Effect of coherent twin boundary in nanotwinned materials on fatigue crack growth based on dislocation emission

Mechanics of Materials 108 (2017) 87–92

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Effect of coherent twin boundary in nanotwinned materials on fatigue crack growth based on dislocation emission Yanan Zhou a, Jianqiu Zhou a,b,∗, Tongyang He a a b

Department of Mechanical Engineering, Nanjing Tech University, Nanjing, Jiangsu Province, PR China, 210009 Department of Mechanical Engineering, Wuhan Institute of Technology, Wuhan, Hubei Province PR China, 430070

a r t i c l e

i n f o

Article history: Received 31 July 2016 Revised 1 March 2017 Available online 9 March 2017 Keywords: Nanocrystalline metal Theoretical model Fatigue crack growth Coherent twin boundary

a b s t r a c t A theoretical model to describe the dislocations emitted from the crack tip and penetrating the coherent twin boundary (CTB) in nanotwinned materials under cyclic far field shear stress was proposed in this paper. The dislocations emitted from the tip are subjected to four stresses: the shear stress on the primary slip plane generated by applied stress, the image stress which drives the dislocation glide to the surface of the crack, the back stress from the other dislocations and the lattice friction stress. Combining the theory of continuously distributed dislocation, we developed a method to analyze the dislocation movement from the crack tip under cyclic far field shear stress and derived the calculation method of fatigue crack growth rate in nanotwinned materials. It can be figured out that the fatigue crack growth rate is dependent on the applied stress, the lamellae thickness and the angle between slip system and CTB. When the critical conditions are satisfied, the dislocation piling up around the CTB will penetrate the CTB, the critical stress can be calculated as a function of the misorientation angle and the lamellae thickness. © 2017 Elsevier Ltd. All rights reserved.

1. Introduction Nanocrystalline (NC) materials with the grain size below 100 nm show high strength, high diffusivity, superplasticity at low temperatures and high strain rates (Belova and Murch, 2003; He et al., 2015; Khan et al., 2008a, b; Marek et al., 2015; Wang et al., 2014). Due to a series of wonderful mechanical properties, nanotwinned metals and alloys are novel emerging materials, attracting a great number of researchers (Alkan et al., 2016; Lu et al., 2009b, 2012; Yuan et al., 2014). Recent years, the fracture and fatigue properties of NC have been observed (Kim et al., 2012; Pan et al., 2013; Shute et al., 2009). The properties of NC materials depend significantly on the interactions between dislocations and two-dimensional defects including conventional grain boundary (GB) and twin boundary (TB) (Lu et al., 2009a, b; Ni et al., 2012). In NC materials, GB has a limited capacity to accumulate dislocations, which leads to low ductility (Lu et al., 2004, 2009a). Otherwise, TB acts not only as barriers to dislocation motion but also as sites for dislocation nucleation and accumulation, which leads to a simultaneous increase in both strength and ductility(Lu et al., 2004, 2009a, b; Wang et al., 2010; Zhao et al., 2006).

∗

Corresponding author. E-mail address: [email protected] (J. Zhou).

http://dx.doi.org/10.1016/j.mechmat.2017.03.006 0167-6636/© 2017 Elsevier Ltd. All rights reserved.

Therefore, it is meaningful to study the factor that affects the interaction between dislocation and TB in materials design. Some models based on the Mott’s assumption (MOTT, 1958) ware proposed to account for the crack growth. These models, however, are rather qualitative and fail to yield any systematic. Mura et al. (1981) and Xie et al. (2015) tried to propose a dislocation model to quantitatively solve the dislocation movement in the slip band. Such studies have a same assumption that the TB blocks the all dislocation movement. However, the interaction between dislocation and TB is complicated and still remain not well understood. For the case of the interaction of dislocation-GB, researchers have done a series of research to propose the dislocation penetrating GB criterion. Livingston and Chalmers (1957) proposed a geometrical criterion which is about the dislocation to penetrate GB. That the angle between the slip surface of the adjacent grain and the intersection line of GB is minimal, meanwhile the angle between the paths of the dislocation slip in the adjacent grain is minimal. Shen et al. (1988) and Lee et al. (1989) put forward the criterion with other conditions: first, the shear stress of the dislocation which penetrating GB should reach the maximum value; second, the Burgers vector of the residual dislocations in GB should be minimal. For nanotwinned materials, grains are subdivided into nanometer thick twin/matrix lamellar structure by the coherent twin

