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Effect of grain size and misorientation angle on fatigue crack growth of nanocrystalline materials
Materials Science & Engineering A 663 (2016) 1–7
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Materials Science & Engineering A journal homepage: www.elsevier.com/locate/msea
Effect of grain size and misorientation angle on fatigue crack growth of nanocrystalline materials Piao Zhou a, Jianqiu Zhou b,n, Zhixiong Ye a, Xu Hong a, Huajun Huang a, Wenjin Xu a a b
School of Mechanical and Electrical Engineering, Wuhan Institute of Technology, Wuhan, Hubei 430070, China School of Mechanical and Power Engineering, Nanjing University of Technology, Nanjing, Jiangsu 210009, China
art ic l e i nf o
a b s t r a c t
Article history: Received 3 March 2016 Received in revised form 22 March 2016 Accepted 23 March 2016 Available online 28 March 2016
It is known that the unique microstructure and deformation mechanism of nanocrystalline (nc) materials make the crack initiation and propagation different from the conventional coarse grained materials. The research of fatigue crack propagation (FCP) in nc metals has remained an empirical field. A theoretical model for I type crack growth was established to address the physical processes. The model describes the crack initiation and propagation and discusses the important topic of the role of the grain size and misorientation angle on the fatigue crack growth on the interface of the nc materials. We make the major research that fatigue crack growth is governed by the irreversibility of displacement at the crack tip and the dislocation glide resistance. The characteristics are observed that the dislocation glide suffer larger resistance from ultrafine grain size and large misorientation angle. It can be found that the critical stress of dislocation penetrating grain boundary and the misorientation angle of the grain boundary are positively correlated. The resent experiment found that grain refinement serves to reduce the extent of crack path tortuosity and cause an increase in fcp. The model results demonstrate the crack growth rate increases with the decrease of the grain size and the misorientation angle increases with the crack growth rate decreasing, in agreement with experimental discovery. & 2016 Published by Elsevier B.V.
Keywords: Nanocrystalline metal Theoretical model Crack initiation Fatigue crack growth Grain size Misorientation angle
1. Introduction Nanocrystalline (Nc) metal materials which the size is 1– 100 nm [1] have been researched in the past few years, because of which have special properties [2–6]. In coarse grain, grain refining which affects the fatigue behavior has attracted a great number of researchers. It is well known that the fatigue endurance will be improved when grain size is decreased with high fatigue stress amplitude for the materials which have higher stack fault energy (SFE) [7,8]. The fatigue endurance will not be affected by changing the grain size when the SFE is low, such as Cu or Al. But for some other materials of brass, possessing low SFE, stress fatigue life will be improved when grain size decrease. In recent years, the fatigue property of ultrafine grained and nc materials have attracted a great number of investigators [9–14]. The research results illustrated that the high cycle fatigue life and fatigue limit stress of ultrafine grained and nc materials are better than those of coarse grain materials, however, the performance of plastic deformation is not well [12,14]. Hanlon [14] researches the fatigue response of nc Ni. The stress life fatigue behavior and fatigue crack growth n
Corresponding author. E-mail address: [email protected] (J. Zhou).
http://dx.doi.org/10.1016/j.msea.2016.03.105 0921-5093/& 2016 Published by Elsevier B.V.
characteristics of pure Ni were studied as a function of grain size, which ranges from tens of nm to tens of um. It was found that grain refinement in the true NC regime generally leads to a significant increase in total life under stress controlled fatigue conditions and cause an increase in fatigue crack growth. The reason is that the volume fraction of grain boundary increases rapidly after grain refining, and the strength of the materials also increases, the resistance of fatigue crack nucleation increases, meanwhile, the fracture surface asperities decrease. Many studies have explained the deformation mechanism of nanocrystalline materials under circulating load [15–19]. In the case of coarse grained materials, the fatigue limit of Nc materials and Twined materials are higher. Singh et al. [20] who researched the fatigue crack propagation of nano-twinned Cu show that the higher the twin density, the lower the crack growth rate is, when the grain size remains constant. Zhang et al. [21–23] who systemically study on the fatigue crack mechanism of the twin boundary found that the fatigue crack can crack in the slip band, and also can crack in the twin boundary. Xie et al. [24] found that the width of the shear band increase with the increase of crack size by observing the change of the shear band width in the nanocrystalline materials. They further concluded that the expansion of the tip crack was mainly influenced by the dislocation deformation of the shear band. Michael [25] characterize the
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P. Zhou et al. / Materials Science & Engineering A 663 (2016) 1–7
Fig. 1. Scanning electron micrographs of mc, ufc, and nc Ni subjected to sinusoidal fatigue loading [16].
microstructure of a NC electro-deposition (ED) Ni–Co alloy that displays superior FCG resistance compared with CG materials. That explains the fatigue irreversibility and damage resistance of this material and observes the FCG of Ni–Co alloy by scanning electron microscopy (SEM) and transmission electron microscopy (TEM). The fatigue life of nc materials mainly include fatigue crack Initiation and fatigue crack growth. Most cycle load is used to promote crack Initiation. Crack growth rate is not representing fatigue life. A fatigue crack initiation is because of stress concentration by the dislocation pill-up. Grain refinement can reduce the stress concentration compared with coarse grain, and improve the resistance of crack Initiation. Hanlon [14] found that a reduction in grain size cause an increase in FCP by compared with nanocrystalline pure Ni, ultrafine crystalline pure Ni and microcrystalline Ni. The reason is that grain refinement serves to reduce the extent of such crack path tortuosity, and the roughness of fatigue fracture drop. The reduced roughness reduces the resistance of FCP, and the FCP rate increase when grain size reduces. The experiment result researched by Hanlon [14] is shown in Fig. 1. The predecessors observed the fatigue growth of nc materials or nano-twined materials by experiment, however, the theoretical model of the fatigue crack propagation problem based on the interface of nc materials is less studied. This paper mainly discusses the fatigue crack growth rate problem which is affected by the grain size and misorientation angle on the interface of the nc materials.
direction of the dislocation slip are minimum. Defining parameter M,
M = (l1li )(g1gi ) + (l1gi )(g1li )
(1)
In the formula: l1 is the normal direction of slip surface and g1 is the slip direction of dislocation slip plane; li and gi identify the slip surface normal direction and slip direction of all potential dislocations in the adjacent grains. the parameters M is minimum when dislocation penetrate grain boundary. Based on above, Shen et al. [27] and Lee et al. [28] put forward two other conditions: one is that the shear stress which is in the position of the dislocation slip should reach the maximum value; the other one is that the burgers vector of the residual dislocations in the grain boundaries should be minimum. Dislocation penetrating through grain boundary are governed by the above three criteria. As shown in Fig. 2, the grain A and B are separated by grain boundaries. There are two slip systems on the arbitrary side of the grain boundary. When the dislocations in the grain A along slip system S1 to the grain boundaries between B and A, the dislocation in the grain A cannot be smoothly slip into the grain B. Grain boundaries play an large obstacle to dislocation slip, and then
2. Materials and methods 2.1. Dislocation penetrating grain boundary A brief summary of the grain boundary penetration is represented. Livingston and Chalmers [26] proposed a geometrical criterion which is about the dislocation to penetrate the grain boundary. That the angle between the slip surface of the adjacent grain and the intersection line of the grain boundary, and the angle between the sliding direction of the dislocation and the
Fig. 2. Two-dimensional sketch map of dislocation penetrating grain boundary.
