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Influence of nanoscale deformation twins near a slant edge crack tip on crack blunting in nanocrystalline metals
Accepted Manuscript Influence of nanoscale deformation twins near a slant edge crack tip on crack blunting in nanocrystalline metals and ceramics Tengwu He, Miaolin Feng PII: DOI: Reference:
S0013-7944(17)30709-9 http://dx.doi.org/10.1016/j.engfracmech.2017.08.007 EFM 5641
To appear in:
Engineering Fracture Mechanics
Received Date: Revised Date: Accepted Date:
6 July 2017 1 August 2017 1 August 2017
Please cite this article as: He, T., Feng, M., Influence of nanoscale deformation twins near a slant edge crack tip on crack blunting in nanocrystalline metals and ceramics, Engineering Fracture Mechanics (2017), doi: http:// dx.doi.org/10.1016/j.engfracmech.2017.08.007
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Influence of nanoscale deformation twins near a slant edge crack tip on crack blunting in nanocrystalline metals and ceramics Tengwu Hea, Miaolin Fenga,* a
State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China
*
Corresponding author: [email protected]; Tel: +862134204539; fax: +862134206334
Abstract A theoretical model is developed that discusses the effect of nanoscale deformation twins on dislocation emission from the tip of slant edge crack in nanocrystalline materials. By combining complex variable method of Muskhelishvili and conformal mapping technique, the explicit solutions of complex potentials are obtained analytically. The critical stress intensity factors (SIFs) for the emission of first lattice edge dislocation from a slant edge crack tip are calculated. The effects of vital parameters such as the crack length, the inclined angle of the crack, the relative thickness of the nanotwin on critical SIFs for dislocation emission are evaluated in detail. The results show that the emission of lattice dislocation from the slant edge crack tip is significantly influenced by nanoscale deformation twinning. The smaller the inclined angle is, the more difficult it is for dislocation emission from the slant edge crack tip. Particularly, the dislocation near the tip of slant edge crack is prone to emit when the inclined angle of the crack is about 45。. As a special case, when the inclined angle =0, the present results will reduce to those of the problem of nanoscale deformation twins interacting with mode I straight crack. Keywords Slant edge crack, Deformation twins, Dislocation emission, Conformal mapping, Nanocrystalline materials, Stress intensity factors
1
1. Introduction Nanocrystalline metals and ceramics are well known for their superior strength, strong hardness, excellent wear resistance, and display widespread and important prospect on micro/nano-technological applications [1-12]. On the other hand, the novel properties are at the expense of extremely low plasticity and cracking resistance at room temperature, which significantly narrows the fields of their practical utilization. However, some recent examples of nanocrystalline materials that combine high strength and high plasticity have been reported [13-16], which evoke tremendous interest of many researchers in the fundamental nature of the special toughening mechanism. Whereafter, plenty of theoretical models such as the local migration of grain boundaries[17], deformation twinning[18], rotation deformation[19, 20] and grain boundary sliding[21, 22] have been developed to elucidate the special deformation mechanism in nanocrystalline materials. In recent years, particular attention has been focused on nanoscale deformation twinning within grains, which effectively operates in nanocrystalline materials with various chemical compositions and structures. Numerous experiments [23], theoretical models [24] and computer simulations [25] have demonstrated that nanoscale deformation twinning can make large contribution to plastic flow in nanocrystalline materials. The nanoscale deformation twinning that often occurs in deformed nanocrystalline materials is initiated mainly through three ways [26-28]: (i) the multiplane nanoscale shear; (ii) the successive events of partial dislocation emission from grain boundaries; (iii) the cooperative emission of partial dislocations
2
from grain boundaries. The above-mentioned works mainly concentrated on the nucleation and growth of nanoscale deformation twins in nanocrystalline materials. However, for nanocrystalline materials with crack defects, when the level of local stresses near the crack tip exceeds that of yield stresses, plastic shear will be induced through the emission of lattice dislocations from the crack tip. The emission of lattice dislocations along a slip plane can give rise to blunting of the crack tip, and further prevent crack propagation and effectively improve fracture toughness of nanocrystalline materials. Hence, it is quite meaningful to study the effects of nanoscale deformation twinning on the dislocation emission from the crack tip. Morozov et al. [29] investigated the influence of the nanotwin generation near crack tip on the fracture toughness of nanomaterials. Ovid'ko and Sheinerman [30] revealed the critical parameters for crack generation at deformation twins stopped by grain boundaries. Li et al. [31] proposed a theoretical model to address the effect of the secondary twin lamellae on the emission of lattice dislocations from the tip of semi-infinite crack in nanotwinned metals. In general, they always simplified the models of cracks to mode I cracks such as straight or flat cracks. Nevertheless, real cracks in nanocrystalline solids are relatively complicated, and their shapes are various. In fact, to some extent, the slant edge crack related to the inclined angle shows a more precise description of real crack as comparing to the straight crack in nanocrystalline materials. In this paper, we aim to study the effect of nanoscale deformation twins near the tip of slant edge crack on the dislocation emission in deformed nanocrystalline metals and ceramics.
3
2. Model and problem formulation Consider the system of nanoscale deformation twinning near the tip of an edge crack of length l situated at an angle (-1/2< <1/2) from the normal of the free surface y=0 in a semi-infinite nanocrystalline specimen x 0 under far-field mode I loadings, as shown in Fig. 1(a). For simplicity, we make some assumptions that the nanocrystalline sample is elastic and isotropic with shear modulus μ and Poisson’s ratio ν, and the defect structure of the solid remains unchanged along the coordinate axis z perpendicular to the x-y plane. We can reduce the mathematical analysis of the problem to the consideration of a two-dimensional structure. Based on the two-dimensional model, we consider a rectangular nanotwin BCDE, and the tip of slant edge crack is supposed to reach the boundary of the twin band, as depicted in Fig. 1(a). For convenience of analysis, we introduce two Cartesian systems ( x, y ), ( x, y ) and a polar coordinate system ( r , ), whose origins are located at L and A, respectively, as shown in Fig. 1(b). Following the theory of disclinations [32], the nanotwin BCDE can be modeled by a quadrupole of wedge disclinations characterized by the same strength of opposite signs
at
z1 lei h / 2ei( /2) , z2 lei h / 2ei( /2) de i , z4 lei h / 2ei( /2 ) and z3 lei h / 2ei( /2 ) dei , respectively. On the other hand, the length, the thickness and the orientation of the nanoscale deformation twin are denoted as d , h and , respectively.
4
y
y
l
N
C
d
B r A E
x
h D
x
L
(a)
(b)
N(0, )
C
D
B A
E
L(0, 1)
(c) Fig. 1 The formation of nanotwins near a slant edge crack tip in a semi-infinite deformed nanocrystalline solid (a) general view (b) the magnified inset highlighting a disclination quadrupole and a lattice dislocation near the tip of slant edge crack (c) The ζ-plane after conformal mapping
For the plane strain problem, the stress fields can be described in terms of two Muskhelishvili’s complex potentials ( z ) and ( z ) [33]
xx yy 2[ (z) (z)]
yy i xy (z) (z) z (z) (z)
(1) (2)
where z x iy , the over-bar represents the complex conjugate of a function, and the prime means differentiation with respect to the argument z . 5
In order to deal with the current problem, the mapping function from the plane in Fig. 1(c) to the z plane in Fig. 1(b) is introduced [34]
z ( )
l
1/2
( i)1/2 ( i )1/2
(3)
with the aid of the mapping function, the surrounding region of the slant edge crack in z-plane is mapped onto the right half of the ζ-plane ( 0 ), where i , as shown in Fig. 1(c). In addition, points L(0,0 ) , N (0, 0 ) and A(l cos , l sin ) in the z-plane correspond to be mapped to points L(0, 1) , N (0, ) and A(0,0) in the plane where (1 2 ) / (1 2 ) . By means of the mapping function in Eq. (3), Eqs. (2) and (3) can be rewritten in the ζ-plane as
xx yy 2[( ) ( )]
yy i xy ( ) ( ) where
( ) ( ) ( ) ( )
( ) ( ) / ( ) , ( ) [ ( )( ) ( )( )] / [( )]2
(4) (5)
and
( ) ( ) / ( ) .
