Theoretical analyses of nanocrack nucleation near the main crack tip in nano and micro crystalline materials

Engineering Fracture Mechanics 221 (2019) 106672

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Theoretical analyses of nanocrack nucleation near the main crack tip in nano and micro crystalline materials Xiaotao Li, Xiaoyu Jiang

T

⁎

School of Mechanics and Engineering, Southwest Jiaotong University, Chengdu 610031, China

ARTICLE INFO

ABSTRACT

Keywords: Dislocation pileup Grain boundary sliding Nanocrack nucleation Crystalline materials

Nano or micro crack nucleation in the process of metallic fracture is a common phenomenon and it affects fracture behaviors of the materials. The mechanisms of nanocrack nucleation are different for metals with nanoscale and microscale grain sizes. In this paper, according to the deformation mechanisms of materials with different grain sizes, theoretical models are established to investigate nanocrack nucleation near a main crack tip under mode I load. The theoretical solution is presented based on the distributed dislocation technique. For nanocrystalline metals, the process of nanocrack nucleation due to grain boundary sliding is analyzed and the effect of grain boundary structure on nanocrack nucleation is studied. The results show that nanocracks may be preferentially nucleated at the triple junctions which are located on the upper or lower sides of the main crack plane, instead of the front of the crack tip. A larger nanocrack may be nucleated at triple junctions with a larger characteristic abutting angle of the triple junctions. For microcrystalline metals, nanocrack nucleation due to lattice dislocation pileup is considered. The problem of inclusion cracking due to lattice dislocation pileup is analyzed. The results show that lattice dislocation pileup causes the entire soft inclusion to crack easily, but a relatively small nanocrack can be only nucleated in the hard inclusion.

1. Introduction Fracture of materials is the most dangerous mode of failure, and it is caused by the main crack propagation. The behaviors of crack propagation in crystalline materials are affected greatly by microstructures [1], such as dislocations, grain boundaries, inclusions and so on. Dislocations are emitted from a crack tip when the stress intensity factor at the crack tip is larger than the critical value for dislocation generation [2]. These emitted dislocations can cause crack blunting [3,4] and relax stress concentration near crack tips [5,6], thus the crack growth is suppressed. Dislocations can also be generated from grain boundaries, microvoids, inclusions, Frank-Read dislocation sources and so on. In the process of manufacturing, it is inevitable to generate grain boundaries and inclusions in polycrystalline materials. In composite materials, there are also many interfaces or inclusions. Grain boundaries, interfaces and inclusions play the role of obstacles for lattice dislocation motion, so dislocations may be piled up at the obstacles and cause stress concentration. For the metallic crystalline materials with micro-scale grain size, lattice dislocation generation is the dominant mechanism of plastic deformation [7]. In this case, the stress concentration due to lattice dislocation pileup is severe. Hence, new nano or micro cracks may be nucleated due to the lattice dislocation pileup near grain boundaries or inclusions. The main crack propagation path and rate can be changed by nucleated cracks. Experimental observation [5] showed lattice dislocation pileup ⁎ Corresponding author at: Southwest Jiaotong University, No. 111, North 1st Section of Second Ring Road, Jinniu District, Chengdu City, Sichuan Province, China. E-mail address: [email protected] (X. Jiang).

https://doi.org/10.1016/j.engfracmech.2019.106672 Received 12 May 2019; Received in revised form 7 September 2019; Accepted 9 September 2019 Available online 10 September 2019 0013-7944/ © 2019 Elsevier Ltd. All rights reserved.