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Fig. 1. Schematic drawings of the proposed model.

boundaries (CTBs) (Lu et al., 20 05, 20 09, 2012; Ovid’Ko, 2011; Yoshida et al., 2014). As the deformation goes ahead, dislocations begin to pile up at the twin boundary (Gutkin et al., 2008). Ovid’Ko (2007) pointed out that if the stress intensity near the crack tip, the crack induces plastic shear through the emission of lattice dislocations from the crack tip. Fang et al. (2013) has investigated the special rotational deformation on the emission of lattice dislocations from the crack tip. In the above-mentioned works, the researchers have separately studied the effect of the twin near the crack tip or the dislocation pile-up. In this paper, we considered the coupled effect of the twin near the crack tip and the further dislocation pile-up under cyclic far field shear stress for the first time. We will study the interaction of dislocation-coherent twin boundary on fatigue crack growth in nanotwinned materials based on the geometrical criterion more comprehensively, which has never been reported by other papers. First, a method to analyze the dislocation movement from the crack tip under cyclic far field shear stress is established. And then, a calculation method of fatigue crack growth rate in nanotwinned materials will be derived. Finally, the critical conditions, which will be used to judge whether the dislocation piling up around the CTB penetrate the CTB or not, will be analyzed and discussed.

Fig. 2. Cyclic farfield shear stress on the primary slip plane.

2. The model of dislocations emitted from crack tip Fig. 1 illustrates the configuration of the nanotwinned crystal. A coherent twin boundary(CTB) are placed ahead of a nanometer sized crack (Ⅰtype crack). We denote the perpendicular distance by d between the CTB and the crack. Dislocations glide along the slip systems which intersect the CTB at the angle of θ . Cyclic farfield shear stress on the primary slip plane is applied on the system, where TⅠ app is the maximum shear stress and TⅡ app is the minimum shear stress as shown in Fig. 2. 2.1. Dislocation accumulation Under the forward flow of the cyclic farfield shear stress, a series of dislocations are emitted from the crack tip and move along the slip band, which is donated as layerⅠ. Finally,the dislocations pileup against the CTB. The reverse flow is expected to take place near the layerⅠ, donated as layer Ⅱ. It is assumed that the positive dislocation movement formed by forward flow is irreversible and the negative dislocations pileup during the reverse flow causes a positive back stress on layerⅠ. The back stress strengthens the pileup of dislocations during the forward flow of next cycle. Considering the independence of slip band, we neglect the distance of two vicinal layers compared with the length of the pileup layers. Fig. 3 shows the dislocation movement in the nanotwinned crystal for the first cycle under the applied shear stress.

Fig. 3. Schematic diagram of dislocations emitted from crack tip (a) three dimensional graph, and (b) two dimensions.

In our model, the theory of continuously distributed dislocation is applied, and the dislocations on the same layer in per circle are regarded as a whole and the dislocations are subjected to several force as follow: (1) shear stress on the primary slip plane, τ app which drives the dislocation away from or toward the crack tip, (2) image stress, τ image which drives the dislocation glide to the surface of the crack, (3) dislocation stress(back stress), τ pileup which hiders the movement of the dislocation, generated from the dislocations of any location(Chowdhury et al., 2014b), (4) lattice frictional stress, τ p , which objects the dislocation moving. When the

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dislocation under the first three stress overcome the lattice frictional (greater than or equal to τ p ), it will start gliding. The equilibrium condition of dislocations inside layerⅠ is expressed as

τIapp − τ image + τIpileup − τIp = 0 where the image stress,

τ image

(1)

is a function of the displacement x.

τ image = A/2x

(2)

A is a constant related to the material properties.

