Simulation on the deformation behaviors of nanotwinned Cu considering the angle effect between loading axis and twin boundary

Computational Materials Science 77 (2013) 322–329

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Simulation on the deformation behaviors of nanotwinned Cu considering the angle effect between loading axis and twin boundary Shu Zhang a, Jianqiu Zhou a,b,⇑, Ying Wang a, Lu Wang a, Shuhong Dong a a b

Department of Mechanical Engineering, Nanjing University of Technology, Nanjing 210009, Jiangsu, People’s Republic of China Department of Mechanical Engineering, Wuhan Institute of Technology, Wuhan 430070, Hubei, People’s Republic of China

a r t i c l e

i n f o

Article history: Received 21 November 2012 Received in revised form 4 March 2013 Accepted 16 April 2013

Keywords: Nanotwinned materials Angle Deformation behaviors Finite-element method

a b s t r a c t The effect of the angle h between loading axis and twin boundary (TB) on the deformation behaviors of nanotwinned Cu was studied by finite-element method (FEM) under tensile deformation. Three different  ¼ 3:7;   ¼ 3:7 and   ¼ 8:8Þ were presented. Evolution of h ¼ 30 ; r h ¼ 30 ; r microstructures ð h ¼ 10 ; r equivalent plastic strain showed that the deformation of the three microstructures were distinct on account of h: the grains with larger h deformed first; compared with the microstructure with   ¼ 3:7, the microstructure with   ¼ 3:7 can effectively delay the onset of shear band. h ¼ 30 ; r h ¼ 10 ; r The results demonstrated that an unusual combination of high strength and ductility can be obtained by making a proper arrangement of h. However, the effect of h on the stress–strain relations was weakening as twin density lamellae thickness decreases. Ó 2013 Elsevier B.V. All rights reserved.

1. Introduction High strength and good ductility are becoming increasingly desirable for the metals and alloys that are widely used in industry applications. However, most of the existing material-strengthening techniques lead to decreased ductility [1,2]. This strength–ductility trade-off can be circumvented by some alternative approaches that involve sophisticated microstructure design, including metals with bimodal grain sizes [3,4], composites with an amorphous phase and nanograins [5], as well as laminated materials [6]. Lu and coworkers [2,7,8] have synthesized electro-deposited polycrystalline Cu with nanoscale growth twins. The TBs are coherent internal boundaries, which has been known as an efficient way to obtain high strength while maintaining substantial ductility [9,10]. Recently, there have been many researches on the mechanical behaviors and deformation mechanisms of nanotwinned materials [11–17]. Zhu et al. [12] provided a quantitative continuum plasticity model capable of describing the variations in strength, ductility and work hardening rate of nanotwinned metals with the twin spacing. Ni et al. [15] reported that the interactions between dislocations and TBs are significantly affected by both intrinsic material properties and extrinsic factors, including stacking fault energy,

⇑ Corresponding author at: Department of Mechanical Engineering, Nanjing University of Technology, Nanjing 210009, Jiangsu, People’s Republic of China. Tel.: +86 25 83588706; fax: +86 25 83374190. E-mail address: [email protected] (J. Zhou). 0927-0256/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.commatsci.2013.04.042

the energy barriers for dislocation reactions at TBs, twin thickness and applied stress. Meanwhile, they concluded that the dislocation density also affected the dislocation–TB interactions. Molecular dynamics (MD) simulations have been performed by Shabib and Miller [17] to study the deformation behaviors of nanotwinned Cu. The key processes observed during the deformation included partial and full dislocation emission from the grain boundaries, TB migration, interaction of the partial dislocations with fixed and migrating TBs, formation of steps, and dislocation emission from the steps. These researches have demonstrated that the nano-meter thick twin/matrix lamellar structure directly affects the mechanical behaviors of nanotwinned materials. Specifically, the twin lamellae thickness plays a significant role. However, the effect of the angle h between the loading axis and TB on the deformation behaviors of nanotwinned materials is neglected. In this paper, we aim to build microstructures of nanotwinned Cu and use FEM to understand the role of h on the deformation behaviors of nanotwinned Cu. FEM has a unique advantage in analyzing the inside microstructure evolution, self-consistent deformation, global–local deformation relationships for nanostructure materials compared with experimental method and molecular dynamics (MD) simulations [18]. In this paper, through the FEM analysis, deformation characteristics in nonuniform deformation, evolution process of the equivalent plastic strain distribution, shear band development, the influence of h on strength and ductility of nanotwinned Cu and the twin lamellae thickness effect will be discussed.