Engineering Fracture Mechanics 221 (2019) 106672

X. Li and X. Jiang

at a grain boundary caused a microcrack to nucleate in molybdenum, and then the main crack propagation direction was changed to merge together with the nucleated microcrack. This phenomenon need be considered theoretically to understand the mechanism. Theoretical modelling studied multiple cracks were nucleated along graphene plates due to lattice dislocation pileup in laminated metal-graphene composites [8]. This work [8] considered the nucleation process of multiple nanocracks due to lattice dislocation pileup. However, the problem of inclusion cracking near a main crack tip did not be analyzed. The interaction between cracks and inclusions has been studied widely [9–11], but inclusion cracking due to lattice dislocation pileup in the process of main crack propagation is analyzed hardly. As grain size decreases, the volume fraction of grain boundaries increases. The hindering effect of grain boundaries on lattice dislocation movement increases and lattice dislocation generation is inhibited. For nanocrystalline materials, it is difficult to form lattice dislocation pileup in grains. In this case, grain boundary sliding has larger contribution to plastic deformation. From the microscopic point of view, grain boundary sliding is caused by the motion of grain boundary dislocations. Nanocracks can be generated by grain boundary sliding, which has been verified by simulated results [12–15], experimental observations [16] and theoretical works [17–19]. Nucleated nanocracks have a significant effect on the behaviors of the main crack propagation. Experiments [16] observed multiple nanocracks were nucleated along grain boundaries or their triple junctions near the main crack tip in nanocrystalline nickel with grain size about 40 nm. And then, the main crack propagated and joined together with nucleated nanocracks. Ovid’ko et al. [17–19] analyzed the mechanism of nanocrack nucleation and investigated nanocrack nucleation at triple junctions near a main crack tip by theoretical methods. Crystal-plasticity simulation results [12] showed nanocracks were generated along grain boundaries near the main crack tip and caused intergranular crack propagation in nanocrystalline nickel with grain size ranging from 14 nm to 61 nm. Molecular dynamics simulation results [13,14] showed nanocrack formation along grain boundaries ahead of a pre-existent crack in nanocrystalline nickel and α-Fe with grain size about 10 nm. Nanocrack nucleation due to grain boundary sliding has been investigated by many previous works. However, the effect of the triple junction structure of grain boundaries on nanocrack nucleation is considered hardly. In fact, different triple junctions have different effects on nanocrack nucleation and crack growth. Not only the volume fraction of grain boundaries but also their structures affect the fracture behaviors of crystalline materials. Therefore, nanocrack nucleation for different structures of triple junctions near the main crack tip need be analyzed in more detail. In crystalline metals, the phenomenon of nano or micro crack nucleation is very common and these nucleated cracks have a great effect on the main propagation. Therefore, the main purpose of this paper is to investigate and reveal the mechanisms of nanocrack nucleation in the process of main crack propagation. In Section 2, nanocrack nucleation induced by grain boundary sliding near a main crack tip in nanocrystalline metals is considered. The theoretical solution is presented by the distributed dislocation technique, and the effect of triple junction structures on nanocrack nucleation is analyzed. In Section 3, nanocrack nucleation due to lattice dislocation pileup at inclusions in microcrystalline metals is studied. The effect of lattice dislocation pileup on nanocrack nucleation is discussed for different cases. In Section 4, important results are summarized. 2. Nanocrack nucleation due to grain boundary sliding in nanocrystalline materials 2.1. Theoretical modelling As shown in Fig. 1(a), a nanocrystalline material contains a pre-existent main crack. This is a two-dimensional model. This paper focuses on nanocrack nucleation in the process of main crack propagation, so the process of main crack formation does not be considered. In nanocrystalline materials, the volume fraction of grain boundaries is relatively large. The grain boundaries serve as the obstacles of crack growth. Hence, the case that the main crack tip is located close to a grain boundary is general and it is considered in the paper. For simplicity, the uniform uniaxial tensile load is considered. In nanocrystalline materials, grain boundary sliding plays an important role in plastic deformation [7,20,21]. Therefore, grain boundary sliding near the main crack tip is considered. From the macroscopic viewpoint, grain boundary sliding refers to the relative movement of two adjacent grains along the grain boundary. From the microscopic perspective, grain boundary sliding is treated as generation and motion of grain boundary dislocations. Under the applied load, grain boundary sliding in multiple grain boundaries is generated due to shear stress concentration near the main crack tip, as shown in Fig. 1(a). Focusing attention on a triple junction near the main crack tip, grain boundary sliding causes dislocation dipoles generation and these dislocations move toward triple junctions under the shear stress acting on the dislocations. The Burgers vectors of dislocations in a grain boundary are parallel to the grain boundary. Triple junctions serve as the obstacles for grain boundary dislocations motion, so the dislocations are piled at triple junctions, as shown in Fig. 1(b). According to existing results [17,22], when grain boundary dislocations arrive at triple junctions, they will come into dislocation reaction and result in the formation of sessile dislocations, as shown in Fig. 1(c). With the process of dislocation reaction repeating, the Burgers vectors magnitude of sessile dislocations increase continuously, as shown in Fig. 1(d). Finally, nanocracks are nucleated to release the strain energy at triple junctions, as shown in Fig. 1(e). To understand the effect of triple junction structure on nanocrack nucleation, the process of nanocrack nucleation due to grain boundary sliding need be analyzed in detail. Next, the theoretical solution will be presented. 2.2. Theoretical solution scheme 2.2.1. Dislocation distribution in grain boundaries The problem of Fig. 1(b) is that high local shear stress near the main crack tip causes grain boundary sliding. Assuming that the 2

Engineering Fracture Mechanics 221 (2019) 106672

X. Li and X. Jiang

Fig. 1. Schematic of nanocrack nucleation due to grain boundary sliding: (a) general view of grain boundary sliding induced by shear stress concentration near the main crack tip, red lines denote the grain boundaries generating severe sliding; (b) enlarged view of grain boundary sliding at a triple junction, and grain boundary sliding is represented by continuously distributed dislocation dipoles; (c) grain boundary dislocations are piled at triple junctions; dislocation reaction occurs and sessile dislocations are generated at triple junctions; (d) the process in (c) occurs repeatedly, Burgers vectors of sessile dislocations increase continuously; (e) nanocrack nucleation to release the strain energy at triple junctions.

3

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X. Li and X. Jiang

Fig. 2. Schematic of grain boundary sliding near a triple junction ahead of a main crack tip.

state in Fig. 1(b) is quasi-equilibrium, grain boundary sliding in a grain boundary is represented by a continuous distribution of dislocations with the infinitesimal Burgers vector. The problem is modelled as Fig. 2. As shown in Fig. 2, the problem of an infinite plane containing a flat main crack and three grain boundaries AD, BD and CD under uniaxial tensile load is considered. A, B C and D are triple junctions. The related geometrical parameters of this model are tagged in Fig. 2. The positive direction of the angles α1, α2, α3 and β are defined as anticlockwise direction. Actually, besides the grain boundaries considered in Fig. 2, there are more grain boundaries generating dislocations due to the shear concentration near the main crack tip. Other dislocations in adjacent grain boundaries affect the distribution of shear stress along grain boundaries AD, BD and CD. However, the shear stress induced by dislocations in adjacent grain boundaries is much smaller than the shear stress induced by the crack. For simplicity, grain boundary sliding in adjacent other grain boundaries is not considered. Based on the distributed dislocation technique [23], the crack can be equivalent to the continuously distributed dislocations, and grain boundary sliding can also be described by a continuous distribution of dislocations. Thus, to obtain the stress field, the problem in Fig. 2 can be divided into two sub-problems. The first sub-problem is that an infinite plane without any defects is subjected to the uniaxial tensile load σ. The second one is that an infinite plane contains four dislocation strips without external loads. The stress field of the first sub-problem can be given by

~ (x , y ) = yy ~ (x , y ) = 0 xy

(1)

The second sub-problem is that an infinite plane contains four dislocation strips without external loads. For the problem of an infinite plane containing two arbitrary dislocation strips, the solution is presented in the section of Appendix A. For convenience, the crack, grain boundaries AD, BD and CD are marked as dislocation strips “1”, “2”, “3” and “4”, respectively. The stress components along the dislocation strip i can be given by

¯ yi yi (x i ) =

2µ ( + 1)

4 j =1

Rj Rj

¯ xi yi (x i ) =

2µ ( + 1)

4 j =1

Rj Rj

[B xj ( j ) Gxjij yi yi ( j, x i , dji , [B xj (

ji j ) Gxj xi yi

( j, x i , dji ,

ji ,

ji )

ji ,

ji )

