压电体中考虑表面效应时开裂孔洞的断裂特征
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摘要
研究了反平面机械载荷和面内电载荷作用下压电体中考虑表面效应时孔边双裂纹问题的断裂特征.基于Gurtin-Murdoch表面理论模型,通过构造映射函数,利用复势电弹理论获得了应力场和电位移场的闭合解答.给出了裂纹尖端应力强度因子、电位移场强因子和能量释放率的解析解.讨论了开裂孔洞几何参数和施加力电载荷对电弹场强因子和能量释放率的影响.
When defects(holes and cracks)in the piezoelectric material are on the order of nanometers,the surface properties of the defects have great influences on the distributions of the stress fields and the electric displacement fields,and the surface effects are not negligible.In this paper,the surface electroelastic constants of the defects are introduced to extend the size of the cracked circular hole in the piezoelectric solids to nanoscale.The fracture characteristics of piezoelectric solids containing two edge cracks emanating from a circular hole with its surface under anti-plane mechanical loads and in-plane electric displacement loads are investigated theoretically.Based on the Gurtin-Murdoch surface model,the closed solutions of the stress field and electric displacement field of the problem are obtained using the complex potential-function electroelastic theory via constructing a conformal mapping function.Analytical expressions of the stress intensity factor,electric displacement field factor and energy release rate at the crack tip are presented.The influences of the geometric parameters of the cracked hole,the applied mechanical load and electrical load on the electroelastic field intensity factors and the energy release rate are discussed.The major results are as follows.The dimensionless stress intensity factor and the dimensionless electric displacement field factor of the nanoscale cracked hole are different,but are both significantly size-dependent.The dimensionless electroelastic field factors increase monotonously with the increase of the relative length of the crack to a fixed value.The dimensionless energy release rate of the nanoscale cracked hole has a significant size effect.The larger is the hole and the longer is the crack,the greater is the normalized energy release rate.The effect of the mechanical loads on the normalized energy release rate is affected by the applied electrical loads.The normalized energy release rate increases first and then decreases with the increase of the electrical loads.
When defects(holes and cracks)in the piezoelectric material are on the order of nanometers,the surface properties of the defects have great influences on the distributions of the stress fields and the electric displacement fields,and the surface effects are not negligible.In this paper,the surface electroelastic constants of the defects are introduced to extend the size of the cracked circular hole in the piezoelectric solids to nanoscale.The fracture characteristics of piezoelectric solids containing two edge cracks emanating from a circular hole with its surface under anti-plane mechanical loads and in-plane electric displacement loads are investigated theoretically.Based on the Gurtin-Murdoch surface model,the closed solutions of the stress field and electric displacement field of the problem are obtained using the complex potential-function electroelastic theory via constructing a conformal mapping function.Analytical expressions of the stress intensity factor,electric displacement field factor and energy release rate at the crack tip are presented.The influences of the geometric parameters of the cracked hole,the applied mechanical load and electrical load on the electroelastic field intensity factors and the energy release rate are discussed.The major results are as follows.The dimensionless stress intensity factor and the dimensionless electric displacement field factor of the nanoscale cracked hole are different,but are both significantly size-dependent.The dimensionless electroelastic field factors increase monotonously with the increase of the relative length of the crack to a fixed value.The dimensionless energy release rate of the nanoscale cracked hole has a significant size effect.The larger is the hole and the longer is the crack,the greater is the normalized energy release rate.The effect of the mechanical loads on the normalized energy release rate is affected by the applied electrical loads.The normalized energy release rate increases first and then decreases with the increase of the electrical loads.
引文
[1]Wang Y J,Gao C F.The mode III cracks originating from the edge of a circular hole in a piezoelectric solid[J].International Journal of Solids and Structures,2008,45(16):4590-4599.
[2]Guo J H,Lu Z X,Han H T,Yang Z Y.Exact solutions for anti-plane problem of two asymmetrical edge cracks emanating from an elliptical hole in a piezoelectric material[J].International Journal of Solids and Structures,2009,46(21):3799-3809.
[3]Guo J H,Lu Z X,Han H T,Yang Z Y.The behavior of two non-symmetrical permeable cracks emanating from an elliptical hole in a piezoelectric solid[J].European Journal of Mechanics A-Solids,2010,29(4):654-663.
