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Effect of rate dependence of crack propagation processes on amorphization in Al
Materials Science & Engineering A 684 (2017) 71–77
Contents lists available at ScienceDirect
Materials Science & Engineering A journal homepage: www.elsevier.com/locate/msea
Effect of rate dependence of crack propagation processes on amorphization in Al
MARK
⁎
Peng-tao Li, Yan-qing Yang , Xian Luo, Na Jin, Gang Liu, Chong-de Kou, Zong-qiang Feng State Key Laboratory of Solidification Processing, Northwestern Polytechnical University, Xi'an 710072, China
A R T I C L E I N F O
A BS T RAC T
Keywords: Crack tip Plastic deformation Twinning Amorphization MD simulation
Uniaxial tension of pure Al is studied at different loading rates by molecular dynamics simulation to understand the initial nucleation processes for amorphous structures and twinning, especially, the competition between twinning partial dislocation and Lomer one. There exists a transition point KI=0.205 eV Å−2.5 between twinning partial dislocation and Lomer one, with Lomer dislocation preferable at high loads. A modified Peierls analytic model is established to describe the transition process. The analytic model reflects that a transition from twinning partial dislocation to Lomer one will occur when stress intensity factor KI reaches 0.205 eV Å−2.5, which is similar to the simulation trends. In addition, amorphization behaviours are observed at crack tip in pure Al by in-situ straining transmission electron microscopy. Both the experiment and simulation results are demonstrated that amorphization behaviours occur at crack tip in FCC Al.
1. Introduction Different crack substructure evolutions are basic complementary mechanisms for energy dissipation and stress deformation in typical metallic systems [1–3]. Numerous experimental and theoretical studies were conducted to reveal mechanisms underlying the process of crack tip cleavage fracture in metal, such as dislocation emission [4], and deformation twin [5–7]. In recent years, an amorphization mechanism during crack propagation process in metallic compound under uniaxial tensile was proposed by literatures [8–12]. It is worth mention that few literatures reported obvious amorphization at crack tip in FCC structure metals through experiments. However, several molecular dynamics (MD) studies focused on the fracture behaviours of pure Al [13], Cu [14,15], Ni [15–17] and Ag [18], their results verified that the amorphization occurs at crack tip. The discrepancy for MD simulation and experiment observation raises an open question whether the amorphization actually exists at crack tip in FCC structure metals. Thus, the crack tip propagation processes in pure Al will observe in experiment to verify the amorphization appears in this kind of metals. Although the amorphization behaviours have been investigated by MD simulation, the question for the origination of amorphous structures at crack tip in FCC metals is still a hot topic. In 2012, Baker and his partners used extended timescale MD simulation in Al to study the nucleation of crack tip amorphization, the results showed that the initial stage of amorphization is Lomer dislocations at high rates, and they also pointed out that the first stage of propagation mechanism
⁎
transform from twining partial dislocation to Lomer one with increasing loading rates [13]. Vijay et al. studied the fracture mechanism of < 100 > Cu, and found that it is favorable to obtain Lomer dislocations under high rates [19,20]. However, the growing competition between twining partial dislocation and Lomer one during early stage of fracture remains unclear and needs to be further studied. The competition between twinning mechanism and amorphization one at atomic level should be discussed. The study for the competition may understand the connection between macroscopic loading rates and complex mechanisms of deformation at crack tip, which apply to nanocrystalline metals [21,22] and dynamic stain ageing [23]. In this manuscript, a two-dimensional (2D) MD simulation, which can describe the transition between twining partial dislocation and Lomer one, is carried out to study the amorphization mechanism at crack tip. Then, Peierls analytical model based on the simulation is performed to help rationalize the competition between the two dislocations [24]. In addition the fracture behaviours at crack tip of FCC Al are investigated by in-situ straining TEM technique, and particular attention is paid to crack-tip substructure evolution such as amorphization. 2. In-situ TEM observation High purity Al thin sheet with a nominal composition of 99.998 (wt %) is studied. Specimens geometry for in-situ TEM tensile experiment were described in detail in Feng's work [25]. The specimen is prepared
Corresponding author. E-mail address: [email protected] (Y.-q. Yang).
http://dx.doi.org/10.1016/j.msea.2016.12.053 Received 8 November 2016; Received in revised form 9 December 2016; Accepted 10 December 2016 Available online 11 December 2016 0921-5093/ © 2016 Published by Elsevier B.V.
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Fig. 1. (a) Main crack and nanovoids ahead in the in-situ tensile process; (b)–(d) the HRTEM images corresponding to the sites 1–3 in (a), respectively. The insets on the right hand are Fast Fourier Transformation (FFT) patterns of (b)–(d). Double arrow in (a) indicates the tensile loading direction.
loading, and high resolution TEM (HRTEM) images are recorded to reveal the deformation behaviours at crack tip from multiple scales. Under uniaxial tensile loading, the crack initiates from the circular edges of the thinned region which are apt to cause stress concentration. As the strain continuously increasing, crack gradually propagates to occur at ahead of the crack tip. Fig. 1a shows nanoviods instantaneously formed at the region ahead of the main crack tip during high
by using electric discharge machining cutting and mechanical grinding to almost 250 µm thick thin foil. And the central part is further thinned by using a twin-jet electropolisher with solution of 15% perchloric acid and 85% ethanol below −15 ℃ at 16 V. In-situ tensile progress is operated on the 300 kV FEG-TEM Tecnai F30 G2 by using a Gatan Model 654 single tilt straining holder with the strain range of 2.0 mm and min step of 1 µm. The whole specimen is subjected to uniaxial 72
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Fig. 2. (a) Main crack and nanovoids ahead in the in-situ tensile process; (b) the HRTEM images corresponding to the sites 1 in (a), (c) and (d) the enlarge imaged corresponding to the sites 2 and 3 in (b), respectively.
