Effect of blunt nanocracks on the splitting transformation of grain boundary dislocation piled up at triple junctions

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Effect of blunt nanocracks on the splitting transformation of grain boundary dislocation piled up at triple junctions Yingxin Zhao a,b,c,∗, Lianyong Xu b,c a

School of Materials Science and Engineering, Tianjin University of Technology, Tianjin 300384, PR China School of Materials Science and Engineering, Tianjin University, Tianjin 300072, PR China c Tianjin Key Laboratory of Advanced Joining Technology, Tianjin 300072, PR China b

a r t i c l e

i n f o

Article history: Received 30 September 2017 Revised 22 January 2018 Available online xxx Keywords: Nanocrack Splitting transformation GBD pile-up Analytic solution Surface stress

a b s t r a c t Considering that grain boundary (GB) deformation is a structural element that hinders crack growth and promotes superplastic deformation in nanocrystalline materials (NCMs), a theoretical model is suggested to describe the effect of the blunt pre-nanocracks with surface stress on the splitting transformation of the first head grain boundary dislocation (GBD) within the pile-up at the triple junctions (TJs) of GBs in mechanically loaded NCMs. The analytic solution of the total energy change that characterizes this process of splitting transformation of GBD is derived quantitatively by the complex variable method, and then, the very beginning plastic deformation occurrence near the nanocrack tip is predicted. We theoretically evaluated the influence of the various parameters of blunt pre-nanocracks, such as nanocrack blunting and length, the characteristics of grain and GBs, such as grain size, the number of GBDs, and GB angles, and the surface stress at critical conditions for such a splitting transformation. Further analyses revealed that the positive (negative) surface stress significantly decreases (increases) the energy and obviously influences the critical conditions for splitting transformation. Our study identified the beneficial role of GB defect structure transformation, namely the splitting transformation, in enhancing the ductility and fracture toughness of NCMs. It also lays the foundation for investigating how the microstructures caused by GB deformation affect the novel mechanical properties of NCMs. © 2018 Elsevier Ltd. All rights reserved.

1. Introduction Owing to the dramatically higher density of grain boundaries (GBs) and their triple junctions (TJs) that inherently accompany a reduction in grain size to the nanometer regime, the use of nanocrystalline materials (NCMs) has resulted in many improvements in vital technological and engineering properties such as susceptibility to corrosion (Telang et al., 2015), strengthening (Aifantis and Willis, 2005; Kedharnath et al., 2017), premelting (Lu et al., 2014), and creep damage (Barai and Weng, 2011; Mohamed, 2016; Roodposhti et al., 2016), which are closely related to the boundary structure. GBs and their TJs (including interphase boundaries between dissimilar phases) are known to occupy higher-energy regions as compared with atoms that occupy lattice sites within interior grains in NCMs and are preferential sites for metallurgical phenomena associated with fracture in polycrystalline materials (Zhang et al., 2017). Thus, different from that for conventional coarse-grained materials, GBs and TJs are crucial for the advanced performance of NCMs and must be compre-

∗

Corresponding author. E-mail address: [email protected] (Y. Zhao).

hensively understood to design, model, and then predict the mechanical properties and novel physical deformation mechanisms of nanosized materials. Extensive studies for decades have confirmed that when the average grain size reduces to the nanometer range, a crossover from intragranular dislocation-based plasticity (which is primary in coarse-grained materials) to GB-mediated plasticity occurs (Luo et al., al., 2009; Kahrobaiyan et al., 2014). The specific structural features of NCMs, such as nanoscopic size of grains, GBs and their TJs with their generic defects, such as grain boundary dislocations (GBDs) and disclinations, can effectively carry plastic flow, and stimulate, under special conditions, the generation and development of such deformation modes, including GB sliding, GB migration, Coble creep, TJ diffusional creep and rotational deformation (Tvergaard, 1985; Yamakov et al., 2002; Wei and Anand, 2004; Pan et al., 2007; Liu and Zhang, 2009; Liu and Ma, 2010; Wei and Kysar, 2011; Padmanabhan et al., 2014; Mompiou and Legros, 2015; Prokoshkina et al., 2017; Bobylev and Ovid’ko, 2017). Furthermore, GBs and their TJs instead become directly involved in the accommodation of strain and serve as active sources for dislocation emission (Shan et al., 2004; Monk et al., 2006; Bobylev et al., 2009; Ovid’ko and Skiba, 2012). Therefore, understanding and quantifying

https://doi.org/10.1016/j.ijsolstr.2018.02.025 0020-7683/© 2018 Elsevier Ltd. All rights reserved.

Please cite this article as: Y. Zhao, L. Xu, Effect of blunt nanocracks on the splitting transformation of grain boundary dislocation piled up at triple junctions, International Journal of Solids and Structures (2018), https://doi.org/10.1016/j.ijsolstr.2018.02.025