3. The force on lattice dislocation emission from the slant edge crack tip For the emission of lattice dislocations from the tip of slant edge crack, we consider a typical situation where the dislocations are of edge character and their Burgers vectors lie along the slip plane making an angle with x'-axis. The force acting on the edge dislocation consists of three parts: (1) the force produced by nanoscale deformation twinning; (2) the image force arising from the slant edge crack; 6
(3) the external loads. Firstly, we calculate the force exerted on the dislocation due to the deformation twinning. Referring to the works of Romanov and Vladimirov [35] and He et al.[20], the elastic fields caused by two wedge disclination dipoles characterizing the deformation twinning can be expressed by the following complex potentials
w ( z ) and w ( z )
w (z)
D 4 (1)k +1 (z zk ) ln(z zk ) w0 (z) 2 k 1
(6)
D 4 (1)k +1 zk ln(z z k ) w0 (z) 2 k 1
(7)
w (z)
where D / [2 (1 )] , w0 (z) and w0 (z) represent the terms due to the interaction of the two wedge disclination dipoles and the slant edge crack in the z-plane. With the substitution of Eq. (3) into Eqs. (6) and (7), the complex potentials in the ζ-plane can be written as
w ( )
D 4 (1)k 1 ( k )( k ) ln( k ) w0 ( ), 0 2 k 1
w ( )
D 4 (1)k 1 zk ln( k ) w0 ( ), 0 2 k 1
(8)
(9)
According to the Riemann-Schwarz symmetry theory, we take into account a new auxiliary function as follows
w ( ) ( )
w ( ) w ( ), 0 ( )
The traction-free boundary condition along the imaginary axis is treated as
7
(10)
w ( )
( ) w ( ) w ( ) 0 ( )
(11)
To solve the free boundary problem on the interface in Eq. (11), it is essential to introduce the following function
w ( ) w ( )
D 4 (1)k 1[ ( ) ln( k ) zk ln( k )] w0 ( ), 0 2 k 1
(12)
In view of Eqs. (10)-(12), we obtain
w (t ) w (t )
(13)
where t denotes the point in the ζ-plane on the imaginary axis. The superscripts and “ ” and “ ” refer to the boundary value as approached from the respective region occupied by the right half ( 0 ) and the left half ( 0 ). Applying the Plemeij formula, we can obtain
w ( )
D 4 (1)k 1[( k )( k ) ln( k ) ( ) ln( k ) zk ln( k )] 2 k 1 (14)
In combination with Eq. (14) and Eqs. (10)-(13), the complex potential w ( ) can be given as
D 4 (1) k 1[ ( k )( k ) ln( k ) zk ln( k )] 2 k 1 zk ( ) D 4 ( ) ( 1) k 1[ ( ) ln( k ) ( k ) ] ( ) 2 k 1 k k
w ( )
(15) The force acting on the edge dislocation caused by two wedge disclination dipoles characterizing the deformation twin can be calculated by means of the Peach-Koehler formula [36] 8
f w f wx if wy xy ( 0 )bx yy ( 0 )by i xx ( 0 )bx xy ( 0 )by (by2 bx2 ) w ( 0 ) w ( 0 ) ( 0 ) w ( 0 ) w ( 0 ) 4 (1 )
(16)
where xx , xy and yy are the components of the perturbation stress produced by two wedge disclination dipoles in the process of deformation twinning, and
( ) w 0 ( ) w ( 0 ) lim w 0 ( ) ( ) ( ) ( ) w ( ) ( ) w0 ( ) ( ) w 0 ( ) ( ) w ( 0 ) lim w 0 [ ( )]3 [ ( )]3 ( ) w 0 ( ) w ( 0 ) lim w 0 ( ) ( ) Secondly, suppose that the first edge dislocation emitted from the tip of slant edge crack is located at z0 lei in the coordinate system. According to the works of Fang and Liu [37], the elastic field of edge dislocation in the ζ-plane can be described by two complex potentials
s ( ) ln( 0 ) s 0 ( )
s ( ) ln( 0 )
K s 0 ( ) 0
(17) (18)
where (by ibx ) / [4 (1 )] and K ( 0 ) / ( 0 ) . Utilizing the similar technique above, we achieve
s 0 ( ) ln( 0 )
s 0 ( ) ln( 0 )
M 0
( ) M [ ] ( ) 0 ( 0 )2
(19)
(20)
where M [ ( 0 ) ( 0 )] / ( 0 ) . By using the Peach-Koehler formula [36], the image force can be evaluated as
9
f s f sx if sy ˆ xy ( 0 )bx ˆ yy ( 0 )by i ˆ xx ( 0 )bx ˆ xy ( 0 )by (by2 bx2 ) s 0 ( 0 ) s 0 ( 0 ) ( 0 )s 0 ( 0 ) s 0 ( 0 ) 4 (1 )
(21)
where ˆ xx , ˆ xy and ˆ yy are the components of stress field resulting from the interaction between the edge dislocation and the free surface of the slant edge crack,
s 0 ( 0 ) s0 ( ) / ( ) , s 0 ( 0 ) s0 ( 0 )( 0 ) s0 ( 0 )( 0 ) / [( 0 )]3
and
s 0 ( 0 ) s0 ( 0 ) / ( 0 ) . Thirdly, the applied load exerted on the edge dislocation can be expressed by the following equation
f b r
b (1Kapp 2 Kapp ) 2 r
(22)
1 3 where 1 sin cos , 2 cos sin2 cos , K app and K app are the 2 2 2 2 2 generalized mode I and mode II SIFs due to the remote loadings, b ( bx iby ) is the Burgers vector of the first edge dislocation. Lastly, the dislocation emission force near the tip of slant edge crack can be calculated by using the superposition principle
f emit f x cos( ) f y sin( ) f Re[ f w f s ]cos( ) Im[ f w f s ]sin( ) f
(23)
where
(by2 bx2 ) 2 Re[ w ( 0 ) s 0 ( 0 )] ( 0 )[ w ( 0 ) s 0 ( 0 )] [ w ( 0 ) s 0 ( 0 )] fw fs 4 (1 )
4. The critical SIFs for the dislocation emission It is a broadly accepted criterion that a new dislocation can be spontaneously emitted from the tip of a crack under the condition that the force acting on it is equal
10
to or larger than zero, and the dislocation distance to the crack free surface is not smaller than the dislocation core radius r0 [38]. Using the Eqs. (14)-(23) together with f emit 0 , we can obtain the following critical mode I and II SIFs for dislocation emission Kapp 0, Kapp
2 r (Im[ f w f s ]sin( ) Re[ f w f s ]cos( )) b 2
(24)
2 r (Im[ f w f s ]sin( ) Re[ f w f s ]cos( )) b1
(25)
for mode II crack, and Kapp 0, Kapp
for mode I crack. It is worth mentioning that when the inclined angle =0, the present results will reduce to those of the problem of nanoscale deformation twins interacting with mode I straight crack. We have completely derived the analytical solutions to the current problem. Then, the effect of deformation twinning on dislocation emission from the tip of slant edge crack can be examined by evaluating the critical SIFs in deformed nanocrystalline materials. In the following calculation, we mainly discuss the influences of typical parameters such as the crack length, the inclined angle of the crack and the relative thickness of the nanoscale twin band on the dislocation emission from the slant edge crack tip. For simplicity, we define the dimensionless critical SIFs as app Kapp / ( b ) c K
and
app Kapp c K / ( b )
.