Fig. 2d. The in-situ TEM observation indicates that amorphization process is a throughout mechanism during crack tip propagation. This finding gives a strong support to the MD predication that amorphization is appears when deformation occurs. The HRTEM and FFT observation corresponding to the initial stage of amorphization is difficult to be carried out as it is hard to perform the static condition to observe the structures evolution at atomic scale and detect the rapid amorphization transformation during the deformation processes. Thus, the MD simulation and corresponding analytic model are the necessary conditions to study the initial stage of amorphization.
crack propagation processes. The distance between main crack (mark as 1) and nanovoids (mark as 2 and 3) are 90 nm and 160 nm, respectively. In order to understand the formation of edge microstructure at crack tip and nanovoids, HRTEM with FFT technique is used to observe the propagation processes at atomic level. As shown in Fig. 1b, the main crack is blunted as a 12 nm semicircle arc. And the corresponding FFT image of Fig. 1b shows diffuse intensity halos, indicating amorphous zones are formed at the crack edge. Similarly, the FFT image for Fig. 1c shows features of scattering, which means an amorphous zone exist between the two dotted lines. Moreover, the FFT result for ellipse nano-void 3 exhibits that amorphous zone is also formed in the place between two dotted lines in Fig. 1d. It is indicates that nano-voids originated from amorphous region and the amorphous zone actual existent at crack tip in FCC Al. As the strain increasing, the nanovoids 2 and 3 in Fig. 1a are merged into a big nanovoid at ahead of crack tip, as shown in Fig. 2a. The nanoviods coalescence zone (marked as sites 1) has been enlarged in Fig. 2b. Sites 2 and 3 are show in Fig. 2c and d, respectively. It can be clearly seen that the edge lattice configuration of nanovoids coalescence zone is amorphous structures in Fig. 2c, and in some zone amorphization seems to be completed across the sample section, as shown in
3. The simulation methods 3.1. Simulation details The simulation of the process of crack tip propagation is performed using the open code LAMMPS [26]. Simulation cells are based on an embedded-atom method (EAM) potential, which was developed for Al with lattice constant a0=4.05 Å on the basis of ab-initio quantum calculations by Mishin and his partners [27–29]. The constant time 73
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Preexistent crack
0.16μm
=70.5
y [111]
2.0GPa
-2.0 GPa
x [11 2 ]
Fig. 3. Diagram of the 2D crystal Al simulation cell showing the plane and crack geometry used in this work. The amplified image in lift picture shows the starting configuration for preexistent crack. The right expanded picture shows the atomistic resolution at the crack tip following the nucleation of the leading partial dislocation. Colors represent the stress in [111] axis.
step Δt is 3 fs (1 fs=10–15 s). The simulation cell is conducted on a flat in single crystal Al with a preexistent crack to observe the possible propagation behaviours in the plane-strain mode I, as shown in Fig. 3. A sharp crack used in this paper is consistent with standard continuum model, which can stably exists in real material, as shown in amplified image in left picture. The crack plane is exhibited in (−110) plane, [11−2] in x axis and [111] in y axis. Moreover, the specific geometry is the optimal configuration for observing crack propagation behaviours, such as deformation twin [30,31] and amorphization [13]. A 2D linear elastic atomistic region with length of 0.160×0.100 µm2, contain 1.91 million atoms. Periodic boundary is applied in z direction to emulate an infinitely thick material. The temperature of the simulation is increased to 300 K using the velocity-rescaled Berebdsen thermostat. After relaxing for 5 ps using a microcanonical (NVE) ensemble, a NoseHoover thermostat is used to maintain the temperature at 300 K for 60 ps in a canonical (NVT) ensemble. Finally, the irradiated simulation cell is performed to uniaxial tension in y direction. Rates are provided by the displacement of mobile boundaries in the y-direction of the simulation cell, corresponding to the mode Ⅰ stress intensity factor, KI, which are in the range of 0.17–0.235 eV Å−2.5. The Mode Ⅰ stress intensity KI at the crack can estimated by using Murakami's method [32]. The time of the nucleation occurrence is considered to be the sum of the simulation on all of the signal-processors. In addition, OVITO is used to visualize the atomic structures [33], where atoms are colored by atom common neighbor analysis (CNA) [34,35].
dislocation, which is constructed according to two Thompson's tetrahedra. Six vertices of the two tetrahedra are labelled by A, C, A′, C′, B and D, respectively. And the middle point of the triangles opposite are denoted as α, γ, α' and γ', respectively. The two Thompson's tetrahedras ABCD and A′BC'D illustrate the slip system of FCC crystal. For the present discussion, one lattice dislocation BD dissociates as Shockley partial dislocation Bα and αD on glide plane α, as shown in Fig. 4a. It is worth mentions that glide plane γ and crack tip surface are in the same plane. When loading rate is larger than the critical value, another lattice dislocation BD dissociates as Shockley partial dislocations Bγ and γD on glide plane γ. Then αD meets γD, a Lomer dislocation αγ is produced at the inter section magenta line of plane α and plane γ, and corresponding atomic configuration is shown in Fig. 3a. The dislocation reaction for Lomer dislocation could be written as
αD + Dγ → αγ . After the Lomer dislocation nucleation, the following growing processes from Lomer dislocation to amorphous zone at high tensile stress are consistent with Baker and co-authors’ results [13]. Fig. 6 shows the nucleation time for all simulation results with different loading rates, which reveals that Lomer dislocation is the main mechanism at high rates KI > 0.205 eV Å−2.5. Both twinning partial and Lomer dislocation are nucleated at the rates 0.195– 0.205 eV Å−2.5. Twinning partial dislocation is the preferable mechanism at low rates of 0.195 eV Å−2.5 > KI > 0.180 eV Å−2.5. The above simulation results suggest that the transition from twinning partial dislocation to Lomer one occurs at KI=0.205 eV Å−2.5. An analytic model is further performed to help rationalize the observed transition from twinning partial dislocation to Lomer one. The model is an 2D plane based on Peierls model developed by Rice and co-authors [24] for understanding the transition process, containing a pre-existent crack extending from −∞ to 0 along the x direction. γ is the distance from the crack tip, δ is the slip displacement, and δ(γ) is slip displacement distribution. The total energy U[δ(γ)] of the model is defined [24], as a sum of four terms
3.2. Simulation results and discussion The simulation results show that there exist two mechanisms at crack tip during the initial stage of crack propagation. One is that a twinning partial dislocation follows a leading partial of the same Burgers vector on the same slip plane, which makes the nucleation of micro-twins, as shown in Fig. 4a and b. The other is that Lomer dislocation occurs after the leading partial dislocation travels under loading, as shown in Fig. 4c, and then amorphization process will appear following the nucleation and migration of the Lomer dislocation. The nucleation processes for Lomer dislocation at crack tip are described as follow. Fig. 5 shows the formation illustrations of Lomer
U [δ (γ )] = U0 +
74
∫0
∞
ϕ [δ (r )] dr +
1 2
⎡
∫0
s [δe (r )] ⎤ ⎡ δe (r )⎤ ⎥⎢ ⎥ ⎢ (1 − v ) s [δs (r )]⎥⋅⎢ δs (r ) ⎥ dr − ⎢⎣ (1 + v ) s [δl (r )] ⎥⎦ ⎢⎣ δl (r ) ⎥⎦
∞⎢
∫0
∞ K eff II
2πr
δe (r ) dr
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A
γ α B
C
?
