应变梯度塑性理论断裂和大变形的研究
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摘要
很多实验发现当材料的特征长度在微米量级,材料表现出很强的尺度效应:越小越强。由于经典塑性理论的本构关系中不包含一个长度参数,所以它不能预测这种尺度效应。因此非常有必要建立一个包含内禀材料长度的微米尺度连续统理论(应变梯度塑性理论)。
本文主要集中在基于细观机制的应变梯度(MSG)形变理论,另外在理论分析时顺带研究了拉伸和旋转梯度(SG)理论。
应用MSG理论研究了微压痕和断裂问题。微压痕结果显示从零点几个微米到十几个微米的范围内,MSG理论预测的结果都和实验结果吻合得非常好。这表明MSG理论可以精确的描述材料在微米和亚微米尺度上的塑性变形。研究断裂的目的是为了通过提高裂尖应力水平来解释在韧性材料断裂实验中观察到的现象。结果显示MSG塑性的应力水平明显高于HRR场的结果,而且它的应力奇异性不仅超过HRR场的,还等于或者超过弹性场的平方根奇异性,并且它的应力奇异性与塑性硬化指数无关。综合考虑其它模型,有限元结果给出了一个韧性材料多尺度断裂描述。
因为原先的MSG和SG理论只能适用于不可压缩材料,所以在重新分解应变梯度张量的基础上,本文把MSG和SG理论拓展到可压缩材料。结果显示在微压痕和断裂问题中可压缩性的影响不可忽略。在断裂问题中最主要的区别是MSG理论裂纹尖端应力奇异性不会超过弹性场的平方根奇异性。
本文建立了MSG和SG的有限变形理论。本构关系建立在初始构形,在初始构形和即时构形间建立了运动学和张量转移关系。在初始和即时构形中给出平衡方程和边界条件。应用有限元方法研究了大变形的微压痕和断裂问题。结果显示通过改变材料常数大变形和小变形都可以和微压痕实验结果吻合好,大小变形理论在断裂问题上的区别仅仅存在于非常靠近裂纹尖端处。
Many experiments have shown that materials display strong size effects when the characteristic length scale is on the order of microns: the smaller, the stronger. The classical plasticity theories can not predict this size dependence of material behavior at the micron scale because their constitutive models possess no internal length scale. Therefore, it is necessary to establish a micron level continuum theory (strain gradient plasticity theory) considering the intrinsic material length.This paper is focussed on the mechanism-based strain gradient (MSG) deformation plasticity theory. Furthermore, the stretch and rotation gradient (SG) theory is mentioned incidentally in some part of the research.Micro-indentation and fracture are studied using MSG theory. The results of micro-indentation show that MSG theory agree very well with experimental data, with indent depth from one tenth of a micron to several microns, indicating that MSG theory are capable of characterizing the plastic behavior rather accurately at the micron and submicron scales. The purpose of the fracture studies is focussed on the increase of stress level around the crack tip in order to explain the observed cleavage fracture in ductile materials. The results show that the stress level in MSG plasticity is significantly higher than the HRR field. MSG plasticity also predicts that the crack tip stress singularity is not only larger than that in the HRR filed, but also exceeds or equals to the square-root singularity. Moreover, the crack tip stress singularity is independent of the plastic work hardening exponent. In conjunction with other model, the results provide a multiscale view of cleavage fracture in ductile materials.MSG and SG theory are generalized to compressible material based on a new decomposition of strain gradient tensor, because the original MSG and SG theories are only suitable for incompressible materials. The results of micro-indentation and fracture show that the influence of compressibility can not be ignored. The main influence in fracture of MSG plasticity is that the crack tip stress singularity does not exceed the square-root singularity.
Finite deformation theories of strain gradient are developed in this paper, for both MSG and SG theory. The constitutive relations are established in the initial configuration by two different methods. The kinematics and tensors transfer relations are provided between the initial and current configurations. The equilibrium equation and boundary conditions are developed in both configurations. The micro-indentation and fracture are studied by finite element method. The micro-indentation results show that both theories for finite and infinitesimal deformation agree well with experimental data by suitably chosen material lengths. The results for fracture show that the difference between finite and infinitesimal deformation only exists in the vicinity very close to crack tip.