A = Gb/2π (1 − υ )

(3)

where G is the shear modulus, b the Burgers vector, and υ Poisson’s ratio. Considering the theory of continuously distributed dislocation, τ Ⅰ pileup is the dislocation stress given by

τIpileup = A



a 0

ρI (x )/(x − x )dx

(4)

where a is the equilibrium position of the first dislocation and we assume a = d/cosθ in our model. ρ Ⅰ (x) is the dislocation density in layer Ⅰ. Eq. (1) can be solved with the use of the inversion formula of Muskhelishvili (Head, 1955). Substituting Eq. (4) into Eq. (1), we can get the dislocation density ρ Ⅰ (x) as

 τIapp − τIp 1 x ρI ( x ) = ( − ) πA 2π x a−x NI =

0

ρI (x )dx =

 a τIapp −τIp 

a

x

1

dx − 0 2π x πA a−x (τIapp −τIp )a 1

0

=

2A

−



x dx a−x

(6)

2

τIIapp − τ image + τIIpileup − τIp − τIIp = 0

(7)

τ app

If denotes the amplitude of fatigue loading, in view of Eqs. (1) and (7), τ Ⅱ pileup is the dislocation stress given by



0

a

ρII (x )/(x − x )dx

(8)

where τ app = τ Ⅰ app -τ Ⅱ app . The following equation of the dislocation density ρ Ⅱ (x) in layer Ⅱ can be written as



ρII (x ) =

−(τ app − τIp − τIIp ) 1 + πA πx



x a−x

(9)

The number of dislocations in layerⅡ between x = 0 and a is

NII =

A 0

ρII (x )dx =

 a −(τ app −τIp −τIIp )  0

=

πA −(τ app −τIp −τIIp )a 2A

x dx a−x

+1

+

a

1 0 πx



x dx a−x

(10)

Then the residual dislocations left at the end of the cycle upon the interaction of positive and negative flow is quantified as N

(τ app − τIp − τIIp )d N = −1 2A cos θ

(11)

(13)

Substituting Eqs. (11) and (13) into Eq. (12), the fatigue crack growth rate can be deduced

da = b cos θ N = dN



 (τ app − τIp − τIIp )d − 1 b cos θ 2A cos θ

(14)

As shown in Fig. 4, the twins are separated by a coherent twin boundary. The slip systems are symmetrical about the CTB. When the dislocations slide along the slip plane S1 and go through the CTB, the CTB obstructs the dislocation movement. So the dislocations piling up around the CTB causes the high dislocation density. With the continuous load of the applied stress, a dislocation near the CTB will penetrate the CTB and slide along the symmetrical slip plane S2 when the stress reaching a critical value and the other dislocations will follow the footsteps of the first dislocation simultaneously. The Burger vector of the first dislocation along the S1 is b1 and the Burger vector of the dislocation along the S2 is b2 . We denote the angle between b1 and b2 as the misorientation angle θ . According to the Burgers vector conservation, the Burgers vector b of the dislocation debris in the CTB can be expressed as b=b1 −b2 (Li et al., 2009). 3.1. The shear stress of the dislocations penetrate the CTB The external loading shear stress τ acting on the incoming dislocations on the S1 will be larger than the critical shear stress τ cr if the dislocations penetrate the CTB mentioned as (Li et al., 2009)

τ b1 2 ≥ τcr b1 2 = Etb b1 + α G(b)2

Fatigue crack growth rate, da/dN is used to describe the crack extension in per circle loading. It can be calculated with the displacement of the crack tip in forward and reverse flow(lⅠ and lⅡ ) along the crack propagation path by Eq. (12) (Wu et al., 1993).

(12)

(15)

Etb is denoted as a unit area energy of CTB, α is material constant and G is shear modulus. According to the reference Li et al. (2009), the relationship between the CTB energy Etb and the misorientation angle θ can be solved by the following formula



2.2. Fatigue crack growth rate

da/dN = (lI − lII ) cos θ

l = lI − lII = bN

3. The dislocation – CTB penetration model

The equilibrium condition of dislocations inside layerⅡ is expressed as

τIIpileup = τ app − τIp − τIIp = A

where θ is the angle between the crack growth paths and the slip plane. The cycle irreversible crack tip displacement in slip plane direction, l, be solved as follow (Alkan et al., 2016).