LDC
Fig. 4. Schematic diagrams of two alternative mechanisms of crack-tip plasticity in Al, A–C, following the nucleation of the leading partial dislocation (A) either micro-twin via nucleation of a twinning partial (B) or amorphization via nucleation of a Lomer dislocation core (LDC) (C) can occur. Atoms are colored by CNA, and green and red ones represent FCC and h.c.p, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
where U0 is the elastic strain energy of the unslipped solid and Φ[δ(γ)] is the energy change gain from the atomic stacking energy ϕ when δ(γ) develops. The third term is an expression of the elastic work required to form the slip distribution δ(γ) in the infinite solid. The slip distribution is decomposed into edge, δe(γ)=|δ(γ)|cosφ, screw, δs(γ)=|δ(γ)|sinφ, and Lomer, δl(γ)=|δ(γ)|(cos2φcos2θ+sin2φ)½, respectively. The stress induced by slip, along the slip plane ahead of the crack tip is required to calculated the elastic work, which can be written for any arbitrary slip f(γ) as the following principle value integral [36].
s [ f (r )] =
μ 2π (1 − v )
∫0
∞
[001] C' A
B α'
γ
ξ df (ξ )/d ξ dξ r r−ξ
α
Where ξ is the distance from the crack tip along the slip plane. The furthest right term is the elastic interaction energy between the slip and the crack tip surface under the far-field applied loading. It can be described by a mode I stress intensity factor, KⅠ. In determining the interaction due to slip, only the relevant component of the applied load needs to be consider, i.e. the effective model II stress intensity factor acting on the slip plane.
[100]
γ'
D
A' [010]
Fig. 5. Schematic illustrations of the formation process for Lomer dislocation. Green lines and magenta one are represented Shockley partial dislocation and lomer dislocation, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
KIIeff = KI cos2 (θ /2)sin(θ /2) where the shear modulus of aluminum，µ=29.2 GPa, and the Poisson's ratio ν=0.319. Taking account of Thompson tetrahedron and simulation geometry [37], the twinning partial slip paths is constrained in the [112] direction. According to Rice and his co-authors’ work [36], the potential Φ is related to the generalized stacking-fault interlinear potential. And the curve for stacking-fault potential along the slip path approximates to Frenkel sinusoid form, The results of Φ dependence on δl/bp at 300 K is shown in Fig. 7 [37]. Moreover, the expression of Φ
for Lomer dislocation is [38]
Φ = 2EK −
T0 d 2 2r
where Ek, the kink formation energy, is described as follow, 75
Materials Science & Engineering A 684 (2017) 71–77
Time (Ps)
P.-t. Li et al.
10
5
10
4
10
3
10
2
10
1
And then the △-Utotal inserts into the Arrhenius expression.
Twinning partial simulation Amorphous simulation Twinning partial analytic Amorphization analytic
t =
0
I
-2.5
)
Fig. 6. Nucleation times of twinning partial dislocation and Lomer one versus applied loads in Al at 300 K. Dashed lines and solid circles denote predictions of the analytic plane model and multiscale simulation results, respectively.
(meV
A-2)
20
15
Leading partial Trailing partial Twinning partial Lomer partial
10
5
0
0.0
0.4
0.8
1.2
1.6
2.0
l/bp Fig. 7. Potential along the leading-to-Lomer, leading-to-twinning and leading-to-trailing slip paths for Al at 300 K. δu represents the discrete slip vector δ projected into the correspondent slip path.
EK = c1 a
Δutotal kT
vf
where kT=0.0258 eV, k is the Boltzmann factor, and the value of attempt frequencies vf=6×1011 s−1, which was obtained from the study for rates dependence of leading partial dislocation emission [37]. The nucleation time curves for twinning partial dislocations and Lomer one versus applied rates are calculated by the analytic model, as shown in Fig. 6. The analytic model indicates that the transition pinot may happen at KI=0.215 eV Å−2.5. The main features of the result reflect that the analytic model predications are similar to the trends in simulation results. The quantitative difference between simulation results and analytic model is partly caused by two factors. One is that the high loading rates at finite temperature will concentrate the strain energy at crack tip, which may decrease the value of transition point [13]. The other one is that the estimate of stress intensity for nucleation at 0 K is less than the atomistic value, Zhu and his co-authors also confirmed this disagreement in the emission of the leading partial dislocation [40]. Despite the analytic model has good capacity to capture the change in deformation mechanism as found by simulation, the prediction process needs complex calculation and does not show the correlation for basic material properties. In the future, a concise expression for material properties dependencies is required to predict the mechanisms transition. From the discussion above, it can be inferred that the in-situ TEM observation provides a new evidence for amorphization process at crack tip in FCC Al under uniaxial tensile. On the other hand, MD simulation in Al is applied to study the mechanism of deformation process, and the results show that amorphization process is initiated by the nucleation and movement of the Lomer dislocation. Furthermore, the initial crack propagation behaviours depend on the competition between twinning partial dislocation and Lomer one. To predict the transition from twinning to amorphization at the initial stage of crack propagation process, an analytic model is developed based on Peierls model. The results indicate that the transition point from analytic model is close to the trends in simulation one. Although the analytic model of crack amorphization does not directly show the amorphization process at crack tip in single or polycrystalline metals, it is possibly apply to nanoscale metals where twinning and amorphization from grain boundaries plays a main role in the fracture process [16,17,41].