本文主要集中在基于细观机制的应变梯度(MSG)形变理论,另外在理论分析时顺带研究了拉伸和旋转梯度(SG)理论。
应用MSG理论研究了微压痕和断裂问题。微压痕结果显示从零点几个微米到十几个微米的范围内,MSG理论预测的结果都和实验结果吻合得非常好。这表明MSG理论可以精确的描述材料在微米和亚微米尺度上的塑性变形。研究断裂的目的是为了通过提高裂尖应力水平来解释在韧性材料断裂实验中观察到的现象。结果显示MSG塑性的应力水平明显高于HRR场的结果,而且它的应力奇异性不仅超过HRR场的,还等于或者超过弹性场的平方根奇异性,并且它的应力奇异性与塑性硬化指数无关。综合考虑其它模型,有限元结果给出了一个韧性材料多尺度断裂描述。
因为原先的MSG和SG理论只能适用于不可压缩材料,所以在重新分解应变梯度张量的基础上,本文把MSG和SG理论拓展到可压缩材料。结果显示在微压痕和断裂问题中可压缩性的影响不可忽略。在断裂问题中最主要的区别是MSG理论裂纹尖端应力奇异性不会超过弹性场的平方根奇异性。
本文建立了MSG和SG的有限变形理论。本构关系建立在初始构形,在初始构形和即时构形间建立了运动学和张量转移关系。在初始和即时构形中给出平衡方程和边界条件。应用有限元方法研究了大变形的微压痕和断裂问题。结果显示通过改变材料常数大变形和小变形都可以和微压痕实验结果吻合好,大小变形理论在断裂问题上的区别仅仅存在于非常靠近裂纹尖端处。
Many experiments have shown that materials display strong size effects when the characteristic length scale is on the order of microns: the smaller, the stronger. The classical plasticity theories can not predict this size dependence of material behavior at the micron scale because their constitutive models possess no internal length scale. Therefore, it is necessary to establish a micron level continuum theory (strain gradient plasticity theory) considering the intrinsic material length.This paper is focussed on the mechanism-based strain gradient (MSG) deformation plasticity theory. Furthermore, the stretch and rotation gradient (SG) theory is mentioned incidentally in some part of the research.Micro-indentation and fracture are studied using MSG theory. The results of micro-indentation show that MSG theory agree very well with experimental data, with indent depth from one tenth of a micron to several microns, indicating that MSG theory are capable of characterizing the plastic behavior rather accurately at the micron and submicron scales. The purpose of the fracture studies is focussed on the increase of stress level around the crack tip in order to explain the observed cleavage fracture in ductile materials. The results show that the stress level in MSG plasticity is significantly higher than the HRR field. MSG plasticity also predicts that the crack tip stress singularity is not only larger than that in the HRR filed, but also exceeds or equals to the square-root singularity. Moreover, the crack tip stress singularity is independent of the plastic work hardening exponent. In conjunction with other model, the results provide a multiscale view of cleavage fracture in ductile materials.MSG and SG theory are generalized to compressible material based on a new decomposition of strain gradient tensor, because the original MSG and SG theories are only suitable for incompressible materials. The results of micro-indentation and fracture show that the influence of compressibility can not be ignored. The main influence in fracture of MSG plasticity is that the crack tip stress singularity does not exceed the square-root singularity.
Finite deformation theories of strain gradient are developed in this paper, for both MSG and SG theory. The constitutive relations are established in the initial configuration by two different methods. The kinematics and tensors transfer relations are provided between the initial and current configurations. The equilibrium equation and boundary conditions are developed in both configurations. The micro-indentation and fracture are studied by finite element method. The micro-indentation results show that both theories for finite and infinitesimal deformation agree well with experimental data by suitably chosen material lengths. The results for fracture show that the difference between finite and infinitesimal deformation only exists in the vicinity very close to crack tip.