Additionally,
the
typical
nanocrystalline material 3C-SiC with =217Gpa, =0.23 is taken as an illustrative example. As a makeshift, the Burgers vector of the edge dislocation and the distance
11
between the tip of slant edge crack and the first emitted dislocation are chosen as b =0.25nm and the core radius of dislocation r0 (= b / 2 ), respectively. app The variations of normalized critical SIFs K ICapp and K IIC versus the dislocation
emission angle 0 with different crack length l are depicted in Figs. 2 and 3, respectively. From Fig. 2, the normalized critical mode I SIFs decrease from a smaller value to negative infinity, sharply shift to positive infinity, then reduce to a minimum, and finally increase to positive infinity. According to the work of Huang and Li [39], the sign of the SIFs depends on the direction of the Burgers vector of the emerging dislocations, which makes the normalized critical SIFs positive or negative. The most probable angle for negative dislocation emission from the slant edge crack tip is always approximately -46。，while for positive dislocation emission it is around 14.5 。, which are both independent on the crack length. Furthermore, it will be more difficult for the dislocation to emit from the slant edge crack tip with increasing the crack length. For the normalized critical mode II SIFs in Fig. 3, it can be observed that the normalized critical SIFs first increase slowly, and then sharply rise with the increment of dislocation emission angle, which implies that it may be easier for negative dislocation to emit from the slant edge crack tip than that of positive dislocation. Besides, it is also more difficult for the dislocation to emit from the tip of slant edge crack with raising the crack length.
12
0.6
l=50nm l=100nm l=200nm
0.4
KIC
app
0.2
0.0
-0.2
-40
-20
0
20
40
(degree)
Fig. 2 Dependence of the normalized critical mode I SIFs on dislocation emission angle θ0 with different crack length l (α=1/5, η=/4, =/6, h=4nm, d=15nm) 0.10
0.08
KIIC
app
0.06
l=50nm l=100nm l=200nm
0.04
0.02
0.00 -20
0
degree
20
40
Fig. 3 Dependence of the normalized critical mode II SIFs on dislocation emission angle θ0 with different crack length l (α=1/5, η=/4, =/6, h=4nm, d=15nm)
The normalized critical mode I and mode II SIFs with respect to the dislocation emission angle 0 with different values of are illustrated in Figs. 4 and 5. For the normalized critical SIFs in Fig. 4, the normalized critical mode I SIFs decrease from a smaller value to negative infinity, dramatically turn to positive infinity, then reduce to a minimum, and finally increase to positive infinity. The most probable angle for negative dislocation emission is always about -30。，while for positive 13
dislocation emission it is nearly 18 。，14.7 。 and 10 。 for =1/4, 1/5 and 1/7, respectively. We can see that the smaller the value of is, the more difficult it is for dislocation emission from the slant edge crack tip. Particularly, when =1/4, the dislocation near the tip of slant edge crack is most prone to emit. For the normalized critical mode II SIFs in Fig. 5, it is found that the normalized critical SIFs first increase slightly, and then abruptly rise with increasing dislocation emission angle, which indicates that it will be easier for negative dislocation to emit from the slant edge crack tip than that of positive dislocation. Additionally, it can be concluded that the smaller the value of is, the more difficult it is for dislocation emission from the slant edge crack tip. Specially, the dislocation near the tip of slant edge crack is prone to emit when =1/4. 1.0 =1/7 =1/5 =1/4
KIC
app
0.5
0.0
-0.5
-1.0
-20
-10
0
10
20
30
40
50
(degree)
Fig. 4 Dependence of the normalized critical mode I SIFs on dislocation emission angle θ0 with different (l=100nm, η=/4, =/6, h=4nm, d=15nm)
14
0.20
0.15 =1/7 =1/5 =1/4
KIIC
app
0.10
0.05
0.00
-0.05 -20
-10
0
10
20
0(degree)
Fig. 5 Dependence of the normalized critical mode II SIFs on dislocation emission angle θ0 with different (l=100nm, η=/4, =/6, h=4nm, d=15nm) 0.8 0.6
h/d=0.1 h/d=0.2 h/d=0.3
0.4
KIC
app
0.2 0.0 -0.2 -0.4 -0.6 -0.8
-20
-10
0
10
20
0(degree)
Fig. 6 Dependence of the normalized critical mode I SIFs on dislocation emission angle θ0 with different h/d (l=100nm, η=/4, =/6, α=1/7, d=15nm)
15
0.6
0.4
Al Ni 3C-SiC
KIC
app
0.2
0.0
-0.2
-0.4
-0.6
-20
-10
0
10
20
0(degree)
Fig. 7 Dependence of the normalized critical mode I SIFs on dislocation emission angle θ0 with different nanocrystalline materials (l=100nm, η=/4, =/6, α=1/7, d=15nm, h=4nm)
Figure 6 plots the variations of the normalized critical mode I SIFs versus the dislocation emission angle with different relative length h/d between the length and the thickness of the twin band. It is found that the most probable angle for negative dislocation emission is always -30。，while for positive dislocation emission it is close to 10.1。. Both angles for dislocation emission are independent of the relative length. At the same time, it can be observed that the larger the relative length is, the easier it is for dislocation emission from the slant edge crack tip, which suggests that nanotwin near the tip of slant edge crack can increase the fracture toughness of brittle nanocrystalline materials. The dependence of the normalized critical mode I SIFs on dislocation emission angle with different nanocrystalline materials is shown in Fig. 7. It means that the nanotwin near the tip of slant crack tip can improve the toughness of nanocrystalline material 3C-SiC more remarkably as comparing to that of nanocrytalline materials Ni and Al.
5. Conclusions 16
The problem of nanoscale deformation twins interacting with a slant edge crack is investigated by establishing a theoretical model in nanocrystalline materials. Using complex variable method of Muskhelishvili and a conformal mapping function, the expression of the critical SIFs for the first lattice dislocation emission from the tip of slant edge crack are obtained. The influences of important parameters such as the crack length, the inclined angle of the crack, the relative length of the twin band on critical SIFs for dislocation emission are discussed at length. Some main conclusions are summarized as follows (1) It is more difficult for the dislocation to emit from the slant edge crack tip with increasing the crack length when the inclined angle of the crack is certain. (2) The smaller the inclined angle of the crack is, the more difficult it is for dislocation emission from the slant edge crack tip. Particularly, when the inclined angle of the crack is about 45。, the dislocation near the tip of slant edge crack is prone to emit. (3) The nanotwin near the tip of slant crack tip can increase the toughness of nanocrystalline material 3C-SiC more remarkably as comparing to that of nanocrytalline materials Ni and Al.
