680燃气轮机拉杆转子轴系临界转速及I型裂纹尖端塑性区的屏蔽效应
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摘要
临界转速是汽轮机和燃气轮机发电机组轴系的重要技术指标,它的准确计算是至关重要的。上海汽轮机有限公司的SIEMENS燃气轮机转子的结构特点是多级轮盘采用轴向拉杆预紧的端齿连接。本文第一部分建立拉杆转子轴系振动计算模型,采用有限元法计算转子的临界转速与轮盘齿间接触特性、预紧力和转速的关系,为680燃机转子的轴系振动特性分析,故障诊断和安全运行提供依据。
本文第二部分根据Eshelby等效夹杂理论,证明裂纹尖端的塑性变形区在力学上可以等同于一个相变夹杂。因此,塑性区的对裂尖场的屏蔽效应可以通过现有的相变增韧理论来定量计算。本文获得小范围屈服条件下,平面应力和平面应变状态时I型裂纹尖端塑性区对裂纹尖端应力强度因子影响的近似解。通过本文解,可以确定裂尖塑性变形时材料的硬化行为,应力状态和T应力对裂尖应力强度因子的影响。
最后证明,本文建立的方法可用于求解材料中空洞和气泡对裂尖应力强度因子的影响。
Critical speed is a key technical indicator of the rotor-shaft system of steam turbine and gas turbine generators, therefore its accurate calculation is essential. The SIEMENS gas turbine of Shanghai Turbine Co., Ltd. (STC) has multi-level wheels connected with each other by transverse teeth subjected to axial pre-tightening forces. The first part of the thesis models the rotor-shaft system, analyzes the vibration characteristics of 680 gas turbine by studying the relationships of the critical speed and the transverse parameters, pre-tightening force and rotational speed by means of Finite Element Method (FEM), and thus provides basis for the vibration analysis, failure diagnosis and safe operation of 680 gas turbine.
In the second part of the thesis, a plastically deformed zone around a stressed crack tip is demonstrated to be mechanically identified with a transformation inclusion by Eshelby inclusion theory. Thus, the shielding effects of the plastic zone can be quantitatively evaluated by the present transformation toughening theory. A closed-form solution to determine the change in the stress intensity factor induced by the plastic zone is given both for plane stress and plane strain mode I cracks under small-scale yielding conditions. By using the present solution, the effects of the strain-hardening behavior of the material, the stress state and the T-stress on the near tip stress intensity factor are identified.
In the third part, it is proved that the established method in the previous part can be used to calculate the effect of the voids and gas bubbles in materials on the crack-tip stress intensity factor.
本文第二部分根据Eshelby等效夹杂理论,证明裂纹尖端的塑性变形区在力学上可以等同于一个相变夹杂。因此,塑性区的对裂尖场的屏蔽效应可以通过现有的相变增韧理论来定量计算。本文获得小范围屈服条件下,平面应力和平面应变状态时I型裂纹尖端塑性区对裂纹尖端应力强度因子影响的近似解。通过本文解,可以确定裂尖塑性变形时材料的硬化行为,应力状态和T应力对裂尖应力强度因子的影响。
最后证明,本文建立的方法可用于求解材料中空洞和气泡对裂尖应力强度因子的影响。
Critical speed is a key technical indicator of the rotor-shaft system of steam turbine and gas turbine generators, therefore its accurate calculation is essential. The SIEMENS gas turbine of Shanghai Turbine Co., Ltd. (STC) has multi-level wheels connected with each other by transverse teeth subjected to axial pre-tightening forces. The first part of the thesis models the rotor-shaft system, analyzes the vibration characteristics of 680 gas turbine by studying the relationships of the critical speed and the transverse parameters, pre-tightening force and rotational speed by means of Finite Element Method (FEM), and thus provides basis for the vibration analysis, failure diagnosis and safe operation of 680 gas turbine.
In the second part of the thesis, a plastically deformed zone around a stressed crack tip is demonstrated to be mechanically identified with a transformation inclusion by Eshelby inclusion theory. Thus, the shielding effects of the plastic zone can be quantitatively evaluated by the present transformation toughening theory. A closed-form solution to determine the change in the stress intensity factor induced by the plastic zone is given both for plane stress and plane strain mode I cracks under small-scale yielding conditions. By using the present solution, the effects of the strain-hardening behavior of the material, the stress state and the T-stress on the near tip stress intensity factor are identified.
In the third part, it is proved that the established method in the previous part can be used to calculate the effect of the voids and gas bubbles in materials on the crack-tip stress intensity factor.
