考虑Taylor位错模型的含非经典应力矢量的应变梯度塑性理论
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摘要
近年来很多实验发现,当材料变形的特征长度在微米或亚微米量级时,材料表现出很强的尺寸效应:越小越硬。经典塑性理论不含任何长度参数,因此无法预测尺寸效应,建立包含内禀材料长度参数的新的本构模型—应变梯度塑性理论就势在必行。
本文致力于低阶和含有非经典应力矢量的应变梯度理论及有限变形研究。
低阶应变梯度理论保留了经典塑性理论的结构,并且不包含附加边界条件。采用非线性偏微分方程的特征线方法研究低阶理论,对于Niordson和Hutchinson的无限大平板剪切问题,我们得到了Bassani低阶理论的“定解域”,同时发现当外加剪应力增加时,“定解域”逐渐缩小直至消失。为得到“定解域”之外的解,需要补充非经典的附加边界条件。在“定解域”内,特征线方法的解与Niordson和Hutchinson的有限差分解吻合得很好。在“定解域”外,解可能不唯一。
基于Fleck和Hutchinson应变梯度理论的框架,考虑Taylor位错模型,建立了一种新的含有非经典应力矢量的应变梯度理论。应用该理论研究了若干具有尺寸效应的典型问题,如无限大平板剪切,细丝扭转,薄梁弯曲,孔洞长大,薄膜—基体双轴加载。
微尺度下,薄膜实验与位错模拟对于研究尺寸效应有重要意义。采用我们的应变梯度塑性理论研究Yu和Spaepen的copper薄膜—Kapton基体单向拉伸的实验结果。结果表明我们的应变梯度理论不仅能较好地预测尺寸效应,还可以得出与Venkatraman和Bravman以及Nix所预言的薄膜屈服应力和薄膜厚度之间的关系相同的趋势。此外,采用该理论研究Shu等的位错模拟无限大平板剪切问题的结果时,可以拟合出内禀材料长度参数的确是在微米量级。
本文最后建立了Fleck和Hutchinson应变梯度理论及我们的应变梯度理论的有限变形理论。采用有限变形理论研究了细丝扭转问题。结果表明,有限变形理论和小变形理论有一定的差别。
Many recent experiments have shown that materials display strong size effects when the characteristic length scale is on the order of microns or submicrons: the smaller, the harder. The classical plasticity theories can not predict this size dependence of material behavior because their constitutive models possess no internal length scale. Therefore, it is necessary to establish new constitutive models—strain gradient plasticity theories which possess the intrinsic material length.This dissertation is focused on lower-order strain gradient plasticity theories and higher-order strain gradient plasticity theories whose non-classical stress is a vector. Furthermore, finite deformation versions of these higher-order theories are also established.The lower-order strain gradient plasticity theory retains the essential structure of classical plasticity theory, and does not seem to require additional, non-classical boundary conditions. The well-posedness of lower-order strain gradient plasticity theory is studied by the method of characteristics for nonlinear partial differential equations. For Niordson and Hutchinson's problem of an infinite layer in shear, the "domain of determinacy" for Bassani's lower-order strain gradient plasticity theory is obtained. It is also established that, as the applied shear stress increases, the "domain of determinacy" shrinks and eventually vanishes. The additional, non-classical boundary conditions are needed for Bassani's lower-order strain gradient plasticity in order to obtain the solution outside the "domain of determinacy". Within the "domain of determinacy", the present results agree well with Niordson and Hutchinson's finite difference solution. Outside the "domain of determinacy", the solution may not be unique.Within Fleck and Hutchinson's theoretical framework of strain gradient plasticity theory and based on the Taylor dislocation model, we develop a new
strain gradient plasticity theory. We also present a few examples that display strong size effects at the micron and submicron scales, including the shear of an infinite layer, torsion of thin wires, bending of thin beams, growth of microvoids, and film-substrate subject to biaxial loading. These examples show that this new strain gradient plasticity theory based on the Taylor dislocation model captures the strong size effects.At the micron and submicron scales, film experiments and dislocation simulations are significant for the studying of size effects. We investigate Yu and Spaepen's copper-Kapton tension experiment by using our strain gradient theory. Our results can not only predict the size effects but also agree well with the relation predicted by Venkatraman et al. and Nix between the yielding stress of a film bonded to a substrate and the thickness of the film. On the other hand, we use the same strain gradient plasticity theory to investigate Shu et al.'s dislocation simulation of an infinite layer in shear, and we find that our intrinsic material length parameter is at the micron scale.We obtain the finite deformation versions of Fleck and Hutchinson's strain gradient plasticity theory as well as ours. We study the torsion of thin wires by using the finite deformation theories. Our results show that there is difference between finite deformation theories and infinitesimal deformation theories.