P. Zhou et al. / Materials Science & Engineering A 663 (2016) 1–7
cause the dislocation accumulation around the grain boundary. With the continuous load of the applied load, when loading to a certain critical value, a dislocation which is around the dislocation pile-up group of grain boundary will penetrate the grain boundaries and slip into the grain B along slip system S2 . Simultaneously, the grain boundary can be left residual dislocation. Setting in the grain A, the Burgers vector of the dislocation along the slip system S1 is b1. The Burgers vector of the first dislocation passing through the grain boundary along the slip system S2 is b2. We refer to the angle between b1 and b2 as the misorientation angle Δθ . Based on the principle of Burgers vector conservation, the Burgers vector Δb of the residual dislocation in grain boundary can expressed as
(2)
Δb = b1 + b2
In FCC crystal, each grain exists a large amount of slip system. With the external force increasing, part slip system is activated in the grain. According to previous studies, the dislocation penetrating grain boundary needs to meet: 1) The distance between the Burgers vector of residual dislocation and the slip system S1 and S2 is the smallest. 2) The shear stress τ acting on the slip system S1 is larger than the critical shear stress τcr of dislocation penetrating grain boundary. From the point of energy law, when the loading stress acting on the slip system is greater than the strain energy of residual dislocation and the energy of grain boundary, dislocation can penetrate grain boundary. If dislocation penetrates the grain boundaries, the relationship between the critical shear stress τcr and the external loading shear stress τ should be:
τb1 ≥ τcr b12 = Egb b1 + αG (Δb)2
(3)
Egb is expressed as a unit area energy of grain boundary, α is numerical constant, G is shear modulus. According to the research of Li [29], the grain boundary energy Egb is closely related to the misorientation angle Δθ between grain A and B. The correlation formula is
⎧ 0° ≤ Δθ ≤ θ1 kΔθ /θ1 ⎪ Egb = ⎨ θ1 ≤ Δθ ≤ θ 2 k ⎪ ⎩ k (90 − Δθ )/90 − θ 2 θ 2 ≤ Δθ ≤ 90°
(4)
constant. Here, we ignore the effect of reversed Hall-Petch relation. Shanmugasundaram [30] researched the strengthening mechanisms of bulk nanocrystalline Al and Al alloys, and revealed the Hall–Petch analysis of friction stress and slope. The values are σ0 = 30 MPa , k = 0.06 MPa/m1/2. The relationship of grain size and the misorientation angle affected on critical shear stress is established. According to semi-quantitative law, The expression of yield stress is:
τtotal = τcr + τy
(7)
2.2. Stress analysis of dislocation passing through grain boundary Under normal circumstances, grain boundaries can hinder the dislocation slip. The slip plane of next emissive dislocation is always the same as the first one. Equilibrium position of new generated dislocations is decided by the balance shear stress and the previous force caused by dislocation pile-up. The last dislocation can hinder the next one to emit, only when the number of dislocation emission is accumulated enough. In NC materials, the first emissive dislocation and grain boundary can hinder the next dislocation to emit. Thus, the emission of dislocations can neglect the influence of the crack blunting. Putting I type crack into X-Y coordinate system, as shown in Fig. 3. Assuming that the first dislocation stays at the grain boundaries which distance from crack tip being grain size d, another dislocation displacement distancing from the crack tip can be carried out from the equilibrium relationship. At the equilibrium position, a dislocation emitting from a crack tip is subjected to the action of three forces: Resolved shear stress τak from the external loads on the crack tip acts on the kth dislocation, which can be expressed [31]:
τak =
KI sin φ cos (φ/2) 2 2π r k
where
Δb b1
(5)
= cos (Δθ ).
The misorientation angle affected the critical shear stress of dislocation penetrating grain boundary has been discussed. The grain size also affects the critical shear stress relating to the penetrability of grain boundary to dislocation. The Hall-Petch equation described the relationship of grain size and yield stress, and the yield stress increases with decreasing grain size. The expression is:
σy = σ 0 + kd−1/2
(6)
where σy is yield stress, d is grain size, σ0 is friction stress, k is
(8)
KI is the stress intensity factor of I type crack, φ is the angle between the slip plane and the X axis, rk is the position of the Kth dislocation. Crack surface will leave a step when the dislocation is emitted from the crack tip. The image force τim which will hinder the outward emission of dislocations is generated between the Kth
Derived from the upper formula, the relationship between the critical shear force τcr and the misorientation Δθ angle is
⎧ ⎛ Δb ⎞2 kΔθ ⎪ 0° ≤ Δθ ≤ θ1 + αG ⎜ ⎟ ⎝ b1 ⎠ θ1b1 ⎪ ⎪ ⎪ ⎛ Δb ⎞2 τcr = ⎨ kb1 + αG ⎜ θ1 ≤ Δθ ≤ θ 2 ⎟ ⎝ b1 ⎠ ⎪ ⎪ ⎛ Δb ⎞2 ⎪ k (90 − Δθ ) + αG ⎜ ⎟ θ 2 ≤ Δθ ≤ 90° ⎪ ⎝ b1 ⎠ ⎩ b1 (90 − θ 2 )
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Fig. 3. Dislocation emission at I type crack tip.
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P. Zhou et al. / Materials Science & Engineering A 663 (2016) 1–7
Fig. 4. (a) Model diagram, different segment in the grain interior express different grain orientations. (b) Dislocations emitted from crack tip penetrate through GB 1 and then stop at GB 2, which leads to the pile-up at grain 2. Table 1 Material properties are used in calculating NC Al [29,30,33]. Parameter
Notation
Value
Shear modulus Poisson's ratio Burgers vector Angle
G υ
27 GPa 0.34 0.286
b φ α σ0 k
Numerical constant friction stress Hall-Petch slope
π/3 0.2 30 MPa 0.06 MPa/m1/2
dislocation and the crack free surfaces. The expression is [31]:
τim = −
Gb 4π (1 − ν ) r k
(9)
ν is Poisson’s ratio. Dislocation pile-up stress τpi on the slip surface can be expressed: τpi = −
Gb ∑ 2π r k i ≠ k
ri 1 rk ri − rk
(10)
The shear stress acting on the kth emitting dislocation is:
τ k = τak + τim + τpi
(11)
Combined with (8)–(11),
τk =
KI sin φ cos (φ/2) Gb Gb − − ∑ 4π (1 − ν ) rk 2π r k i ≠ k 2 2π r k
ri 1 rk ri − rk
(12)
At the equilibrium positions, the equilibrium conditions for kth dislocation is τk = 0. The final equilibrium positions of all dislocations can be solved. This equation is complex to be solved, we study an approximate solution. The result is
ri =
Dπ 2 (i − 1)2 8nτak
here, D =
Gb . 2π
(13)
Fig. 5. The relationship between the critical stress τcr and (a) the orientation deviation angle Δθ (b) grain size d .
