1. Applied Mechanics and Structure Safety Key Laboratory of Sichuan Province, School of Mechanics and Engineering, Southwest Jiaotong University, Chengdu 610031, China 2. State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi'an Jiaotong University, Xi'an 710049, China
In recent years, many gradient materials have been studied. The metal materials with gradient microstructure mainly include: grain size distribution gradient, twin density gradient, dislocation density gradient, solute or precipitate density gradient, or combinations thereof. There are many studies of gradient nanograined material, but few studies of the dislocation density gradient. In fact, the dislocation density gradient structure is ubiquitous. The Taylor relation is only applicable to reveal the relationship between dislocation density and plastic flow stress, without the description of its dependence on dislocation density gradient. Discrete dislocation dynamics (DDD) has its advantage in describing plastic deformation in terms of dislocation motion and dislocation interactions. In this work, Threedimensional discrete dislocation dynamics (3DDDD) simulation was performed to investigate the compression behavior of single crystal copper micropillar with dislocation density gradient structure. The effects of loading direction perpendicular and parallel to the direction of dislocation density gradient on the anisotropic responses of micropillar compression were analyzed. The compressional stressstrain response shows that, when the loading direction is parallel to the gradient direction, the critical stress of elasticplastic transition is higher. However, the plastic flow stress is not affected by the loading direction when the strain is relative larger. Further analysis of spatialtemporal evolution of plastic strain and dislocation density indicate that: when the loading direction is perpendicular to the dislocation density gradient direction, the dislocation sources are firstly activated in the region with the lowest dislocation density, then the dislocations in the region with higher dislocation density are activated subsequently; and the whole deformation process is accompanied with multiple slip bands, then the deformation of the whole model is relatively more uniform. When the loading direction is parallel to the dislocation density gradient direction, the dislocation sources start to activate in the middle layer of the model, then expand to the two adjacent ends; and the plastic deformation of the whole model mainly concentrates in only one slip band.
Fund: National Natural Science Foundation of China Nos(11672251);National Natural Science Foundation of China Nos(11872321);and Opening Fund of State Key Laboratory for Strength and Vibration of Mechanical Structures(SV2018KF10)
XIONG Jian,WEI Dean,LU Songjiang,KAN Qianhua,KANG Guozheng,ZHANG Xu. A ThreeDimensional Discrete Dislocation Dynamics Simulation on Micropillar Compression of Single Crystal Copper with Dislocation Density Gradient. Acta Metall Sin, 2019, 55(11): 14771486.
Fig.1 Dislocation discretization in the discrete dislocation dynamics simulation (The red node is the physical node, representing the real dislocation node; the blue nodes are the discretization nodes, which are generated according to certain rules; the green point is the surface node, which connects a dislocation segment; the black line is the dislocation segment)Color online
Layer
Number of source
Dislocation density
10^{12} m^{2}
Dislocation spacing
nm
Bottom layer
100
38.0
162
Middle layer
200
76.5
114
Top layer
300
114.0
93
Table 1 Number of sources, dislocation density and dislocation spacing in different layers for the model configurations
Parameter
Value
Unit
Burgers vector (b)
0.256
nm
Mean dislocation density (ρ)
76.5
10^{12} m^{2}
Mean dislocation source length (l_{FRS})
400
nm
Poisson's ratio (ν)
0.324

Shear modulus (μ)
54.6
GPa
Drag coefficient (B)
1×10^{4}
Pa·s