+ B yj ( j ) Gyjij yi yi ( j, x i , dji , + B yj (

ji j ) Gyj xi yi

( j, x i , dji,

ji ,

ji )]d j

ji ,

ji )]d j

(2)

Here, μ is the shear modulus; κ is Kolosov’s constant; Rj is the half length of the dislocation strip j. The dislocation influence function G(ξj, xi, dji, θji, αji) has been derived in the Appendix A. To obtain the dislocation influence functions in Eq. (2), the related parameters Rj, dji, θji and αji need be only replaced, and these parameters for this theoretical model are given by 11 = 0 = arctan(yE / xE) 13 = arctan(yF / xF ) 14 = arctan(yH / xH )

d11 = 0

R1 = a R2 = R3 = R4 =

l1 2 l2 ; 2 l3 2

d12 =

xE2 + yE2

d13 =

xF2 + yF2

d14 =

xH2 + yH2

12

22

d22 = 0 d23 = d24 =

(xF (xH

xE) 2 + (yF xE)2 + (yH

yE )2 yE )2

;

23

24

= =

+ arccos

d33 = 0 d34 =

(xH

xF

)2

+ (yH

33

yF

)2

34

=0

arccos

=

arccos

d44 = 0

(0.5l2)2

l2 d23 2 + (0.5l )2 d24 1

(0.5l3)2

l2 d24

=0 (0.5l3)2

;

1 1

2

l3 d34 44

4

2 + (0.5l )2 d23 1

2 + (0.5l )2 d34 2

=0 = 1 13 = 2 14 = 3 22 = 0 23 = 2 24 = 3 33 = 0 34 = 3 44 = 0 11

12

=0

(3)

Engineering Fracture Mechanics 221 (2019) 106672

X. Li and X. Jiang

The midpoints of grain boundaries AD, BD and CD are defined as E, F and G, respectively. In formula (3), the coordinates of the points E, F and H can be given by

xE = a + d cos + 0.5l1 cos yE = sin + 0.5l1 sin 1 xF = a + d cos 0.5l2 cos yF = sin 0.5l2 sin 2 xH = a + d cos + 0.5l3 cos yH = sin + 0.5l3 sin 3

1

2

3

(4)

In the quasi-equilibrium state shown in Fig. 2, several conditions must be satisfied: (1) the crack plane is a free surface, so the stress components along the crack plane are equal to zero; (2) the shear stress along grain boundaries is equal to the resistance of grain boundaries to dislocation motion τ0. (3) there is no net dislocation on the crack; dislocation dipoles are generated in grain boundaries due to grain boundary sliding, so the total number of dislocations in every grain boundary is equal to zero. The stress field can be obtained by the superposition of the two sub-problems. According to the conditions above, several integral equations about dislocation density functions can be established, as follows:

¯ y1 y1 (x1) + ~yy = 0 ¯ x1 y1 (x1) + ~xy = 0

condition (1)

¯ x2 y2 (x2 ) + ~yy sin ¯ (x ) + ~ sin

condition (2)

x3 y3

3

yy

¯ x 4 y4 (x 4 ) + ~yy sin a a

condition (3)

0.5l1 0.5l1

B x2 ( 2 )d

along the crack

1 cos 1

=

0

along grain boundary AD

2 cos

2

=

0

along grain boundary BD

3 cos 3

=

0

B x1 ( 1)d 1 =

2

0.5l2 0.5l2

=

along grain boundary CD

a a

B y1 ( 1)d 1 = 0

B x3 ( 3)d

3

0.5l3 0.5l3

=

B x 4 ( 4 )d

4

=0

(5)

It is difficult to obtain the analytic solution of Eq. (5), so Gauss-Chebyshev quadrature method is applied to obtain its numerical solution. Letting ξj = Rj s and xj = Rj t, the integral interval [-Rj, Rj] can be normalized to interval [-1, 1]. Due to the singularity of crack tips and dislocation pileups at triple junctions, the dislocation density can be set as

B (s ) =

(6)

s2

(s )/ 1

Here, φ(s) is an unknown function. Eq. (5) can be discretized as a series of linear equations based on Gauss-Chebyshev quadrature method, and it can be written as Rj 4 j=1 n

2µ +1

n I=1

Rj 4 j=1 n

2µ +1

[

n I=1

ji xj (sI ) Gxj yi yi

[

(sI , tK , dji,

ji xj (sI ) Gxj xi yi

ji ,

(sI , tK , dji,

ji ) ji ,

ji yj (sI ) Gyj yi yi

+ ji )

(sI , tK , dji ,

ji yj (sI ) Gyj xi yi

+

ji ,

(sI , tK , dji ,

ji )] ji ,

+

ji )]

=0 =0

i=1

2µ +1

Rj 4 j=1 n

n I=1

[

ji xj (sI ) Gxj xi yi

(sI , tK , dji,

ji ,

ji )

+

ji yj (sI ) Gyj xi yi

(sI , tK , dji,

ji ,

ji )]

+

sin 2 1 2

=

0

i=2

2µ +1

Rj 4 j=1 n

n I=1

[

ji xj (sI ) Gxj xi yi

(sI , tK , dji,

ji ,

ji )

+

ji yj (sI ) Gyj xi yi

(sI , tK , dji,

ji ,

ji )]

+

sin 2 2 2

=

0

i=3

2µ +1

Rj

n I=1

[

ji xj (sI ) Gxj xi yi

(sI , tK , dji,

ji ,

ji )

+

ji yj (sI ) Gyj xi yi

(sI , tK , dji,

ji ,

ji )]

+

sin 2 3 2

=

0

i=4

4 j =1 n

n I=1

{

x 1 (sI )

n I=1

=

y1 (sI )

=

n I=1

x2 (sI )

=

n I=1

x3 (sI )

=

n I=1

x 4 (sI )

=0

(7)

Here, n is the number of discrete integration points. sI is discrete integration point and tK is collocation point. They are given by

sI = cos tK = cos

2I 1 2n K n

I = 1, 2, ...,n

K = 1, 2, ...,n

1

(8)