P. Zhou et al. / Materials Science & Engineering A 663 (2016) 1–7
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Fig. 6. Relationship between the fatigue crack growth rate da/dN and the stress intensity factor ΔKI of I type crack. (a–c) Effect of grain size; (d) experiment comparative data.
Combined with (12) and (14), we can get
2.3. Fatigue crack propagation The current work researched the role of misorientation angle and grain size on the fatigue crack propagation under I type crack. As shown in Fig. 4, the crack tip emits a large number of screw dislocations under monotonic cycle load. The role of slip resistances by grain boundaries break the equilibrium position of dislocation, and the dislocation slip is irreversible. The dislocation under the applied external load will produce irreversible continuous slip. Crack propagates with plastic displacement continuous accumulation at crack tip. We consider establish the expression of the fatigue crack growth rate da/dN of the NC materials based on grain size d, distance between crack tip and the equilibrium position of dislocation emission r and stress intensity factor of I type crack ΔKI . With continuous emission of the crack tip dislocations, the continuous accumulation of plastic deformation will cause the FCP. The expression of FCP is [32]
da r = max dN 2G
k
∑ τk i=1
(14)
rmax is the maximum distance from the crack tip to the equilibrium position of dislocation emitting. τk is the shear stress of the kth dislocation.
ΔKI sin φ cos (φ/2) da r = max dN 2G 2 2π +
brmax 4π
k
∑∑ i=1 j≠i
k
∑ i=1
rj 1 ri r j − ri
brmax 1 − ri 8π (1 − ν )
k
∑1 i=1
ri
(15)
We research the influence factors of crack growth rate on the interface of the NC materials by the theoretical model. For a given grain size d and misorientation Δθ , we can predict the corresponding values of da/dN at certain ΔkI . The theoretical model can make some theoretical contributions to avoid FCG of NC materials. (Table 1).
3. Results and discussion This paper selects the NC materials Al as the research object. The range of misorientation angle Δθ is selected as Δθ = 5° , 10° , 15° , the parameters of NC materials are as follows: 3.1. The influence factors of the critical stress of dislocation penetrating grain boundary From the Eqs. (5) to (7), we know that the critical stress τcr of dislocation penetrating grain boundary is closely related to the
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3.2. Influence factors of FCP rate The crack initiation is observed on the interface of the nc metals with continuum load. We neglect the problem of crack initiation. This paper mainly focuses on the content that NC material itself exist crack (I type crack). To determine the role of the FCP under monotonic cycle load, we look at the evolution of da/dN with the vary of grain size and misorientation angle. Selecting grain size d = 40, 60, 80 nm , misorientation angle Δθ = 5° , 10° , 15° , the approach of modeling FCP will be illustrated based on above theoretical model. In the present work, we employed the da/dN formulations to quantitatively research the microstructure of grain size d and misorientation angle Δθ affected on FCP. In this model, the variation of the microstructure and the resistance of dislocation emission at crack tip play an important role. The discussion up has researched the consequences of crack path vary induced by different grain size to FCP mechanisms. The experiment results show that grain refinement serves to reduce the extent of crack path tortuosity. Fig. 6(a–c) shows effects of grain size on the variation of FCP rate da/dN as a function of ΔKI for nc material Al. Here, the variation of grain size from 40 nm to 80 nm leads to a reduction of FCP rate da/dN at an misorientation angle Δθ of constant. The mainly reason is that the fracture surface asperities increase with grain size increase. The larger grain size exhibit apparently greater resistance to fatigue crack growth. The analytical result is in agreement with recent experimental findings, as shown in Fig. 6 (d). The effect of orientation deviation angle on the fatigue crack growth of nc Al is shown in Fig. 7(a–b). It is evident that the fatigue crack growth is less affected by misorientation angle. However, the fcp of nc Al is significantly decrease with Δθ greatly increase. This is because the increase of the misorientation angle can increase the energy barrier of the dislocation penetrating the grain boundary, which hinder dislocation emission at the crack tip.
4. Conclusion Fig. 7. Relationship between the fatigue crack growth rate da/dN and the stress intensity factor ΔKI of I type crack. (a–b) Effect of orientation deviation angle.
orientation deviation angle Δθ and grain size d . Here, we ignored the materials with larger misorientation angle. The relatively small misorientation angle Δθ = 5° , 10° , 15° and a range of grain size d = 20, 40, 60, 80 nm are discussed. It is clear at a glance that the critical stress of dislocation penetrating grain boundary is enhanced with the increase of misorientation angle, as shown in Fig. 5a. The increasing rate slows of critical stress τcr when the misorientation angle continuously increases. This can be tentatively explained as follows: dislocation penetrating grain boundary is mainly affected by the energy of grain boundary and residual Burgers vector. The energy of grain boundary is constant. The larger the value of residual burgers vector is, the easier the dislocation to penetrate grain boundary. The value of residual burgers vector will decrease with the increase of misorientation angle Δθ . This means that the dislocation pill-up increase if dislocation penetrates grain boundary. All grain boundaries with larger orientation deviation angle Δθ are difficult to dislocation to penetrate. As a result, the critical stress increase. From Fig. 5b, it can be seen that the critical stress decreases with increasing grain size and correlates the Hall-Petch relationship. The mainly reason is that grain boundary increases in unit volume with grain size decrease. Meanwhile, grain boundaries hinder the dislocation to slip. The result basically agree with the research of Li [29] and Embury [34].
The effect of grain size and misorientation angle on the interface of fatigue crack growth in nc metals were investigated. The varieties of grain size and misorientation angle have been shown to have a substantial effect on fatigue crack growth behavior. The extents of crack face roughness decided the fatigue crack growth. We establish the theoretical model of misorientation angle and grain size on the fatigue crack propagation under I type crack, and the calculated conclusions are: (1) The critical stress of dislocation penetrating grain boundary and the misorientation angle of the grain boundary are positively correlated. the critical stress decreases with increasing grain size and correlates the Hall-Petch relationship. (2) The grain size will influence the crack face roughness of nc metals, and then the crack growth rate is vary. The crack growth rate increases with the increase of the stress intensity factor KI , and the grain size and crack growth rate were negatively correlated when the grain boundary misorientation angle is invariant. The fcp of nc Al is significantly decrease with Δθ greatly increase when the grain size is invariant. This paper only considers the single cycle loading condition for the theoretical model of NC materials in I type crack propagation, and neglected the problems of inverse Hall-Petch relationship. There is no further study on the fatigue crack growth of type II and III. This is all problems to be considered in the future.
