晶体材料循环变形的微尺度效应及其离散位错模拟
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摘要
近年来,微纳米科技方兴未艾。各种微结构器件、微机械电子、微机电系统(MEMS)相继出现并得到广泛应用。在各种微机电系统中,各种构件的特征尺度与晶粒尺度处在同一量级,由于微几何及微结构约束,材料的力学行为呈现强烈的微尺度效应。另一方面,服役中的微构件通常承受循环载荷作用。基于传统尺度无关本构理论的疲劳强度设计和寿命预测方法将不再适用。因此,对微尺度下多晶材料的循环塑性行为及其内在机理进行研究,不但具有重要的理论意义而且具有实际工程价值。针对此问题,本文进行了如下研究:
(1)基于二维离散位错动力学,研究了单晶薄梁在循环拉压载荷作用下的循环塑性响应。重点研究了不同循环载荷下尺度效应的内在机制。结果表明:在纯拉-压和纯弯曲循环作用下,单晶循环塑性呈现明显的尺度效应。但是,两种载荷下尺度效应的内在机制是不同的。当试样受循环拉压载荷时,“位错饥饿机制”起主导作用;当试样受循环弯曲载荷时,“几何必需位错机制”起主导作用。当试样承受拉弯组合载荷作用时,循环弯矩-转角响应呈现明显的尺度效应;但是,拉压循环载荷下,应力-应变响应呈现较弱甚至没有尺度效应。
(2)在多晶体中,晶界的位错可穿透性对循环塑性变形起着十分重要的作用。基于位错-晶界交互作用模型,扩展了二维离散位错动力学计算框架。在此基础上,通过离散位错模拟,探讨了位错与晶界相互作用对多晶循环塑性行为的影响及其内在机制。模拟结果表明:循环变形行为与晶界相对于位错的可穿透性密切相关。对于位错不可穿透晶界,循环应力应变响应不存在循环硬化现象;而对于位错可穿透晶界,由于位错的不断累积及位错间交互作用不断增强,循环塑性响应呈现出明显的硬化现象并逐渐趋于循环饱和。除此之外,还对晶粒大小、应变幅大小及加载历史等因素对循环强化和循环饱和行为的影响进行了模拟分析。
(3)基于元胞模型,采用双周期边界条件,对单晶及多晶材料在拉-压非比例载荷作用下的应力应变响应进行了离散位错模拟。结果表明:①无论是单晶还是多晶材料,其本构响应都呈现了明显的附加强化现象;②对于多晶材料,应力-应变响应虽然与所选取的元胞大小无关,但却与晶粒大小密切相关,存在明显的晶粒尺度效应;③在双向非比例载荷作用下,两个方向的载荷比对材料的塑性变形有重要影响,在不同载荷比下,多晶的循环塑性行为呈现出应变强化或应变软化现象;④与单向拉压载荷不同,非比例载荷作用下,多晶循环硬化行为不明显。
Classic Constitutive Relation of Continuum Mechanics has achieved great success in mechanics, aerospace, building construction, electric power, traffic vessel and other traditional engineering field. With the rapid development of the computer science and technology, finite element methods have been widely used in structure strength checking, optimal design, safety life evaluation and analysis of structure reliability, which greatly improved the technology used both in product design and science research. It is of no doubt that the traditional continuum mechanics theory has been successful in macro scale. However, the recent micro-nano technology is at the ascendant. Micro-structure apparatus, micro mechanical electronic, micro electromechanical systems and those photoelectric apparatus with optoisolator, optical conductor, photo semiconductor as the representatives emerged in secession and has been widely employed. These apparatus usually work under cyclic loading. On the other hand, with the material size down to micron or even below, their mechanical behaviors show great micro size effect, thus strength analysis based on traditional mechanical constitutive relations is no longer applicable. Therefore, research on the cyclic plastic behaviors of polycrystal materials under micro-scale and its underlying mechanism is not only of great significance in mechanic theory but also of important material value in real engineering projects. Toward this end, the main research contents and results acquired are as follows:
(1) The cyclic plastic response of a single crystalline thin beam subject to combined cyclic tension and bending is analyzed using two-dimensional discrete dislocation plasticity. In this contribution, special attention is paid to the difference in the inherent mechanism of the size effect for different cyclic loads. Results show that the cyclic plastic response has a strong size effect for both cyclic pure tension-compression and pure bending. However, the inherent mechanisms are different. The dislocation starvation mechanism dominates the cyclic tension-compression while the geometrically necessary dislocation dominates the cyclic pure bending. When the combined cyclic tension and bending are applied to the thin beam, the cyclic moment-rotation response shows strong size effect while the stress-strain response shows weak or even no size effect. In addition, it is also found that the cyclic loading paths have considerable influences on the shape of the cyclic stress-strain loops.
(2) To simulate the dislocation transmission across grain boundary, a dislocation-grain boundary penetration model is proposed and then integrated into the two-dimensional discrete dislocation dynamics (DDD) framework by Giessen and Needleman (1995). By this extended DDD technology, cyclic plastic response of polycrystals is analyzed. Special attentions are paid to significant influence of dislocation-penetrable grain boundaries (GBs) on micro-plastic cyclic responses of polycrystals and the underlying dislocation mechanisms. Results show that, when GBs are penetrable to dislocations, continual dislocation accumulation and enhanced dislocation-dislocation interactions give rise to the cyclic hardening behavior; on the other hand, when a dynamic balance among dislocation nucleation, penetration and annihilation is approximately established, cyclic stress gradually tends to saturation. In addition, other factors, including the grain size, cyclic strain amplitude and its history, have considerable influences on the cyclic hardening and saturation.
(3) The cyclic plastic response of periodic crystalline material subjected to non-proportional loads are analyzed using two-dimensional discrete dislocation plasticity. It is found that①the additional hardening presents for both the single crystal and polycrystals;②the cyclic plastic response has a weak dependence of cell size but a strong grain size effect;③the load radios have a significant influence on cyclic plastic deformation. With different load radios, it can lead to either strain hardening or strain softening;④under the nonproportional loads, the cyclic hardening behavior is not present obviously.
(1)基于二维离散位错动力学,研究了单晶薄梁在循环拉压载荷作用下的循环塑性响应。重点研究了不同循环载荷下尺度效应的内在机制。结果表明:在纯拉-压和纯弯曲循环作用下,单晶循环塑性呈现明显的尺度效应。但是,两种载荷下尺度效应的内在机制是不同的。当试样受循环拉压载荷时,“位错饥饿机制”起主导作用;当试样受循环弯曲载荷时,“几何必需位错机制”起主导作用。当试样承受拉弯组合载荷作用时,循环弯矩-转角响应呈现明显的尺度效应;但是,拉压循环载荷下,应力-应变响应呈现较弱甚至没有尺度效应。
(2)在多晶体中,晶界的位错可穿透性对循环塑性变形起着十分重要的作用。基于位错-晶界交互作用模型,扩展了二维离散位错动力学计算框架。在此基础上,通过离散位错模拟,探讨了位错与晶界相互作用对多晶循环塑性行为的影响及其内在机制。模拟结果表明:循环变形行为与晶界相对于位错的可穿透性密切相关。对于位错不可穿透晶界,循环应力应变响应不存在循环硬化现象;而对于位错可穿透晶界,由于位错的不断累积及位错间交互作用不断增强,循环塑性响应呈现出明显的硬化现象并逐渐趋于循环饱和。除此之外,还对晶粒大小、应变幅大小及加载历史等因素对循环强化和循环饱和行为的影响进行了模拟分析。
(3)基于元胞模型,采用双周期边界条件,对单晶及多晶材料在拉-压非比例载荷作用下的应力应变响应进行了离散位错模拟。结果表明:①无论是单晶还是多晶材料,其本构响应都呈现了明显的附加强化现象;②对于多晶材料,应力-应变响应虽然与所选取的元胞大小无关,但却与晶粒大小密切相关,存在明显的晶粒尺度效应;③在双向非比例载荷作用下,两个方向的载荷比对材料的塑性变形有重要影响,在不同载荷比下,多晶的循环塑性行为呈现出应变强化或应变软化现象;④与单向拉压载荷不同,非比例载荷作用下,多晶循环硬化行为不明显。
Classic Constitutive Relation of Continuum Mechanics has achieved great success in mechanics, aerospace, building construction, electric power, traffic vessel and other traditional engineering field. With the rapid development of the computer science and technology, finite element methods have been widely used in structure strength checking, optimal design, safety life evaluation and analysis of structure reliability, which greatly improved the technology used both in product design and science research. It is of no doubt that the traditional continuum mechanics theory has been successful in macro scale. However, the recent micro-nano technology is at the ascendant. Micro-structure apparatus, micro mechanical electronic, micro electromechanical systems and those photoelectric apparatus with optoisolator, optical conductor, photo semiconductor as the representatives emerged in secession and has been widely employed. These apparatus usually work under cyclic loading. On the other hand, with the material size down to micron or even below, their mechanical behaviors show great micro size effect, thus strength analysis based on traditional mechanical constitutive relations is no longer applicable. Therefore, research on the cyclic plastic behaviors of polycrystal materials under micro-scale and its underlying mechanism is not only of great significance in mechanic theory but also of important material value in real engineering projects. Toward this end, the main research contents and results acquired are as follows:
(1) The cyclic plastic response of a single crystalline thin beam subject to combined cyclic tension and bending is analyzed using two-dimensional discrete dislocation plasticity. In this contribution, special attention is paid to the difference in the inherent mechanism of the size effect for different cyclic loads. Results show that the cyclic plastic response has a strong size effect for both cyclic pure tension-compression and pure bending. However, the inherent mechanisms are different. The dislocation starvation mechanism dominates the cyclic tension-compression while the geometrically necessary dislocation dominates the cyclic pure bending. When the combined cyclic tension and bending are applied to the thin beam, the cyclic moment-rotation response shows strong size effect while the stress-strain response shows weak or even no size effect. In addition, it is also found that the cyclic loading paths have considerable influences on the shape of the cyclic stress-strain loops.
