The longitudinal free vibration problem of a micro-scaled bar is formulated using the strain gradient elasticity theory. The equation of motion together with initial conditions, classical and non-classical corresponding boundary conditions for a micro-scaled elastic bar is derived via Hamilton’s principle. The resulting higher-order equation is solved for clamped-clamped and clamped-free boundary conditions. Effects of the additional length scale parameters on the frequencies are investigated. It is observed that size effect is more significant when the ratio of the microbar diameter to the additional length scale parameter is small. It is also observed that the difference between natural frequencies predicted by current and classical models becomes more prominent for both lower values of slenderness ratio of the microbar and for higher modes.
AkgözBCivalekÖ (2011) Strain gradient elasticity and modified couple stress models for buckling analysis of axially loaded micro-scaled beams. International Journal of Engineering Science49: 1268–1280.
2.
AltanBSAifantisEC (1997) On some aspects in the special theory of gradient elasticity. Journal of the Mechanical Behavior of Materials8: 231–282.
3.
AltanBSEvensenHAifantisEC (1996) Longitudinal vibrations of a beam; a gradient elasticity approach. Mechanics Research Communications23: 35–40.
4.
AnsariRGholamiRRouhiH (2012) Various gradient elasticity theories in predicting the vibrational response of single-walled carbon nanotubes with arbitrary boundary conditions. Journal of Vibration and Control. DOI: 10.1177/1077546312439223.
5.
AydogduM (2009) Axial vibration of nanorods with the nonlocal continuum rod model. Physica E41: 861–864.
6.
ChongACMLamDCC (1999) Strain gradient plasticity effect in indentation hardness of polymers. Journal of Materials Research14: 4103–4110.
7.
CivalekÖAkgözB (2010) Free vibration analysis of microtubules as cytoskeleton components: nonlocal Euler-Bernoulli beam modeling. Scientia Iranica: Transaction B-Mechanical Engineering17: 367–375.
8.
CivalekÖDemirC (2011) Bending analysis of microtubules using nonlocal Euler-Bernoulli beam theory. Applied Mathematical Modeling35: 2053–2067.
9.
CivalekÖDemirCAkgözB (2010) Free vibration and bending analyses of cantilever microtubules based on nonlocal continuum model. Mathematical and Computational Applications15: 289–298.
10.
CosseratECosseratF (1909) Theory of Deformable Bodies. (Translated by D.H. Delphenich), Paris: Scientific Library, A. Hermann and Sons.
11.
DemirCCivalekÖAkgözB (2010) Free vibration analysis of carbon nanotubes based on shear deformable beam theory by discrete singular convolution ttechnique. Mathematical and Computational Applications15: 57–65.
12.
EringenAC (1972) Nonlocal polar elastic continua. International Journal of Engineering Science10: 1–16.
13.
EringenAC (1983) On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. Journal of Applied Physics54: 4703–4710.
14.
EringenACSuhubiES (1964) Nonlinear theory of simple microelastic solid – I. International Journal of Engineering Science2: 189–203.
15.
FleckNAMullerGMAshbyMFHutchinsonJW (1994) Strain gradient plasticity: theory and experiment. Acta Metallurgica et Materialia42: 475–487.
JomehzadehENooriHRSaidiAR (2011) The size-dependent vibration analysis of micro-plates based on a modified couple stress theory. Physica E43: 877–883.
18.
KahrobaiyanMHTajalliSAMovahhedyMRAkbariJAhmadianMT (2011) Torsion of strain gradient bars. International Journal of Engineering Science49: 856–866.
19.
KeLLWangYSYangJKitipornchaiS (2012) Free vibration of size-dependent Mindlin microplates based on the modified couple stress theory. Journal of Sound and Vibration331: 94–106.
20.
KoiterWT (1964) Couple stresses in the theory of elasticity: I and II: 67. Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen, Series B, Physical Sciences, 17–44.
21.
KongSZhouSNieZWangK (2008) The size-dependent natural frequency of Bernoulli–Euler micro-beams. International Journal of Engineering Science46: 427–437.
22.
KongSZhouSNieZWangK (2009) Static and dynamic analysis of micro beams based on strain gradient elasticity theory. International Journal of Engineering Science47: 487–498.
23.
