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Orientation effects on the tensile properties of single crystal nickel with nanovoid: Atomistic simulation
Computational Materials Science 132 (2017) 116–124
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Orientation effects on the tensile properties of single crystal nickel with nanovoid: Atomistic simulation J.P. Wang, Z.F. Yue, Z.X. Wen ⇑, D.X. Zhang, C.Y. Liu School of Mechanics, Civil Engineering and Architecture, Northwestern Polytechnical University, Xi’an 710072, PR China
a r t i c l e
i n f o
Article history: Received 16 May 2016 Received in revised form 19 February 2017 Accepted 20 February 2017
Keywords: Molecular Dynamics (MD) simulation Lattice orientation Nanovoid Dislocation nucleation
a b s t r a c t The tensile behavior of monocrystalline nickel with nano-void was studied via molecular dynamics simulation (MD) considering different lattice orientations. A series of simulations were performed to analyze the effect of system size, void volume fraction and lattice orientation on mechanical properties and microstructure evolution. The influence of size of sample on the incipient yield stress is discussed. The results show that void volume fractions have significant effects on Young’s modulus, incipient yield stress and incipient yield strain. Dislocation structure begins to nucleate in the stress concentration area. The critical stress of [1 0 0], [1 1 0] and [1 1 1] orientation is 6.97 GPa, 6.77 GPa and 7.31 GPa, respectively. The results of dislocation propagation and stress-strain responses reveal the elastic-plastic properties of different orientations. The fracture strain is the elongation of the specimen. The elongation of the sample along [1 1 1] orientation is greatest, which illustrate that the sample has good ductility performance along [1 1 1] orientation in the same initial damage. The reaction between the dislocations and the boundary leads to the rearrangement of the boundary atoms, which proves that the dislocation can not be stopped inside the crystal, or closed or terminated at the crystal surface or grain boundary. The difference of the rearrangement of the boundary atoms is due to the difference of the sliding systems of different orientations. Ó 2017 Elsevier B.V. All rights reserved.
1. Introduction The failure modes of materials have significant influence on the design of material properties in materials science. Rapture failure at the macroscopic scale can be attributed to nucleation, growth and propagation of cracks, but at the microscopic scale cracks are initially easily formed at defects in the casting process, such as voids and inclusions [1]. These defects are known to play a fundamental role in the deformation of the material. Nucleation, growth and coalescence of voids are deemed as the primary mechanism of ductile material fracture, in which void growth is particularly important. Therefore, it is necessary to study the deformation response of porous materials with the consideration of microstructure evolution. Most previous work on void growth studies, both experimental and simulation approaches, have been used to better describe ductile fracture processes. Furthermore, some numerical methods, such as crystal plasticity finite element method [2–7], dislocation dynamics and molecular dynamics [8–19], have been used to reveal the mechanism of void growth. ⇑ Corresponding author. E-mail address: [email protected] (Z.X. Wen). http://dx.doi.org/10.1016/j.commatsci.2017.02.024 0927-0256/Ó 2017 Elsevier B.V. All rights reserved.
Molecular dynamics (MD) is an effective tool for studying the fracture mechanism of micro-scale materials. It can directly observe the motion of atoms and obtain the details of atomic deformation. MD simulations in monocrystalline and bicrystalline copper were carried out with LAMMPS (large-scale atomic/Molecular Massively Parallel Simulator) to reveal void growth mechanisms. The results confirm that the emission of dislocation shear loops is the primary mechanism of void growth [20–24], consistent with the analytical results of [21] and [25–28]. Tang and his colleagues simulated the growth behavior of spherical nanovoids in c-TiAl single crystal [17]. They found that the emission of dislocation loop is the main reason for the growth of void, and the continuous dislocation nucleation and the increase of the shear loop prompt the growth of voids. In addition, extensive work has been performed on other factors affecting void growth, such as size effect [13,14,16,26], initial void volume fraction [2,5,8,9,20,24,26] And strain rate [16,26,28]. The initial stage of the growth of nanocrystalline voids in monocrystalline aluminum was studied by MD simulation. The simulation results showed that the dependence of the critical negative pressure on void radius, system size and temperature was obtained at high rate tension [24]. The decrease in the ratio of the radius of the void to the system size leads to an increase in the tensile strength of the system. Tang
J.P. Wang et al. / Computational Materials Science 132 (2017) 116–124
et al. [29] based on a new analytical method for the emission energy of the dislocation loop, the reduction of stress with increasing void size is analyzed. Bhatia et al. [26] investigated nanovoid growth using MD and revealed its dependence on void size, strain rate, crystallographic loading orientation, initial nanovoid volume fraction, and simulation cell size. Dislocations and shear loops nucleate on different slip systems with different loading orientations, which include orientation that favors single slip and multiple slip. It is well known that the mechanical properties of monocrystalline metallic materials have a strong correlation with their crystal orientation. The mechanisms of void growth and failure modes are significantly different in different lattice orientations. Therefore, it is significant to study the void growth behavior of single crystal metallic materials with different orientations. A number of studies have been carried out in this area [3,7,9,10,13,15,30,31]. Potirniche et al. [15] interpreted that plasticity anisotropy caused by initial lattice orientation has only a minor effect on void growth in the microscale regions. Stress triaxiality is the main factor controlling the growth and coalescence of voids. The numerical results of Sangyul Ha and KiTae Kim [3] showed that the stress triaxiality and the deformation pattern specified by the crystal orientation have competitive effects on the evolution of voids. For low-level stress triaxiality, the deformation mode is mainly determined by the crystal orientation. Eduardo et al. [13] investigated the effect of loading orientation on void initiation in FCC metals. The results showed that there is a significant effect of the loading orientation on the sequence where the loops form and interact. Based on the above studies, it has been found that dislocation nucleation and morphology of different crystal orientations have not been systematically studied in detail. In this study, uniaxially stretched MD simulation was used to analyze the effect of lattice orientation on tensile fracture in thin nickel specimens with infinite cylindrical voids. Nickel is a typical FCC material. In addition, its alloys, such as B2-type NiAl, LI2-type Ni3Al, are used in the aerospace industry due to their excellent high temperature performance. The study will examine the microstructural evolution and the mechanism of void growth through stress-strain responses and morphological changes of voids in different crystal orientations. At the same time, size effects and void volume fraction effects are taken into account. 2. MD simulation presentation 2.1. Atomistic model and simulation conditions In this simulation, a system of size 64:416 27:0336 5:28 nm3 was chosed to investigate the orientation effect, as shown in Fig. 1. Considering that the size of the model must be a multiple of the lattice spacing, the lattice spacing in the [1 1 1]-oriented z is different from that of [1 0 0] and [1 1 0]. A [1 1 1]-oriented simulation system of size 64:416 27:0336 5:4758 nm3 is used
117
herein. The radius of the cylindrical nano-voids was 5.28 nm. The size in z direction is greater than the potential cut-off radius 0.58 nm. Model with three different orientations was used to simulate the uniaxial tensile behavior of single crystal nickel and nano-voids. The parameters are shown in Table 1. Periodic boundary conditions (PBC) was applied to the zdirection, x and y were the shrink-wrapped boundary. Microcanonical ensemble was taken during the simulation process, and the temperature of the thermostat atom is controlled by rescaling the atomic velocity by 40 picoseconds in 1 fs time step. Therefore, the model relaxes to the equilibrium state at the beginning of the simulation. These models hold the left boundary and are uniaxially loaded at a constant strain rate of 5 108 s1 at the right boundary, as shown in Fig. 1. All simulations were performed by LAMMPS at an invariable temperature of 300 K. The atomic stress tensor is calculated using the virial definition, which can be expressed as: N X f a ði; jÞrb ði; jÞ
rt ðiÞ ¼
ð1Þ
j–ðiÞ
re ðiÞ ¼
n X
rt ðiÞ=n
ð2Þ
t¼1
where f a is the interatomic force between atom i and j in the direction a， r b is the distance in direction b. Considering average over the volume around atom i within the cut-off distance, re is the average atomic stress tensor obtained by function (2). 2.2. EAM potential In recent years, the embedded atomic method [32,33] (EAM) interatomic potential has been widely used for crack propagation in metal materials and many other simulations. Compared with the F-S and L/J potentials, it can calculate the elastic properties of materials, energy changes, and the position of atoms in the whole simulation process of metal materials. Many examples have shown that EAM is an accurate representation of the interatomic forces in a metal lattice. In this study, the embedded atom method of Ni developed by Mishin et al. [34] was used to simulate the uniaxial tensile beha vior of single-crystal nickel with nanovoid. A reasonable simulation of fracture and damage can be obtained considering that the potential energy can well describe the bonding capacity in a metal system and the dependence of the strength of a single bond on the local environment (e.g., surface and defects). In addition, this EAM potential, based on the first principles, can reproduce many of the fundamental properties such as vacancy migration energy, unstable and stable stacking fault energy. Many studies [35–38] have used this EAM potential to investigate the deformation behavior of materials, which further validates the accuracy and reliability of this interatomic potential.