(2) To simulate the dislocation transmission across grain boundary, a dislocation-grain boundary penetration model is proposed and then integrated into the two-dimensional discrete dislocation dynamics (DDD) framework by Giessen and Needleman (1995). By this extended DDD technology, cyclic plastic response of polycrystals is analyzed. Special attentions are paid to significant influence of dislocation-penetrable grain boundaries (GBs) on micro-plastic cyclic responses of polycrystals and the underlying dislocation mechanisms. Results show that, when GBs are penetrable to dislocations, continual dislocation accumulation and enhanced dislocation-dislocation interactions give rise to the cyclic hardening behavior; on the other hand, when a dynamic balance among dislocation nucleation, penetration and annihilation is approximately established, cyclic stress gradually tends to saturation. In addition, other factors, including the grain size, cyclic strain amplitude and its history, have considerable influences on the cyclic hardening and saturation.
(3) The cyclic plastic response of periodic crystalline material subjected to non-proportional loads are analyzed using two-dimensional discrete dislocation plasticity. It is found that①the additional hardening presents for both the single crystal and polycrystals;②the cyclic plastic response has a weak dependence of cell size but a strong grain size effect;③the load radios have a significant influence on cyclic plastic deformation. With different load radios, it can lead to either strain hardening or strain softening;④under the nonproportional loads, the cyclic hardening behavior is not present obviously.
引文
[1] Connolley T, McHugh PE, Bruzzi M. A review of deformation and fatigue of metals at small size scales. Fatigue & Fracture of Engineering Materials & Structures, 2005,28(12): 1119-1152
[2] Geers MGD, Brekelmans WAM, Bayley CJ. Second-order crystal plasticity: internal stress effects and cyclic loading. Modelling and Simulation in Materials Science and Engineering, 2007,15(1): S133-S145
[3] Atkinson M. Further analysis of the size effect in indentation hardness tests of some metals. Journal of Materials Research, 1995,10(11): 2908-2915
[4] Ma Q, Clarke DR. Size dependent hardness of silver single crystals. Journal of Materials Research, 1995,10(4): 853-863
[5] Poole WJ, Ashby MF, Fleck NA. Micro-hardness of annealed and work-hardened copper polycrystals. Scripta Materialia, 1996,34(4): 559-564
[6] Nix WD. Elastic and plastic properties of thin films on substrates: nanoindentation techniques. 11th International Conference on the Strength of Materials (ICSMA-11). Prague, Czech Republic: Elsevier Science Sa Lausanne, 1997:37-44.
[7] McElhaney KW, Nix WD, Vlassak JJ. Determination of indenter tips geometry and indentation contact area for depth-sensing indentation experiments. Journal of Materials Research, 1998,13(5): 1300-1306
[8] Suresh S, Nieh TG, Choi BW. Nano-indentation of copper thin films on silicon substrates. Scripta Materialia, 1999,41(9): 951-957
[9] Tymiak NI, Kramer DE, Bahr DF, et al. Plastic strain and strain gradients at very small indentation depths. Acta Materialia, 2001,49(6): 1021-1034
[10] Wei Y, Hutchinson JW. Hardness trends in micron scale indentation. Journal of the Mechanics and Physics of Solids, 2003,51(11-12): 2037-2056
[11] Abu Al-Rub RK, Voyiadjis GZ. Analytical and experimental determination of the material intrinsic length scale of strain gradient plasticity theory from micro- and nano-indentation experiments. International Journal of Plasticity, 2004,20(6): 1139-1182
[12] Fleck NA, Muller GM, Ashby MF, et al. Strain gradient plasticity: Theory and experiment. Acta Metallurgica et Materialia, 1994,42(2): 475-487
[13] Stolken JS, Evans AG. A microbend test method for measuring the plasticity length scale. Acta Materialia, 1998,46(14): 5109-5115
[14] Wang W, Huang Y, Hsia KJ, et al. A study of microbend test by strain gradient plasticity. International Journal of Plasticity, 2003,19(3): 365-382
[15] Haque MA, Saif MTA. Strain gradient effect in nanoscale thin films. Acta Materialia, 2003,51(11): 3053-3061
[16] Espinosa HD, Prorok BC, Peng B. Plasticity size effects in free-standing submicron polycrystalline FCC films subjected to pure tension. Journal of the Mechanics and Physics of Solids, 2004,52(3): 667-689
[17] Nye JF. Some geometrical relations in dislocated crystals. Acta Metallurgica, 1953,1(2): 153-162
[18] Ashby MF. The deformation of plastically non-homogeneous materials. Philosophical Magazine, 1970,21(170): 399 - 424
[19] Arzt E. Size effects in materials due to microstructural and dimensional constraints: a comparative review. Acta Materialia, 1998,46(16): 5611-5626
[20] Gao H, Huang Y. Geometrically necessary dislocation and size-dependent plasticity. Scripta Materialia, 2003,48(2): 113-118
[21] Xiang Y, Vlassak JJ. Bauschinger and size effects in thin-film plasticity. Acta Materialia, 2006,54(20): 5449-5460
[22] Fleck NA, Hutchinson JW. A phenomenological theory for strain gradient effects in plasticity. Journal of the Mechanics and Physics of Solids, 1993,41(12): 1825-1857
[23] Fleck NA, Hutchinson JW. Strain gradient plasticity. Advances in Applied Mechanics, Vol 33. San Diego: Academic Press Inc, 1997:295-361.
[24] Shu JY, Fleck NA. Strain gradient crystal plasticity: size-dependentdeformation of bicrystals. Journal of the Mechanics and Physics of Solids, 1999,47(2): 297-324
[25] Nix WD, Gao H. Indentation size effects in crystalline materials: A law for strain gradient plasticity. Journal of the Mechanics and Physics of Solids, 1998,46(3): 411-425
[26] Gao H, Huang Y, Nix WD, et al. Mechanism-based strain gradient plasticity— I. Theory. Journal of the Mechanics and Physics of Solids, 1999,47(6): 1239-1263
[27] Huang Y, Gao H, Nix. WD, et al. Mechanism-based strain gradient plasticity-II. Analysis. Journal of the Mechanics and Physics of Solids, 2000,48(1): 99-128
[28] Jiang H, Huang Y, Zhuang Z, et al. Fracture in mechanism-based strain gradient plasticity. Journal of the Mechanics and Physics of Solids, 2001,49(5): 979-993
[29] Aifantis EC. on the microstructural origin of certain inelastic models. Journal of Engineering Materials and Technology, 1984,106(44): 326-330
[30] Aifantis EC. Strain gradient interpretation of size effects. International Journal of Fracture, 1999,95(1-4): 299-314
[31] Taylor MB, Zbib HM, Khaleel MA. Damage and size effect during superplastic deformation. International Journal of Plasticity, 2002,18(3): 415-442
[32] Acharya A, Bassani JL. Lattice incompatibility and a gradient theory of crystal plasticity. Journal of the Mechanics and Physics of Solids, 2000,48(8): 1565-1595
[33] Evers LP, Parks DM, Brekelmans WAM, et al. Crystal plasticity model with enhanced hardening by geometrically necessary dislocation accumulation. Journal of the Mechanics and Physics of Solids, 2002,50(11): 2403-2424
[34] Tvergaard V, Niordson C. Nonlocal plasticity effects on interaction of different size voids. International Journal of Plasticity, 2004,20(1): 107-120
[35] Gurtin ME. On the plasticity of single crystals: free energy, microforces, plastic-strain gradients. Journal of the Mechanics and Physics of Solids, 2000,48(5): 989-1036
[36] Gurtin ME. A gradient theory of single-crystal viscoplasticity that accounts for geometrically necessary dislocations. Journal of the Mechanics and Physics of Solids, 2002,50(1): 5-32
[37] Gurtin ME. On a framework for small-deformation viscoplasticity: free energy, microforces, strain gradients. International Journal of Plasticity, 2003,19(1): 47-90
[38] Evers LP, Brekelmans WAM, Geers MGD. Scale dependent crystal plasticity framework with dislocation density and grain boundary effects. International Journal of Solids and Structures, 2004,41(18-19): 5209-5230
[39] Uchic MD, Dimiduk DM, Florando JN, et al. Sample dimensions influence strength and crystal plasticity. Science, 2004,305(5686): 986-989
[40] Dimiduk DM, Uchic MD, Parthasarathy TA. Size-affected single-slip behavior of pure nickel microcrystals. Acta Materialia, 2005,53(15): 4065-4077
[41] Greer JR, Oliver WC, Nix WD. Size dependence of mechanical properties of gold at the micron scale in the absence of strain gradients. Acta Materialia, 2005,53(6): 1821-1830
[42] Greer JR, Nix WD. Size dependence of mechanical properties of gold at the sub-micron scale. Applied Physics a-Materials Science & Processing, 2005,80(8): 1625-1629
[43] Greer JR, Nix WD. Nanoscale gold pillars strengthened through dislocation starvation. Physical Review B, 2006,73(24): 6
[44] Nix WD, Greer JR, Feng G, et al. Deformation at the nanometer and micrometer length scales: Effects of strain gradients and dislocation starvation. Thin Solid Films, 2007,515(6): 3152-3157