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the relationship between these mechanisms and GB structure are critically important and a great step toward future GB engineering (Li and Chew, 2017). In recent years, it has been clearly recognized that the features of GB deformations mentioned above are intensively related to TJs of GBs (Wu and He, 1998; Gutkin et al., 2005; Shi and Zikry, 2009; Zisman, 2017). TJs of GBs, where GB planes change their orientations, serve as obstacles for GBD motion and plastic flow occurring through GB deformation (Ovid’ko and Sheinerman, 2012). The transformation of GB structure into a neighboring energetically stable boundary must occur as GBDs are split from TJs. That is, in these circumstances, the splitting transformation of GB defects can occur at TJs and result in the formation of sessile, gliding or climbing GBDs, which can provide a further GB deformation, extremely affect the dominant deformation mechanism realized in mechanically loaded NCMs, and then play a constructive role in both the unique structure and properties of NCMs (Ovid’ko et al., 20 0 0; Fedorov et al., 20 03). From the viewpoint of thermodynamics, several works have suggested theoretical models to address the importance of motion and splitting transformation of GB defects at TJs to superplastic deformation of NCMs. For instance, Gutkin et al. (2003) and Gutkin and Ovid’ko (2003) considered a model of crossover from GB sliding to rotational deformation, which is realized by the very beginning transformation of a pile-up of gliding GBDs stopped by a TJ of GBs, into two walls of climbing GBDs (treated as the dipoles of partial wedge disclinations). The conditions necessary for such a transformation were then determined and discussed. Han et al. (2014) proposed a new composite model to describe the competition of deformation mechanism between stress-driven migration of GB and GB sliding, which is characterized by GBDs traversing through super TJs in mechanically loaded NCMs. Shimokawa et al. (2011) and Shimokawa and Tsuboi (2015) investigated the atomic-scale intergranular crack tip plasticity in tilt GBs, which act as an effective dislocation source. Ovid’ko and Sheinerman (2017) suggested a theoretical model to describe GB sliding and its accommodation through dislocation slip in ultrafine-grained and NC metals. The emission of lattice dislocations from TJs into interior grains characterizes the initial stage of dislocation slip accommodation. In general, GBs are recognized as one of the major barriers to crack growth in most engineering materials, and grain refinement increases the fracture toughness of materials (Xiao and Chen, 20 01b; Pook, 20 07; Legros et al., 2008; Zhou et al., 2008; Jin et al., 2015; Fang et al., 2016; Vetterick et al., 2018). Thus, the role of GBs and their TJs as dislocation sources must be known to elucidate the fracture phenomenon in fine-grained materials (Shimokawaa and Tsuboi, 2015). Bobylev et al. (2010) theoretically described two scenarios for the evolution of GBDs at a TJ near a microcrack tip in creep, similar to that in tests specified by constant and comparatively low applied stresses in a NCM. In the first case, such dislocations are immobile, and their stress fields compensate, in part, for local stresses near the microcrack tip. In the second case, GBDs at the TJ climb along adjacent GBs and promote microcrack blunting. Both these cases suppress microcrack growth and enhance ductility and toughness of NCMs. Kim et al. (2013) performed in situ tensile experiments to investigate the interaction of an advancing crack with GBs in thin copper foils. They found that certain GBs are effective in arresting the crack growth, and the stress field of the arrested crack then activates multiple deformation modes in the grain ahead of the crack tip. Hosseinian et al. (2018) performed in situ TEM experiments to investigate the combined effects of thickness (30 vs. 100 nm) and average grain size (40 vs. 70 nm for the thicker films) on the crack propagation mechanisms in ultrathin NC gold microbeams. They found that for the thinner specimens, secondary nanocracks are generated (as a result of GB sliding) ahead of the main crack and coalesce together. Instead, secondary nanoc-

racks do not form ahead of the main crack for the thicker specimens; the main crack extends as a result of sustained GB sliding at the crack tip. From the above data, it can be well established that GBs could interact with cracks during the process of fracture in polycrystalline materials and NCMs. GBDs piled up at TJs can shield the stress field near the crack tip. Consequently, a mechanical response of the splitting transformations of GBD is different from that without cracks. However, the aforementioned theoretical models of splitting transformations of GBDs piled up at TJs operate without pre-existent cracks. Furthermore, GB defect structure transformation, namely the splitting transformation, can shield the stress intensity factor near the nanocrack tip. This implies that when dislocation emission from the crack tip stops because of the presence of neighboring GBs, the dislocation reactions at TJs can additionally reduce the stress intensity factor, thereby hindering crack growth. Therefore, hereinafter, the main aim of this work is to theoretically describe in detail the splitting transformation of GBDs piled up at TJs of GBs near a pre-existent or growing blunt nanocrack with surface stress in plastically deformed and mechanically loaded NCMs, with focus on one of the transformations of GBD, which serves as an elementary act of the very beginning of the GB deformation, and is relatively most energetically favorable (Fedorov et al., 2003). In the framework of the model, stress concentration near nanocrack initiates the splitting transformation of GBD. The total energy change is analytically derived, and the critical conditions for splitting transformation are quantitatively predicted in plastically deformed NCMs. Special attention is paid to quantify the effect of blunt pre-nanocracks, the characteristics of grain and GB, and the surface stress on the presented type of splitting transformation of GBD piled up at TJ. The results provide a greater depth of understanding the GB-crack-dislocation interaction, which is essential for investigating how the microstructures formed because of the splitting transformation of GBDs affect the crack propagation behavior and the fracture toughness and then for designing better GB engineered materials.

2. Model The geometric features of a blunt nanocrack and GBD pile-up at TJ A are critical to understand the splitting transformation of GBD piled up at TJ in plastically deformed NC solids. In this section, a deformed elastically isotropic NC specimen with shear modulus μ and Poisson ratio υ is considered under the action of remote tensile loading σ 0 applied perpendicular to the nanocrack growth direction. This specimen consists of nanoscale grains divided by GBs and contains a pre-existent mode I blunt nanocrack, schematically shown in Fig. 1. Here, we assume that the nanocrack has already been blunted because of previous consecutive processes of dislocation emission and nanocrack advance (Ovid’ko and Sheinerman, 2010). Thus, for more reasonable and realistic description of crack stability and growth, an elliptically pre-cracked twodimensional grain structure of a typical fragment of the solid is schematically shown in Fig. 1a. In doing so, we model the blunt nanocrack as an elongated ellipse, which is oriented along x and y axis of the Cartesian coordinate system (xOy) with the origin at the ellipse center (Fig. 1a). The curvature radius ρ ( = q2 /p) characterizes the “bluntness” of the nanocrack tip, and at the nanocrack tip, the radius is much smaller than the crack half-lengths p and q. To read and analyze easily, using the relationship z = z1 + p − ρ /2, the stress field in the Cartesian coordinate system (xOy) can be transited to the coordinate system (x1 O1 y1 ), whose origin is of ρ /2 away from the tip of the elliptically blunt nanocrack. The elliptical representation will lead to a quantitative estimate of nanocrack blunting effect.

Please cite this article as: Y. Zhao, L. Xu, Effect of blunt nanocracks on the splitting transformation of grain boundary dislocation piled up at triple junctions, International Journal of Solids and Structures (2018), https://doi.org/10.1016/j.ijsolstr.2018.02.025

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Fig. 1. GBDs piled up at TJ near a nanocrack in a deformed NCM. (a) General view. (b) Magnified inset highlights GBDs within pile-up is stopped at TJ A near an elliptically blunt nanocrack.

