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Int. J. Nanomanufacturing, Vol. 13, No. 1, 2017
Dislocation dynamic simulation of dislocation pattern evolution in the early fatigue stages of aluminium single crystal Jinxuan Bai*, Qingshun Bai, Xin He and Qingchun Zhang School of Mechanical and Electrical Engineering, Harbin Institute of Technology, 92 West Dazhi Street, Nan Gang District, Harbin 150001, China Email:
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[email protected] *Corresponding author Abstract: The two-dimensional discrete dislocation dynamics simulations under fully periodic boundary conditions has been employed to study the dislocation pattern evolution in the early stages of fatigue in aluminium single crystal. Long-range force among of dislocations is solved by line elasticity model, and short-range force is obtained by constitutive equations of dislocation nucleation, slip, pileup and annihilate. Dislocation movement mechanisms of single-slip-oriented and multi-slip systems are simulated and the evolution process of fatigue pattern is revealed. The result shows that dislocation quantity and microstructure strongly depend on external load and internal configuration. The dislocation pattern of single-slip-oriented generate matrix wall, and positive dislocations are vertical alignment and negative dislocations are at the angel of 45° with slip oriented in the initial stage. For multi-slip system, maze structure of dislocations is produced during dislocation multiplication, which eventually transforms into persistent slip band. The result is consistent with the existing experiment. Keywords: the aluminium single crystal; single crystal aluminium; early fatigue stages; dislocation pattern; discrete dislocation dynamics. Reference to this paper should be made as follows: Bai, J., Bai, Q., He, X. and Zhang, Q. (2017) ‘Dislocation dynamic simulation of dislocation pattern evolution in the early fatigue stages of aluminium single crystal’, Int. J. Nanomanufacturing, Vol. 13, No. 1, pp.12–22. Biographical notes: Jinxuan Bai is a Doctoral candidate at the School of Mechanical and Electrical Engineering, Harbin Institute of Technology, Harbin, China. His research orientation is discrete dislocation dynamic simulation of metal and diamond. Qingshun Bai is an Associate Professor at the School of Mechanical and Electrical Engineering, Harbin Institute of Technology, Harbin, China. He received his PhD in Mechanical from HIT.
Copyright © 2017 Inderscience Enterprises Ltd.
Dislocation dynamic simulation of dislocation pattern
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Xin He is a Master’s candidate at the School of Mechanical and Electrical Engineering, Harbin Institute of Technology, Harbin, China. His research orientation is the mechanics performance of grapheme. Qingchun Zhang is a Master Tutor at the School of Mechanical and Electrical Engineering, Harbin Institute of Technology, Harbin, China. His research orientation is the ultra-precision machining technology.
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Introduction
Aluminium alloys are used extensively in various precision and optical components, such as mirrors reflector and hard-disk (To et al., 1997). This imposes great demands on the surface quality of aluminium component, and ultraprecision diamond is an important technology to obtain a super mirror surface directly on non-ferrous metals. As the cutting depth is always less than the average grain size, the improvement of machining precision is restricted to a certain extent by the uncertainty of workpiece materials mechanics characteristic (Luo et al., 2000). Studies shows that the surface finish of single crystal Al is much worse than general crystallised in grind process, because the plastic deformation of Al is very complex than other non-ferrous metals (Zhao et al., 2004). Currently, the research of single crystal Cu is extensive, however the plastic mechanism of Al is seldom seen. The cutting behaviour and mechanism of machined surface are knew to depend on the crystallographic factors, such as crystal orientation, the slip system and the mobile dislocation density of the workpiece, as well as the plastic deformation in crystalline materials occurs through the motion of defects. The point defects may be involved as in diffusional creep, and interfaces may be involved as in twining and stress-induced martensitic transformations (Benat et al., 2013). The plastic deformation of Al through the generation and motion of dislocation is the sole concern in this article. During the cyclic deformation, the crystalline materials forms dislocation structure of high ordering which is called dislocation configuration or dislocation pattern, such as the vein, ladder, cell, maze (Yang et al., 2001). The plastic deformation is the irreversible process which is far from equilibrium, thus there is a mass of dislocation group on the microscopic scale, and each other of them occurs the strongly nonlinearity couple and coordination so that dislocation distributions occur instability, bifurcate and pattern. Since the establishment of dislocation theory in 1930, the study of single dislocation has been approach to mature. However, the research about materials dislocation pattern of single crystal Al were still in the start stage, the process study of aluminium dislocation pattern forming, instability and characteristics is a basic research for the materials processing and materials design, this will introduce the crystal plastic theory into a new stage and provide a theoretical foundation for explaining the aluminium hardening behaviour during ultra-precision processing (Qian et al., 1993). Comparing with the molecular dynamics and the traditional finite element method, the dislocation dynamic has the significant advantages in time and space scales for the research of materials pattern. In 1989, the Los Alamos Laboratory firstly used the dislocation dynamics to get the dislocation distributions information in two dimensions,