[45] Deshpande VS, Needleman A, Van der Giessen E. Plasticity size effects in tension and compression of single crystals. Journal of the Mechanics and Physics of Solids, 2005,53(12): 2661-2691
[46] Benzerga AA, Shaver NF. Scale dependence of mechanical properties of single crystals under uniform deformation. Scripta Materialia, 2006,54(11): 1937-1941
[47] Tang H, Schwarz KW, Espinosa HD. Dislocation escape-related size effects in single-crystal micropillars under uniaxial compression. Acta Materialia, 2007,55(5): 1607-1616
[48] Hasegawa T, Yakou T, Kocks UF. Forward and reverse rearrangements of dislocations in tangled walls. Materials Science and Engineering, 1986,81: 189-199
[49] Margolin H, Hazaveh F, Yaguchi H. The grain boundary contribution to the bauschinger effect. Scripta Metallurgica, 1978,12(12): 1141-1145
[50] Ono N, Tsuchikawa T, Nishimura S, et al. Intergranular constraint and the Bauschinger effect. Materials Science and Engineering, 1983,59(2): 223-233
[51] Xiang Y, Vlassak JJ. Bauschinger effect in thin metal films. Scripta Materialia, 2005,53(2): 177-182
[52] Brown LM, Stobbs WM. The work-hardening of copper-silica. Philosophical Magazine, 1971,23(185): 1185-1199
[53] Atkinson JD, Brown LM, Stobbs WM. The work-hardening of copper-silica: IV. The Bauschinger effect and plastic relaxation. Philosophical Magazine, 1974,30(6): 1247 - 1280
[54] Aran A, Demirkol M, Karabulut A. Bauschinger effect in precipitation-strengthened aluminium alloy 2024. Materials Science and Engineering, 1987,89: L35-L39
[55] Stoltz R, Pelloux R. The Bauschinger effect in precipitation strengthened aluminum alloys. Metallurgical and Materials Transactions A, 1976,7(8): 1295-1306
[56] Ham RK, Broom T. The mechanism of fatigue softening. Philosophical Magazine, 1962,7(73): 95-103
[57] Winter AT. The effect of work-hardening on the low strain amplitude fatigue of copper crystals. Philosophical Magazine, 1975,31(2): 411 - 417
[58] Mughrabi H. The cyclic hardening and saturation behaviour of copper single crystals. Materials Science and Engineering, 1978,33(2): 207-223
[59] Yan B, Laird C. Matrix hardening behavior and the nucleation stress for persistent slip bands in fatigued monocrystalline copper. Materials Science and Engineering, 1986,80(1): 59-64
[60] Gong B, Wang Z, Chen D, et al. Investigation of macro deformation bands in fatigued [001] Cu single crystals by electron channeling contrast technique. Scripta Materialia, 1997,37(10): 1605-1610
[61] Figueroa JC, Bhat SP, De La Veaux R, et al. The cyclic stress-strain response of copper at low strains—i. Constant amplitude testing. Acta Metallurgica, 1981,29(10): 1667-1678
[62] Wang Z, Laird C. Cyclic stress—strain response of polycrystalline copper under fatigue conditions producing enhanced strain localization. Materials Science and Engineering, 1988,100: 57-68
[63] Wang Z, Sun Z, Laird C. Effect of initial ramp loading on fatigue behavior in polycrystalline copper. Materials Science and Engineering: A, 1992,151(2): 121-130
[64] Morrison DJ. Influence of grain size and texture on the cyclic stress-strain response of nickel. Materials Science and Engineering: A, 1994,187(1): 11-21
[65] El-Madhoun Y, Mohamed A, Bassim MN. Cyclic stress-strain response and dislocation structures in polycrystalline aluminum. Materials Science and Engineering A, 2003,359(1-2): 220-227
[66] Li Y, Laird C. Cyclic response and dislocation structures of AISI 316L stainless steel. Part 2: polycrystals fatigued at intermediate strain amplitude. Materials Science and Engineering: A, 1994,186(1-2): 87-103
[67] Feltner CE, Laird C. Cyclic stress-strain response of F.C.C. metals and alloys—I Phenomenological experiments. Acta Metallurgica, 1967,15(10): 1621-1632
[68] Hancock JR, Grosskreutz JC. Mechanisms of fatigue hardening in copper single crystals. Acta Metallurgica, 1969,17(2): 77-97
[69] Woods PJ. Low-amplitude fatigue of copper and copper-5 at. % aluminium single crystals. Philosophical Magazine, 1973,28(1): 155 - 191
[70] Finney JM, Laird C. Strain localization in cyclic deformation of copper single crystals. Philosophical Magazine, 1975,31(2): 339 - 366
[71] Kuhlmann-Wilsdorf D. Dislocation behavior in fatigue IV. Quantitative interpretation of friction stress and back stress derived from hysteresis loops. Materials Science and Engineering, 1979,39(2): 231-245
[72] Blochwitz C, Veit U. Plateau Behaviour of Fatigued FCC Single Crystals. Crystal Research and Technology, 1982,17(5): 529-551
[73] Laird C, Buchinger L. Hardening behavior in fatigue. Metallurgical and Materials Transactions A, 1985,16(12): 2201-2214
[74] Laird C, Charsley P, Mughrabi H. Low energy dislocation structures produced by cyclic deformation. Materials Science and Engineering, 1986,81: 433-450
[75] Basinski ZS, Basinski SJ. Fundamental aspects of low amplitude cyclic deformation in face-centred cubic crystals. Progress in Materials Science, 1992,36: 89-148
[76] Feltner CE. A debris mechanism of cyclic strain hardening for F.C.C. metals. Philosophical Magazine, 1965,12(120): 1229 - 1248
[77] Bhat SP, Laird C. The cyclic stress-strain curves in monocrystalline and polycrystalline metals. Scripta Metallurgica, 1978,12(8): 687-692
[78] Rasmussen KV, Pedersen OB. Fatigue of copper polycrystals at low plastic strain amplitudes. Acta Metallurgica, 1980,28(11): 1467-1478
[79] Kuokkala VT, Kettunen P. Cyclic stress-strain response of polycrystalline copper in constant and variable amplitude fatigue. Acta Metallurgica, 1985,33(11): 2041-2047
[80] Mughrabi H. Plateaus in the cyclic stress-strain curves of single- and polycrystalline metals. Scripta Metallurgica, 1979,13(6): 479-484
[81] Lukas P, Kunz L. Is there a plateau in the cyclic stress-strain curves of polycrystalline copper? Materials Science and Engineering, 1985,74(1): L1-L5
[82] Peralta P, Llanes L, Czapka A, et al. Effect of texture and grain size as independent factors in the cyclic behavior of polycrystalline copper. Scripta Metallurgica et Materialia, 1995,32(11): 1877-1881
[83] Llanes L, Rollett AD, Laird C, et al. Effect of grain size and annealing texture on the cyclic response and the substructure evolution of polycrystalline copper. Acta Metallurgica et Materialia, 1993,41(9): 2667-2679
[84] Liu CD, You DX, Bassim MN. Cyclic strain hardening in polycrystalline copper. Acta Metallurgica et Materialia, 1994,42(5): 1631-1638
[85] Morrison DJ, Chopra V. Cyclic stress-strain response of polycrystalline nickel. Materials Science and Engineering: A, 1994,177(1-2): 29-42
[86] Boutin J, Marchand N, Bailon JP, et al. An intermediate plateau in the cyclic stress-strain curve of [alpha] brass. Materials Science and Engineering, 1984,67(2): L23-L27
[87] Morrison DJ, Chopra V, Jones JW. Effects of grain size on cyclic strain localization in polycrystalline nickel. Scripta Metallurgica et Materialia, 1991,25(6): 1299-1304
[88] Lukas P, Kunz L. Effect of grain size on the high cycle fatigue behaviour of polycrystalline copper. Materials Science and Engineering, 1987,85: 67-75
[89] Liang FL, Laird C. Control of intergranular fatigue cracking by slip homogeneity in copper I: Effect of grain size. Materials Science and Engineering: A, 1989,117: 95-102
[90] Llanes L, Laird C. Effect of grain size and ramp loading on the low amplitude cyclic stress-strain curve of polycrystalline copper. Materials Science and Engineering: A, 1990,128(2): L1-L4
[91] Mughrabi H. Dislocation wall and cell structures and long-range internal stresses in deformed metal crystals. Acta Metallurgica, 1983,31(9): 1367-1379
[92] Calabrese C, Laird C. Cyclic stress—strain response of two-phase alloys Part I. Microstructures containing particles penetrable by dislocations. Materials Science and Engineering, 1974,13(2): 141-157
[93] Laird C, Wang Z, Ma BT, et al. Low energy dislocation structures produced by cyclic softening. Materials Science and Engineering: A, 1989,113: 245-257
[94] Nian L, Bai-ping D. The effect of low-stress high-cycle fatigue on themicrostructure and fatigue threshold of a 40Cr steel. International Journal of Fatigue, 1995,17(1): 43-48
[95] Lamba HS, Sidebottom OM. Cyclic plasticity for nonproportional paths. Journal of Engineering Materials and Technology, Transactions of the ASME, 1978,100(1): 96-111
[96] Doong S-H, Socie DF, Robertson IM. Dislocation Substructures and Nonproportional Hardening. Journal of Engineering Materials and Technology, 1990,112(4): 456-464
[97] Cailletaud G, Doquet V, Pineau A. Cyclic multiaxial behavior of an austenitic stainless steel microstructural observations and macromechanical modeling. Fatigue under Biaxial and Multiaxial Loading London: ESIS Publication, 1991:131-149.