Suppose that the blunt nanocrack tip intersects GB AG at a point O2 distant by a from the nearest TJ A, and it is not directly related to the grain size (boundary length) d. To establish the relationship between the grain size d and the critical conditions of splitting transformation of GBDs within the pile-up, we give a = λd, where 0 < λ ≤ 1. GBO2 A plane makes an angle α with the x2 axis. A Cartesian coordinate system (x2 O2 y2 ) with origin O2 at the right edge of the ellipse is introduced. According to the theory of defects in solids, GB sliding or other plastic deformation in a NC specimen under the action of mechanical loading gives rise to the formation of GBD pile-up stopped by TJ A of GBO2 A. This pile-up is composed of N (N > 1) GBDs each characterized by the Burgers vector bA . It should be noted that in this situation, the magnitude bA is

3

small, and GBDs do not create local stresses, which can initiate local cracking (Bobylev et al., 2010; Ovid’ko and Sheinerman, 2012). The combined action of GBD pile-up stresses, the blunt nanocrack stresses and the external stress causes the splitting transformation of GBDs piled up at TJ A in plastically deformed NCMs. The force and energy characteristics of this process are suggested using the models in Figs. 1 and 2. If the local force of head GBD within the pile-up at TJ A operating near the blunt nanocrack reaches a certain high value, the head GBD (with Burgers vector bA ) will split into two dislocations (with Burgers vectors bT1 and bT2 , respectively) that move along the adjacent GBs AB and AC, and then, intergrain deformation is initiated. The detailed analysis model is shown in Fig. 2. Here, the theoretical study of all the accommodation mechanisms for the splitting transformation of GBD pile-up near the blunt nanocrack tip is beyond the scope of our work. Below, for definiteness, according to the work of Fedorov et al. (2003), Gutkin et al. (2003), and Gutkin and Ovid’ko (2003), we consider only the most likely situation that can happen, which is theoretically described by the following basic scenario in plastically deformed NCMs, as shown in Fig. 2. Head dislocation with Burgers vector bA within pile-up splits into two mobile GBDs with Burgers vectors bT1 and bT2 , respectively, which move along adjacent GBs AB and AC (Fig. 2b). To estimate the key issues easily, suppose that no difference in GBD remains at TJ A in this process. That is, the Burgers vectors of all the three GBDs, participating in the split reaction, satisfy the equation bA = bT1 + bT2 . This process may repeatedly occur, which then result in the consequent splitting transformation of all GBDs within pile-up (Fig. 2c) and the formation of two walls of GBDs, which glide along the GBs adjacent to TJ A. Fig. 2 illustrates a typical two-dimensional model of the splitting transformation of GBD pile-up, which represents the elementary act of the very beginning of the grain and GB deformation. Suppose that no dislocation dipole emission occurs from the blunt nanocrack in the process. GB sliding, migration and other deformations, which are treated to be the dominant modes of superplasticity in NCMs, occur through the motion of gliding GBDs. They have Burgers vectors that are parallel to the corresponding GB planes along which these GBDs glide. Thus, we use the angle ω(η) to characterize the geometry of the splitting between the Burgers vector bA of head GBD and the Burgers vector bT1 (bT2 ) of the first (second) resultant GBD, which plays the role of the angle between GB planes in some situations discussed here and is shown in Fig. 3. Herein, we propose the following relation for facilitating the calculation and study: the Burgers vectors bT1 andbT2 of the first and second resultant GBDs have magnitudes of bT1 = bA sin η/sin (ω + η) and bT2 = bA sin ω/sin (ω + η), respectively, which satisfy the Burgers vector conservation. Assuming that η = ω, bT1 = bT2 = bA /(2cos ω). In addition, new coordinate systems (x3 Ay3 ), (x4 Ay4 ), (x5 Ay5 ) and (x6 Ay6 ) are introduced to facilitate the following calculation. 3. Splitting transformation of GBD piled up at TJ near the blunt nanocrack To make possible the splitting transition from the initial GBD to the new split configuration, the total energy of the initial GBD must be higher than that of the new split configuration. Accordingly, one has to calculate and compare these energies to evaluate the critical conditions for such a splitting transformation. In addition, the critical stress, which is necessary and energetically favorable for splitting transformation of the first head GBD within pileup at TJ A in plastic deformation processes, is a well-established and widely used parameter. According to the Rice–Thomson model, the energy and critical stress characteristics of the splitting transformation of GBD are derived using the complex variable method

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Fig. 2. Schematic of analysis model for the splitting transformation of GBD piled up at TJ A. (a) The initial state of GBD pile-up stopped at TJ A. (b) Head GBD within pile-up splits into two mobile GBDs, which move along adjacent GBs. (c) Second head GBD within pile-up splits into two mobile GBDs, which move along adjacent GBs, leading to the repeatedly occurring splitting transformation of GBD pile-up.

tion E = 0 gives a set of critical values of parameters for the defect configuration, at which the splitting transformation becomes energetically favorable. When no crack tip plasticity is involved, that is, there is no emission of dislocation dipole from the blunt nanocrack, Einitial of the initial defect configuration, representing a pile-up ofNGBDs near the blunt nanocrack under the remote tensile loading σ 0 , consists of six terms as follows: pu hd−pu hd−sur f ace hd c hd−c Einitial = Esel + Eint + Eint f + Esum + Esel f + Eint

(1)

hd denotes the self energy of the head GBD within pilewhere Esel f pu

up; Esum is the sum energy of the unchanged GBDs during the splitting transformation of the head GBD near the blunt nanocrack tip under the remote tensile loading σ 0 and contains the self pu energy Esel f of the unchanged GBDs during the splitting transforFig. 3. Coordinate position transformations of (xOy) into different coordinate systems.

in the isotropic linear elasticity, which are needed for the athermal realization of this stage. Hence, the significant effect of nanocrack tip plasticity on this phenomenon can be studied, and the very beginning plastic deformation occurrence near the crack tip can be predicted in this theoretical model. In the energy and critical stress characteristic calculations of the splitting transformation, for definiteness and simplicity, the following model assumptions are made: (1) The plane of the stress action due to remote tensile loadings σ 0 is parallel to the plane of the GB where the pile-up is located. (2) After the splitting of the head GBD within pile-up, the rest GBDs composing the pile-up do not move from their initial positions (herein and in the following, we denote such GBDs as ‘‘unchanged GBDs’’). These assumptions in fact do not affect the quantitative results of our consideration and, simultaneously, allow us to find analytical expression for critical stress to the splitting transformations of GBD piled up at TJ near the blunt nanocrack (Gutkin et al., 2003). In the framework of the model considered, the energy change E = Efinal –Einitial (Einitial and Efinal are, respectively, the energies of the initial and final states of the defect configuration containing the blunt nanocrack) is produced by the splitting transformation process in which head GBD within pile-up (Fig. 2a) splits into two dislocations (Fig. 2b). The splitting transformation process is energetically favorable (unfavorable), if E < 0( E > 0). The equa-

