuCGoswamiS, et al. (2020) An energy approach to the solution of partial differential equations in computational mechanics via machine learning: concepts, implementation and applications. Computer Methods in Applied Mechanics and Engineering362: 29.
54.
ScottJF (1968) Lattice perturbations in CAWO4 and CAMOO4. The Journal of Chemical Physics48: 874–876.
55.
ShariyatM (2012) Nonlinear transient stress and wave propagation analyses of the FGM thick cylinders, employing a unified generalized thermoelasticity theory. International Journal of Mechanical Sciences65: 24–37.
56.
SharmaNDLandisCMSharmaP (2010) Piezoelectric thin-film superlattices without using piezoelectric materials. Journal of Applied Physics108: 024304.
57.
SharmaNDMarangantiRSharmaP (2007) On the possibility of piezoelectric nanocomposites without using piezoelectric materials. Journal of the Mechanics and Physics of Solids55: 2328–2350.
58.
ShenSHuS (2010) A theory of flexoelectricity with surface effect for elastic dielectrics. Journal of the Mechanics and Physics of Solids58: 665–677.
59.
ShuJYKingWEFleckNA (1999) Finite elements for materials with strain gradient effects. International Journal for Numerical Methods in Engineering44: 373–391.
60.
SiGSunLZhangZ, et al. (2021) Design, fabrication, and testing of a novel 3d 3-fingered electrothermal microgripper with multiple degrees of freedom. Micromachines12: 444.
61.
SolyaevYLurieS (2018) Numerical predictions for the effective size-dependent properties of piezoelectric composites with spherical inclusions. Composite Structures202: 1099–1108.
62.
SuYZhouZ (2020) Electromechanical analysis of flexoelectric nanosensors based on nonlocal elasticity theory. Micromachines11: 1077.
63.
Tadi BeniY (2016) Size-dependent electromechanical bending, buckling, and free vibration analysis of functionally graded piezoelectric nanobeams. Journal of Intelligent Material Systems and Structures27: 2199–2215.
64.
VaghefpourH (2021) Nonlinear vibration and tip tracking of cantilever flexoelectric nanoactuators. Iranian Journal of Science and Technology Transactions of Mechanical Engineering45: 879–889.
65.
WangDHYangQDongHM (2013) A monolithic compliant piezoelectric-driven microgripper: design, modeling, and testing. IEEE/ASME Transactions on Mechatronics18: 138–147.
66.
WangZLYangRZhouJ, et al. (2010) Lateral nanowire/nanobelt based nanogenerators, piezotronics and piezo-phototronics. Materials Science & Engineering R-Reports70: 320–329.
67.
XinLDuiGYangS, et al. (2014) An elasticity solution for functionally graded thick-walled tube subjected to internal pressure. International Journal of Mechanical Sciences89: 344–349.
68.
XinLDuiGYangS, et al. (2015) Solutions for behavior of a functionally graded thick-walled tube subjected to mechanical and thermal loads. International Journal of Mechanical Sciences98: 70–79.
69.
XinLYangSZhouD, et al. (2016) An approximate analytical solution based on the mori–tanaka method for functionally graded thick-walled tube subjected to internal pressure. Composite Structures135: 74–82.
70.
XueJChenLChangL, et al. (2021) A wideband flexoelectric energy harvester based on graphene substrate. Engineering Structures231: 111779.
71.
YanZJiangLY (2013) Flexoelectric effect on the electroelastic responses of bending piezoelectric nanobeams. Journal of Applied Physics113: 194102.
72.
YudinPVTagantsevAK (2013) Fundamentals of flexoelectricity in solids. Nanotechnology24: 432001.
73.
YvonnetJLiuLP (2017) A numerical framework for modeling flexoelectricity and maxwell stress in soft dielectrics at finite strains. Computer Methods in Applied Mechanics and Engineering313: 450–482.
74.
ZhaoXZhengSLiZ (2022) Bending, free vibration and buckling analyses of afg flexoelectric nanobeams based on the strain gradient theory. Mechanics of Advanced Materials and Structures29: 548–563.
75.
ZhengYChuLDuiG, et al. (2021a) Modeling and simulation of functionally graded flexoelectric micro-cylinders based on the mixed finite element method. Applied Physics A127: 228.
76.
ZhengYChuLDuiG, et al. (2021b) Numerical predictions for the effective electrical properties of flexoelectric composites with a single inclusion. Applied Physics A127: 686.
77.
ZhengYChuLDuiG, et al. (2022) Numerical predictions for the effective properties of flexoelectric composites with spherical inclusion. Acta Mechanica233: 2093–2106.
78.
ZhouYNybergTRXiongG, et al. (2017) Analysis of finite deformation of curved beams bonded with piezoelectric actuating layers. Journal of Intelligent Material Systems and Structures28: 1553–1564.
79.
ZhuangXGuoHAlajlanN, et al. (2021) Deep autoencoder based energy method for the bending, vibration, and buckling analysis of kirchhoff plates with transfer learning. European Journal of Mechanics A-Solids87: 18.
80.
ZubkoPCatalanGTagantsevAK (2013) Flexoelectric effect in solids. Annual Review of Materials Research43: 387–421.