[98] Itoh, Sakane M, Ohnami M. Effect of stacking fault energy on cyclic constitutive relation under nonproportional loading. Journal of the Society of Materials Science, 1992,41(468): 1361-1367
[99] Kanazawa K, Miller KJ, Brown MW. Low cycle fatigue under out of phase loading conditions. Journal of Engineering Materials and Technology, 1977,99: 222-228
[100] Mroz Z. On the description of anisotropic workhardening. Journal of the Mechanics and Physics of Solids, 1967,15(3): 163-175
[101] Krieg RD. A practical two surface plasticity theory. Journal of Applied Mechanics 1975,42: 641-646
[102] Dafalias YF, Popov EP. Plastic internal variables formalism of cyclic plasticity. Journal of Applied Mechanics, 1976,43: 645-651
[103] Tseng NT, Lee GC. Simple Plasticity Model of Two-Surface Type. Journal of Engineering Mechanics, 1983,109(3): 795-810
[104] McDowell DL. A two surface model for transient nonproportional cyclic plasticity. Journal of Applied Mechanics, 1985,54: 323-334
[105] Ellyin F, Xia Z. A rate-independent constitutive model for transient non-proportional loading. Journal of the Mechanics and Physics of Solids, 1989,37(1): 71-91
[106] Ohno N, Takahashi Y, Kuwabara K. Constitutive modeling of anisothermal cyclic plasticity of 304 stainless steel. Journal of Engineering Materials and Technology, 1989,111: 106-113
[107] Jiang Y, Sehitoglu H. Multiaxial cyclic ratchetting under multiple step loading. International Journal of Plasticity, 1994,10(8): 849-870
[108] Armstrong PJ, Frederick CO. A Mathematical Representation of the Multiaxial Bauschinger Effect. CEGB Report RD/B/N731: Berkeley Nuclear Laboratories, 1966.
[109] Chaboche JL, Nouailhas D, Pacou D, et al. Modeling of the cyclic response and ratcheting effects on inconel 718 alloy. European Journal of Mechanics - A/Solids, 1991,10: 101-112
[110] Ohno N, Wang JD. Kinematic hardening rules with critical state of dynamic recovery, part I: formulation and basic features for ratchetting behavior. International Journal of Plasticity, 1993,9(3): 375-390
[111] Abdel-Karim M, Ohno N. Kinematic hardening model suitable for ratchetting with steady-state. International Journal of Plasticity, 2000,16(3-4): 225-240
[112] Delobelle P, Robinet P, Bocher L. Experimental study and phenomenological modelization of ratchet under uniaxial and biaxial loading on an austenitic stainless steel. International Journal of Plasticity, 1995,11(4): 295-330
[113] Burlet H, Cailletaud G. numerical techniques for cyclic plasticity at variable temperature. Engineering Computations, 1986,3(2): 143-153
[114] Alder BJ, Wainwright TE. Phase transition for a hard sphere system. J Chem Phys, 1957,27(5): 1208-1209
[115] Needleman A. Computational mechanics at the mesoscale. Acta Materialia, 2000,48(1): 105-124
[116] Buehler MJ, Hartmaier A, Gao H, et al. Atomic plasticity: description and analysis of a one-billion atom simulation of ductile materials failure. Computer Methods in Applied Mechanics and Engineering, 2004,193(48-51): 5257-5282
[117] Shilkrot LE, Miller RE, Curtin WA. Multiscale plasticity modeling: coupled atomistics and discrete dislocation mechanics. Journal of the Mechanics and Physics of Solids, 2004,52(4): 755-787
[118] Amodeo RJ, Ghoniem NM. Dislocation dynamics. I. A proposed methodology for deformation micromechanics. Physical Review B, 1990,41(10): 6958-6967
[119] Kubin LP, Canova G. The modelling of dislocation patterns. Scripta Metallurgica et Materialia, 1992,27(8): 957-962
[120] Kubin LP, Canova G, Condat M. Dislocation microstructures and plastic flow: a 3D simulation. Solid State Phenomena, 1992,23-24: 455-472
[121] Zbib HM, Rhee M, Hirth JP. On plastic deformation and the dynamics of 3D dislocations. International Journal of Mechanical Sciences, 1998,40(2-3): 113-127
[122] Ghoniem NM, Tong SH, Sun LZ. Parametric dislocation dynamics: A thermodynamics-based approach to investigations of mesoscopic plastic deformation. Physical Review B, 2000,61(2): 913-927
[123] Van der Giessen E, Needleman A. discrete dislocation plasticity: A simple planar model. Modelling and Simulation in Materials Science and Engineering, 1995,3: 689-735
[124] Fivel M, Verdier M, Canova G. 3D simulation of a nanoindentation test at a mesoscopic scale. Materials Science and Engineering A, 1997,234-236: 923-926
[125] Weygand D, Freiedman LH, Van der Giessen E. Aspects of boundary-value problem solutions with three-dimensional dislocation dynamics. Modelling and Simulation in Materials Science and Engineering, 2002,10(4): 437-468
[126] Lemarchand C, Devincre B, Kubin LP. Homogenization method for a discrete-continuum simulation of dislocation dynamics. Journal of the Mechanics and Physics of Solids, 2001,49(9): 1969-1982
[127] Brechet YJM, Canova GR, Kubin LP. Static versus propagative plastic strain localisations. Scripta Metallurgica et Materialia, 1993,29(9): 1165-1170
[128] Brechet Y, Canova G, Kubin LP. Strain softening, slip localization and propagation: From simulations to continuum modelling. Acta Materialia, 1996,44(11): 4261-4271
[129] Tang M, Kubin LP, Canova GR. Dislocation mobility and the mechanical response of b.c.c. single crystals: A mesoscopic approach. Acta Materialia, 1998,46(9): 3221-3235
[130] Tang M, Fivel M, Kubin LP. From forest hardening to strain hardening in body centered cubic single crystals: simulation and modeling. Materials Science and Engineering A, 2001,309-310: 256-260
[131] Zbib HM, Diaz de la Rubia T. A multiscale model of plasticity. International Journal of Plasticity, 2002,18(9): 1133-1163
[132] Ghoniem NM, Tong SH, Huang J, et al. Mechanisms of dislocation-defect interactions in irradiated metals investigated by computer simulations. Journal of Nuclear Materials, 2002,307-311 (Part 2): 843-851
[133] Cleveringa HHM, Van der Giessen E, Needleman A. A discrete dislocation analysis of bending. International Journal of Plasticity, 1999,15(8): 837-868
[134] Deshpande VS, Needleman A, Van der Giessen E. Discrete dislocation plasticity analysis of static friction. Acta Materialia, 2004,52(10): 3135-3149
[135] Fivel MC, Robertson CF, Canova GR, et al. Three-dimensional modeling of indent-induced plastic zone at a mesoscale. Acta Materialia, 1998,46(17): 6183-6194
[136] Leger RW, Khraishi TA, Shen YL. A dislocation dynamics study of strength differential in particle-containing metals during cyclic loading. Journal of Materials Science, 2004,39(11): 3593-3604
[137] Arsenlis A, Cai W. Enabling strain hardening simulations with dislocation dynamics. Modelling and Simulation in Materials Science and Engineering, 2007,15(6): 553-595
[138] Devincre B, Kubin LP, Lemarchand C, et al. Mesoscopic simulations of plastic deformation. Materials Science and Engineering A, 2001,309-310: 211-219
[139] Benzerga AA, Brechet Y, Needleman A, et al. Incorporating three-dimensional mechanisms into two-dimensional dislocation dynamics. Modelling and Simulation in Materials Science and Engineering, 2004,12(1): 159-196
[140] Balint DS, Deshpande VS, Needleman A, et al. A discrete dislocation plasticity analysis of grain-size strengthening. Materials Science and Engineering A, 2005,400-401: 186-190
[141] Biner SB, Morris JR. A two-dimensional discrete dislocation simulation of the effect of grain size on strengthening behavior. Modelling and Simulation in Materials Science and Engineering, 2002,10(6): 617-635
[142] Biner SB, Morris JR. The effects of grain size and dislocation source density on the strengthening behaviour of polycrystals:a two-demensional discrete dislocation simulation. Philosophical Magazine, 2003,83: 3677-3690
[143] Nicola L, Van der Giessen E, Needleman A. Size effects in polycrystalline thin films analyzed by discrete dislocation plasticity. Thin Solid Films, 2005,479(1-2): 329-338
[144] Shi MX, Huang Y, Gao H. The J-integral and geometrically necessary dislocations in nonuniform plastic deformation. International Journal of Plasticity, 2004,20(8-9): 1739-1762
[145] Nicola L, Van der Giessen E, Needleman A. Discrete dislocation analysis of size effects in thin films. Journal of Applied Physics, 2003,93(10): 5920-5928
[146] Nicola L, Xiang Y, Vlassak JJ, et al. Plastic deformation of freestanding thin films: Experiments and modeling. Journal of the Mechanics and Physics of Solids, 2006,54(10): 2089-2110
[147] Chaboche JL. Time-independent constitutive theories for cyclic plasticity. International Journal of Plasticity, 1986,2(2): 149-188
[148] Ohno N, Wang JD. Transformation of a nonlinear kinematic hardening rule to a multisurface form under isothermal and nonisothermal conditions. International Journal of Plasticity, 1991,7(8): 879-891
[149] Besseling JF. A theory of elastic, plastic and creep deformations of an initially isotropic material showing anisotropic strain hardening, creep recovery and secondary creep. Journal of Applied Mechanics, 1958,25: 529-536
[150] Lefebvre S, Devincre B, Hoc T. Simulation of the Hall-Petch effect in ultra-fine grained copper. Intenational Conference on Fundamentals of Plastic Deformation. La Colle sur Loup, FRANCE: Elsevier Science Sa, 2004:150-153.