Acknowledgments The authors would like to deeply appreciate the support from the National Natural Sciences Foundation of China (11572191 and 51601112) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (20130073110057). 17
References [1] Kuntz JD, Zhan GD, Mukherjee AK. Nanocrystalline-Matrix Ceramic Composites for Improved Fracture Toughness. MRS Bulletin. 2004;29:22-7. [2] Yang F, Yang W. Crack growth versus blunting in nanocrystalline metals with extremely small grain size. Journal of the Mechanics & Physics of Solids. 2009;57:305-24. [3] Aifantis EC. Deformation and failure of bulk nanograined and ultrafine-grained materials. Materials Science & Engineering A. 2009;503:190-7. [4] Weng GJ. A homogenization scheme for the plastic properties of nanocrystalline materials. Reviews on Advanced Materials Science. 2009;19:41-62. [5] Wei Y, Bower AF, Gao H. Enhanced strain-rate sensitivity in fcc nanocrystals due to grain-boundary diffusion and sliding. Acta Materialia. 2008;56:1741-52. [6] Gurtin ME, Anand L. Nanocrystalline grain boundaries that slip and separate: A gradient theory that accounts for grain-boundary stress and conditions at a triple-junction. Journal of the Mechanics & Physics of Solids. 2008;56:184-99. [7] Barai P, Weng GJ. The competition of grain size and porosity in the viscoplastic response
of
nanocrystalline
solids.
International
Journal
of
Plasticity.
2008;24:1380-410. [8] Barai P, Weng GJ. Mechanics of creep resistance in nanocrystalline solids. Acta Mechanica. 2008;195:327-48. [9] Mukhopadhyay A, Basu B. Consolidation–microstructure–property relationships in bulk nanoceramics and ceramic nanocomposites: a review. International Materials
18
Reviews. 2007;52:257-88. [10] Dao M, Lu L, Asaro RJ, Hosson JTMD, Ma E. Toward a quantitative understanding of mechanical behavior of nanocrystalline metals. Acta Materialia. 2007;55:4041-65. [11] Wolf D, Yamakov V, Phillpot SR, Mukherjee A, Gleiter H. Deformation of nanocrystalline materials by molecular-dynamics simulation: relationship to experiments? Acta Materialia. 2005;53:1-40. [12] Ovid'Ko IA. Deformation and Diffusion Modes in Nanocrystalline Materials. International Materials Reviews. 2005;50:65-82. [13] Ovid’ko IA, Sheinerman AG. Special strain hardening mechanism and nanocrack generation in nanocrystalline materials. Applied Physics Letters. 2007;90:171927--3. [14] Ovid’Ko IA, Sheinerman AG, Aifantis EC. Stress-driven migration of grain boundaries and fracture processes in nanocrystalline ceramics and metals. Acta Materialia. 2008;56:2718-27. [15] Zhao Y, Qian J, Daemen LL, Pantea C, Zhang J, Voronin GA, et al. Enhancement of fracture toughness in nanostructured diamond–SiC composites. Applied Physics Letters. 2004;84:1356-8. [16] Youssef KM, Scattergood RO, Murty KL, Koch CC. Nanocrystalline Al–Mg alloy with ultrahigh strength and good ductility. Scripta Materialia. 2006;54:251-6. [17] Kolesnikova AL, Ovid’Ko IA, Fedorov AA. Local migration of grain boundaries in polycrystalline materials under plastic deformation conditions. Technical Physics Letters. 2003;29:488-90.
19
[18] Ovid'Ko IA, Sheinerman AG. DEFORMATION TWINNING THROUGH NANOSCALE
IDEAL SHEARS
IN
NANO-
AND
POLYCRYSTALLINE
MATERIALS AT ULTRA HIGH STRESSES. 2011;27:189-94. [19] Ovid'Ko IA, Sheinerman AG. Nanoscale rotational deformation near crack tips in nanocrystalline solids. Journal of Physics D Applied Physics. 2012;45:335301-8(8). [20] He T, Xiao W, Zhang Y, Zhu H. Effect of special rotational deformation on the dislocation emission from a branched crack tip in deformed nanocrystalline materials. Acta Mechanica. 2017;228:823-36. [21] Bobylev SV, Mukherjee AK, Ovid’Ko IA, Sheinerman AG. Effects of intergrain sliding on crack growth in nanocrystalline materials. International Journal of Plasticity. 2010;26:1629-44. [22] He T, Xiao W, Zhang Y, Zhu H. Effect of cooperative grain boundary sliding and migration on dislocation emission from a branched crack tip in deformed nanocrystalline solids. International Journal of Fracture. 2017;206:1-10. [23] Zhu YT, Liao XZ, Wu XL. Deformation twinning in bulk nanocrystalline metals: Experimental observations. JOM. 2008;60:60. [24] Wu XL, Zhu YT. Partial-dislocation-mediated processes in nanocrystalline Ni with nonequilibrium grain boundaries. Applied Physics Letters. 2006;89:58. [25] Wang YM, Hodge AM, Biener J, Hamza AV. Deformation twinning during nanoindentation of nanocrystalline Ta. Applied Physics Letters. 2005;86:44. [26] Ovid, Apos IA, ko. Nanoscale multiplane shear and twin deformation in nanowires and nanocrystalline solids. Applied Physics Letters. 2011;99:1403.
20
[27] Ovid’Ko IA, Skiba NV. Generation of nanoscale deformation twins at locally distorted grain boundaries in nanomaterials. International Journal of Plasticity. 2014;62:50-71. [28] Ovid’Ko IA, Skiba NV. Nanotwins induced by grain boundary deformation processes in nanomaterials. Scripta Materialia. 2014;71:33-6. [29] Morozov NF, Ovid’Ko IA, Skiba NV, Sheinerman AG. Influence of nanotwin generation near crack twins on the fracture toughness of nanomaterials. Doklady Physics. 2013;58:544-7. [30] Ovid'ko IA, Sheinerman AG. Generation of Nanocracks at Deformation Twins in Nanomaterials. Materials Research Letters. 2013;1:168-73. [31] Li J, Ni Y, Soh AK, Wu XL. Strong Crack Blunting by Hierarchical Nanotwins in Ultrafine/Nano-grained Metals. Materials Research Letters. 2015;3:190-6. [32] Romanov AE, Vladimirov VI. Disclinations in solids. Physica Status Solidi. 1983;78:11-34. [33] Muskhelishvili NI. Some Basic Problems of the Mathematical Theory of Elasticity. Mathematical Gazette. 1955;48:351. [34] Wang SD, Lee S. Screw dislocation near a slant edge crack. Mechanics of Materials. 1999;31:63-70. [35] Romanov A, Vladimirov V. Dislocations in Solids vol 9, ed FRN Nabarro. Amsterdam: North-Holland; 1992. [36] Hirth JP, Lothe J. Theory of dislocations: J. Wiley; 1968. [37] Fang QH, Liu YW. Size-dependent interaction between an edge dislocation and a
21
nanoscale inhomogeneity with interface effects. Acta Materialia. 2006;54:4213-20. [38] Rice JR, Thomson R. Ductile versus brittle behaviour of crystals. Philosophical Magazine. 1974;29:73-97. [39] Huang M, Li Z. Dislocation emission criterion from a blunt crack tip. Journal of the Mechanics & Physics of Solids. 2004;44:1991-2003.
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Highlights 1. Interaction between a slant edge crack and nanoscale deformation twin 2. Expressions of the critical SIFs for the first lattice dislocation emission from the slant edge crack tip are obtained 3. Analyze the effects of the crack length, the inclined angle of the crack, the relative thickness of nanoscale twin band and different nanocrystalline materials on the SIFs.
