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磁电弹性材料中含有带四条纳米裂纹的正4n边形纳米孔的反平面断裂问题

杨东升 刘官厅

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磁电弹性材料中含有带四条纳米裂纹的正4n边形纳米孔的反平面断裂问题

杨东升, 刘官厅

Anti-plane fracture problem of four nano-cracks emanating from a regular 4n-polygon nano-hole in magnetoelectroelastic materials

Yang Dong-Sheng, Liu Guan-Ting
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  • 基于Gurtin-Murdoch表面理论和保角变换技术, 研究了磁电弹性材料中含有带四条纳米裂纹的正4n边形纳米孔的反平面断裂问题. 得出了考虑表面效应时磁电非渗透边界条件下的应力强度因子、电位移强度因子、磁感应强度因子和能量释放率的精确解. 数值算例展示了表面效应和孔口尺寸对磁电非渗透边界条件下应力强度因子、电位移强度因子、磁感应强度因子和能量释放率的影响. 研究发现, 考虑表面效应时的应力场强度因子、电位移强度因子和磁感应强度因子具有明显的尺寸依赖, 并且当孔口尺寸增加到一定程度后, 表面效应的影响开始减小, 最终趋于经典弹性理论.
    According to the conformal mapping from the exterior region of the regular n-polygon hole to the exterior region of a unit circle and from the exterior region of four cracks emanating from a circle to the interior region of a unit circle, a new conformal mapping is constructed to map the exterior region of four cracks emanating from a regular 4n-polygon hole to the interior of a unit circle. Then, based on the Gurtin-Murdoch surface/interface model and complex method, the anti-plane fracture of four nano-cracks emanating from a regular 4n-polygon nano-hole in magnetoelectroelastic material is studied. The exact solutions of stress intensity factor, electric displacement intensity factor, magnetic induction intensity factor, and energy release rate are obtained under the boundary condition of magnetoelectrically impermeable with considering the surface effect. Without considering the effect of the surface effect, the exact solution of four cracks emanating from a regular 4n-polygon hole in a magnetoelectroelastic material can be obtained. The numerical results show the influences of surface effect and the size of defect on the stress intensity factor, electric displacement intensity factor, magnetic induction intensity factor and energy release rate under the magnetoelectrically impermeable boundary condition. It can be seen that the stress intensity factor, electric displacement intensity factor, and magnetic induction intensity factor are significantly size-dependent when considering the surface effects of the nanoscale defects. And when the size of defect increases to a certain extent, the influence of surface effect begins to decrease and finally tends to follow the classical elasticity theory. When the distance between the center and the vertex of the regular 4n-polygon nano-hole is constant, the dimensionless field intensity factor decreases gradually with the increase of the number of edges, and approaches to the conclusion of a circular hole with four cracks. With the increase of the relative size of the crack, the dimensionless field intensity factor increases gradually. The dimensionless energy release rate of the nanoscale cracked hole has a significant size effect. The increase of mechanical load will increase the normalized energy release rate. The normalized energy release rate first decreases and then increases with electrical load increasing. The normalized energy release rate decreases with magnetic load increasing.
      通信作者: 刘官厅, guantingliu@imnu.edu.cn
    • 基金项目: 国家重点研发计划(批准号: 2017YFC1405605)、自然资源部海洋遥测技术创新中心创新青年基金(批准号: 21k20190088)、内蒙古自然科学基金(批准号: 2018MS01005) 和内蒙古师范大学研究生科研创新基金(批准号: CXJJS19098)资助的课题.
      Corresponding author: Liu Guan-Ting, guantingliu@imnu.edu.cn
    • Funds: Project supported by the National Key R&D Program of China (Grant No. 2017YFC1405605), the Innovation Youth Fund of the Ocean Telemetry Technology Innovation Center of the Ministry of Natural Resources, China (Grant No. 21k20190088), the Natural Science Foundation of Inner Mongolia, China (Grant No. 2018MS01005), and the Graduate Students’ Scientific Research Innovation Fund Program of Inner Mongolia Normal University, China (Grant No. CXJJS19098)
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    刘鑫, 郭俊宏, 于静 2016 内蒙古大学学报(自然科学版) 41 37Google Scholar

    Liu X, Guo J H, Yu J 2016 J. Inner Monglia Univ. (Natural Science Edition) 41 37Google Scholar

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    Gao C F, Kessler H, Balke H 2003 Int. J. Eng. Sci. 41 969Google Scholar

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    Gao C F, Kessler H, Balke H 2003 Int. J. Eng. Sci. 41 983Google Scholar

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    齐敏 2005 硕士学位论文 (石家庄: 石家庄铁道大学)

    Qi M 2005 M. S. Thesis (Shijiazhuang: Shijiazhuang Tiedao University) (in Chinese)

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    Lv X, Liu G T 2018 Chin. Phys. B 27 074601Google Scholar