[151] Lefebvre S, Devincre B, Hoc T. Yield stress strengthening in ultrafine-grained metals: A two-dimensional simulation of dislocation dynamics. Journal of the Mechanics and Physics of Solids, 2007,55(4): 788-802
[152] Balint DS, Deshpande VS, Needleman A, et al. Discrete dislocation plasticity analysis of the grain size dependence of the flow strength of polycrystals. International Journal of Plasticity, 2008,24(12): 2149-2172
[153] Shen Z, Wagoner RH, Clark WAT. Dislocation pile-up and grain boundary interactions in 304 stainless steel. Scripta Metallurgica 1986,20(6): 921-926
[154] Shen Z, Wagoner RH, Clark WAT. Dislocation and grain boundary interactions in metals. Acta Metallurgica, 1988,36(12): 3231-3242
[155] Brentnall WD, Rostoker W. Some observations on microyielding. Acta Metallurgica, 1965,13(3): 187-198
[156] Hauser JJ, Chalmers B. The plastic deformation of bicrystals of f.c.c. metals. Acta Metallurgica, 1961,9(9): 802-818
[157] Worthington PJ, Smith E. The formation of slip bands in polycrystalline 3% silicon iron in the pre-yield microstrain region. Acta Metallurgica, 1964,12(11): 1277-1281
[158] Carrington WE, McLean D. Slip nuclei in silicon-iron. Acta Metallurgica, 1965,13(5): 493-499
[159] Livingston JD, Chalmers B. Multiple slip in bicrystal deformation. Acta Metallurgica, 1957,5(6): 322-327
[160] Lee TC, Robertson IM, Birnbaum HK. Prediction of slip transfer mechanisms across grain boundaries. Scripta Metallurgica, 1989,23(5): 799-803
[161] Lee TC, Robertson IM, Birnbaum HK. TEM in situ deformation study of the interaction of lattice dislocations with grain boundaries in metals. Philosophical Magazine, 1990,62(1): 131 - 153
[162] Clark WAT, Wagoner RH, Shen ZY, et al. On the criteria for slip transmission across interfaces in polycrystals. Scripta Metallurgica et Materialia, 1992,26(2): 203-206
[163] de Koning M, Kurtz RJ, Bulatov VV, et al. Modeling of dislocation-grain boundary interactions in FCC metals. Journal of Nuclear Materials, 2003,323(2-3): 281-289
[164] de Koning M, Miller R, Bulatov VV, et al. Modelling grain-boundary resistance in intergranular dislocation slip transmission. Philosophical Magazine a-Physics of Condensed Matter Structure Defects and Mechanical Properties, 2002,82(13): 2511-2527
[165] Cheng Y, Mrovec M, Gumbsch P. Atomistic simulations of interactions between the 1/2(111) edge dislocation and symmetric tilt grain boundaries in tungsten. Philosophical Magazine, 2008,88(4): 547-560
[166] Dewald MP, Curtin WA. Multiscale modelling of dislocation/grain-boundary interactions: I. Edge dislocations impinging on Sigma 11 (113) tilt boundary in Al. IUTAM Symposium on Plasticity at the Micron Scale. Lyngby, DENMARK: Iop Publishing Ltd, 2006:S193-S215.
[167] Li Z, Hou C, Huang M, et al. Strengthening mechanism in micro-polycrystals with penetrable grain boundaries by discrete dislocation dynamics simulation and Hall - Petch effect. Computational Materials Science, Accepted for publication:
[168] Hasson GC, Goux C. Interfacial energies of tilt boundaries in aluminium. Experimental and theoretical determination. Scripta Metallurgica, 1971,5(10): 889-894
[169] Kato M, Fujii T, Onaka S. Dislocation Bow-Out Model for Yield Stress of Ultra-Fine Grained Materials. Materials Transactions, 2008,49(6): 1278-1283
[170] Hussein MI, Borg U, Niordson CF, et al. Plasticity size effects in voided crystals. Journal of the Mechanics and Physics of Solids, 2008,56(1): 114-131
[171] Greengard L, Rokhlin V. A fast algorithm for particle simulations. Journal of Computational Physics, 1987,73(2): 325-348
[2] Geers MGD, Brekelmans WAM, Bayley CJ. Second-order crystal plasticity: internal stress effects and cyclic loading. Modelling and Simulation in Materials Science and Engineering, 2007,15(1): S133-S145
[3] Atkinson M. Further analysis of the size effect in indentation hardness tests of some metals. Journal of Materials Research, 1995,10(11): 2908-2915
[4] Ma Q, Clarke DR. Size dependent hardness of silver single crystals. Journal of Materials Research, 1995,10(4): 853-863
[5] Poole WJ, Ashby MF, Fleck NA. Micro-hardness of annealed and work-hardened copper polycrystals. Scripta Materialia, 1996,34(4): 559-564
[6] Nix WD. Elastic and plastic properties of thin films on substrates: nanoindentation techniques. 11th International Conference on the Strength of Materials (ICSMA-11). Prague, Czech Republic: Elsevier Science Sa Lausanne, 1997:37-44.
[7] McElhaney KW, Nix WD, Vlassak JJ. Determination of indenter tips geometry and indentation contact area for depth-sensing indentation experiments. Journal of Materials Research, 1998,13(5): 1300-1306
[8] Suresh S, Nieh TG, Choi BW. Nano-indentation of copper thin films on silicon substrates. Scripta Materialia, 1999,41(9): 951-957
[9] Tymiak NI, Kramer DE, Bahr DF, et al. Plastic strain and strain gradients at very small indentation depths. Acta Materialia, 2001,49(6): 1021-1034
[10] Wei Y, Hutchinson JW. Hardness trends in micron scale indentation. Journal of the Mechanics and Physics of Solids, 2003,51(11-12): 2037-2056
[11] Abu Al-Rub RK, Voyiadjis GZ. Analytical and experimental determination of the material intrinsic length scale of strain gradient plasticity theory from micro- and nano-indentation experiments. International Journal of Plasticity, 2004,20(6): 1139-1182
[12] Fleck NA, Muller GM, Ashby MF, et al. Strain gradient plasticity: Theory and experiment. Acta Metallurgica et Materialia, 1994,42(2): 475-487
[13] Stolken JS, Evans AG. A microbend test method for measuring the plasticity length scale. Acta Materialia, 1998,46(14): 5109-5115
[14] Wang W, Huang Y, Hsia KJ, et al. A study of microbend test by strain gradient plasticity. International Journal of Plasticity, 2003,19(3): 365-382
[15] Haque MA, Saif MTA. Strain gradient effect in nanoscale thin films. Acta Materialia, 2003,51(11): 3053-3061
[16] Espinosa HD, Prorok BC, Peng B. Plasticity size effects in free-standing submicron polycrystalline FCC films subjected to pure tension. Journal of the Mechanics and Physics of Solids, 2004,52(3): 667-689
[17] Nye JF. Some geometrical relations in dislocated crystals. Acta Metallurgica, 1953,1(2): 153-162
[18] Ashby MF. The deformation of plastically non-homogeneous materials. Philosophical Magazine, 1970,21(170): 399 - 424
[19] Arzt E. Size effects in materials due to microstructural and dimensional constraints: a comparative review. Acta Materialia, 1998,46(16): 5611-5626
[20] Gao H, Huang Y. Geometrically necessary dislocation and size-dependent plasticity. Scripta Materialia, 2003,48(2): 113-118
[21] Xiang Y, Vlassak JJ. Bauschinger and size effects in thin-film plasticity. Acta Materialia, 2006,54(20): 5449-5460
[22] Fleck NA, Hutchinson JW. A phenomenological theory for strain gradient effects in plasticity. Journal of the Mechanics and Physics of Solids, 1993,41(12): 1825-1857
[23] Fleck NA, Hutchinson JW. Strain gradient plasticity. Advances in Applied Mechanics, Vol 33. San Diego: Academic Press Inc, 1997:295-361.
