层状磁电复合材料界面共线裂纹平面问题分析
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摘要
本文研究了面内电磁势载荷作用下双层压电压磁复合材料中共线界面裂纹问题.考虑了压电材料的导磁性质和压磁材料的介电性质,引入了界面电位移和磁感强度的连续性条件.利用Fourier变换得到一组第二类Cauchy型奇异积分方程.进一步导出了相应问题的应力强度因子、电位移强度因子和磁感强度强度因子的表达式,给出了应力强度因子的数值结果.结果表明电磁载荷会导致界面裂纹尖端I、II混合型应力奇异性,同时还伴随着电位移和磁感强度的奇异性.比较了双裂纹左右端的应力强度因子,发现在面内极化方向上施加面内磁势载荷时共线裂纹内侧尖端区域的两个法向应力场发生互相干涉增强.
A plane fracture problem of collinear interfacial cracks in layered magnetoelectric composites is studied under in-plane magnetic or electric loading. The magnetic permittivity of the piezoelectric material and the dielectric constant of the piezomagnetic material are considered. A system of singular integral equation of the second kind with Cauchy kernel is obtained by means of Fourier transform and further solved by using Jacobi polynomials. The primary interfacial fracture parameters, such as the stress intensity factors, the electric displacement intensity factors, the magnetic induction intensity factors are then obtained. It is found that a magnetic or electric loading normal to the crack surfaces can lead to a mixture of mode I and mode II type stress singularities at crack tips. The stress intensity factors at the right and left ends of the two cracks are compared. It is found that when the in-plane magnetization load is applied in the in-plane polarization direction, the two normal stress fields in the inner tip region of the collinear cracks interfere with each other.
A plane fracture problem of collinear interfacial cracks in layered magnetoelectric composites is studied under in-plane magnetic or electric loading. The magnetic permittivity of the piezoelectric material and the dielectric constant of the piezomagnetic material are considered. A system of singular integral equation of the second kind with Cauchy kernel is obtained by means of Fourier transform and further solved by using Jacobi polynomials. The primary interfacial fracture parameters, such as the stress intensity factors, the electric displacement intensity factors, the magnetic induction intensity factors are then obtained. It is found that a magnetic or electric loading normal to the crack surfaces can lead to a mixture of mode I and mode II type stress singularities at crack tips. The stress intensity factors at the right and left ends of the two cracks are compared. It is found that when the in-plane magnetization load is applied in the in-plane polarization direction, the two normal stress fields in the inner tip region of the collinear cracks interfere with each other.
引文
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[2]LI X F,TANG G J.Electroelastic analysis of an interface anti-plane shear crack in a layered piezoelectric plate[J].International Journal of Engineering Science,2003,41:1405-1422.
[3]LIU H T,ZHOU Z G,WU J G.Non-singular anti-plane fracture theory of two parallel cracks within non-local functionally graded piezoelectric materials[J].Strength Fracture and Complexity,2017,10:23-38.
[4]LIANG J.Non-local theory solution of two collinear mode-I cracks in piezoelectric materials[J].Applied Mathematical Modelling,2008,32:1126-1142.
[5]CAO R K,WAN Y P.Electro-elastic analysis of bonded piezoelectric finite wedges with two collinear interface cracks under anti-plane deformation[J].Mathematics and Mechanics of Solids,2017,22:1068-1093.
[6]ABE M,IKEDA T,KOGANEMARU M,et al.Stress intensity factor analysis of a three-dimensional interfacial corner between anisotropic piezoelectric multi-materials under several boundary conditions on the corner surfaces[J].Engineering Fracture Mechanics,2017,171:1-21.
[7]金燕,万永平.多孔压电驻极体材料的孔洞微结构优化研究[J].力学季刊,2016,37(1):96-103.
[8]王永健,宋豪鹏,高存法,等.双压电材料内含椭圆孔孔边界面裂纹的反平面问题[J].力学季刊,2015,36(3):416-426.
[9]TIAN W Y,GABBERT U.Parallel crack near the interface of magnetoelectroelastic bimaterials[J].Computational Materials Science,2005,32:562-567.
