研究纯金属弹性常数的一种新的力学模型
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摘要
弹性常数是用于表征物体抵抗弹性变形能力的物理量,是材料力学行为的主要指标。以弹性力学、连续介质力学、固体物理学、势函数理论等为基础,以一个由同种的、可用球模型近似的粒子组成的、单层粒子的平面固体薄板为研究对象,建立基于粒子间作用力的连续介质二维模型。基于该模型,推导出模型中任意一个三角形区域的弹性常数的计算公式;提出理想微结构的概念,推导出模型为理想微结构时的整体弹性常数、弹性模量、泊松比的计算公式,其中泊松比是一个等于1/3的常数。求解纯铜的弹性常数和弹性模量,并验证该模型的合理性和正确性。
The elastic constants are physical quantities that are used to characterize the ability of object to resist elastic deformation,it is the main indexes to describe material mechanical behaviors.On the ground of elasticity,continuum mechanics,solid state physics,potential function theory and so on,orienting the plane solid thin plate toward research object,the plate is composed of a single layer of particles that are homologous and can be replaced by ball model approximately,the model of the continuous medium two-dimensional model based on interparticle force is built.Based on the model,the calculation formulas of the elastic constants are deduced to whichever triangle regional of the model.The concept of the ideal micro structure is presented,and the calculation formulas of the totality elastic constants,the elastic modulus and the poisson ratio are deduced while the micro structure of the model is ideal micro structure.From these calculation formulas,it can be known that the poisson ratio is a constant of 1/3.The elastic constants and the elastic modulus of pure copper are solved,the rationality and correctness of the model is verified.
The elastic constants are physical quantities that are used to characterize the ability of object to resist elastic deformation,it is the main indexes to describe material mechanical behaviors.On the ground of elasticity,continuum mechanics,solid state physics,potential function theory and so on,orienting the plane solid thin plate toward research object,the plate is composed of a single layer of particles that are homologous and can be replaced by ball model approximately,the model of the continuous medium two-dimensional model based on interparticle force is built.Based on the model,the calculation formulas of the elastic constants are deduced to whichever triangle regional of the model.The concept of the ideal micro structure is presented,and the calculation formulas of the totality elastic constants,the elastic modulus and the poisson ratio are deduced while the micro structure of the model is ideal micro structure.From these calculation formulas,it can be known that the poisson ratio is a constant of 1/3.The elastic constants and the elastic modulus of pure copper are solved,the rationality and correctness of the model is verified.
引文
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[2]Dienes GJ.Atheoretical estimate of the effect of radiation onthe elasticconstants of simple metals[J].Phys.Rev.,1952(86):228-234.
[3]汪永江.空位所引起的晶体弹性常数的改变[J].物理学报,1966,22(2):214-222.WANG YongJiang.The change of the elastic constants of crystals due tovacancies[J].Acta Physica Sinica,1966,22(2):214-222(In Chi-nese).
[4]赵伊君,张志杰.金属力学性质的物理力学计算Ⅰ———用Morse势计算弹性常数[J].国防科技大学学报,1980(3):51-79.ZHAO YiJun,ZHANG ZhiJie.Physical mechanical calculation of me-chanical properties of metals I———using morse potential to calculate theelastic constants[J].Journal of National University of Defense Technolo-gy,1980(3):51-79(In Chinese).
[5]丁屹.晶体弹性常数的数值计算方法[J].声学学报,1989,14(1):68-72.DING Yi.Numerical calculation of elastic constants of crystals[J].ActaAcustica,1989,14(1):68-72(In Chinese).
[6]Panda C V,Vyas P R,Pandya TC,et al.Animprovedlattice mechani-cal model for FCCtransition metals[J].Physica B Condensed Matter,2001,307(01):138-149.
[7]Zhang M,ShenJ,He J W.Elastic constants of Al and TiNcalculated byinitio method[J].Trans.Nonferrous Met.Soc.China,2001,11(2):244-248.
[8]Tabar H R.Modelling the nano-scale phenomena in condensed matterphysics via computer-based numerical simulation[J].Physics Reports,2000,325:239-310.
[9]陈丽.FCC晶体弹性常数的分子动力学模拟及其实用性[J].机械工程学报,2005,41(9):46-50.CHENLi.Calculation and applicability analysis for elastic constants ofFCCcrystal[J].Chinese Journal of Mechanical Engineering,2005,41(9):46-50(In Chinese).