[24] Shu JY, Fleck NA. Strain gradient crystal plasticity: size-dependentdeformation of bicrystals. Journal of the Mechanics and Physics of Solids, 1999,47(2): 297-324
[25] Nix WD, Gao H. Indentation size effects in crystalline materials: A law for strain gradient plasticity. Journal of the Mechanics and Physics of Solids, 1998,46(3): 411-425
[26] Gao H, Huang Y, Nix WD, et al. Mechanism-based strain gradient plasticity— I. Theory. Journal of the Mechanics and Physics of Solids, 1999,47(6): 1239-1263
[27] Huang Y, Gao H, Nix. WD, et al. Mechanism-based strain gradient plasticity-II. Analysis. Journal of the Mechanics and Physics of Solids, 2000,48(1): 99-128
[28] Jiang H, Huang Y, Zhuang Z, et al. Fracture in mechanism-based strain gradient plasticity. Journal of the Mechanics and Physics of Solids, 2001,49(5): 979-993
[29] Aifantis EC. on the microstructural origin of certain inelastic models. Journal of Engineering Materials and Technology, 1984,106(44): 326-330
[30] Aifantis EC. Strain gradient interpretation of size effects. International Journal of Fracture, 1999,95(1-4): 299-314
[31] Taylor MB, Zbib HM, Khaleel MA. Damage and size effect during superplastic deformation. International Journal of Plasticity, 2002,18(3): 415-442
[32] Acharya A, Bassani JL. Lattice incompatibility and a gradient theory of crystal plasticity. Journal of the Mechanics and Physics of Solids, 2000,48(8): 1565-1595
[33] Evers LP, Parks DM, Brekelmans WAM, et al. Crystal plasticity model with enhanced hardening by geometrically necessary dislocation accumulation. Journal of the Mechanics and Physics of Solids, 2002,50(11): 2403-2424
[34] Tvergaard V, Niordson C. Nonlocal plasticity effects on interaction of different size voids. International Journal of Plasticity, 2004,20(1): 107-120
[35] Gurtin ME. On the plasticity of single crystals: free energy, microforces, plastic-strain gradients. Journal of the Mechanics and Physics of Solids, 2000,48(5): 989-1036
[36] Gurtin ME. A gradient theory of single-crystal viscoplasticity that accounts for geometrically necessary dislocations. Journal of the Mechanics and Physics of Solids, 2002,50(1): 5-32
[37] Gurtin ME. On a framework for small-deformation viscoplasticity: free energy, microforces, strain gradients. International Journal of Plasticity, 2003,19(1): 47-90
[38] Evers LP, Brekelmans WAM, Geers MGD. Scale dependent crystal plasticity framework with dislocation density and grain boundary effects. International Journal of Solids and Structures, 2004,41(18-19): 5209-5230
[39] Uchic MD, Dimiduk DM, Florando JN, et al. Sample dimensions influence strength and crystal plasticity. Science, 2004,305(5686): 986-989
[40] Dimiduk DM, Uchic MD, Parthasarathy TA. Size-affected single-slip behavior of pure nickel microcrystals. Acta Materialia, 2005,53(15): 4065-4077
[41] Greer JR, Oliver WC, Nix WD. Size dependence of mechanical properties of gold at the micron scale in the absence of strain gradients. Acta Materialia, 2005,53(6): 1821-1830
[42] Greer JR, Nix WD. Size dependence of mechanical properties of gold at the sub-micron scale. Applied Physics a-Materials Science & Processing, 2005,80(8): 1625-1629
[43] Greer JR, Nix WD. Nanoscale gold pillars strengthened through dislocation starvation. Physical Review B, 2006,73(24): 6
[44] Nix WD, Greer JR, Feng G, et al. Deformation at the nanometer and micrometer length scales: Effects of strain gradients and dislocation starvation. Thin Solid Films, 2007,515(6): 3152-3157
[45] Deshpande VS, Needleman A, Van der Giessen E. Plasticity size effects in tension and compression of single crystals. Journal of the Mechanics and Physics of Solids, 2005,53(12): 2661-2691
[46] Benzerga AA, Shaver NF. Scale dependence of mechanical properties of single crystals under uniform deformation. Scripta Materialia, 2006,54(11): 1937-1941
[47] Tang H, Schwarz KW, Espinosa HD. Dislocation escape-related size effects in single-crystal micropillars under uniaxial compression. Acta Materialia, 2007,55(5): 1607-1616
[48] Hasegawa T, Yakou T, Kocks UF. Forward and reverse rearrangements of dislocations in tangled walls. Materials Science and Engineering, 1986,81: 189-199
[49] Margolin H, Hazaveh F, Yaguchi H. The grain boundary contribution to the bauschinger effect. Scripta Metallurgica, 1978,12(12): 1141-1145
[50] Ono N, Tsuchikawa T, Nishimura S, et al. Intergranular constraint and the Bauschinger effect. Materials Science and Engineering, 1983,59(2): 223-233
[51] Xiang Y, Vlassak JJ. Bauschinger effect in thin metal films. Scripta Materialia, 2005,53(2): 177-182
[52] Brown LM, Stobbs WM. The work-hardening of copper-silica. Philosophical Magazine, 1971,23(185): 1185-1199
[53] Atkinson JD, Brown LM, Stobbs WM. The work-hardening of copper-silica: IV. The Bauschinger effect and plastic relaxation. Philosophical Magazine, 1974,30(6): 1247 - 1280
[54] Aran A, Demirkol M, Karabulut A. Bauschinger effect in precipitation-strengthened aluminium alloy 2024. Materials Science and Engineering, 1987,89: L35-L39
[55] Stoltz R, Pelloux R. The Bauschinger effect in precipitation strengthened aluminum alloys. Metallurgical and Materials Transactions A, 1976,7(8): 1295-1306
[56] Ham RK, Broom T. The mechanism of fatigue softening. Philosophical Magazine, 1962,7(73): 95-103
[57] Winter AT. The effect of work-hardening on the low strain amplitude fatigue of copper crystals. Philosophical Magazine, 1975,31(2): 411 - 417
[58] Mughrabi H. The cyclic hardening and saturation behaviour of copper single crystals. Materials Science and Engineering, 1978,33(2): 207-223
[59] Yan B, Laird C. Matrix hardening behavior and the nucleation stress for persistent slip bands in fatigued monocrystalline copper. Materials Science and Engineering, 1986,80(1): 59-64
[60] Gong B, Wang Z, Chen D, et al. Investigation of macro deformation bands in fatigued [001] Cu single crystals by electron channeling contrast technique. Scripta Materialia, 1997,37(10): 1605-1610
[61] Figueroa JC, Bhat SP, De La Veaux R, et al. The cyclic stress-strain response of copper at low strains—i. Constant amplitude testing. Acta Metallurgica, 1981,29(10): 1667-1678
[62] Wang Z, Laird C. Cyclic stress—strain response of polycrystalline copper under fatigue conditions producing enhanced strain localization. Materials Science and Engineering, 1988,100: 57-68
[63] Wang Z, Sun Z, Laird C. Effect of initial ramp loading on fatigue behavior in polycrystalline copper. Materials Science and Engineering: A, 1992,151(2): 121-130
[64] Morrison DJ. Influence of grain size and texture on the cyclic stress-strain response of nickel. Materials Science and Engineering: A, 1994,187(1): 11-21
[65] El-Madhoun Y, Mohamed A, Bassim MN. Cyclic stress-strain response and dislocation structures in polycrystalline aluminum. Materials Science and Engineering A, 2003,359(1-2): 220-227
[66] Li Y, Laird C. Cyclic response and dislocation structures of AISI 316L stainless steel. Part 2: polycrystals fatigued at intermediate strain amplitude. Materials Science and Engineering: A, 1994,186(1-2): 87-103
[67] Feltner CE, Laird C. Cyclic stress-strain response of F.C.C. metals and alloys—I Phenomenological experiments. Acta Metallurgica, 1967,15(10): 1621-1632
[68] Hancock JR, Grosskreutz JC. Mechanisms of fatigue hardening in copper single crystals. Acta Metallurgica, 1969,17(2): 77-97
[69] Woods PJ. Low-amplitude fatigue of copper and copper-5 at. % aluminium single crystals. Philosophical Magazine, 1973,28(1): 155 - 191
[70] Finney JM, Laird C. Strain localization in cyclic deformation of copper single crystals. Philosophical Magazine, 1975,31(2): 339 - 366
[71] Kuhlmann-Wilsdorf D. Dislocation behavior in fatigue IV. Quantitative interpretation of friction stress and back stress derived from hysteresis loops. Materials Science and Engineering, 1979,39(2): 231-245
[72] Blochwitz C, Veit U. Plateau Behaviour of Fatigued FCC Single Crystals. Crystal Research and Technology, 1982,17(5): 529-551
[73] Laird C, Buchinger L. Hardening behavior in fatigue. Metallurgical and Materials Transactions A, 1985,16(12): 2201-2214
[74] Laird C, Charsley P, Mughrabi H. Low energy dislocation structures produced by cyclic deformation. Materials Science and Engineering, 1986,81: 433-450