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which open the dislocation pattern computer simulation (Gulluoglu et al., 1989). Yang et al. (2003) reveal the effect of boundary conditions on the dislocation pattern in fatigued copper single crystal. The journal Nature published papers to indicate that the dislocation dynamics simulation had been probed very effective in the study of dislocation patterning, and the authors obtained intermittent dislocation flow in viscoplastic deformation (Miguel et al., 2010). With the development of dislocation dynamics, this method was used more and more extensive. Philippa et al. exploit a program using two-dimensional dislocation dynamics with anisotropic strain equations to simulate the dimensional change of irradiated graphite (Young et al., 2013). Cheng and Shehadeh (2006) took the multiscale dislocation dynamics to analyse laser shock peening (LSP) in Si single crystals, and obtained the dislocation mechanism of hard brittle materials. Baranov et al. (2014) investigate the distribution of dislocation in oxide nuclear fuel under irradiation using the values of dislocation density from experiments, and they consider the synergistic action of gliding and climbing. The dislocations dynamic can directly revealing the changes of mechanical properties. Hartmailer employed the 2D-DD to study the relation between microstructure and strength of material (Ahmed and Hartmaier, 2010). With the simulation of the 2D-DD model, it was found the material yield behaviour was consistent with the classical theory. Ahmed studied the different mechanisms of plastic deformation of ultrafine-grained metals at different temperatures. From the investigations, it was found that the metal followed the Hall-Petch law at low temperatures, however a high strain-rate sensitivity was found with the increasing of temperatures (Ahmed and Hartmaier, 2011). In the present work, the dislocation pattern evolution process for single-slip-oriented and the multi-slip systems of Al single crystal has been revealed, and the influence of boundary conditions and dislocation initial configuration to plastic hardening are analysed. The article is organised as follows: the follow section will introduce the fundamental principles of dislocation dynamic and the method of simulating dislocation motion. Subsequently, numerical results which obtained for various forms is presented and discussed.
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Dislocation dynamics method
As the dislocation loop shuttles back and forth, and the screw dislocation remove from the slip plane, only the edge dislocations which are taken into the simulation. The edge dislocation is straight lines and the Burgers vector is parallel to the slip system. The dislocation dynamic method hypothesis the positive and negative dislocations are equal, and the sum of burgers vectors is zero.
2.1 Dislocation movement equation As the simulation environment is in the normal temperature, the dislocation climb not being mainly considered in this article. Dislocation motion does not conserve energy, there are dissipative forces on the dislocations that arise from the generation of phonons, the equation of dislocation movement is the equal (1). m
d 2 xi = Fx (i ) − Bvi dt 2
(1)
Dislocation dynamic simulation of dislocation pattern
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where x is the displacement of dislocation i, B is the friction coefficient and vi is the velocity of the dislocation in ± xˆ. If the damping of material is large, then the terminal velocity which is rapidly increased in a time is small compared to the time step of the simulation, and this is called the over-damped limit, thus allowing us to ignore the initial inertial movement. The slip velocity vslip of dislocation is described as the equal (2). vslip = bM ( τ slip − τ 0 )
(2)
where τslip is the stress vector which parallel to the slip direction, and M denotes the dislocation mobility, which is the materials constant (M = 1/B which is the drag coefficient, B is 10–4 Pas), τ0 is the minimum shear stress, b is the burgers vector. If the velocities of all the dislocation have been calculated, the movement equations of dislocation could be obtained by the Euler method with an explicit time step, as the equal (3). xi (t + Δt ) = xi (t ) + vi (t )Δt
(3)
where Δt is the time step, Δt is 1 ns in the present research.
2.2 Peach-Koehler force equation The long-range force among of dislocations is solved by the dislocation elasticity model, and the short-range force is solved by the interaction effect of dislocation nucleation, slip, pile-up, annihilate. Using linear elasticity properties, the effective stress applied on each dislocation is evaluated as the superposition of the internal stress which induced by other dislocations in the simulated volume and the applied stress imposed by the loading. This induces a force given by the Peach-Koehler equation (4). FPK = σb × ξ
(4)
where ξ is the unit vector of the line direction. The component of force fgl along the slip direction is f gl( k ) = τ rss b
(5)
The Peach-Koehler force on dislocation k is written as app + f gl( k ) = ⎛ τ rss ⎜ ⎝
∑τ j≠k
(i ) rss
⎞b ⎟ ⎠
(6)
( j) app where the τ rss denotes the internal resolved by dislocation j, and the τ rss is applied shear stress on the plane.