Table 2 Material parameters used in the discrete dislocation dynamics simulation
Fig.2 Initial dislocation configurations of dislocation density gradient structure model under different loading directions (The colors of dislocation lines in the graph represents different types of Burgers vectors of dislocations)Color online(a) the loading direction is perpendicular to the direction of dislocation density gradient, the arrows indicate loading direction(b) the loading direction is parallel with the direction of dislocation density gradient, the arrows indicate loading direction
Fig.3 Equivalent stressstrain curves and dislocation density evolution curves of single crystal copper micropillar with dislocation density gradient structure under different loading directions
Slip system
X direction [1$\stackrel{\u02c9}{1}$0]
Z direction [$\stackrel{\u02c9}{1}$$\stackrel{\u02c9}{1}$2]
Table 3 Schmid factors of 12 slip systems in single crystalline copper under different loading directions
Fig.4 Primary slip systems under two different loading directions (The planes ACD (normal vector is ($\stackrel{\u02c9}{\mathrm{1}}$11)) and BCD (normal vector is (1$\stackrel{\u02c9}{\mathrm{1}}$1)) are the primary slip planes, and the red edges of the Thompson tetrahedron are the primary slip directions, AC and AD represent the slip directions of [0$\stackrel{\u02c9}{\mathrm{1}}$1] and [101] in the slip plane ACD, respectively; BC and BD represent the slip directions of [$\stackrel{\u02c9}{\mathrm{1}}$01] and [011] in the slip plane BCD, respectively)Color online(a) along X axis (b) along Z axis
Fig.5 The snapshots of evolved dislocation structures corresponding to strains of 0.05% (a), 0.10% (b), 0.15% (c), 0.20% (d) and 0.25% (e) in the compression process along the Xaxis direction which is perpendicular to the dislocation density gradient directionColor online
Fig.6 Distributions of plastic strain on the surface corresponding to strains of 0.05% (a), 0.10% (b), 0.15% (c), 0.20% (d) and 0.25% (e) in the compression process along the Xaxis direction which is perpendicular to the dislocation density gradient directionColor online
Fig.7 The evolutions of plastic strain (a) and the dislocation density (b) in bottom, middle and top layer, which have different dislocation densities, when loading direction is along Xaxis which is perpendicular to the dislocation density gradient direction
Fig.8 The snapshots of evolved dislocation structure corresponding to strains of 0.05% (a), 0.10% (b), 0.15% (c), 0.20% (d) and 0.25% (e) in the compression process along the Zaxis direction which is parallel with the dislocation density gradient directionColor online
Fig.9 Distributions of plastic strain on the surface corresponding to strains of 0.05% (a), 0.10% (b), 0.15% (c), 0.20% (d) and 0.25% (e) in the compression process along the Zaxis direction which is parallel with the dislocation density gradient directionColor online
Fig.10 The evolutions of plastic strain (a) and the dislocation density (b) in bottom, middle and top layer, which have different dislocation densities, when loading direction is along Zaxis which is parallel with the dislocation density gradient direction
[1]
LiuZ Q, MeyersM A, ZhangZ F, et al. Functional gradients and heterogeneities in biological materials: Design principles, functions, and bioinspired applications [J]. Prog. Mater. Sci., 2017, 88: 467
[2]
LiY. Research progress on gradient metallic materials [J]. Mater. China, 2016, 35: 658
SakaiT, OhashiM, ChibaK, et al. Recovery and recrystallization of polycrystalline nickel after hot working [J]. Acta Metall., 1988, 36: 1781
[11]
SakaiT, OhashiM. Dislocation substructures developed during dynamic recrystallisation in polycrystalline nickel [J]. Mater. Sci. Technol., 1990, 6: 1251
[12]
HeD, HuanY, LiQ. Effect of dislocation density gradient on deformation behavior of pure nickel subjected to torsion [A]. DEStech Transactions on Materials Science and Engineering [C]. Nanjing: DEStech Publications, 2017