[75] Basinski ZS, Basinski SJ. Fundamental aspects of low amplitude cyclic deformation in face-centred cubic crystals. Progress in Materials Science, 1992,36: 89-148
[76] Feltner CE. A debris mechanism of cyclic strain hardening for F.C.C. metals. Philosophical Magazine, 1965,12(120): 1229 - 1248
[77] Bhat SP, Laird C. The cyclic stress-strain curves in monocrystalline and polycrystalline metals. Scripta Metallurgica, 1978,12(8): 687-692
[78] Rasmussen KV, Pedersen OB. Fatigue of copper polycrystals at low plastic strain amplitudes. Acta Metallurgica, 1980,28(11): 1467-1478
[79] Kuokkala VT, Kettunen P. Cyclic stress-strain response of polycrystalline copper in constant and variable amplitude fatigue. Acta Metallurgica, 1985,33(11): 2041-2047
[80] Mughrabi H. Plateaus in the cyclic stress-strain curves of single- and polycrystalline metals. Scripta Metallurgica, 1979,13(6): 479-484
[81] Lukas P, Kunz L. Is there a plateau in the cyclic stress-strain curves of polycrystalline copper? Materials Science and Engineering, 1985,74(1): L1-L5
[82] Peralta P, Llanes L, Czapka A, et al. Effect of texture and grain size as independent factors in the cyclic behavior of polycrystalline copper. Scripta Metallurgica et Materialia, 1995,32(11): 1877-1881
[83] Llanes L, Rollett AD, Laird C, et al. Effect of grain size and annealing texture on the cyclic response and the substructure evolution of polycrystalline copper. Acta Metallurgica et Materialia, 1993,41(9): 2667-2679
[84] Liu CD, You DX, Bassim MN. Cyclic strain hardening in polycrystalline copper. Acta Metallurgica et Materialia, 1994,42(5): 1631-1638
[85] Morrison DJ, Chopra V. Cyclic stress-strain response of polycrystalline nickel. Materials Science and Engineering: A, 1994,177(1-2): 29-42
[86] Boutin J, Marchand N, Bailon JP, et al. An intermediate plateau in the cyclic stress-strain curve of [alpha] brass. Materials Science and Engineering, 1984,67(2): L23-L27
[87] Morrison DJ, Chopra V, Jones JW. Effects of grain size on cyclic strain localization in polycrystalline nickel. Scripta Metallurgica et Materialia, 1991,25(6): 1299-1304
[88] Lukas P, Kunz L. Effect of grain size on the high cycle fatigue behaviour of polycrystalline copper. Materials Science and Engineering, 1987,85: 67-75
[89] Liang FL, Laird C. Control of intergranular fatigue cracking by slip homogeneity in copper I: Effect of grain size. Materials Science and Engineering: A, 1989,117: 95-102
[90] Llanes L, Laird C. Effect of grain size and ramp loading on the low amplitude cyclic stress-strain curve of polycrystalline copper. Materials Science and Engineering: A, 1990,128(2): L1-L4
[91] Mughrabi H. Dislocation wall and cell structures and long-range internal stresses in deformed metal crystals. Acta Metallurgica, 1983,31(9): 1367-1379
[92] Calabrese C, Laird C. Cyclic stress—strain response of two-phase alloys Part I. Microstructures containing particles penetrable by dislocations. Materials Science and Engineering, 1974,13(2): 141-157
[93] Laird C, Wang Z, Ma BT, et al. Low energy dislocation structures produced by cyclic softening. Materials Science and Engineering: A, 1989,113: 245-257
[94] Nian L, Bai-ping D. The effect of low-stress high-cycle fatigue on themicrostructure and fatigue threshold of a 40Cr steel. International Journal of Fatigue, 1995,17(1): 43-48
[95] Lamba HS, Sidebottom OM. Cyclic plasticity for nonproportional paths. Journal of Engineering Materials and Technology, Transactions of the ASME, 1978,100(1): 96-111
[96] Doong S-H, Socie DF, Robertson IM. Dislocation Substructures and Nonproportional Hardening. Journal of Engineering Materials and Technology, 1990,112(4): 456-464
[97] Cailletaud G, Doquet V, Pineau A. Cyclic multiaxial behavior of an austenitic stainless steel microstructural observations and macromechanical modeling. Fatigue under Biaxial and Multiaxial Loading London: ESIS Publication, 1991:131-149.
[98] Itoh, Sakane M, Ohnami M. Effect of stacking fault energy on cyclic constitutive relation under nonproportional loading. Journal of the Society of Materials Science, 1992,41(468): 1361-1367
[99] Kanazawa K, Miller KJ, Brown MW. Low cycle fatigue under out of phase loading conditions. Journal of Engineering Materials and Technology, 1977,99: 222-228
[100] Mroz Z. On the description of anisotropic workhardening. Journal of the Mechanics and Physics of Solids, 1967,15(3): 163-175
[101] Krieg RD. A practical two surface plasticity theory. Journal of Applied Mechanics 1975,42: 641-646
[102] Dafalias YF, Popov EP. Plastic internal variables formalism of cyclic plasticity. Journal of Applied Mechanics, 1976,43: 645-651
[103] Tseng NT, Lee GC. Simple Plasticity Model of Two-Surface Type. Journal of Engineering Mechanics, 1983,109(3): 795-810
[104] McDowell DL. A two surface model for transient nonproportional cyclic plasticity. Journal of Applied Mechanics, 1985,54: 323-334
[105] Ellyin F, Xia Z. A rate-independent constitutive model for transient non-proportional loading. Journal of the Mechanics and Physics of Solids, 1989,37(1): 71-91
[106] Ohno N, Takahashi Y, Kuwabara K. Constitutive modeling of anisothermal cyclic plasticity of 304 stainless steel. Journal of Engineering Materials and Technology, 1989,111: 106-113
[107] Jiang Y, Sehitoglu H. Multiaxial cyclic ratchetting under multiple step loading. International Journal of Plasticity, 1994,10(8): 849-870
[108] Armstrong PJ, Frederick CO. A Mathematical Representation of the Multiaxial Bauschinger Effect. CEGB Report RD/B/N731: Berkeley Nuclear Laboratories, 1966.
[109] Chaboche JL, Nouailhas D, Pacou D, et al. Modeling of the cyclic response and ratcheting effects on inconel 718 alloy. European Journal of Mechanics - A/Solids, 1991,10: 101-112
[110] Ohno N, Wang JD. Kinematic hardening rules with critical state of dynamic recovery, part I: formulation and basic features for ratchetting behavior. International Journal of Plasticity, 1993,9(3): 375-390
[111] Abdel-Karim M, Ohno N. Kinematic hardening model suitable for ratchetting with steady-state. International Journal of Plasticity, 2000,16(3-4): 225-240
[112] Delobelle P, Robinet P, Bocher L. Experimental study and phenomenological modelization of ratchet under uniaxial and biaxial loading on an austenitic stainless steel. International Journal of Plasticity, 1995,11(4): 295-330
[113] Burlet H, Cailletaud G. numerical techniques for cyclic plasticity at variable temperature. Engineering Computations, 1986,3(2): 143-153
[114] Alder BJ, Wainwright TE. Phase transition for a hard sphere system. J Chem Phys, 1957,27(5): 1208-1209
[115] Needleman A. Computational mechanics at the mesoscale. Acta Materialia, 2000,48(1): 105-124
[116] Buehler MJ, Hartmaier A, Gao H, et al. Atomic plasticity: description and analysis of a one-billion atom simulation of ductile materials failure. Computer Methods in Applied Mechanics and Engineering, 2004,193(48-51): 5257-5282
[117] Shilkrot LE, Miller RE, Curtin WA. Multiscale plasticity modeling: coupled atomistics and discrete dislocation mechanics. Journal of the Mechanics and Physics of Solids, 2004,52(4): 755-787
[118] Amodeo RJ, Ghoniem NM. Dislocation dynamics. I. A proposed methodology for deformation micromechanics. Physical Review B, 1990,41(10): 6958-6967
[119] Kubin LP, Canova G. The modelling of dislocation patterns. Scripta Metallurgica et Materialia, 1992,27(8): 957-962
[120] Kubin LP, Canova G, Condat M. Dislocation microstructures and plastic flow: a 3D simulation. Solid State Phenomena, 1992,23-24: 455-472
[121] Zbib HM, Rhee M, Hirth JP. On plastic deformation and the dynamics of 3D dislocations. International Journal of Mechanical Sciences, 1998,40(2-3): 113-127
[122] Ghoniem NM, Tong SH, Sun LZ. Parametric dislocation dynamics: A thermodynamics-based approach to investigations of mesoscopic plastic deformation. Physical Review B, 2000,61(2): 913-927
[123] Van der Giessen E, Needleman A. discrete dislocation plasticity: A simple planar model. Modelling and Simulation in Materials Science and Engineering, 1995,3: 689-735
[124] Fivel M, Verdier M, Canova G. 3D simulation of a nanoindentation test at a mesoscopic scale. Materials Science and Engineering A, 1997,234-236: 923-926