本文致力于低阶和含有非经典应力矢量的应变梯度理论及有限变形研究。
低阶应变梯度理论保留了经典塑性理论的结构,并且不包含附加边界条件。采用非线性偏微分方程的特征线方法研究低阶理论,对于Niordson和Hutchinson的无限大平板剪切问题,我们得到了Bassani低阶理论的“定解域”,同时发现当外加剪应力增加时,“定解域”逐渐缩小直至消失。为得到“定解域”之外的解,需要补充非经典的附加边界条件。在“定解域”内,特征线方法的解与Niordson和Hutchinson的有限差分解吻合得很好。在“定解域”外,解可能不唯一。
基于Fleck和Hutchinson应变梯度理论的框架,考虑Taylor位错模型,建立了一种新的含有非经典应力矢量的应变梯度理论。应用该理论研究了若干具有尺寸效应的典型问题,如无限大平板剪切,细丝扭转,薄梁弯曲,孔洞长大,薄膜—基体双轴加载。
微尺度下,薄膜实验与位错模拟对于研究尺寸效应有重要意义。采用我们的应变梯度塑性理论研究Yu和Spaepen的copper薄膜—Kapton基体单向拉伸的实验结果。结果表明我们的应变梯度理论不仅能较好地预测尺寸效应,还可以得出与Venkatraman和Bravman以及Nix所预言的薄膜屈服应力和薄膜厚度之间的关系相同的趋势。此外,采用该理论研究Shu等的位错模拟无限大平板剪切问题的结果时,可以拟合出内禀材料长度参数的确是在微米量级。
本文最后建立了Fleck和Hutchinson应变梯度理论及我们的应变梯度理论的有限变形理论。采用有限变形理论研究了细丝扭转问题。结果表明,有限变形理论和小变形理论有一定的差别。
Many recent experiments have shown that materials display strong size effects when the characteristic length scale is on the order of microns or submicrons: the smaller, the harder. The classical plasticity theories can not predict this size dependence of material behavior because their constitutive models possess no internal length scale. Therefore, it is necessary to establish new constitutive models—strain gradient plasticity theories which possess the intrinsic material length.This dissertation is focused on lower-order strain gradient plasticity theories and higher-order strain gradient plasticity theories whose non-classical stress is a vector. Furthermore, finite deformation versions of these higher-order theories are also established.The lower-order strain gradient plasticity theory retains the essential structure of classical plasticity theory, and does not seem to require additional, non-classical boundary conditions. The well-posedness of lower-order strain gradient plasticity theory is studied by the method of characteristics for nonlinear partial differential equations. For Niordson and Hutchinson's problem of an infinite layer in shear, the "domain of determinacy" for Bassani's lower-order strain gradient plasticity theory is obtained. It is also established that, as the applied shear stress increases, the "domain of determinacy" shrinks and eventually vanishes. The additional, non-classical boundary conditions are needed for Bassani's lower-order strain gradient plasticity in order to obtain the solution outside the "domain of determinacy". Within the "domain of determinacy", the present results agree well with Niordson and Hutchinson's finite difference solution. Outside the "domain of determinacy", the solution may not be unique.Within Fleck and Hutchinson's theoretical framework of strain gradient plasticity theory and based on the Taylor dislocation model, we develop a new
strain gradient plasticity theory. We also present a few examples that display strong size effects at the micron and submicron scales, including the shear of an infinite layer, torsion of thin wires, bending of thin beams, growth of microvoids, and film-substrate subject to biaxial loading. These examples show that this new strain gradient plasticity theory based on the Taylor dislocation model captures the strong size effects.At the micron and submicron scales, film experiments and dislocation simulations are significant for the studying of size effects. We investigate Yu and Spaepen's copper-Kapton tension experiment by using our strain gradient theory. Our results can not only predict the size effects but also agree well with the relation predicted by Venkatraman et al. and Nix between the yielding stress of a film bonded to a substrate and the thickness of the film. On the other hand, we use the same strain gradient plasticity theory to investigate Shu et al.'s dislocation simulation of an infinite layer in shear, and we find that our intrinsic material length parameter is at the micron scale.We obtain the finite deformation versions of Fleck and Hutchinson's strain gradient plasticity theory as well as ours. We study the torsion of thin wires by using the finite deformation theories. Our results show that there is difference between finite deformation theories and infinitesimal deformation theories.