10 0.17 0.18 0.19 0.20 0.21 0.22 0.23
K (eV/A
e
⎛ ⎛ πr ⎞ ⎞ ⎛ 2a E ⎞ ⎛r⎞ arctg ⎜exp ⎜ ⎟ ⎟ ⎜EL + d 2 l2 ⎟ ln ⎜ ⎟ ⎝ w0 ⎠ ⎠ ⎝ π dφ ⎠ ⎝ r0 ⎠ ⎝
4. Conclusions
where the T0d2/(2γ) denotes the long range interaction potential [39], d is the distance between Peierls valleys, EL is dislocation line energy, T0 is the orientation dependence of EL, c1 is the shape factor of Peierls model, w0 and r0 are represent the width and cut-off radius in the equilibrium configuration, respectively. The calculated results of Φ dependence on δl/bp for Lomer dislocation is shown by red line in Fig. 5. The energy difference △U between stable slip statuses δst(γ) and maximal point value δex(γ) represents the interplanar activation energy per unit length at any applied load, which can be obtained from the simulation cell,
In summary, MD simulation and analytic model in Al are performed to study the competition between twinning partial dislocation and Lomer one during the crack propagation process. The simulation results show that there exists a transition point KI=0.205 eV Å−2.5 between twinning partial dislocation and Lomer one, with Lomer dislocation is preferable one at high loads. Similarly, the analytic model also predicts that the transition point is at KI=0.215 eV Å−2.5, which influences the crack propagation mechanism from twinning to amorphization. In addition, the amorphization process at crack tip during the crack propagation in FCC Al alloy is observed by HRTEM. The results rationalize the amorphization assumption that the amorphous zone is likely at crack tip in FCC structures at high strain rates.
ΔU = U [δex (r )] − U [δst (r )] The solutions for the equation are obtained through the calculation by Rice's method [24], which consider the adaptive discretization effect of the slip distribution. Next, in order to compare with the simulation results, two calculation steps are required. Firstly, the total activation energy for an actual dislocation line with an effective lateral length L is calculated.
Acknowledgments Thanks are given to the financial supports of the Natural Science Foundation of China (Nos. 51271147, 51201134, 51201135), the Fundamental Research Funds for the Central Universities (3102014JCQ01023) and the Research Fund of the State Key Laboratory of Solidification Processing (NWPU), China (Grant no. 115-QP-2014).
ΔUtotal = LΔU In this work, L=10bp=16.5 Å, bp is the length of the Burgers vector. 76
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1975–1983. [22] T. Zhu, J. Li, A. Samanta, H.G. Kim, S. Suresh, Proc. Natl. Acad. Sci. USA 104 (2007) 3031–3036. [23] W.A. Curtin, D.L. Olmsted, L.G. Hector, Nat. Mater. 5 (2006) 875–880. [24] J.R. Rice, G.E. Beltz, J. Mech. Phys. Solids 42 (1994) 333–360. [25] Z.Q. Feng, Y.Q. Yang, Y.X. Chen, B. Huang, M.S. Fu, M.H. Li, J.G. Ru, Mater. Sci. Eng.: A 586 (2013) 259–266. [26] S. Plimpton, J. Comput. Phys. 117 (1995) 1–19. [27] Y. Mishin, D. Farkas, M.J. Mehl, D.A. Papaconstantopoulos, Phys. Rev. B 59 (1999) 3393–3407. [28] D. Farkas, M. Duranduru, W.A. Curtin, C. Ribbens, Philos. Mag. A 81 (2001) 1241–1255. [29] Y. Cheng, M.X. Shi, Y.W. Zhang, Int. J. Solids Struct. 49 (2012) 3345–3354. [30] E.B. Tadmor, S. Hai, J. Mech. Phys. Solids 51 (2003) 765–793. [31] E. Tadmor, J. Mech. Phys. Solids 52 (2004) 2507–2519. [32] Y. Murakami, L.M. Keer, J. Appl. Mech. 60 (1993) (1063-1063). [33] A. Stukowski, Model. Simul. Mater. Sci. Eng. 18 (2010) 015012. [34] D. Faken, H. Jónsson, Comput. Mater. Sci. 2 (1994) 279–286. [35] H. Tsuzuki, P.S. Branicio, J.P. Rino, Comput. Phys. Commun. 177 (2007) 518–523. [36] J.R. Rice, J. Mech. Phys. Solids 40 (1992) 239–271. [37] D.H. Warner, W.A. Curtin, S. Qu, Nat. Mater. 6 (2007) 876–881. [38] D. Rodney, L. Proville, Phys. Rev. B 79 (2009) 094108. [39] G. Schoeck, Philos. Mag. 87 (2007) 1631–1647. [40] T. Zhu, J. Li, S. Yip, Phys. Rev. Lett. 93 (2004) 025503. [41] J. Wang, F. Sansoz, J. Huang, Y. Liu, S. Sun, Z. Zhang, S.X. Mao, Nat. Commun. 4 (2013) 1742.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
J.R. Rice, Math. Fundam. 2 (1968) 191–311. J.R. Rice, Mech. Mater. 6 (1987) 317–335. J.R. Rice, R. Thomson, Philos. Mag. 29 (1974) 73–97. A. Kelly, W.R. Tyson, A.H. Cottrell, Philos. Mag. 15 (1967) 567–586. Y. a, a, D.W.a., S.R.P.a., H.G.b., 2002. S. Hai, E.B. Tadmor, Acta Mater. 51 (2003) 117–131. V. Yamakov, D. Wolf, S.R. Phillpot, H. Gleiter, Acta Mater. 51 (2003) 4135–4147. X.-T. Luo, G.-J. Yang, C.-J. Li, K. Kondoh, Scr. Mater. 65 (2011) 581–584. P.R. Okamoto, J.K. Heuer, N.Q. Lam, S. Ohnuki, Y. Matsukawa, K. Tozawa, J.F. Stubbins, Appl. Phys. Lett. 73 (1998) 473–475. I.A. Ovid'ko, J. Phys. D: Appl. Phys. 24 (1991) 2190. I.A. Ovid'ko, J. Phys. D: Appl. Phys. 27 (1994) 999. M. Chen, J.W. McCauley, K.J. Hemker, Science 299 (2003) 1563–1566. K.L. Baker, D.H. Warner, Model. Simul. Mater. Sci. Eng. 20 (2012) 065005. A.M. Dongare, A.M. Rajendran, B. LaMattina, M.A. Zikry, D.W. Brenner, Phys. Rev. B 80 (2009) 104108. B. Hess, B.J. Thijsse, E. Van der Giessen, Phys. Rev. B 71 (2005). P.S. Branício, J.-P. Rino, Phys. Rev. B 62 (2000) 16950–16955. H. Ikeda, Y. Qi, T. Çagin, K. Samwer, W.L. Johnson, W.A. Goddard, Phys. Rev. Lett. 82 (1999) 2900–2903. R.M. Lynden-Bell, J. Phys.: Condens. Matter 7 (1995) 4603. S. Vijay Kumar, D.R. Mahapatra, J. Phys.: Condens. Matter 20 (2008) 335206. S. Vijay Kumar, D.R. Mahapatra, Nanotechnology 20 (2009) 045701. H. Van Swygenhoven, P.M. Derlet, A.G. Frøseth, Acta Mater. 54 (2006)
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Effect of rate dependence of crack propagation processes on amorphization in Al
MARK
⁎
Peng-tao Li, Yan-qing Yang , Xian Luo, Na Jin, Gang Liu, Chong-de Kou, Zong-qiang Feng State Key Laboratory of Solidification Processing, Northwestern Polytechnical University, Xi'an 710072, China
A R T I C L E I N F O
A BS T RAC T
Keywords: Crack tip Plastic deformation Twinning Amorphization MD simulation
Uniaxial tension of pure Al is studied at different loading rates by molecular dynamics simulation to understand the initial nucleation processes for amorphous structures and twinning, especially, the competition between twinning partial dislocation and Lomer one. There exists a transition point KI=0.205 eV Å−2.5 between twinning partial dislocation and Lomer one, with Lomer dislocation preferable at high loads. A modified Peierls analytic model is established to describe the transition process. The analytic model reflects that a transition from twinning partial dislocation to Lomer one will occur when stress intensity factor KI reaches 0.205 eV Å−2.5, which is similar to the simulation trends. In addition, amorphization behaviours are observed at crack tip in pure Al by in-situ straining transmission electron microscopy. Both the experiment and simulation results are demonstrated that amorphization behaviours occur at crack tip in FCC Al.