引文
[1] Fleck N. A., Muller G. M., Ashby M. F., and Hutchinson J. W. Strain gradient plasticity: theory and experiments. Acta Metall. Mater., 1994, 42: 475～487
[2] Stolken J. S. and Evans A. G. A mierobend test method for measuring the plasticity length scale. Acta Mater., 1998, 46: 5109～5115
[3] Lloyd D. J. Particle reinforced aluminum and magnesium matrix composites. Int. Mater. Rev. 1994, 39: 1～23
[4] Nix W. D. Mechanical properties of thin films. Met. Trans., 1989, A, 20A: 2217～2245
[5] De Guzman M. S., Neubauer G., Flinn P., and Nix W. D. The role of indentation depth on the measured hardness of materials. Materials Research Symposium Proceedings, 1993, 308: 613～618
[6] Stelmashenko N. A., Walls M. G., Brown L. M., and Milman Y. V. Microindentation on W and Mo oriented single crystals: an STM study. Acta Metall. Mater., 1993, 41: 2855～2865
[7] Atkinson M. Further analysis of the size effect in indentation hardness tests of some metals. J. Mater. Res., 1995, 10: 2908～2915
[8] Ma Q. and Clarke D. R. Size dependent hardness of silver single crystals. J. Mater. Resh., 1995, 10: 853～863
[9] Poole W. J., Ashby M. F., and Fleck N. A. Micro-hardness of annealed and work-hardened copper polycrystals. Scripta Metall. Mater. 1996, 34: 559～564
[10] McElhaney K. W., Vlassak J. J., and Nix W. D. Determination of indenter tip geometry and indentation contact area for depth-sensing indentation experiments. J. Mat. Res., 1998, 13: 1300～1306
[11] Elssner G., Korn D., and Ruehle M. The influence of interface impurities on fracture energy of UHV diffusion bonded metal-ceramic bicrystals. Scripta Metall. Mater., 1994, 31: 1037-1042
[12]Hutchinson J. W. Linking scales in mechanics. In: Karihaloo B. L., Mai Y. W., Ripley M. I., and Ritchie R. O, eds. Advances in Fracture Research. Amsterdam: Pergamon Press, 1997, 1-14
[13]Toupin R. A. Elastic materials with couple stresses. Arch Ration. Mech. Anal., 1962,11:385-414
[14]Mindlin R. D. Micro-structure in linear elasticity. Arch. Ration. Mech. Anal., 1964, 16: 51-78
[15]Mindlin R. D. Second gradient of strain and surface tension in linear elasticity. Int. J. Solids Struct., 1965,1: 417-438
[16]Ashby M. F. The deformation of plastically non-homogeneous alloys. Phil. Mag., 1970,21:399-424
[17]Nye J. F. Some geometrical relations in dislocated crystal. Acta Metall., 1953, 1: 153-162
[18]Cottrell A. H. The Mechanical Properties of Matter. New York: Wiley, 1964. 277-277
[19] Fleck N. A. and Hutchinson J. W. A phenomenological theory for strain gradient effects in plasticity. J. Mech. Phys. Solids, 1993,41: 1825-1857
[20]Koiter W. T. Couple stresses in the theory of elasticity. I and II. Proc K. Ned. Akad. Wet. (B). 1964,67: 17-44
[21] Shu J. Y. and Fleck N. A. The prediction of a size effect in micro indentation. Int. J. Solids Struct. 1998, 35: 1363-1383
[22]Xia Z. C. and Hutchinson J. W. Crack tip fields in strain gradient plasticity. J. Mech. Phys. Solids, 1996, 44: 1621-1648
[23] Huang Y, Zhang L., Guo T. F., and Hwang K. C. Fracture of materials with strain gradient effects. In: Karihaloo, B. L., Mai, Y. W., Ripley, M. I., and Ritchie, R. O., eds. Advances in Fracture Research. Amsterdam: Pergamon Press, 1997.2275-2286
[24] Huang Y., Zhang L., Guo T. F., and Hwang K. C. Near-tip fields for cracks in
??materials with strain-gradient effects. In: Willis J. R., eds. Proceedings of IUTAM Symposium on Nonlinear Analysis of Fracture. Cambridge, England: Kluwer Academic Publishers, 1995. 231~242
[25] Huang Y., Zhang L., Guo T. R, and Hwang K. C. Mixed mode near-tip fields for cracks in materials with strain-gradient effects. J. Mech. Phys. of Solids, 1997,45:439-465
[26] Fleck N. A. and Hutchinson J. W. Strain gradient plasticity. In: Hutchinson, J. W. and Wu, T. Y., eds. Advances in Applied Mechanics. New York: Academic Press, 1997,33:295-361
[27] Symshlyaev V. P. And Fleck N. A. The role of strain gradients in the grain size effect for polycrystal. J. Mech. Phys. Solids, 1996, 44: 465-495
[28] Chen J. Y, Wei Y, Huang Y, Hutchinson J. W, and Hwang K. C. The crack tip fields in strain gradient plasticity: the asymptotic and numerical analyses. Eng. Fracture Mech, 1999, 64: 625-648
[29] Wei Y. and Hutchinson J. W. Steady-state crack growth and work of fracture for solids characterized by strain gradient plasticity. J. Mech. Phys. Solids, 1997,45: 1253-1273
[30] Shi M. X., Huang Y. and Hwang K. C. Fracture in the higher-order elastic continuum. J. Mech. Phys. Solids, 2000, 48: 2513-2538
[31]Begley M. R. and Hutchinson J. W. The mechanics of size-dependent indentation. J. Mech. Phys. Solids, 1998, 46: 2049-2068
[32]Nix W. D. and Gao H. Indentation size effects in crystalline materials: a law for strain gradient plasticity. J. Mech. Phys. Solids, 1998, 46: 411-425
[33] Gao H., Huang Y, Nix W. D., and Hutchinson J. W. Mechanism-based strain gradient plasticity -Ⅰ. Theory. J. Mech. Phys. Solids, 1999,47: 1239-1263
[34] Huang Y, Gao H., Nix W.D. and Hutchinson J.W. Mechanism-based strain gradient plasticity - Ⅱ. analysis. J. Mech. Phys. Solids, 2000,48: 99-128
[35]Huang Y, Xue Z., Gao H., Nix W. D. and Xia Z. C. A study of micro-indentation hardness tests by mechanism-based strain gradient plasticity. J.