Fig. 1. Simulation Model (left boundary is fixed, right is loading at a constant strain rate).
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Table 1 Parameters for specimens with different orientations.
15.0
300
X
Y
Z
Number of atoms
[1 0 0] [1 1 0] [1 1 1]
[1 0 0] [1 1 0] [1 1 1]
[0 1 0] 1 0 ½1 ½1 1 2
[0 0 1] [0 0 1] 1 0 ½1
805,440 804,525 832,348
2.3. Post processing The open visualization tool (Ovito) of Stukowski [39] is used to observe and analyze the atomic configuration throughout the stretching process. Ovito uses the centrasymmetry parameter defined by Kelchner et al. [40] to highlight defective atoms and observe the evolution of slip bands, partial dislocations, stacking faults and other defects during rapture failure. It is noted that the slip system can be observed by controlling the centrosymmetry parameter.
250
Young's modulus (GPa)
Crystal orientation
[100] E [110] E [111] E [100] σc [110] σc [111] σc
12.5
200
10.0
150
100 7.5 50
5.0 --
0 Model-1
Model-2
model-3
σc (Incipient Yield Stress / Gpa)
118
model-4
Model Fig. 2. The change of E and rc of variable system size for different orientations.
3. Results and discussion 0.2
3.1. System size effects
Lubarda Model with Rcore=b
Table 2 Parameters for specimens with different size.
Normalized Yield Stress σy /G
Lubarda Model with Rcore=2b
The length scale significantly affects the mechanical properties of the material, such as the stress-strain response [14,16,41]. To study the effect of the specimen size，MD simulation was conducted with four systems of different sizes varying from 21:472 9:0112 1:76 nm3 to 85:888 36:0448 7:04 nm3 as shown in Table 2 (the model referred in Section 2.1), while void volume fraction was fixed as 5%. The loading constant strain rate was 5 108 s1, and the temperature was 300 K. The simulated results of different orientations interpret that the size effect is evident. All the curves (can be saw at Section 3.3) exhibit a linear stress-strain behavior before up to their peak stress points where an abrupt stress drop is observed. The peak stress increases with the decrease of specimen size for a fixed VVF. In addition, the slope of the line before point (a) was defined as the Young’s modulus by Zhao et al. [19]. They defined the peak stress at point (a) of stress-strain curve as the incipient yield strength as well. In Fig. 2, the initial slope of all these stress-strain curves, remains unchanged with the increasing of specimen size from Model-1 to Model-4. This means that the Young’s modulus is almost insensitive to the specimen size. Comparatively speaking, size effect on the incipient yield stress is quite notable. The trend of incipient yield stress conforms to the analytical results by literature [20,21]. And the reason can be concluded as that the development of dislocation intersection and the propagation occurs more easily before the dislocation reaches the boundary of the simulation ‘‘box” for larger system size model. Furthermore, the increase of plastic flow solved shear stress at decreasing of system cell size is concluded from Fig. 2, which is consistent with the gradient plasticity of experiment measurement. Another reason for this phenomenon proposed by Bhatia [26] is the volume decreases faster than surface area as system cell size decreasing with fixed VVF, which leads to a higher stress concentration. In Fig. 3, the yield stress obtained from MD simulations is compared with the Lubarda model [21],
[111] Atomistic Calculations [100] Atomistic Calculations [110] Atomistic Calculations
0.1
υ=0.291 G=73GPa 0.0
0
5
10
15
20
25
30
Normalized Void Radius R/b Fig. 3. Comparison of the critical stress obtained from present MD simulations and the prediction by the Lubarda model.
4 pffiffi 1 þ R2 Rcore þ 1 b 1 ¼ pffiffiffi 4 pffiffi R 2ð1 mÞ G 1 þ R2 Rcore 1
ry
ð3Þ
where ry is the yield stress, G is the shear modulus, R is the void radius, Rcore is the dislocation core size, v is the Possion’s ratio, and b is the Burgers vector. As shown in Fig. 3, compared with the MD simulation results, the Rcore = b and Rcore = 2b curves define the upper and lower bounds, respectively. Thus, this parametric study suggests a change of the core structure of shear loop with the increase of void size. The stress required to emit dislocation from a larger void is lower than from a smaller void. It is noted that the force on the dislocation at a given equilibrium distance from the void due to a remote stress increases more rapidly with the ratio R/b than the force due to attraction from the void surface. This is why the critical stress for dislocation emission decreases with increasing the R/ b ratio. 3.2. Effects of void volume fraction
Model
Lx (nm)
Ly (nm)
Lz (nm)
Number of atoms
Model-1 Model-2 Model-3 Model-4
21.472 42.944 64.416 85.888
9.0112 18.0224 27.0336 36.0448
1.76 3.52 5.28 7.04
30,430 239,830 805,440 1,904,500
In order to investigate the effect of void volume fraction on mechanical properties, MD simulations with different void volume fractions were carried out. The model size is consistent with the previous simulation system model-3, but the radius of the nano-
J.P. Wang et al. / Computational Materials Science 132 (2017) 116–124
void has changed. Three different void radii are 3.168 nm, 5.28 nm and 7.392 nm, corresponding to void volume fraction of 1.8%, 5% and 9.9%, respectively. Initial dislocation nucleate at the stress concentration region in different void volume fraction tensile simulations. The stress concentration area where the stress first reaches the threshold strength of the interatomic bond has a larger stress concentration factor than other locations. Therefore, motion of atoms can easily occur in this region. At the convergence of several factors, the high Schmidt factor and the free surface attracting dislocations at the ends of the slip plane, dislocations tend to slip in the slip plane at low applied stresses. Fig. 4 shows that the void volume fraction has remarkable influence on the Young’s modulus and the incipient yield strain, in particular the Young’s modulus. It has been proved that the initial defect has obvious effect on mechanical properties of materials. Our results show that the higher the VVF is, the lower the Young’s modulus and fracture strains is. It was confirmed that with the increase of void defects, the deformation resistance and damage resistance of materials were decreased. The presence of micro void equivalent to an initial damage in the material, taking into account the damage mechanics framework. The material may be considered to have an initial damage with the pre-void adding. According to the damage mechanics, the Young’s modulus can be expressed as:
E ¼ Eð1 DÞ
ð4Þ
where D is the damage variable of the material, in this paper, the void volume fraction (vvf) is chosen to define it. A damage variable as a function of the void volume fraction can be expressed as:
D ¼ gðrÞ
ð5Þ
E is the Young’s modulus of undamaged material and E is the actual modulus of damaged material. D becomes larger, as the radius of the void (r) increases. Moreover, the change in D results in an increase in the Young’s modulus. This is consistent with experiments. Our results are consistent with the above damage mechanics model. As the radius of the void increases, the damage of the material increases gradually. In the case of constant sample size, the larger the damage is, the smaller the Young’s modulus is. We can conclude that during uniaxial tensile simulation process of different void volume fraction, the slip plane of [1 0 0] orienta-
300 [100] E [110] E [111] E [100] εc [110] εc [111] εc
0.12
0.10
200 0.08
εc
Young's modulus (GPa)
250
150
0.06
100
119
and ½1 1 1, [1 1 0] orientation simution tensile simulation is ½1 1 1 1 1, [1 1 1] orientation is ½1 1 1, respectively. lation is [1 1 1] and ½1 3.3. Effects of lattice orientation Lattice orientation plays an important role in the microstructure evolution and mechanical properties of anisotropic monometallic crystals [41–43]. In this section, we discuss the influence of lattice orientation on dislocation nucleation, emission, propagation, stress-strain response and dislocation morphology. 3.3.1. Stress concentration ahead dislocation nucleation Fig. 5(a), (c) and (e) shows the stress distribution around the void before nucleation of the dislocations in the different orientation drawing processes. As observed in these snapshots, the atomic stress tensor rxx is symmetrically distributed along the loading direction before the dislocation nucleation (t = 80 ps for [1 0 0] orientation, t = 41 ps for [1 1 0] orientation and t = 26 ps for [1 1 1] orientation). The atomic stress tensor is calculated using the formula (2) defined in Section 2. And the stress rxx at the void edge perpendicular to the sample loading direction shows significant stress concentration. The distribution of the atomic stress tensor rxx (Fig. 5) reveals that stress concentration is a common phenomenon occurring near nano-void during the stretching of different orientations. Fig. 5(a) , (c) and (e) clearly show that the trend in stress distribution is very similar in different orientations. After dislocation nucleation, we can find three dislocation glide structures from Fig. 5 for the difference of lattice orientation and the slip system activated. According to the stress distribution of three different orientations after dislocation nucleating, Fig. 5(b), (d) and (f) show clearly stress redistribution. The evolution of the microstructure prompt the stress state to change significantly nearby dislocation, and decrease in stress concentration area attributing to the emission of dislocations. This phenomenon indicate that nucleation and glide of dislocations make stress relax in the concentration filed. The same conclusion that the stress decreases as a result of dislocation loop nucleation at the surface of void was also obtained by Traiviratana et al. [20]. The greater the atomic stress, the easier it is to move the atoms. The same results are reported in [30,40], suggesting that dislocations tend to nucleate in the stress concentration region for all single crystal models. At the same time, Wu et al. [44] reported that the microstructural changes of the model due to the forward dislocation emission of the crack tip could lead to a further change in the stress field around the crack tip. The atoms at the site of dislocation emission have the highest tensile stress levels, and have a one-to-one relationship between the stress distribution and microstructure characteristic around the crack tip. Our results of simulation confirmed previous works that dislocation structure initially start to nucleate at the field of stress concentration. The reason is that the stress concentration field have the larger resolved shear stress than other region at the same time. The stress in this area arrives the critical resolved shear stress firstly, and slip system activated. Therefore, dislocation initially nucleate at the stress concentration area and propagation along the activated slip systems.