引文
[1] Park, H.B., Kim, K. M., Lee, B.W., Plastic zone size in fatigue cracking[J]. Int. J. Ves. & Piping, 1996(68): 279-285.
[2] McClung, R.C., Crack closure and plastic zone sizes in fatigue[J]. Fatigue Frac. Eng. Mater. Struct., 1991(14): 455-468.
[3] Kang, K.J., Beom H.G., Plastic zone size near the crack tip in a constrained ductile layer under mixed mode loading[J]. Eng. Fract. Mech., 2000(66): 257-268.
[4] Betagon, C., Hancock, J.W., Two parameter characterization of elastic-plastic crack-tip fields[J]. J. Appl. Mech., 1991(58): 104-110.
[5] O’Dowd, N.P., Shih, C.F., Family of crack-tip fields characterized by a triaxiality parameter-II Fracture applications[J]. J. Mech. Phys. Solids, 1992(40): 939-963.
[6]Jiang, Y., Kurath, P., 1996. Characteristics of the Armstrong–Frederick type plasticity models. Int. J. Plasticity 12, 387–415.
[7] Ayatollahi, M.R., Pavier, M.J., Smith D.J., Mode I cracks subjected to large T-stresses[J]. Int. J. Fract., 2002(117): 159-174.
[8] Jiang, Y., Feng, M., Ding, F., 2005. A reexamination of plasticity-induced crack closure in fatigue crack propagation. Int. J. Plasticity 21, 1720–1740.
[9] Weertman, J., 1996. Dislocation based fracture mechanics, Word Scientific, Singapore.
[10] Hutchinson, J.W., Plastic stress and strain fields at a crack tip[J]. J. Mech. Phys. Solids, 1968(16):337-347.
[11] Rice, J.R. and Rosengren, G.F., Plane strain deformation near a crack tip in a power-law hardening material[J]. J. Mech. Phys. Solids, 1968(16): 1-12.
[12] Rice, J.R., Johnson, M.A., Inelastic behavior of Solids[M]. (edited by Kanninen, M. F., Adler, W.F., Rosenfield, A. R. and Jaffee, R. I.). New York: McGraw-Hill, 1970:641.
[13] McMeeking, R.M., Finite deformation analysis of crack-tip opening in elastic-plastic materials and implication for fracture[J]. J. Mech. Phys. Solids, 1977(25): 357-381.
[14] Robert, H., Dodds, J.R., Ted L. Anderson and Mark T., Kirk, 1991. A framework to correlate a/w ratio effects on elastic-plastic fracture toughness (Jc), Int. J. Fract. 48, 1-22.
[15] Irwin, G.R., Plastic zone near a crack and fracture toughness[C]. Prov. Seventh Sagomore Ordnance Mater. Res. Conf., 1960( IV): 63-78.
[16] Ding, F., Feng, M., and Jiang, Y., Modeling of fatigue crack growth from a notch[J]. Int. J. Plasticity, 2007(23):1167-1188.
[17] Dugdale, D.S., 1960. Yielding of steel sheets containing slits. J. Mech. Phys. Solids 8,100-108.
[18] Barenblatt, G.I., 1962. The mathematical theory of equilibrium cracks in brittle fracture. Adv. Appl. Mech. 5, 55-129.
[19] Becker, W., Gross, D., About the Dugdale crack under mixed-mode loading[J]. Int. J. Fract., 1968(37): 163-171.
[20] Panasyuk, V.V., Sayruk, M.P., Plastic strip model in elastic-plastic problems of fracture mechanics[J]. Advances in Mechanics, 1992, 15(3-4):123-147.
[21] Neimitz, A, 2004. Modification of Dugdale model to include the work hardening and in and out-of-plane constraints. Eng. Fract. Mech. 71, 1585-1600.
[22] Reinhart, H.W., Fracture mechanics of fictitious crack propagation in concrete[M]. Heron, 1984(29): 3-42.
[23] Hillerborg, A., The theoretical basis of a method to determine the fracture energy GF of concrete[J]. Mater. Struct., 1985(18): 291-296.
[24] Foote, R.M.L., Mai, Y.W., Cotterell B., Crack growth resistance curves in strain softening materials[J]. J. Mech. Phys. Solids., 1986(34): 593-607.
[25] Li, Z., Zhao, Y. and Schmauder, S., A Cohesion Model of Microcrack Toughening[J]. Eng. Fract. Mech., 1993(44): 257-265.