1. Introduction Different crack substructure evolutions are basic complementary mechanisms for energy dissipation and stress deformation in typical metallic systems [1–3]. Numerous experimental and theoretical studies were conducted to reveal mechanisms underlying the process of crack tip cleavage fracture in metal, such as dislocation emission [4], and deformation twin [5–7]. In recent years, an amorphization mechanism during crack propagation process in metallic compound under uniaxial tensile was proposed by literatures [8–12]. It is worth mention that few literatures reported obvious amorphization at crack tip in FCC structure metals through experiments. However, several molecular dynamics (MD) studies focused on the fracture behaviours of pure Al [13], Cu [14,15], Ni [15–17] and Ag [18], their results verified that the amorphization occurs at crack tip. The discrepancy for MD simulation and experiment observation raises an open question whether the amorphization actually exists at crack tip in FCC structure metals. Thus, the crack tip propagation processes in pure Al will observe in experiment to verify the amorphization appears in this kind of metals. Although the amorphization behaviours have been investigated by MD simulation, the question for the origination of amorphous structures at crack tip in FCC metals is still a hot topic. In 2012, Baker and his partners used extended timescale MD simulation in Al to study the nucleation of crack tip amorphization, the results showed that the initial stage of amorphization is Lomer dislocations at high rates, and they also pointed out that the first stage of propagation mechanism
⁎
transform from twining partial dislocation to Lomer one with increasing loading rates [13]. Vijay et al. studied the fracture mechanism of < 100 > Cu, and found that it is favorable to obtain Lomer dislocations under high rates [19,20]. However, the growing competition between twining partial dislocation and Lomer one during early stage of fracture remains unclear and needs to be further studied. The competition between twinning mechanism and amorphization one at atomic level should be discussed. The study for the competition may understand the connection between macroscopic loading rates and complex mechanisms of deformation at crack tip, which apply to nanocrystalline metals [21,22] and dynamic stain ageing [23]. In this manuscript, a two-dimensional (2D) MD simulation, which can describe the transition between twining partial dislocation and Lomer one, is carried out to study the amorphization mechanism at crack tip. Then, Peierls analytical model based on the simulation is performed to help rationalize the competition between the two dislocations [24]. In addition the fracture behaviours at crack tip of FCC Al are investigated by in-situ straining TEM technique, and particular attention is paid to crack-tip substructure evolution such as amorphization. 2. In-situ TEM observation High purity Al thin sheet with a nominal composition of 99.998 (wt %) is studied. Specimens geometry for in-situ TEM tensile experiment were described in detail in Feng's work [25]. The specimen is prepared
Corresponding author. E-mail address: [email protected] (Y.-q. Yang).
http://dx.doi.org/10.1016/j.msea.2016.12.053 Received 8 November 2016; Received in revised form 9 December 2016; Accepted 10 December 2016 Available online 11 December 2016 0921-5093/ © 2016 Published by Elsevier B.V.
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Fig. 1. (a) Main crack and nanovoids ahead in the in-situ tensile process; (b)–(d) the HRTEM images corresponding to the sites 1–3 in (a), respectively. The insets on the right hand are Fast Fourier Transformation (FFT) patterns of (b)–(d). Double arrow in (a) indicates the tensile loading direction.
loading, and high resolution TEM (HRTEM) images are recorded to reveal the deformation behaviours at crack tip from multiple scales. Under uniaxial tensile loading, the crack initiates from the circular edges of the thinned region which are apt to cause stress concentration. As the strain continuously increasing, crack gradually propagates to occur at ahead of the crack tip. Fig. 1a shows nanoviods instantaneously formed at the region ahead of the main crack tip during high
by using electric discharge machining cutting and mechanical grinding to almost 250 µm thick thin foil. And the central part is further thinned by using a twin-jet electropolisher with solution of 15% perchloric acid and 85% ethanol below −15 ℃ at 16 V. In-situ tensile progress is operated on the 300 kV FEG-TEM Tecnai F30 G2 by using a Gatan Model 654 single tilt straining holder with the strain range of 2.0 mm and min step of 1 µm. The whole specimen is subjected to uniaxial 72
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Fig. 2. (a) Main crack and nanovoids ahead in the in-situ tensile process; (b) the HRTEM images corresponding to the sites 1 in (a), (c) and (d) the enlarge imaged corresponding to the sites 2 and 3 in (b), respectively.