0.04
50
0.02
0
0.00 9a
15a
21a
Micro-void Radius Fig. 4. The change of E and nc of variable void radius (a is the lattice constant equaling 0.352 nm).
3.3.2. Orientation effect on the dislocation propagation and stressstrain response For the purpose of researching effects of orientation on dislocation propagation and stress-strain response, three different models (refer to Section 2.1) were adopted. The snapshots and stress-strain response curve were summarized from results as following. The snapshots are corresponding with the strain level as points (a), (b), (c), (d), (e) and (f) showing in stress-strain curve, respectively.
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Fig. 5. The contour plots of the atomic tensile stress field rxx of void growth for different orientation at (a) [1 0 0] orientation e = 0.045 t = 80 ps; (b) [1 0 0] orientation e = 0.05 t = 100 ps; (c) [1 1 0] orientation e = 0.032 t = 41 ps; (d) [1 1 0] orientation e = 0.034 t = 64 ps; (e) [1 1 1] orientation e = 0.030 t = 56 ps; (f) [1 1 1] orientation e = 0.031 t = 60 ps; (h is angle of horizontal direction).
Fig. 6. Snapshots of the microstructure evolution of [1 0 0] orientation. (a, b) Dislocation nucleate at the surface of void and generate by semi-circular shear loop. (e) Crack is main cause for fracture of metals. (a) Dislocation nucleation, (b, c, d, e) intermediate states e = 5.4%, 5.8%, 6.2%, 12.4% respectively, (f) rapture failure.
Fig. 6(a) shows that initial dislocation emanate from the free void surface. As seen in Fig. 5, this area is also stress concentration region. For [1 0 0] orientation specimen, the dislocation nucleated around the void due to stress concentration effects. Correspondingly, the stress response shown in Fig. 7 then dropped abruptly after point (a) due to the emission of dislocation. The stressstrain curve (Fig. 7) obviously appears as linear relationship ahead of point (a), where the stress increases linearly with the increasing in the applied strain. It elucidates that the single crystal nickel specimen undergoes purely elastic deformation before emission of initial dislocation and the slope of the line was not changed. That means the deformation of specimen abides the Hooke’s law in e = 0–ec. This can be interpreted as that the stress only induces the elasticity distortion of lattice from micro-scale, and cannot break the metal-metal bonds. The propagation and expansion of initial dislocation nucleation after loading some time can be found from Fig. 6(b). In addition, the snapshots of microstructure evolution in the tensile process show that generation of dislocation is conducted by semi-circular shear loops as shown in Fig. 6(b). The dislocation source and glide clearly present in the snapshot. Fur-
ther step of dislocation propagation was shown in Fig. 6(c), because of the randomness of the atomic motion, dislocation glide only occurred at the half bottom of the simulation specimen. With the constant loading, upper half of specimens occurred plastic deformation and the dislocation nucleated at the free surface of void as shown in Fig. 6(d). As observed in Fig. 7, the stress kept decreasing from point (a) to point (d) with the dislocation nucleation and propagation of Fig. 6(b), (c), (d). The fluctuation of stress-strain carve between point (d) and point (e) can be attributed to the piling up of dislocations and formation of the micro voids (Fig. 6(d)) and cracks (Fig. 6(e)). The dislocation piling up made stress increase. Moreover, the formation of micro voids and cracks relax the stress field around them. Then stress maintained a sustained downward trend with the propagation of cracks. When loading is further increased, the crack extension was achieved through void nucleation, void growth and coalescence, which could be verified by the snapshots of point (d), (e) and (f) in Fig. 7. Similar to [1 0 0] orientation simulation, dislocation nucleate in the stress concentration region, and propagate by the semi-circular shear ring in the sliding plane during tensile simulation along
J.P. Wang et al. / Computational Materials Science 132 (2017) 116–124
(a) Dislocation Nucleation
7
Von-Mises Stress (GPa)
6
(b)
(a) εc=0.05 σc=6.97
5
(c)
(f) εf=0.147
4
σf=0.80
3
(e)
(d)
2
(f) Fracture Point 1 0 0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0.20
Strain Fig. 7. Stress-Strain response curve of [1 0 0] orientation.
[1 1 0] orientation. These phenomena can be observed in Fig. 8 (a) and (b). We can observe the stress-strain curve sharply dropping-off in Fig. 9 after point (a). As shown in Fig. 8, the [1 1 0] orientation has a lower initial yield stress than that of [1 0 0], possibly because the [1 1 0] direction is the close-packed orientation of the crystal structure, dislocation nucleation along [1 1 0] orientation easier and Faster than the [1 0 0] orientation. After dislocation nucleation, the stress decreases with increasing strain due to the propagation of dislocations. The snapshots of microstructural evolution clearly show a view of the change in atomic position. The peak stress between point (c) and (d) is caused by the accumulation and cross slip of dislocations. The reaction of partial dislocation effectively hinders the movement of dislocations, which is the main cause of stress increasing. In addition, the stress field can be changed by the evolution of the microstructure. The nucleation and growth of the micro-voids change the stress state and lower the stress. The mechanism of micro-void nucleation is the cross slip caused by the second slip system. With the continuous loading, more and more dislocation nucleation and slip will increase the dislocation density. Dislocation density has great influence on dislocation movement. The higher the dislocation density is, the more difficult the dislocation movement is.
121
The deformation of the metal material increases, the dislocation density increase, leading to a rapid increase in the plastic deformation resistance of the metal. The mechanical properties of the material: hardness and strength significantly increased, plasticity and toughness decreased, which is so-called ‘‘Work hardening” phenomenon. The numerical results show that the increase of applied stress will aggravate the dislocation activity. It is necessary to apply a larger stress to obtain further deformation for increasing of the difficulty of dislocation movement. So we can find a narrow stress increase between point (d) and (e) in Fig. 9. As for the stress decrease between point (d) and (e), it attributes to the growth and coalescence of micro voids, and the nucleation and propagation of crack. The void coalescence and crack propagation, which enhances the plastic deformation and affects the fracture mechanism of the specimen, can observe in Fig. 8(e) as the tensile simulation progresses. The stretching process of [1 1 1] orientation is shown in Fig. 10, and the stress-strain response is shown in Fig. 11. Similar with the tensile simulation along [1 0 0] and [1 1 0] orientation, the dislocations nucleate in the stress-concentrated region and propagate through the semi-circular shear loops in [1 1 1] simulation. However, the evolution of microstructures is different, as shown in Fig. 10. Such as the presence of twin and Lomer-Cottrell dislocations, retard the initial dislocation slip and increase the difficulty of plastic deformation. Due to planar faults’ propagation, the two faults can meet with each other to form a Lomer-Cottrell stairrod lock. The Lomer-Cottrell dislocation is one of the major hardening mechanisms during plastic deformation [41]. The stress between points (c) and (d) in the stress-strain curve first decreases and then increases. The cause of the stress reduction is the generation of slip and dislocations, and the increase in stress is due to the activation of the second slip system. The intersection of these two slip systems hinders the movement of the displacement. The micro crack nucleation on the interstitial surface can be seen in Fig. 7(d) due to dislocation stacking. In addition, micro crack began to propagate and grow as observed in Fig. 10(e), which can be used to explain the appearances of stress peak between point (d) and (e) in stress-strain response. The stress-strain curve tended to decrease as the result of further plastic deformation. The mechanical properties of different orientation samples were found by the stress-strain curves obtained by [1 0 0], [1 1 0] and [1 1 1] tensile simulations. The results are shown in Table 3.
Fig. 8. Snapshots of the microstructure evolution of [1 1 0] orientation. (a, b) Dislocation nucleate at the surface of void and generate by semi-circular shear loop. (d) Void nucleation cause by dislocation interaction. (e) Crack form void coalescence. (a) Dislocation nucleation, (b, c, d, e) intermediate states e = 3.8%, 4.8%, 8.2%, 10.4% respectively, (f) rapture failure.