The Burgers vectors of dislocations in every grain boundary are parallel to the grain boundary, so the dislocations in grain boundaries are glide dislocations. Extra equations can be established as y2 (sI )

=

y3 (sI )

=

y4 (sI )

=0

(9)

Combining Eqs. (7) and (9), the unknownφ(sI) can be obtained. According to the book [23], the dislocation density function along grain boundaries can be obtained by Eq. (6), where

(s ) =

1 n

n I=1

1 + 2

n 1

cos j=1

2I 1 j 2n

cos(j arccos(s ))

(sI )

5

(10)

Engineering Fracture Mechanics 221 (2019) 106672

X. Li and X. Jiang 8.0x108

(b)

4.0x108

Dislocation density (m-1)

d=60 nm d=75 nm d=90 nm

0.0

-4.0x108

-8.0x108

D

-20

-10

0

10

20

Grain boundary DA (nm)

(c)

8.0x108

dislocation density (m-1)

Dislocation density (m-1)

(a)

4.0x108

8.0x108

d=60 nm d=75 nm d=90 nm

4.0x108

0.0

-4.0x108

-8.0x108

A

B

-20

-10

0

10

Grain boundary BD (nm)

20

D

d=60 nm d=75 nm d=90 nm

0.0

-4.0x108

-8.0x108

D

-20

-10

0

10

Grain boundary DC (nm)

20

C

Fig. 3. Dislocation distribution in grain boundaries, the calculation parameters are a = 5000 nm, σ = 200 MPa, α1 = -15°, α2 = 60°, α3 = -65°, β = 70° and l1 = l2 = l3 = 50 nm. (a) dislocation distribution in grain boundary AD; (b) dislocation distribution in grain boundary BD; (c) dislocation distribution in grain boundary CD.

In this paper, as an example of calculation, the nanocrystalline aluminum is studied, and the material parameters are bg = 0.1 nm, ν = 0.34 and μ = 27 GPa [24]. Here, bg is the Burgers vector magnitude of grain boundary dislocations. Previous results [25,26] showed the resistance of grain boundary to dislocation motion in nanocrystalline aluminum was very small, so the resistance in our calculations is neglected, that is τ0 = 0. The dislocation density along grain boundaries can be plotted in Fig. 3. As shown in Fig. 3, the dislocation density in grain boundaries for different distances between the triple junction D and the crack tip is depicted. The results show dislocations are piled at triple junctions. The closer the distance to triple junctions is, the larger the dislocation density is. The inhibition effect of triple junctions on grain boundary dislocation motion cause dislocation pileups. As the distance d increases, the dislocation density will decrease. This is because the shear stress along the grain boundaries induced by the crack decreases with d increasing. Next, the configurational force for nanocrack nucleation along grain boundaries will be calculated. Firstly, the Burgers vector magnitude of the sessile dislocation at the triple junction D is calculated. Secondly, the stress components along grain boundaries are calculated. And then, the configurational force for nanocrack nucleation can be obtained. 2.2.2. Configurational force for nanocrack nucleation The dislocation density functions have been obtained in Section 2.2.1. The dislocations are generated in pairs. The number of negative dislocations and positive dislocations is equal. Thus, the number of dislocation dipoles is equal to the number of positive or negative dislocations. The number of dislocation dipoles in grain boundaries AD, BD and CD can be calculated by

NAD =

0.5l1 B x2 ( 2 )d 2 x20

where B x2 ( 2 )| 2= x20 = 0

NBD =

0.5l2 x30

B x3 ( 3)d

3

where B x3 ( 3)| 3= x30 = 0

NCD =

0.5l3 x 40

B x 4 ( 4 )d

4

where B x 4 ( 4 )| 4 = x 40 = 0

(11)

Here, N is the number of dislocation dipoles and its subscript denotes the grain boundary. Under applied load, the dislocations will move toward triple junctions continuously in the process of dislocation reaction. Therefore, it is assumed that all of the dislocations in 6

Engineering Fracture Mechanics 221 (2019) 106672

X. Li and X. Jiang

Fig. 4. Schematic of nanocrack nucleation due to the energy accumulation of a sessile dislocation.

grain boundaries come into dislocation reaction. When dislocation reaction at the triple junction D is finished completely, the components of Burgers vector of the sessile dislocation at the triple junction D can be given by

bx = bg (NBD cos

2

+ NAD cos

1

NCD cos

3)

b y = bg (NBD sin

2

NAD sin

1

NCD sin

3)

(12)

After finishing the process of dislocation reaction, the problem can be described as an infinite plane containing a crack and a sessile dislocation under tensile load, as shown in Fig. 4. Because the stress field near the triple junction D is mainly contributed by the crack and the sessile dislocation at the triple junction D, the effect of sessile dislocations at other triple junctions on the stress field near the triple junction D can be neglected. The stress field near the triple junction D can be calculated by the superposition of the stress field induced by the sessile dislocation and the crack. For convenience, a local coordinate system xD-oD-yD is established. The origin of the coordinate system is the triple junction D and the directions of the coordinate axis are same as the global coordinate. The stress field in the local coordinate system due to the dislocation can be given by 2µ {b G ( + 1) x xij

¯ij (xD, yD ) = yD

Gxxx =

r4

Gyyy =

+ b y Gyij} ij = xx , xy or yy

where x (3xD2 + yD2 ), Gyxx = D4 (xD2

xD (xD2 r4

+

3yD2 ),

yD2 ), r xD 2 (xD yD2 ), r4 xD2 + yD2 .