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Fig. 1. View of the 85 polycrystalline microstructure.

2. Numerical simulation method Fig. 3. (a) Schematic three-dimensional mixture framework of nanotwinned Cu and (b) dislocation movement within TBAZ.

2.1. Microstructure constructing and constitutive model Fig. 1 shows the polycrystal assembly containing 85 grains, where each grain in the microstructure is numbered. Within each grain, on account of the two-dimensional characteristic of our model, only three slip systems are considered and the slip directions are constrained to be in the plane, as shown in Fig. 2a. Once nucleated, dislocations glide along the slip system which runs parallel to the TBs and along the alternating mirrored slip systems which intersect the TB at an angle of 70.53°. As illustrated in Fig. 2b, the three slip systems are arranged in an isosceles triangle. The reference base vectors i1 and i2 of the local coordinates as well as j1 and j2 of the global coordinates are aligned as shown. h is the angle from j1 to i1, where j1 is along  direction. According the loading direction and i1 is along ½1 1 2 to the Schmid law, the resolved shear stress ss for the sth slip system is defined as

ss ¼ ms  r

ð1Þ

where r is the applied stress, and ms is the Schmid factor given by

ms ¼ cos /ðsÞ  cos kðsÞ

ð2Þ

(a)

Here, /(s) is the angle between the slip plane normal and the loading axis whereas kðsÞ is the angle between the slip direction and the loading axis. During plastic deformation, the displacement of a point in crystal is the sum of the displacements caused by all three slip systems at the point and it can be written as follows:

u¼

3 X us ¼ u1 i1 þ u2 i2

ð3Þ

s¼1

where i1, i2 are the reference base vectors of the local coordinates, u1, u2 are the shadows that u casts on the local coordinates, us is the displacement on the sth slip system, which can be expressed as

us ¼ cðsÞ ðRnðsÞ Þb

ðsÞ

ð4Þ

where R is the radius vector of X which is a casual point, c(s) is the shear strain on the sth slip system which is defined by orthogonal pair of unit vectors (b(s), n(s)). b(s) is along the sth slip direction and n(s) is normal to the sth slip plane.

(b)

Fig. 2. Model setup (a) Illustration of dislocation–TB interaction. (b) Arrangement of the three slip systems.

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Further more, the equivalent strain ee can be expressed as

ee

pffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ¼ ðe1  e2 Þ2 þ e21 þ e22 þ 6e212 3

ð6Þ

2.1.1. Shear deformation across the TBs During the deformation process, the evolution of the dislocation density qs takes into account the athermal storage of dislocations qþs as well as the annihilation of dislocations qs

qs ¼ qþs ðcÞ þ qs ðcÞ

ð7Þ

where c is the plastic strain. Based on the statistical approach proposed by Kocks and Mecking [19], the evolution of dislocation density with the increasing of c [20] can be written as follows:

   1=m pffiffiffiffiffi dqs dqþs dqs M 1 c_ ¼ þ ¼ þ n qs  C 20  qs b k dc dc dc c_ 0

Fig. 4. Calculated r–e curves with different h.

Therefore, the strain tensor could be obtained by Eqs. (3) and (4):

eij ¼

  1 @ui @uj ði; j ¼ 1; 2Þ þ 2 @xj @xi

ð5Þ

ð8Þ

where M, b, k and n are the Taylor factor, Burger vector, twin lamellae thickness and proportionality factor, respectively; C20 is a constant as well as c_ 0 , m is inversely proportional to the temperature T. The shear flow stress s is associated with the dislocation density q by [21–23]

 ¼ 3:7, (b)   ¼ 3:7, (c)   ¼ 8:8, and (d) finite element meshed of the whole mode. Fig. 5. Schematic of distribution of the angle: (a)  h ¼ 30 ; r h ¼ 10 ; r h ¼ 30 ; r