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(a) εc=0.034
(b)
6
Von-Mises Stress (GPa)
8
(a) Dislocation Nucleation
σc=6.77
5
(c)
(f) εf=0.137
4
σf=0.61 3
(e)
2
(d)
1
(a) Dislocation Nucleation
7
Von-Mises Stress (GPa)
7
(a) εc=0.031
(b)
6
σc=7.31 (d)
(e)
(c)
5
Fracture Point ε=0.161 σ=0.92
4
(f)
3 2
Fracture Point
(f) Fracture Point 1
0 0.00
0.05
0.10
0.15
0.20
0.25
0.30
0 0.00
0.05
0.10
0.20
0.25
0.30
Fig. 11. Stress-Strain response curve of [1 1 1] orientation.
Fig. 9. Stress-strain response curve of [1 1 0] orientation.
The stress-strain curves of [1 0 0], [1 1 0] and [1 1 1] orientation simulations (Figs. 6, 8 and 10) were found to follow a linear relationship before the incipient yield stress point, which indicates that the material is in the elastic deformation stage, and the Young’s modulus of different orientations are: E[1 1 1]>E[1 1 0]>E[1 0 0]. The differences behaved in other mechanic properties is shown in Table 3, where critical yield stress is: rc [1 1 1]>rc[1 0 0]>rc[1 1 0], critical yield strain is: nc[1 1 1]nf[1 0 0]>nf[1 1 0]. The fracture strain is the elongation of the specimen. The elongation of the sample along 111 orientation is greatest, which illustrate that the sample has good ductility performance along [1 1 1] orientation in the same initial damage. The trend of these mechanical properties make good agreement with literature [45]. In fcc crystal, dislocation motion is observed commonly to obey Schmidt’s law along the {1 1 1}h1 1 0i family of slip systems. Because of different resolved stresses and active slip system caused by loading orientation, nucleation, propagation, and interaction of dislocations result in shape distortion of the initially cylinder void. The consequence of tensile simulation with different orientation shows that [1 0 0], [1 1 0] and [1 1 1] showed apparently difference of dislocation slip and evolution of microstructure when
0.15
Strain
Strain
Table 3 Mechanical property for [1 0 0], [1 1 0] and [1 1 1] orientations.
E (GPa) rc (GPa) nc nf np = nf-nc
[1 0 0]
[1 1 0]
[1 1 1]
134.6 7.17 0.053 0.147 0.094
198.2 6.77 0.034 0.137 0.103
234.1 7.31 0.031 0.161 0.130
crystal material transform deformation mechanism from elastic to ductile and begin incipient yielding. This can be attributed to the difference in activation of the slip system as well as the differences between single and multiple slips. Slip is a plastic deformation process produced by dislocation motion, and the analysis of atomic microstructure displays the generation of slip bands along the [1 1 0] direction. Bringa et al. [13] mentioned that the following number of slip systems with highest Schmidt factor: [1 0 0] [1 1 0] [1 1 1]
eight slip system four slip system six slip system
Fig. 10. Snapshots of the microstructure evolution of [1 1 1] orientation. (a, b) Dislocation nucleate at the surface of void and generate by semi-circular shear loop. (e) Crack form owing to dislocation piling up. (f) Lomer-Cottrell dislocation. (a) Dislocation nucleation, (b, c, d, e, f) intermediate states e = 3.6%, 4.4%, 7.2%, 10.6%, 12.4% respectively.
J.P. Wang et al. / Computational Materials Science 132 (2017) 116–124
123
Experiment proved that only the {1 1 1}h1 1 0i slip system shown in Fig. 12 was activated under the temperature lower than 900 °C. The Schmidt factor of the {1 1 1}h1 1 0i slip system is concluded in Table 4. The lower yield stress for [1 0 0] orientation is connected to the greater availability of slip systems (eight) with the highest Schmidt factor (0.408), in comparison with four systems with the Schmidt factor of 0.408 for [1 1 0] orientation and six systems with 0.2722 for [1 1 1]. Typically, the resolved shear stress is important because it is well known that dislocation motion in fcc single crystal is governed by the critical resolved shear stress via Schmidt’s law. Therefore, crystallographic orientation shows an important effect on how the Schmidt stress and stress normal to the slip plane contribute to nucleate dislocations. 3.3.3. Orientation effect on the dislocation morphology The visualization of stacking faults and dislocations is accomplished using filters with centrosymmetry parameters. Morphology of characteristic slip bands in the tensile process for different orientation can be found referring to Figs. 13–15. For instance, the variety of dislocation nucleation, distribution of dislocation slip band and the slip direction of different orientations. The [1 0 0] oriented slip band is perpendicular to the loading direction and is oriented at an angle of 45° to the [0 1 0] orientation. For a more detailed MD view selected by CSP > 4 expression, the internal dislocation slip profile of the sample is shown in Fig. 13 (b). Dislocations at the boundary altered the arrangement of the boundary atoms, as shown in Fig. 13(c). In Fig. 13(b), dislocations are represented in the (1 1 1) and (1 1 1) planes, which are projected on a single surface for clarity. The perfect dislocations
Fig. 13. Morphology of slips (a) [1 0 0] orientation overall sample morphology (b) dislocation characteristic by deleting the atoms with CSP < 4 (c) dislocation network on boundary.
Fig. 14. Morphology of slips (a) [1 1 0] orientation overall sample morphology, (b) dislocation characteristic by deleting the atoms with CSP < 4, (c) dislocation network on boundary.
[1 1 0] and [1 0 1], with Burgers vectors aligned with the intersections of the planes, can be decomposed into partials in (1 1 1) along the following reactions: Fig. 12. {1 1 1} h1 1 0i slip system.
a=2½1 1 0 ! a=6½2 1 1 þ a=6½1 2 1 a=2½1 0 1 ! a=6½2 1 1 þ a=6½1 1 2
Table 4 Schmidt factor of the {1 1 1} h1 1 0i slip system. Orientation
[1 0 0] [1 1 0] [1 1 1]
{111}h1 1 0i Number
Schmidt factor
8 4 4 8 6 6
0.4082 0.0000 0.4082 0.0000 0.2722 0.0000
The energy criterion states that a dislocation reaction should lead to lower energy configuration. The above reaction is energetically favorable, so reaction could conduct. As shown in Fig. 14(b), the plane of the [1 1 0] crystal orientation is the topography (1 1 1) and (1 1 1), which is consistent with the result of Bringa [13]. As the result of dislocation gliding, we can find dislocation piling up in the junction region between Slip plane (1 1 1) and (1 1 1) in the tensile process. The new voids nucleation can be observed at that region as shown in the Fig. 8.
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elongation of the specimen. Among the three orientations, the elongation of the sample along [1 1 1] orientation is the greatest, which illustrate that the sample has good ductility performance along [1 1 1] orientation in the same initial damage. The mechanism of this phenomenon is also analyzed by molecular dynamics. The reaction between the dislocations and the boundary leads to the rearrangement of the boundary atoms, which proves that the dislocation can not stop inside the crystal, or closed or terminated at the crystal surface or grain boundary. The difference of the rearrangement of the boundary atoms is due to the difference of the sliding systems of different orientations. Acknowledgements This work is supported by the National Natural Science Foundation of China (Nos. 51375388 and 51210008) and the Natural Science Basic Research Plan in Shaanxi Province of China (No. 2016JM1020).
Fig. 15. Morphology of slips (a) [1 1 1] orientation overall sample morphology, (b) dislocation characteristic by deleting the atoms with CSP < 4, (c) dislocation network on boundary.