[4]Guo J H,Lu Z X,Feng X.The fracture behavior of multiple cracks emanating from a circular hole in piezoelectric materials[J].Acta Mechanica,2010,215(1-4):119-134.
[5]Wang Y J,Gao C F,Song H P.The anti-plane solution for the edge cracks originating from an arbitrary hole in a piezoelectric material[J].Mechanics Research Communications,2015,65:17-23.
[6]王永健,高存法.含等边三角形孔孔边裂纹横观各向同性压电弹性体的反平面问题研究[J].应用力学学报,2015,32(6):973-978.(Wang Y J,Gao C F.The anti-plane solution for the cracked equilateral triangle hole in transverse isotropic piezoelectric materials[J].Chinese Journal of Applied Mechanics,2015,32(6):973-978.(in Chinese))
[7]王永健,宋豪鹏,高存法,邢时超.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.(Wang Y J,Song H P,Gao CF,Xing S C.The anti-plane problem for a cracked elliptical hole at the interface of bi-materials[J].Chinese Quarterly of Mechanics,2015,36(3):416-426.(in Chinese))
[8]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.(Wang W H,Guo JH,Xing Y M.Anti-plane problem analysis of edge crack emanating from regular triangle hole with smooth vertices inpiezoelectroelastic solids[J].Acta Materiae Compositae Sinica,2015,32(2):601-607.(in Chinese))
[9]Fan S W,Guo J H,Yu J.Anti-plane problem of four edge cracks emanating from a square hole in piezoelectric solids[J].Chinese Journal of Aeronautics,2017,30(1):461-468.
[10]Yang J,Zhou Y T,Ma H L,Ding S H,Li X.The fracture behavior of two asymmetrical limited permeable cracks emanating from an elliptical hole in one-dimensional hexagonal quasicrystals with piezoelectric effect[J].International Journal of Solids and Structures,2017,108:175-185.
[11]Fu X L,Wang G F.Surface effects on elastic fields around surface defects[J].Acta Mechanica Solida Sinica,2010,23(3):248-254.
[12]Duan H L,Wang J,Huang Z P,Karihaloo B L.Size-dependent effective elastic constants of solids containing nano-inhomogeneities with interface stress[J].Journal of the Mechanics and Physics of Solids,2005,53(7):1574-1596.
[13]Gurtin M E,Murdoch A I.A continuum theory of elastic material surfaces[J].Archive for Rational Mechanics and Analysis,1975,57(4):291-323.
[14]Gurtin M E,Murdoch A I.Surface stress in solids[J].International Journal of Solids and Structures,1978,14(6):431-440.
[15]Gurtin M E,Weissmuller J,Larche F.A general theory of curved deformable interfaces in solids at equilibrium[J].Philosophical Magazine A,1998,78(5):1093-1109.
[16]Wang G F,Feng X Q,Wang T J,Gao W.Surface effects on the near-tip stresses for mode-I and modeIII cracks[J].Journal of Applied Mechanics,2008,75(1):011001.
[17]Fu X L,Wang G F,Feng X Q.Surface effects on the near-tip stress fields of a mode-II crack[J].International Journal of Fracture,2008,151(2):95-106.
[18]Fu X L,Wang G F,Feng X Q.Surface effects on mode-I crack tip fields:a numerical study[J].Engineering Fracture Mechanics,2010,77(7):1048-1057.
[19]Nan H S,Wang B L.Effect of residual surface stress on the fracture of nanoscale materials[J].Mechanics Research Communications,2012,44:30-34.
[20]Nan H S,Wang B L.Effect of interface stress on the fracture behavior of a nanoscale linear inclusion along the interface of biomaterials[J].International Journal of Solids and Structures,2014,51(23-24):4094-4100.
[21]Xu J Y,Dong C Y.Surface and interface stress effects on the interaction ofnano-inclusions and nanocracks in an infinite domain under anti-plane shear[J].International Journal of Mechanical Sciences,2016,111-112:12-23.