Fig. 2d. The in-situ TEM observation indicates that amorphization process is a throughout mechanism during crack tip propagation. This finding gives a strong support to the MD predication that amorphization is appears when deformation occurs. The HRTEM and FFT observation corresponding to the initial stage of amorphization is difficult to be carried out as it is hard to perform the static condition to observe the structures evolution at atomic scale and detect the rapid amorphization transformation during the deformation processes. Thus, the MD simulation and corresponding analytic model are the necessary conditions to study the initial stage of amorphization.
crack propagation processes. The distance between main crack (mark as 1) and nanovoids (mark as 2 and 3) are 90 nm and 160 nm, respectively. In order to understand the formation of edge microstructure at crack tip and nanovoids, HRTEM with FFT technique is used to observe the propagation processes at atomic level. As shown in Fig. 1b, the main crack is blunted as a 12 nm semicircle arc. And the corresponding FFT image of Fig. 1b shows diffuse intensity halos, indicating amorphous zones are formed at the crack edge. Similarly, the FFT image for Fig. 1c shows features of scattering, which means an amorphous zone exist between the two dotted lines. Moreover, the FFT result for ellipse nano-void 3 exhibits that amorphous zone is also formed in the place between two dotted lines in Fig. 1d. It is indicates that nano-voids originated from amorphous region and the amorphous zone actual existent at crack tip in FCC Al. As the strain increasing, the nanovoids 2 and 3 in Fig. 1a are merged into a big nanovoid at ahead of crack tip, as shown in Fig. 2a. The nanoviods coalescence zone (marked as sites 1) has been enlarged in Fig. 2b. Sites 2 and 3 are show in Fig. 2c and d, respectively. It can be clearly seen that the edge lattice configuration of nanovoids coalescence zone is amorphous structures in Fig. 2c, and in some zone amorphization seems to be completed across the sample section, as shown in
3. The simulation methods 3.1. Simulation details The simulation of the process of crack tip propagation is performed using the open code LAMMPS [26]. Simulation cells are based on an embedded-atom method (EAM) potential, which was developed for Al with lattice constant a0=4.05 Å on the basis of ab-initio quantum calculations by Mishin and his partners [27–29]. The constant time 73
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Preexistent crack
0.16μm
=70.5
y [111]
2.0GPa
-2.0 GPa
x [11 2 ]
Fig. 3. Diagram of the 2D crystal Al simulation cell showing the plane and crack geometry used in this work. The amplified image in lift picture shows the starting configuration for preexistent crack. The right expanded picture shows the atomistic resolution at the crack tip following the nucleation of the leading partial dislocation. Colors represent the stress in [111] axis.
step Δt is 3 fs (1 fs=10–15 s). The simulation cell is conducted on a flat in single crystal Al with a preexistent crack to observe the possible propagation behaviours in the plane-strain mode I, as shown in Fig. 3. A sharp crack used in this paper is consistent with standard continuum model, which can stably exists in real material, as shown in amplified image in left picture. The crack plane is exhibited in (−110) plane, [11−2] in x axis and [111] in y axis. Moreover, the specific geometry is the optimal configuration for observing crack propagation behaviours, such as deformation twin [30,31] and amorphization [13]. A 2D linear elastic atomistic region with length of 0.160×0.100 µm2, contain 1.91 million atoms. Periodic boundary is applied in z direction to emulate an infinitely thick material. The temperature of the simulation is increased to 300 K using the velocity-rescaled Berebdsen thermostat. After relaxing for 5 ps using a microcanonical (NVE) ensemble, a NoseHoover thermostat is used to maintain the temperature at 300 K for 60 ps in a canonical (NVT) ensemble. Finally, the irradiated simulation cell is performed to uniaxial tension in y direction. Rates are provided by the displacement of mobile boundaries in the y-direction of the simulation cell, corresponding to the mode Ⅰ stress intensity factor, KI, which are in the range of 0.17–0.235 eV Å−2.5. The Mode Ⅰ stress intensity KI at the crack can estimated by using Murakami's method [32]. The time of the nucleation occurrence is considered to be the sum of the simulation on all of the signal-processors. In addition, OVITO is used to visualize the atomic structures [33], where atoms are colored by atom common neighbor analysis (CNA) [34,35].
dislocation, which is constructed according to two Thompson's tetrahedra. Six vertices of the two tetrahedra are labelled by A, C, A′, C′, B and D, respectively. And the middle point of the triangles opposite are denoted as α, γ, α' and γ', respectively. The two Thompson's tetrahedras ABCD and A′BC'D illustrate the slip system of FCC crystal. For the present discussion, one lattice dislocation BD dissociates as Shockley partial dislocation Bα and αD on glide plane α, as shown in Fig. 4a. It is worth mentions that glide plane γ and crack tip surface are in the same plane. When loading rate is larger than the critical value, another lattice dislocation BD dissociates as Shockley partial dislocations Bγ and γD on glide plane γ. Then αD meets γD, a Lomer dislocation αγ is produced at the inter section magenta line of plane α and plane γ, and corresponding atomic configuration is shown in Fig. 3a. The dislocation reaction for Lomer dislocation could be written as
αD + Dγ → αγ . After the Lomer dislocation nucleation, the following growing processes from Lomer dislocation to amorphous zone at high tensile stress are consistent with Baker and co-authors’ results [13]. Fig. 6 shows the nucleation time for all simulation results with different loading rates, which reveals that Lomer dislocation is the main mechanism at high rates KI > 0.205 eV Å−2.5. Both twinning partial and Lomer dislocation are nucleated at the rates 0.195– 0.205 eV Å−2.5. Twinning partial dislocation is the preferable mechanism at low rates of 0.195 eV Å−2.5 > KI > 0.180 eV Å−2.5. The above simulation results suggest that the transition from twinning partial dislocation to Lomer one occurs at KI=0.205 eV Å−2.5. An analytic model is further performed to help rationalize the observed transition from twinning partial dislocation to Lomer one. The model is an 2D plane based on Peierls model developed by Rice and co-authors [24] for understanding the transition process, containing a pre-existent crack extending from −∞ to 0 along the x direction. γ is the distance from the crack tip, δ is the slip displacement, and δ(γ) is slip displacement distribution. The total energy U[δ(γ)] of the model is defined [24], as a sum of four terms
3.2. Simulation results and discussion The simulation results show that there exist two mechanisms at crack tip during the initial stage of crack propagation. One is that a twinning partial dislocation follows a leading partial of the same Burgers vector on the same slip plane, which makes the nucleation of micro-twins, as shown in Fig. 4a and b. The other is that Lomer dislocation occurs after the leading partial dislocation travels under loading, as shown in Fig. 4c, and then amorphization process will appear following the nucleation and migration of the Lomer dislocation. The nucleation processes for Lomer dislocation at crack tip are described as follow. Fig. 5 shows the formation illustrations of Lomer
U [δ (γ )] = U0 +
74
∫0
∞
ϕ [δ (r )] dr +
1 2
⎡
∫0
s [δe (r )] ⎤ ⎡ δe (r )⎤ ⎥⎢ ⎥ ⎢ (1 − v ) s [δs (r )]⎥⋅⎢ δs (r ) ⎥ dr − ⎢⎣ (1 + v ) s [δl (r )] ⎥⎦ ⎢⎣ δl (r ) ⎥⎦
∞⎢
∫0
∞ K eff II
2πr
δe (r ) dr
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A
γ α B
C
?