[26] Williams, J.G., Fracture mechanics of polymers[M]. Ellis Horwood, 1984
[27] Tijssens, M.G.A., Vander Gissen E., Sluys L.J., Modeling of crazing using a cohesive surface methodology[J]. Mech. Mater, 2000(32): 19-35.
[28] Wei, Y., Xu, G., A multiscale model for the ductile fracture of crystalline materials[J]. Int. J. Plasticity, 2005(21): 2123-2149.
[29] Kang, K.J., Beom H.G., Plastic zone size near the crack tip in a constrained ductile layer under mixed mode loading[J]. Eng. Fract. Mech., 2000(66): 257-268.
[30] Li, Z., Duan, J., The effect of a plastically deformed zone near crack tip on the stress intensity factors[J]. Int. J. Fract., 2002(117): L29-L34.
[31]孙庆平,黄克智和余寿文,结构陶瓷增韧研究评述.力学进展,1990(20): 289-302.
[32] Hutchinson.J.W., Report DEAP[R]. Harvard University, 1974:5-8.
[33] Budiansky, B., Hutchinson, J.W. and Lambropoulos, J.C., Contiuum Theory of Dilatant Transformation Toughening in Ceramics[J]. Int. J. Solids Struct., 1983(19):337-355.
[34] Eshelby, J.D., The determination of the elastic fields of an ellipsoidal inclusion and related problem[J]. London: Proceeding of the Royal Society, 1957, 241(A): 376-396.
[35]王超,王延荣,徐星仲,李宝清.应用三维有限元法计算汽轮机转子临界转速和模化长叶片[J].动力工程,2007(6):840-844
[36] Evans, A.G., Faber, K.T., Crack growth resistance of microcracking brittle materials[J]. J. ofthe American Ceramic Society, 1983(67): 255-260.
[37] Lambropoulous, J.C., Shear, Shape and Orientation Effects in Transformation Toughening in Ceramics[J]. Int. J. Solids Struct., 1986(22):1083-1106.
[38] Hutchinson, J.W., Crack tip shielding by micracracking in brittle solids[J]. Acta Metall, 1987(35): 1605-1619.
[39] Li, Z., Yang, L., The application of the Eshelby equivalent inclusion method for unifying modulus and transformation toughening[J]. Int. J. Solids Struct., 2002(39): 5225-5240.
[40] Withers, D.J., Stobbs, W.M., Pederson, O.B., The application of the Eshelby Method of internal stress determination to short fibre metal matrix composites[J]. Acta Metall, 1989(37): 3061-3084.
[41] Moschobidis, Z.A., Mura, T., Two ellipsoidal inhomegeneities by the equivalent inclusion method[J]. J. Appl. Mech., 1975(42): 847-852.
[42] Taya, M., Chou, T.W., On two kinds ellipsoidal inhomegeneities in an infinite elastic body: An Application to a Hybrid Composites[J]. Int. J. Solids Struct., 1981(17): 553-563.
[43] Johnson, W. C., Earmme, Y.Y. and Lee, J.K., Approximation of the strain field associated with an inhomegeneous precipitate[J]. J. Appl. Mech., 1980(47): 775-780.
[44] Li, Z., Yang, L., The near-tip stress intensity factor for a crack partially penetrating an inclusion[J]. J. Appl. Mech., 2004(71): 465-469.
[45] Li, Z., Chen, Q., Crack-inclusion for mode I crack analyzed by Eshelby equivalent inclusion method[J]. Int. J. Fract., 2002(118): 29-40.
[46] Yang, L., Chen, Q., Li, Z., Crack-inclusion interaction for mode II crack analyzed by Eshelby equivalent inclusion method[J]. Eng. Fract. Mech., 2004(71): 1421-1433.
[47] Li, Z., Yang, L., Li, S., Sun, J., 2007. The stress intensity factors for a short crack partially penetrating an inclusion of arbitrary shape, Int. J. Fruct. 148, 243-250.
[48] Ippolito, M., Mattoni, A., Colombo, L., Cleri, F., Fracture toughness of nanostructured silicon carbide[J]. Appl. Phys. Lett., 2005(87), 141912.
[49] Ippolito, M., Mattoni, A., Colombo, L., Pugno., Role of lattice discreteness on brittle fracture: Atomistic simulations versus analytical models[J]. Phys. Review B, 2006 (73): 104111.
[50] Mura, T., Micromechanics of Defects in Solids[M]. Second Revised Edition. Dordrecht, 1987.