The slip bands of [1 1 1] crystal orientation is about 75° with the loading direction, and the main slip plane is (1 1 1). Since the dislocations are the boundaries of the slip and nonslip regions, the dislocation lines cannot be stopped within the bulk, and can only stop at the surface of the crystal or grain boundary. When dislocation glide to the boundary, the atomic redistribution can be found by observing the reaction of dislocation and boundary atoms. Whereas the different form of slip cause the different motion of the boundary atoms. Fig. 13(c) shows that the boundary atoms slide by approximately 45° along the (1 1 1) and (1 1 1) planes due to the dislocations, and the boundary atoms without dislocations are regular distributions. Fig. 14(c) shows additional semi-atomic surface embedding in the boundary region of dislocation stacking. The reaction part of the dislocation and the boundary looks like a wave, as shown in Fig. 14(c). A layer atoms was embedded in the boundary as the dislocation reacting with the boundary of [1 1 1] orientation in Fig. 15(c). The first zone have one layer atoms but the second have two layers. The former is the initial boundary and the latter is boundary reacting with the dislocation slip band. 4. Conclusion MD simulation was conducted to study the effects of crystal orientation on the initial stress distribution, dislocation nucleation and emission, typical dislocation morphology and stress-strain response of single crystal nickel with nanovoid. The system effect and void volume fraction effect were simulated as well. Based above the computations and discussions, the conclusions were summarized as following: The results of different sizes and void volume fractions show significant effects on mechanical properties such as Young’s modulus, incipient yield stress and incipient yield strain. The results of different orientation simulation show that the dislocation structure begins to nucleate in the stress concentration region. The critical stress of [1 0 0], [1 1 0] and [1 1 1] orientation is 6.97 GPa, 6.77 GPa and 7.31 GPa, respectively. The main reason for the above phenomenon is that the shear stress reaches the threshold stress of dislocation nucleation. The results of dislocation propagation and stress-strain responses reveal three different orientations of elastic-plastic properties. The fracture strain is the
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Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Orientation effects on the tensile properties of single crystal nickel with nanovoid: Atomistic simulation J.P. Wang, Z.F. Yue, Z.X. Wen ⇑, D.X. Zhang, C.Y. Liu School of Mechanics, Civil Engineering and Architecture, Northwestern Polytechnical University, Xi’an 710072, PR China
a r t i c l e
i n f o
Article history: Received 16 May 2016 Received in revised form 19 February 2017 Accepted 20 February 2017
Keywords: Molecular Dynamics (MD) simulation Lattice orientation Nanovoid Dislocation nucleation
a b s t r a c t The tensile behavior of monocrystalline nickel with nano-void was studied via molecular dynamics simulation (MD) considering different lattice orientations. A series of simulations were performed to analyze the effect of system size, void volume fraction and lattice orientation on mechanical properties and microstructure evolution. The influence of size of sample on the incipient yield stress is discussed. The results show that void volume fractions have significant effects on Young’s modulus, incipient yield stress and incipient yield strain. Dislocation structure begins to nucleate in the stress concentration area. The critical stress of [1 0 0], [1 1 0] and [1 1 1] orientation is 6.97 GPa, 6.77 GPa and 7.31 GPa, respectively. The results of dislocation propagation and stress-strain responses reveal the elastic-plastic properties of different orientations. The fracture strain is the elongation of the specimen. The elongation of the sample along [1 1 1] orientation is greatest, which illustrate that the sample has good ductility performance along [1 1 1] orientation in the same initial damage. The reaction between the dislocations and the boundary leads to the rearrangement of the boundary atoms, which proves that the dislocation can not be stopped inside the crystal, or closed or terminated at the crystal surface or grain boundary. The difference of the rearrangement of the boundary atoms is due to the difference of the sliding systems of different orientations. Ó 2017 Elsevier B.V. All rights reserved.
1. Introduction The failure modes of materials have significant influence on the design of material properties in materials science. Rapture failure at the macroscopic scale can be attributed to nucleation, growth and propagation of cracks, but at the microscopic scale cracks are initially easily formed at defects in the casting process, such as voids and inclusions [1]. These defects are known to play a fundamental role in the deformation of the material. Nucleation, growth and coalescence of voids are deemed as the primary mechanism of ductile material fracture, in which void growth is particularly important. Therefore, it is necessary to study the deformation response of porous materials with the consideration of microstructure evolution. Most previous work on void growth studies, both experimental and simulation approaches, have been used to better describe ductile fracture processes. Furthermore, some numerical methods, such as crystal plasticity finite element method [2–7], dislocation dynamics and molecular dynamics [8–19], have been used to reveal the mechanism of void growth. ⇑ Corresponding author. E-mail address: [email protected] (Z.X. Wen). http://dx.doi.org/10.1016/j.commatsci.2017.02.024 0927-0256/Ó 2017 Elsevier B.V. All rights reserved.
Molecular dynamics (MD) is an effective tool for studying the fracture mechanism of micro-scale materials. It can directly observe the motion of atoms and obtain the details of atomic deformation. MD simulations in monocrystalline and bicrystalline copper were carried out with LAMMPS (large-scale atomic/Molecular Massively Parallel Simulator) to reveal void growth mechanisms. The results confirm that the emission of dislocation shear loops is the primary mechanism of void growth [20–24], consistent with the analytical results of [21] and [25–28]. Tang and his colleagues simulated the growth behavior of spherical nanovoids in c-TiAl single crystal [17]. They found that the emission of dislocation loop is the main reason for the growth of void, and the continuous dislocation nucleation and the increase of the shear loop prompt the growth of voids. In addition, extensive work has been performed on other factors affecting void growth, such as size effect [13,14,16,26], initial void volume fraction [2,5,8,9,20,24,26] And strain rate [16,26,28]. The initial stage of the growth of nanocrystalline voids in monocrystalline aluminum was studied by MD simulation. The simulation results showed that the dependence of the critical negative pressure on void radius, system size and temperature was obtained at high rate tension [24]. The decrease in the ratio of the radius of the void to the system size leads to an increase in the tensile strength of the system. Tang
J.P. Wang et al. / Computational Materials Science 132 (2017) 116–124
et al. [29] based on a new analytical method for the emission energy of the dislocation loop, the reduction of stress with increasing void size is analyzed. Bhatia et al. [26] investigated nanovoid growth using MD and revealed its dependence on void size, strain rate, crystallographic loading orientation, initial nanovoid volume fraction, and simulation cell size. Dislocations and shear loops nucleate on different slip systems with different loading orientations, which include orientation that favors single slip and multiple slip. It is well known that the mechanical properties of monocrystalline metallic materials have a strong correlation with their crystal orientation. The mechanisms of void growth and failure modes are significantly different in different lattice orientations. Therefore, it is significant to study the void growth behavior of single crystal metallic materials with different orientations. A number of studies have been carried out in this area [3,7,9,10,13,15,30,31]. Potirniche et al. [15] interpreted that plasticity anisotropy caused by initial lattice orientation has only a minor effect on void growth in the microscale regions. Stress triaxiality is the main factor controlling the growth and coalescence of voids. The numerical results of Sangyul Ha and KiTae Kim [3] showed that the stress triaxiality and the deformation pattern specified by the crystal orientation have competitive effects on the evolution of voids. For low-level stress triaxiality, the deformation mode is mainly determined by the crystal orientation. Eduardo et al. [13] investigated the effect of loading orientation on void initiation in FCC metals. The results showed that there is a significant effect of the loading orientation on the sequence where the loops form and interact. Based on the above studies, it has been found that dislocation nucleation and morphology of different crystal orientations have not been systematically studied in detail. In this study, uniaxially stretched MD simulation was used to analyze the effect of lattice orientation on tensile fracture in thin nickel specimens with infinite cylindrical voids. Nickel is a typical FCC material. In addition, its alloys, such as B2-type NiAl, LI2-type Ni3Al, are used in the aerospace industry due to their excellent high temperature performance. The study will examine the microstructural evolution and the mechanism of void growth through stress-strain responses and morphological changes of voids in different crystal orientations. At the same time, size effects and void volume fraction effects are taken into account. 2. MD simulation presentation 2.1. Atomistic model and simulation conditions In this simulation, a system of size 64:416 27:0336 5:28 nm3 was chosed to investigate the orientation effect, as shown in Fig. 1. Considering that the size of the model must be a multiple of the lattice spacing, the lattice spacing in the [1 1 1]-oriented z is different from that of [1 0 0] and [1 1 0]. A [1 1 1]-oriented simulation system of size 64:416 27:0336 5:4758 nm3 is used
117
herein. The radius of the cylindrical nano-voids was 5.28 nm. The size in z direction is greater than the potential cut-off radius 0.58 nm. Model with three different orientations was used to simulate the uniaxial tensile behavior of single crystal nickel and nano-voids. The parameters are shown in Table 1. Periodic boundary conditions (PBC) was applied to the zdirection, x and y were the shrink-wrapped boundary. Microcanonical ensemble was taken during the simulation process, and the temperature of the thermostat atom is controlled by rescaling the atomic velocity by 40 picoseconds in 1 fs time step. Therefore, the model relaxes to the equilibrium state at the beginning of the simulation. These models hold the left boundary and are uniaxially loaded at a constant strain rate of 5 108 s1 at the right boundary, as shown in Fig. 1. All simulations were performed by LAMMPS at an invariable temperature of 300 K. The atomic stress tensor is calculated using the virial definition, which can be expressed as: N X f a ði; jÞrb ði; jÞ
rt ðiÞ ¼
ð1Þ
j–ðiÞ
re ðiÞ ¼
n X
rt ðiÞ=n
ð2Þ
t¼1
where f a is the interatomic force between atom i and j in the direction a， r b is the distance in direction b. Considering average over the volume around atom i within the cut-off distance, re is the average atomic stress tensor obtained by function (2). 2.2. EAM potential In recent years, the embedded atomic method [32,33] (EAM) interatomic potential has been widely used for crack propagation in metal materials and many other simulations. Compared with the F-S and L/J potentials, it can calculate the elastic properties of materials, energy changes, and the position of atoms in the whole simulation process of metal materials. Many examples have shown that EAM is an accurate representation of the interatomic forces in a metal lattice. In this study, the embedded atom method of Ni developed by Mishin et al. [34] was used to simulate the uniaxial tensile beha vior of single-crystal nickel with nanovoid. A reasonable simulation of fracture and damage can be obtained considering that the potential energy can well describe the bonding capacity in a metal system and the dependence of the strength of a single bond on the local environment (e.g., surface and defects). In addition, this EAM potential, based on the first principles, can reproduce many of the fundamental properties such as vacancy migration energy, unstable and stable stacking fault energy. Many studies [35–38] have used this EAM potential to investigate the deformation behavior of materials, which further validates the accuracy and reliability of this interatomic potential.