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S0013-7944(17)30709-9 http://dx.doi.org/10.1016/j.engfracmech.2017.08.007 EFM 5641
To appear in:
Engineering Fracture Mechanics
Received Date: Revised Date: Accepted Date:
6 July 2017 1 August 2017 1 August 2017
Please cite this article as: He, T., Feng, M., Influence of nanoscale deformation twins near a slant edge crack tip on crack blunting in nanocrystalline metals and ceramics, Engineering Fracture Mechanics (2017), doi: http:// dx.doi.org/10.1016/j.engfracmech.2017.08.007
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Influence of nanoscale deformation twins near a slant edge crack tip on crack blunting in nanocrystalline metals and ceramics Tengwu Hea, Miaolin Fenga,* a
State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China
*
Corresponding author: [email protected]; Tel: +862134204539; fax: +862134206334
Abstract A theoretical model is developed that discusses the effect of nanoscale deformation twins on dislocation emission from the tip of slant edge crack in nanocrystalline materials. By combining complex variable method of Muskhelishvili and conformal mapping technique, the explicit solutions of complex potentials are obtained analytically. The critical stress intensity factors (SIFs) for the emission of first lattice edge dislocation from a slant edge crack tip are calculated. The effects of vital parameters such as the crack length, the inclined angle of the crack, the relative thickness of the nanotwin on critical SIFs for dislocation emission are evaluated in detail. The results show that the emission of lattice dislocation from the slant edge crack tip is significantly influenced by nanoscale deformation twinning. The smaller the inclined angle is, the more difficult it is for dislocation emission from the slant edge crack tip. Particularly, the dislocation near the tip of slant edge crack is prone to emit when the inclined angle of the crack is about 45。. As a special case, when the inclined angle =0, the present results will reduce to those of the problem of nanoscale deformation twins interacting with mode I straight crack. Keywords Slant edge crack, Deformation twins, Dislocation emission, Conformal mapping, Nanocrystalline materials, Stress intensity factors
1
1. Introduction Nanocrystalline metals and ceramics are well known for their superior strength, strong hardness, excellent wear resistance, and display widespread and important prospect on micro/nano-technological applications [1-12]. On the other hand, the novel properties are at the expense of extremely low plasticity and cracking resistance at room temperature, which significantly narrows the fields of their practical utilization. However, some recent examples of nanocrystalline materials that combine high strength and high plasticity have been reported [13-16], which evoke tremendous interest of many researchers in the fundamental nature of the special toughening mechanism. Whereafter, plenty of theoretical models such as the local migration of grain boundaries[17], deformation twinning[18], rotation deformation[19, 20] and grain boundary sliding[21, 22] have been developed to elucidate the special deformation mechanism in nanocrystalline materials. In recent years, particular attention has been focused on nanoscale deformation twinning within grains, which effectively operates in nanocrystalline materials with various chemical compositions and structures. Numerous experiments [23], theoretical models [24] and computer simulations [25] have demonstrated that nanoscale deformation twinning can make large contribution to plastic flow in nanocrystalline materials. The nanoscale deformation twinning that often occurs in deformed nanocrystalline materials is initiated mainly through three ways [26-28]: (i) the multiplane nanoscale shear; (ii) the successive events of partial dislocation emission from grain boundaries; (iii) the cooperative emission of partial dislocations
2
from grain boundaries. The above-mentioned works mainly concentrated on the nucleation and growth of nanoscale deformation twins in nanocrystalline materials. However, for nanocrystalline materials with crack defects, when the level of local stresses near the crack tip exceeds that of yield stresses, plastic shear will be induced through the emission of lattice dislocations from the crack tip. The emission of lattice dislocations along a slip plane can give rise to blunting of the crack tip, and further prevent crack propagation and effectively improve fracture toughness of nanocrystalline materials. Hence, it is quite meaningful to study the effects of nanoscale deformation twinning on the dislocation emission from the crack tip. Morozov et al. [29] investigated the influence of the nanotwin generation near crack tip on the fracture toughness of nanomaterials. Ovid'ko and Sheinerman [30] revealed the critical parameters for crack generation at deformation twins stopped by grain boundaries. Li et al. [31] proposed a theoretical model to address the effect of the secondary twin lamellae on the emission of lattice dislocations from the tip of semi-infinite crack in nanotwinned metals. In general, they always simplified the models of cracks to mode I cracks such as straight or flat cracks. Nevertheless, real cracks in nanocrystalline solids are relatively complicated, and their shapes are various. In fact, to some extent, the slant edge crack related to the inclined angle shows a more precise description of real crack as comparing to the straight crack in nanocrystalline materials. In this paper, we aim to study the effect of nanoscale deformation twins near the tip of slant edge crack on the dislocation emission in deformed nanocrystalline metals and ceramics.
3
2. Model and problem formulation Consider the system of nanoscale deformation twinning near the tip of an edge crack of length l situated at an angle (-1/2< <1/2) from the normal of the free surface y=0 in a semi-infinite nanocrystalline specimen x 0 under far-field mode I loadings, as shown in Fig. 1(a). For simplicity, we make some assumptions that the nanocrystalline sample is elastic and isotropic with shear modulus μ and Poisson’s ratio ν, and the defect structure of the solid remains unchanged along the coordinate axis z perpendicular to the x-y plane. We can reduce the mathematical analysis of the problem to the consideration of a two-dimensional structure. Based on the two-dimensional model, we consider a rectangular nanotwin BCDE, and the tip of slant edge crack is supposed to reach the boundary of the twin band, as depicted in Fig. 1(a). For convenience of analysis, we introduce two Cartesian systems ( x, y ), ( x, y ) and a polar coordinate system ( r , ), whose origins are located at L and A, respectively, as shown in Fig. 1(b). Following the theory of disclinations [32], the nanotwin BCDE can be modeled by a quadrupole of wedge disclinations characterized by the same strength of opposite signs
at
z1 lei h / 2ei( /2) , z2 lei h / 2ei( /2) de i , z4 lei h / 2ei( /2 ) and z3 lei h / 2ei( /2 ) dei , respectively. On the other hand, the length, the thickness and the orientation of the nanoscale deformation twin are denoted as d , h and , respectively.
4
y
y
l
N
C
d
B r A E
x
h D
x
L
(a)
(b)
N(0, )
C
D
B A
E
L(0, 1)
(c) Fig. 1 The formation of nanotwins near a slant edge crack tip in a semi-infinite deformed nanocrystalline solid (a) general view (b) the magnified inset highlighting a disclination quadrupole and a lattice dislocation near the tip of slant edge crack (c) The ζ-plane after conformal mapping
For the plane strain problem, the stress fields can be described in terms of two Muskhelishvili’s complex potentials ( z ) and ( z ) [33]
xx yy 2[ (z) (z)]
yy i xy (z) (z) z (z) (z)
(1) (2)
where z x iy , the over-bar represents the complex conjugate of a function, and the prime means differentiation with respect to the argument z . 5
In order to deal with the current problem, the mapping function from the plane in Fig. 1(c) to the z plane in Fig. 1(b) is introduced [34]
z ( )
l
1/2
( i)1/2 ( i )1/2
(3)
with the aid of the mapping function, the surrounding region of the slant edge crack in z-plane is mapped onto the right half of the ζ-plane ( 0 ), where i , as shown in Fig. 1(c). In addition, points L(0,0 ) , N (0, 0 ) and A(l cos , l sin ) in the z-plane correspond to be mapped to points L(0, 1) , N (0, ) and A(0,0) in the plane where (1 2 ) / (1 2 ) . By means of the mapping function in Eq. (3), Eqs. (2) and (3) can be rewritten in the ζ-plane as
xx yy 2[( ) ( )]
yy i xy ( ) ( ) where
( ) ( ) ( ) ( )
( ) ( ) / ( ) , ( ) [ ( )( ) ( )( )] / [( )]2
(4) (5)
and
( ) ( ) / ( ) .