2.3 Dislocation multiplication During the 2-D dislocation dynamics, the dislocation initiation is usually simulated by the nucleation mechanism of famous Frank-Read source. The Frank-Read sources which
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have the specific nucleation strength and distribution density are random arrangement in the crystal slip system. If the shear force on the dislocation source is greater than the critical stress τnuc at a set interval, then there will be a pair of dislocation dipole that has the different Burgers vector. The distance Lnuc between the new initiation of dislocation dipole must accord the equation (7). Lnuc =
G b 2π (1 − ν) τ nuc
(7)
2.4 Dislocation pile-up and annihilate In addition to the slip obstacles which caused by the grain boundary and other dislocation, the sediment and impurity in real crystal also will cause dislocation pile-up so that the dislocation obstacles are random distribute in the slip system. If the kinetic dislocation contact the obstacles, they will be pinned. Meanwhile the dislocation could get ride the bondage ones they get specific stress value. If two dislocations of opposite sign are approaching each other closer than a cut-off distance of 6b they annihilate and are consequently removed from the simulation. This annihilation process is considered to be spontaneous and take place, even if the dislocation move on different slip plane.
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Analysis of dislocation simulation
The research of formation and characteristics of fatigue dislocation pattern could promote the profound understanding of materials hardening and fatigue mechanism, and the study can explain some particular phenomena in the ultra-precision. Al single crystal is simplified as the isotropy materials. The Young modulus of Al is 70 GPa and the Poisson’s ratio of Al is 0.33. By the development of discrete dislocation dynamics programs, revealing the dislocation motion law and obtaining the inherent relation between dislocation evolution and mechanical properties.
3.1 Single-slip plane The boundary condition is very important to study the phenomenon of dislocation self-organisation, the free-space boundary condition cannot yield a reasonable result (Yang et al., 2013). Therefore, the article employs the periodic discrete dislocation dynamic boundary. In the present work, there are two different types of initial configuration are used to research the plastic behaviour in the single-slip. The first one contains no sources, but random distributions of dislocation, the initial relax without the extra load is as Figure 1. From Figure 1(a), we get that the dislocation is rambling, and the positive and negative dislocations are entangled each other in uncertain ways.
Dislocation dynamic simulation of dislocation pattern Figure 1
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Pre-existing dislocation free evolution without the extra load, (a) the pre-existing dislocation (b) 500 ns dislocation relax (see online version for colours)
It is easy to observe that there are many main features in the microstructure [see Figure 1(b)], i.e., the elastic interaction forces with all other dislocations are in balance and dislocation dipoles are appeared at the slip plane. The simulation area has arose four dislocation walls and the sign of the Burgers vector alternates from wall to wall. The result is very similar to a polygonised structure which is consistent with the Gulluoglu et al. (1989). The previous research verifies the correctness of 2D-DDD code. The second configurations contain dislocation source. The dislocation patterns will produce a continuous change under the circulation of external load. Although the dislocation density of pure aluminium after annealing is general 1010m–2, the dislocation pattern evolution is usual start with the higher dislocation density. So, the present study lays 20 positive and negative dislocations on the xOy plane randomly. The multiplicative dislocations are also random distribution. The strain rate equation (8) is used to control dislocation multiplicative mechanism. γ p = φρbv
(8)
where the γP is the plastic strain rate, φ is the correction factor, ρ is the movability dislocation density, v the average velocity of dislocation. If γP is under 2 × 10–4, the dislocation will be generated (Yang et al., 2003). The dislocation pattern change with external load is shown in Figure 2. Figure 2 shows that dislocations pattern forms the matrix walls at the beginning of the loop, at this moment the matrix walls has present irregular arrangement, and the angle between the matrix walls and slip direction are various values. When the simulation time increases to 250 ns, it is easy to obtain that the same sign matrix dislocations which established by the negative dislocations are perpendicular to the slip direction, but the positive matrix dislocations are nearly ±45° with the slip direction. The result accords the dislocation theory (Feng and Qiu, 1987), this means the dislocations get an equilibrium distribution. In the late stage of simulation [Figure 2(c)], the perpendicular matrix walls fade away, and the negative dislocations come into being the 45° dislocation band. Finally, positive and negative dislocation get mutual parallel, they has formed the persistent slip band (PSB which is named ladder structure). The result has been
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confirmed by Qian et al. (1993), as Figure 3. If the simulation will not introduce the new external environment mechanism, the result could be quite stable. Figure 2
Dislocation free evolution with the external load, (a) 0 ns pre-existing dislocation (b) 250 ns dislocation pattern (c) 500 ns dislocation pattern (see online version for colours)
Figure 3
The formation of the PSB
Source: Qian et al. (1993)
The change of dislocation quantity is shown in Figure 4. The number of dislocations rapidly increases in the initial stage because the single crystal Al is in rapid hardening zone. Thereafter, it subsequently tends to be stable which marks the dislocation arrives the plateau region. As the dislocation annihilation, the curve is fluctuation within a narrow range.