Fig. 1. Simulation Model (left boundary is fixed, right is loading at a constant strain rate).
J.P. Wang et al. / Computational Materials Science 132 (2017) 116–124
Table 1 Parameters for specimens with different orientations.
15.0
300
X
Y
Z
Number of atoms
[1 0 0] [1 1 0] [1 1 1]
[1 0 0] [1 1 0] [1 1 1]
[0 1 0] 1 0 ½1 ½1 1 2
[0 0 1] [0 0 1] 1 0 ½1
805,440 804,525 832,348
2.3. Post processing The open visualization tool (Ovito) of Stukowski [39] is used to observe and analyze the atomic configuration throughout the stretching process. Ovito uses the centrasymmetry parameter defined by Kelchner et al. [40] to highlight defective atoms and observe the evolution of slip bands, partial dislocations, stacking faults and other defects during rapture failure. It is noted that the slip system can be observed by controlling the centrosymmetry parameter.
250
Young's modulus (GPa)
Crystal orientation
[100] E [110] E [111] E [100] σc [110] σc [111] σc
12.5
200
10.0
150
100 7.5 50
5.0 --
0 Model-1
Model-2
model-3
σc (Incipient Yield Stress / Gpa)
118
model-4
Model Fig. 2. The change of E and rc of variable system size for different orientations.
3. Results and discussion 0.2
3.1. System size effects
Lubarda Model with Rcore=b
Table 2 Parameters for specimens with different size.
Normalized Yield Stress σy /G
Lubarda Model with Rcore=2b
The length scale significantly affects the mechanical properties of the material, such as the stress-strain response [14,16,41]. To study the effect of the specimen size，MD simulation was conducted with four systems of different sizes varying from 21:472 9:0112 1:76 nm3 to 85:888 36:0448 7:04 nm3 as shown in Table 2 (the model referred in Section 2.1), while void volume fraction was fixed as 5%. The loading constant strain rate was 5 108 s1, and the temperature was 300 K. The simulated results of different orientations interpret that the size effect is evident. All the curves (can be saw at Section 3.3) exhibit a linear stress-strain behavior before up to their peak stress points where an abrupt stress drop is observed. The peak stress increases with the decrease of specimen size for a fixed VVF. In addition, the slope of the line before point (a) was defined as the Young’s modulus by Zhao et al. [19]. They defined the peak stress at point (a) of stress-strain curve as the incipient yield strength as well. In Fig. 2, the initial slope of all these stress-strain curves, remains unchanged with the increasing of specimen size from Model-1 to Model-4. This means that the Young’s modulus is almost insensitive to the specimen size. Comparatively speaking, size effect on the incipient yield stress is quite notable. The trend of incipient yield stress conforms to the analytical results by literature [20,21]. And the reason can be concluded as that the development of dislocation intersection and the propagation occurs more easily before the dislocation reaches the boundary of the simulation ‘‘box” for larger system size model. Furthermore, the increase of plastic flow solved shear stress at decreasing of system cell size is concluded from Fig. 2, which is consistent with the gradient plasticity of experiment measurement. Another reason for this phenomenon proposed by Bhatia [26] is the volume decreases faster than surface area as system cell size decreasing with fixed VVF, which leads to a higher stress concentration. In Fig. 3, the yield stress obtained from MD simulations is compared with the Lubarda model [21],
[111] Atomistic Calculations [100] Atomistic Calculations [110] Atomistic Calculations
0.1
υ=0.291 G=73GPa 0.0
0
5
10
15
20
25
30
Normalized Void Radius R/b Fig. 3. Comparison of the critical stress obtained from present MD simulations and the prediction by the Lubarda model.
4 pffiffi 1 þ R2 Rcore þ 1 b 1 ¼ pffiffiffi 4 pffiffi R 2ð1 mÞ G 1 þ R2 Rcore 1
ry
ð3Þ
where ry is the yield stress, G is the shear modulus, R is the void radius, Rcore is the dislocation core size, v is the Possion’s ratio, and b is the Burgers vector. As shown in Fig. 3, compared with the MD simulation results, the Rcore = b and Rcore = 2b curves define the upper and lower bounds, respectively. Thus, this parametric study suggests a change of the core structure of shear loop with the increase of void size. The stress required to emit dislocation from a larger void is lower than from a smaller void. It is noted that the force on the dislocation at a given equilibrium distance from the void due to a remote stress increases more rapidly with the ratio R/b than the force due to attraction from the void surface. This is why the critical stress for dislocation emission decreases with increasing the R/ b ratio. 3.2. Effects of void volume fraction
Model
Lx (nm)
Ly (nm)
Lz (nm)
Number of atoms
Model-1 Model-2 Model-3 Model-4
21.472 42.944 64.416 85.888
9.0112 18.0224 27.0336 36.0448
1.76 3.52 5.28 7.04
30,430 239,830 805,440 1,904,500
In order to investigate the effect of void volume fraction on mechanical properties, MD simulations with different void volume fractions were carried out. The model size is consistent with the previous simulation system model-3, but the radius of the nano-
J.P. Wang et al. / Computational Materials Science 132 (2017) 116–124
void has changed. Three different void radii are 3.168 nm, 5.28 nm and 7.392 nm, corresponding to void volume fraction of 1.8%, 5% and 9.9%, respectively. Initial dislocation nucleate at the stress concentration region in different void volume fraction tensile simulations. The stress concentration area where the stress first reaches the threshold strength of the interatomic bond has a larger stress concentration factor than other locations. Therefore, motion of atoms can easily occur in this region. At the convergence of several factors, the high Schmidt factor and the free surface attracting dislocations at the ends of the slip plane, dislocations tend to slip in the slip plane at low applied stresses. Fig. 4 shows that the void volume fraction has remarkable influence on the Young’s modulus and the incipient yield strain, in particular the Young’s modulus. It has been proved that the initial defect has obvious effect on mechanical properties of materials. Our results show that the higher the VVF is, the lower the Young’s modulus and fracture strains is. It was confirmed that with the increase of void defects, the deformation resistance and damage resistance of materials were decreased. The presence of micro void equivalent to an initial damage in the material, taking into account the damage mechanics framework. The material may be considered to have an initial damage with the pre-void adding. According to the damage mechanics, the Young’s modulus can be expressed as:
E ¼ Eð1 DÞ
ð4Þ
where D is the damage variable of the material, in this paper, the void volume fraction (vvf) is chosen to define it. A damage variable as a function of the void volume fraction can be expressed as:
D ¼ gðrÞ
ð5Þ
E is the Young’s modulus of undamaged material and E is the actual modulus of damaged material. D becomes larger, as the radius of the void (r) increases. Moreover, the change in D results in an increase in the Young’s modulus. This is consistent with experiments. Our results are consistent with the above damage mechanics model. As the radius of the void increases, the damage of the material increases gradually. In the case of constant sample size, the larger the damage is, the smaller the Young’s modulus is. We can conclude that during uniaxial tensile simulation process of different void volume fraction, the slip plane of [1 0 0] orienta-
300 [100] E [110] E [111] E [100] εc [110] εc [111] εc
0.12
0.10
200 0.08
εc
Young's modulus (GPa)
250
150
0.06
100
119
and ½1 1 1, [1 1 0] orientation simution tensile simulation is ½1 1 1 1 1, [1 1 1] orientation is ½1 1 1, respectively. lation is [1 1 1] and ½1 3.3. Effects of lattice orientation Lattice orientation plays an important role in the microstructure evolution and mechanical properties of anisotropic monometallic crystals [41–43]. In this section, we discuss the influence of lattice orientation on dislocation nucleation, emission, propagation, stress-strain response and dislocation morphology. 3.3.1. Stress concentration ahead dislocation nucleation Fig. 5(a), (c) and (e) shows the stress distribution around the void before nucleation of the dislocations in the different orientation drawing processes. As observed in these snapshots, the atomic stress tensor rxx is symmetrically distributed along the loading direction before the dislocation nucleation (t = 80 ps for [1 0 0] orientation, t = 41 ps for [1 1 0] orientation and t = 26 ps for [1 1 1] orientation). The atomic stress tensor is calculated using the formula (2) defined in Section 2. And the stress rxx at the void edge perpendicular to the sample loading direction shows significant stress concentration. The distribution of the atomic stress tensor rxx (Fig. 5) reveals that stress concentration is a common phenomenon occurring near nano-void during the stretching of different orientations. Fig. 5(a) , (c) and (e) clearly show that the trend in stress distribution is very similar in different orientations. After dislocation nucleation, we can find three dislocation glide structures from Fig. 5 for the difference of lattice orientation and the slip system activated. According to the stress distribution of three different orientations after dislocation nucleating, Fig. 5(b), (d) and (f) show clearly stress redistribution. The evolution of the microstructure prompt the stress state to change significantly nearby dislocation, and decrease in stress concentration area attributing to the emission of dislocations. This phenomenon indicate that nucleation and glide of dislocations make stress relax in the concentration filed. The same conclusion that the stress decreases as a result of dislocation loop nucleation at the surface of void was also obtained by Traiviratana et al. [20]. The greater the atomic stress, the easier it is to move the atoms. The same results are reported in [30,40], suggesting that dislocations tend to nucleate in the stress concentration region for all single crystal models. At the same time, Wu et al. [44] reported that the microstructural changes of the model due to the forward dislocation emission of the crack tip could lead to a further change in the stress field around the crack tip. The atoms at the site of dislocation emission have the highest tensile stress levels, and have a one-to-one relationship between the stress distribution and microstructure characteristic around the crack tip. Our results of simulation confirmed previous works that dislocation structure initially start to nucleate at the field of stress concentration. The reason is that the stress concentration field have the larger resolved shear stress than other region at the same time. The stress in this area arrives the critical resolved shear stress firstly, and slip system activated. Therefore, dislocation initially nucleate at the stress concentration area and propagation along the activated slip systems.