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S. Zhang et al. / Computational Materials Science 77 (2013) 322–329 Table 1 Material parameters used in calculation for nanotwinned Cu. Description

Notation

Value (Cu)

Average thickness of GBAZ Empirical constant Magnitude of the burgers vector Shear modulus of copper Proportionality factor Numerical constant Numerical constant Geometrical factor Loading strain rate Reference strain rate

dTBAZ

3  109 m 0.2–0.5 0.256  109 m 42.1 GPa 0.2 18.5 12.5 0.5–1.5 6  101 s1 1 s1

a b

l n C20 m / c_ c_ 0

structural characteristics (entirely different properties between TBAZ and TI), we present a model which introduces the strain gradient theory to study the mechanical behavior parallel to the TBs. In Fig. 3a, the shaded parts are TBAZ and the others are TI. Assuming that twins and matrix are equally spaced, this leads to 50% of twinned volume. The thickness of TBAZ is dTBAZ, while a twin lamellar thickness is kT. Then the volume fraction of TBAZ region, denoted as fTBAZ, can be written as:

fTBAZ ¼

2dTBAZ kT

ð10Þ

So the volume fraction of TI region will be 1  fTBAZ:

fTI ¼

kT  2dTBAZ ¼ 1  fTBAZ kT

ð11Þ

TBAZs, as interfaces between adjacent twin lamellaes, are affected by twins on both sides in the deformation process. The dislocation accumulation leads to the nonuniform deformation in the form of local strain gradient in TBAZ. Consequently, strain gradient is taken into account. Hence, in addition to the statistical storage dislocation with respect to uniform deformation is generated owing to the regular atomic arrangement of TI, we suppose that geometrically necessary dislocation in consonance with shape change is generated in TBAZ. Statistical storage dislocation density of TI can also be calculated according to the method in the Section 2.1.1:

   1=m pffiffiffiffiffi dqs dqþs dqs M 1 c_ ¼ þ ¼ þ n qs  C 20  qs b d dcTI dcTI dcTI c_ 0

 ¼ 3:7, (b)   ¼ 3:7 and (c) Fig. 6. Statistical distribution of h: (a)  h ¼ 30 ; r h ¼ 10 ; r   ¼ 8:8. h ¼ 30 ; r

pffiffiffiffi pffiffiffiffiffi s ¼ alb q ¼ alb qS

where cTI is the shear strain of TI under external stress. d is the grain size. The meaning of the other parameters can be seen in Section 2.2.1. A dislocation model is adopted as shown in Fig. 3b, which is subjected to an applied shear stress s. cTBAZ is the nonhomogeneous plastic strain of TBAZ, and cTI is the uniform strain of TI. Here we denote that the plastic strain of TBAZ near the TI surface is identical with that of TI, and gradually increases as the individual slip step moves far away from the interface. The shear strain produced by geometrically necessary dislocation is taken as:

cGNDS ¼ cTBAZ  cTI ¼ ð9Þ

where s, a, and l are the plasticity flow stress, empirical constant and shear modulus, respectively. 2.1.2. Shear deformation along the TBs Dao et al. [24] proposed the concept of twin boundary affected zone (TBAZ), which is plastically softer than the predominantly elastic twin interior (TI) region between TBAZs. Considering the

ð12Þ

/nb d

ð13Þ

where / and n are the geometrical factor and the number of geometrically necessary dislocation, respectively. The deformation volume of dislocation evolution is the volume of TBAZ: 2

V ¼ d  dTBAZ

ð14Þ

Therefore the geometrically necessary dislocation density within TBAZ can be expressed as:

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Fig. 7. Contour plots of the equivalent plastic strain at a strain rate of 6  101 s1 (a) initial grain configuration and boundary conditions (b–f) strain evolution process for the  ¼ 8:8. microstructure with  h ¼ 30 ; r

qG ¼

nk 1 cTBAZ  cTI ¼  V /b dTBAZ

ð15Þ

where k is the length of dislocation line. The geometrical mean strain gradient associated with the geometrically necessary dislocations within TBAZ can be defined as g = (cTBAZ  cTI)/dTBAZ. Thus, the geometrically necessary dislocation density in individual grain can be obtained:

qG ¼

2dTBAZ 1 cTBAZ  cTI 2dTBAZ   ¼ g /b kT dTBAZ /bkT

ð16Þ

Both the statistical storage dislocations and the geometrically necessary dislocations contribute to the overall work hardening through Taylor’s theory of the flow stress:

pffiffiffiffi

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

s ¼ alb q ¼ alb qS þ qG

ð17Þ

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with small h are located nearly in the hard orientations to the loading axis, and therefore are the least deformed grains. 3.2. Shear band development

 ¼ 8:8. Fig. 8. h Distribution for the microstructure with  h ¼ 30 ; r

2.3. Numerical simulation on mechanical behaviors of nanotwinned Cu The variations of stress with strain at a fixed kT(kT = 35 nm) are plotted in Fig. 4. It can be seen that in the range of 0 6 h 6 90 ; h ¼ 45 is a turning point and the deformation responses on either side of the point are symmetric. To understand the h effect in detail, three different microstructures have been investigated by varying the arithmetic mean angle  h and the vari ). The value, in degrees, of each h is indicated and five ance of  h (r angle ranges are classified by five different colors respectively, as shown in Fig. 5a–c. The angles in each microstructure are randomly generated in the range of 0–50°, and as shown in Fig. 6, the numbers of grains with various angles can be represented by a normal  ¼ 3:7, (b) distribution. They are referred to as: (a)  h ¼ 30 , r   ¼ 3:7 and (c)   ¼ 8:8. h ¼ 10 , r h ¼ 30 , r The model was firstly imported into ABAQUS/CAE under plane stress condition and the finite-element mesh including about 5653 elements was generated in the software as shown in Fig. 5d. The parameters used in the calculation are as seen in Table 1. In order to obtain a quasi-static loading status, a smooth step using numerical velocity dissipation at a regular time interval was set to reduce the influence of inertial effect [22].

Fig. 9a–d shows four contour plots of the equivalent plastic  ¼ 3:7 at a strain rate h ¼ 10 , r strain for the microstructure with  1 1 of 6  10 s , whose corresponding overall strain levels are 0.02, 0.06, 0.108 and 0.11, respectively. Fig. 9e–h display strain  ¼ 3:7, evolution process for the microstructure with  h ¼ 30 , r which is deformed to the overall strain of 0.01, 0.04, 0.08, 0.1 and 0.11 at a strain rate of 6  101 s1. From Fig. 9a, b, e and f, for both of the two microstructures, there is no clear evidence of shear band. However, for the microstructure with  h ¼ 10 , r ¼ 3:7, as shown in Fig. 9c, a shear band begins to develop compared with the previous figures. Especially, in Fig. 9d, the shear strain along the shear band has mobilized significantly, and almost all strain activities outside the shear band have ceased. For the  ¼ 3:7, the definitive shear band is microstructure with  h ¼ 30 , r still not yet evidenced in Fig. 9h. The upper concentrated zone forms two shear bands and which one of the two shear bands is eventually dominant in deformation is still an unknown factor.  ¼ 3:7, the Compared with the microstructure with  h ¼ 10 , r  ¼ 3:7 can effectively delay the onmicrostructure with  h ¼ 30 , r set of shear band.It shows that h will influence the deformation and failure behaviors of nanotwinned materials. As shown in Fig. 4, with the increase of h, the strength first decreases and then increases. At 2% strain, the stress of h ¼ 0 ð90 Þ is 858 MPa, which is about 120 MPa higher than the stress of h ¼ 45 (740 MPa). At 6% strain, the stress of h ¼ 0 ð90 Þ is 954 MPa, while the stress of h ¼ 45 is 882 MPa. This shows that even if the twin lamellae thickness is a fixed value, the r–e curves are different because of the variation of h. Besides, even though the ultrafine-grained or nanocrystalline materials may be intrinsically ductile, they tend to suffer from plastic instabilities upon deformation [25,26]. One instance is the concentration of large deformation in shear bands. The undesirable inhomogeneous deformation can result in the onset of failure, severely limiting the useful ductility [27]. Fig. 9 demonstrates that if make proper arrangement of h, the onset of failure can be delayed, which leads to an unusual combination of high tensile strength and ductility. In production process, this measure can improve the mechanical performance of nanotwinned materials. 3.3. Twin lamellae thickness effect