[22]肖俊华,韩彬,徐耀玲,张福成.考虑表面弹性效应时正三角形孔边裂纹反平面剪切问题的断裂力学分析[J].固体力学学报,2017,38(6):530-536.(Xiao J H,Han B,Xu Y L,Zhang F C.Fracture analysis of cracked equilateral triangle hole with surface elasticity effect underantiplane shear[J].Chinese Journal of Solid Mechanics,2017,38(6):530-536.(in Chinese))
[23]肖俊华,崔友强,徐耀玲,张福成.纳米尺度孔边裂纹裂尖Ⅲ型应力强度因子研究[J].中国机械工程,2018,29(19):2347-2352.(Xiao J H,Cui Y Q,Xu Y L,Zhang F C.Study on mode III stress intensity factor at tip of nano crack emanating from a circular hole[J].China Mechanical Engineering,2018,29(19):2347-2352.(in Chinese))
[24]肖俊华,徐耀玲,张福成.基于表面弹性理论的开裂椭圆孔断裂力学研究[J].固体力学学报,2019,40(1):82-89.(Xiao J H,Xu Y L,Zhang F C.Fracture mechanics of a cracked elliptical hole based on the surface elasticity theory[J].Chinese Journal of Solid Mechanics,2019,40(1):82-89.(in Chinese))
[25]Xiao J H,Xu Y L,Zhang F C.A rigorous solution for the piezoelectric materials containing elliptic cavity or crack with surface effect[J].Zeitschrift für Angewandte Mathematik und Mechanik,2016,96(5):633-641.
[26]Wang X,Zhou K.A crack with surface effects in a piezoelectric material[J].Mathematics and Mechanics of Solids,2017,22(1):3-19.
[27]Xiao J H,Xu Y L,Zhang F C.Fracture characteristics of a cracked equilateral triangle hole with surface effect in piezoelectric materials[J].Theoretical and Applied Fracture Mechanics,2018,96:476-482.
[28]Zhang T Y,Wang Z J,Chan W K.Eigenstress model for surface stress of solids[J].Physical Review B,2010,81(19):195427.
[29]Zhou X Y,Ren H,Huang B L,Zhang T Y.Size-dependent elastic properties of thin films:surface anisotropy and surface bonding[J].Science China-Technological Sciences,2014,57(4):680-691.
[30]Sun H L,Chen L Y,Sun S,Zhang T Y.Size-and temperature-dependent Young’s modulus and size-dependent thermal expansion coefficient of nanowires[J].Science China-Technological Sciences,2018,61(5):687-698.
[31]Chen S H,Yao Y.Elastic theory of nanomaterials based on surface-energy density[J].Journal of Applied Mechanics,2014,81(12):121002.
[32]Yao Y,Chen S H.Surface effect in the bending of nanowires[J].Mechanics of Materials,2016,100:12-21.
[33]Jia N,Yao Y,Yang Y Z,Chen S H.Surface effect on the resonant frequency of Timoshenko nanobeams[J].International Journal of Mechanical Sciences,2017,133:21-27.
[34]Huang Z X.Lagrangian formulism of elasticity with relevance to surface energy[J].Acta Mechanica,2013,224(8):1813-1821.
[35]Huang Z X.Torsional wave and vibration subjected to constraint of surface elasticity[J].Acta Mechanica,2018,229(3):1171-1182.
[36]Chen T,Chiu M S,Weng C N.Derivation of the generalized Young-Laplace equation of curved interfaces in nanoscaled solids[J].Journal of Applied Physics,2006,100(7):074308.
[37]Muskhelishvili N I.Some Basic Problems of the Mathematical Theory of Elasticity[M].Groningen:Noordhoff,1953.
[38]Pak Y E.Crack extension force in a piezoelectric material[J].Journal of Applied Mechanics,1990,57(3):647-653.
[39]Chen T.Exact size-dependent connections between effective moduli of fibrous piezoelectric nanocomposites with interface effects[J].Acta Mechanica,2008,196(3-4):205-217.
[40]Huang G Y,Yu S W.Effect of surface piezoelectricity on the electromechanicalbehaviour of a piezoelectric ring[J].Physica Status Solidi B,2006,243(4):R22-R24.
[2]Guo J H,Lu Z X,Han H T,Yang Z Y.Exact solutions for anti-plane problem of two asymmetrical edge cracks emanating from an elliptical hole in a piezoelectric material[J].International Journal of Solids and Structures,2009,46(21):3799-3809.
