The crack nucleation in hierarchically nanotwinned metals

Engineering Fracture Mechanics 201 (2018) 29–35

Contents lists available at ScienceDirect

Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech

The crack nucleation in hierarchically nanotwinned metals Feng Zhanga,c, Yaqian Liua,c, Jianqiu Zhoua,b,c,

⁎

T

a

Department of Mechanical Engineering, Nanjing Tech University, Nanjing, Jiangsu 210009, China Department of Mechanical Engineering, Wuhan Institute of Technology, Wuhan, Hubei 430070, China c Key Lab of Design and Manufacture of Extreme Pressure Equipment, Jiangsu Province, China b

A R T IC LE I N F O

ABS TRA CT

Keywords: Nanocrystalline metals Hierarchical twin lamellae Dislocations Crack nucleation

The nanocrystalline (NC) metals with hierarchical twin lamellae exhibit excellent mechanical properties. However, recent efforts are mainly focused on the effects of primary twin boundaries (TB) and grain boundaries (GB) on crack nucleation. Few investigations have been performed on secondary twin boundary (STB) effect on crack nucleation in hierarchically nanotwinned metal. A relevant model is established to describe the crack nucleation criteria quantitatively. Actually, the crack advance mainly depends on the energy of crack nucleation and the energy of pile-up dislocations. Furthermore, predictions about the site of crack nucleation are made for different hierarchically nanotwinned metals. This work will help us better understand the deformation mechanism and the failure behavior in NC materials with hierarchically nanostructures.

1. Introduction The hierarchically nanotwinned materials exhibit unique mechanic behaviors, such as good ductility, high strength and high hardness [1–9]. Secondary twin lamellae are observed when the primary twin boundary spacing (TBS) reaches to the critical TBS. A self-consistent model was used to understand the global and local mechanical behaviors of the hierarchical materials [10,11]. The hierarchically nanotwinned materials derive high strength from unique deformation mechanism [12]. Also, the hierarchically twinned structures were observed during the martensitic transformation [13,14]. Various researches have confirmed that the hierarchical twin has a significant role in mechanical behaviors. Tao and Qu observed the primary and secondary lamellae in the nanotwinned Cu and Cu-Al alloy by equal-channel angular pressing [15,16]. Besides the experimental achievements, molecular dynamic (MD) simulations provide the possibility of analyzing mechanical behaviors of hierarchical metals quantitatively [17,18]. In the present contribution, a corresponding model was proposed to explore the crack nucleation at GB-TB intersections by Zhang [19]. Moreover, a theoretical model was described to analyze the effects of TBSs and grain size in face-centered cubic (FCC) metals by Zhu [20]. However, Zhang only put the TB-GB intersections into consideration. In the hierarchically nanotwinned materials, dislocations may also pile up at the primary-secondary twin boundary. In this paper, the energy of pile-up dislocations considering the secondary twin boundary effect will be calculated accurately to predict the site of crack nucleation (e.g. TB-GB or TB-STB intersection). 2. The model for boundary intersections As shown in Fig. 1(a), lots of TB-GB and TB-STB intersections exist in the hierarchically nanotwinned model. During the plastic

⁎

Corresponding author at: Department of Mechanical Engineering, Nanjing Tech University, Nanjing, Jiangsu 210009, China. E-mail address: [email protected] (J. Zhou).

https://doi.org/10.1016/j.engfracmech.2018.08.027 Received 14 May 2018; Received in revised form 24 July 2018; Accepted 28 August 2018 Available online 30 August 2018 0013-7944/ © 2018 Elsevier Ltd. All rights reserved.

Engineering Fracture Mechanics 201 (2018) 29–35

F. Zhang et al.

Fig. 1. Schematic drawing of pile-up dislocations at the TB-GB or TB-STB intersection.

deformation process, dislocations pile up on the boundary p , q and o . At the same time, a stress concentration is produced at the TBGB intersection or TB-STB intersection. When reaching the condition of crack nucleation, the crack will propagate and stress concentration is released. Fig. 1(b) shows the geometric relationship between each boundary. φqp is the angle between boundary p and q and φop is the angle between boundary p and o . As shown in Fig. 1(c), a global rectangular coordinate x-y axes colored by red is defined. The vertical direction is along the direction of loading stress, and the angle between the horizontal direction and boundary p is marked as φp . Then we establish three local coordinate systems on each boundary. Fig. 1(d) draws the diagram of pile-up dislocations on each boundary. A wedge crack is at the TB-GB intersection or TB-STB intersection, and the crack advance direction is along the boundary p or q respectively. In the local frame x ip−yi p , the in-plane stress components of the dislocation i can be written by [21],