P. Zhou et al. / Materials Science & Engineering A 663 (2016) 1–7
Acknowledgements This work was supported by National Natural Science Foundation of China (10502025, 10872087 and 11272143) and the Program for Chinese New Century Excellent Talents in University (NCET-12-0712).
References [1] H. Gleiter, Nanocrystalline Materials, Springer, Berlin Heidelberg, 1991. [2] M.A. Meyers, A. Mishra, D.J. Benson, Mechanical properties of nanocrystalline materials, Prog. Mater. Sci. 51 (2006) 427–556. [3] S. Benkassem, L. Capolungo, M. Cherkaoui, Mechanical properties and multiscale modeling of nanocrystalline materials, Acta Mater. 55 (2007) 3563–3572. [4] L. Capolungo, M. Cherkaoui, J. Qu, On the elastic–viscoplastic behavior of nanocrystalline materials, Int. J. Plast. 23 (2007) 561–591. [5] N. Naga Krishna, A.K. Akash, K. Sivaprasad, R. Narayanasamy, Studies on void coalescence analysis of nanocrystalline cryorolled commercially pure aluminium formed under different stress conditions, Mater. Des. 31 (2010) 3578–3584. [6] M.A. Meyersm, E. Ashworth, A model for the effect of grain size on the yield stress of metals, Philos. Mag. A 46 (1982) 737–759. [7] A.W. Thompson, W.A. Backofen, The effect of grain size on fatigue, Acta Metall. 19 (1971) 597–606. [8] P. Lukáš, L. Kunz, Effect of grain size on the high cycle fatigue behaviour of polycrystalline copper, Mater. Sci. Eng. 85 (1987) 67–75. [9] S.R. Agnew, A.Y. Vinogradov, S. Hashimoto, J.R. Weertman, Overview of fatigue performance of Cu processed by severe plastic deformation, J. Electron Mater. 28 (1999) 1038–1044. [10] A. Vinogradov, S. Hashimoto, Multiscale phenomena in fatigue of ultrafine grain materials – an overview, Mater. Trans. 42 (2001) 74–84. [11] S.D. Wu, Z.G. Wang, C.B. Jiang, G.Y. Li, I.V. Alexandrov, R.Z. Valiev, Shear bands in cyclically deformed ultrafine grained copper processed by ECAP, Mater. Sci. Eng.: A. 387–389 (2004) 560–564. [12] L. Kunz, P. Lukáš, M. Svoboda, Fatigue strength, microstructural stability and strain localization in ultrafine-grained copper, Mater. Sci. Eng.: A 424 (2006) 97–104. [13] C. Xu, Q. Wang, M. Zheng, J. Li, M. Huang, Q. Jia, J. Zhu, L. Kunz, M. Buksa, Fatigue behavior and damage characteristic of ultra-fine grain low-purity copper processed by equal-channel angular pressing (ECAP), Mater. Sci. Eng.: A 475 (2008) 249–256. [14] T. Hanlon, E.D. Tabachnikova, S. Suresh, Fatigue behavior of nanocrystalline metals and alloys, Int. J. Fatigue 27 (2005) 1147–1158. [15] R.Z. Valiev, T.G. Langdon, Principles of equal-channel angular pressing as a
7
processing tool for grain refinement, Prog. Mater. Sci. 51 (2006) 881–981. [16] R.Z. Valiev, R.K. Islamgaliev, I.V. Alexandrov, Bulk nanostructured materials from severe plastic deformation, Prog. Mater. Sci. 45 (2000) 103–189. [17] L. Lu, R. Schwaiger, Z.W. Shan, M. Dao, K. Lu, S. Suresh, Nano-sized twins induce high rate sensitivity of flow stress in pure copper, Acta Mater. 53 (2005) 2169–2179. [18] Z.F. Zhang, Z.G. Wang, Dependence of intergranular fatigue cracking on the interactions of persistent slip bands with grain boundaries, Acta Mater. 51 (2003) 347–364. [19] Z.F. Zhang, Z.G. Wang, Grain boundary effects on cyclic deformation and fatigue damage, Prog. Mater. Sci. 53 (2008) 1025–1099. [20] A. Singh, L. Tang, M. Dao, L. Lu, S. Suresh, Fracture toughness and fatigue crack growth characteristics of nanotwinned copper, Acta Mater. 59 (2011) 2437–2446. [21] S. Qu, P. Zhang, S.D. Wu, Q.S. Zang, Z.F. Zhang, Twin boundaries: strong or weak? Scr. Mater. 59 (2008) 1131–1134. [22] P. Zhang, Z.J. Zhang, L.L. Li, Z.F. Zhang, Twin boundary: stronger or weaker interface to resist fatigue cracking? Scr. Mater. 66 (2012) 854–859. [23] Z.J. Zhang, P. Zhang, L.L. Li, Z.F. Zhang, Fatigue cracking at twin boundaries: effects of crystallographic orientation and stacking fault energy, Acta Mater. 60 (2012) 3113–3127. [24] J. Xie, X. Wu, Y. Hong, Shear bands at the fatigue crack tip of nanocrystalline nickel, Scr. Mater. 57 (2007) 5–8. [25] M.D. Sangid, G.J. Pataky, H. Sehitoglu, R.G. Rateick, T. Niendorf, H.J. Maier, Superior fatigue crack growth resistance, irreversibility, and fatigue crack growth–microstructure relationship of nanocrystalline alloys, Acta Mater. 59 (2011) 7340–7355. [26] J.D. Livingston, B. Chalmers, Multiple slip in bicrystal deformation, Acta Metall. 5 (1957) 322–327. [27] Z. Shen, R.H. Wagoner, W.A.T. Clark, Dislocation and grain boundary interactions in metals, Acta Metall. 36 (1988) 3231–3242. [28] T.C. Lee, I.M. Robertson, H.K. Birnbaum, Prediction of slip transfer mechanisms across grain boundaries, Scr. Metall. 23 (1989) 799–803. [29] Z. Li, C. Hou, M. Huang, C. Ouyang, Strengthening mechanism in micro-polycrystals with penetrable grain boundaries by discrete dislocation dynamics simulation and Hall–Petch effect, Comput. Mater. Sci. 46 (2009) 1124–1134. [30] T. Shanmugasundaram, M. Heilmaier, B.S. Murty, V.S. Sarma, On the Hall– Petch relationship in a nanostructured Al–Cu alloy, Mater. Sci. Eng.: A 527 (2010) 7821–7825. [31] R. Thomson, T.J. Chuang, I.H. Lin, The role of surface stress in fracture, Acta Metall. 34 (1986) 1133–1143. [32] P.B. Chowdhury, H. Sehitoglu, R.G. Rateick, H.J. Maier, Modeling fatigue crack growth resistance of nanocrystalline alloys, Acta Mater. 61 (2013) 2531–2547. [33] G.C. Hasson, C. Goux, Interfacial energies of tilt boundaries in aluminium. Experimental and theoretical determination, Scr. Metall. 5 (1971) 889–894. [34] J. Embury, Strengthening mechanisms in Al alloys - an overview of natural limits and engineering possibilities, Mater. Sci. Forum 217–222 (1996) 57–70.
