二维十次对称压电准晶含Griffith裂纹的平面问题
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摘要
利用Stroh公式结合半逆解法,得到裂纹尖端附近的场解和场强度因子解,并利用权函数方法求解裂纹尖端的能量释放率.结合算例,探讨不同集中载荷作用下场强度因子和能量释放率的变化规律,并分析无限远处均匀载荷作用时裂尖附近的应力和位移,与椭圆孔以及相应的退化结果对比验证.结果表明,在裂尖附近作用集中载荷,对力强度因子以及电位移强度因子有显著影响,能量释放率是电场、声子场、相位子场、声子场-相位子场耦合效应以及电场-声子场耦合效应共同作用的结果,且应力强度因子、电位移强度因子和能量释放率共同表征了裂纹扩展过程中的应力集中以及扩展的大致方向.
The analytical expressions for the entire fields and field intensity factors in the coupled fields were obtained by utilizing the generalized Stroh formalism combined with semi-inverse method;the energy release rate of the crack tip was solved with the weight function method.What's more,through numerical examples,the change rules of field intensity factors derived from the concentrated loadings were discussed;the stress and displacement around crack tip with remote uniform loading were analyzed,and the results were compared with elliptical hole and degradation results.Results show that concentrated loadings near the crack tip have obvious influence for stress intensity factors and electric displacement intensity factor.Energy release rate is a combined result because of electric field,phonon field,phase field,phonon-phase coupling field and electric-phonon coupling effect.Intensity factors and energy release rate jointly characterize some rules of the stress concentration and the direction of crack propagation.
The analytical expressions for the entire fields and field intensity factors in the coupled fields were obtained by utilizing the generalized Stroh formalism combined with semi-inverse method;the energy release rate of the crack tip was solved with the weight function method.What's more,through numerical examples,the change rules of field intensity factors derived from the concentrated loadings were discussed;the stress and displacement around crack tip with remote uniform loading were analyzed,and the results were compared with elliptical hole and degradation results.Results show that concentrated loadings near the crack tip have obvious influence for stress intensity factors and electric displacement intensity factor.Energy release rate is a combined result because of electric field,phonon field,phase field,phonon-phase coupling field and electric-phonon coupling effect.Intensity factors and energy release rate jointly characterize some rules of the stress concentration and the direction of crack propagation.
引文
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[3]FAN T Y.Mathematical theory of elasticity of quasicrystals and its applications[M].Berlin:Springer,2011:67-69.
[4]STADNIK Z M.Physical properties of quasicrystals[M].New York:Springer Science&Business Media,1999.
[5]董闯.准晶材料的形成机制,性能及应用前景[J].材料研究学报,1994,8(6):482-490.DONG Chuang.The formation mechanism,properties and application potentials of quasicrystalline materials[J].Chinese Journal of Materials Research,1994,8(6):482-490.
[6]ALTAY G,DKMECI M C.On the fundamental equations of piezoelasticity of quasicrystal media[J].International Journal of Solids and Structures,2012,49(23):3255-3262.
[7]HU C Z,WANG R H,DING D H,YANG W G.Piezoelectric effects in quasicrystals[J].Physical Review B,1997,56(5):2463-2469.
[8]YU J,GUO J H,PAN E N,et al.General solutions of plane problem in one-dimensional quasicrystal piezoelectric materials and its application on fracture mechanics[J].Applied Mathematics and MechanicsEnglish Edition,2015,36(6):793-814.
[9]GAO C F,WANG M Z.Green’s functions for generalized 2D problems in piezoelectric media with an elliptic hole[J].Mechanics Research Communications,1998,25(6):685-693.
[10]GAO Y,XU B X.Method on holomorphic vector functions and applications in two-dimensional quasicrystals[J].International Journal of Modern Physics B,2008,22(6):635-643.
[11]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.FAN Shi-wang,GUO Jun-hong.The antiplane problem of one-dimensional hexagonal quasicrystals with an edge crack emanating from a triangle hole[J].Chinese Journal of Applied Mechanics,2016,33(3):421-426.
[12]HWU C.Anisotropic Elastic Plates[M].New York:Springer,2010:.
[13]SUO Z,KUO C M,BARNETT D M,et al.Fracture mechanics for piezoelectric ceramics[J].Journal of the Mechanics and Physics of Solids,1992,40(4):739-765.
[14]SOSA H,KHUTORYANSKY N.New developments concerning piezoelectric materials with defects[J].International Journal of Solids and Structures,1996,33(23):3399-3414.
[15]GUO Y C,FAN T Y.A mode-II Griffith crack in decagonal quasicrystals[J].Applied Mathematics and Mechanics-English Edition,2001,22(11):1311-1317.
[16]FU R,ZHANG T Y.Effects of an electric field on the fracture toughness of poled lead zirconate titanate ceramics[J].Journal of the American Ceramic Society,2000,83(5):1215-1218.
[17]MCMEEKING R,RICOEUR A.The weight function for cracks in piezoelectrics[J].International Journal of Solids and Structures,2003,40(22):6143-6162.
[18]FAN T Y.Mathematical theory and methods of mechanics of quasicrystalline materials[J].Engineering,2013,5(4):407-448.
[19]LEE J S,JIANG L Z.Exact electroelastic analysis of piezoelectric laminae via state space approach[J].International Journal of Solids and Structures,1996,33(7):977-990.
[20]YANG L Z,GAO Y,PAN E N,et al.An exact solution for a multilayered two-dimensional decagonal quasicrystal plate[J].International Journal of Solids and Structures,2014,51(9):1737-1749.
[21]高存法,樊蔚勋.压电介质内裂纹问题的精确解[J].应用数学与力学,1999,20(1):47-53.GAO Cun-fa,FAN Wei-xun.A exact solution of crack problems in piezoelectric materials[J].Applied Mathematics and Mechanics,1999,20(1):47-53.