3(x ip )2 + (yi p )2 ⎤ σxx = −Ab pyi p ⎡ ⎢ ((x p )2 + (y p )2)2 ⎥ i ⎦ ⎣ i

(1a)

(x ip )2−(yi p )2 ⎤ σyy = Ab pyi p ⎡ ⎢ ((x p )2 + (y p )2)2 ⎥ i ⎦ ⎣ i

(1b)

(x ip )2−(yi p )2 ⎤ σxy = Ab px ip ⎡ ⎢ ((x p )2 + (y p )2)2 ⎥ i ⎦ ⎣ i

(1c)

In the local

frame x jq−yjq

or

xko−yko ,

the in-plane stress components of the dislocation j or k can be written in the similar form,

q, o 2 q, o 2 ⎡ 3(x j, k ) + (yj, k ) ⎤ σxx = −Abq, oyjq,,ko ⎢ q, o 2 q, o 2 2 ⎥ ((x ) + (yj, k ) ) ⎣ j, k ⎦

(2a) 30

Engineering Fracture Mechanics 201 (2018) 29–35

F. Zhang et al. q, o 2 q, o 2 ⎡ (x j, k ) −(yj, k ) ⎤ σyy = Abq, oyjq,,ko ⎢ q, o 2 ((x ) + (yjq,,ko )2)2 ⎥ ⎣ j, k ⎦

(2b)

q, o 2 q, o 2 ⎡ (x j, k ) −(yj, k ) ⎤ Abq, ox jq,,ko ⎢ q, o 2 ((x ) + (yjq,,ko )2)2 ⎥ ⎣ j, k ⎦

(2c)

σxy =

G . [2π (1 − υ)]

bp

bq

bo

, and are the Burgers vector magnitude of the dislocations on the boundary p , q and o , respectively. The Here A = shear stress of dislocation i on the boundary q caused by dislocation j on the boundary q can be written in the dislocation j as

2(x ijpq )2yijpq x ijpq ((x ijpq )2−(yijpq )2) ⎡ ⎤ cos(2φqp) ⎥ sin(2φqp) + σxypq, ij = Ab p ⎢ pq 2 pq 2 2 ((x ijpq )2 + (yijpq )2)2 ((x ij ) + (yij ) ) ⎣ ⎦

(3)

Similarly, combining the Eq. (2a), Eq. (2b) and Eq. (2c), the shear stress of dislocation i on the boundary p imposed by dislocation k on the boundary o can be written as

2(x ikpo )2yikpo x ikpo ((x ikpo )2−(yikpo )2) ⎤ σxypo, ik = Ab p ⎡ ⎢ ((x po )2 + (y po )2)2 sin(2φop) + ((x po )2 + (y po )2)2 cos(2φop) ⎥ ik ik ik ⎦ ⎣ ik

(x ijpq , yijpq )

(4)

and(x ikpo , yikpo )

are the coordinates of dislocation j and k in the local coordinate system of dislocation i . Moreover, the where equilibrium equations of dislocations on each boundary are given by

∑p ∑i Fijp + ∑p ∑i Fikp + τ pb p = 0, i = 1, 2, ...,n p

(5)

τp

denotes the applied stress resolved on the boundary p and where Moreover, the nonlinear equilibrium equations can be written as

np

is the number of emissive dislocations on the boundary p .

np

4

∑ ∑ σxyp1,ij b1 + τ 1b1 = 0, j = 1, 2. ...n1 (6a)

p=1 i=1 4

n

p

∑ ∑ σxyp2,ij b2 + τ 2b2 = 0, j = 1, 2. ...n2 (6b)

p=1 i=1 4

n

p

∑ ∑ σxyp3,ij b3 + τ 3b3 = 0, j = 1, 2. ...n3 (6c)

p=1 i=1 4

n

p

∑ ∑ σxyp4,ik b4 + τ 4b4 = 0, k = 1, 2. ...n4

(6d)

p=1 i=1

x np1,4

= λ /sin φqp , x nP2 = 2λSTB /sin(φop−φqp) and x no3 = λ / sin (φop−φqp) . By solving Eqs. (6a)–(6d), one can obtain the number (N) and with equilibrium positions of dislocations on each boundary. 2.1. Energy criterion of the crack nucleate based on pile-up dislocations According to Wu [22], the energy W of pile-up dislocations is written as 4