引文
[1] Fleck N. A., Muller G. M., Ashlby M. F. and Hutchinson J. W. Strain gradient plasticity: theory and experiment. Acta Metall. Mater. 1994, 42: 475-487
[2] Stolken, J. S. and Evans, A. G. A microbend 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., A, 1989, 20A: 2217～2245
[5] 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 Metali. Mater., 1993, 41: 2855～2865
[6] Ma Q. and Clarke D. R. Size dependent hardness of silver single crystals. J. Mater. Resh., 1995, 10: 853～863
[7] Poole W. J., Ashby M. F., and Fleck N. A. Micro-hardness of annealed and work-hardenedcopper polycrystals. Scripta Metall. Mater., 1996, 34: 559～564
[8] 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
[9] Elssner, G., Korn, D. and Ruhle, M. The influence of interface impurities on fractureenergy of UHV diffusion-bonded metal-ceramic bierystals. Scripta Metall. Mater., 1994, 31: 1037-1042
[10] Hutchinson J. W. Linking scales in mechanics. In: Karihaloo B. L., Mai Y. W., Ripley M.I., and Ritchie Ro O, eds. Advances in Fracture Research. Amsterdam: Pergamon Press, 1997, 1～14
[11] Toupin R. A. Elastic materials with couple stresses. Arch Ration. Mech. Anal., 1962, 11:385～414
[12] Mindlin R. D. Micro-structure in linear elasticity. Arch. Ration. Mech. Anal., 1964, 16:51～78
[13] Mindlin R. D. Second gradient of strain and surface tension in linear elasticity. Int. J.Solids Struct., 1965, 1: 417～438
[14] Acharya, A., Bassani, J. L. Lattice incompatibility and a gradient theory of crystal plasticity.J. Mech. Phys. Solids, 2000, 48: 1565-1595
[15] Acharya, A. and Beaudoin, A. J. Grain-size effect in viscoplastic polycrystals at moderate strains. J. Mech. Phys. Solids, 2000,48:2213-2230
[16] Busso, E. P., Meissonneir, F. T., O'Dowd, N. P. Gradient-dependent deformation of two-phase single crystals. J. Mech. Phys. Solids, 2000, 48: 2333-2362
[17] Bassani, J. L. Incompatibility and a simple gradient theory of plasticity. J. Mech. Phys. Solids, 2001,49: 1983-1996
[18] Gao, H. and Huang, Y. Taylor-based nonlocal theory of plasticity. Int. J. Solids Struct., 2001,38:2615-2637
[19] Guo, Y., Huang, Y., Gao, H., Zhuang, Z. and Hwang, K. C. Taylor-based nonlocal theory of plasticity: numerical studies of micro-indentation experiments and crack tip fields. Int. J. Solids Struct, 2001,38:7447-7460
[20] Huang, Y., Qu, S., Hwang, K. C, Li, M. and Gao, H. A conventional theory of mechanism-based strain gradient plasticity. Int. J. Plasticity, 2004,20: 753-782
[21 ] Hwang, K. C, Guo, Y., Jiang, H., Huang, Y. and Zhuang, Z. The finite deformation theory of Taylor-based nonlocal plasticity. Int. J. Plasticity, 2004,20: 831-839
[22] Aifantis, E. C. On the microstructural origin of certain inelastic models. Trans. ASME J. Engrg. Mater. Technol., 1984, 106:326-330
[23] Fleck, N. A., Hutchinson, J. W. A reformulation of strain gradient plasticity. J. Mech. Phys. Solids, 2001,49:2245-2271
[24] Fleck, N. A., Hutchinson, J. W. A phenomenological theory for strain gradient effects in plasticity. J. Mech. Phys. Solids, 1993,41:1825-1857
[25] Fleck, N. A., Hutchinson, J. W. Strain gradient plasticity. In: Hutchinson, J. W., Wu, T. Y. (Eds.), Adv. Appl. Mech., vol. 33. Academic Press, New York, 1997,295-361
[26] Gao, H., Huang, Y., Nix, W. D. and Hutchinson, J. W. Mechanism-based strain gradient plasticity-I. Theory. J. Mech. Phys. Solids, 1999b, 47:1239-1263