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    Zhong X C, Li C F 2008 Arch. Appl. Mech. 78 117Google Scholar

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    Gurtin M E, Murdoch A I 1975 Arch. Ration. Mech. Anal. 57 291Google Scholar

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    Gurtin M E, Murdoch A I 1978 Int. J. Solids Struct. 14 431Google Scholar

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    Gurtin M E, Weissmuller J, Larche F 1998 Philos. Mag. A 78 1093Google Scholar

    [14]

    Xiao J H, XU Y L, Zhang F C 2018 Acta. Mech. 229 4915Google Scholar

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    肖俊华, 崔友强, 徐耀玲, 张福成 2018 中国机械工程 29 2347Google Scholar

    Xiao J H, Cui Y Q, Xu Y L, Zhang F C 2018 China Mech. Eng. 29 2347Google Scholar

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    Xiao J H, Cui Y Q, Xu Y L, Zhang F C 2018 Theor. Appl. Fract. Mech. 96 476Google Scholar

    [17]

    Guo J H, Li X F 2018 Acta Mech. 229 4251Google Scholar

    [18]

    Liu Y Z, Guo J H, Zhang X Y 2019 Z. Angew. Math. Mech. 99 e201900043

    [19]

    Guo J H, He L T, Liu Y Z, Li L H 2020 Theor. Appl. Fract. Mech. 107 102553Google Scholar

    [20]

    Guo J H, Lu Z X 2011 Appl. Math. Comput. 217 9397Google Scholar

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    王永健 2012 博士学位论文 (南京: 南京航空航天大学)

    Wang Y J 2012 Ph. D. Dissertation (Nanjing: Nanjing University of Aeronautics and Astronautics) (in Chinese)

    [22]

    Fan S W, Guo J H, Yu J 2017 Chin. J. Aeronaut. 30 461Google Scholar

    [23]

    Dharmendra S, Sharma 2014 Int. J. Mech. Sci. 78 177Google Scholar

    [24]

    Wang Y B, Guo J H 2018 Appl. Math. Mech. -Engl. 39 797Google Scholar

    [25]

    Fang X Q, Gupta V, Liu J X 2013 Philos. Mag. Lett. 93 58Google Scholar

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    穆斯海里什维里 著(赵惠元 译) 1958 数学弹性力学的几个基本问题 (北京: 科学出版社) 第233页

    Muskhelishvili N I (translated by Zhao H Y) 1958 Some Basic Problems of the Mathematical Theory of Elasticity (Beijing: Science Press) p233 (in Chinese)

  • 图 1  磁电弹性材料中含有带四条裂纹的正方形孔

    Fig. 1.  Four cracks emanating from a square hole in magnetoelectroelastic materials

    图 2  保角变换

    Fig. 2.  Conformal mapping

    图 3  只受机械载荷作用时表面效应对应力强度因子的影响

    Fig. 3.  Surface effect on the stress intensity factor near the crack tip induced by anti-plane mechanical load $\tau_{32}^{\infty}$ only

    图 4  只受电载荷作用时表面效应对应力强度因子的影响

    Fig. 4.  Surface effect on the stress intensity factor near the crack tip induced by in-plane electrical load $D_2^{\infty}$ only

    图 5  只受磁载荷作用时表面效应对应力强度因子的影响

    Fig. 5.  Surface effect on the stress intensity factor near the crack tip induced by in-plane magnetic load $B_2^{\infty}$ only

    图 6  只受机械荷作用时表面效应对电位移强度因子的影响

    Fig. 6.  Surface effect on the electric displacement intensity factor near the crack tip induced by anti-plane mechanical load $\tau_{32}^{\infty}$ only

    图 7  只受电载荷作用时表面效应对电位移强度因子的影响

    Fig. 7.  Surface effect on the electric displacement intensity factor near the crack tip induced by in-plane electrical load $D_2^{\infty}$ only

    图 8  只受磁载荷作用时表面效应对电位移强度因子的影响

    Fig. 8.  Surface effect on the electric displacement intensity factor near the crack tip induced by in-plane magnetic load $B_2^{\infty}$ only

    图 9  只受机械载荷作用时表面效应对磁感应强度因子的影响

    Fig. 9.  Surface effect on the magnetic induction intensity factor near the crack tip induced by anti-plane mechanical load $\tau_{32}^{\infty}$ only

    图 10  只受电载荷作用时表面效应对磁感应强度因子的影响

    Fig. 10.  Surface effect on the magnetic induction intensity factor near the crack tip induced by in-plane electrical load $D_2^{\infty}$ only

    图 11  只受磁载荷作用时表面效应对磁感应强度因子的影响

    Fig. 11.  Surface effect on the magnetic induction intensity factor near the crack tip induced by in-plane magnetic load $B_2^{\infty}$ only