3. The force on lattice dislocation emission from the slant edge crack tip For the emission of lattice dislocations from the tip of slant edge crack, we consider a typical situation where the dislocations are of edge character and their Burgers vectors lie along the slip plane making an angle with x'-axis. The force acting on the edge dislocation consists of three parts: (1) the force produced by nanoscale deformation twinning; (2) the image force arising from the slant edge crack; 6
(3) the external loads. Firstly, we calculate the force exerted on the dislocation due to the deformation twinning. Referring to the works of Romanov and Vladimirov [35] and He et al.[20], the elastic fields caused by two wedge disclination dipoles characterizing the deformation twinning can be expressed by the following complex potentials
w ( z ) and w ( z )
w (z)
D 4 (1)k +1 (z zk ) ln(z zk ) w0 (z) 2 k 1
(6)
D 4 (1)k +1 zk ln(z z k ) w0 (z) 2 k 1
(7)
w (z)
where D / [2 (1 )] , w0 (z) and w0 (z) represent the terms due to the interaction of the two wedge disclination dipoles and the slant edge crack in the z-plane. With the substitution of Eq. (3) into Eqs. (6) and (7), the complex potentials in the ζ-plane can be written as
w ( )
D 4 (1)k 1 ( k )( k ) ln( k ) w0 ( ), 0 2 k 1
w ( )
D 4 (1)k 1 zk ln( k ) w0 ( ), 0 2 k 1
(8)
(9)
According to the Riemann-Schwarz symmetry theory, we take into account a new auxiliary function as follows
w ( ) ( )
w ( ) w ( ), 0 ( )
The traction-free boundary condition along the imaginary axis is treated as
7
(10)
w ( )
( ) w ( ) w ( ) 0 ( )
(11)
To solve the free boundary problem on the interface in Eq. (11), it is essential to introduce the following function
w ( ) w ( )
D 4 (1)k 1[ ( ) ln( k ) zk ln( k )] w0 ( ), 0 2 k 1
(12)
In view of Eqs. (10)-(12), we obtain
w (t ) w (t )
(13)
where t denotes the point in the ζ-plane on the imaginary axis. The superscripts and “ ” and “ ” refer to the boundary value as approached from the respective region occupied by the right half ( 0 ) and the left half ( 0 ). Applying the Plemeij formula, we can obtain
w ( )
D 4 (1)k 1[( k )( k ) ln( k ) ( ) ln( k ) zk ln( k )] 2 k 1 (14)
In combination with Eq. (14) and Eqs. (10)-(13), the complex potential w ( ) can be given as
D 4 (1) k 1[ ( k )( k ) ln( k ) zk ln( k )] 2 k 1 zk ( ) D 4 ( ) ( 1) k 1[ ( ) ln( k ) ( k ) ] ( ) 2 k 1 k k
w ( )
(15) The force acting on the edge dislocation caused by two wedge disclination dipoles characterizing the deformation twin can be calculated by means of the Peach-Koehler formula [36] 8
f w f wx if wy xy ( 0 )bx yy ( 0 )by i xx ( 0 )bx xy ( 0 )by (by2 bx2 ) w ( 0 ) w ( 0 ) ( 0 ) w ( 0 ) w ( 0 ) 4 (1 )
(16)
where xx , xy and yy are the components of the perturbation stress produced by two wedge disclination dipoles in the process of deformation twinning, and
( ) w 0 ( ) w ( 0 ) lim w 0 ( ) ( ) ( ) ( ) w ( ) ( ) w0 ( ) ( ) w 0 ( ) ( ) w ( 0 ) lim w 0 [ ( )]3 [ ( )]3 ( ) w 0 ( ) w ( 0 ) lim w 0 ( ) ( ) Secondly, suppose that the first edge dislocation emitted from the tip of slant edge crack is located at z0 lei in the coordinate system. According to the works of Fang and Liu [37], the elastic field of edge dislocation in the ζ-plane can be described by two complex potentials
s ( ) ln( 0 ) s 0 ( )
s ( ) ln( 0 )
K s 0 ( ) 0
(17) (18)
where (by ibx ) / [4 (1 )] and K ( 0 ) / ( 0 ) . Utilizing the similar technique above, we achieve
s 0 ( ) ln( 0 )
s 0 ( ) ln( 0 )
M 0
( ) M [ ] ( ) 0 ( 0 )2
(19)
(20)
where M [ ( 0 ) ( 0 )] / ( 0 ) . By using the Peach-Koehler formula [36], the image force can be evaluated as
9
f s f sx if sy ˆ xy ( 0 )bx ˆ yy ( 0 )by i ˆ xx ( 0 )bx ˆ xy ( 0 )by (by2 bx2 ) s 0 ( 0 ) s 0 ( 0 ) ( 0 )s 0 ( 0 ) s 0 ( 0 ) 4 (1 )
(21)
where ˆ xx , ˆ xy and ˆ yy are the components of stress field resulting from the interaction between the edge dislocation and the free surface of the slant edge crack,
s 0 ( 0 ) s0 ( ) / ( ) , s 0 ( 0 ) s0 ( 0 )( 0 ) s0 ( 0 )( 0 ) / [( 0 )]3
and
s 0 ( 0 ) s0 ( 0 ) / ( 0 ) . Thirdly, the applied load exerted on the edge dislocation can be expressed by the following equation
f b r
b (1Kapp 2 Kapp ) 2 r
(22)
1 3 where 1 sin cos , 2 cos sin2 cos , K app and K app are the 2 2 2 2 2 generalized mode I and mode II SIFs due to the remote loadings, b ( bx iby ) is the Burgers vector of the first edge dislocation. Lastly, the dislocation emission force near the tip of slant edge crack can be calculated by using the superposition principle
f emit f x cos( ) f y sin( ) f Re[ f w f s ]cos( ) Im[ f w f s ]sin( ) f
(23)
where
(by2 bx2 ) 2 Re[ w ( 0 ) s 0 ( 0 )] ( 0 )[ w ( 0 ) s 0 ( 0 )] [ w ( 0 ) s 0 ( 0 )] fw fs 4 (1 )
4. The critical SIFs for the dislocation emission It is a broadly accepted criterion that a new dislocation can be spontaneously emitted from the tip of a crack under the condition that the force acting on it is equal
10
to or larger than zero, and the dislocation distance to the crack free surface is not smaller than the dislocation core radius r0 [38]. Using the Eqs. (14)-(23) together with f emit 0 , we can obtain the following critical mode I and II SIFs for dislocation emission Kapp 0, Kapp
2 r (Im[ f w f s ]sin( ) Re[ f w f s ]cos( )) b 2
(24)
2 r (Im[ f w f s ]sin( ) Re[ f w f s ]cos( )) b1
(25)
for mode II crack, and Kapp 0, Kapp
for mode I crack. It is worth mentioning that when the inclined angle =0, the present results will reduce to those of the problem of nanoscale deformation twins interacting with mode I straight crack. We have completely derived the analytical solutions to the current problem. Then, the effect of deformation twinning on dislocation emission from the tip of slant edge crack can be examined by evaluating the critical SIFs in deformed nanocrystalline materials. In the following calculation, we mainly discuss the influences of typical parameters such as the crack length, the inclined angle of the crack and the relative thickness of the nanoscale twin band on the dislocation emission from the slant edge crack tip. For simplicity, we define the dimensionless critical SIFs as app Kapp / ( b ) c K
and
app Kapp c K / ( b )
.
