一维六方压电准晶中三角形孔边快速传播裂纹的解析解
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摘要
根据一维六方压电准晶的本构方程、几何方程和运动平衡方程,导出了一维六方压电准晶在运动坐标系下的反平面弹性问题的控制方程,并利用复变函数方法,构造保角映射函数,将物理平面的三角形孔边裂纹外部映射到数学平面的单位圆内部,从而研究了一维六方压电准晶中三角形孔边快速传播裂纹的反平面剪切问题.并在电不可通与电可通两种边界条件下,给出了裂纹以速度v传播时的Ⅲ型裂纹的动态应力强度因子和电位移强度因子的解析解.当裂纹传播速度趋于零时,三角形孔边快速传播裂纹的动力学问题可以还原为静力学问题,通过极限运算,可以得到在裂纹尖端处的应力强度因子和电位移强度因子的解析解,所得结果与已有的静力学结果一致,这些解析解在科学与工程断裂中有着潜在的应用价值,为工程力学分析提供了理论依据.
According to the constitutive equations,geometric equations and motion equilibrium equations of 1D hexagonal piezoelectric quasicrystals, the governing equations of anti-plane elastic problem in kinematic coordinate system were derived. The anti-plane shear problem of 1D hexagonal piezoelectric quasicrystals was analyzed through adopting the construction of conformal mapping with the complex variable function method. Under the boundary conditiona of electrically unpassable and passable, the analytic solutions of the dynamic stress intensity factors and the electric displacement intensity factors of the crack propagating at velocity v were obtained. When the crack velocity approaches zeros,dynamic problem reduced to static problem, the analytic solutions of the stress intensity factor and electric displacement intensity factor at the crack tip could be obtained, it is consistent with the existing static results, which has potential application value in scientific and engineering fracture and provided a theoretical basis for engineering mechanics analysis.
According to the constitutive equations,geometric equations and motion equilibrium equations of 1D hexagonal piezoelectric quasicrystals, the governing equations of anti-plane elastic problem in kinematic coordinate system were derived. The anti-plane shear problem of 1D hexagonal piezoelectric quasicrystals was analyzed through adopting the construction of conformal mapping with the complex variable function method. Under the boundary conditiona of electrically unpassable and passable, the analytic solutions of the dynamic stress intensity factors and the electric displacement intensity factors of the crack propagating at velocity v were obtained. When the crack velocity approaches zeros,dynamic problem reduced to static problem, the analytic solutions of the stress intensity factor and electric displacement intensity factor at the crack tip could be obtained, it is consistent with the existing static results, which has potential application value in scientific and engineering fracture and provided a theoretical basis for engineering mechanics analysis.
引文
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[14]樊世旺,郭俊宏.一维六方压电准晶三角形孔边裂纹反平面问题[J].应用力学学报,2016, 33(3):421-426.
[15]郭俊宏,于静.多场耦合材料断裂力学[M].北京:科学出版社,2016.
[2]薛存,周又和,雍华东.压电材料反平面Ⅲ型裂纹电塑性区的分析[J].固体力学学报,2012, 33(5):449-455.
[3]方岱宁,刘金喜.压电与铁电体的断裂力学[M].北京:清华大学出版社,2008.
[4]GAO C F, WANG M Z. Periodical cracks in piezoelectric media[J]. Mechanics Research Communications, 1999, 26(4):427-432.
[5]WANG X. The general solution of one-dimensional hexagonal quasicrystal[J]. Mechanics Research Communications, 2006, 33(4):76-580.
[6]Wang X, Pan E. Analytical solutions for some defect problems in 1D hexagonal and 2D octagonal quasicrystals[J]. Pramana, 2008, 70(5):911-933.
[7]Zhang Liang-liang, Zhang Yi-ming, Gao Yang. General solutions of plane elasticity of one-dimensional orthorhombic quasicrystals with piezoelectric effect[J]. Physics Letters A, 2014,(378):2768-2776.
[8]李星,霍华颂,时朋朋.一维六方压电准晶对称条形体中共线双半无限快速传播裂纹的解析解[J].力学学报,2014, 35(2):135-141.
[9]张峰,李星.一维六方压电准晶中唇形匀速移动裂纹问题的解析解[J].力学季刊,2015, 36(2):213-220.
[10]王玮华,郭俊宏,邢永明.压电弹性体中光滑顶点的正三角形孔边裂纹的反平面问题分析[J].复合材料学报,2015, 32(2):601-607.
[11]FAN T Y. Mathematical theory of elasticity of quasicrystals and its applications[M]. Beijing:Springer,2011.
[12]郭俊宏,刘官厅.一维六方准晶中具有不对称裂纹的圆形孔口问题的解析解[J].应用数学学报,2007,30(6):1066-1075.
[13]杨娟,李星.压电效应下一维六方准晶中孔边多裂纹反平面断裂间题研究[D].宁夏大学,2015.
[14]樊世旺,郭俊宏.一维六方压电准晶三角形孔边裂纹反平面问题[J].应用力学学报,2016, 33(3):421-426.
[15]郭俊宏,于静.多场耦合材料断裂力学[M].北京:科学出版社,2016.