基于表面弹性理论的开裂椭圆孔断裂力学研究
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摘要
研究纳米尺度时开裂椭圆孔的Ⅲ型断裂性能.基于表面弹性理论和保角映射技术,利用复势函数理论获得了缺陷(裂纹和椭圆孔)周围应力场和裂纹尖端应力强度因子的闭合解答.所得结果具有一般性,许多已有和新的解答可由论文退化的特殊情形得到.利用解析结果讨论了缺陷的绝对尺寸、椭圆孔的形状比以及裂纹的相对尺寸对应力强度因子的影响.结果表明:考虑表面效应且缺陷尺寸在纳米尺度时,应力强度因子具有显著的尺寸依赖效应;应力强度因子随椭圆孔形状比的变化规律受缺陷表面常数的影响;缺陷表面效应的影响取决于椭圆孔的形状比,非常大的形状比屏蔽了表面效应的影响;裂纹相对尺寸非常小时表面效应影响较弱,裂纹相对尺寸较大时表面效应较为明显.
The mode Ⅲ fracture characteristics of a nano-sized cracked elliptical hole is investigated.Based on the surface elasticity theory and the conformal mapping technique,the closed-form solutions of the stress fields around the defects(the crack and the elliptical hole)and the stress intensity factor at the crack tip are presented by using the complex potential function.The presented solutions are so general that many existed and new results can be regarded as special degenerated cases.By applying the analytical solutions to typical examples,effects of the size of defects,the shape ratio of elliptical hole and the relative size of crack on the stress intensity factor are discussed.The results show that the stress intensity factor is significantly size-dependent when considering the surface effect of the defects at the nanometer scale.The size effect of the defects decreases with the increase of the defect size,and gradually approaches the case of classical theory of elasticity.The variation of the stress intensity factor with the shape ratio of the elliptical hole is related to the surface constant of the defects.With the increase of the shape ratio of the elliptical hole,the dimensionless stress intensity factor increases slightly and then decreases for the case of classical theory of elasticity and a positive surface constant.When the surface constant is negative,the dimensionless stress intensity factor monotonously decreases to a stable value.The surface effect of defects depends on the shape ratio of the elliptical hole,and a very high shape ratio shields the contribution of the surface effect.With the increase of the relative size of the crack,the dimensionless stress intensity factor first increases to the maximum and then decreases.The surface effect is weak when the relative size of the crack is very small,whereas the surface effect is obvious when the relative size of the crack is large.
The mode Ⅲ fracture characteristics of a nano-sized cracked elliptical hole is investigated.Based on the surface elasticity theory and the conformal mapping technique,the closed-form solutions of the stress fields around the defects(the crack and the elliptical hole)and the stress intensity factor at the crack tip are presented by using the complex potential function.The presented solutions are so general that many existed and new results can be regarded as special degenerated cases.By applying the analytical solutions to typical examples,effects of the size of defects,the shape ratio of elliptical hole and the relative size of crack on the stress intensity factor are discussed.The results show that the stress intensity factor is significantly size-dependent when considering the surface effect of the defects at the nanometer scale.The size effect of the defects decreases with the increase of the defect size,and gradually approaches the case of classical theory of elasticity.The variation of the stress intensity factor with the shape ratio of the elliptical hole is related to the surface constant of the defects.With the increase of the shape ratio of the elliptical hole,the dimensionless stress intensity factor increases slightly and then decreases for the case of classical theory of elasticity and a positive surface constant.When the surface constant is negative,the dimensionless stress intensity factor monotonously decreases to a stable value.The surface effect of defects depends on the shape ratio of the elliptical hole,and a very high shape ratio shields the contribution of the surface effect.With the increase of the relative size of the crack,the dimensionless stress intensity factor first increases to the maximum and then decreases.The surface effect is weak when the relative size of the crack is very small,whereas the surface effect is obvious when the relative size of the crack is large.
引文
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[2]Zhou H,Wen J,Wang Z,Zhang Y,Du X.Fatigue crack initiation prediction of cope hole details in orthotropic steel deck using the theory of critical distances[J].Fatigue&Fracture of Engineering Materials&Structures,2016,39(9):1051-1066.
[3]Bo A,Watz V,Olsson E.Fatigue crack initiation and growth at holes in a high strength bainitic roller bearing steel when loaded with non-proportional shear and compressive cycles[J].International Journal of Fatigue,2011,33(9):1244-1256.
[4]Chaves V,Beretta G,Navarro A.Biaxial fatigue limits and crack directions for stainless steel specimens with circular holes[J].Engineering Fracture Mechanics,2017,174:139-154.
[5]Boljanovic S, Maksimovic S, Djuric M.Fatigue strength assessment of initial semi-elliptical cracks located at a hole[J].International Journal of Fatigue,2016,92:548-556.