TangH, SchwarzK W, EspinosaH D. Dislocation escaperelated size effects in singlecrystal micropillars under uniaxial compression [J]. Acta Mater., 2007, 55: 1607
[15]
SengerJ, WeygandD, GumbschP, et al. Discrete dislocation simulations of the plasticity of micropillars under uniaxial loading [J]. Scr. Mater., 2008, 58: 587
[16]
MotzC, WeygandD, SengerJ, et al. Initial dislocation structures in 3D discrete dislocation dynamics and their influence on microscale plasticity [J]. Acta Mater., 2009, 57: 1744
[17]
LuS J, ZhangB, LiX Y, et al. Grain boundary effect on nanoindentation: A multiscale discrete dislocation dynamics model [J]. J. Mech. Phys. Solids, 2019, 126: 117
[18]
WeiD A, FanH D, TangJ, et al. Effect of a vertical twin boundary on the mechanical property of bicrystalline copper micropillars [A]. TMS 148th Annual Meeting & Exhibition Supplemental Proceedings [C]. Cham: Springer, 2019: 1305
[19]
WeinbergerC R, CaiW. Surfacecontrolled dislocation multiplication in metal micropillars [J]. Proc. Natl. Acad. Sci. USA, 2008, 105: 14304
[20]
SengerJ, WeygandD, MotzC, et al. Aspect ratio and stochastic effects in the plasticity of uniformly loaded micrometersized specimens [J]. Acta Mater., 2011, 59: 2937
[21]
ElawadyJ A. Unravelling the physics of sizedependent dislocationmediated plasticity [J]. Nat. Commun., 2015, 6: 5926
[22]
LiZ H, HouC T, HuangM S, et al. Strengthening mechanism in micropolycrystals with penetrable grain boundaries by discrete dislocation dynamics simulation and HallPetch effect [J]. Comput. Mater. Sci., 2009, 46: 1124
[23]
HuangM S, LiangS, LiZ H. An extended 3D discretecontinuous model and its application on single and bicrystal micropillars [J]. Model. Simul. Mater. Sci. Eng., 2017, 25: 035001
[24]
HuangM S, LiZ H. Coupled DDDFEM modeling on the mechanical behavior of microlayered metallic multilayer film at elevated temperature [J]. J. Mech. Phys. Solids, 2015, 85: 74
[25]
HuangS, HuangM S, LiZ H. Effect of interfacial dislocation networks on the evolution of matrix dislocations in nickelbased superalloy [J]. Int. J. Plast., 2018, 110: 1
[26]
LiuZ L, LiuX M, ZhuangZ, et al. A multiscale computational model of crystal plasticity at submicrontonanometer scales [J]. Int. J. Plast., 2009, 25: 1436
[27]
CuiY N, LiuZ L, ZhuangZ. Quantitative investigations on dislocation based discretecontinuous model of crystal plasticity at submicron scale [J]. Int. J. Plast., 2015, 69: 54
[28]
FanH D, AubryS, ArsenlisA, et al. Discrete dislocation dynamics simulations of twin sizeeffects in magnesium [A]. MRS Online Proceedings Library [C]. Cambridge: Cambridge University Press, 2015: 1741
[29]
GuoX R, SunC Y, WangC H, et al. Investigation of strain rate effect by threedimensional discrete dislocation dynamics for fcc single crystal during compression process [J]. Acta Metall. Sin., 2018, 54: 1322
BulatovV, CaiW, FierJ, et al. Scalable line dynamics in ParaDiS [A]. Proceedings of 2004 ACM/IEEE Conference on Supercomputing [C]. Pittsburgh: IEEE, 2004
[31]
WeygandD, FriedmanL H, Van der GiessenE, et al. Aspects of boundaryvalue problem solutions with threedimensional dislocation dynamics [J]. Model. Simul. Mater. Sci. Eng., 2002, 10: 437
[32]
VattréA, DevincreB, FeyelF, et al. Modelling crystal plasticity by 3D dislocation dynamics and the finite element method: The discretecontinuous model revisited [J]. J. Mech. Phys. Solids, 2014, 63: 491
[33]
SengerJ, WeygD, MotzC. Aspect ratio and stochastic effects in the plasticity of uniformly loaded micrometersized specimens [J]. Acta Mater., 2011, 59: 2937
[34]
ZhangX, AifantisK E, SengerJ, et al. Internal length scale and grain boundary yield strength in gradient models of polycrystal plasticity: How do they relate to the dislocation microstructure? [J]. J. Mater. Res., 2014, 29: 2116
[35]
WuZ J, ZhaoE J, XiangH P, et al. Crystal structures and elastic properties of superhard IrN2 and IrN3 from first principles [J]. Phys. Rev., 2007, 76B: 054115