Gxxy = r2 =

Gxyy = Gyxy =

yD

r4 yD

(xD2

yD2 ),

(xD2 r4

yD2 ), (13)

The stress field induced by the crack can be obtained by the linear elastic fracture mechanics. Because the distance d is much less than the crack length, the high orders of the stress field can be neglected. So the stress components in the global coordinate system due to the crack can be given by xx (x ,

y ) = kI (2

yy (x , xy (x ,

)

y ) = kI (2 y ) = kI (2

0.5cos

2

(1

)

0.5cos

)

0.5cos

2

sin 2 sin

(1 + sin

2

sin 2 cos

3 2

2

where = arctan[y /(x

=

a)] (x

3 2

sin

(x

a);

) 3 2

kII (2

)

)+k

II (2

)

a )2

and

+

)

= arctan[y/(x

2

(2 + cos

0.5cos

2

2

sin 2 sin

a)] +

(x < a )

2

cos

sin 2 cos

(1

0.5cos

+ kII (2 y2

0.5sin

3 2

)

3 2

3 2

) (14)

Here, the local stress intensity factors at the right tip of the crack can be given by

kI =

2µ +1

a

y ( + 1)

and kII =

2µ +1

a

x (+ 1)

where i ( +1)

is

=

1 n

n I=1

sin

(2I

1)(2n 1) 4n 2I 1 sin 4n

i (sI )

i = x or y

(15)

The stress components due to the crack in the local coordinate system can be obtained by coordinate system transformation, that

x = a + d cos

+ xD and y = d sin

(16)

+ yD

Substituting formula (16) into (14), The stress components due to the crack in the local coordinate system are obtained. Thus, the total stress components in the local coordinate system can be given by the superposition. 7

Engineering Fracture Mechanics 221 (2019) 106672

X. Li and X. Jiang ij (xD ,

yD ) = ¯ij (xD, yD ) +

ij (xD ,

(17)

yD ) ij = xD xD , xD yD or yD yD

The stress components along grain boundaries AD, BD and CD can be calculated by the Mohr stress transformation, that is xx xy yy

( , xD , yD )

cos2 sin2 sin cos

( , xD , yD ) = ( , xD , yD )

sin2 cos2 sin cos

sin 2 sin 2 cos 2

x DxD

and

x D yD

= =

1

+

2

=

yD yD

3

for AD for BD for CD

(18)

Nanocracks can be nucleated and grow when the configurational force is larger than the surface energy of two free surfaces [8,17,18,27,28], and the criterion of crack nucleation can be given by

F

2

(19)

e

Here, γe = γ for a nanocrack nucleation in a grain interior, γ is the surface energy. γe = γ-γb/2 for a nanocrack nucleation along a grain boundary, the term γb is the releasing energy of a grain boundary cracking. In this paper, nanocrack nucleation along grain boundaries is only analyzed. The configurational force along an arbitrary angle can be obtained by [27]

F( ) =

1 µ lc

lc 0

xy

( , xD , yD = 0)

xD dxD lc xD

2

lc

+

0

yy

( , xD , yD = 0)

xD dxD lc xD

2

(20)

Here, lc is the length of nucleated nanocrack in the process of nanocrack growth. When F > 2γ-γb, the nanocrack can be nucleated, and it will continue to grow until F = 2γ-γb. Solving the equation F = 2γ-γb, the critical length of nanocrack nucleation along grain boundaries can be obtained. Next, the results about nanocrack nucleation will be shown and analyzed.

(a)

5

4

3

2

d=60 nm d=75 nm d=90 nm

3

2

1

1

0

5

4

F (J.m-2)

-2

F (J.m )

(b)

d=60 nm d=75 nm d=90 nm

0

10

20

30

40

0

50

0

10

20

lc (nm) (c)

40

50

5

d=60 nm d=75 nm d=90 nm

4

F (J.m-2)

30

lc (nm)

3

2

1 0

10

20

30

40

50

lc (nm) Fig. 5. Configurational force F versus lc, the calculation parameters are a = 5000 nm, σ = 200 MPa, α1 = -15°, α2 = 60°, α3 = -65°, β = 70° and l1 = l2 = l3 = 50 nm. (a) F along grain boundary AD; (b) F along grain boundary BD; (c) F along grain boundary CD. 8

Engineering Fracture Mechanics 221 (2019) 106672

X. Li and X. Jiang

(a)

2.5

(b)

2.0

α=20° α=40° α=60° α=80° α=100° α=120°

1.5

1.0

F (J.m-2)

F (J.m-2)

2.0

0.5

0.0

0

5

10

15

1.0

0.0

20

0

5

10

15

20

lc (nm)

5

α=20° α=40° α=60° α=80° α=100° α=120°

4

F (J.m-2)

α=20° α=40° α=60° α=80° α=100° α=120°

1.5

0.5

lc (nm)

(c)

2.5

3

2

1

0

0

5

10

15

20

lc (nm) Fig. 6. Configurational force F versus lc, the calculation parameters are a = 5000 nm, σ = 200 MPa, α2 = 60°, β = 70°, d = 60 nm and l1 = l2 = l3 = 50 nm. (a) F along grain boundary AD; (b) F along grain boundary BD; (c) F along grain boundary CD; (d) model and geometry parameters in this calculation.

2.3. Results and discussions of nanocrack nucleation along grain boundaries 2.3.1. Effect of the distance d on nanocrack nucleation The variation of the configurational force along grain boundaries versus lc for different distances is plotted in Fig. 5. For nanocrystalline aluminum [29], γ = 1.2 J·m−2 and γb = 0.5 J·m−2. The horizontal dashed line denotes the required energy for grain boundary cracking, that is 2γe = γ-2γb = 1.9 J·m−2. The results show that the configurational force for nanocrack nucleation decreases first and then increases with lc increasing. The configurational force increases with the distance d decreasing. This is because the shear stress near the crack tip is relatively large. In the following discussions, d will be taken as 60 nm due to the greater effect of the distance. The length of nanocrack nucleation can be determined by the equation F(lc) = 1.9 J·m−2. For d = 60 nm, the length of nanocrack nucleation is about 4 nm for AD, 5 nm for BD and 13 nm for CD. This means there is a greater possibility of nanocrack nucleation along grain boundary CD. 2.3.2. Effect of the characteristic abutting angle of triple junction on nanocrack nucleation As shown in Fig. 6(d), α is the characteristic abutting angle of the triple junction structure. The length of nanocrack nucleation for different α can be obtained by Fig. 6(a)–(c). The results in Fig. 6(a)–(c) show the length of nanocrack nucleation increases with the angle α increasing. This means the triple junction with a larger α is a more favorable structure for nanocrack nucleation. When α is relatively small, the dislocations from different grain boundaries cancel out during dislocation reaction, so the Burgers vector magnitude of the sessile dislocation is relatively small. The Burgers vector magnitude of the sessile dislocation increases with α increasing. Thus, larger Burgers vector magnitude causes a larger nanocrack to nucleate. The length of nanocrack nucleation is smaller than 1 nm at about α < 40°. In this case, a nanocrack cannot be nucleated. Actually, previous results [30] showed that dislocation splitting was a more possible deformation mechanism at a triple junction with a relatively small α. Different structures of grain boundaries have different deformation mechanisms, which affects the main crack propagation. The results also show a nanocrack is more likely to be nucleated along the grain boundary CD, which is consistent with the results in Fig. 5. This illustrates nanocracks always are nucleated and grow forward. For the grain boundary CD, the length of nanocrack 9