LDC
Fig. 4. Schematic diagrams of two alternative mechanisms of crack-tip plasticity in Al, A–C, following the nucleation of the leading partial dislocation (A) either micro-twin via nucleation of a twinning partial (B) or amorphization via nucleation of a Lomer dislocation core (LDC) (C) can occur. Atoms are colored by CNA, and green and red ones represent FCC and h.c.p, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
where U0 is the elastic strain energy of the unslipped solid and Φ[δ(γ)] is the energy change gain from the atomic stacking energy ϕ when δ(γ) develops. The third term is an expression of the elastic work required to form the slip distribution δ(γ) in the infinite solid. The slip distribution is decomposed into edge, δe(γ)=|δ(γ)|cosφ, screw, δs(γ)=|δ(γ)|sinφ, and Lomer, δl(γ)=|δ(γ)|(cos2φcos2θ+sin2φ)½, respectively. The stress induced by slip, along the slip plane ahead of the crack tip is required to calculated the elastic work, which can be written for any arbitrary slip f(γ) as the following principle value integral [36].
s [ f (r )] =
μ 2π (1 − v )
∫0
∞
[001] C' A
B α'
γ
ξ df (ξ )/d ξ dξ r r−ξ
α
Where ξ is the distance from the crack tip along the slip plane. The furthest right term is the elastic interaction energy between the slip and the crack tip surface under the far-field applied loading. It can be described by a mode I stress intensity factor, KⅠ. In determining the interaction due to slip, only the relevant component of the applied load needs to be consider, i.e. the effective model II stress intensity factor acting on the slip plane.
[100]
γ'
D
A' [010]
Fig. 5. Schematic illustrations of the formation process for Lomer dislocation. Green lines and magenta one are represented Shockley partial dislocation and lomer dislocation, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
KIIeff = KI cos2 (θ /2)sin(θ /2) where the shear modulus of aluminum，µ=29.2 GPa, and the Poisson's ratio ν=0.319. Taking account of Thompson tetrahedron and simulation geometry [37], the twinning partial slip paths is constrained in the [112] direction. According to Rice and his co-authors’ work [36], the potential Φ is related to the generalized stacking-fault interlinear potential. And the curve for stacking-fault potential along the slip path approximates to Frenkel sinusoid form, The results of Φ dependence on δl/bp at 300 K is shown in Fig. 7 [37]. Moreover, the expression of Φ
for Lomer dislocation is [38]
Φ = 2EK −
T0 d 2 2r
where Ek, the kink formation energy, is described as follow, 75
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Time (Ps)
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10
5
10
4
10
3
10
2
10
1
And then the △-Utotal inserts into the Arrhenius expression.
Twinning partial simulation Amorphous simulation Twinning partial analytic Amorphization analytic
t =
0
I
-2.5
)
Fig. 6. Nucleation times of twinning partial dislocation and Lomer one versus applied loads in Al at 300 K. Dashed lines and solid circles denote predictions of the analytic plane model and multiscale simulation results, respectively.
(meV
A-2)
20
15
Leading partial Trailing partial Twinning partial Lomer partial
10
5
0
0.0
0.4
0.8
1.2
1.6
2.0
l/bp Fig. 7. Potential along the leading-to-Lomer, leading-to-twinning and leading-to-trailing slip paths for Al at 300 K. δu represents the discrete slip vector δ projected into the correspondent slip path.
EK = c1 a
Δutotal kT
vf
where kT=0.0258 eV, k is the Boltzmann factor, and the value of attempt frequencies vf=6×1011 s−1, which was obtained from the study for rates dependence of leading partial dislocation emission [37]. The nucleation time curves for twinning partial dislocations and Lomer one versus applied rates are calculated by the analytic model, as shown in Fig. 6. The analytic model indicates that the transition pinot may happen at KI=0.215 eV Å−2.5. The main features of the result reflect that the analytic model predications are similar to the trends in simulation results. The quantitative difference between simulation results and analytic model is partly caused by two factors. One is that the high loading rates at finite temperature will concentrate the strain energy at crack tip, which may decrease the value of transition point [13]. The other one is that the estimate of stress intensity for nucleation at 0 K is less than the atomistic value, Zhu and his co-authors also confirmed this disagreement in the emission of the leading partial dislocation [40]. Despite the analytic model has good capacity to capture the change in deformation mechanism as found by simulation, the prediction process needs complex calculation and does not show the correlation for basic material properties. In the future, a concise expression for material properties dependencies is required to predict the mechanisms transition. From the discussion above, it can be inferred that the in-situ TEM observation provides a new evidence for amorphization process at crack tip in FCC Al under uniaxial tensile. On the other hand, MD simulation in Al is applied to study the mechanism of deformation process, and the results show that amorphization process is initiated by the nucleation and movement of the Lomer dislocation. Furthermore, the initial crack propagation behaviours depend on the competition between twinning partial dislocation and Lomer one. To predict the transition from twinning to amorphization at the initial stage of crack propagation process, an analytic model is developed based on Peierls model. The results indicate that the transition point from analytic model is close to the trends in simulation one. Although the analytic model of crack amorphization does not directly show the amorphization process at crack tip in single or polycrystalline metals, it is possibly apply to nanoscale metals where twinning and amorphization from grain boundaries plays a main role in the fracture process [16,17,41].
10 0.17 0.18 0.19 0.20 0.21 0.22 0.23
K (eV/A
e
⎛ ⎛ πr ⎞ ⎞ ⎛ 2a E ⎞ ⎛r⎞ arctg ⎜exp ⎜ ⎟ ⎟ ⎜EL + d 2 l2 ⎟ ln ⎜ ⎟ ⎝ w0 ⎠ ⎠ ⎝ π dφ ⎠ ⎝ r0 ⎠ ⎝
4. Conclusions
where the T0d2/(2γ) denotes the long range interaction potential [39], d is the distance between Peierls valleys, EL is dislocation line energy, T0 is the orientation dependence of EL, c1 is the shape factor of Peierls model, w0 and r0 are represent the width and cut-off radius in the equilibrium configuration, respectively. The calculated results of Φ dependence on δl/bp for Lomer dislocation is shown by red line in Fig. 5. The energy difference △U between stable slip statuses δst(γ) and maximal point value δex(γ) represents the interplanar activation energy per unit length at any applied load, which can be obtained from the simulation cell,
In summary, MD simulation and analytic model in Al are performed to study the competition between twinning partial dislocation and Lomer one during the crack propagation process. The simulation results show that there exists a transition point KI=0.205 eV Å−2.5 between twinning partial dislocation and Lomer one, with Lomer dislocation is preferable one at high loads. Similarly, the analytic model also predicts that the transition point is at KI=0.215 eV Å−2.5, which influences the crack propagation mechanism from twinning to amorphization. In addition, the amorphization process at crack tip during the crack propagation in FCC Al alloy is observed by HRTEM. The results rationalize the amorphization assumption that the amorphous zone is likely at crack tip in FCC structures at high strain rates.