[51] Li, R., Chudnovsky, A., 1993. Energy analysis of crack interaction with an elastic inclusion. Int. J. Fract. 63, 247-261.
[52] Sun, X., Li, H., Chang, C., Measurement and research on plane stress fracture toughness Kc (in Chinese)[J]. Aerospace Materials & Technology , 1994(2): 53-56.
[53] Pandey, A.B., Majumdar, B.S., Miracle, D.B., 1998. Effects of thickness and precracking onthe fracture toughness of particle-reinforced Al-alloy composites. Metall. Mater. Trans. 29A, 1237-1243.
[54] Larsson, S.G., Carisson, A.J., Influence of non-singular stress terms and specimen geometry on small-scale yielding at crack-tip in elastic-plastic materials[J]. J. Mech. Phys. Solids, 1973(21): 447-473.
[55] Rice, J.R., Limitations to the small scale yielding approximation for crack tip plasticity[J]. J. Mech. Phys. Solids, 1974(22):17-26.
[56] Ayatollahi, M.R., Pavier, M.J., Smith D.J., Mode I cracks subjected to large T-stresses[J]. Int. J. Fract., 2002(117): 159-174.
[57] Wang, X., Elastic T-stress solutions for semi-elliptical surface cracks in finite thickness plates[J]. Eng. Fract. Mech., 2003(70): 731-756.
[58] Zhou, C., Li. Z., The effect of T-stress on crack-inclusion interaction under mode I loading[J]. Mech. Res. Communications, 2007(34): 283-288.
[59] Suo, Z., Shih, C.F., Varias, A.G., A theory for cleavage cracking in the presence of plastic flow[J]. Acta Metall. Mater., 1993(41): 151-1557.
[60] Lipkin, D.M., ClarKe, D.R., Beltz, G.E., A strain gradient model of cleavage fracture in plastically deforming materials[J]. Acta Mater. ,1996(44): 4051-4058.
[61] Bonfoh, N., Lipinski, P., Carmasol, A., Tiem, S., Micromechanical modeling of ductile damage of polycrystalline materials with heterogeneous particles[J]. Int. J. Plasticity, 2004(20): 85-106.
[62] Khan, S.M.A., Khraisheh, M.K., A new criterion for mixed mode fracture initiation based on the crack tip plastic core region[J]. Int. J. Plasticity, 2004(20): 55-84.
[2] McClung, R.C., Crack closure and plastic zone sizes in fatigue[J]. Fatigue Frac. Eng. Mater. Struct., 1991(14): 455-468.
[3] Kang, K.J., Beom H.G., Plastic zone size near the crack tip in a constrained ductile layer under mixed mode loading[J]. Eng. Fract. Mech., 2000(66): 257-268.
[4] Betagon, C., Hancock, J.W., Two parameter characterization of elastic-plastic crack-tip fields[J]. J. Appl. Mech., 1991(58): 104-110.
[5] O’Dowd, N.P., Shih, C.F., Family of crack-tip fields characterized by a triaxiality parameter-II Fracture applications[J]. J. Mech. Phys. Solids, 1992(40): 939-963.
[6]Jiang, Y., Kurath, P., 1996. Characteristics of the Armstrong–Frederick type plasticity models. Int. J. Plasticity 12, 387–415.
[7] Ayatollahi, M.R., Pavier, M.J., Smith D.J., Mode I cracks subjected to large T-stresses[J]. Int. J. Fract., 2002(117): 159-174.
[8] Jiang, Y., Feng, M., Ding, F., 2005. A reexamination of plasticity-induced crack closure in fatigue crack propagation. Int. J. Plasticity 21, 1720–1740.
[9] Weertman, J., 1996. Dislocation based fracture mechanics, Word Scientific, Singapore.
[10] Hutchinson, J.W., Plastic stress and strain fields at a crack tip[J]. J. Mech. Phys. Solids, 1968(16):337-347.
[11] Rice, J.R. and Rosengren, G.F., Plane strain deformation near a crack tip in a power-law hardening material[J]. J. Mech. Phys. Solids, 1968(16): 1-12.
[12] Rice, J.R., Johnson, M.A., Inelastic behavior of Solids[M]. (edited by Kanninen, M. F., Adler, W.F., Rosenfield, A. R. and Jaffee, R. I.). New York: McGraw-Hill, 1970:641.
[13] McMeeking, R.M., Finite deformation analysis of crack-tip opening in elastic-plastic materials and implication for fracture[J]. J. Mech. Phys. Solids, 1977(25): 357-381.
