力學(xué)大會(huì)2015集從微觀到連續(xù)體表面效應(yīng)及其彈性性質(zhì)的確

才引起巨大的研究。另一方面，隨著現(xiàn)代納米科技的發(fā)展，納米結(jié)構(gòu)極大的表面積后，表面彈性性能可以通過(guò)使原子模型的總能量????????(??)與連續(xù)體模型的總應(yīng)變能 H∈3× (H∈這 假設(shè)對(duì)于一個(gè)固體自由表面Γ有一個(gè)變形梯度那么它的表面彈性特性的由連續(xù)體理論Gurtin-Murdoch理論表示為如下形式(2????(??)是四階表面

表面彈來(lái)源于兩個(gè)半無(wú)限 間體之原子鍵的破壞和拉 提出在不發(fā)生表面 構(gòu)情況的原子結(jié)構(gòu)模型并 此推導(dǎo)

???

Ω0,??0,??0,?]有????.由此我們定義這個(gè)晶體中各原子的參考位置1原

????=???????0∽???

和

Oppenheimer近似，我們可以寫出該有限晶體的總內(nèi)能作為它所有原子位其中????∈??3和??????∈??3×3

(3(4(5(6 ??={??0∈Ω:????????(?Ω,??0)>

??=

∈??Ω:??????

??

為”下性質(zhì)

為“表面原子”

使得我們進(jìn)一步(7原子結(jié)構(gòu)模型中的能量變化等價(jià)于增加的應(yīng)變能我們可得該晶體的體彈性剛度張量(8接下來(lái)我們?cè)敿?xì)推導(dǎo)該晶體的表面彈性性質(zhì)圖2層數(shù)??1,2,…,Θ}標(biāo)示出平行于自由表面的晶并且在第??晶層的原子集合可表示從假設(shè)2）我們可推得當(dāng)作用任意一個(gè)非對(duì)稱應(yīng)變??∈??33，各原子的新位置可寫這里??1Θ→??表示第??層晶層在表面形成時(shí)所對(duì)應(yīng)應(yīng)力

(9(10

(11(12(13(14我們可將式12

(15(16因此式(15)中展開分量可簡(jiǎn)化(17為了簡(jiǎn)化式(15)我們定義以下??×??矩(18

(19衡狀態(tài)，須最小化式(15)，也即使下列二次型??:??Θ???→??最小化:(20

??:????×??→????× ?? ??????，那么通過(guò)用 ???1連續(xù)對(duì)(20)由因?yàn)榉€(wěn)定的晶體結(jié)構(gòu)要求????2(??1)在??→∞時(shí)有界則可得??(ΛΛ值得注意的是式(22)可轉(zhuǎn)化為一個(gè)低階（??×??）algebraic其中??由下式確定