[3]Guo J H,Lu Z X,Han H T,Yang Z Y.The behavior of two non-symmetrical permeable cracks emanating from an elliptical hole in a piezoelectric solid[J].European Journal of Mechanics A-Solids,2010,29(4):654-663.
[4]Guo J H,Lu Z X,Feng X.The fracture behavior of multiple cracks emanating from a circular hole in piezoelectric materials[J].Acta Mechanica,2010,215(1-4):119-134.
[5]Wang Y J,Gao C F,Song H P.The anti-plane solution for the edge cracks originating from an arbitrary hole in a piezoelectric material[J].Mechanics Research Communications,2015,65:17-23.
[6]王永健,高存法.含等边三角形孔孔边裂纹横观各向同性压电弹性体的反平面问题研究[J].应用力学学报,2015,32(6):973-978.(Wang Y J,Gao C F.The anti-plane solution for the cracked equilateral triangle hole in transverse isotropic piezoelectric materials[J].Chinese Journal of Applied Mechanics,2015,32(6):973-978.(in Chinese))
[7]王永健,宋豪鹏,高存法,邢时超.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.(Wang Y J,Song H P,Gao CF,Xing S C.The anti-plane problem for a cracked elliptical hole at the interface of bi-materials[J].Chinese Quarterly of Mechanics,2015,36(3):416-426.(in Chinese))
[8]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.(Wang W H,Guo JH,Xing Y M.Anti-plane problem analysis of edge crack emanating from regular triangle hole with smooth vertices inpiezoelectroelastic solids[J].Acta Materiae Compositae Sinica,2015,32(2):601-607.(in Chinese))
[9]Fan S W,Guo J H,Yu J.Anti-plane problem of four edge cracks emanating from a square hole in piezoelectric solids[J].Chinese Journal of Aeronautics,2017,30(1):461-468.
[10]Yang J,Zhou Y T,Ma H L,Ding S H,Li X.The fracture behavior of two asymmetrical limited permeable cracks emanating from an elliptical hole in one-dimensional hexagonal quasicrystals with piezoelectric effect[J].International Journal of Solids and Structures,2017,108:175-185.
[11]Fu X L,Wang G F.Surface effects on elastic fields around surface defects[J].Acta Mechanica Solida Sinica,2010,23(3):248-254.
[12]Duan H L,Wang J,Huang Z P,Karihaloo B L.Size-dependent effective elastic constants of solids containing nano-inhomogeneities with interface stress[J].Journal of the Mechanics and Physics of Solids,2005,53(7):1574-1596.
[13]Gurtin M E,Murdoch A I.A continuum theory of elastic material surfaces[J].Archive for Rational Mechanics and Analysis,1975,57(4):291-323.
[14]Gurtin M E,Murdoch A I.Surface stress in solids[J].International Journal of Solids and Structures,1978,14(6):431-440.
[15]Gurtin M E,Weissmuller J,Larche F.A general theory of curved deformable interfaces in solids at equilibrium[J].Philosophical Magazine A,1998,78(5):1093-1109.
[16]Wang G F,Feng X Q,Wang T J,Gao W.Surface effects on the near-tip stresses for mode-I and modeIII cracks[J].Journal of Applied Mechanics,2008,75(1):011001.
[17]Fu X L,Wang G F,Feng X Q.Surface effects on the near-tip stress fields of a mode-II crack[J].International Journal of Fracture,2008,151(2):95-106.
[18]Fu X L,Wang G F,Feng X Q.Surface effects on mode-I crack tip fields:a numerical study[J].Engineering Fracture Mechanics,2010,77(7):1048-1057.
[19]Nan H S,Wang B L.Effect of residual surface stress on the fracture of nanoscale materials[J].Mechanics Research Communications,2012,44:30-34.
[20]Nan H S,Wang B L.Effect of interface stress on the fracture behavior of a nanoscale linear inclusion along the interface of biomaterials[J].International Journal of Solids and Structures,2014,51(23-24):4094-4100.
[21]Xu J Y,Dong C Y.Surface and interface stress effects on the interaction ofnano-inclusions and nanocracks in an infinite domain under anti-plane shear[J].International Journal of Mechanical Sciences,2016,111-112:12-23.