W= ∑ p=1 2

3

n pD (b p)2 2

ln

3

np np

( ) + ∑ ∑ ∑ D (b ) ln ⎛⎝ R r0

p 2

⎜

p=1 i=1 j≻1

np np

pq

y ′ij ∑ ∑ ∑ ∑ D (b p)2ln ⎛⎜ p=1 q≻p i=1 j=1 ⎝ np np

+ ∑ ∑ ∑ D (b p)2ln ⎛ p=4 i=1 j=1 ⎝

⎜

(s (−1) p + 1R + cyij′ pq ) R2 + (yij′ pq )2

yik ′ po (s ′ (−1) pR + c ′ yik′ po ) R2 + (yik ′ po )2

−

−

np np

(−1) p + 1R ⎞ ⎟ x ij′ pq

p

(−1) R + ∑ ∑ ∑ D (b p)2ln ⎛ po ⎞+ x ik ′ ⎠ ⎝ p = 4 i = 1 k ≻ 2 ⎠

yij′ pq (sx ij′ pq + cyij′ pq ) (x ij′ pq )2 + (yij′ pq )2

yik ′ po (s ′ x ′ikpo + c ′ yik′ po ) (x ik ′ po )2 + (yik′ po )2

2

pq 2

+

c 2

R + (yij′ ) ⎞ ⎞⎟ ln ⎛ pq 2 (x ′ ) + (yij′ pq )2 ⎝ ij ⎠⎠

+

c′ 2

R + (y ′ ) ln ⎛ po 2 ik po 2 ⎞ ⎞ (x ik ) + (yik ′ ′ ) ⎠ ⎝ ⎠

⎜

⎟

2

po 2

⎟

(7)

where R is the crack propagation length in the dislocation stress field and r0 is the radius of a dislocation core. The relative positions of dislocation j and k marked as (x ij′ pq , yij′ pq ) and (x ik′ po , yik′ po ) can be expressed by

x ij′ pq = cx ijpq + syijpq , yij′ pq = −sx ijpq + cyijpq x ′ikpo

=

−c′x ikpo

+

s′yikpo ,

y′ikpo

=

s′x ikpo

+

(8)

c′yikpo

where s = sin φqp , c = cos φqp , s′ = sin φop and c′ = cos φop . In Eq. (7), the first part stands for the energy of the dislocations itself, the following two parts arise from the interaction energy of the dislocations on the same boundary, and the last two parts represents the 31

Engineering Fracture Mechanics 201 (2018) 29–35

F. Zhang et al.

interaction energy on the different boundaries. Assume the pile-up dislocations on boundaries as strength vectors with the magnitude B = nb . As mentioned in Section 2, there are two grain boundaries, one primary twin boundary and one secondary twin boundary in our model. These four pile-up dislocation groups may influence each other in the stress fields. Through summing these four strength vectors, the net effect of pile-dislocation groups on crack is written as

⎡Wn ⎤ = ⎣ Wt ⎦

3

sin φ

sin φ

∑ B q ⎡⎢ cos φpq ⎤⎥ + B 4 ⎡⎢ cos φpo ⎤⎥

q=1

⎣

pq ⎦

⎣

po ⎦

(10)

where Wn and Wt represent the normal and tangential wedge strength. For the plane strain condition, the energy of the crack Ec can be derived based on fracture mechanics. The expression is

∂Ec = 2γ −m ∂c m=

(11)

(1−ν 2) 2 (KI + KII2 ) E

(12)

where ν , γ , and E are Poisson’s ratio, surface energy and Young’s modulus, respectively. KI and KII are the mode I and mode II stress intensity factors, which can be expressed as [23,24]

KI 2 =

EWn σn∞ π 1 EWn 2 ⎞ ·⎛ c (σn∞ )2 + + 2 2(1−ν 2) 8πc ⎝ 1−ν 2 ⎠

(13)

