Elastic-Stresses-Hertz-Contact(赫兹接触)

ElasticStresses-HertzianContact弹性应力——赫兹接触Acontinuumanalysisofcontactforcesandthedeformationsthatarisefromquasistaticcompressionofelastic,elastic-plasticorperfectlyplasticbodiescanbeusedtodevelopatheoryofimpactforhardbodiescomposedofrate-independentmaterials.由弹性体、弹塑性体或者绝对塑性体的准静态压缩引起的接触力与变形的连续分析，能够用来发展由比率独立的材料组成的坚硬体的冲击理论。Inthistheorydeformationsarenegligibleoutsideasmallcontactregion,andthedeformingregionactsasanonlinearinelasticspringbetweentworigidbodies;themassofthedeformingregionisassumedtobenegligible.这一理论中小的接触区域之外的变形是微不足道，两个刚体之间的变形区域由一个非线性弹簧替换；变形区域的质量被认为是微不足道的。Hertz(1882)firstdevelopedthisquasistatictheoryforelasticdeformationlocalizednearthecontactpatchandappliedittothecollisionofsolidbodieswithsphericalcontactsurfaces.Hertz'stheoryprovidesaverygoodapproximationforcollisionsbetweenhardcompactbodieswherethecontactregionremainssmallincomparisonwiththesizeofeitherbody.赫兹首次对接触面附近的弹性变形使用准静态理论，并且将之应用于两球形体表面的接触中。赫兹理论为坚硬紧凑体间的碰撞提供了一个很好的近似值，当然接触区域比起碰撞体本身非常小。LetnonconformingelasticbodiesBandB'comeintocontactatapointC;inaneighborhoodofCthesurfacesofthebodieshaveradiiofcurvatureRBandRB',asdescribedinFig.6.1.两不相容弹性体B与B'接触于C点；曲率半径分别为RB和RB'的物体的接触表面C点附近。IfthesebodiesarecompressedbyforceF=F3inthenormaldirection,Hertzshowedthatthecontactregionspreadstoradiusaandwithinthecontactareathereisanellipticaldistributionofcontactpressure如果物体受法向压力F作用，接触区域将会是半径为a的区域，并且接触区域内接触压力为椭圆分布。ararprp≤−=,)/1()(2/1220(6.1)whererisaradialcoordinateoriginatingatthecenterandp0=p(0)isthepressureatthecenterofthecontactarea.式中r为径向坐标，在接触区中心处压力p0=p(0)。Thiscontactpressuregenerateslocalelasticdeformationsandsurfacedisplacementsthatcauseinitiallynonconformingsurfacestotouchorconformwithinacontactarea.接触压力导致局部弹性变形和表面位移，而这一位移使得接触区开始不相容的表面开始接触或是重叠。ThispressuredistributionresultsinacompressivereactionforceFoneachbody.压力分布导致碰撞体间产生压缩反力。200322)(apdrrpFaππ==∫(6.2)Themeanpressurepistwo-thirdsofthepressureatthecenterofthecontactcircle,3/20pp=.平均压力p等于接触中心处压力的2/3。ForthepressuredistributiongiveninEq.(6.1),Hertzobtainedthenormaldisplacement)(rwiatthesurfaceofbodyi(i=B,B')fromtheBoussinesqsolutionforaforceappliednormaltothesurfaceofanelastichalfspace(TimoshenkoandGoodier,1970):如（6.1）压力表达式所示，从弹性半空间体表面施加法向力后得到的Boussinesq解中，Hertz获得碰撞体表面的法向位移)(rwi。ararEaprwiii≤−−=−),/2()1(25.0)(22102πν(6.3)wherecompressivedisplacementsarepositive.InthisexpressiontheelasticmoduliofbodyiaregivenasYoung'smodulusEiandPoisson'sratioiν.式中的压缩量为正值。Ei为弹性模量，iν为波松比。ThecompressionofeachbodyiδisequivalenttotherelativedisplacementbetweentheinitialcontactpointCandthecenterofmass,)0(iiw=δ.物体的压缩量等于接触点C相对于质量中心的位移，)0(iiw=δ。Thusforaxisymmetricbodieswithconvexcontactsurfacesofcurvature1−iR,ifthecontactareaissmallincomparisonwiththecross-section,theradialdistributionofthenormaldisplacementcanbeexpressedas这样对于曲率为1−iR的外凸面轴对称物体，如果接触区域比起横截面非常小，那么沿径向分布的法向位移就可以表达成：iiiRrrw2/)(2−≈δThetotalindentationfromcompressionBB′+=δδδcanberelatedtothepressuremagnitude0patthecenterofthecontactareabysummingtheindividualeffectsexpressedbyEq.(6.3):通过表达式（6.3），总的压缩量BB′+=δδδ与接触中心的压力大小0p有关系。*02/Eapπδ=(6.4)whereaneffectiveradius*Randmodulus*Ehavebeendefinedas*R为等效半径，*E为等效模量。111*)(−−′−+=BBRRR11212*])1()1[(−−′′−−+−=BBBBEEEννThissizeofthecontactareacanbedeterminedfromEqs.(6.3)and(6.4);thecontactradiusaisthenrelatedtocontactforceFusingEq.(6.2):这类型的接触区域可由(6.3)与(6.4)确定；接触半径a与接触力F的关系由式(6.2)可得：**2*2*43RaEFRaR==δRearranging,weobtain重新整理，有3/12***)43(REFRa=(6.5)3/22**2*2*)43(REFRaR==δ(6.6)3/12**3*2*0)6(23REFEaFEpππ==(6.7)Themeanpressureinthecontactregionpandacompliancerelationforinteractionforce)(δFareobtainedfromEqs.(6.6)and(6.7):接触区的平均压力p与力相互作用F可由式(6.6)与(6.7)得出：**34REpδπ=,2/3*2**)(34RREFδ=(6.8)where*Ristheeffectiveradiusofcurvatureinthecontactareabeforecompression.式中*R为压缩前接触区的等效半径。ThisforcecanbeintegratedtoobtaintheworkWdonebythenormalcontactforceincompressingthesmalldeformingregiontoanyindentationδ,对力积分可以获得法向接触力压缩小变形区域时所作的功W与压缩量间的关系。2/5*/0*2***3**)(158)/()/(*RRdRERFREWRδδδδ∫=′′=(6.9)