pu−c

mation of the head GBD, Eint of the interaction between the unpu−σ changed GBDs and blunt nanocrack, Eint of the interaction between the unchanged GBDs and the remote tensile loading σ 0 , pu−sur f ace Eint of the interaction between the unchanged GBDs and the stress created in the solid with a blunt nanocrack by its surpu face stress, and the sum of Eint that characterizes pair elastic inc teractions between unchanged GBDs; Esel is the self-energy of the f stress field induced by the applied load and the unchanged GBDs hd−pu in the cracked solid; Eint is the energy of the elastic interaction of the head dislocation and the unchanged GBDs within pile-up hd−c near blunt nanocrack; Eint is the energy of the interaction behd−sur f ace

tween head GBD and blunt nanocrack; Eint is the energy of head GBD interaction with the stress created in the solid with a blunt nanocrack by its surface stress. In the following, let us calculate both the energy and critical stress characteristics of the splitting transformation of the head GBD within pile-up according to the proposed model (Fig. 2a and b) under the combined effect of external stress and blunt nanocrack, which play important roles in inducing the evolution of GBD pile-up. In the splitting transformation process, shown in Fig. 2b, head GBD with Burgers vector magnitude bA ( = bAx + ibAy ) within pileup splits into two mobile GBDs with Burgers vector magnitude bT1 ( = bT1x + ibT1y ) and bT2 ( = bT2x + ibT2y ) that move the same distances ε from TJ A along the corresponding GBs AB and AC, respectively. Thus, the positions of the first and second resultant GBDs can be expressed as zT1 = p + aeiα + ε eiU and zT2 = p + aeiα + ε eiQ ,

Please cite this article as: Y. Zhao, L. Xu, Effect of blunt nanocracks on the splitting transformation of grain boundary dislocation piled up at triple junctions, International Journal of Solids and Structures (2018), https://doi.org/10.1016/j.ijsolstr.2018.02.025

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where Q = α + κ − π and U = α + ω, and the angle U(Q) is the difference in the direction between GB AB (AC) and the horizontalx3 axis (Fig. 3). The pile-up after the splitting transformation of the first head GBD (Fig. 2b) contains (N − 1) residual GBDs. Based on the assumption, (N − 1) GBDs stay in the original position during the splitting transformation of the head GBD. This supposition is reasonable in at least the situation where the GBDs resulted from the splitting transformation are close to TJ A. Thus, the energy of the defect configuration resulted from the splitting of the first head GBD in its final state (Fig. 2b) is given as follows:

E f inal =

pu Esum

+

c Esel f

+

T d1 Esel f

+

T d2 Esel f

+

T d1−T d2 Eint

+

T d2−pu T d1−sur f ace T d2−sur f ace T d1−c T d2−c +Eint + Eint + Eint + Eint + Eint T d1−σ T d2−σ −Eint − Eint

(2)

T d1 (E T d2 ) is the self-energy of where Esel f sel f T d1−T d2 tant GBD; Eint is the elastic energy

where

(ζ ) = ϕ  (ζ )/w (ζ ),  (ζ ) = ψ  (ζ )/w (ζ ),

 ( ζ ) = [ϕ   (ζ )w (ζ ) − ϕ  (ζ )w  (ζ )]/[w (ζ )]3 . To evaluate the elastic fields σ GBD produced by the interaction between GBD pile-up composed of N edge dislocations with Burgers vector magnitude bA ( = bAx + ibAy ) in the initial stage located at their equilibrium positions of zGBDj (j = 1, 2, ... N) near TJ A and blunt nanocrack, the equilibrium positions of GBDs within pile-up are found according to the theory of the Laguerre polynom LN (x) (Eshelby et al., 1951), which are given by the solution of the following equation:

x2

T d1−pu Eint

the first (second) resul-

associated with the elastic interaction between the first and second resultant dislocation; T d1−pu T d2−pu Eint (Eint ) is the energy of the elastic interaction of the first (second) resultant GBD and the unchanged GBDs near the T d1−c T d2−c blunt nanocrack; Eint (Eint ) is the energy of the elastic interaction of the first (second) resultant GBD and the blunt nanocrack; T d1−sur f ace T d2−sur f ace Eint (Eint ) is the energy of the first (second) resultant dislocation interaction with the stress created in the solid with the T d1−σ T d2−σ blunt nanocrack by its surface stress; Eint (Eint ) is the effective work of the first (second) resultant GBD interaction with the stress created in the solid with the blunt nanocrack by the applied load σ 0 that is spent to transfer the first (second) resultant GBD over the distance ε along the GB AB (AC) adjacent to TJ A (Fig. 2b). The splitting transformation of the head GBD within pile-up into the two climbing GBDs is characterized by the energy change E (per unit length along the axis perpendicular to the plane in Fig. 2a and b), which can be expressed as follows:

5

d2 y dy + (1 − x ) + Ny = 0 dx d x2

(9)

The first derivative of the Laguerre polynom LN (x ) is the solution of the equation

x

d2 y dy + (2 − x ) + (N − 1 )y = 0 dx d x2

(10)

Thus, LN (x ) can be given as follows:

LN (x ) = −

N−1  k=0

N!(−x )k k!(k + 1 )!(N − k − 1 )!

(11)

The roots of LN (x ) characterize the spacing dGBDj (j = 2, 3, ... N) of GBDs between the jth and the head dislocation in a discrete GBD pile-up. Thus, the equilibrium positions of zGBDk (k = 1, 2, ... N) can be given as zGBD1 = p + aeiα and zGBDk = p + (a − dGBDk )eiα (k = 2, 3, ... N). The elastic fields σ GBDk produced by the interaction between the kth(k = 1, 2, 3, ... N) GBD within pile-up with Burgers vector magnitude bA and blunt nanocrack can be estimated by using two complex potentials ϕkGBD (ζ ) and ψkGBD (ζ ) using the mapping function in Eq. (6).

ϕkGBD (ζ ) = γA ln (ζ − ζGBDk ) + ϕ0GBD k (ζ )

(12)

E = E f inal − Einitial T d1−pu T d2−pu hd−pu T d1 T d2 hd T d1−T d2 = Esel + Eint + Eint − Eint f + Esel f − Esel f + Eint T d1−c T d2−c hd−c +Eint + Eint − Eint + hd−sur f ace −Eint

−

T d1−σ Eint

−

T d1−sur f ace Eint

T d2−σ Eint

+

T d2−sur f ace Eint

(3)

To obtain a detailed analytical expression of the energy change E, the following calculations are performed by using the complex method of elasticity. For the plane strain problem, the stress field along the slip plane is expressed in terms of Muskhelishvili’s complex potentials (z) and  (z) (Muskhelishvili, 1975):

  σxx + σyy = 2 (z ) + (z )

(4)

  σyy − σxx + 2iσxy = 2 z  (z ) − (z )

(5)

where z = x + iy, and the overbar represents the complex conjugate. For mathematical convenience, we use

z = w(ζ ) = R(ζ + m/ζ )

(6)

to map the elliptical hole with major semi-axispand minor semi-axis q in thez-plane to a unit circle in theζ plane, where R = (p + q)/2 and m = (p − q)/(p + q). Hence, ζ =   √ 2 z m[1 + 1 − (l/z ) ]/l, where l = p2 − q2 . Considering the mapping function in Eq. (6), the stress fields can be given by using two complex potentials (z) and  (z) as in the work by Muskhelishvili (1975).