Additionally,
the
typical
nanocrystalline material 3C-SiC with =217Gpa, =0.23 is taken as an illustrative example. As a makeshift, the Burgers vector of the edge dislocation and the distance
11
between the tip of slant edge crack and the first emitted dislocation are chosen as b =0.25nm and the core radius of dislocation r0 (= b / 2 ), respectively. app The variations of normalized critical SIFs K ICapp and K IIC versus the dislocation
emission angle 0 with different crack length l are depicted in Figs. 2 and 3, respectively. From Fig. 2, the normalized critical mode I SIFs decrease from a smaller value to negative infinity, sharply shift to positive infinity, then reduce to a minimum, and finally increase to positive infinity. According to the work of Huang and Li [39], the sign of the SIFs depends on the direction of the Burgers vector of the emerging dislocations, which makes the normalized critical SIFs positive or negative. The most probable angle for negative dislocation emission from the slant edge crack tip is always approximately -46。，while for positive dislocation emission it is around 14.5 。, which are both independent on the crack length. Furthermore, it will be more difficult for the dislocation to emit from the slant edge crack tip with increasing the crack length. For the normalized critical mode II SIFs in Fig. 3, it can be observed that the normalized critical SIFs first increase slowly, and then sharply rise with the increment of dislocation emission angle, which implies that it may be easier for negative dislocation to emit from the slant edge crack tip than that of positive dislocation. Besides, it is also more difficult for the dislocation to emit from the tip of slant edge crack with raising the crack length.
12
0.6
l=50nm l=100nm l=200nm
0.4
KIC
app
0.2
0.0
-0.2
-40
-20
0
20
40
(degree)
Fig. 2 Dependence of the normalized critical mode I SIFs on dislocation emission angle θ0 with different crack length l (α=1/5, η=/4, =/6, h=4nm, d=15nm) 0.10
0.08
KIIC
app
0.06
l=50nm l=100nm l=200nm
0.04
0.02
0.00 -20
0
degree
20
40
Fig. 3 Dependence of the normalized critical mode II SIFs on dislocation emission angle θ0 with different crack length l (α=1/5, η=/4, =/6, h=4nm, d=15nm)
The normalized critical mode I and mode II SIFs with respect to the dislocation emission angle 0 with different values of are illustrated in Figs. 4 and 5. For the normalized critical SIFs in Fig. 4, the normalized critical mode I SIFs decrease from a smaller value to negative infinity, dramatically turn to positive infinity, then reduce to a minimum, and finally increase to positive infinity. The most probable angle for negative dislocation emission is always about -30。，while for positive 13
dislocation emission it is nearly 18 。，14.7 。 and 10 。 for =1/4, 1/5 and 1/7, respectively. We can see that the smaller the value of is, the more difficult it is for dislocation emission from the slant edge crack tip. Particularly, when =1/4, the dislocation near the tip of slant edge crack is most prone to emit. For the normalized critical mode II SIFs in Fig. 5, it is found that the normalized critical SIFs first increase slightly, and then abruptly rise with increasing dislocation emission angle, which indicates that it will be easier for negative dislocation to emit from the slant edge crack tip than that of positive dislocation. Additionally, it can be concluded that the smaller the value of is, the more difficult it is for dislocation emission from the slant edge crack tip. Specially, the dislocation near the tip of slant edge crack is prone to emit when =1/4. 1.0 =1/7 =1/5 =1/4
KIC
app
0.5
0.0
-0.5
-1.0
-20
-10
0
10
20
30
40
50
(degree)
Fig. 4 Dependence of the normalized critical mode I SIFs on dislocation emission angle θ0 with different (l=100nm, η=/4, =/6, h=4nm, d=15nm)
14
0.20
0.15 =1/7 =1/5 =1/4
KIIC
app
0.10
0.05
0.00
-0.05 -20
-10
0
10
20
0(degree)
Fig. 5 Dependence of the normalized critical mode II SIFs on dislocation emission angle θ0 with different (l=100nm, η=/4, =/6, h=4nm, d=15nm) 0.8 0.6
h/d=0.1 h/d=0.2 h/d=0.3
0.4
KIC
app
0.2 0.0 -0.2 -0.4 -0.6 -0.8
-20
-10
0
10
20
0(degree)
Fig. 6 Dependence of the normalized critical mode I SIFs on dislocation emission angle θ0 with different h/d (l=100nm, η=/4, =/6, α=1/7, d=15nm)
15
0.6
0.4
Al Ni 3C-SiC
KIC
app
0.2
0.0
-0.2
-0.4
-0.6
-20
-10
0
10
20
0(degree)
Fig. 7 Dependence of the normalized critical mode I SIFs on dislocation emission angle θ0 with different nanocrystalline materials (l=100nm, η=/4, =/6, α=1/7, d=15nm, h=4nm)
Figure 6 plots the variations of the normalized critical mode I SIFs versus the dislocation emission angle with different relative length h/d between the length and the thickness of the twin band. It is found that the most probable angle for negative dislocation emission is always -30。，while for positive dislocation emission it is close to 10.1。. Both angles for dislocation emission are independent of the relative length. At the same time, it can be observed that the larger the relative length is, the easier it is for dislocation emission from the slant edge crack tip, which suggests that nanotwin near the tip of slant edge crack can increase the fracture toughness of brittle nanocrystalline materials. The dependence of the normalized critical mode I SIFs on dislocation emission angle with different nanocrystalline materials is shown in Fig. 7. It means that the nanotwin near the tip of slant crack tip can improve the toughness of nanocrystalline material 3C-SiC more remarkably as comparing to that of nanocrytalline materials Ni and Al.
5. Conclusions 16
The problem of nanoscale deformation twins interacting with a slant edge crack is investigated by establishing a theoretical model in nanocrystalline materials. Using complex variable method of Muskhelishvili and a conformal mapping function, the expression of the critical SIFs for the first lattice dislocation emission from the tip of slant edge crack are obtained. The influences of important parameters such as the crack length, the inclined angle of the crack, the relative length of the twin band on critical SIFs for dislocation emission are discussed at length. Some main conclusions are summarized as follows (1) It is more difficult for the dislocation to emit from the slant edge crack tip with increasing the crack length when the inclined angle of the crack is certain. (2) The smaller the inclined angle of the crack is, the more difficult it is for dislocation emission from the slant edge crack tip. Particularly, when the inclined angle of the crack is about 45。, the dislocation near the tip of slant edge crack is prone to emit. (3) The nanotwin near the tip of slant crack tip can increase the toughness of nanocrystalline material 3C-SiC more remarkably as comparing to that of nanocrytalline materials Ni and Al.