0.04
50
0.02
0
0.00 9a
15a
21a
Micro-void Radius Fig. 4. The change of E and nc of variable void radius (a is the lattice constant equaling 0.352 nm).
3.3.2. Orientation effect on the dislocation propagation and stressstrain response For the purpose of researching effects of orientation on dislocation propagation and stress-strain response, three different models (refer to Section 2.1) were adopted. The snapshots and stress-strain response curve were summarized from results as following. The snapshots are corresponding with the strain level as points (a), (b), (c), (d), (e) and (f) showing in stress-strain curve, respectively.
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Fig. 5. The contour plots of the atomic tensile stress field rxx of void growth for different orientation at (a) [1 0 0] orientation e = 0.045 t = 80 ps; (b) [1 0 0] orientation e = 0.05 t = 100 ps; (c) [1 1 0] orientation e = 0.032 t = 41 ps; (d) [1 1 0] orientation e = 0.034 t = 64 ps; (e) [1 1 1] orientation e = 0.030 t = 56 ps; (f) [1 1 1] orientation e = 0.031 t = 60 ps; (h is angle of horizontal direction).
Fig. 6. Snapshots of the microstructure evolution of [1 0 0] orientation. (a, b) Dislocation nucleate at the surface of void and generate by semi-circular shear loop. (e) Crack is main cause for fracture of metals. (a) Dislocation nucleation, (b, c, d, e) intermediate states e = 5.4%, 5.8%, 6.2%, 12.4% respectively, (f) rapture failure.
Fig. 6(a) shows that initial dislocation emanate from the free void surface. As seen in Fig. 5, this area is also stress concentration region. For [1 0 0] orientation specimen, the dislocation nucleated around the void due to stress concentration effects. Correspondingly, the stress response shown in Fig. 7 then dropped abruptly after point (a) due to the emission of dislocation. The stressstrain curve (Fig. 7) obviously appears as linear relationship ahead of point (a), where the stress increases linearly with the increasing in the applied strain. It elucidates that the single crystal nickel specimen undergoes purely elastic deformation before emission of initial dislocation and the slope of the line was not changed. That means the deformation of specimen abides the Hooke’s law in e = 0–ec. This can be interpreted as that the stress only induces the elasticity distortion of lattice from micro-scale, and cannot break the metal-metal bonds. The propagation and expansion of initial dislocation nucleation after loading some time can be found from Fig. 6(b). In addition, the snapshots of microstructure evolution in the tensile process show that generation of dislocation is conducted by semi-circular shear loops as shown in Fig. 6(b). The dislocation source and glide clearly present in the snapshot. Fur-
ther step of dislocation propagation was shown in Fig. 6(c), because of the randomness of the atomic motion, dislocation glide only occurred at the half bottom of the simulation specimen. With the constant loading, upper half of specimens occurred plastic deformation and the dislocation nucleated at the free surface of void as shown in Fig. 6(d). As observed in Fig. 7, the stress kept decreasing from point (a) to point (d) with the dislocation nucleation and propagation of Fig. 6(b), (c), (d). The fluctuation of stress-strain carve between point (d) and point (e) can be attributed to the piling up of dislocations and formation of the micro voids (Fig. 6(d)) and cracks (Fig. 6(e)). The dislocation piling up made stress increase. Moreover, the formation of micro voids and cracks relax the stress field around them. Then stress maintained a sustained downward trend with the propagation of cracks. When loading is further increased, the crack extension was achieved through void nucleation, void growth and coalescence, which could be verified by the snapshots of point (d), (e) and (f) in Fig. 7. Similar to [1 0 0] orientation simulation, dislocation nucleate in the stress concentration region, and propagate by the semi-circular shear ring in the sliding plane during tensile simulation along
J.P. Wang et al. / Computational Materials Science 132 (2017) 116–124
(a) Dislocation Nucleation
7
Von-Mises Stress (GPa)
6
(b)
(a) εc=0.05 σc=6.97
5
(c)
(f) εf=0.147
4
σf=0.80
3
(e)
(d)
2
(f) Fracture Point 1 0 0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0.20
Strain Fig. 7. Stress-Strain response curve of [1 0 0] orientation.
[1 1 0] orientation. These phenomena can be observed in Fig. 8 (a) and (b). We can observe the stress-strain curve sharply dropping-off in Fig. 9 after point (a). As shown in Fig. 8, the [1 1 0] orientation has a lower initial yield stress than that of [1 0 0], possibly because the [1 1 0] direction is the close-packed orientation of the crystal structure, dislocation nucleation along [1 1 0] orientation easier and Faster than the [1 0 0] orientation. After dislocation nucleation, the stress decreases with increasing strain due to the propagation of dislocations. The snapshots of microstructural evolution clearly show a view of the change in atomic position. The peak stress between point (c) and (d) is caused by the accumulation and cross slip of dislocations. The reaction of partial dislocation effectively hinders the movement of dislocations, which is the main cause of stress increasing. In addition, the stress field can be changed by the evolution of the microstructure. The nucleation and growth of the micro-voids change the stress state and lower the stress. The mechanism of micro-void nucleation is the cross slip caused by the second slip system. With the continuous loading, more and more dislocation nucleation and slip will increase the dislocation density. Dislocation density has great influence on dislocation movement. The higher the dislocation density is, the more difficult the dislocation movement is.
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The deformation of the metal material increases, the dislocation density increase, leading to a rapid increase in the plastic deformation resistance of the metal. The mechanical properties of the material: hardness and strength significantly increased, plasticity and toughness decreased, which is so-called ‘‘Work hardening” phenomenon. The numerical results show that the increase of applied stress will aggravate the dislocation activity. It is necessary to apply a larger stress to obtain further deformation for increasing of the difficulty of dislocation movement. So we can find a narrow stress increase between point (d) and (e) in Fig. 9. As for the stress decrease between point (d) and (e), it attributes to the growth and coalescence of micro voids, and the nucleation and propagation of crack. The void coalescence and crack propagation, which enhances the plastic deformation and affects the fracture mechanism of the specimen, can observe in Fig. 8(e) as the tensile simulation progresses. The stretching process of [1 1 1] orientation is shown in Fig. 10, and the stress-strain response is shown in Fig. 11. Similar with the tensile simulation along [1 0 0] and [1 1 0] orientation, the dislocations nucleate in the stress-concentrated region and propagate through the semi-circular shear loops in [1 1 1] simulation. However, the evolution of microstructures is different, as shown in Fig. 10. Such as the presence of twin and Lomer-Cottrell dislocations, retard the initial dislocation slip and increase the difficulty of plastic deformation. Due to planar faults’ propagation, the two faults can meet with each other to form a Lomer-Cottrell stairrod lock. The Lomer-Cottrell dislocation is one of the major hardening mechanisms during plastic deformation [41]. The stress between points (c) and (d) in the stress-strain curve first decreases and then increases. The cause of the stress reduction is the generation of slip and dislocations, and the increase in stress is due to the activation of the second slip system. The intersection of these two slip systems hinders the movement of the displacement. The micro crack nucleation on the interstitial surface can be seen in Fig. 7(d) due to dislocation stacking. In addition, micro crack began to propagate and grow as observed in Fig. 10(e), which can be used to explain the appearances of stress peak between point (d) and (e) in stress-strain response. The stress-strain curve tended to decrease as the result of further plastic deformation. The mechanical properties of different orientation samples were found by the stress-strain curves obtained by [1 0 0], [1 1 0] and [1 1 1] tensile simulations. The results are shown in Table 3.