ΔU = U [δex (r )] − U [δst (r )] The solutions for the equation are obtained through the calculation by Rice's method [24], which consider the adaptive discretization effect of the slip distribution. Next, in order to compare with the simulation results, two calculation steps are required. Firstly, the total activation energy for an actual dislocation line with an effective lateral length L is calculated.
Acknowledgments Thanks are given to the financial supports of the Natural Science Foundation of China (Nos. 51271147, 51201134, 51201135), the Fundamental Research Funds for the Central Universities (3102014JCQ01023) and the Research Fund of the State Key Laboratory of Solidification Processing (NWPU), China (Grant no. 115-QP-2014).
ΔUtotal = LΔU In this work, L=10bp=16.5 Å, bp is the length of the Burgers vector. 76
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1975–1983. [22] T. Zhu, J. Li, A. Samanta, H.G. Kim, S. Suresh, Proc. Natl. Acad. Sci. USA 104 (2007) 3031–3036. [23] W.A. Curtin, D.L. Olmsted, L.G. Hector, Nat. Mater. 5 (2006) 875–880. [24] J.R. Rice, G.E. Beltz, J. Mech. Phys. Solids 42 (1994) 333–360. [25] Z.Q. Feng, Y.Q. Yang, Y.X. Chen, B. Huang, M.S. Fu, M.H. Li, J.G. Ru, Mater. Sci. Eng.: A 586 (2013) 259–266. [26] S. Plimpton, J. Comput. Phys. 117 (1995) 1–19. [27] Y. Mishin, D. Farkas, M.J. Mehl, D.A. Papaconstantopoulos, Phys. Rev. B 59 (1999) 3393–3407. [28] D. Farkas, M. Duranduru, W.A. Curtin, C. Ribbens, Philos. Mag. A 81 (2001) 1241–1255. [29] Y. Cheng, M.X. Shi, Y.W. Zhang, Int. J. Solids Struct. 49 (2012) 3345–3354. [30] E.B. Tadmor, S. Hai, J. Mech. Phys. Solids 51 (2003) 765–793. [31] E. Tadmor, J. Mech. Phys. Solids 52 (2004) 2507–2519. [32] Y. Murakami, L.M. Keer, J. Appl. Mech. 60 (1993) (1063-1063). [33] A. Stukowski, Model. Simul. Mater. Sci. Eng. 18 (2010) 015012. [34] D. Faken, H. Jónsson, Comput. Mater. Sci. 2 (1994) 279–286. [35] H. Tsuzuki, P.S. Branicio, J.P. Rino, Comput. Phys. Commun. 177 (2007) 518–523. [36] J.R. Rice, J. Mech. Phys. Solids 40 (1992) 239–271. [37] D.H. Warner, W.A. Curtin, S. Qu, Nat. Mater. 6 (2007) 876–881. [38] D. Rodney, L. Proville, Phys. Rev. B 79 (2009) 094108. [39] G. Schoeck, Philos. Mag. 87 (2007) 1631–1647. [40] T. Zhu, J. Li, S. Yip, Phys. Rev. Lett. 93 (2004) 025503. [41] J. Wang, F. Sansoz, J. Huang, Y. Liu, S. Sun, Z. Zhang, S.X. Mao, Nat. Commun. 4 (2013) 1742.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
J.R. Rice, Math. Fundam. 2 (1968) 191–311. J.R. Rice, Mech. Mater. 6 (1987) 317–335. J.R. Rice, R. Thomson, Philos. Mag. 29 (1974) 73–97. A. Kelly, W.R. Tyson, A.H. Cottrell, Philos. Mag. 15 (1967) 567–586. Y. a, a, D.W.a., S.R.P.a., H.G.b., 2002. S. Hai, E.B. Tadmor, Acta Mater. 51 (2003) 117–131. V. Yamakov, D. Wolf, S.R. Phillpot, H. Gleiter, Acta Mater. 51 (2003) 4135–4147. X.-T. Luo, G.-J. Yang, C.-J. Li, K. Kondoh, Scr. Mater. 65 (2011) 581–584. P.R. Okamoto, J.K. Heuer, N.Q. Lam, S. Ohnuki, Y. Matsukawa, K. Tozawa, J.F. Stubbins, Appl. Phys. Lett. 73 (1998) 473–475. I.A. Ovid'ko, J. Phys. D: Appl. Phys. 24 (1991) 2190. I.A. Ovid'ko, J. Phys. D: Appl. Phys. 27 (1994) 999. M. Chen, J.W. McCauley, K.J. Hemker, Science 299 (2003) 1563–1566. K.L. Baker, D.H. Warner, Model. Simul. Mater. Sci. Eng. 20 (2012) 065005. A.M. Dongare, A.M. Rajendran, B. LaMattina, M.A. Zikry, D.W. Brenner, Phys. Rev. B 80 (2009) 104108. B. Hess, B.J. Thijsse, E. Van der Giessen, Phys. Rev. B 71 (2005). P.S. Branício, J.-P. Rino, Phys. Rev. B 62 (2000) 16950–16955. H. Ikeda, Y. Qi, T. Çagin, K. Samwer, W.L. Johnson, W.A. Goddard, Phys. Rev. Lett. 82 (1999) 2900–2903. R.M. Lynden-Bell, J. Phys.: Condens. Matter 7 (1995) 4603. S. Vijay Kumar, D.R. Mahapatra, J. Phys.: Condens. Matter 20 (2008) 335206. S. Vijay Kumar, D.R. Mahapatra, Nanotechnology 20 (2009) 045701. H. Van Swygenhoven, P.M. Derlet, A.G. Frøseth, Acta Mater. 54 (2006)
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