KII 2 =

EWt σt∞ π 1 EWt 2 ⎞ c (σt∞ )2 + ·⎛ + 2 2 2(1−ν ) 8πc ⎝ 1−ν 2 ⎠

(14)

σn∞

σt∞

and are the normal and tangential components of the remote stress applied on the crack. Combing the Eqs. (11), (12), where (13) and (14), the derivative of the crack energy can be written by

G (Wn2 + Wt2 ) σn∞ Wn + σt∞ Wt ∂Ec π (1−ν ) c [(σn∞ )2−(σt∞ )2]− − = 2γ − ∂c 4G 4π (1−ν ) c 2

(15)

Via integrating the crack length from zero to c , one can obtain the expression of crack nucleation energy

Ec = 2γc−

σ ∞ Wn + σt∞ Wt ⎞ G (Wn2 + Wt2 ) c π (1−ν ) c 2 ln − [(σn∞ )2 + (σt∞ )2]−⎛ n c 4π (1−ν ) b 8G 2 ⎝ ⎠

(16)

The energy criterion is based on the difference between W and Ec , whenW > Ec , a wedge crack may nucleate. One can calculate the value of W and Ec by solving Eqs. (7) and (16). In doing so, the typical values of parameters of Cu are used, which are listed in Table 1. 3. Results and discussion 3.1. STB effect on crack nucleation at TB-GB The difference between W and Ec and the twin lamellae spacing λ under two various applied stress (σ = 0.01G and σ = 0.03G ) in hierarchically nanotwinned Ni was investigated, as shown in Fig. 2. The red curve represents the energy of pile-up dislocations without considering secondary twin boundaries, the black one take the secondary twin boundaries into account and the energy of crack nucleation level are colored in green. In Fig. 2(a), it is found that when λ < λlim the black curve coincides with the red one, while the black one is higher than the red one when λ > λlim . The reason is that in the hierarchically nanotwinned materials the secondary twin boundaries are only formed when λ > λlim , and then dislocations can pile up on the secondary twin boundary. According to Eq. (7), the effect of secondary twin boundary on the value of W should not be ignored. The intersection λ1 between the black curve and the green straight line is smaller than the intersection λ2 between the red curve and the green straight line. The results reveal that the crack nucleation appears more easily due to the pile-up dislocations at secondary twin boundaries. Similar behaviors can be seen in Fig. 2(b), for the same λ , the increasing applied stress enlarges the value of W . It is due to that the number of dislocations on each boundary increases. Therefore, the crack nucleates more easily. Fig. 2(c)–(d) illustrate the effect of Effect of boundary orientation on crack nucleation under different applied stress. The three cases are: (1) φp = 30°, φqp = 115°, φop = 170°; (1) φp = 45°, φqp = 130°, φop = 170°; (1) φp = 60°, φqp = 145°, φop = 170°. For all cases, Table 1 Values of parameters of Cu used in calculation. Properties

G (GPa)

ν

b (nm)

γ (J / m2)

λlim (nm)

φp

φqp

φop

Value

27

0.34

0.286

0.56

40 nm

30°

115°

170°

32

Engineering Fracture Mechanics 201 (2018) 29–35

F. Zhang et al.

Fig. 2. Effect of STB on crack nucleation under different applied stress (a) σ = 0.01G and (b)σ = 0.03G ; Effect of boundary orientation on crack nucleation under different applied stress (c) σ = 0.01G and (d)σ = 0.03G .