(5)

The number of dislocations in layerⅠ between x = 0 and a is

a

Fig. 4. The dislocation-CTB interaction model.

Etb =

k θ / θ 1 k k(90 − θ )/90 − θ2

0 ◦ < θ ≤ θ 1 θ 1 < θ ≤ θ 2 θ2 < θ ≤ 90◦

(16)

The Hall–Petch slope k and the divided angle θ 1 ,θ 2 of polycrystalline aluminum were measured (Shanmugasundaram et al., 2010), where k≈0.06 MPa•m1/2 , θ 1 ≈20° and θ 2 ≈70°.Combined

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with ((15)–(16), the relationship between the critical shear stress τ cr and the misorientation angle θ can be solved as

⎧ k θ b 2 ⎪ ⎨ θ1 b 1 + α G ( b 1 ) τcr = kb1 + α G( b b )2 ⎪ ⎩ k(90−θ ) 1 b 2 + α G( b ) b1 (90−θ2 ) 1

0 ◦ < θ ≤ θ 1

θ 1 < θ ≤ θ 2 θ2 < θ ≤ 90◦

(17)

According to our model, θ =2θ and b/b1 =sin (θ ). Not only the misorientation angle, but also the lamellae thickness of the twin, λ, affects the critical shear stress. As well-known Hall–Petch relation σ = σ 0 +kdn , with n(normally around n = 0.5) is the Hall–Petch exponent and k is the Hall–Petch slope(Li et al., 2009). We changed the former formula to adapt to our model. The yield or flow strength σ y of polycrystals is enhanced as

τy = τ0 + kλ−0.5

(18)

where τ 0 is friction stress, τ 0 = 30 MPa(Shanmugasundaram et al., 2010) and λ is the lamellae thickness of the twin. Then we can get the relationship between the misorientation angle and the lamellae thickness of the twin on the critical shear stress for dislocation penetrating the CTB. According to semi-quantitative law, the expression of the final yield stress is(Zhou et al., 2016):

τtotal

⎧ k θ 0 ◦ < θ ≤ θ 1 ⎨ θ1 b1 + α G( b1b )2 +τ0 + kλ−0.5 b 2 −0.5 θ 1 < θ ≤ θ 2 = τcr + τy = kb1 + α G( b1 ) +τ0 + kλ ⎩ k(90−θ ) b 2 −0.5 + α G ( ) + τ + k λ θ2 < θ ≤ 90◦ 0 b1 (90−θ2 ) b1 (19)

4. Results and discussion The calculations are carried out for nanotwinned aluminum: the shear modulus G = 26.13 GPa, the Burgers vector b = 0.255 nm, and Poisson’s ratio υ = 0.345 (Zhang et al., 2013). The lattice frictional stress τ Ⅰ p in layerⅠand τ Ⅱ p in layerⅡwere resolved by Chowdhury et al. (2014a), where τ Ⅰ p = 1.1 MPa and τ Ⅱ p = 2.41 MPa. 4.1. The effect of the distance between CTB and crack on the fatigue crack growth rate We employed the fatigue crack growth rate formulations to quantitatively elucidate the effect on fatigue crack growth. Focused on the formula(14), we can get that the applied stress τ , the distance between CTB and crack d and the angle θ between slip system and CTB will have a chance to effect fatigue crack growth together. It can be seen that the angle θ between slip system and CTB play a little influence to the fatigue crack growth. Select the distance between CTB and crack d = 5, 25, 40, 55 nm. We can find clearly in Fig. 5(a) that the fatigue crack growth rate increases with the increasing of d. The fatigue crack growth rate is more sensitive when d is larger. The main reason is that the CTB is the barrier to the dislocation slip. When tip close to the CTB, dislocation movement will be hindered. Here, we assume the d as the lamellae thickness of the twin. The thinner of twinned, the more dislocation be stored on the CTB and the fracture toughness of the nanocrystalline materials could be enhanced by reducing the lamellae thickness of the twin. The analytical result is in agreement with recent experimental findings, as shown in Fig. 5(b) (Chowdhury et al., 2016). 4.2. The influence of misorientation angle and lamellae thickness to the critical stress For the former formula (17–19), it can be known that the misorientation angle θ and the lamellae thickness λ of the twin may