Contents lists available at ScienceDirect
Materials Science & Engineering A journal homepage: www.elsevier.com/locate/msea
Effect of grain size and misorientation angle on fatigue crack growth of nanocrystalline materials Piao Zhou a, Jianqiu Zhou b,n, Zhixiong Ye a, Xu Hong a, Huajun Huang a, Wenjin Xu a a b
School of Mechanical and Electrical Engineering, Wuhan Institute of Technology, Wuhan, Hubei 430070, China School of Mechanical and Power Engineering, Nanjing University of Technology, Nanjing, Jiangsu 210009, China
art ic l e i nf o
a b s t r a c t
Article history: Received 3 March 2016 Received in revised form 22 March 2016 Accepted 23 March 2016 Available online 28 March 2016
It is known that the unique microstructure and deformation mechanism of nanocrystalline (nc) materials make the crack initiation and propagation different from the conventional coarse grained materials. The research of fatigue crack propagation (FCP) in nc metals has remained an empirical field. A theoretical model for I type crack growth was established to address the physical processes. The model describes the crack initiation and propagation and discusses the important topic of the role of the grain size and misorientation angle on the fatigue crack growth on the interface of the nc materials. We make the major research that fatigue crack growth is governed by the irreversibility of displacement at the crack tip and the dislocation glide resistance. The characteristics are observed that the dislocation glide suffer larger resistance from ultrafine grain size and large misorientation angle. It can be found that the critical stress of dislocation penetrating grain boundary and the misorientation angle of the grain boundary are positively correlated. The resent experiment found that grain refinement serves to reduce the extent of crack path tortuosity and cause an increase in fcp. The model results demonstrate the crack growth rate increases with the decrease of the grain size and the misorientation angle increases with the crack growth rate decreasing, in agreement with experimental discovery. & 2016 Published by Elsevier B.V.
Keywords: Nanocrystalline metal Theoretical model Crack initiation Fatigue crack growth Grain size Misorientation angle
1. Introduction Nanocrystalline (Nc) metal materials which the size is 1– 100 nm [1] have been researched in the past few years, because of which have special properties [2–6]. In coarse grain, grain refining which affects the fatigue behavior has attracted a great number of researchers. It is well known that the fatigue endurance will be improved when grain size is decreased with high fatigue stress amplitude for the materials which have higher stack fault energy (SFE) [7,8]. The fatigue endurance will not be affected by changing the grain size when the SFE is low, such as Cu or Al. But for some other materials of brass, possessing low SFE, stress fatigue life will be improved when grain size decrease. In recent years, the fatigue property of ultrafine grained and nc materials have attracted a great number of investigators [9–14]. The research results illustrated that the high cycle fatigue life and fatigue limit stress of ultrafine grained and nc materials are better than those of coarse grain materials, however, the performance of plastic deformation is not well [12,14]. Hanlon [14] researches the fatigue response of nc Ni. The stress life fatigue behavior and fatigue crack growth n
Corresponding author. E-mail address: [email protected] (J. Zhou).
http://dx.doi.org/10.1016/j.msea.2016.03.105 0921-5093/& 2016 Published by Elsevier B.V.
characteristics of pure Ni were studied as a function of grain size, which ranges from tens of nm to tens of um. It was found that grain refinement in the true NC regime generally leads to a significant increase in total life under stress controlled fatigue conditions and cause an increase in fatigue crack growth. The reason is that the volume fraction of grain boundary increases rapidly after grain refining, and the strength of the materials also increases, the resistance of fatigue crack nucleation increases, meanwhile, the fracture surface asperities decrease. Many studies have explained the deformation mechanism of nanocrystalline materials under circulating load [15–19]. In the case of coarse grained materials, the fatigue limit of Nc materials and Twined materials are higher. Singh et al. [20] who researched the fatigue crack propagation of nano-twinned Cu show that the higher the twin density, the lower the crack growth rate is, when the grain size remains constant. Zhang et al. [21–23] who systemically study on the fatigue crack mechanism of the twin boundary found that the fatigue crack can crack in the slip band, and also can crack in the twin boundary. Xie et al. [24] found that the width of the shear band increase with the increase of crack size by observing the change of the shear band width in the nanocrystalline materials. They further concluded that the expansion of the tip crack was mainly influenced by the dislocation deformation of the shear band. Michael [25] characterize the
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Fig. 1. Scanning electron micrographs of mc, ufc, and nc Ni subjected to sinusoidal fatigue loading [16].
microstructure of a NC electro-deposition (ED) Ni–Co alloy that displays superior FCG resistance compared with CG materials. That explains the fatigue irreversibility and damage resistance of this material and observes the FCG of Ni–Co alloy by scanning electron microscopy (SEM) and transmission electron microscopy (TEM). The fatigue life of nc materials mainly include fatigue crack Initiation and fatigue crack growth. Most cycle load is used to promote crack Initiation. Crack growth rate is not representing fatigue life. A fatigue crack initiation is because of stress concentration by the dislocation pill-up. Grain refinement can reduce the stress concentration compared with coarse grain, and improve the resistance of crack Initiation. Hanlon [14] found that a reduction in grain size cause an increase in FCP by compared with nanocrystalline pure Ni, ultrafine crystalline pure Ni and microcrystalline Ni. The reason is that grain refinement serves to reduce the extent of such crack path tortuosity, and the roughness of fatigue fracture drop. The reduced roughness reduces the resistance of FCP, and the FCP rate increase when grain size reduces. The experiment result researched by Hanlon [14] is shown in Fig. 1. The predecessors observed the fatigue growth of nc materials or nano-twined materials by experiment, however, the theoretical model of the fatigue crack propagation problem based on the interface of nc materials is less studied. This paper mainly discusses the fatigue crack growth rate problem which is affected by the grain size and misorientation angle on the interface of the nc materials.
direction of the dislocation slip are minimum. Defining parameter M,
M = (l1li )(g1gi ) + (l1gi )(g1li )
(1)
In the formula: l1 is the normal direction of slip surface and g1 is the slip direction of dislocation slip plane; li and gi identify the slip surface normal direction and slip direction of all potential dislocations in the adjacent grains. the parameters M is minimum when dislocation penetrate grain boundary. Based on above, Shen et al. [27] and Lee et al. [28] put forward two other conditions: one is that the shear stress which is in the position of the dislocation slip should reach the maximum value; the other one is that the burgers vector of the residual dislocations in the grain boundaries should be minimum. Dislocation penetrating through grain boundary are governed by the above three criteria. As shown in Fig. 2, the grain A and B are separated by grain boundaries. There are two slip systems on the arbitrary side of the grain boundary. When the dislocations in the grain A along slip system S1 to the grain boundaries between B and A, the dislocation in the grain A cannot be smoothly slip into the grain B. Grain boundaries play an large obstacle to dislocation slip, and then
2. Materials and methods 2.1. Dislocation penetrating grain boundary A brief summary of the grain boundary penetration is represented. Livingston and Chalmers [26] proposed a geometrical criterion which is about the dislocation to penetrate the grain boundary. That the angle between the slip surface of the adjacent grain and the intersection line of the grain boundary, and the angle between the sliding direction of the dislocation and the
Fig. 2. Two-dimensional sketch map of dislocation penetrating grain boundary.