[27] Huang, Y., Gao, H., Nix, W. D., Hutchinson, J. W. Mechanism-based strain gradient plasticity - II. Analysis. J. Mech. Phys. Solids, 2000a, 48:99-128
[28] Huang, Y., Xue, Z., Gao, H., Nix, W. D., Xia, Z. C. A study of microindentation hardness tests by mechanism-based strain gradient plasticity. J. Mater. Res. 2000b, 15:1786-1796
[29] Gurtin, M.E. On the plasticity of single crystals: free energy, microforces, plastic-strain gradient. J. Mech. Phys. Solids, 2000,48:989-1036
[30] Gurtin, M.E. A gradient theory of single-crystal viscoplasticity that accounts for geometrically necessary dislocations. J. Mech. Phys. Solids, 2002, 50: 5-32
[31] Qiu, X., Huang, Y., Nix, W. D., Hwang, K. C. and Gao, H. Effect of intrinsic lattice resistance in strain gradient plasticity. Acta Mater., 2001,49: 3949-3958
[32] Qiu, X., Huang, Y., Wei, Y., Gao, H. and Hwang, K. C. The flow theory of mechanism-based strain gradient plasticity. Mechanics of Materials, 2003, 35: 245-258
[33] Hwang, K. C, Jiang, H., Huang, Y., Gao, H. and Hu, N. A finite deformation theory of strain gradient plasticity. J. Mech. Phys. Solids, 2002, 50: 81-99
[34] Hwang, K. C, Jiang, H., Huang, Y. and Gao, H. Finite deformation analysis of mechanism-based strain gradient plasticity: torsion and crack tip field. Int. J. Plasticity, 2003, 19:235-251
[35] Needleman, A. Computational mechanics at the mesoscale. Acta Materialia, 2000, 48: 105-124
[36] Shu, J. Y., Fleck, N. A., Van der Giessen, E. and Needleman, A. Boundary layers in constrained plastic flow: comparison of nonlocal and discrete dislocation plasticity. J. Mech. Phys. Solids, 2001,49: 1361-1395
[37] Niordson, C. F., Hutchinson, J. W. On lower order strain gradient plasticity theories. Eur. J. Mech. A Solids, 2003,22: 771-778
[38] Taylor, G. I. The mechanism of plastic deformation of crystals. Part I. —theoretical. Proc. Roy. Soc. London A, 1934,145:362-387
[39] Taylor, G. I. Plastic strain in metals. J. Inst Metals, 1938,6:307-324
[40] Yu, D.Y.W., Spaepen, F. The yield strength of thin copper films on Kapton. Journal of applied physics, 2004,95:2991-2997
[41] Volokh, K. Y., Hutchinson, J. W. Are lower order gradient theories of plasticity really lower order. Trans. ASME J. Appl. Mech., 2002,69:862-864
[42] Courant, R., Hilbert, D. Methods of Mathematical Physics, vol. II. Wiley, New York. 1962
[43] Acharya, A., Bassani, J. L., Beaudoin, A. Geometrically necessary dislocations, hardening, and a simple gradient theory of crystal plasticity. Scripta Mater., 2003,48:167-172
[44] Bailey, J. E. and Hirsch, P. B. The dislocation distribution, flow stress, and stored energy in cold-worked polycrystalline silver. Phil. Mag., 1960,5: 485-497
[45] Nabarro, F. R. N., Basinski, Z. S. and Holt, D. B. The plasticity of pure single crystals. Adv. Phys., 1964,13: 193-323
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[48] Cottrell, A. H. The mechanical Properties of Materials. J. Wiley, New York., 1964
[49] Gao, H. and Huang, Y. Geometrically necessary dislocation and size-dependent plasticity. Scripta Mater., 2003,48: 113-118
[50] Arsenlis, A. and Parks, D. M. Crystallographic aspects of geometrically-necessary and statistically-stored dislocation density. Acta Mater. 1999, 47: 1597-1611
