Bending analysis of micro cantilevers based on the Chen-Wang strain gradient plasticity theory
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1 mme.modares.ac.ir - * r_ansari@guilan.ac.ir3756 * : : : - Bendinganalysisofmicro cantileversbasedonthechen-wangstrain gradientplasticitytheory HakimehAlizadeh,RezaAnsari * DepartmentofMechanicalEngineering,GuilanUniversity,Rasht,Iran P.O.B.3756,Rasht,Iran,r_ansari@guilan.ac.ir ARTICLEINFORMATION ABSTRACT OriginalResearchPaper Received28August204 Accepted5December204 AvailableOnline03January205 Keywords: StrainGradientPlasticity MicroCantileverBeam Bending IntrinsicLengthScale Chen-WangTheory Whenthecantileverbeamthicknessisscaleddowntomicron,thedimensionofmaterialandthe intrinsiclengthscaleaffectthemechanicalbehaviorofthebeam.thepurposeofthispaperisto analyzethebendingofcantilevermicro-beamandpresentrelationforthebeamdeflectionusing Chen-Wanggradientplasticitytheory.Tothisend,theEuler-Bernoullibeammodelisutilizedand threecasesincludingelastic,rigid-plasticandelasto-plasticbeamsareconsidered.clearrelations for elastic and plastic strains are given. For all mentioned cases, the beam deflection is determinedfordifferentintrinsiclengthsandtheobtainedresultsarecomparedwitheachother andexperimentaldataandsomeexplanationsarepresented.theresultsobtainedfromclassical theoryarealsoshownintheresultssectiontoprovethatclassicaltheoriesdonothavetheability topredictbehaviorofmicron-sizestructuresprecisely.numericalresultsclarifythedependence ofresponsestotherangeofdimensionsandintrinsiclengths.thecomparisonbetweenthe presentresultsandthoseobservedfromexperimentaltestsauthenticatethereliabilityofutilized gradienttheory. []. [2] [4] [3] Classiccontinuummechanictheories Pleasecitethisarticleusing: : H.Alizadeh, R. Ansari,Bending analysis of micro cantileversbased on the Chen-Wangstrain gradient plasticity theory,modares Mechanical EngineeringVol.5,No.2,pp ,205(InPersian
2 [] - = ( = 3 2 : ( ( (2 = + = (2 = = 2 3 = 2 3 (3 ( (4 [] (5 = ( = (5 (6 = ( (6 (7 = ( + = (, (7 = ( ( + (8 0 ( (9 [8] = ( Stretchgradient - Rotationgradient -...[5] [6] 4. 5 [7]. [0-8] -.[] [2]. - [7].[6-3] [8] [9]. 7 [20]. 8. [2] 9 -Microstructure 2-Non-Local 3-Lowerorder 4-Yieldsurface 5-Higherorder 6-Von-mises 7-Couplestress 8-Stress-Straincurve 9-Mechanism-Based
3 - = ( (22 - (3. = 2 = 2 (23 = 2 3 (23 (24 = 2 = 2 (24 = + ( 0 ( 0 + ( ( ( = 0 (25 (- 26 (-26 = 8 [ ln + 3 = [ ln (27 =0. (0 = 0, (0 = 0 (27 = ( -28 = = ( = 0-28 (25 ( = 0 (29 (30 (29 = 3 6 (30 (3 = (2 2 (3 (32 = ( ( =0 (33 (0 = (, = 0, = ( (0 ( = ( - = = = = = 0 - (2 (3 = + + = ( (2 = 3-3 = 2 = 2 = 2 3 ( -3 (4 = = 2 3 ( -4 (5 = ( (5 (6 =, -6 = ( -6 (7 = 2 3 = 2 3 (7 [8] (8 ( ( = 2 5 ( (8 (7 = (9 (9 (8 = 2 5 ( ( (20 (20 [3] (20. (2 = 2 3 (2 (9 200
4 = -42 = ( -42 = ( ( ( < = ( ( (3 2 6 (43 ( = + ( 0 ( 0 + ( ( ( = 0 (44 - (27 =0 = ( + = + = -45 = = ( = 0-45 ( = 0 (46 (27 (47 = 2 (47 ( =0 (48 = 2 (48 (49 =0 = 2, = 2 (49. [9] - (3 = 6-33 = [ ln (34 (2 = 3 (34 (35 =0 = (3, = 6 2 (35 (36 ( = ( (30.. = (37 (37 (36 (32 = + (38 (40 (39 (3 (30 = (2 2 (39 = (3 3 6 (40. =0 ( = + + (4. (42-5 (
5 ln ln (53 4 = 8 2 (54 (54 /03 [23] 4/ /200 /2 40 /200 2/3 h. - 0/229 0/83 0/57 0/455 ([22] 0/33 0/8 0/6 0/ /36 /20 /0 /055 ([23] /5 /35 /22 /09 0/025 0/054 0/088 0/ ( [22]. [23] D. [22] = (50 = (50 [22] h =. (5 = = 2 (5 (48 - = ln + 3. (52 (52 /44[22] - 8/ = 2 [23] ( - 4 = (53 (47 -Lam 2-Shrotriya 202
6 = 0.4 = Stiffness