P. Zhou et al. / Materials Science & Engineering A 663 (2016) 1–7
cause the dislocation accumulation around the grain boundary. With the continuous load of the applied load, when loading to a certain critical value, a dislocation which is around the dislocation pile-up group of grain boundary will penetrate the grain boundaries and slip into the grain B along slip system S2 . Simultaneously, the grain boundary can be left residual dislocation. Setting in the grain A, the Burgers vector of the dislocation along the slip system S1 is b1. The Burgers vector of the first dislocation passing through the grain boundary along the slip system S2 is b2. We refer to the angle between b1 and b2 as the misorientation angle Δθ . Based on the principle of Burgers vector conservation, the Burgers vector Δb of the residual dislocation in grain boundary can expressed as
(2)
Δb = b1 + b2
In FCC crystal, each grain exists a large amount of slip system. With the external force increasing, part slip system is activated in the grain. According to previous studies, the dislocation penetrating grain boundary needs to meet: 1) The distance between the Burgers vector of residual dislocation and the slip system S1 and S2 is the smallest. 2) The shear stress τ acting on the slip system S1 is larger than the critical shear stress τcr of dislocation penetrating grain boundary. From the point of energy law, when the loading stress acting on the slip system is greater than the strain energy of residual dislocation and the energy of grain boundary, dislocation can penetrate grain boundary. If dislocation penetrates the grain boundaries, the relationship between the critical shear stress τcr and the external loading shear stress τ should be:
τb1 ≥ τcr b12 = Egb b1 + αG (Δb)2
(3)
Egb is expressed as a unit area energy of grain boundary, α is numerical constant, G is shear modulus. According to the research of Li [29], the grain boundary energy Egb is closely related to the misorientation angle Δθ between grain A and B. The correlation formula is
⎧ 0° ≤ Δθ ≤ θ1 kΔθ /θ1 ⎪ Egb = ⎨ θ1 ≤ Δθ ≤ θ 2 k ⎪ ⎩ k (90 − Δθ )/90 − θ 2 θ 2 ≤ Δθ ≤ 90°
(4)
constant. Here, we ignore the effect of reversed Hall-Petch relation. Shanmugasundaram [30] researched the strengthening mechanisms of bulk nanocrystalline Al and Al alloys, and revealed the Hall–Petch analysis of friction stress and slope. The values are σ0 = 30 MPa , k = 0.06 MPa/m1/2. The relationship of grain size and the misorientation angle affected on critical shear stress is established. According to semi-quantitative law, The expression of yield stress is:
τtotal = τcr + τy
(7)
2.2. Stress analysis of dislocation passing through grain boundary Under normal circumstances, grain boundaries can hinder the dislocation slip. The slip plane of next emissive dislocation is always the same as the first one. Equilibrium position of new generated dislocations is decided by the balance shear stress and the previous force caused by dislocation pile-up. The last dislocation can hinder the next one to emit, only when the number of dislocation emission is accumulated enough. In NC materials, the first emissive dislocation and grain boundary can hinder the next dislocation to emit. Thus, the emission of dislocations can neglect the influence of the crack blunting. Putting I type crack into X-Y coordinate system, as shown in Fig. 3. Assuming that the first dislocation stays at the grain boundaries which distance from crack tip being grain size d, another dislocation displacement distancing from the crack tip can be carried out from the equilibrium relationship. At the equilibrium position, a dislocation emitting from a crack tip is subjected to the action of three forces: Resolved shear stress τak from the external loads on the crack tip acts on the kth dislocation, which can be expressed [31]:
τak =
KI sin φ cos (φ/2) 2 2π r k
where
Δb b1
(5)
= cos (Δθ ).
The misorientation angle affected the critical shear stress of dislocation penetrating grain boundary has been discussed. The grain size also affects the critical shear stress relating to the penetrability of grain boundary to dislocation. The Hall-Petch equation described the relationship of grain size and yield stress, and the yield stress increases with decreasing grain size. The expression is:
σy = σ 0 + kd−1/2
(6)
where σy is yield stress, d is grain size, σ0 is friction stress, k is
(8)
KI is the stress intensity factor of I type crack, φ is the angle between the slip plane and the X axis, rk is the position of the Kth dislocation. Crack surface will leave a step when the dislocation is emitted from the crack tip. The image force τim which will hinder the outward emission of dislocations is generated between the Kth
Derived from the upper formula, the relationship between the critical shear force τcr and the misorientation Δθ angle is
⎧ ⎛ Δb ⎞2 kΔθ ⎪ 0° ≤ Δθ ≤ θ1 + αG ⎜ ⎟ ⎝ b1 ⎠ θ1b1 ⎪ ⎪ ⎪ ⎛ Δb ⎞2 τcr = ⎨ kb1 + αG ⎜ θ1 ≤ Δθ ≤ θ 2 ⎟ ⎝ b1 ⎠ ⎪ ⎪ ⎛ Δb ⎞2 ⎪ k (90 − Δθ ) + αG ⎜ ⎟ θ 2 ≤ Δθ ≤ 90° ⎪ ⎝ b1 ⎠ ⎩ b1 (90 − θ 2 )
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Fig. 3. Dislocation emission at I type crack tip.
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Fig. 4. (a) Model diagram, different segment in the grain interior express different grain orientations. (b) Dislocations emitted from crack tip penetrate through GB 1 and then stop at GB 2, which leads to the pile-up at grain 2. Table 1 Material properties are used in calculating NC Al [29,30,33]. Parameter
Notation
Value
Shear modulus Poisson's ratio Burgers vector Angle
G υ
27 GPa 0.34 0.286
b φ α σ0 k
Numerical constant friction stress Hall-Petch slope
π/3 0.2 30 MPa 0.06 MPa/m1/2
dislocation and the crack free surfaces. The expression is [31]:
τim = −
Gb 4π (1 − ν ) r k
(9)
ν is Poisson’s ratio. Dislocation pile-up stress τpi on the slip surface can be expressed: τpi = −
Gb ∑ 2π r k i ≠ k
ri 1 rk ri − rk
(10)
The shear stress acting on the kth emitting dislocation is:
τ k = τak + τim + τpi
(11)
Combined with (8)–(11),
τk =
KI sin φ cos (φ/2) Gb Gb − − ∑ 4π (1 − ν ) rk 2π r k i ≠ k 2 2π r k
ri 1 rk ri − rk
(12)
At the equilibrium positions, the equilibrium conditions for kth dislocation is τk = 0. The final equilibrium positions of all dislocations can be solved. This equation is complex to be solved, we study an approximate solution. The result is
ri =
Dπ 2 (i − 1)2 8nτak
here, D =
Gb . 2π
(13)
Fig. 5. The relationship between the critical stress τcr and (a) the orientation deviation angle Δθ (b) grain size d .