Mater. Res. 2000 (in press)
[36] Taylor G. I. Plastic strain in metal. J. Inst. Matals., 1938, 62: 307-324
[37]Nix W. D. and Gibeling J.C. Mechanism of time-dependent flow and fracture of metals. Metals/Materials Technology Series 8313-004. ASM, Metals Park, OH. 1985
[38] Bishop J. F. W. and Hill R. A theory of plastic distortion of a polycrystalline aggregate under combined stresses. Philosophical Magazine, 1951,42: 414
[39] Bishop J. F. W. and Hill R. A theoretical derivation of the plastic properties of a polycrystalline face-centered matal. Philosophical Magazine, 1951, 42: 1298
[40] Kocks U. F. The relation between polycrystal deformation and single crystal deformation. Metall. Trans., 1970,1: 1121-1144
[41]Arsenlis A. and Parks D. M. Crystallographic aspects of geometrically- necessary and statistically-stored dislocation density. Acta Mater., 1999, 47: 1597-1611
[42] Huang Y., Chen J. Y., Guo T. F., Zhang L. and Hwang K. C. Analytic and numerical studies on mode I and mode II fracture in elastic-plastic materials with strain gradient effects. Int. J. Fracture, 1999,100: 1~27
[43] Shu J. Y, King W. E. And Fleck N. A. Finite edlements for materials with strain gradient effects. Int. J. Numer. Meth. Engng., 1999,44: 373-391
[44]Specht B. Modified shape functions for the three node plate bending element passing the patch test. International Journal of Numerical methods in Engineering. 1988,26: 705-715
[45]Zienkiewicz O. C. and Taylor R. L. The Finite Element Method, 4~(th) edn. New York: McGraw-Hill, 1989.
[46] Hutchinson J. W. Singular behavior at the end of a tensile crack in a hardening material. J. Mech. Phys. Solids, 1968,16: 13-31
[47] Rice J. R. and Rosengren G. F. Plane strain deformation near a crack tip in a power law hardening material. J. Mech. Phys. Solids, 1968, 16: 1-12
[48] Chen J. Y, Huang Y., and Hwang K. C. Mode Ⅰ and mode Ⅱ plane-stress near-tip fields for cracks in materials with strain gradient effects. Key Eng. Mater., 1998, 145: 19～28
[49] Shi M. X., Huang Y., Gao H. and Hwang K. C. Non-existence of separable crack tip field in mechanism-based strain gradient plasticity. Int. J. Solids Struct. 2000 (in press)
[50] 杨卫.宏微观断裂力学.北京:国防工业出版社,1995.
