一维六方准晶压电材料中多缺陷的相互作用
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摘要
基于准晶压电材料的基本方程,利用解析函数理论和复变方法,研究了一维六方准晶压电材料中多缺陷的相互作用问题,建立了多条平行位错以及它们与半无限裂纹相互作用的断裂力学模型,给出了一维六方准晶压电材料中n条平行位错相互作用的Peach-Koehler公式和n条平行位错的等效作用点,得到了n条平行位错与半无限裂纹相互作用下电弹性场的解析解,数值算例给出了位错位置对裂纹面上应力的影响和位错Burgers矢量的大小对裂纹面上应力的影响,为讨论裂纹尖端的位错发射、位错屏蔽和裂纹钝化奠定了理论基础.这些结果均为本文首次给出,丰富和发展了经典弹性理论中的相应结果.
Like the classical materials,there are a variety of defects such as cracks and dislocations in quasicrystalline materials.Based on the fundamental equations of piezoelectricity of quasicrystalline material,by means of the analytic function theory and the complex variable method,the interactions among multi-defects in the piezoelectric material of one-dimensional hexagonal quasicrystals are studied.First,the models of fracture mechanics of the interactions among n parallel dislocations and a semi-infinite crack in the material are established,and the interaction forces and the equivalent action point of the nparallel dislocations are obtained,which are the versions of the well-known Peach-Koehler formula in the piezoelectric material of one-dimensional hexagonal quasicrystals with nparallel dislocations.Second,the analytic solutions of electric-elastic fields of the interactions among n parallel dislocations and a semi-infinite crack in the piezoelectric material of one-dimensional hexagonal quasicrystals are derived.Finally,some numerical examples show that the stress and electric displacement of crack surface vary with the position of dislocation and the size of Burgers vector.These results offer the basis of theory to discuss the dislocation emission from a crack tip,screening for dislocation and crack shielding in the piezoelectric material of one-dimensional hexagonal quasicrystals.As the development of the corresponding parts of classical elasticity,these are all firstly given in the present paper.When the electric fields or phason fields disappear,the results of this paper degenerate into those of the classical one.
Like the classical materials,there are a variety of defects such as cracks and dislocations in quasicrystalline materials.Based on the fundamental equations of piezoelectricity of quasicrystalline material,by means of the analytic function theory and the complex variable method,the interactions among multi-defects in the piezoelectric material of one-dimensional hexagonal quasicrystals are studied.First,the models of fracture mechanics of the interactions among n parallel dislocations and a semi-infinite crack in the material are established,and the interaction forces and the equivalent action point of the nparallel dislocations are obtained,which are the versions of the well-known Peach-Koehler formula in the piezoelectric material of one-dimensional hexagonal quasicrystals with nparallel dislocations.Second,the analytic solutions of electric-elastic fields of the interactions among n parallel dislocations and a semi-infinite crack in the piezoelectric material of one-dimensional hexagonal quasicrystals are derived.Finally,some numerical examples show that the stress and electric displacement of crack surface vary with the position of dislocation and the size of Burgers vector.These results offer the basis of theory to discuss the dislocation emission from a crack tip,screening for dislocation and crack shielding in the piezoelectric material of one-dimensional hexagonal quasicrystals.As the development of the corresponding parts of classical elasticity,these are all firstly given in the present paper.When the electric fields or phason fields disappear,the results of this paper degenerate into those of the classical one.
引文
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[11]Gao Y.Governing equations and general solutions of plane elasticity of cubic quasicrystals[J].Physics Letters A,2009,373:885-889.
[12]Guo J H,Lu Z X.Exact solution of four cracks originating from an elliptical hole in one-dimensional hexagonal quasicrystals[J].Applied Mathematics and Computation,2011,217:9397-9403.
[13]Li L H,Liu G T.Stroh formalism for icosahedral quasicrystal and its application[J].Physics Letters A,2012,376:987-990.
[14]Altay G,Domeci M C.On the fundamental equations of piezoelasticity of quasicrystal media[J].Int J Solids Struct,2012,49:3255-3262.
[15]Li X Y,Li P D,Wu T H,Shi M X,Zhu Z W.Three-dimensional fundamental solutions for one-dimensional hexagonal quasicrystal with piezoelectric effect[J].Phys Lett A,2014,378:826-834.
[16]Zhang L L,Zhang Y M,Gao Y.General solutions of plane elasticity of one-dimensional orthorhombic quasicrystals with piezoelectric effect[J].Phys Lett A,2014,378:2768-2776.
[17]Yang J,Li X.The Anti-plane shear problem of two symmetric cracks originating from an elliptical hole in1Dhexagonal piezoelectric QCs[J].Advanced Materials Research,2014,936:127-135.
[18]Yu J,Guo J H,Pan E N,Xing Y M.General solutions of plane problem in one-dimensional quasicrystal piezoelectric materials and its application on fracture mechanics[J].Applied Mathematics and Mechanics,2015,36(6):793-814.
[19]Yu J,Guo J H,Xing Y M.Complex variable method for an anti-plane elliptical cavity of one-dimensional hexagonal piezoelectric quasicrystals[J].Chinese Journal of Aeronautics,2015,28(4):1287-1295.
[20]周彦斌,刘官厅.一维六方准晶压电材料反平面Ⅲ型裂纹的电塑性分析[J].固体力学学报,2015,36(1):63-68.(Zhou Y B,Liu G T.Analysis of the electric yielding zone of the anti-plane modeⅢcrack in piezoelectric materials of one-dimensional hexagonal quasicrystals[J].Chinese Journal of solid mechanics,2015,36(1):63-68.(in Chinese))
[21]钟海英,刘官厅.一维六方准晶压电材料中唇形裂纹反平面问题的解析解[J].固体力学学报,2015,36(2):179-184.(Zhong H Y,Liu G T.Anti-plane Analysis of a lip-shape crack of piezoelectricity of onedimensional hexagonal quasicrystals[J].Chinese Journal of solid mechanics,2015,36(2):179-184.(in Chinese))
[22]Lin I H,Thomson R.Cleavage,dislocation emission and shielding for cracks under general loading[J].Acta Metall,1986,34(2):187-206.