[10]SOH A K,Liu J X.Mode III interfacial edge crack in a magnetoelectroelastic bimaterial[J].Key Engineering Materials,2004,261:393-398.
[11]WANG B L,SUN Y G,HAN J C,et al.Interfacial fracture of layered magnetoelectroelastic materials[J].Mechanics of Composite Materials,2008,44:95-108.
[12]LI Y D,LEE K Y.Effects of magneto-electric loadings and piezomagnetic/piezoelectric stiffening on multiferroic interface fracture[J].Engineering Fracture Mechanics,2010,77:856-866.
[13]XIONG T,LI Y D,SONG Z K.Interfacial sliding in a bi-layered multiferroic ceramics:localized sliding-prevention/promotion based mechanism of intra-layer fracture[J].International Journal of Solids and Structures,2015,77:15-27.
[14]胡晓娜,岳岩鹏.集中荷载作用下的压电压磁双材料Ⅲ型界面裂纹问题[J].力学季刊,2015,36(3):534-540.
[15]时朋朋,霍华颂,李星.功能梯度压电/压磁双材料的周期界面裂纹问题[J].力学季刊,2013,34(2):191-198.
[16]WAN Y,YUE Y,ZHONG Z.Multilayered piezomagnetic/piezoelectric composite with periodic interface cracks under magnetic or electric field[J].Engineering Fracture Mechanics,2012,84:132-145.
[17]TIAN W,ZHONG Z,LI Y.Multilayered piezomagnetic/piezoelectric composites with periodic interfacial cracks subject to in-plane loading[J].Smart Materials and Structures,2016,25:015029.
[18]LIANG J.Two collinear interface cracks between two dissimilar functionally graded piezoelectric/piezomagnetic material layers under anti-plane shear loading[J].JSME International Journal Series A-Solid Mechanics and Material Engineering,2006,49:559-569.
[19]SINGH B,ROKNE J,DHALIWAL R.Closed-form solutions for two anti-plane collinear cracks in a magnetoelectroelastic layer[J].European Journal of Mechanics-A/Solids,2009,28:599-609.
[20]MA P,FENG W J,JING J.Two dimensional interfacial cracks analysis between piezoelectric and piezomagnetic layers[J].Engineering Mechanics,2011,28:163-169.
[21]LANDIS C M.Energetically consistent boundary conditions for electromechanical fracture[J].International Journal of Solids and Structures,2004,41:6291-6315.
[22]PARTON V Z.Fracture mechanics of piezoelectric materials[J].Acta Astronautica,1976,3:671-683.
[23]HAO T H,SHEN Z Y.A new electric boundary condition of electric fracture mechanics and its applications[J].Engineering Fracture Mechanics,1994,47(6):793-802.
[24]WANG B L,MAI Y W.Applicability of the crack-face electromagnetic boundary conditions for fracture of magnetoelectroelastic materials[J].International Journal of Solids and Structures,2007,44:387-398.
[25]WILSON Z A,BORDEN M J,LANDIS C M.A phase-field model for fracture in piezoelectric ceramics[J].International Journal of Fracture,2013,183:135-153.
[26]SHEN S,KUANG Z B,NISHIOKA T.Wave scattering from an interface crack in multilayered piezoelectric plate[J].European Journal of Mechanics Series A Solids,2000,19:547-560.
[27]ERDOGAN F.Approximate solutions of systems of singular integral equations[J].SIAM Journal on Applied Mathematics,1969,17:1041-1059.
[28]ERDéLYI A.Higher transcendental functions:vol.1[M].New York:Mc Graw-Hill Interamericana,1953.
[29]GU B,YU S W,FENG X Q.Transient response of an insulating crack between dissimilar piezoelectric layers under mechanical and electrical impacts[J].Archive of Applied Mechanics,2002,72:615-629.
[30]ZHONG Y F,CHEN L,YU W B,et al.Asymptotical construction of a fully coupled,Reissner-Mindlin model for piezoelectric and piezomagnetic laminates[J].Composite Structures,2012,94:3583-3591.