LamDCCYangFChongACMWangJTongP (2003) Experiments and theory in strain gradient elasticity. Journal of the Mechanics and Physics of Solids51: 1477–1508.
24.
LazopoulosAK (2012) Dynamic response of thin strain gradient elastic beams. International Journal of Mechanical Sciences58: 27–33.
25.
LazopoulosKALazopoulosAK (2010) Bending and buckling of thin strain gradient elastic beams. European Journal of Mechanics A/Solids29: 837–843.
26.
LazopoulosKALazopoulosAK (2011) On a strain gradient elastic Timoshenko beam model. Journal of Applied Mathematics and Mechanics91: 875–882.
27.
LiewKMHeXQKitipornchaiS (2006) Predicting nano vibration of multi-layered graphene sheets embedded in an elastic matrix. Acta Materialia54: 4229–4236.
28.
LiewKMHuYAHeXQ (2008) Flexural wave propagation in single-walled carbon nanotubes. Journal of Computational and Theoretical Nanoscience5: 581–586.
29.
LimCWLiCYuJL (2012) Free torsional vibration of nanotubes based on nonlocal stress theory. Journal of Sound and Vibration331: 2798–2808.
30.
LiuGRChengYMiDLiZR (2005) A study on self-insertion of peptides into single-walled carbon nanotubes based on molecular dynamics simulation. International Journal of Modern Physics C16: 1239–1250.
31.
LouJShrotriyaPAllamehSBuchheitTSoboyejoWO (2006) Strain gradient plasticity length scale parameters for LIGA Ni MEMs thin films. Materials Science and Engineering: A441: 299–307.
32.
MaHMGaoXLReddyJN (2008) A microstructure-dependent Timoshenko beam model based on a modified couple stress theory. Journal of the Mechanics and Physics of Solids56: 3379–3391.
33.
MaHMGaoXLReddyJN (2011) A non-classical Mindlin plate model based on a modified couple stress theory. Acta Mechanica220: 217–235.
34.
MindlinRD (1964) Micro-structure in linear elasticity. Archive for Rational Mechanics and Analysis16: 51–78.
35.
MindlinRD (1965) Second gradient of strain and surface tension in linear elasticity. International Journal of Solids and Structures1: 417–438.
36.
MindlinRDTierstenHF (1962) Effects of couple-stresses in linear elasticity. Archive for Rational Mechanics and Analysis11: 415–448.
37.
Papargyri-BeskouSPolyzosDBeskosDE (2003a) Dynamic analysis of gradient elastic flexural beams. Structural Engineering and Mechanics15: 705–716.
38.
Papargyri-BeskouSTsepouraKGPolyzosDBeskosDE (2003b) Bending and stability analysis of gradient elastic beams. International Journal of Solids and Structures40: 385–400.
39.
ParkSKGaoXL (2006) Bernoulli-Euler beam model based on a modified couple stress theory. Journal of Micromechanics and Microengineering16: 2355–2359.
40.
PeddiesonJBuchananGRMcNittRP (2003) Application of nonlocal continuum models to nanotechnology. International Journal of Engineering Science41: 305–312.
41.
RaoSS (2007) Vibration of Continuous Systems, Hoboken, NJ, USA: John Wiley & Sons, Inc.
42.
ReddyJN (2007a) Nonlocal theories for bending, buckling and vibration of beams. International Journal of Engineering Science45: 288–307.
43.
ReddyJN (2007b) Theory and Analysis of Elastic Plates and Shells, 2nd ed. Philadelphia: Taylor & Francis.
ShenHS (2010a) Nonlocal shear deformable shell model for bending buckling of microtubules embedded in an elastic medium. Physics Letters A374: 4030–4039.
46.
ShenHS (2010b) Buckling and postbuckling of radially loaded microtubules by nonlocal shear deformable shell model. Journal of Theoretical Biology264: 386–394.
47.
ShenHS (2011) Nonlinear vibration of microtubules in living cells. Current Applied Physics11: 812–821.
48.
ShenHSZhangCL (2010) Nonlocal shear deformable shell model for post-buckling of axially compressed double-walled carbon nanotubes embedded in an elastic matrix. ASME Journal of Applied Mechanics77: 41–46.
49.
Ş imş ekM (2010) Dynamic analysis of an embedded microbeam carrying a moving microparticle based on the modified couple stress theory. International Journal of Engineering Science48: 1721–1732.