Engineering Fracture Mechanics 221 (2019) 106672

X. Li and X. Jiang 3.5

β=90° β=70° β=50° β=30° β=10°

3.0

F (J.m-2)

2.5 2.0 1.5 1.0 0.5 0

5

10

15

20

lc (nm) Fig. 7. Configurational force F along the grain boundary CD versus lc, the calculation parameters are a = 5000 nm, σ = 200 MPa, α1 = -20°, α2 = 60°, α3 = -40°, d = 60 nm and l1 = l2 = l3 = 50 nm.

nucleation is smaller than 8 nm at about α < 100°. But for α = 120°, the configurational force is always > 1.9 J·m−2, which means a nanocrack can be nucleated along CD and it will propagate continuously. In this case, the nanocrack maybe propagates into a relatively long crack. 2.3.3. Effect of the orientation β on nanocrack nucleation The variation of the configurational force for nanocrack nucleation along grain boundary CD versus lc for different β is plotted in Fig. 7. β is the orientation of the triple junction D, as shown in Fig. 2. According to the results in Figs. 5 and 6, a nanocrack is more likely to be nucleated along CD, so the configurational force along CD is only plotted in Fig. 7. The length of nanocrack nucleation is < 1 nm at β ≤ 30°, and it is about 2 nm at β = 50°, 7.5 nm at β = 70°, 4 nm at β = 90°. In this section, nanocrack nucleation due to grain boundary sliding is only considered. For the shear stress field near a crack tip under mode I load, the shear stress is relatively small when β is relatively small; it increases with β increasing and reaches its maximum at about β = 70°; and then it decreases. Grain boundary sliding is closely linked with local shear stress. So a larger nanocrack can be nucleated at the triple junction located at about β = 70°, and it cannot be nucleated at the triple junction located at about β ≤ 30°. For a main crack under mode I load, nanocracks are more likely to be nucleated on the upper and lower sides of the crack plane, instead of the front of the crack tip. Lattice dislocation generation is suppressed in nanocrystalline metals. In this case, grain boundary dislocation motion has a larger contribution to plastic deformation. In this section, nanocrack nucleation due to grain boundary sliding in nanocrystalline aluminum is studied. In fact, by the same way, the nanocrystalline ceramics can also be considered. In ceramics at room temperature, the plastic deformation is contributed mainly by grain boundary dislocations. The deformation mechanism of ceramics is similar to nanocrystalline metals. Thus, the theoretical solution in this paper can be applied to analyze crack nucleation due to grain boundary sliding in ceramics, and the material parameters need be only changed. 3. Nanocrack nucleation due to lattice dislocation pileup in microcrystalline metals 3.1. Theoretical modelling In microcrystalline metals, the dominant mechanism of plastic deformation is lattice dislocation generation, so the lattice dislocation generation is only considered in this section. As shown in Fig. 8, a two-dimensional model is considered and a microcrystalline metal contains a flat main crack under uniaxial tensile load. Edge dislocations are emitted from the crack tip and move along a slip plane. When moving dislocations encounter with a grain boundary or inclusion, the dislocations will be piled up. In fact, besides crack tips, there are many other dislocation sources in microcrystalline metals, such as Frank-Read dislocation source, grain boundaries and so on. As shown in Fig. 8, there are dislocations emitted from the crack tip in a grain and dislocation dipoles generated from Frank-Read dislocation sources in multiple grains. In this paper, the dislocations emitted from the crack tip is only considered. Certainly, this is an approximate treatment, but maybe it is reasonable to consider emitted dislocations in one grain. Because dislocation generation in other grains is expected to have much smaller effects on the crack growth: (1) dislocation generation in other grains does not contribute to crack blunting; (2) intensity of dislocation motion in other grains is smaller; (3) the shielding effect of dislocations on crack-tip stress field in other grains is smaller due to the longer distance to the crack tip. Therefore, the problem can be simplified into an infinite plane containing a crack and a dislocation pileup, as shown in Fig. 9. Next, the theoretical solution will be presented.

10

Engineering Fracture Mechanics 221 (2019) 106672

X. Li and X. Jiang

Fig. 8. Schematic of dislocation generation near the main crack tip in microcrystalline materials.

Fig. 9. Schematic of inclusion cracking due to lattice dislocation pileup.

3.2. Theoretical solution scheme In this section, the problem of inclusion cracking due to lattice dislocation pileup under uniaxial tensile load is considered. Compared with the length of the crack and dislocation pileup, the size of inclusions is assumed to be relatively small. So the effect of the inclusion on the stress field near the crack tip can be neglected. In doing so, the problem in Fig. 9 can be divided into two subproblems based on the distributed dislocation technique. The first one is an infinite plane without any defects under uniaxial tensile load. The second one is an infinite plane containing two dislocation strips without external loads. The first sub-problem has been solved in Section 2 and the second sub-problem has been presented in Appendix A. Thus, the total stress field can be obtained by superposition of this two sub-problems. In this quasi-equilibrium state, three conditions must be satisfied, that is: (1) the crack plane is free-traction; (2) the shear stress along the slip plane is equal to the lattice friction τf. (3) there is no net dislocation on the crack. For convenience, the crack and slip plane are marked as dislocation strips “1” and “2”, respectively. According to the conditions above, several equations can be established:

condition (1) condition (2)