[6]Moure M M,Garcia-Castillo S K,Sanchez-Saez S,Barbero E.Matrix cracking evolution in open-hole laminates subjected to thermo-mechanical loads[J].Composite Structures,2018,183:510-520.
[7]Yu X,Chen J,Chen J.Interaction effect of cracks and anisotropic influence on degradation of edge stretchability in hole-expansion of advanced high strength steel[J].International Journal of Mechanical Sciences,2016,105:348-359.
[8]Chen Y D,Feng Q,Zheng Y R,Ding X F.Formation of hole-edge cracks in a combustor liner of an aero engine[J].Engineering Failure Analysis,2015,55:148-156.
[9]Albedah A,Benyahia F,Dinar H,Bouiadjra B B.Analytical formulation of the stress intensity factor for crack emanating from central holes and repaired with bonded composite patch in aircraft structures[J].Composites Part B,2013,45(1):852-857.
[10] Liu J P,Li Y H,Xu S D,Xu S,Jin C Y,Liu Z S.Moment tensor analysis of acoustic emission for cracking mechanisms in rock with a pre-cut circular hole under uniaxial compression[J].Engineering Fracture Mechanics,2015,135(6):206-218.
[11] Liu J P,Li Y H,Xu S D.Estimation of cracking and damage mechanisms of rock specimens with precut holes by moment tensor analysis of acoustic emission[J].International Journal of Fracture,2014,188(1):1-8.
[12] 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.
[13] Fu X L,Wang G F.Surface effects on elastic fields around surface defects[J].Acta Mechanica Solida Sinica,2010,23(3):248-254.
[14] 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.
[15] Gurtin M E,Murdoch A I.Surface stress in solids[J].International Journal of Solids and Structures,1978,14(6):431-440.
[16] Gurtin M E,Weissmuller J,Larche F.A general theory of curved deformable interfaces in solids at equilibrium[J].Philosophical Magazine A:Physics of Condensed Matter,Structure,Defects and Mechanical Properties,1998,78(5):1093-1109.
[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] Luo J,Wang X.On the anti-plane shear of an elliptic nano inhomogeneity[J].European Journal of Mechanics,A/Solids,2009,28(5):926-934.
[19] Kim C I,Schiavone P,Ru C Q.The effects of surface elasticity on an elastic solid with mode-III crack:complete solution[J].Journal of Applied Mechanics,2010,77(2):021011-1-7.
[20] Fu X L,Wang G F,Feng X Q.Surface effects on modeI crack tip fields:a numerical study[J].Engineering Fracture Mechanics,2010,77(7):1048-1057.
[21] Kim C I,Schiavone P,Ru C Q.The effect of surface elasticity on a mode-III interface crack[J].Archives of Mechanics,2011,63(3):267-286.
[22] Kim C I,Schiavone P,Ru C Q.Effect of surface elasticity on an interface crack in plane deformations[J].Proceedings Mathematical Physical&Engineering Sciences,2011,467(2136):3530-3549.
[23] 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.
[24] Wang G F,Li Y.Influence of surface tension on mode-I crack tip field[J].Engineering Fracture Mechanics,2013,109(3):290-301.
[25] 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.
[26] Nguyen T B,Rungamornrat J,Senjuntichai T,Wijeyewickrema A C.FEM-SGBEM coupling for modeling of mode-I planar cracks in three-dimensional elastic media with residual surface tension effects[J].Engineering Analysis with Boundary Elements,2015,55:40-51.
[27]Wang X,Fan H.Interaction between a nanocrack with surface elasticity and a screw dislocation[J].Mathematics and Mechanics of Solids,2015,22(2):1-13.
[28]Wang X.Interaction of a screw dislocation with an interface and a nanocrack incorporating surface elasticity[J].Zeitschrift für Angewandte Mathematik und Physik,2015,66(6):3645-3661.
[29] 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.
[30] Nguyen T B,Rungamornrat J,Senjuntichai T.Analysis of planar cracks in 3Delastic media with consideration of surface elasticity[J].International Journal of Fracture,2016,202(1):51-77.
[31] Xu J Y,Dong C Y.Surface and interface stress effects on the interaction of nano-inclusions and nanocracks in an infinite domain under anti-plane shear[J].International Journal of Mechanical Sciences,2016,111-112:12-23.
[32] Intarit P,Senjuntichai T, RungamornratJ, Rajapakse RKND.Penny-shaped crack in elastic medium with surface energy effects[J].Acta Mechanica,2017,228(2):617-630.
[33] Fan M,Zhang Y M,Xiao Z M.The interface effect of a nano-inhomogeneity on the fracture behavior of a crack and the nearby edge dislocation[J].International Journal of Damage Mechanics,2017,26(3):480-497.