[51] Shi, M. X., Huang, Y., Gao, H. The J-integral and geometrically necessary dislocations in nonuniform plastic deformation. International Journal of Plasticity, 2004, 20: 1739-1762
[52] Bishop, J. F. W. and Hill, R. A theory of plastic distortion of a polyerystalline aggregate under combined stresses. Phil. Mag., 1951a, 42: 414-427
[53] Bishop, J. F. W. and Hill, R. A theoretical derivation of the plastic properties of apolycrystalline face-centered metal. Phil. Mag., 1951b, 42: 1298-1307
[54] Kocks, U. F. The relation between polycrystal deformation and single crystal deformation.Metall. Mater. Trans., 1970, 1: 1121-1144
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[58] Haque, M. A. and Saif, M. T. A. Strain gradient effect in nanoscale thin films. Acta Mater., 2003, 51, 3053-3061
[59] Liu, B., Qiu, X., Huang, Y., Hwang, K. C., Li, M. and Liu, C. The size effect on void growth in ductile materials. J. Mech. Phys. Solids, 2003, 51: 1171-1187
[60] Rice, J. R., Tracey, D. M. On the ductile enlargement of voids in triaxial stress fields. J. Mech. Phys. Solids, 1969, 17: 201-217
[61] Venkatraman, R., Bravman, J. C. Separation of film thickness and grain boundary strengthening effects in AI thin films on Si. J. Mater. Res., 1992, 7: 2040-2048
[62] Nix, W. D. Elastic and plastic properties of thin films on substrates: nanoindentationtechniques. Mater. Sci. Eng. A, 1997, 37: A234-236
[63] Nix, W. D. Yielding and strain hardening of thin metal films on substrates. Seripta Mater., 1998, 39: 545-554
[64] Hill, R., On uniqueness and stability in the theory of finite elastic strain, J. Mech.Phys. Solids, 1957, 5: 229-241
[2] Stolken, J. S. and Evans, A. G. A microbend 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., A, 1989, 20A: 2217～2245
[5] 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 Metali. Mater., 1993, 41: 2855～2865
[6] Ma Q. and Clarke D. R. Size dependent hardness of silver single crystals. J. Mater. Resh., 1995, 10: 853～863
[7] Poole W. J., Ashby M. F., and Fleck N. A. Micro-hardness of annealed and work-hardenedcopper polycrystals. Scripta Metall. Mater., 1996, 34: 559～564
[8] 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
[9] Elssner, G., Korn, D. and Ruhle, M. The influence of interface impurities on fractureenergy of UHV diffusion-bonded metal-ceramic bierystals. Scripta Metall. Mater., 1994, 31: 1037-1042
[10] Hutchinson J. W. Linking scales in mechanics. In: Karihaloo B. L., Mai Y. W., Ripley M.I., and Ritchie Ro O, eds. Advances in Fracture Research. Amsterdam: Pergamon Press, 1997, 1～14
[11] Toupin R. A. Elastic materials with couple stresses. Arch Ration. Mech. Anal., 1962, 11:385～414
[12] Mindlin R. D. Micro-structure in linear elasticity. Arch. Ration. Mech. Anal., 1964, 16:51～78
[13] Mindlin R. D. Second gradient of strain and surface tension in linear elasticity. Int. J.Solids Struct., 1965, 1: 417～438
[14] Acharya, A., Bassani, J. L. Lattice incompatibility and a gradient theory of crystal plasticity.J. Mech. Phys. Solids, 2000, 48: 1565-1595
[15] Acharya, A. and Beaudoin, A. J. Grain-size effect in viscoplastic polycrystals at moderate strains. J. Mech. Phys. Solids, 2000,48:2213-2230
[16] Busso, E. P., Meissonneir, F. T., O'Dowd, N. P. Gradient-dependent deformation of two-phase single crystals. J. Mech. Phys. Solids, 2000, 48: 2333-2362