Fig. 5. (a) the relationship between the fatigue crack growth rate da/dN and the distance between CTB and crack d (θ = 15°). (b) experiment data [33] and calculation result.

Fig. 6. The relationship between the critical stress τ and the misorientation angle.

play a certain role to the critical stress τ total when dislocations penetrate CTB. In our research, the misorientation angle θ will be suitable for the range θ 1 < θ ≤ θ 2 , where θ = {30°, 40°, 50°, 60°}. We selected the lamellae thickness λ = 5, 15, 30, 50 nm of the twin in our model. As shown in Fig. 6, the critical stress increases with the misorientation angle θ . It can be explained as: according to the barrier of the CTB, Burgers vector b1 in S1 become to be the residual

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3 The critical stress increases with the misorientation angle θ . The critical stress decreases with the increasing of the lamellae thickness λ of the twin and it is sensitive when λ < 25 nm. Decreasing the λ can be an effective way to raise the ductility and fracture toughness of nanotwinned metals and the result basically agree with the research of Xie et al. (2015).

Acknowledgments This work was supported by Key Project of Chinese Ministry of Education (211061), National Natural Science Foundation of China (10502025, 10872087, 11272143), Program for Chinese New Century Excellent Talents in university (NCET-12-0712). PhD programs Foundation of Ministry of Education of China (20133221110 0 08). Fig. 7. The relationship between the critical stress τ and the lamellae thickness λ of the twin.

Burgers vector b2 in S2 when dislocations penetrate CTB. The CTB change the orientation and shorten the value of Burgers vector b1 which we denote as the b. The larger of the b is, the more difficult of dislocations penetrating the CTB will be. With the increasing misorientation angle θ , the value of the b increases. As a result, the critical stress increases. As shown in Fig. 7, the critical stress decreases with the increasing of the lamellae thickness λ of the twin. The rate of the critical stress decreases with the increasing of λ, and it is sensitive when λ < 25 nm. With decreasing λ, the volume fraction of CTB in the grain will increase. It has been pointed out that the dislocation will be more difficult to emit with the increasing thickness of twin lamellae (Zhang et al., 2013). So, decreasing the λ can be an effective way to raise the ductility and fracture toughness of nanotwinned metals. It is also supported by experiments (Dao et al., 2006; Lu et al., 2009b) showing that the electro-deposited Cu sample composed of nano-scale twins has a substantial increase in tensile elongation and strain hardening when the twin lamellae decreases. Finally, the result basically agree with the research of Xie et al. (2015). 5. Conclusions In our theoretical model, CTB is obstacle for fatigue crack growth and the crack growth depends on the applied stress, the lamellae thickness, and the angle between slip band and CTB. According to our model, we can develop some expressions such as the dislocation density, the number of dislocations, the fatigue crack growth rate and the critical shear stress of the dislocation penetrating the CTB. In this study, we have analyzed the fatigue crack growth behavior of nanotwinned aluminum theoretically. Some conclusions are summarized as follows. 1 The applied stress τ , the distance between CTB and crack d and the angle θ between slip system and CTB will have a chance to effect fatigue crack growth together. When the first dislocation doesn’t penetrate the CTB, the angle θ between slip system and CTB play a little influence to the fatigue crack growth and the fracture toughness of the nanocrystalline materials could be enhanced by reducing the lamellae thickness of the twin. 2 When the critical conditions are satisfied, the dislocation piling up around the CTB will penetrate the CTB. The misorientation angle θ and the lamellae thickness λ of the twin may play a certain role to the critical stress when dislocations penetrate CTB.

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