[34]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,2017,6:530-536.(Xiao J H,Han B,Xu Y L,Zhang F C.Fracture analysis of cracked equilateral triangle hole with surface elasticity effect under antiplane shear[J].Chinese Journal of Solid Mechanics,2017,6:530-536.(in Chinese))
[35]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))
[36] 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.
[37] Chen S H,Yao Y.Elastic theory of nanomaterials based on surface-energy density[J].Journal of Applied Mechanics,2014,81(12):121002.
[38] Yao Y,Chen S H.Surface effect on resonant properties of nanowires predicted by an elastic theory for nanomaterials[J].Journal of Applied Physics,2015,118(4):044303.
[39] Yao Y,Chen S H.Buckling behavior of nanowires predicted by a new surface energy density model[J].Acta Mechanica,2016,227(7):1799-1811.
[40] Yao Y,Chen S H.Surface effect in the bending of nanowires[J].Mechanics of Materials,2016,100:12-21.
[41] Yao Y,Yang Y Z,Chen S H.Size-dependent elasticity of nanoporous materials predicted by surface energy density-based theory[J].Journal of Applied Mechanics,2017,84(6):061004.
[42] 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.
[43] Huang Z X.Lagrangian formulism of elasticity with relevance to surface energy[J].Acta Mechanica,2013,224(8):1813-1821.
[44] Huang Z X.On a new version of the boundary condition associated with surface energy[J].Philosophical Magazine Letters,2014,94(5):303-310.
[45] Huang Z X.Torsional wave and vibration subjected to constraint of surface elasticity[J].Acta Mechanica,2018,229(3):1171-1182.
[46] 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.
[47] Muskhelishvili N I.Some Basic Problems of the Mathematical Theory of Elasticity[M].Groningen:Noordhoff,1953.
[48] Xiao J H,Shi C F,Xu Y L,Zhang F C.Interface stress of orthotropic materials with a nanodefect under antiplane shear loading[J].Journal of Mechanics of Materials and Structures,2016,11(5):491-504.
[2]Zhou H,Wen J,Wang Z,Zhang Y,Du X.Fatigue crack initiation prediction of cope hole details in orthotropic steel deck using the theory of critical distances[J].Fatigue&Fracture of Engineering Materials&Structures,2016,39(9):1051-1066.
[3]Bo A,Watz V,Olsson E.Fatigue crack initiation and growth at holes in a high strength bainitic roller bearing steel when loaded with non-proportional shear and compressive cycles[J].International Journal of Fatigue,2011,33(9):1244-1256.
[4]Chaves V,Beretta G,Navarro A.Biaxial fatigue limits and crack directions for stainless steel specimens with circular holes[J].Engineering Fracture Mechanics,2017,174:139-154.
[5]Boljanovic S, Maksimovic S, Djuric M.Fatigue strength assessment of initial semi-elliptical cracks located at a hole[J].International Journal of Fatigue,2016,92:548-556.
[6]Moure M M,Garcia-Castillo S K,Sanchez-Saez S,Barbero E.Matrix cracking evolution in open-hole laminates subjected to thermo-mechanical loads[J].Composite Structures,2018,183:510-520.
[7]Yu X,Chen J,Chen J.Interaction effect of cracks and anisotropic influence on degradation of edge stretchability in hole-expansion of advanced high strength steel[J].International Journal of Mechanical Sciences,2016,105:348-359.
[8]Chen Y D,Feng Q,Zheng Y R,Ding X F.Formation of hole-edge cracks in a combustor liner of an aero engine[J].Engineering Failure Analysis,2015,55:148-156.
[9]Albedah A,Benyahia F,Dinar H,Bouiadjra B B.Analytical formulation of the stress intensity factor for crack emanating from central holes and repaired with bonded composite patch in aircraft structures[J].Composites Part B,2013,45(1):852-857.
[10] Liu J P,Li Y H,Xu S D,Xu S,Jin C Y,Liu Z S.Moment tensor analysis of acoustic emission for cracking mechanisms in rock with a pre-cut circular hole under uniaxial compression[J].Engineering Fracture Mechanics,2015,135(6):206-218.
[11] Liu J P,Li Y H,Xu S D.Estimation of cracking and damage mechanisms of rock specimens with precut holes by moment tensor analysis of acoustic emission[J].International Journal of Fracture,2014,188(1):1-8.
[12] 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.
[13] Fu X L,Wang G F.Surface effects on elastic fields around surface defects[J].Acta Mechanica Solida Sinica,2010,23(3):248-254.
[14] 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.
[15] Gurtin M E,Murdoch A I.Surface stress in solids[J].International Journal of Solids and Structures,1978,14(6):431-440.
[16] Gurtin M E,Weissmuller J,Larche F.A general theory of curved deformable interfaces in solids at equilibrium[J].Philosophical Magazine A:Physics of Condensed Matter,Structure,Defects and Mechanical Properties,1998,78(5):1093-1109.