P. Zhou et al. / Materials Science & Engineering A 663 (2016) 1–7
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Fig. 6. Relationship between the fatigue crack growth rate da/dN and the stress intensity factor ΔKI of I type crack. (a–c) Effect of grain size; (d) experiment comparative data.
Combined with (12) and (14), we can get
2.3. Fatigue crack propagation The current work researched the role of misorientation angle and grain size on the fatigue crack propagation under I type crack. As shown in Fig. 4, the crack tip emits a large number of screw dislocations under monotonic cycle load. The role of slip resistances by grain boundaries break the equilibrium position of dislocation, and the dislocation slip is irreversible. The dislocation under the applied external load will produce irreversible continuous slip. Crack propagates with plastic displacement continuous accumulation at crack tip. We consider establish the expression of the fatigue crack growth rate da/dN of the NC materials based on grain size d, distance between crack tip and the equilibrium position of dislocation emission r and stress intensity factor of I type crack ΔKI . With continuous emission of the crack tip dislocations, the continuous accumulation of plastic deformation will cause the FCP. The expression of FCP is [32]
da r = max dN 2G
k
∑ τk i=1
(14)
rmax is the maximum distance from the crack tip to the equilibrium position of dislocation emitting. τk is the shear stress of the kth dislocation.
ΔKI sin φ cos (φ/2) da r = max dN 2G 2 2π +
brmax 4π
k
∑∑ i=1 j≠i
k
∑ i=1
rj 1 ri r j − ri
brmax 1 − ri 8π (1 − ν )
k
∑1 i=1
ri
(15)
We research the influence factors of crack growth rate on the interface of the NC materials by the theoretical model. For a given grain size d and misorientation Δθ , we can predict the corresponding values of da/dN at certain ΔkI . The theoretical model can make some theoretical contributions to avoid FCG of NC materials. (Table 1).
3. Results and discussion This paper selects the NC materials Al as the research object. The range of misorientation angle Δθ is selected as Δθ = 5° , 10° , 15° , the parameters of NC materials are as follows: 3.1. The influence factors of the critical stress of dislocation penetrating grain boundary From the Eqs. (5) to (7), we know that the critical stress τcr of dislocation penetrating grain boundary is closely related to the
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3.2. Influence factors of FCP rate The crack initiation is observed on the interface of the nc metals with continuum load. We neglect the problem of crack initiation. This paper mainly focuses on the content that NC material itself exist crack (I type crack). To determine the role of the FCP under monotonic cycle load, we look at the evolution of da/dN with the vary of grain size and misorientation angle. Selecting grain size d = 40, 60, 80 nm , misorientation angle Δθ = 5° , 10° , 15° , the approach of modeling FCP will be illustrated based on above theoretical model. In the present work, we employed the da/dN formulations to quantitatively research the microstructure of grain size d and misorientation angle Δθ affected on FCP. In this model, the variation of the microstructure and the resistance of dislocation emission at crack tip play an important role. The discussion up has researched the consequences of crack path vary induced by different grain size to FCP mechanisms. The experiment results show that grain refinement serves to reduce the extent of crack path tortuosity. Fig. 6(a–c) shows effects of grain size on the variation of FCP rate da/dN as a function of ΔKI for nc material Al. Here, the variation of grain size from 40 nm to 80 nm leads to a reduction of FCP rate da/dN at an misorientation angle Δθ of constant. The mainly reason is that the fracture surface asperities increase with grain size increase. The larger grain size exhibit apparently greater resistance to fatigue crack growth. The analytical result is in agreement with recent experimental findings, as shown in Fig. 6 (d). The effect of orientation deviation angle on the fatigue crack growth of nc Al is shown in Fig. 7(a–b). It is evident that the fatigue crack growth is less affected by misorientation angle. However, the fcp of nc Al is significantly decrease with Δθ greatly increase. This is because the increase of the misorientation angle can increase the energy barrier of the dislocation penetrating the grain boundary, which hinder dislocation emission at the crack tip.
4. Conclusion Fig. 7. Relationship between the fatigue crack growth rate da/dN and the stress intensity factor ΔKI of I type crack. (a–b) Effect of orientation deviation angle.
orientation deviation angle Δθ and grain size d . Here, we ignored the materials with larger misorientation angle. The relatively small misorientation angle Δθ = 5° , 10° , 15° and a range of grain size d = 20, 40, 60, 80 nm are discussed. It is clear at a glance that the critical stress of dislocation penetrating grain boundary is enhanced with the increase of misorientation angle, as shown in Fig. 5a. The increasing rate slows of critical stress τcr when the misorientation angle continuously increases. This can be tentatively explained as follows: dislocation penetrating grain boundary is mainly affected by the energy of grain boundary and residual Burgers vector. The energy of grain boundary is constant. The larger the value of residual burgers vector is, the easier the dislocation to penetrate grain boundary. The value of residual burgers vector will decrease with the increase of misorientation angle Δθ . This means that the dislocation pill-up increase if dislocation penetrates grain boundary. All grain boundaries with larger orientation deviation angle Δθ are difficult to dislocation to penetrate. As a result, the critical stress increase. From Fig. 5b, it can be seen that the critical stress decreases with increasing grain size and correlates the Hall-Petch relationship. The mainly reason is that grain boundary increases in unit volume with grain size decrease. Meanwhile, grain boundaries hinder the dislocation to slip. The result basically agree with the research of Li [29] and Embury [34].
The effect of grain size and misorientation angle on the interface of fatigue crack growth in nc metals were investigated. The varieties of grain size and misorientation angle have been shown to have a substantial effect on fatigue crack growth behavior. The extents of crack face roughness decided the fatigue crack growth. We establish the theoretical model of misorientation angle and grain size on the fatigue crack propagation under I type crack, and the calculated conclusions are: (1) The critical stress of dislocation penetrating grain boundary and the misorientation angle of the grain boundary are positively correlated. the critical stress decreases with increasing grain size and correlates the Hall-Petch relationship. (2) The grain size will influence the crack face roughness of nc metals, and then the crack growth rate is vary. The crack growth rate increases with the increase of the stress intensity factor KI , and the grain size and crack growth rate were negatively correlated when the grain boundary misorientation angle is invariant. The fcp of nc Al is significantly decrease with Δθ greatly increase when the grain size is invariant. This paper only considers the single cycle loading condition for the theoretical model of NC materials in I type crack propagation, and neglected the problems of inverse Hall-Petch relationship. There is no further study on the fatigue crack growth of type II and III. This is all problems to be considered in the future.
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Acknowledgements This work was supported by National Natural Science Foundation of China (10502025, 10872087 and 11272143) and the Program for Chinese New Century Excellent Talents in University (NCET-12-0712).