  σxx + σyy = 2 (ζ ) + (ζ )

(7)

  σyy − σxx + 2iσxy = 2 w(ζ )  (ζ ) + (ζ )

(8)

ψkGBD (ζ ) = γA ln (ζ − ζGBDk )   2  2   −γA ζGBDk + m/ζGBDk ζGBDk / ζGBDk − m (ζ − ζGBDk ) +ψ0GBD k (ζ )

(13)

 √ 2 where ζGBDk (= zGBDk m[1 + 1 − (l/zGBDk ) ]/l ) is the mapped position of GBD pile-up composed of N edge dislocations, which are related to zGBDk through Eq. (6), γA = μ(bAy − ibAx )/[4π (1 − υ )] for the plane strain state problem, ϕ0GBD (ζ ) and ψ0GBD (ζ ) are holomork k phic in a NCM. Considering the works of Xiao and Chen (2001a), Li and Li (2007), and Zhao et al. (2014a), two complex potentials ϕ0GBD (ζ ) k and ψ0GBD ( ζ ) denoting the interaction between the k thGBD and the k blunt nanocrack can be obtained.

  ϕ0GBD k (ζ ) = γA ln ζ − γA ln ζ − 1/ζGBDk − γA ln (ζ − m/ζGBDk ) γ A + 2 ζGBDk ζGBDk ζGBDk −m   2  2 ζGBDk 1 + mζGBDk − ζGBDk ζGBDk +m ×

ζ − 1/ζGBDk

(14)

  ψ0GBD k (ζ ) = γA ln ζ − γA ln ζ − 1/ζGBDk −γA ln (ζ − m/ζGBDk ) − ζ

1 + mζ 2 GBD ϕ (ζ ) ζ 2 − m 0k

Please cite this article as: Y. Zhao, L. Xu, Effect of blunt nanocracks on the splitting transformation of grain boundary dislocation piled up at triple junctions, International Journal of Solids and Structures (2018), https://doi.org/10.1016/j.ijsolstr.2018.02.025

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+

×

γA  2  ζGBDk ζGBDk ζGBDk −m   2 2 ζGBDk m3 + ζGBDk − mζGBDk ζGBDk +m

+ (15)

ζ − m/ζGBDk

From the above equations, the energy terms in Eq. (3) for a mechanically loaded NC specimen can be calculated as follows: (1) Following the work of Hirth and Lothe (1982), the self enerhd , E T d1 and E T d2 can be given as follows: gies Esel f sel f sel f hd Esel f b2A

=

T d1 Esel f b2T 1

=

T d2 Esel f b2T 2

=

 D 2

ln

R0 + Z0 rc

(16)

where D = μ/[2π (1 − υ )], R0 is the screening length of the dislocation long-range stress field, rc is the core radius of the stress field of the corresponding dislocation, and Z0 is the factor considering the contribution of dislocation core to the self energy. In the next section of numerical analyses, we consider R0 = 107 bA , rc = bA , and Z0 = 1. (2) Considering the general method (Mura, 1968), the energy of the interaction between two dislocations with the Burgers vector magnitudes b1 and b2 can be calculated as the work spent for the generation of the dislocation with the Burgers vector magnitude b2 in the stress field of the dislocation with Burgers vector magnitude b1 . According to Eqs. (7) and (8), the stresses σ xx , σ yy and σ xy can be given as follows:

σxx = (ζ ) + (ζ )   −1/2 w(ζ )  (ζ ) + w(ζ )  (ζ ) + (ζ ) + (ζ )

(17)

σyy = (ζ ) + (ζ )   +1/2 w(ζ )  (ζ ) + w(ζ )  (ζ ) + (ζ ) + (ζ )

(18)

  σxy = −i/2 w(ζ )  (ζ ) − w(ζ )  (ζ ) + (ζ ) − (ζ )

(19)

Because the kth resultant GBD (k = 1, 2) is located near the blunt nanocrack, its stress field is screened by the nanocrack. The proper energies of such resultant dislocations are calculated using a formula that considers the image forces that characterize the screening effect Bobylev et al., 2010). In doing so, on the basis of the analogous derivation process in Eqs. (12)–((15) and the b

b

work of Zhao et al. (2014b), the stress components σxxT k , σyyT k and

−ζ

ϕ Tdk (ζ ) = γT k ln (ζ − ζT k ) + γT k ln ζ   −γT k ln ζ − 1/ζT k − γT k ln (ζ − m/ζT k )    1 + mζT2k − ζT k ζT2k + m ζ T k γT k  + (20) ζ − 1/ζT k ζT k ζT k ζ 2 − m Tk

ϕ0Tdk (ζ ) = ϕ Tdk (ζ ) − γT k ln (ζ − ζT k )

(21)

ψ Tdk (ζ ) = γT k ln (ζ − ζT k )      −γT k ζT k + m/ζT k ζT2k / ζT2k − m (ζ − ζT k )   +γT k ln ζ − γT k ln ζ − 1/ζT k − γT k ln (ζ − m/ζT k )

1 + mζ ζ2 − m

2

ζ − m/ζT k



ϕ0Tdk (ζ )

(22) √ ζT k (= zT k m[1 +

γT k = μ(bT ky − ibT kx )/[4π (1 − υ )], 2 1 − (l/zT k ) ]/l ) is the mapped position of the kth resultant

where



GBD. Substituting Eqs. (20)–(22) into Eqs. (17)–(19), the stress comb

b

b

ponents σxxT k , σyyT k and σxyT k (k = 1, 2)of the kth resultant GBD can be obtained. In view of the work of Muskhelishvili (1975), the b

b

b

stress components σx3Txk3 , σy3Tyk3 and σx3Tyk3 of the kth resultant GBD near the blunt nanocrack in the coordinate system (x3 Ay3 ) are easily transformed from the coordinate system (xOy) by the substitution of z = p + aeiα + z3 , here z3 = x3 + iy3 . By using the revolution movement of the coordinate axes, the b shear stress component σx4Ty24 induced by the second resultant GBD acting on the gliding plane y4 = 0 for the first resultant GBD can be indicated as follows:

  σxb4Ty24 = σxb3Ty23 cos (2U ) + σyb3Ty23 − σxb3Tx23 sin (2U )/2

(23)

with x3 = x4 cos U − y4 sin U and y3 = x4 sin U + y4 cos U. T d1−T d2 Eint associated with the elastic interaction between the first and second resultant GBDs can be calculated under the introduced coordinate system (x4 Ay4 ), withx4 -axis parallel to the gliding T d1−T d2 plane of the first resultant GBD. In doing so, Eint can be given as follows: T d1−T d2 Eint = bT 1



xT 42

R0

σxb4Ty24 (x4 , yT 42 )dx4

(24)

where xT42 and yT42 are the parameters characterizing the distances between the first and second resultant GBDs, and xT42 = ε − ε cos (ω + η) and yT42 = ε sin (ω + η). In addition, by using the revolution movement of the coordinate −c T d2−c axes, the shear stress component σxT4d1 y4 (σx5 y5 ) acting on its gliding plane y4 = 0 (y5 = 0) for the first (second) resultant GBD yields

  −c σxT4d1 = σxb3Ty13 cos (2U ) + σyb3Ty13 − σxb3Tx13 sin (2U )/2 y4

(25)

with x3 = x4 cos U − y4 sin U and y3 = x4 sin U + y4 cos U.

  −c σxT5d2 = σxb3Ty23 cos (2Q ) + σyb3Ty23 − σxb3Tx23 sin (2Q )/2 y5

(26)

with x3 = x5 cos Q − y5 sin Q and y3 = x5 sin Q + y5 cos Q. T d1−c T d2−c Thus, we have the energies Eint and Eint by integrating the following formulae. T d1−c Eint = bT 1

bT k σxy (k = 1,

2) of the kth resultant dislocation near the blunt nanocrack in the coordinate system (xOy) are given by using two comT T plex potentials ϕ0dk (ζ ) and ψ0dk (ζ )

γ  Tk  ζT k ζT k ζT2k − m

  ζT k m3 + ζT2k − mζT k ζT2k + m

T d2−c Eint = bT 2



xT 41 R0 xT 52 R0

−c σxT4d1 y4 (x4 , yT 41 )d x4

(27)

−c σxT5d2 y5 (x5 , yT 52 )d x5

(28)

where xT41 = ε , yT41 = 0, xT52 = ε , and yT52 = 0 denoting the positions of the first and second resultant GBDs. T d1−pu

T d2−pu

(3) To find Eint (Eint ) denoting the sum energy of the elastic interaction of the first (second) resultant GBD and the (N − 1) unchanged GBDs within pile-up, the elastic enT d1−pu T d2−pu ergy Eint j (Eint j ) of the interaction of the first (second) resultant GBD with the jth (j = 2, 3, ... N) unchanged GBD needs to be obtained, which can be calculated as the work spent for the generation of the jth dislocation in the shear stress created by the first (second) resultant GBD. T d1−pu Eint = −bA j T d2−pu Eint = −bA j



−x61 j −R0 −x62 j −R0

σxb6Ty16 (x6 , y61 )dx6

(29)

σxb6Ty26 (x6 , y62 )dx6

(30)

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where x61j = ε cos ω + dGBDj and y61 = ε sin ω are the distances between the first resultant GBD and thejthunchanged GBD alongx6 andy6 - axes, respectively. x62j = ε cos η + dGBDj and y62 = −ε sin η are the distances between the second resultant GBD and thejthunchanged GBD along x6 - and y6 - axes, respectively. b b The shear stress component σx6Ty16 (σx6Ty26 ) acting on the gliding plane y6 = 0 for the first (second) resultant GBD can be indicated as follows:

  σxb6Ty16 = σxb3Ty13 cos (2α ) + σyb3Ty13 − σxb3Tx13 sin (2α )/2 σ

bT 2 x6 y6

=σ

bT 2 x3 y3

cos (2α ) −



σ

bT 2 y3 y3

−σ

bT 2 x3 x3



(31) (32)

with x3 = x6 cos α − y6 sin α and y3 = x6 sin α + y6 cos α . Substitution of Eq. (31) with the existing stress components T d1−pu σxb3Tx13 , σyb3Ty13 and σxb3Ty13 into Eq. (29) gives the elastic energy Eint , j T d1−pu

T d1−pu

and then, Eint can be found as the sum of energies Eint j given by Eq. (29). T d1−pu Eint =

N 

T d1−pu Eint j

as

(33)

j=2

T d2−pu

By using the above same method, the total energy Eint be expressed as follows T d2−pu Eint

=

N 

T d2−pu Eint j

can

(34)

j=2

(4) From Eqs. (12)–(15) in the case of k = 1, we have two complex potentials ϕ1GBD (ζ ) and ψ1GBD (ζ ) characterizing the interaction between head GBD within pile-up and nanocrack. Substituting them into Eqs. (17)–(19), the stress comGBD1 , σ GBD1 and σ GBD1 of the head GBD can be ponents σxx yy xy given in the coordinate system (xOy). By the substitution of 1, z = p + aeiα + z3 (z3 = x3 + iy3 ), the stress components σxGBD 3 x3 1 and σ GBD1 are given in the coordinate system (x Ay ). σyGBD 3 3 x3 y3 3 y3

By using the revolution movement of the coordinate axes, 1 acting the head GBD-induced shear stress component σxGBD 6 y6 on the gliding plane y6 = 0 for the head GBD can be given as follows:



 GBD1

1 1 1 σxGBD = σxGBD cos (2α ) + σyGBD − σx 3 x 3 6 y6 3 y3 3 y3

sin (2α )/2 (35)

with x3 = x6 cos α − y6 sin α and y3 = x6 sin α + y6 cos α . hd−c Eint characterizing the elastic interaction of head GBD with the blunt nanocrack can be obtained by the following integration:

hd−c Eint = bA

0 R0

1 σxGBD (x6 , 0 )dx6 6 y6

(36)

hd−pu

hd−pu

Eint is the sum of the energies Eint j characterizing interaction between the jth(j = 2, 3, ... N) GBD within pile-up and head GBD with Burgers vector magnitude bA . First, the calculation of hd−pu Eint j is carried out as follows: hd−pu Eint = −bA j