    图 12  无量纲应力强度因子随右侧裂纹长度变化

    Fig. 12.  Variations of the dimensionless stress intensity factor at crack tip with the crack length $L_1$

    图 13  无量纲应力强度因子随n的变化

    Fig. 13.  Variations of the dimensionless stress intensity factor at crack tip with n

    图 14  无量纲能量释放率随孔口尺寸的变化

    Fig. 14.  Variations of the dimensionless energy release rate with the size of the cracked hole

    图 15  正则化能量释放率随机械载荷的变化

    Fig. 15.  Variations of the normalized energy release rate with the applied mechanical load

    图 16  正则化能量释放率随电载荷的变化

    Fig. 16.  Variations of the normalized energy release rate with the applied eletrical load

    图 17  正则化能量释放率随磁载荷的变化

    Fig. 17.  Variations of the normalized energy release rate with the applied magnetic load

  • [1]

    Nan C W 1994 Phys. Rev. B 50 6082Google Scholar

    [2]

    Guo J H, Lu Z X 2010 Int. J. solids Struct. 47 1847Google Scholar

    [3]

    Rogowski B 2011 Arch. Appl. Mech. 81 1607Google Scholar

    [4]

    刘鑫, 郭俊宏, 于静 2016 内蒙古大学学报(自然科学版) 41 37Google Scholar

    Liu X, Guo J H, Yu J 2016 J. Inner Monglia Univ. (Natural Science Edition) 41 37Google Scholar

    [5]

    Gao C F, Kessler H, Balke H 2003 Int. J. Eng. Sci. 41 969Google Scholar

    [6]

    Gao C F, Kessler H, Balke H 2003 Int. J. Eng. Sci. 41 983Google Scholar

    [7]

    Liu X, Guo J H 2016 Theor. Appl. Fract. Mech. 86 225Google Scholar

    [8]

    齐敏 2005 硕士学位论文 (石家庄: 石家庄铁道大学)

    Qi M 2005 M. S. Thesis (Shijiazhuang: Shijiazhuang Tiedao University) (in Chinese)

    [9]

    Lv X, Liu G T 2018 Chin. Phys. B 27 074601Google Scholar

    [10]

    Zhong X C, Li C F 2008 Arch. Appl. Mech. 78 117Google Scholar

    [11]

    Gurtin M E, Murdoch A I 1975 Arch. Ration. Mech. Anal. 57 291Google Scholar

    [12]

    Gurtin M E, Murdoch A I 1978 Int. J. Solids Struct. 14 431Google Scholar

    [13]

    Gurtin M E, Weissmuller J, Larche F 1998 Philos. Mag. A 78 1093Google Scholar

    [14]

    Xiao J H, XU Y L, Zhang F C 2018 Acta. Mech. 229 4915Google Scholar

    [15]

    肖俊华, 崔友强, 徐耀玲, 张福成 2018 中国机械工程 29 2347Google Scholar

    Xiao J H, Cui Y Q, Xu Y L, Zhang F C 2018 China Mech. Eng. 29 2347Google Scholar

    [16]

    Xiao J H, Cui Y Q, Xu Y L, Zhang F C 2018 Theor. Appl. Fract. Mech. 96 476Google Scholar

    [17]

    Guo J H, Li X F 2018 Acta Mech. 229 4251Google Scholar

    [18]

    Liu Y Z, Guo J H, Zhang X Y 2019 Z. Angew. Math. Mech. 99 e201900043

    [19]

    Guo J H, He L T, Liu Y Z, Li L H 2020 Theor. Appl. Fract. Mech. 107 102553Google Scholar

    [20]

    Guo J H, Lu Z X 2011 Appl. Math. Comput. 217 9397Google Scholar

    [21]

    王永健 2012 博士学位论文 (南京: 南京航空航天大学)

    Wang Y J 2012 Ph. D. Dissertation (Nanjing: Nanjing University of Aeronautics and Astronautics) (in Chinese)

    [22]

    Fan S W, Guo J H, Yu J 2017 Chin. J. Aeronaut. 30 461Google Scholar

    [23]

    Dharmendra S, Sharma 2014 Int. J. Mech. Sci. 78 177Google Scholar

    [24]

    Wang Y B, Guo J H 2018 Appl. Math. Mech. -Engl. 39 797Google Scholar

    [25]

    Fang X Q, Gupta V, Liu J X 2013 Philos. Mag. Lett. 93 58Google Scholar

    [26]

    穆斯海里什维里 著(赵惠元 译) 1958 数学弹性力学的几个基本问题 (北京: 科学出版社) 第233页

    Muskhelishvili N I (translated by Zhao H Y) 1958 Some Basic Problems of the Mathematical Theory of Elasticity (Beijing: Science Press) p233 (in Chinese)

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出版历程
  • 收稿日期:  2020-06-04
  • 修回日期:  2020-07-26
  • 上网日期:  2020-11-27
  • 刊出日期:  2020-12-20

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