Fig. 8. Snapshots of the microstructure evolution of [1 1 0] orientation. (a, b) Dislocation nucleate at the surface of void and generate by semi-circular shear loop. (d) Void nucleation cause by dislocation interaction. (e) Crack form void coalescence. (a) Dislocation nucleation, (b, c, d, e) intermediate states e = 3.8%, 4.8%, 8.2%, 10.4% respectively, (f) rapture failure.
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(a) εc=0.034
(b)
6
Von-Mises Stress (GPa)
8
(a) Dislocation Nucleation
σc=6.77
5
(c)
(f) εf=0.137
4
σf=0.61 3
(e)
2
(d)
1
(a) Dislocation Nucleation
7
Von-Mises Stress (GPa)
7
(a) εc=0.031
(b)
6
σc=7.31 (d)
(e)
(c)
5
Fracture Point ε=0.161 σ=0.92
4
(f)
3 2
Fracture Point
(f) Fracture Point 1
0 0.00
0.05
0.10
0.15
0.20
0.25
0.30
0 0.00
0.05
0.10
0.20
0.25
0.30
Fig. 11. Stress-Strain response curve of [1 1 1] orientation.
Fig. 9. Stress-strain response curve of [1 1 0] orientation.
The stress-strain curves of [1 0 0], [1 1 0] and [1 1 1] orientation simulations (Figs. 6, 8 and 10) were found to follow a linear relationship before the incipient yield stress point, which indicates that the material is in the elastic deformation stage, and the Young’s modulus of different orientations are: E[1 1 1]>E[1 1 0]>E[1 0 0]. The differences behaved in other mechanic properties is shown in Table 3, where critical yield stress is: rc [1 1 1]>rc[1 0 0]>rc[1 1 0], critical yield strain is: nc[1 1 1]
0.15
Strain
Strain
Table 3 Mechanical property for [1 0 0], [1 1 0] and [1 1 1] orientations.
E (GPa) rc (GPa) nc nf np = nf-nc
[1 0 0]
[1 1 0]
[1 1 1]
134.6 7.17 0.053 0.147 0.094
198.2 6.77 0.034 0.137 0.103
234.1 7.31 0.031 0.161 0.130
crystal material transform deformation mechanism from elastic to ductile and begin incipient yielding. This can be attributed to the difference in activation of the slip system as well as the differences between single and multiple slips. Slip is a plastic deformation process produced by dislocation motion, and the analysis of atomic microstructure displays the generation of slip bands along the [1 1 0] direction. Bringa et al. [13] mentioned that the following number of slip systems with highest Schmidt factor: [1 0 0] [1 1 0] [1 1 1]
eight slip system four slip system six slip system
Fig. 10. Snapshots of the microstructure evolution of [1 1 1] orientation. (a, b) Dislocation nucleate at the surface of void and generate by semi-circular shear loop. (e) Crack form owing to dislocation piling up. (f) Lomer-Cottrell dislocation. (a) Dislocation nucleation, (b, c, d, e, f) intermediate states e = 3.6%, 4.4%, 7.2%, 10.6%, 12.4% respectively.
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Experiment proved that only the {1 1 1}h1 1 0i slip system shown in Fig. 12 was activated under the temperature lower than 900 °C. The Schmidt factor of the {1 1 1}h1 1 0i slip system is concluded in Table 4. The lower yield stress for [1 0 0] orientation is connected to the greater availability of slip systems (eight) with the highest Schmidt factor (0.408), in comparison with four systems with the Schmidt factor of 0.408 for [1 1 0] orientation and six systems with 0.2722 for [1 1 1]. Typically, the resolved shear stress is important because it is well known that dislocation motion in fcc single crystal is governed by the critical resolved shear stress via Schmidt’s law. Therefore, crystallographic orientation shows an important effect on how the Schmidt stress and stress normal to the slip plane contribute to nucleate dislocations. 3.3.3. Orientation effect on the dislocation morphology The visualization of stacking faults and dislocations is accomplished using filters with centrosymmetry parameters. Morphology of characteristic slip bands in the tensile process for different orientation can be found referring to Figs. 13–15. For instance, the variety of dislocation nucleation, distribution of dislocation slip band and the slip direction of different orientations. The [1 0 0] oriented slip band is perpendicular to the loading direction and is oriented at an angle of 45° to the [0 1 0] orientation. For a more detailed MD view selected by CSP > 4 expression, the internal dislocation slip profile of the sample is shown in Fig. 13 (b). Dislocations at the boundary altered the arrangement of the boundary atoms, as shown in Fig. 13(c). In Fig. 13(b), dislocations are represented in the (1 1 1) and (1 1 1) planes, which are projected on a single surface for clarity. The perfect dislocations
Fig. 13. Morphology of slips (a) [1 0 0] orientation overall sample morphology (b) dislocation characteristic by deleting the atoms with CSP < 4 (c) dislocation network on boundary.
Fig. 14. Morphology of slips (a) [1 1 0] orientation overall sample morphology, (b) dislocation characteristic by deleting the atoms with CSP < 4, (c) dislocation network on boundary.
[1 1 0] and [1 0 1], with Burgers vectors aligned with the intersections of the planes, can be decomposed into partials in (1 1 1) along the following reactions: Fig. 12. {1 1 1} h1 1 0i slip system.
a=2½1 1 0 ! a=6½2 1 1 þ a=6½1 2 1 a=2½1 0 1 ! a=6½2 1 1 þ a=6½1 1 2
Table 4 Schmidt factor of the {1 1 1} h1 1 0i slip system. Orientation
[1 0 0] [1 1 0] [1 1 1]
{111}h1 1 0i Number
Schmidt factor
8 4 4 8 6 6
0.4082 0.0000 0.4082 0.0000 0.2722 0.0000
The energy criterion states that a dislocation reaction should lead to lower energy configuration. The above reaction is energetically favorable, so reaction could conduct. As shown in Fig. 14(b), the plane of the [1 1 0] crystal orientation is the topography (1 1 1) and (1 1 1), which is consistent with the result of Bringa [13]. As the result of dislocation gliding, we can find dislocation piling up in the junction region between Slip plane (1 1 1) and (1 1 1) in the tensile process. The new voids nucleation can be observed at that region as shown in the Fig. 8.
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elongation of the specimen. Among the three orientations, the elongation of the sample along [1 1 1] orientation is the greatest, which illustrate that the sample has good ductility performance along [1 1 1] orientation in the same initial damage. The mechanism of this phenomenon is also analyzed by molecular dynamics. The reaction between the dislocations and the boundary leads to the rearrangement of the boundary atoms, which proves that the dislocation can not stop inside the crystal, or closed or terminated at the crystal surface or grain boundary. The difference of the rearrangement of the boundary atoms is due to the difference of the sliding systems of different orientations. Acknowledgements This work is supported by the National Natural Science Foundation of China (Nos. 51375388 and 51210008) and the Natural Science Basic Research Plan in Shaanxi Province of China (No. 2016JM1020).
Fig. 15. Morphology of slips (a) [1 1 1] orientation overall sample morphology, (b) dislocation characteristic by deleting the atoms with CSP < 4, (c) dislocation network on boundary.
The slip bands of [1 1 1] crystal orientation is about 75° with the loading direction, and the main slip plane is (1 1 1). Since the dislocations are the boundaries of the slip and nonslip regions, the dislocation lines cannot be stopped within the bulk, and can only stop at the surface of the crystal or grain boundary. When dislocation glide to the boundary, the atomic redistribution can be found by observing the reaction of dislocation and boundary atoms. Whereas the different form of slip cause the different motion of the boundary atoms. Fig. 13(c) shows that the boundary atoms slide by approximately 45° along the (1 1 1) and (1 1 1) planes due to the dislocations, and the boundary atoms without dislocations are regular distributions. Fig. 14(c) shows additional semi-atomic surface embedding in the boundary region of dislocation stacking. The reaction part of the dislocation and the boundary looks like a wave, as shown in Fig. 14(c). A layer atoms was embedded in the boundary as the dislocation reacting with the boundary of [1 1 1] orientation in Fig. 15(c). The first zone have one layer atoms but the second have two layers. The former is the initial boundary and the latter is boundary reacting with the dislocation slip band. 4. Conclusion MD simulation was conducted to study the effects of crystal orientation on the initial stress distribution, dislocation nucleation and emission, typical dislocation morphology and stress-strain response of single crystal nickel with nanovoid. The system effect and void volume fraction effect were simulated as well. Based above the computations and discussions, the conclusions were summarized as following: The results of different sizes and void volume fractions show significant effects on mechanical properties such as Young’s modulus, incipient yield stress and incipient yield strain. The results of different orientation simulation show that the dislocation structure begins to nucleate in the stress concentration region. The critical stress of [1 0 0], [1 1 0] and [1 1 1] orientation is 6.97 GPa, 6.77 GPa and 7.31 GPa, respectively. The main reason for the above phenomenon is that the shear stress reaches the threshold stress of dislocation nucleation. The results of dislocation propagation and stress-strain responses reveal three different orientations of elastic-plastic properties. The fracture strain is the
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