(21(22(23(24(25lgebraicRiccati方程而非整個(gè)分子動(dòng)力學(xué)模擬。EAM勢(shì)能可寫(26)彈性特性。下列所有計(jì)算均采用JohnsonEAM勢(shì)能。表1由EAM勢(shì)能計(jì)算所Δ12------------Δ23--表2由?????(??0)11(??0)22?------計(jì)算結(jié)果及與實(shí)驗(yàn)和分子動(dòng)力學(xué)模擬的比較在表量(??0)12都為0，這與面對(duì)對(duì)稱性的要求一致。表面彈性張量的計(jì)算結(jié)果在表3中顯示。(????)1111????)2222，(??0)11??0)22也得到了驗(yàn)證表3由????-------????--------------------2。類似的結(jié)果也能適用于多原子的多晶體結(jié)構(gòu)。除了顯式地寫出了確定表面彈性性質(zhì)方程，這一分析還體現(xiàn)了1）表面松弛總是使得表面張力和表面彈性張量變小，2）Bauer,C.E.,Speiser,R.,Hirth,J.P.Surfaceenergyofcopperasafunctionofoxygenactivity.Metall.Trans.A,1976,7(1),75–709.Buerger,M.J.ElementaryCrystallography.Wiley,NewYork,Cuenot,S.,Fretigny,C.,Demoustier-Champagne,S.,Nysten,B.Surfacetensioneffectonthemechanicalpropertiesofnanomaterialsmeasuredbyatomicmicroscopy.Phys.Rev.B,2004,69,165410.Demuth,J.E.,Marcus,P.M.,Jepsen,D.W.ysisoflow-energy-electron-diffractionintensityspectrafor(001),(110),and(111)nickel.Phys.Rev.B,1975,11(4),1460–1474.Diao,J.,Gall,K.,Dunn,M.L.Surface-stress-inducedphasetransformationinmetalnanowires.Nat.Mater.,2003,2,Duan,H.L.,Wang,J.,Huang,Z.P.,Karihaloo,B.L.Size-dependanteffectiveelasticconstantsofsolidscontainingnano-inhomogeneitieswithinterfacestress.J.Mech.Phys.Solids,2005,53,1574–1596.Foiles,S.M.,Baskes,M.I.,Daw,M.S.Embedded-atom-methodfunctionsforthefccmetalscu,ag,au,ni,pd,pt,andtheiralloys.Phys.Rev.B,1986,33(12),7983–7991.Goudeau,P.,Renault,P.O.,Villain,P.,Coupeau,C.,Pelosin,V.,Boubeker,B.,Badawi,K.F.,Thiaudiere,D.,Gailhanou,M.CharacterizationofthinfilmelasticpropertiesusingX-raydiffractionandmechanicalmethods:applicationtopolycrystallinestainlesssteel.ThinSolidFi,2001,398,496–500.Greer,J.R.,Nix,W.D.Nanoscalegoldpillarsstrengthenedthroughdislocationstarvation.Phys.Rev.B,2006,73,Gurtin,M.E.,Murdoch,A.I.Acontinuumtheoryofelasticmaterialsurfaces.Arch.Rat.Mech..,1975,57(4),Gurtin,M.E.,Murdoch,A.I.Surfacestressinsolids.Int.J.SolidsStruct.,1978,14,Hu,L.X.,Liu,L.P.Interfacialwaveswithsurfaceelasticity.J.Appl.Mech.,2014,81,Hurley,D.C.,Tewarya,V.K.,Richards,A.J.Thin-filmelastic-propertymeasurementswithlaser-ultrasonicsawspectrometry.ThinSolidFi,2001,398,326–330.Jing,G.Y.,Duan,H.L.,Sun,X.M.,Zhang,Z.S.,Xu,J.,Li,Y.D.,Wang,J.X.,Yu,D.P.Surfaceeffectsonelasticpropertiesofsilvernanowires:contactatomic-microscopy.Phys.Rev.B,2006,73,235409.Johnson,R.A.Alloymodelswiththeembedded-atommethod.Phys.Rev.B,1989,39(17),Kim,Ju-Young,Greer,JuliaR.Tensileandcompressivebehaviorofgoldandmolybdenumsinglecrystalsatthenano-scale.ActaMater.,2009,57(17),5245–5253.Kumikov,V.K.,Khokonov,Kh.B.Onthemeasurementofsurfaceenergyandsurfacetensionofsolidmetals.J.Appl.Phys.,1983,54,1346–1350.Liang,Wuwei,Zhou,Min,Ke,Fujiu.Shapememoryeffectincunanowires.NanoLett.,2005,5(10),Lindgren,S.A.,Wallden,L.,Rundgren,J.,Westrin,P.Low-energyelectrondiffractionfromcu(111):subthresholdeffectandenergy-dependentinnerpotential;surfacerelaxationandmetricdistancesbetweenspectra.Phys.Rev.B,1984,29(2),576–588.Maranganti,R.,Sharma,P.Lengthscalesatwhichclassicalelasticitybreaksdownforvariousmaterials.Phys.Rev.Lett.,2007,98,195504.Mi,C.,Jun,S.,Kouris,D.A.Atomisticcalculationsofinterfaceelasticpropertiesinnoncoherentmetallicbilayers.Phys.Rev.B,2008,77(7),075425.Ojaghnezhad,F.,Shodja,H.M.Acombinedfirstprinciplesandyticaltreatmentfordeterminationofthesurfaceelasticconstants:applicationtosi(001)idealandreconstructedsurfaces.Philos.Mag.Lett.,2012,92(1),7–19.23Park,H.S.,Klein,P.A.SurfaceCauchy–Bornysisofsurfacestresseffectsonmetallicnanowires.Phys.Rev.75,Park,H.S.,Klein,P.A.,Wagner,G.J.AsurfaceCauchy–Bornmodelfornanoscalematerials.Int.J.Numer.Meth.Eng.,2006,68,1072–1095.Pathak,S.,Shenoy,V.B.Sizedependenceofthermalexpansionofnanostructures.Phys.Rev.B,2005,72,Price,C.W.,Hirth,J.P.Surfaceenergyandsurfacestresstensorinanatomisticmodel.Surf.Sci.,1976,57,509–Renault,P.O.,LeBourhis,E.,Villain,P.,Goudeau,Ph.,Badawi,K.F.,Faurie,D.MeasurementoftheelasticconstantsoftexturedanisotropicthinfifromX-raydiffractiondata.Appl.Phys.Lett.,2003,83(3),473–475.Shenoy,V.B.Atomisticcalculationsofelasticpropertiesofmetallicfcccrystalsurfaces.Phys.Rev.B,2005,71(9),094104.Statiris,P.,Lu,H.C.,Gustafsson,T.Temperaturedependentsignreversalofthesurfacecontractionofag(111).Phys.Rev.Lett.,1994,72(22),3574–3577.Steigmann,D.J.,Ogden,R.W.nedeformationsofelasticsolidswithintrinsicboundaryelasticity.Proc.R.Soc.A-Math.Phys.,1997,453,853–877.Steigmann,D.J.,Ogden,R.W.Elasticsurface–substratein 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LIULi(1BoschAutomotiveProducts(Suzhou)Co.,Suzhou,215021(2DepartmentofMathematics,DepartmentofMechanicalandAerospaceEngineering,RutgersUniversity,Piscataway,08854,Wepresentan ysisofsurfaceelasticityfromtheBorn–Oppenheimerapproximationfor24monatomiccrystals.The ysisshowsthattherelaxationsofcrystalnespar