−dGBD j −R0

1 σxGBD (x6 , 0 )dx6 6 y6

(37)

where dGBDj (j = 2, 3, ... N) characterizes the spacing of GBDs between thejthand the head dislocation in a discrete GBD pile-up. hd−pu Substituting Eq. (35) into Eq. (37), we can obtain Eint j first, hd−pu

and then, the total energy Eint integral formula. hd−pu Eint =

N  j=2

hd−pu Eint j

hd−sur f ace

is given by using the following

(38)

T d1−sur f ace

T d2−sur f ace

(5) The energies Eint , Eint and Eint indicate the interaction of head GBD, the first and second resultant GBDs with the stress created in the solid with the blunt nanocrack by its surface stress. In view of the previous works of Wang and Wang (2006) and sur f ace sur f ace sur f ace Zhao et al. (2014a), the stresses σxx , σyy and σxy in the coordinate system (xOy) induced by the nanocrack surface stress can be expressed as follows:

σxxsur f ace − iσxysur f ace =

sin (2α )/2

7

τs 2R

G0 ( z )

(39)

σyysur f ace = −σxxsur f ace

(40)

G0 ( z ) =

where

(1−m2 )G2 (z ) ,G ( z ) = z + {[1−mG2 (z )](z2 −l 2 )}1/2 [G2 (z )−m]2/3

(z2 − l 2 )1/2 , and τ s is the residual surface stress.

Referring to the stresses in Eqs. (39) and (40), and then by the substitution of z = p + aeiα + z3 into these stress fields, we can eassur f ace sur f ace sur f ace ily obtain the transformed stresses σx3 x3 , σy3 y3 and σx3 y3 in the coordinate system (x3 Ay3 ). By using the revolution movement of the coordinate axes, the sur f ace sur f ace sur f ace shear stress components σx6 y6 , σx4 y4 and σx5 y5 acting on the gliding planes of head GBD, and on the first and second resultant GBDs (y6 = 0, y4 = 0 and y5 = 0) can be revealed as follows:

  f ace f ace f ace f ace σxsur = σxsur cos (2α ) + σysur − σxsur sin (2α )/2 (41) 6 y6 3 y3 3 y3 3 x3 with x3 = x6 cos α − y6 sin α and y3 = x6 sin α + y6 cos α .

  f ace f ace f ace f ace σxsur = σxsur cos (2U ) + σysur − σxsur sin (2U )/2 (42) 4 y4 3 y3 3 y3 3 x3 with x3 = x4 cos U − y4 sin U and y3 = x4 sin U + y4 cos U.

  f ace f ace f ace f ace σxsur = σxsur cos (2Q ) + σysur − σxsur sin (2Q )/2 (43) 5 y5 3 y3 3 y3 3 x3 with x3 = x5 cos Q − y5 sin Q and y3 = x5 sin Q + y5 cos Q. hd−sur f ace T d1−sur f ace T d2−sur f ace Thus, the energies Eint , Eint and Eint can be given as follows: hd−sur f ace Eint = bA



0 R0

T d1−sur f ace Eint = bT 1

T d2−sur f ace Eint = bT 2



f ace σxsur (x6 , 0 )dx6 6 y6 xT 41 R0



xT 52 R0

(44)

f ace σxsur (x4 , yT 41 )dx4 4 y4

(45)

f ace σxsur (x5 , yT 52 )dx5 5 y5

(46)

T d1−σ T d2−σ (6) Eint (Eint ) is the effective work spent to transfer the first (second) resultant GBD by distance ε under the stress created in the solid with the blunt nanocrack by the applied load σ 0 along the GBs AB (AC) adjacent to the TJ A. The stress fields created in the solid with the blunt nanocrack by the applied load σ 0 in the coordinate system (rO1 θ ) can be given as follows (Neuber, 1958; Creager and Pari, 1967):

σ

external x1 x1

 √ pσ0 θ 1 3θ ρn 3θ = √ cos − sin θ sin − cos 2 2 2 2r 2 2 2r (47)

 √ pσ0 θ 1 3θ ρn 3θ σyexternal = cos + sin θ sin + cos √ 1 y1 2

2 2r

2

2

2r

2

(48)

 √ pσ0 1 3θ ρn 3θ σxexternal = sin θ cos − sin √ 1 y1 2 2r

2

2

2r

2

(49)

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By substitution of Eqs. (47)–(49) with z1 = r eiθ = ρn /2 + aeiα + z3 , we can obtain the stresses σxexternal , σyexternal and σxexternal cre3 x3 3 y3 3 y3 ated in the solid with the blunt nanocrack by the external applied load σ 0 in the coordinate system (x3 Ay3 ). By using the revolution movement of the coordinate axes, the shear stress component σxexternal (σxexternal ) acting on the gliding plane for the first (second) 4 y4 5 y5 resultant GBD can be given as follows:

  σxexternal = σxexternal cos (2U ) + σyexternal − σxexternal sin (2U )/2 4 y4 3 y3 3 y3 3 x3

(50) with z3 = z4

eiU .

  σxexternal = σxexternal cos (2Q ) + σyexternal − σxexternal sin (2Q )/2 5 y5 3 y3 3 y3 3 x3 (51) eiQ .

with z3 = z5 T d1−σ T d2−σ Thus, the effective works Eint and Eint spent to transfer the first and second resultant GBDs by distance ε under the stress created in the solid with a blunt nanocrack by the applied load σ 0 along the GBs AB and AC adjacent to the TJ A can be written as T d1−σ Eint = bT 1 T d2−σ Eint = bT 2



xT 41 0



xT 52 0

σxexternal (x4 , yT 41 )dx4 4 y4

(52)

σxexternal (x5 , yT 52 )dx5 5 y5

(53)

Fig. 4. Energy difference E/μb2A versus the normalized distance ε /bA without the influence of the blunt nanocrack for σ 0 /μ = 0.002 and ω = η = 20° in the case of N = 5 in the initial pile-up.