[51] Suo Z., Shih C. E and Varias A. G. A theory for cleavage cracking in the presence of plastic flow. Acta Metall. Mater., 1993, 41: 1551～1557
[52] Beltz G. E., Rice J. R., Shih C. E and Xia L. A self-consistent method for cleavage in the presence of plastic flow. Acta Mater., 1996, 44: 3943～3954
[53] Hwang K. C. and Inoue T. Recent advances in strain gradient plasticity. Materials Science Research International, 1998, 4, No. 4: 227～238
[54] Jiang H. and Hwang K. C. The influence of compressibility in strain gradient plasticity. In: B. Xu B., Tokuda M. and Wang X. eds. Microstructures and Mechanical Properties of New Engineering Material-Proceedings of the fourth International Symposium on Microstructures and Mechanical Properties of New Engineering Materials. Beijing: International Academic Publishers, 1999. 79～86
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[2] Stolken J. S. and Evans A. G. A mierobend test method for measuring the plasticity length scale. Acta Mater., 1998, 46: 5109～5115
[3] Lloyd D. J. Particle reinforced aluminum and magnesium matrix composites. Int. Mater. Rev. 1994, 39: 1～23
[4] Nix W. D. Mechanical properties of thin films. Met. Trans., 1989, A, 20A: 2217～2245
[5] De Guzman M. S., Neubauer G., Flinn P., and Nix W. D. The role of indentation depth on the measured hardness of materials. Materials Research Symposium Proceedings, 1993, 308: 613～618
[6] Stelmashenko N. A., Walls M. G., Brown L. M., and Milman Y. V. Microindentation on W and Mo oriented single crystals: an STM study. Acta Metall. Mater., 1993, 41: 2855～2865
[7] Atkinson M. Further analysis of the size effect in indentation hardness tests of some metals. J. Mater. Res., 1995, 10: 2908～2915
[8] Ma Q. and Clarke D. R. Size dependent hardness of silver single crystals. J. Mater. Resh., 1995, 10: 853～863
[9] Poole W. J., Ashby M. F., and Fleck N. A. Micro-hardness of annealed and work-hardened copper polycrystals. Scripta Metall. Mater. 1996, 34: 559～564
[10] McElhaney K. W., Vlassak J. J., and Nix W. D. Determination of indenter tip geometry and indentation contact area for depth-sensing indentation experiments. J. Mat. Res., 1998, 13: 1300～1306
[11] Elssner G., Korn D., and Ruehle M. The influence of interface impurities on fracture energy of UHV diffusion bonded metal-ceramic bicrystals. Scripta Metall. Mater., 1994, 31: 1037-1042
[12]Hutchinson J. W. Linking scales in mechanics. In: Karihaloo B. L., Mai Y. W., Ripley M. I., and Ritchie R. O, eds. Advances in Fracture Research. Amsterdam: Pergamon Press, 1997, 1-14
[13]Toupin R. A. Elastic materials with couple stresses. Arch Ration. Mech. Anal., 1962,11:385-414
[14]Mindlin R. D. Micro-structure in linear elasticity. Arch. Ration. Mech. Anal., 1964, 16: 51-78
[15]Mindlin R. D. Second gradient of strain and surface tension in linear elasticity. Int. J. Solids Struct., 1965,1: 417-438
[16]Ashby M. F. The deformation of plastically non-homogeneous alloys. Phil. Mag., 1970,21:399-424
[17]Nye J. F. Some geometrical relations in dislocated crystal. Acta Metall., 1953, 1: 153-162
[18]Cottrell A. H. The Mechanical Properties of Matter. New York: Wiley, 1964. 277-277
[19] Fleck N. A. and Hutchinson J. W. A phenomenological theory for strain gradient effects in plasticity. J. Mech. Phys. Solids, 1993,41: 1825-1857
[20]Koiter W. T. Couple stresses in the theory of elasticity. I and II. Proc K. Ned. Akad. Wet. (B). 1964,67: 17-44
[21] Shu J. Y. and Fleck N. A. The prediction of a size effect in micro indentation. Int. J. Solids Struct. 1998, 35: 1363-1383
[22]Xia Z. C. and Hutchinson J. W. Crack tip fields in strain gradient plasticity. J. Mech. Phys. Solids, 1996, 44: 1621-1648
[23] Huang Y, Zhang L., Guo T. F., and Hwang K. C. Fracture of materials with strain gradient effects. In: Karihaloo, B. L., Mai, Y. W., Ripley, M. I., and Ritchie, R. O., eds. Advances in Fracture Research. Amsterdam: Pergamon Press, 1997.2275-2286
[24] Huang Y., Zhang L., Guo T. F., and Hwang K. C. Near-tip fields for cracks in