50.
Ş imsekM (2011a) Nonlocal effects in the forced vibration of an elastically connected double-carbon nanotube system under a moving nanoparticle. Computational Materials Science50: 2112–2123.
51.
Ş imsekM (2011b) Forced vibration of an embedded singlewalled carbon nanotube traversed by a moving load using nonlocal Timoshenko beam theory. Steel and Composite Structures11: 59–76.
52.
ToupinRA (1962) Elastic materials with couple stresses. Archive for Rational Mechanics and Analysis11: 385–414.
53.
ToupinRA (1964) Theory of elasticity with couple stresses. Archive for Rational Mechanics and Analysis17: 85–112.
54.
TsepouraKGPapargyri-BeskouSPolyzosDBeskosDE (2002) Static and dynamic analysis of a gradient-elastic bar in tension. Archive of Applied Mechanics72: 483–497.
55.
TsiatasGC (2009) A new Kirchhoff plate model based on a modified couple stress theory. International Journal of Solids and Structures46: 2757–2764.
56.
VardoulakisISulemJ (1995) Bifurcation Analysis in Geomechanics, London: Blackie/Chapman and Hall.
57.
WangCMKitipornchaiSLimCWEisenbergerM (2008) Beam bending solutions based on nonlocal Timoshenko beam theory. ASCE Journal of Engineering Mechanics134: 475–481.
58.
WangQLiewKM (2007) Application of nonlocal continuum mechanics to static analysis of micro- and nano-structures. Physics Letters A363: 236–242.
59.
WangQShindoY (2006) Nonlocal continuum models for carbon nanotubes subjected to static loading. Journal of Mechanics of Materials and Structures1: 663–680.
60.
WangQLiewKMDuanWH (2008) Modeling of the mechanical instability of carbon nanotubes. Carbon46: 285–290.
61.
WangQVaradanVKQuekST (2006) Small scale effect on elastic buckling of carbon nanotubes with nonlocal continuum models. Physics Letters A357: 130–135.
62.
WangBZhaoJZhouS (2010) A micro scale Timoshenko beam model based on strain gradient elasticity theory. European Journal of Mechanics A/Solids29: 591–599.
63.
WangBZhouSZhaoJChenX (2011) A size-dependent Kirchhoff micro-plate model based on strain gradient elasticity theory. European Journal of Mechanics A/Solids30: 517–524.
64.
XiangPLiewKM (2012) Free vibration analysis of microtubules based on an atomistic-continuum model. Journal of Sound and Vibration331: 213–230.
65.
XiangYWangCMKitipornchaiSWangQ (2010) Dynamic instability of nanorods/nanotubes subjected to an end follower force. ASCE Journal of Engineering Mechanics136: 1054–1058.
66.
YangFChongACMLamDCCTongP (2002) Couple stress based strain gradient theory for elasticity. International Journal of Solids and Structures39: 2731–2743.
67.
YangYZhangLLimCW (2011) Wave propagation in double-walled carbon nanotubes on a novel analytically nonlocal Timoshenko beam-model. Journal of Sound and Vibration330: 1704–1717.
68.
YangYZhangLLimCW (2012) Wave propagation in fluid-filled single-walled carbon nanotube on analytically nonlocal Euler-Bernoulli beam model. Journal of Sound and Vibration331: 1567–1579.
69.
YinLQianQWangLXiaW (2010) Vibration analysis of microscale plates based on modified couple stress theory. Acta Mechanica Solida Sinica23: 386–393.
70.
ZhangYQLiuGRHanX (2006a) Effect of small length scale on elastic buckling of multi-walled carbon nanotubes under radial pressure. Physics Letters A349: 370–376.
71.
ZhangYQLiuGRQiangHFLiGY (2006b) Investigation of buckling of double-walled carbon nanotubes embedded in an elastic medium using the energy method. International Journal of Mechanical Sciences48: 53–61.
72.
ZhangYQLiuGRHanX (2005a) Transverse vibrations of double-walled carbon nanotubes under compressive axial load. Physics Letters A340: 258–266.
73.
ZhangYQLiuGRXieXY (2005b) Free transverse vibrations of double-walled carbon nanotubes using a theory of nonlocal elasticity. Physical Review B71: 195404–195404.