7 - [2] J.S.Stolken,A.G.Evans,microbendtestmethodformeasuringthe plasticitylengthscale,actamaterialiavol.46,pp ,998. [3] R. K. Abu Al-Rub, G. Z. Voyiadjis, Analytical and Experimental Determination of the Material Intrinsic Length Scale of Strain Gradient Plasticity Theoty from Micro- and Nano-Indentation Experiments, InternationalJournalofPlasticityVol.20(6,pp.39-82,2004. [4] M.B.Taylor,H.M.Zabib,M.A.Khaleel,Damageandsizeeffectdurind superplastic deformation, InternationalJournalofPlasticityVol.8,pp ,2002. [5] W.D.Nix,H.Gao,Indentationsizeeffectincrystallinematerials:lowfor strain gradient plasticity, JournalofmechanicsandphysicsofSolidsVol. 46(3,pp.4-425,998. [6] A.Acharya,A.J.Beaudoin,Grain-sizeeffectinviscoplasticpolycrystalsat moderate strains, JournalofmechanicsandphysicsofSolidsVol.48,pp ,2000. [7] E. C. Aifantis, Strain gradient interpretation of size effect, International JournalofFractureVol.95,pp ,999. [8] N.A.Fleck,J.W.Hutchinson,phenomenologicaltheoryforstrain gradient effects in plasticity, JournalofmechanicsandphysicsofSolids Vol.4,pp ,993. [9] N.A.Fleck,J.W.Hutchinson,Straingradientplasticity,Advances in AppliedMechanicsVol.33,pp ,997. [0] N.A.Fleck,J.W.Hutchinson,reformulationofstraingradientplasticity, JournalofmechanicsandphysicsofSolidsVol.49,pp ,200. [] S.H.Chen,T.C.Wang,newhardeninglawforstraingradientplasticity, ActaMaterialiaVol.48,pp ,2000. [2] S. H. Chen, T. C. Wang, Interface crack problems with strain gradient effectsinternationaljournaloffracturevol.7,pp.25-37,2002. [3] S.K.Park,X.L.Gao,Bernoulli-Eulerbeammodelbasedonmodied couple stress theory, JournalofmechanicsandphysicsofSolids Vol. 6, pp ,2006. [4] S.H. Chen, B. Feng, Size effect in micro-scale cantilever beam bending, ActaMechanicaVol.29,pp ,20. [5] N. Challamel, C. M. Wang, The small length scale effect for non-local cantileverbeam:paradoxsolved,nanotechnologyvol.9,pp , [6] W.Wang,Y.Huang,K.J.Hsia,K.X.Hu,A.Chandra,studyofmicrobend test by strain gradient plasticity, InternationalJournalofPlasticityVol. 9,pp [7] M.I.Idiart,V.S.Deshpande,N.A.Fleck,J.R.Willis,Sizeeffectinthe bendingofthinfoils,internationaljournalofengineeringsciencevol.47, pp ,2009. [8] N.A.Fleck,J.R.Willis,mathematicalbasisforstraingradientplasticity theory. Part II: tensorial plastic multiplier, Journal of mechanics and physicsofsolidsvol.57,pp ,2009. [9] B.Ji,W.J.Chen,J.Zhao,Measurementoflength-scaleandsolutionof cantilever beam in couple stress elasto-plasticity, ActaMechanica Vol. 25,pp ,2009. [20] Z.F.Shi,B.Huang,H.Tan,Y.Huang,T.Y.Zhang,P.D.Wu,K.C.Hwang,H. Gao, Determination of the microscale stress-strain curve and strain gradienteffectfromthe micro-bend ofultra-thinbeams,international JournalofPlasticityVol.24,pp ,2008. [2] Y.Q.Mao,S.G.Ai,D.N.Fang,Y.M.Fu,C.P.Chen,Elassto-plasticanalysis microfgmbeambasingon mechanism-based strain gradientplasticity theory,compositestructurevol.0,pp.68-79,203. [22] D.C.C.Lam,F.Yang,A.C.Chong,J.Wang,P.Tong,Experimentsand theory in strain gradient elasticity, Journalofmechanicsandphysicsof SolidsVol.5,pp ,2003. [23] P.Shrotriya,S.M.Allameh,J.Lou,T.Buchheit,W.O.Soboyejo,Onthe measurementoftheplasticitylengthscaleparameterinliganickelfoils, MechicsofMaterialsVol.35,pp , [] N.A.Fleck,G.M.Muler,M.F.Ashby,J.W.Hutchinson,StrainGradient Plasticity: Theory and Experiment, ActaMetallorgicaetMaterialia Vol. 42,pp ,
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