the energy of pile-up dislocations with STB increases with the twin lamellae spacing. The generation of crack at TB-GB intersection is more likely to nucleate for the curves above the green horizontal line. As shown in Fig. 2(c), crack nucleation becomes energetically favorable for λ > 60 nm in case 1 and for λ > 100 nm in case 2, while nucleation is impossible for all investigated values of twin lamellae spacing. The results indicate that boundary orientation has a significant effect on crack nucleation. That is because the resolved shear stress is related to boundary orientation, which directly affects the pile-up dislocation emission. Moreover, comparing Fig. 2(c) with Fig. 2(d), one can find that the increase of applied stress makes the crack nucleate at a smaller twin lamellae spacing, which further confirms the results in Fig. 2(a) and (b). Fig. 3(a) depicts the equilibrium positions of dislocations in each boundary for a given twin lamellae spacing, λ = 60 nm. One can find that the dislocation arrangement is not uniform and most of dislocations pile up at the GBs, especially the boundary marked ‘P = 1’. Fig. 3(b) shows the relation between the number of pile-up dislocations and primary twin boundary spacing. As mentioned above, increasing the primary twin boundary spacing can increase the number of pile-up dislocations. For the same λ , the applied stress increasing leads to the increasing of N. In addition, it is found that when λ > 40 nm, λ and N follow a linear relationship approximately, however, when λ < 40 nm, the two black points deviate the black dashed straight line. The reason is that, under σ = 0.01G , the value of λlim is around 40 and the secondary twin lamellae may not appear, so the number of pile-up dislocations decreases more quickly. Similarly phenomenon has been shown at σ = 0.03G . As shown in Fig. 4, the TEM images of the dislocations at the boundary for two different primary twin boundary spacings were presented by Cheng and Zhao [25]. It was shown that when λ = 40 nm the dislocations only pile up near the primary twin-grain boundary whileλ = 60 nm the dislocations can not only pile up near the primary twin-grain boundary, also can pile up near the primary-secondary twin boundary. This fact was consistent with the black point as shown in Fig. 3. The crack nucleation was determined by the grain size and the energy of pile-up dislocations, as mentioned by Zhang [26].

3.2. The site of crack nucleation in hierarchical nanotwinned metals In Fig. 5, the values of W at the TB-GB and TB-STB intersection have been calculated for different alloys with hierarchically nanotwinned structures. It is found that the value of W at the TB-STB intersection increases more quickly than the one at the TB-GB 33

Engineering Fracture Mechanics 201 (2018) 29–35

F. Zhang et al.

Fig. 3. (a) Equilibrium positions of dislocations in each boundary; (b) the relationship between the number of pile-up dislocations and primary twin boundary spacing.

Fig. 4. Our further analysis on the image of dislocation near the boundary in nanotwinned Ni-Fe alloys [25]; (a) λ = 40 nm, (b) λ = 60 nm.

Fig. 5. The energy of pile-up dislocations at different sites for (a) Ni-Fe and (b) Ni-Cu alloys.

intersection. The reason is that more dislocations can pile up on the boundary q as a result of the λSTB increasing. Also it shows that the value of W dislocations at the primary twin-grain boundary intersection is not zero for λSTB = 0 . That is because that when the secondary twin lamellae appears, dislocations have already piled up near the boundary p and q . Comparing Fig. 5(a) with Fig. 5(b), the crack nucleation first appears at the primary twin-grain boundary intersection for Ni-Fe alloy. This fact is due to the energy of pile-up dislocations at the primary twin-grain boundary intersection is larger than at the primary-secondary twin boundary 34

Engineering Fracture Mechanics 201 (2018) 29–35

F. Zhang et al.

intersection. While for Ni-Cu alloy the value of Ec is large, the crack nucleation first appears at the primary-secondary twin boundary intersection, as marked by the purple dashed line. 4. Conclusion Based on pile-up dislocations model, we analyzed the effect of secondary twin boundary on the critical condition of crack nucleation. Due to the emissive dislocations at secondary twin boundary, the crack nucleates more easily at the site of TB-GB intersection. Crack nucleation is strongly influenced by the boundary orientation. In addition, the applied stress increasing can promote the crack nucleation through increasing dislocations. As shown in Fig. 5, the crack nucleation can occurs not only at the TB-GB intersection, but also at the TB-STB intersection. The crack nucleation mainly depends on the energy of crack nucleation and the energy of pile-up dislocations. These findings could give us better understanding of failure in hierarchically nanotwinned metals. Acknowledgments This project was supported by National Natural Science Foundation of China (10502025, 10872087, 11272143), Key Project of Chinese Ministry of Education (211061), Key University Science Research Project of Jiangsu Province (17KJA130002). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]