[17] Bassani, J. L. Incompatibility and a simple gradient theory of plasticity. J. Mech. Phys. Solids, 2001,49: 1983-1996
[18] Gao, H. and Huang, Y. Taylor-based nonlocal theory of plasticity. Int. J. Solids Struct., 2001,38:2615-2637
[19] Guo, Y., Huang, Y., Gao, H., Zhuang, Z. and Hwang, K. C. Taylor-based nonlocal theory of plasticity: numerical studies of micro-indentation experiments and crack tip fields. Int. J. Solids Struct, 2001,38:7447-7460
[20] Huang, Y., Qu, S., Hwang, K. C, Li, M. and Gao, H. A conventional theory of mechanism-based strain gradient plasticity. Int. J. Plasticity, 2004,20: 753-782
[21 ] Hwang, K. C, Guo, Y., Jiang, H., Huang, Y. and Zhuang, Z. The finite deformation theory of Taylor-based nonlocal plasticity. Int. J. Plasticity, 2004,20: 831-839
[22] Aifantis, E. C. On the microstructural origin of certain inelastic models. Trans. ASME J. Engrg. Mater. Technol., 1984, 106:326-330
[23] Fleck, N. A., Hutchinson, J. W. A reformulation of strain gradient plasticity. J. Mech. Phys. Solids, 2001,49:2245-2271
[24] Fleck, N. A., Hutchinson, J. W. A phenomenological theory for strain gradient effects in plasticity. J. Mech. Phys. Solids, 1993,41:1825-1857
[25] Fleck, N. A., Hutchinson, J. W. Strain gradient plasticity. In: Hutchinson, J. W., Wu, T. Y. (Eds.), Adv. Appl. Mech., vol. 33. Academic Press, New York, 1997,295-361
[26] Gao, H., Huang, Y., Nix, W. D. and Hutchinson, J. W. Mechanism-based strain gradient plasticity-I. Theory. J. Mech. Phys. Solids, 1999b, 47:1239-1263
[27] Huang, Y., Gao, H., Nix, W. D., Hutchinson, J. W. Mechanism-based strain gradient plasticity - II. Analysis. J. Mech. Phys. Solids, 2000a, 48:99-128
[28] Huang, Y., Xue, Z., Gao, H., Nix, W. D., Xia, Z. C. A study of microindentation hardness tests by mechanism-based strain gradient plasticity. J. Mater. Res. 2000b, 15:1786-1796
[29] Gurtin, M.E. On the plasticity of single crystals: free energy, microforces, plastic-strain gradient. J. Mech. Phys. Solids, 2000,48:989-1036
[30] Gurtin, M.E. A gradient theory of single-crystal viscoplasticity that accounts for geometrically necessary dislocations. J. Mech. Phys. Solids, 2002, 50: 5-32
[31] Qiu, X., Huang, Y., Nix, W. D., Hwang, K. C. and Gao, H. Effect of intrinsic lattice resistance in strain gradient plasticity. Acta Mater., 2001,49: 3949-3958
[32] Qiu, X., Huang, Y., Wei, Y., Gao, H. and Hwang, K. C. The flow theory of mechanism-based strain gradient plasticity. Mechanics of Materials, 2003, 35: 245-258
[33] Hwang, K. C, Jiang, H., Huang, Y., Gao, H. and Hu, N. A finite deformation theory of strain gradient plasticity. J. Mech. Phys. Solids, 2002, 50: 81-99
[34] Hwang, K. C, Jiang, H., Huang, Y. and Gao, H. Finite deformation analysis of mechanism-based strain gradient plasticity: torsion and crack tip field. Int. J. Plasticity, 2003, 19:235-251
[35] Needleman, A. Computational mechanics at the mesoscale. Acta Materialia, 2000, 48: 105-124
[36] Shu, J. Y., Fleck, N. A., Van der Giessen, E. and Needleman, A. Boundary layers in constrained plastic flow: comparison of nonlocal and discrete dislocation plasticity. J. Mech. Phys. Solids, 2001,49: 1361-1395
[37] Niordson, C. F., Hutchinson, J. W. On lower order strain gradient plasticity theories. Eur. J. Mech. A Solids, 2003,22: 771-778
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