[125] Weygand D, Freiedman LH, Van der Giessen E. Aspects of boundary-value problem solutions with three-dimensional dislocation dynamics. Modelling and Simulation in Materials Science and Engineering, 2002,10(4): 437-468
[126] Lemarchand C, Devincre B, Kubin LP. Homogenization method for a discrete-continuum simulation of dislocation dynamics. Journal of the Mechanics and Physics of Solids, 2001,49(9): 1969-1982
[127] Brechet YJM, Canova GR, Kubin LP. Static versus propagative plastic strain localisations. Scripta Metallurgica et Materialia, 1993,29(9): 1165-1170
[128] Brechet Y, Canova G, Kubin LP. Strain softening, slip localization and propagation: From simulations to continuum modelling. Acta Materialia, 1996,44(11): 4261-4271
[129] Tang M, Kubin LP, Canova GR. Dislocation mobility and the mechanical response of b.c.c. single crystals: A mesoscopic approach. Acta Materialia, 1998,46(9): 3221-3235
[130] Tang M, Fivel M, Kubin LP. From forest hardening to strain hardening in body centered cubic single crystals: simulation and modeling. Materials Science and Engineering A, 2001,309-310: 256-260
[131] Zbib HM, Diaz de la Rubia T. A multiscale model of plasticity. International Journal of Plasticity, 2002,18(9): 1133-1163
[132] Ghoniem NM, Tong SH, Huang J, et al. Mechanisms of dislocation-defect interactions in irradiated metals investigated by computer simulations. Journal of Nuclear Materials, 2002,307-311 (Part 2): 843-851
[133] Cleveringa HHM, Van der Giessen E, Needleman A. A discrete dislocation analysis of bending. International Journal of Plasticity, 1999,15(8): 837-868
[134] Deshpande VS, Needleman A, Van der Giessen E. Discrete dislocation plasticity analysis of static friction. Acta Materialia, 2004,52(10): 3135-3149
[135] Fivel MC, Robertson CF, Canova GR, et al. Three-dimensional modeling of indent-induced plastic zone at a mesoscale. Acta Materialia, 1998,46(17): 6183-6194
[136] Leger RW, Khraishi TA, Shen YL. A dislocation dynamics study of strength differential in particle-containing metals during cyclic loading. Journal of Materials Science, 2004,39(11): 3593-3604
[137] Arsenlis A, Cai W. Enabling strain hardening simulations with dislocation dynamics. Modelling and Simulation in Materials Science and Engineering, 2007,15(6): 553-595
[138] Devincre B, Kubin LP, Lemarchand C, et al. Mesoscopic simulations of plastic deformation. Materials Science and Engineering A, 2001,309-310: 211-219
[139] Benzerga AA, Brechet Y, Needleman A, et al. Incorporating three-dimensional mechanisms into two-dimensional dislocation dynamics. Modelling and Simulation in Materials Science and Engineering, 2004,12(1): 159-196
[140] Balint DS, Deshpande VS, Needleman A, et al. A discrete dislocation plasticity analysis of grain-size strengthening. Materials Science and Engineering A, 2005,400-401: 186-190
[141] Biner SB, Morris JR. A two-dimensional discrete dislocation simulation of the effect of grain size on strengthening behavior. Modelling and Simulation in Materials Science and Engineering, 2002,10(6): 617-635
[142] Biner SB, Morris JR. The effects of grain size and dislocation source density on the strengthening behaviour of polycrystals:a two-demensional discrete dislocation simulation. Philosophical Magazine, 2003,83: 3677-3690
[143] Nicola L, Van der Giessen E, Needleman A. Size effects in polycrystalline thin films analyzed by discrete dislocation plasticity. Thin Solid Films, 2005,479(1-2): 329-338
[144] Shi MX, Huang Y, Gao H. The J-integral and geometrically necessary dislocations in nonuniform plastic deformation. International Journal of Plasticity, 2004,20(8-9): 1739-1762
[145] Nicola L, Van der Giessen E, Needleman A. Discrete dislocation analysis of size effects in thin films. Journal of Applied Physics, 2003,93(10): 5920-5928
[146] Nicola L, Xiang Y, Vlassak JJ, et al. Plastic deformation of freestanding thin films: Experiments and modeling. Journal of the Mechanics and Physics of Solids, 2006,54(10): 2089-2110
[147] Chaboche JL. Time-independent constitutive theories for cyclic plasticity. International Journal of Plasticity, 1986,2(2): 149-188
[148] Ohno N, Wang JD. Transformation of a nonlinear kinematic hardening rule to a multisurface form under isothermal and nonisothermal conditions. International Journal of Plasticity, 1991,7(8): 879-891
[149] Besseling JF. A theory of elastic, plastic and creep deformations of an initially isotropic material showing anisotropic strain hardening, creep recovery and secondary creep. Journal of Applied Mechanics, 1958,25: 529-536
[150] Lefebvre S, Devincre B, Hoc T. Simulation of the Hall-Petch effect in ultra-fine grained copper. Intenational Conference on Fundamentals of Plastic Deformation. La Colle sur Loup, FRANCE: Elsevier Science Sa, 2004:150-153.
[151] Lefebvre S, Devincre B, Hoc T. Yield stress strengthening in ultrafine-grained metals: A two-dimensional simulation of dislocation dynamics. Journal of the Mechanics and Physics of Solids, 2007,55(4): 788-802
[152] Balint DS, Deshpande VS, Needleman A, et al. Discrete dislocation plasticity analysis of the grain size dependence of the flow strength of polycrystals. International Journal of Plasticity, 2008,24(12): 2149-2172
[153] Shen Z, Wagoner RH, Clark WAT. Dislocation pile-up and grain boundary interactions in 304 stainless steel. Scripta Metallurgica 1986,20(6): 921-926
[154] Shen Z, Wagoner RH, Clark WAT. Dislocation and grain boundary interactions in metals. Acta Metallurgica, 1988,36(12): 3231-3242
[155] Brentnall WD, Rostoker W. Some observations on microyielding. Acta Metallurgica, 1965,13(3): 187-198
[156] Hauser JJ, Chalmers B. The plastic deformation of bicrystals of f.c.c. metals. Acta Metallurgica, 1961,9(9): 802-818
[157] Worthington PJ, Smith E. The formation of slip bands in polycrystalline 3% silicon iron in the pre-yield microstrain region. Acta Metallurgica, 1964,12(11): 1277-1281
[158] Carrington WE, McLean D. Slip nuclei in silicon-iron. Acta Metallurgica, 1965,13(5): 493-499
[159] Livingston JD, Chalmers B. Multiple slip in bicrystal deformation. Acta Metallurgica, 1957,5(6): 322-327
[160] Lee TC, Robertson IM, Birnbaum HK. Prediction of slip transfer mechanisms across grain boundaries. Scripta Metallurgica, 1989,23(5): 799-803
[161] Lee TC, Robertson IM, Birnbaum HK. TEM in situ deformation study of the interaction of lattice dislocations with grain boundaries in metals. Philosophical Magazine, 1990,62(1): 131 - 153
[162] Clark WAT, Wagoner RH, Shen ZY, et al. On the criteria for slip transmission across interfaces in polycrystals. Scripta Metallurgica et Materialia, 1992,26(2): 203-206
[163] de Koning M, Kurtz RJ, Bulatov VV, et al. Modeling of dislocation-grain boundary interactions in FCC metals. Journal of Nuclear Materials, 2003,323(2-3): 281-289
[164] de Koning M, Miller R, Bulatov VV, et al. Modelling grain-boundary resistance in intergranular dislocation slip transmission. Philosophical Magazine a-Physics of Condensed Matter Structure Defects and Mechanical Properties, 2002,82(13): 2511-2527
[165] Cheng Y, Mrovec M, Gumbsch P. Atomistic simulations of interactions between the 1/2(111) edge dislocation and symmetric tilt grain boundaries in tungsten. Philosophical Magazine, 2008,88(4): 547-560
[166] Dewald MP, Curtin WA. Multiscale modelling of dislocation/grain-boundary interactions: I. Edge dislocations impinging on Sigma 11 (113) tilt boundary in Al. IUTAM Symposium on Plasticity at the Micron Scale. Lyngby, DENMARK: Iop Publishing Ltd, 2006:S193-S215.
[167] Li Z, Hou C, Huang M, et al. Strengthening mechanism in micro-polycrystals with penetrable grain boundaries by discrete dislocation dynamics simulation and Hall - Petch effect. Computational Materials Science, Accepted for publication:
[168] Hasson GC, Goux C. Interfacial energies of tilt boundaries in aluminium. Experimental and theoretical determination. Scripta Metallurgica, 1971,5(10): 889-894
[169] Kato M, Fujii T, Onaka S. Dislocation Bow-Out Model for Yield Stress of Ultra-Fine Grained Materials. Materials Transactions, 2008,49(6): 1278-1283
[170] Hussein MI, Borg U, Niordson CF, et al. Plasticity size effects in voided crystals. Journal of the Mechanics and Physics of Solids, 2008,56(1): 114-131
[171] Greengard L, Rokhlin V. A fast algorithm for particle simulations. Journal of Computational Physics, 1987,73(2): 325-348