[23]Muskhelishvili N I.Translated by Radok J R M.Some Basic Problems of the Mathematical Theory of Elasticity[M].Groningen-Holland:P.Noordholff Ltd.,1953.
[24]Li X Y,Li P D,et al.Three-dimensional fundamental solutions for one-dimensional hexagonal quasicrystal with piezoelectric effect[J].Phys Lett A,2014,378:826-834.
[2]Zhang Z,Urban K.Transmission electron microscope observation of dislocation and stackling fanlts in a decagonal Al-Cu-Co alloy[J].Phil Mag Lett,1989,60(1):97-102.
[3]Ding D H,Yang W G,Hu C Z,et al.Generalized elasticity theory of quasicrystals[J].Phys Rev B,1993,48(10):7003-7010.
[4]Ding D H,Wang R H,Yang W G et al.Elasticity theory of straight dislocation in quasicrystals[J].Phil Mag Lett,1995,72(5):353-359.
[5]Wang R H,Yang W G,Hu C Z,et al.Point and space groups and elastic behaviours of onedimensional quasicrytals[J].J Phys Condens Matter,1997,9:2411-2422.
[6]Fan T Y,Li X F,Sun Y F.A moving screw dislocation in a one-dimensional hexagonal quasicrystals[J].Acta Phys Sinica(Overseas Edition),1999,8(4):288-295.
[7]Li X F,Fan T Y.A straight dislocation in one dimensional hexagonal quasicrystals[J].Phys Stat B,1999,212(1):19-26.
[8]Liu G T,Guo R P,Fan T Y.On the interaction between dislocations and cracks in one-dimensional hexagonal quasicrystals[J].Chinese Physics,2003,2(10):1149-1155.
[9]Liu G T,Fan T Y,Guo R P.Governing equations and general solutions of plane elasticity of one-dimensional quasicrystals[J].Int J Solids and Structures,2004,41(14):3949-3959.
[10]Guo J H,Liu G T.Exact analytic solutions for an elliptic hole with two straight cracks in one-dimensional hexagonal quasicrystal[J].Chin Phys B,2008,17,2610-2620.
[11]Gao Y.Governing equations and general solutions of plane elasticity of cubic quasicrystals[J].Physics Letters A,2009,373:885-889.
[12]Guo J H,Lu Z X.Exact solution of four cracks originating from an elliptical hole in one-dimensional hexagonal quasicrystals[J].Applied Mathematics and Computation,2011,217:9397-9403.
[13]Li L H,Liu G T.Stroh formalism for icosahedral quasicrystal and its application[J].Physics Letters A,2012,376:987-990.
[14]Altay G,Domeci M C.On the fundamental equations of piezoelasticity of quasicrystal media[J].Int J Solids Struct,2012,49:3255-3262.
[15]Li X Y,Li P D,Wu T H,Shi M X,Zhu Z W.Three-dimensional fundamental solutions for one-dimensional hexagonal quasicrystal with piezoelectric effect[J].Phys Lett A,2014,378:826-834.
[16]Zhang L L,Zhang Y M,Gao Y.General solutions of plane elasticity of one-dimensional orthorhombic quasicrystals with piezoelectric effect[J].Phys Lett A,2014,378:2768-2776.
[17]Yang J,Li X.The Anti-plane shear problem of two symmetric cracks originating from an elliptical hole in1Dhexagonal piezoelectric QCs[J].Advanced Materials Research,2014,936:127-135.
[18]Yu J,Guo J H,Pan E N,Xing Y M.General solutions of plane problem in one-dimensional quasicrystal piezoelectric materials and its application on fracture mechanics[J].Applied Mathematics and Mechanics,2015,36(6):793-814.
[19]Yu J,Guo J H,Xing Y M.Complex variable method for an anti-plane elliptical cavity of one-dimensional hexagonal piezoelectric quasicrystals[J].Chinese Journal of Aeronautics,2015,28(4):1287-1295.
[20]周彦斌,刘官厅.一维六方准晶压电材料反平面Ⅲ型裂纹的电塑性分析[J].固体力学学报,2015,36(1):63-68.(Zhou Y B,Liu G T.Analysis of the electric yielding zone of the anti-plane modeⅢcrack in piezoelectric materials of one-dimensional hexagonal quasicrystals[J].Chinese Journal of solid mechanics,2015,36(1):63-68.(in Chinese))
[21]钟海英,刘官厅.一维六方准晶压电材料中唇形裂纹反平面问题的解析解[J].固体力学学报,2015,36(2):179-184.(Zhong H Y,Liu G T.Anti-plane Analysis of a lip-shape crack of piezoelectricity of onedimensional hexagonal quasicrystals[J].Chinese Journal of solid mechanics,2015,36(2):179-184.(in Chinese))
[22]Lin I H,Thomson R.Cleavage,dislocation emission and shielding for cracks under general loading[J].Acta Metall,1986,34(2):187-206.
[23]Muskhelishvili N I.Translated by Radok J R M.Some Basic Problems of the Mathematical Theory of Elasticity[M].Groningen-Holland:P.Noordholff Ltd.,1953.
[24]Li X Y,Li P D,et al.Three-dimensional fundamental solutions for one-dimensional hexagonal quasicrystal with piezoelectric effect[J].Phys Lett A,2014,378:826-834.