[22]肖俊华,韩彬,徐耀玲,张福成.考虑表面弹性效应时正三角形孔边裂纹反平面剪切问题的断裂力学分析[J].固体力学学报,2017,38(6):530-536.(Xiao J H,Han B,Xu Y L,Zhang F C.Fracture analysis of cracked equilateral triangle hole with surface elasticity effect underantiplane shear[J].Chinese Journal of Solid Mechanics,2017,38(6):530-536.(in Chinese))
[23]肖俊华,崔友强,徐耀玲,张福成.纳米尺度孔边裂纹裂尖Ⅲ型应力强度因子研究[J].中国机械工程,2018,29(19):2347-2352.(Xiao J H,Cui Y Q,Xu Y L,Zhang F C.Study on mode III stress intensity factor at tip of nano crack emanating from a circular hole[J].China Mechanical Engineering,2018,29(19):2347-2352.(in Chinese))
[24]肖俊华,徐耀玲,张福成.基于表面弹性理论的开裂椭圆孔断裂力学研究[J].固体力学学报,2019,40(1):82-89.(Xiao J H,Xu Y L,Zhang F C.Fracture mechanics of a cracked elliptical hole based on the surface elasticity theory[J].Chinese Journal of Solid Mechanics,2019,40(1):82-89.(in Chinese))
[25]Xiao J H,Xu Y L,Zhang F C.A rigorous solution for the piezoelectric materials containing elliptic cavity or crack with surface effect[J].Zeitschrift für Angewandte Mathematik und Mechanik,2016,96(5):633-641.
[26]Wang X,Zhou K.A crack with surface effects in a piezoelectric material[J].Mathematics and Mechanics of Solids,2017,22(1):3-19.
[27]Xiao J H,Xu Y L,Zhang F C.Fracture characteristics of a cracked equilateral triangle hole with surface effect in piezoelectric materials[J].Theoretical and Applied Fracture Mechanics,2018,96:476-482.
[28]Zhang T Y,Wang Z J,Chan W K.Eigenstress model for surface stress of solids[J].Physical Review B,2010,81(19):195427.
[29]Zhou X Y,Ren H,Huang B L,Zhang T Y.Size-dependent elastic properties of thin films:surface anisotropy and surface bonding[J].Science China-Technological Sciences,2014,57(4):680-691.
[30]Sun H L,Chen L Y,Sun S,Zhang T Y.Size-and temperature-dependent Young’s modulus and size-dependent thermal expansion coefficient of nanowires[J].Science China-Technological Sciences,2018,61(5):687-698.
[31]Chen S H,Yao Y.Elastic theory of nanomaterials based on surface-energy density[J].Journal of Applied Mechanics,2014,81(12):121002.
[32]Yao Y,Chen S H.Surface effect in the bending of nanowires[J].Mechanics of Materials,2016,100:12-21.
[33]Jia N,Yao Y,Yang Y Z,Chen S H.Surface effect on the resonant frequency of Timoshenko nanobeams[J].International Journal of Mechanical Sciences,2017,133:21-27.
[34]Huang Z X.Lagrangian formulism of elasticity with relevance to surface energy[J].Acta Mechanica,2013,224(8):1813-1821.
[35]Huang Z X.Torsional wave and vibration subjected to constraint of surface elasticity[J].Acta Mechanica,2018,229(3):1171-1182.
[36]Chen T,Chiu M S,Weng C N.Derivation of the generalized Young-Laplace equation of curved interfaces in nanoscaled solids[J].Journal of Applied Physics,2006,100(7):074308.
[37]Muskhelishvili N I.Some Basic Problems of the Mathematical Theory of Elasticity[M].Groningen:Noordhoff,1953.
[38]Pak Y E.Crack extension force in a piezoelectric material[J].Journal of Applied Mechanics,1990,57(3):647-653.
[39]Chen T.Exact size-dependent connections between effective moduli of fibrous piezoelectric nanocomposites with interface effects[J].Acta Mechanica,2008,196(3-4):205-217.
[40]Huang G Y,Yu S W.Effect of surface piezoelectricity on the electromechanicalbehaviour of a piezoelectric ring[J].Physica Status Solidi B,2006,243(4):R22-R24.