[10]谢根全,龙述尧.考虑小尺度效应影响的金属纳米丝弹性模量的计算[J].苏州科技学院学报:工程技术版,2005,18(2):27-31.XIE GenQuan,LONG ShuYao.Calculation of the elastic modulus ofmetal nanorod based on small size effect[J].J.of University of Scienceand Technology of Suzhou:Engineering and Technology Edition,2005,18(2):27-31(In Chinese).
[11]王刚锋,冯西桥,余寿文.纳米晶体材料的有效弹性模量与界面效应[J].科学通报,2002,47(14):1062-1065.WANG GangFeng,FENG XiQiao,YUShouWen.The effective elasticmodulus andthe interface effect of nanocrystalline materials[J].ChineseScience Bulletin,2002,47(14):1062-1065(In Chinese).
[12]邸玉贤,计欣华,李林安,等.纳米金属材料宏观弹性模量的数值模拟研究[J].机械强度,2007,29(1):16-20.DI YuXian,JI XinHua,LI LinAn,et al.Computational simulation ofthe elastic modulus of nanocrystalline materials[J].Journal of Mechani-cal Strength,2007,29(1):16-20(In Chinese).
[13]张克从.近代晶体学基础:上册[M].北京:科学出版社,1998:157-162.ZHANG KeCong.The Basis of Modern Crystallography:Volume 1[M].Beijing:Science Press,1998:157-162(In Chinese).
[14]王俊奎,丁立祚.弹性固体力学[M].北京:中国铁道出版社,1990:68-69.WHANGJunKui,DINGLiZuo.Elasticitysolid mechanics[M].Beijing:China Railway Press,1990:68-69(In Chinese).
[15]程昌钧.弹性力学[M].兰州:兰州大学出版社,1995:93-96.CHENG ChangJun.Elastic mechanics[M].Lanzhou:Lanzhou Universi-ty Press,1995:93-96(In Chinese).
[16]施振东,韩耀新.弹性力学教程[M].北京:北京航空学院出版社,1987:77-80.SHI ZhenDong,HAN YaoXin.Elastic mechanics course[M].Beijing:Beijing College of Aviation Press,1987:77-80(In Chinese).
[17]尹祥础.固体力学[M].北京:地震出版社,1985:68-70.YI XiangChu.Solid mechanics[M].Beijing:Earthquake PublishingHouse,1985:68-70(In Chinese).
[18]李同林,殷绥域.弹塑性力学[M].武汉:中国地质大学出版社,2006:50-51.LI TongLin,YINSuiYu.Elasticity and Plasticity Mechanics[M].Wu-han:China University of Geosciences Press,2006:50-51(In Chinese).
[19]陈长乐.固体物理学[M].西安:西北工业大学出版社,1998:1;1998:37.CHEN ChangLe.Solid State Physics[M].Xi′an:Northwestern Poly-technical University Press,1998:1;1998:37(In Chinese).
[20]伐因斯坦,鲍里斯康斯坦丁.现代晶体学:第2卷[M].合肥:中国科学技术大学出版社,1992:75.Weinstein,Boris Konstantinovich.Modern crystallography:Volume 2[M].Hefei:Press of University of Science and Technology of China,1992:75(In Chinese).
[21]杨顺华.晶体位错理论基础:第1卷[M].北京:科学出版社,1998:435.YANGShunHua.The theory basis of crystal dislocation:Volume 1[M].Beijing:Science Press,1998:435(In Chinese).
[22]万纾民.固体中原子间相互作用势能函数与碱金属、碱土金属弹性的电子理论[J].中国科学:A辑,1987(2):176.WANShuMin.Solidinteraction potential function between atoms and al-kali metal,alkaline earth metal electrontheory of elasticity[J].Sciencein China:ASeries,1987(2):176(In Chinese).
[23]谢佑卿.固体中多原子相互作用的新势能函数[J].中国科学:A辑,1992(8):887.XIE YouQing.Anewpotential energyfunction of solid multi-atominter-action[J].Science in China:ASeries,1992(8):887(In Chinese).
[24]于长丰,杨新铁,阎坤,王玲.由微观参量表示的金属单晶体杨氏模量的解析计算式[J].物理实验,2004,24(8):46.YUChangFeng,YANG XinTie,YAN Kun,WANG Ling.Analytic ex-pression dependent of microscopic parameters on the Young modulus ofmetallic single crystal[J].Physics Experimentation,2004,24(8):46(In Chinese).
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