¯ y1 y1 (x1) +

¯ x2 y2 (x2 ) +

condition (3)

=0

on the crack

¯ x1 y1 (x1) = 0 a a

sin

s cos s

B x1 ( 1)d 1 =

= a a

f

on the slip plane

B y1 ( 1)d 1 = 0

(21)

Because the numerical procedure of Eq. (21) is very similar to Eq. (5), the procedure is omitted. In this section, the microcrystalline aluminum is considered, and the material parameters are τf = 20 MPa, bl = 0.258 nm, ν = 0.34 and μ = 27 GPa [5,24]. Here, bl is the Burgers vector magnitude of lattice dislocations. Previous experiments [31–33] showed there was a dislocation-free zone between the dislocation zone and crack tip. The size of dislocation-free zone decreases with increasing applied load [34], and reaches a fixed value when the applied load is relatively large [35]. Hence the dislocation-free zone size is taken as 100 nm in present calculation. Dislocation density along the slip plane for different dislocation zone sizes is shown in Fig. 10. The vertical dashed lines denote the locations of the inclusion. As shown in Fig. 10, a large number of dislocations are piled up against the inclusion. The closer the distance to the inclusion is, 11

Engineering Fracture Mechanics 221 (2019) 106672

X. Li and X. Jiang

Dislocation density (m-1)

1.5x109

ds=900 nm ds=1900 nm 1.0x109

5.0x108

0.0

0

500

1000

1500

2000

Dislocation zone (nm) Fig. 10. Dislocation distribution of the pileup, the calculation parameters are a = 100 μm, σ = 50 MPa, and θs = 55°.

the larger the dislocation density is. Most of lattice dislocations are very close to the inclusion. For simplicity, an array of dislocations in the slip plane are approximately equivalent to an superdislocation located near the inclusion. The direction of Burgers vector of the superdislocation is parallel to the slip plane, and the magnitude is equal to Nsbl. Here, Ns is the number of dislocation in the slip plane, and it can be given by

Ns =

0.5ds 0.5ds

B x2 ( 2)d

(22)

2

Further, the problem can described as an infinite plane containing a crack and a superdislocation under uniaxial tensile load. This problem of an infinite plane containing a crack and an arbitrary dislocation has been solved in Section 2.2.2. By the same way, the configurational force for inclusion cracking can be obtained. Next, the results of inclusion cracking will be analyzed. 3.3. Results and discussions of inclusion cracking The configurational force for inclusion cracking versus lc along different directions βa is plotted in Fig. 11. The results show that the configurational force reaches its maximum at about βa = -10° or −30°. This means inclusion cracking along the directions of βa = -10° or −30° is most likely. According to previous results [36], there was a positive relationship between the surface energy and the modulus of elasticity, that was γ = Ea0/10. Here, E is the modulus of elasticity and a0 is the atomic plane spacing. Generally, the material with a larger elastic modulus has a larger surface energy. For microcrystalline aluminum, γ = 1.2 J·m−2, so it is expected that γ < 1.2 J·m−2 for a soft inclusion and γ > 1.2 J·m−2 for a hard inclusion. The variation of nanocrack length versus the surface energy of the inclusion is plotted in Fig. 12. βa is taken as −30°, because the 8

8

F (J.m-2)

7

βa=-90°

7

βa=-70°

6

6

5

5

4

4 50

βa=-50° βa=-30° βa=-10° 60

70

80

90

100

110

βa=10°

120

βa=30°

3

βa=50°

2

βa=70° βa=90°

1 0

0

200

400

600

800

1000

lc (nm) Fig. 11. Configurational force F versus lc, the calculation parameters are a = 100 μm, σ = 50 MPa, ds = 900 nm and θs = 55°. 12

Engineering Fracture Mechanics 221 (2019) 106672

Length of nanocrack nucleation (nm)

X. Li and X. Jiang 400 350 300 250 200 150 100 50 2

4

6

8

10

Surface energy of the inclusion (J.m ) -2

Fig. 12. Length of the inclusion cracking versus the surface energy of the inclusion, the calculation parameters are a = 100 μm, σ = 50 MPa, ds = 900 nm, βa = -30° and θs = 55°.

configurational force at about βa = -30° is maximum. For a soft inclusion (γ < 1.2 J·m−2), the length of nucleated crack is very large, so the dislocation pileup will cause the entire soft inclusion to crack. After the entire inclusion cracking, plastic deformation will generate further and dislocations will be emitted from the nucleated crack tip. These new emitted dislocations may be piled up at a new inclusion. If the intensity of dislocation pileup is enough, the new soft inclusion can be cracked. Actually, this process improves the ductility and fracture resistance due to the shielding effect of emitted dislocations. For a hard inclusion (γ > 1.2 J·m−2), the length of nucleated nanocrack is relatively small. For example, it is 70 nm at γ = 4 J·m−2, 48 nm at γ = 6 J·m−2 and 30 nm at γ = 8 J·m−2. Thus, for a hard inclusion with a relatively large surface energy, a small nanocrack can be nucleated in the inclusion. The lattice dislocation pileup is more difficult to cause the entire hard inclusion crack. The hard inclusion hinders the motion of lattice dislocations, so the crack-tip dislocation emission is suppressed. The mode of crack propagation is transformed from ductility to brittlement. The strength of the material is improved by hard inclusions, but the resistance of fracture is decreased. 4. Conclusions This paper analyzes the mechanisms of crack nucleation near a main crack tip in nano and micro crystalline materials. For nanocrystalline materials, nanocrack nuclaeation due to grain boundary sliding is considered. For microcrystalline materials, nanocrack nucleation induced by dislocation pileup is considered. Further, theoretical solutions are presented based on the distributed dislocation technique. The configurational force for crack nucleation is calculated and the length of nucleated crack is obtained by an energy criterion. Important conclusions are summarized as follows: (1) Nanocracks are more likely to be nucleated along the grain boundary CD, and nanocracks are preferentially nucleated at the triple junctions with a larger characteristic abutting angle. (2) A larger nanocrack can be nucleated at the triple junctions located at around β = 70°, and it is difficult to be nucleated at the triple junctions located at about β ≤ 30°. (3) Lattice dislocation pileup can cause the entire soft inclusion crack easily, while it can only cause a small nanocrack to generate in the hard inclusion. The theoretical models reveal the mechanisms of nanocrack nucleation in the fracture process of nano and micro crystalline materials. The results are helpful to analyze the behaviors of nanocrack nucleation at different triple junctions of grain boundaries in nanocrystalline materials and explain related fracture phenomenon and behaviors of microcrystalline metals containing inclusions. Declaration of Competing Interest The authors declared that there is no conflict of interest. Acknowledgement This work was supported by the National Natural Science Foundation of China (11472230) and Doctoral Innovation Fund Program of Southwest Jiaotong University (D-CX201836). The authors would like to thank Dr. A.G. Sheinerman for his suggestions and discussions.