[14] Robert, H., Dodds, J.R., Ted L. Anderson and Mark T., Kirk, 1991. A framework to correlate a/w ratio effects on elastic-plastic fracture toughness (Jc), Int. J. Fract. 48, 1-22.
[15] Irwin, G.R., Plastic zone near a crack and fracture toughness[C]. Prov. Seventh Sagomore Ordnance Mater. Res. Conf., 1960( IV): 63-78.
[16] Ding, F., Feng, M., and Jiang, Y., Modeling of fatigue crack growth from a notch[J]. Int. J. Plasticity, 2007(23):1167-1188.
[17] Dugdale, D.S., 1960. Yielding of steel sheets containing slits. J. Mech. Phys. Solids 8,100-108.
[18] Barenblatt, G.I., 1962. The mathematical theory of equilibrium cracks in brittle fracture. Adv. Appl. Mech. 5, 55-129.
[19] Becker, W., Gross, D., About the Dugdale crack under mixed-mode loading[J]. Int. J. Fract., 1968(37): 163-171.
[20] Panasyuk, V.V., Sayruk, M.P., Plastic strip model in elastic-plastic problems of fracture mechanics[J]. Advances in Mechanics, 1992, 15(3-4):123-147.
[21] Neimitz, A, 2004. Modification of Dugdale model to include the work hardening and in and out-of-plane constraints. Eng. Fract. Mech. 71, 1585-1600.
[22] Reinhart, H.W., Fracture mechanics of fictitious crack propagation in concrete[M]. Heron, 1984(29): 3-42.
[23] Hillerborg, A., The theoretical basis of a method to determine the fracture energy GF of concrete[J]. Mater. Struct., 1985(18): 291-296.
[24] Foote, R.M.L., Mai, Y.W., Cotterell B., Crack growth resistance curves in strain softening materials[J]. J. Mech. Phys. Solids., 1986(34): 593-607.
[25] Li, Z., Zhao, Y. and Schmauder, S., A Cohesion Model of Microcrack Toughening[J]. Eng. Fract. Mech., 1993(44): 257-265.
[26] Williams, J.G., Fracture mechanics of polymers[M]. Ellis Horwood, 1984
[27] Tijssens, M.G.A., Vander Gissen E., Sluys L.J., Modeling of crazing using a cohesive surface methodology[J]. Mech. Mater, 2000(32): 19-35.
[28] Wei, Y., Xu, G., A multiscale model for the ductile fracture of crystalline materials[J]. Int. J. Plasticity, 2005(21): 2123-2149.
[29] Kang, K.J., Beom H.G., Plastic zone size near the crack tip in a constrained ductile layer under mixed mode loading[J]. Eng. Fract. Mech., 2000(66): 257-268.
[30] Li, Z., Duan, J., The effect of a plastically deformed zone near crack tip on the stress intensity factors[J]. Int. J. Fract., 2002(117): L29-L34.
[31]孙庆平,黄克智和余寿文,结构陶瓷增韧研究评述.力学进展,1990(20): 289-302.
[32] Hutchinson.J.W., Report DEAP[R]. Harvard University, 1974:5-8.
[33] Budiansky, B., Hutchinson, J.W. and Lambropoulos, J.C., Contiuum Theory of Dilatant Transformation Toughening in Ceramics[J]. Int. J. Solids Struct., 1983(19):337-355.
[34] Eshelby, J.D., The determination of the elastic fields of an ellipsoidal inclusion and related problem[J]. London: Proceeding of the Royal Society, 1957, 241(A): 376-396.
[35]王超,王延荣,徐星仲,李宝清.应用三维有限元法计算汽轮机转子临界转速和模化长叶片[J].动力工程,2007(6):840-844
[36] Evans, A.G., Faber, K.T., Crack growth resistance of microcracking brittle materials[J]. J. ofthe American Ceramic Society, 1983(67): 255-260.
[37] Lambropoulous, J.C., Shear, Shape and Orientation Effects in Transformation Toughening in Ceramics[J]. Int. J. Solids Struct., 1986(22):1083-1106.
[38] Hutchinson, J.W., Crack tip shielding by micracracking in brittle solids[J]. Acta Metall, 1987(35): 1605-1619.
[39] Li, Z., Yang, L., The application of the Eshelby equivalent inclusion method for unifying modulus and transformation toughening[J]. Int. J. Solids Struct., 2002(39): 5225-5240.