Dislocation dynamic simulation of dislocation pattern Figure 4
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The change of dislocation quantity during simulation process (see online version for colours)
3.2 Multi-slip system The dislocation slip models which the pre-existing dislocations in the multi-slip systems are appeared as Figure 5. Figure 5
Schematic diagram of discrete dislocation dynamic (see online version for colours)
In Figure 5, GB is the grain boundary, P is pile up, L is the lock of dislocations, D is dipoles of dislocation, S is dislocation source. As mentioned in Shishvan and Giessen (2013), the strength of each source is randomly selected from the Gaussian distribution. The nucleation stress is 50 MPa, and the standard deviation is 10 MPa, as well as obstacle stress is 150 MPa, the time of nucleate and annihilate are 10 ns and 6 ns. The distance between adjacent slip planes is 100 b. Slip system Angle is 45°, 90°, 45°, dislocation density is kept constant. The dislocation interaction force of multi-slip system is equal (9). σ xx =
⎡ y ( 3x 2 + y 2 ) x ( y2 − x2 ) ⎤ −G + by bx ⎢ ⎥ 2 4π (1 − v 2 ) ⎢ ( x2 + y2 ) ( x 2 + y 2 )2 ⎥⎦ ⎣
σ yy =
⎡ y ( x2 − y 2 ) x (3 y2 + x2 ) ⎤ G + bx b ⎢ ⎥ y 4π (1 − ν 2 ) ⎢ ( x 2 + y 2 )2 ( x 2 + y 2 )2 ⎥⎦ ⎣
σ xy =
⎡ y ( x2 − y 2 ) y ( y 2 − x2 ) ⎤ G − bx b ⎢ ⎥ y 4π (1 − ν 2 ) ⎢ ( x 2 + y 2 )2 ( x 2 + y 2 )2 ⎦⎥ ⎣
(9)
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Initially, a single crystal is used to study the dislocation pattern evolution in multi-slip system, the dislocations density is kept constant by the dislocation annihilate. The pattern process is shown in Figure 6. Additionally, an infinite crystal is achieved by applying periodic boundary condition. In the case of an initially, dislocation quantity gradually increase. Figure 6(a) and Figure 6 (b) show that the dislocations slip mainly occurs on the ±45° direction, this is because single crystal has formed the work hardening so that the dislocation produced maze structure. The essence of the phenomenon is two groups of orthogonal dislocation walls, and the Burgers vector angle of two group of dislocation is nearly 90° (Qian et al., 1993). With simulation time gets the 400 ns, the edge dislocations slip in the slip system has made the broken of primary dislocation maze, and a mass of dislocations are destroy so that the crystal internal occurs alternate of multiplication and annihilation. Finally, single crystal produces the slip band and dislocation matrix walls in Figure 6(d). Figure 6
Multi-slip system dislocation pattern evolution and experiment image, (a) 100 ns dislocation pattern (b) 200 ns dislocation pattern (c) 400 ns dislocation pattern (d) 500 ns dislocation pattern (e) the dislocation image of SAE (see online version for colours)
Dislocation dynamic simulation of dislocation pattern Figure 6
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Multi-slip system dislocation pattern evolution and experiment image, (a) 100 ns dislocation pattern (b) 200 ns dislocation pattern (c) 400 ns dislocation pattern (d) 500 ns dislocation pattern (e) the dislocation image of SAE (continued) (see online version for colours)
The simulation result indicates that dislocation pattern occurs a mass of PSB which has been verified by Figure 6(e), the dislocation image of SAE in Qian et al. (1993). It is easy to know that the PSB volume fraction is gradually increase along with the load. The Taylor relation which describes the strength model. τ = τ 0 + α Gbρ1 2
(10)
where τ is the flow stress, τ0 is the yield strength of materials without dislocation, G is the shear modulus, b is the norm of Burgers vector, and ρ is materials dislocation density. According to the equal (10), we know the local strain of PSB is higher than the average strain in workpiece. Therefore, the Al more easily occurs plastic hardening than other materials during the formation and extension of single PSB.
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Conclusions
In this study, a two dimensional dislocation dynamic program for single slip system and multi-slip system was established. The numerical results for the dislocation evolution response of 2D-DDD model materials were presented and the dislocation movement process of the fatigue pattern was achieved. With the analysis, the characteristics variation of dislocation density and dislocation microstructure was revealed under various simulation conditions. For the single system without external load, the sign of the Burgers vector alternates among four dislocation walls. There is PSB in the xOy system when the simulation condition is quasi-static loading. During the multi-slip system simulation, the dislocation pattern occurs dislocation maze in the initial stage. With the simulation, the single crystal produces the slip band and dislocation matrix walls.
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Acknowledgements This research work was jointly supported by the National Science Fund for Distinguished Young Scholars of China (Grant No. 50925521) and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry of China (Grant year 2013).
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