??materials with strain-gradient effects. In: Willis J. R., eds. Proceedings of IUTAM Symposium on Nonlinear Analysis of Fracture. Cambridge, England: Kluwer Academic Publishers, 1995. 231~242
[25] Huang Y., Zhang L., Guo T. R, and Hwang K. C. Mixed mode near-tip fields for cracks in materials with strain-gradient effects. J. Mech. Phys. of Solids, 1997,45:439-465
[26] Fleck N. A. and Hutchinson J. W. Strain gradient plasticity. In: Hutchinson, J. W. and Wu, T. Y., eds. Advances in Applied Mechanics. New York: Academic Press, 1997,33:295-361
[27] Symshlyaev V. P. And Fleck N. A. The role of strain gradients in the grain size effect for polycrystal. J. Mech. Phys. Solids, 1996, 44: 465-495
[28] Chen J. Y, Wei Y, Huang Y, Hutchinson J. W, and Hwang K. C. The crack tip fields in strain gradient plasticity: the asymptotic and numerical analyses. Eng. Fracture Mech, 1999, 64: 625-648
[29] Wei Y. and Hutchinson J. W. Steady-state crack growth and work of fracture for solids characterized by strain gradient plasticity. J. Mech. Phys. Solids, 1997,45: 1253-1273
[30] Shi M. X., Huang Y. and Hwang K. C. Fracture in the higher-order elastic continuum. J. Mech. Phys. Solids, 2000, 48: 2513-2538
[31]Begley M. R. and Hutchinson J. W. The mechanics of size-dependent indentation. J. Mech. Phys. Solids, 1998, 46: 2049-2068
[32]Nix W. D. and Gao H. Indentation size effects in crystalline materials: a law for strain gradient plasticity. J. Mech. Phys. Solids, 1998, 46: 411-425
[33] Gao H., Huang Y, Nix W. D., and Hutchinson J. W. Mechanism-based strain gradient plasticity -Ⅰ. Theory. J. Mech. Phys. Solids, 1999,47: 1239-1263
[34] Huang Y, Gao H., Nix W.D. and Hutchinson J.W. Mechanism-based strain gradient plasticity - Ⅱ. analysis. J. Mech. Phys. Solids, 2000,48: 99-128
[35]Huang Y, Xue Z., Gao H., Nix W. D. and Xia Z. C. A study of micro-indentation hardness tests by mechanism-based strain gradient plasticity. J.
Mater. Res. 2000 (in press)
[36] Taylor G. I. Plastic strain in metal. J. Inst. Matals., 1938, 62: 307-324
[37]Nix W. D. and Gibeling J.C. Mechanism of time-dependent flow and fracture of metals. Metals/Materials Technology Series 8313-004. ASM, Metals Park, OH. 1985
[38] Bishop J. F. W. and Hill R. A theory of plastic distortion of a polycrystalline aggregate under combined stresses. Philosophical Magazine, 1951,42: 414
[39] Bishop J. F. W. and Hill R. A theoretical derivation of the plastic properties of a polycrystalline face-centered matal. Philosophical Magazine, 1951, 42: 1298
[40] Kocks U. F. The relation between polycrystal deformation and single crystal deformation. Metall. Trans., 1970,1: 1121-1144
[41]Arsenlis A. and Parks D. M. Crystallographic aspects of geometrically- necessary and statistically-stored dislocation density. Acta Mater., 1999, 47: 1597-1611
[42] Huang Y., Chen J. Y., Guo T. F., Zhang L. and Hwang K. C. Analytic and numerical studies on mode I and mode II fracture in elastic-plastic materials with strain gradient effects. Int. J. Fracture, 1999,100: 1~27
[43] Shu J. Y, King W. E. And Fleck N. A. Finite edlements for materials with strain gradient effects. Int. J. Numer. Meth. Engng., 1999,44: 373-391
[44]Specht B. Modified shape functions for the three node plate bending element passing the patch test. International Journal of Numerical methods in Engineering. 1988,26: 705-715
[45]Zienkiewicz O. C. and Taylor R. L. The Finite Element Method, 4~(th) edn. New York: McGraw-Hill, 1989.
[46] Hutchinson J. W. Singular behavior at the end of a tensile crack in a hardening material. J. Mech. Phys. Solids, 1968,16: 13-31
[47] Rice J. R. and Rosengren G. F. Plane strain deformation near a crack tip in a power law hardening material. J. Mech. Phys. Solids, 1968, 16: 1-12
[48] Chen J. Y, Huang Y., and Hwang K. C. Mode Ⅰ and mode Ⅱ plane-stress near-tip fields for cracks in materials with strain gradient effects. Key Eng. Mater., 1998, 145: 19～28
[49] Shi M. X., Huang Y., Gao H. and Hwang K. C. Non-existence of separable crack tip field in mechanism-based strain gradient plasticity. Int. J. Solids Struct. 2000 (in press)
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