[40] Withers, D.J., Stobbs, W.M., Pederson, O.B., The application of the Eshelby Method of internal stress determination to short fibre metal matrix composites[J]. Acta Metall, 1989(37): 3061-3084.
[41] Moschobidis, Z.A., Mura, T., Two ellipsoidal inhomegeneities by the equivalent inclusion method[J]. J. Appl. Mech., 1975(42): 847-852.
[42] Taya, M., Chou, T.W., On two kinds ellipsoidal inhomegeneities in an infinite elastic body: An Application to a Hybrid Composites[J]. Int. J. Solids Struct., 1981(17): 553-563.
[43] Johnson, W. C., Earmme, Y.Y. and Lee, J.K., Approximation of the strain field associated with an inhomegeneous precipitate[J]. J. Appl. Mech., 1980(47): 775-780.
[44] Li, Z., Yang, L., The near-tip stress intensity factor for a crack partially penetrating an inclusion[J]. J. Appl. Mech., 2004(71): 465-469.
[45] Li, Z., Chen, Q., Crack-inclusion for mode I crack analyzed by Eshelby equivalent inclusion method[J]. Int. J. Fract., 2002(118): 29-40.
[46] Yang, L., Chen, Q., Li, Z., Crack-inclusion interaction for mode II crack analyzed by Eshelby equivalent inclusion method[J]. Eng. Fract. Mech., 2004(71): 1421-1433.
[47] Li, Z., Yang, L., Li, S., Sun, J., 2007. The stress intensity factors for a short crack partially penetrating an inclusion of arbitrary shape, Int. J. Fruct. 148, 243-250.
[48] Ippolito, M., Mattoni, A., Colombo, L., Cleri, F., Fracture toughness of nanostructured silicon carbide[J]. Appl. Phys. Lett., 2005(87), 141912.
[49] Ippolito, M., Mattoni, A., Colombo, L., Pugno., Role of lattice discreteness on brittle fracture: Atomistic simulations versus analytical models[J]. Phys. Review B, 2006 (73): 104111.
[50] Mura, T., Micromechanics of Defects in Solids[M]. Second Revised Edition. Dordrecht, 1987.
[51] Li, R., Chudnovsky, A., 1993. Energy analysis of crack interaction with an elastic inclusion. Int. J. Fract. 63, 247-261.
[52] Sun, X., Li, H., Chang, C., Measurement and research on plane stress fracture toughness Kc (in Chinese)[J]. Aerospace Materials & Technology , 1994(2): 53-56.
[53] Pandey, A.B., Majumdar, B.S., Miracle, D.B., 1998. Effects of thickness and precracking onthe fracture toughness of particle-reinforced Al-alloy composites. Metall. Mater. Trans. 29A, 1237-1243.
[54] Larsson, S.G., Carisson, A.J., Influence of non-singular stress terms and specimen geometry on small-scale yielding at crack-tip in elastic-plastic materials[J]. J. Mech. Phys. Solids, 1973(21): 447-473.
[55] Rice, J.R., Limitations to the small scale yielding approximation for crack tip plasticity[J]. J. Mech. Phys. Solids, 1974(22):17-26.
[56] Ayatollahi, M.R., Pavier, M.J., Smith D.J., Mode I cracks subjected to large T-stresses[J]. Int. J. Fract., 2002(117): 159-174.
[57] Wang, X., Elastic T-stress solutions for semi-elliptical surface cracks in finite thickness plates[J]. Eng. Fract. Mech., 2003(70): 731-756.
[58] Zhou, C., Li. Z., The effect of T-stress on crack-inclusion interaction under mode I loading[J]. Mech. Res. Communications, 2007(34): 283-288.
[59] Suo, Z., Shih, C.F., Varias, A.G., A theory for cleavage cracking in the presence of plastic flow[J]. Acta Metall. Mater., 1993(41): 151-1557.
[60] Lipkin, D.M., ClarKe, D.R., Beltz, G.E., A strain gradient model of cleavage fracture in plastically deforming materials[J]. Acta Mater. ,1996(44): 4051-4058.
[61] Bonfoh, N., Lipinski, P., Carmasol, A., Tiem, S., Micromechanical modeling of ductile damage of polycrystalline materials with heterogeneous particles[J]. Int. J. Plasticity, 2004(20): 85-106.
[62] Khan, S.M.A., Khraisheh, M.K., A new criterion for mixed mode fracture initiation based on the crack tip plastic core region[J]. Int. J. Plasticity, 2004(20): 55-84.
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