Acknowledgments The authors would like to deeply appreciate the support from the National Natural Sciences Foundation of China (11572191 and 51601112) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (20130073110057). 17
References [1] Kuntz JD, Zhan GD, Mukherjee AK. Nanocrystalline-Matrix Ceramic Composites for Improved Fracture Toughness. MRS Bulletin. 2004;29:22-7. [2] Yang F, Yang W. Crack growth versus blunting in nanocrystalline metals with extremely small grain size. Journal of the Mechanics & Physics of Solids. 2009;57:305-24. [3] Aifantis EC. Deformation and failure of bulk nanograined and ultrafine-grained materials. Materials Science & Engineering A. 2009;503:190-7. [4] Weng GJ. A homogenization scheme for the plastic properties of nanocrystalline materials. Reviews on Advanced Materials Science. 2009;19:41-62. [5] Wei Y, Bower AF, Gao H. Enhanced strain-rate sensitivity in fcc nanocrystals due to grain-boundary diffusion and sliding. Acta Materialia. 2008;56:1741-52. [6] Gurtin ME, Anand L. Nanocrystalline grain boundaries that slip and separate: A gradient theory that accounts for grain-boundary stress and conditions at a triple-junction. Journal of the Mechanics & Physics of Solids. 2008;56:184-99. [7] Barai P, Weng GJ. The competition of grain size and porosity in the viscoplastic response
of
nanocrystalline
solids.
International
Journal
of
Plasticity.
2008;24:1380-410. [8] Barai P, Weng GJ. Mechanics of creep resistance in nanocrystalline solids. Acta Mechanica. 2008;195:327-48. [9] Mukhopadhyay A, Basu B. Consolidation–microstructure–property relationships in bulk nanoceramics and ceramic nanocomposites: a review. International Materials
18
Reviews. 2007;52:257-88. [10] Dao M, Lu L, Asaro RJ, Hosson JTMD, Ma E. Toward a quantitative understanding of mechanical behavior of nanocrystalline metals. Acta Materialia. 2007;55:4041-65. [11] Wolf D, Yamakov V, Phillpot SR, Mukherjee A, Gleiter H. Deformation of nanocrystalline materials by molecular-dynamics simulation: relationship to experiments? Acta Materialia. 2005;53:1-40. [12] Ovid'Ko IA. Deformation and Diffusion Modes in Nanocrystalline Materials. International Materials Reviews. 2005;50:65-82. [13] Ovid’ko IA, Sheinerman AG. Special strain hardening mechanism and nanocrack generation in nanocrystalline materials. Applied Physics Letters. 2007;90:171927--3. [14] Ovid’Ko IA, Sheinerman AG, Aifantis EC. Stress-driven migration of grain boundaries and fracture processes in nanocrystalline ceramics and metals. Acta Materialia. 2008;56:2718-27. [15] Zhao Y, Qian J, Daemen LL, Pantea C, Zhang J, Voronin GA, et al. Enhancement of fracture toughness in nanostructured diamond–SiC composites. Applied Physics Letters. 2004;84:1356-8. [16] Youssef KM, Scattergood RO, Murty KL, Koch CC. Nanocrystalline Al–Mg alloy with ultrahigh strength and good ductility. Scripta Materialia. 2006;54:251-6. [17] Kolesnikova AL, Ovid’Ko IA, Fedorov AA. Local migration of grain boundaries in polycrystalline materials under plastic deformation conditions. Technical Physics Letters. 2003;29:488-90.
19
[18] Ovid'Ko IA, Sheinerman AG. DEFORMATION TWINNING THROUGH NANOSCALE
IDEAL SHEARS
IN
NANO-
AND
POLYCRYSTALLINE
MATERIALS AT ULTRA HIGH STRESSES. 2011;27:189-94. [19] Ovid'Ko IA, Sheinerman AG. Nanoscale rotational deformation near crack tips in nanocrystalline solids. Journal of Physics D Applied Physics. 2012;45:335301-8(8). [20] He T, Xiao W, Zhang Y, Zhu H. Effect of special rotational deformation on the dislocation emission from a branched crack tip in deformed nanocrystalline materials. Acta Mechanica. 2017;228:823-36. [21] Bobylev SV, Mukherjee AK, Ovid’Ko IA, Sheinerman AG. Effects of intergrain sliding on crack growth in nanocrystalline materials. International Journal of Plasticity. 2010;26:1629-44. [22] He T, Xiao W, Zhang Y, Zhu H. Effect of cooperative grain boundary sliding and migration on dislocation emission from a branched crack tip in deformed nanocrystalline solids. International Journal of Fracture. 2017;206:1-10. [23] Zhu YT, Liao XZ, Wu XL. Deformation twinning in bulk nanocrystalline metals: Experimental observations. JOM. 2008;60:60. [24] Wu XL, Zhu YT. Partial-dislocation-mediated processes in nanocrystalline Ni with nonequilibrium grain boundaries. Applied Physics Letters. 2006;89:58. [25] Wang YM, Hodge AM, Biener J, Hamza AV. Deformation twinning during nanoindentation of nanocrystalline Ta. Applied Physics Letters. 2005;86:44. [26] Ovid, Apos IA, ko. Nanoscale multiplane shear and twin deformation in nanowires and nanocrystalline solids. Applied Physics Letters. 2011;99:1403.
20
[27] Ovid’Ko IA, Skiba NV. Generation of nanoscale deformation twins at locally distorted grain boundaries in nanomaterials. International Journal of Plasticity. 2014;62:50-71. [28] Ovid’Ko IA, Skiba NV. Nanotwins induced by grain boundary deformation processes in nanomaterials. Scripta Materialia. 2014;71:33-6. [29] Morozov NF, Ovid’Ko IA, Skiba NV, Sheinerman AG. Influence of nanotwin generation near crack twins on the fracture toughness of nanomaterials. Doklady Physics. 2013;58:544-7. [30] Ovid'ko IA, Sheinerman AG. Generation of Nanocracks at Deformation Twins in Nanomaterials. Materials Research Letters. 2013;1:168-73. [31] Li J, Ni Y, Soh AK, Wu XL. Strong Crack Blunting by Hierarchical Nanotwins in Ultrafine/Nano-grained Metals. Materials Research Letters. 2015;3:190-6. [32] Romanov AE, Vladimirov VI. Disclinations in solids. Physica Status Solidi. 1983;78:11-34. [33] Muskhelishvili NI. Some Basic Problems of the Mathematical Theory of Elasticity. Mathematical Gazette. 1955;48:351. [34] Wang SD, Lee S. Screw dislocation near a slant edge crack. Mechanics of Materials. 1999;31:63-70. [35] Romanov A, Vladimirov V. Dislocations in Solids vol 9, ed FRN Nabarro. Amsterdam: North-Holland; 1992. [36] Hirth JP, Lothe J. Theory of dislocations: J. Wiley; 1968. [37] Fang QH, Liu YW. Size-dependent interaction between an edge dislocation and a
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nanoscale inhomogeneity with interface effects. Acta Materialia. 2006;54:4213-20. [38] Rice JR, Thomson R. Ductile versus brittle behaviour of crystals. Philosophical Magazine. 1974;29:73-97. [39] Huang M, Li Z. Dislocation emission criterion from a blunt crack tip. Journal of the Mechanics & Physics of Solids. 2004;44:1991-2003.
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Highlights 1. Interaction between a slant edge crack and nanoscale deformation twin 2. Expressions of the critical SIFs for the first lattice dislocation emission from the slant edge crack tip are obtained 3. Analyze the effects of the crack length, the inclined angle of the crack, the relative thickness of nanoscale twin band and different nanocrystalline materials on the SIFs.
23