References [1] H. Gleiter, Nanocrystalline Materials, Springer, Berlin Heidelberg, 1991. [2] M.A. Meyers, A. Mishra, D.J. Benson, Mechanical properties of nanocrystalline materials, Prog. Mater. Sci. 51 (2006) 427–556. [3] S. Benkassem, L. Capolungo, M. Cherkaoui, Mechanical properties and multiscale modeling of nanocrystalline materials, Acta Mater. 55 (2007) 3563–3572. [4] L. Capolungo, M. Cherkaoui, J. Qu, On the elastic–viscoplastic behavior of nanocrystalline materials, Int. J. Plast. 23 (2007) 561–591. [5] N. Naga Krishna, A.K. Akash, K. Sivaprasad, R. Narayanasamy, Studies on void coalescence analysis of nanocrystalline cryorolled commercially pure aluminium formed under different stress conditions, Mater. Des. 31 (2010) 3578–3584. [6] M.A. Meyersm, E. Ashworth, A model for the effect of grain size on the yield stress of metals, Philos. Mag. A 46 (1982) 737–759. [7] A.W. Thompson, W.A. Backofen, The effect of grain size on fatigue, Acta Metall. 19 (1971) 597–606. [8] P. Lukáš, L. Kunz, Effect of grain size on the high cycle fatigue behaviour of polycrystalline copper, Mater. Sci. Eng. 85 (1987) 67–75. [9] S.R. Agnew, A.Y. Vinogradov, S. Hashimoto, J.R. Weertman, Overview of fatigue performance of Cu processed by severe plastic deformation, J. Electron Mater. 28 (1999) 1038–1044. [10] A. Vinogradov, S. Hashimoto, Multiscale phenomena in fatigue of ultrafine grain materials – an overview, Mater. Trans. 42 (2001) 74–84. [11] S.D. Wu, Z.G. Wang, C.B. Jiang, G.Y. Li, I.V. Alexandrov, R.Z. Valiev, Shear bands in cyclically deformed ultrafine grained copper processed by ECAP, Mater. Sci. Eng.: A. 387–389 (2004) 560–564. [12] L. Kunz, P. Lukáš, M. Svoboda, Fatigue strength, microstructural stability and strain localization in ultrafine-grained copper, Mater. Sci. Eng.: A 424 (2006) 97–104. [13] C. Xu, Q. Wang, M. Zheng, J. Li, M. Huang, Q. Jia, J. Zhu, L. Kunz, M. Buksa, Fatigue behavior and damage characteristic of ultra-fine grain low-purity copper processed by equal-channel angular pressing (ECAP), Mater. Sci. Eng.: A 475 (2008) 249–256. [14] T. Hanlon, E.D. Tabachnikova, S. Suresh, Fatigue behavior of nanocrystalline metals and alloys, Int. J. Fatigue 27 (2005) 1147–1158. [15] R.Z. Valiev, T.G. Langdon, Principles of equal-channel angular pressing as a
7
processing tool for grain refinement, Prog. Mater. Sci. 51 (2006) 881–981. [16] R.Z. Valiev, R.K. Islamgaliev, I.V. Alexandrov, Bulk nanostructured materials from severe plastic deformation, Prog. Mater. Sci. 45 (2000) 103–189. [17] L. Lu, R. Schwaiger, Z.W. Shan, M. Dao, K. Lu, S. Suresh, Nano-sized twins induce high rate sensitivity of flow stress in pure copper, Acta Mater. 53 (2005) 2169–2179. [18] Z.F. Zhang, Z.G. Wang, Dependence of intergranular fatigue cracking on the interactions of persistent slip bands with grain boundaries, Acta Mater. 51 (2003) 347–364. [19] Z.F. Zhang, Z.G. Wang, Grain boundary effects on cyclic deformation and fatigue damage, Prog. Mater. Sci. 53 (2008) 1025–1099. [20] A. Singh, L. Tang, M. Dao, L. Lu, S. Suresh, Fracture toughness and fatigue crack growth characteristics of nanotwinned copper, Acta Mater. 59 (2011) 2437–2446. [21] S. Qu, P. Zhang, S.D. Wu, Q.S. Zang, Z.F. Zhang, Twin boundaries: strong or weak? Scr. Mater. 59 (2008) 1131–1134. [22] P. Zhang, Z.J. Zhang, L.L. Li, Z.F. Zhang, Twin boundary: stronger or weaker interface to resist fatigue cracking? Scr. Mater. 66 (2012) 854–859. [23] Z.J. Zhang, P. Zhang, L.L. Li, Z.F. Zhang, Fatigue cracking at twin boundaries: effects of crystallographic orientation and stacking fault energy, Acta Mater. 60 (2012) 3113–3127. [24] J. Xie, X. Wu, Y. Hong, Shear bands at the fatigue crack tip of nanocrystalline nickel, Scr. Mater. 57 (2007) 5–8. [25] M.D. Sangid, G.J. Pataky, H. Sehitoglu, R.G. Rateick, T. Niendorf, H.J. Maier, Superior fatigue crack growth resistance, irreversibility, and fatigue crack growth–microstructure relationship of nanocrystalline alloys, Acta Mater. 59 (2011) 7340–7355. [26] J.D. Livingston, B. Chalmers, Multiple slip in bicrystal deformation, Acta Metall. 5 (1957) 322–327. [27] Z. Shen, R.H. Wagoner, W.A.T. Clark, Dislocation and grain boundary interactions in metals, Acta Metall. 36 (1988) 3231–3242. [28] T.C. Lee, I.M. Robertson, H.K. Birnbaum, Prediction of slip transfer mechanisms across grain boundaries, Scr. Metall. 23 (1989) 799–803. [29] Z. Li, C. Hou, M. Huang, C. Ouyang, Strengthening mechanism in micro-polycrystals with penetrable grain boundaries by discrete dislocation dynamics simulation and Hall–Petch effect, Comput. Mater. Sci. 46 (2009) 1124–1134. [30] T. Shanmugasundaram, M. Heilmaier, B.S. Murty, V.S. Sarma, On the Hall– Petch relationship in a nanostructured Al–Cu alloy, Mater. Sci. Eng.: A 527 (2010) 7821–7825. [31] R. Thomson, T.J. Chuang, I.H. Lin, The role of surface stress in fracture, Acta Metall. 34 (1986) 1133–1143. [32] P.B. Chowdhury, H. Sehitoglu, R.G. Rateick, H.J. Maier, Modeling fatigue crack growth resistance of nanocrystalline alloys, Acta Mater. 61 (2013) 2531–2547. [33] G.C. Hasson, C. Goux, Interfacial energies of tilt boundaries in aluminium. Experimental and theoretical determination, Scr. Metall. 5 (1971) 889–894. [34] J. Embury, Strengthening mechanisms in Al alloys - an overview of natural limits and engineering possibilities, Mater. Sci. Forum 217–222 (1996) 57–70.