3. Results and discussion 3.1. Nonuniform deformation The reference configuration and the loading status of the two ¼ 8:8 are shown in dimensional microstructure with  h ¼ 30 , r Fig. 7a. The equivalent plastic strain evolution contour plots during the uniaxial tension at the strain rate of 6  101 s1 are reported in Fig. 7b–f. Some grains firstly display large plastic deformation even if the overall level of strain is still low. With increasing strain, the neighboring grains begin to accommodate their plastic deformation with each other and strain localization becomes significant. Consequently, these high strain locations become combined together to form a shear band, as shown in Fig. 7f. It is clear that, even in the neighboring grains, there is great different strain distribution on account of h. As seen in Fig. 8, the green points denote the early deformed grains. In the range of 0– 50°, the grains having larger h deform first compared to the other grains. That is to say, grains with large h are optimally oriented with respect to the loading axis for soft orientations; the grains

Fig. 10 shows the r–e relationships of nanotwinned Cu with h ¼ 0 and h ¼ 30 at kT of 15 and 35 nm, respectively. It is shown that smaller kT leads to higher strength. During deformation, TBs serve as effective barriers to dislocation slip transmission. The magnitude of stress concentration depends on the applied external stress and the number of piled-up dislocations behind the leading dislocation. As kT decreases, fewer dislocations are expected to pile up. That means a higher external stress is required for the dislocations to cross TBs. The Dr–e curves are also shown in the inset of Fig. 10. Note that Dr of kT = 35 nm drops sharply as e increases, while Dr–e of kT = 15 nm has a slight drop. This is attributed to the influence of strain hardening. Dislocations moving along the TBs experience no barriers before stopping at the GBs, accompanied by the migration of TBs during deformation [28,29]. As kT decreases, the number of dislocations stopping at the GBs would increase, as shown in Fig. 11. These dislocations give rise to a back stress that hinders the movement of other dislocations, leading to kinematic strain hardening [28,29]. In short, the effect of h on the r–e relationships is weakening as kT decreases.

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 ¼ 3:7 deformed to the Fig. 9. Contour plots of the equivalent plastic strain at a strain rate of 6  101 s1 (a–d) strain evolution process for the microstructure with  h ¼ 10 ; r  ¼ 3:7 deformed to the overall strain of 0.02, 0.06, 0.108 and overall strain of 0.02, 0.06, 0.108 and 0.11 (e–h) strain evolution process for the microstructure with  h ¼ 30 , r 0.11.

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strength first decreased and then increased. Even if the twin lamellae thickness was a fixed value (kT = 35 nm), the largest difference was about 120 MPa in strength for the r–e curves with various angles. Besides, the shear band formation limited the uniform tensile  ¼ 3:7 was beneelongation, but the microstructure with  h ¼ 30 ; r ficial to improve the homogeneous plastic deformation and made the shear bands less evident compared with the microstructure  ¼ 3:7. However, the influence of h began to wear with  h ¼ 10 ; r off as twin lamellae thickness decreased. Acknowledgments

Fig. 10. r–e curves and Dr–e curves of nanotwinned Cu.

This work was supported by National Natural Science Foundation of China (10502025, 10872087, and 11272143), Key Project of Chinese Ministry of Education (211061), Natural Science Foundation of Hubei Province (Q20111501) and Research Innovation Program for College Graduates of Jiangsu Province (CXZZ11_0342). References

Fig. 11. Schematic drawings of piled-up dislocations after deformation (a) large kT and (b) small kT.

4. Conclusions A constitutive model considering the angle h between the loading axis and TB was developed. Three different microstructures are also proposed to better understand the deformation behaviors of nanotwinned Cu under uniaxial tensile load. The FEM simulation results revealed that h directly affected the deformation and failure behaviors of nanotwinned Cu. In the range of 0 6 h 6 90 , the

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