[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] Luo J,Wang X.On the anti-plane shear of an elliptic nano inhomogeneity[J].European Journal of Mechanics,A/Solids,2009,28(5):926-934.
[19] Kim C I,Schiavone P,Ru C Q.The effects of surface elasticity on an elastic solid with mode-III crack:complete solution[J].Journal of Applied Mechanics,2010,77(2):021011-1-7.
[20] Fu X L,Wang G F,Feng X Q.Surface effects on modeI crack tip fields:a numerical study[J].Engineering Fracture Mechanics,2010,77(7):1048-1057.
[21] Kim C I,Schiavone P,Ru C Q.The effect of surface elasticity on a mode-III interface crack[J].Archives of Mechanics,2011,63(3):267-286.
[22] Kim C I,Schiavone P,Ru C Q.Effect of surface elasticity on an interface crack in plane deformations[J].Proceedings Mathematical Physical&Engineering Sciences,2011,467(2136):3530-3549.
[23] 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.
[24] Wang G F,Li Y.Influence of surface tension on mode-I crack tip field[J].Engineering Fracture Mechanics,2013,109(3):290-301.
[25] 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.
[26] Nguyen T B,Rungamornrat J,Senjuntichai T,Wijeyewickrema A C.FEM-SGBEM coupling for modeling of mode-I planar cracks in three-dimensional elastic media with residual surface tension effects[J].Engineering Analysis with Boundary Elements,2015,55:40-51.
[27]Wang X,Fan H.Interaction between a nanocrack with surface elasticity and a screw dislocation[J].Mathematics and Mechanics of Solids,2015,22(2):1-13.
[28]Wang X.Interaction of a screw dislocation with an interface and a nanocrack incorporating surface elasticity[J].Zeitschrift für Angewandte Mathematik und Physik,2015,66(6):3645-3661.
[29] 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.
[30] Nguyen T B,Rungamornrat J,Senjuntichai T.Analysis of planar cracks in 3Delastic media with consideration of surface elasticity[J].International Journal of Fracture,2016,202(1):51-77.
[31] Xu J Y,Dong C Y.Surface and interface stress effects on the interaction of nano-inclusions and nanocracks in an infinite domain under anti-plane shear[J].International Journal of Mechanical Sciences,2016,111-112:12-23.
[32] Intarit P,Senjuntichai T, RungamornratJ, Rajapakse RKND.Penny-shaped crack in elastic medium with surface energy effects[J].Acta Mechanica,2017,228(2):617-630.
[33] Fan M,Zhang Y M,Xiao Z M.The interface effect of a nano-inhomogeneity on the fracture behavior of a crack and the nearby edge dislocation[J].International Journal of Damage Mechanics,2017,26(3):480-497.
[34]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,2017,6:530-536.(Xiao J H,Han B,Xu Y L,Zhang F C.Fracture analysis of cracked equilateral triangle hole with surface elasticity effect under antiplane shear[J].Chinese Journal of Solid Mechanics,2017,6:530-536.(in Chinese))
[35]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))
[36] 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.
[37] Chen S H,Yao Y.Elastic theory of nanomaterials based on surface-energy density[J].Journal of Applied Mechanics,2014,81(12):121002.
[38] Yao Y,Chen S H.Surface effect on resonant properties of nanowires predicted by an elastic theory for nanomaterials[J].Journal of Applied Physics,2015,118(4):044303.
[39] Yao Y,Chen S H.Buckling behavior of nanowires predicted by a new surface energy density model[J].Acta Mechanica,2016,227(7):1799-1811.
[40] Yao Y,Chen S H.Surface effect in the bending of nanowires[J].Mechanics of Materials,2016,100:12-21.
[41] Yao Y,Yang Y Z,Chen S H.Size-dependent elasticity of nanoporous materials predicted by surface energy density-based theory[J].Journal of Applied Mechanics,2017,84(6):061004.
[42] 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.
[43] Huang Z X.Lagrangian formulism of elasticity with relevance to surface energy[J].Acta Mechanica,2013,224(8):1813-1821.
[44] Huang Z X.On a new version of the boundary condition associated with surface energy[J].Philosophical Magazine Letters,2014,94(5):303-310.
[45] Huang Z X.Torsional wave and vibration subjected to constraint of surface elasticity[J].Acta Mechanica,2018,229(3):1171-1182.
[46] 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.
[47] Muskhelishvili N I.Some Basic Problems of the Mathematical Theory of Elasticity[M].Groningen:Noordhoff,1953.
[48] Xiao J H,Shi C F,Xu Y L,Zhang F C.Interface stress of orthotropic materials with a nanodefect under antiplane shear loading[J].Journal of Mechanics of Materials and Structures,2016,11(5):491-504.