Chen XH, Lu L. Work hardening of ultrafine-grained copper with nanoscale twins. Scr Mater 2007;57:133–6. Chen XH, Lu L, Lu K. Grain size dependence of tensile properties in ultrafine-grained Cu with nanoscale twins. Scr Mater 2011;64:311–4. Singh A, Tang L, Dao M, Lu L, Suresh S. Fracture toughness and fatigue crack growth characteristics of nanotwinned copper. Acta Mater 2011;59:2437–46. Chen XH, Lu L, Lu K. Electrical resistivity of ultrafine-grained copper with nanoscale growth twins. J Appl Phys 2007;102:223. Chen KC, Wu WW, Liao CN, Chen LJ, Tu KN. Observation of atomic diffusion at twin-modified grain boundaries in copper. Science 2008;321:1066–9. Anderoglu O, Misra A, Wang H, Zhang X. Thermal stability of sputtered Cu films with nanoscale growth twins. J Appl Phys 2008;103:1357. Shen YF, Chen XH, Wu B, Lu L. Microstructure and tensile strength of Cu with nano-scale twins. Solid Mech Appl 2007;144:19–24. Zhou H, Qu S. The effect of nanoscale twin boundaries on fracture toughness in nanocrystalline Ni. Nanotechnology. 2010;21:035706. Zhou H, Qu S, Yang W. Toughening by nano-scaled twin boundaries in nanocrystals. Model Simulmaterscieng 2010;18:065002. Zhang S, Zhou J, Wang L, Wang Y. The effect of the angle between loading axis and twin boundary on the mechanical behaviors of nanotwinned materials. Mater Des 2013;45:292–9. Zhang S, Zhou J, Wang L, Wang Y, Dong S. Effect of twin boundaries on nanovoid growth based on dislocation emission. Materi Sci Eng A Struct Mater Propert Microstruct Process 2013;582:29–35. Liao XZ, Zhao YH, Srinivasan SG, Zhu YT, Valiev RZ, Gunderov DV. Deformation twinning in nanocrystalline copper at room temperature and low strain rate. Appl Phys Lett 2004;84:592–4. Saburi T, Wayman C, Takata K, Nenno S. The shape memory mechanism in 18R martensitic alloys. Acta Metall 1980;28:15–32. Müllner P, Chernenko V, Kostorz G. A microscopic approach to the magnetic-field-induced deformation of martensite (magnetoplasticity). J Magn Magn Mater 2003;267:325–34. Hu C, Wang R, Ding D-H. Symmetry groups, physical property tensors, elasticity and dislocations in quasicrystals. Rep Prog Phys 2000;63:1. Qu S, An X, Yang H, Huang C, Yang G, Zang Q, et al. Microstructural evolution and mechanical properties of Cu–Al alloys subjected to equal channel angular pressing. Acta Mater 2009;57:1586–601. Wang J, Anderoglu O, Hirth J, Misra A, Zhang X. Dislocation structures of Σ 3 112 twin boundaries in face centered cubic metals. Appl Phys Lett 2009;95:021908. Zhu L, Ruan H, Li X, Dao M, Gao H, Lu J. Modeling grain size dependent optimal twin spacing for achieving ultimate high strength and related high ductility in nanotwinned metals. Acta Mater 2011;59:5544–57. Zhang S, Zhou J, Wang L, Liu H, Dong S. Crack nucleation due to dislocation pile-ups at twin boundary–grain boundary intersections. Mater Sci Eng, A 2015;632:78–81. Zhu L, Qu S, Guo X, Lu J. Analysis of the twin spacing and grain size effects on mechanical properties in hierarchically nanotwinned face-centered cubic metals based on a mechanism-based plasticity model. J Mech Phys Solids 2015;76:162–79. Wang L, Zhou J, Zhang S, Liu H, Dong S. Effect of dislocation–GB interactions on crack blunting in nanocrystalline materials. Mater Sci Eng, A 2014;592:128–35. Wu MS. Crack nucleation due to dislocation pile-ups at I-, U-and amorphized triple lines. Mech Mater 1997;25:215–34. Wu MS, Niu J. A theoretical investigation into the nucleation of stable crack precursors in polycrystalline ice. Int J Fract 1994;68:151–81. Stroh AN. A theoretical calculation of the stored energy in a work-hardened material. Pro Roy Soc A: Math Phys Eng Sci 1953;218:391–400. Cheng S, Zhao Y, Wang Y, Li Y, Wang X-L, Liaw PK, et al. Structure modulation driven by cyclic deformation in nanocrystalline NiFe. Phys Rev Lett 2010;104:255501. Zhang Z, Zhang P, Li L, Zhang Z. Fatigue cracking at twin boundaries: effects of crystallographic orientation and stacking fault energy. Acta Mater 2012;60:3113–27.

35