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X. Li and X. Jiang

Appendix A .1. ASolution of an infinite plane containing two dislocation strips As shown in Fig. A1(a), the lengths of the two dislocation strips are 2li and 2lj, respectively. The distance between the centers of dislocation strips i and j is dij. θij and αij are called as the angle and orientation of the dislocation strip j, respectively, and their positive values are defined as anticlockwise directions. In order to obtain the stress filed, two coordinate systems need be established, as shown in Fig. A1(b). The solution of an infinite plane containing two arbitrarily oriented dislocation strips will be presented. The stress field in the coordinate system xi-oi-yi induced by an edge dislocation located at the point (ξi, 0) can be given by [23]

Fig. A1. Two arbitrarily located dislocation strips in an infinite plane: (a) related geometry parameters; (b) coordinate systems along dislocation strips. 2µ {b xi Gxi IJ ( i, ( + 1)

¯IJ ( i, x i , yi ) =

x i , yi ) + b yi Gyi IJ ( i, x i , yi )} IJ = xi yi or yi yi where

yi

Gxi yi yi (x i , yi , i ) =

1

Gxi xi yi (xi , yi , i ) =

xi r14

i

yi2 ]; Gyi yi yi (x i , yi , i ) =

2 i)

[(x i r4 [(x i

i)

2

xi r14

yi2 ]; Gyi xi yi (x i , yi , i ) =

r12 = (x i

i)

2

i

[(x i

yi

r 14

i)

[(x i

+ 3yi2 ]

2

i)

2

yi2 ]

+ yi2

(A1)

Here, μ is the shear modulus; κ is Kolosov’s constant; κ=(3-ν)/(1 + ν) for plane strain and κ = 3-4ν for plane stress (ν is the Poisson's ratio); b is the component of Burgers vector. The stress components along the dislocation strip j induced by the dislocation can be obtained by the coordinate transformation.

¯ IJij (x j , i, dij ,

ij ,

ij )

=

2µ [b xi Gxiji IJ (xj , i, ( + 1)

dij ,

ij ,

ij )

+ b yi Gyiji IJ (xj , i, dij ,

ij ,

ij )]

IJ = xj yj or yj yj ij ,

2 ij ) = Gxi xi xi (x i , yi , i )sin

where, 2 ij + Gxi yi yi (x i , yi , i )cos

ij

Gxi xi yi (x i , yi , i ) sin 2

ij

(xj , i, dij ,

ij ,

ij )

= Gyi xi xi (x i , yi ,

)sin2

)cos2

ij

Gyi xi yi (xi , yi , i ) sin 2

ij

Gxiji xj yj (xj , i, dij ,

ij ,

ij )

= 0.5[Gxi yi yi (x i , yi , i )

Gxi xi xi (x i , yi , i )] sin 2

ij

+ Gxi xi yi (x i , yi , i ) cos 2

ij

Gyiji xj yj

ij ,

ij )

= 0.5[Gyi yi yi (x i , yi , i )

Gyi xi xi (xi , yi , i )] sin 2

ij

+ Gyi xi yi (x i , yi , i ) cos 2

ij

Gxiji yj yj (xj , i, dij , Gyiji yj yj

(xj , i, dij ,

i

ij

x i = dij cos yi = dij sin

+ Gyi yi yi (x i , yi ,

ij

+ x j cos

ij

ij

+ xj sin

ij

i

(A2)

Here, denotes the stress along dislocation strip j induced by the dislocation located in dislocation strip i. Therefore, the stress along dislocation strip j induced by the total dislocations in dislocation strip i can be calculated by integrating from -li to li.

¯ ij

¯ IJij (xj , dij ,

ij ,

ij )

=

2µ ( + 1)

li li

[B xii ( i ) Gxiji IJ (xj , i, dij ,

ij ,

ij )

+ B yii ( i ) Gyiji IJ (x j , i, dij ,

IJ = x j yj or yj yj

ij ,

ij )]d i

(A3)

Here, B(ξi) is the dislocation density; its superscript denotes the dislocation strip and subscript denotes the component of Burgers vector. To obtain the solution of an infinite plane containing two dislocation strips, the stress ¯ ii , ¯ ji and ¯ jj need been derived. The 14

Engineering Fracture Mechanics 221 (2019) 106672

X. Li and X. Jiang

stress can be obtained by substituting related parameters into the parameters xj, dij, θij and αij in formula (A3), that is

¯ IJii (x i ) = ¯IJij (xj x i , dij ji ij x i , dij , ij ¯ IJ (x i ) = ¯IJ (xj ¯IJjj (xj ) = ¯ IJij (xj , dij

0, ij

0,

0,

ij

,

ij ij

0,

ij

0)

ij ij

ij )

(A4)

0)

So the stress components along dislocation strips i and j induced by the total dislocations in the two dislocation strips can be calculated by

¯IJ (x i ) = ¯ IJii (x i ) + ¯IJji (x i ) IJ = x i yi or yi yi on dislocation strip i ¯IJ (xj ) = ¯IJij (xj ) + ¯IJjj (xj ) IJ = xj yj or yj yj on dislocation strip j

(A5)

The stress components along dislocation strips have been given by formula (A5). For the problem of an infinite plane containing multiple dislocation strips, the stress components can be obtained by a similar superposition way.

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