By combining the above equations of all energies, the total energy change E that characterizes the process of the splitting transformations of GBD piled up at TJ A near the blunt nanocrack, shown in Fig. 2a and b, can be determined using Eq. (3), but its expression is not shown here because it is long. The critical characteristic energy change

E = 0

(54)

gives a set of critical values of parameters for the defect configuration, at which the splitting transformation of the first head GBD within pile-up becomes energetically favorable, when no crack tip plasticity is involved. Consider that N = 5, σ0 /μ = 0.002 and ω = η = 20° and that no crack is present, the problem corresponds to describe the splitting transformation of GBD piled up at TJ of GBs in (super) plastically deformed NCM, without the influence of the blunt nanocrack. Thus, the critical characteristic energy change of the splitting transformations of GBD is

E = E f inal − Einitial T d1−pu T d1 T d2 hd T d1−T d2 = Esel + Eint f + Esel f − Esel f + Eint T d2−pu hd−pu T d1−σ T d2−σ +Eint − Eint − Eint − Eint

(55)

According to Eq. (55), the influence of the normalized distance ε /bA from TJ A along the corresponding GBs AB and AC on the normalized energy difference E/μb2A is shown in Fig. 4, which is completely consistent with the results obtained by Fedorov et al. (2003). 4. Numerical results and discussion In view of the mechanical model and data presented in Sections 2 and 3, the critical condition ( E = 0), especially the critical stress σ c , can be determined theoretically by formula (54). To facilitate quantitative analyses in the following illustration, the surface of nanocrack and matrix properties are assumed to be isotropic. The material parameters used in the following analyses are as follows: Burgers vector bA of the GBDs has a small magnitude of 0.1 nm (Ovid’ko and Sheinerman, 2005; Sutton and Balluffi, 1995), shear modulus μ = 82 GPa and Poisson ration υ = 0.29

Fig. 5. Normalized critical stress σ 0c /μ versus normalized distance ε /d for the different surface stresses τ s0 for α = 45°, ω = 30°, ρ /p = 10 − 5 , d = 25 nm, N = 5 and p = 50 nm.

(Kuhn and Medlin, 20 0 0) for BCC Fe. To determine the surface stress, the residual surface tension τ s must be known. Former studies have shown that the absolute value of normalized intrinsic length τ s0 ( = τ s /μ) is nearly 0.01 nm (Miller and Shenoy, 20 0 0), which plays a very important role in characterizing the nanoscale surface properties. Furthermore, in the following calculations, let a = 0.3d and η = ω, unless otherwise specified. Thus, the influence of various parameters of the blunt pre-nanocrack and the characteristics of grain and GB under the critical condition ( E = 0), especially the critical external stress, above which the splitting transformation can occur, can be determined easily in NCM. Fig. 5 depicts a plot of the normalized critical stress σ 0c /μ initiating the splitting transformation of the first head GBD within pile-up with respect to the normalized distance ε /d, at which ε represents the distance moved by the first and second resultant GBDs, for different surface stresses. In our analyses, we suppose that the distance α moved by the resultant GBDs cannot exceed the nanograin boundary length d (nanograin size). The figure shows that an increase in distance ε makes the normalized critical stress σ 0c /μ rapidly decrease and promotes the splitting transformation, which is consistent with the result of the work of

Please cite this article as: Y. Zhao, L. Xu, Effect of blunt nanocracks on the splitting transformation of grain boundary dislocation piled up at triple junctions, International Journal of Solids and Structures (2018), https://doi.org/10.1016/j.ijsolstr.2018.02.025

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Fig. 6. Normalized critical stress σ 0c /μ versus angle α for different surface stresses τ s0 for ω = 30°, N = 5, ρ /p = 10−5 , p = 50 nm, d = 25 nm and ε = 0.5d.

Gutkin and Ovid’ko (2003), Fedorov et al. (2003), and Han et al. (2014). In addition, the positive (negative) surface stress characterized by positive (negative) residual surface stress decreases (increases) the critical stress σ 0c /μ, and contributes to (holds back) the splitting transformation. This implies that the surface stress characterized by positive (negative) residual surface stress significantly decreases (rises) the energy of the splitting transformation. As it follows from these dependences, we can see that the impact of surface stress is increasingly obvious as the resultant GBDs are far away from TJ A. From the detailed analysis, it can be observed that there exists a critical distance ε c , at which the splitting transformation can occur in a non-barrier way with the positive surface stress, for example, ε c /d ≈ 0.87 in this case. The dependence of the normalized critical stress σ 0c /μ on the angle α of GBO2 A plane with the x2 -axis is presented in Fig. 6 for different surface stresses τ s0 = τ s /μ. It can be seen that with the increase in angle α , the normalized critical stress σ 0c /μ first obviously grows, reaches its maximum, and then reduces gradually, leading to a critical value of the angle α c , which makes the critical stress reach the maximum value and the splitting transformation most difficult to occur for the other given parameters. However, the effect of angle α on the critical stress σ 0c /μ is inconsistent with that reported in the work of Fedorov et al. (2003), Gutkin and Ovid’ko (2003), and Gutkin et al. (2003), largely because of the effect of the interaction between GBDs within pile up at TJ A and nanocrack. Interestingly, regardless of whether the surface stress is considered or not, all the curves of normalized critical stress σ 0c /μ almost intersect at a point α 0 , at which the effect of the surface stress on critical stress can be negligible. When α < α 0 , the critical stress σ 0c /μ reduces (rises) with the positive (negative) surface stress, and the splitting transformation occurs more easily (difficultly). However, when the angleα is larger than the critical value α 0 , a contrary effect of surface stress on the critical stress is presented. Furthermore, when the negative surface stress appears, only the external load is applied, and for a certain value of stress, the splitting transformation may occur in a certain range of the angle α , in this case 32° < α < 77°. That is, if the angle α is small or large enough, the barrierless splitting transformation of GBD occurs effectively, and the transition from the barrierless splitting to the barrier one occurs at a certain critical value. These findings theoretically and clearly indicate that material hardening or softening of nanocrack surfaces due to residual surface stress has a significant effect on critical stress and other critical conditions for

9

Fig. 7. Normalized critical stress σ 0c /μ versus angle α for different nanocrack lengths for ω = 30°, N = 5, ρ /p = 10−5 , d = 25 nm and ε = 0.5d.

Fig. 8. Normalized critical stress σ 0c /μ versus grain size d for different surface stresses τ s0 for α = 55°, ω = 30°, N = 5, ρ /p = 10−5 , p = 50 nm and ε = 0.5d.

such a splitting transformation of GBD piled up at TJ, and its influence is also affected by the angle α . Fig. 7 illustrates the dependence of the normalized critical stress σ 0c /μ on the angle α between the GBO2 A plane and thex2 axis for different nanocrack lengths without surface stress effect. As shown in Fig. 7, a growing nanocrack makes the normalized critical stress σ 0c /μ higher for a given angle α , indicating that the splitting transformation occurs more easily for a short nanocrack. It may be because that as a nanocrack grows, the local stress concentration diminishes, which is due to the interaction of the applied load and GBDs piled up at TJ A with the nanocrack. It is obvious that the effect of nanocrack length on the critical stress is strongly dependent on the angle α . Fig. 8 plots the dependence of the normalized critical stress σ 0c /μ on the grain size d for different surface stresses τ s0 . It reveals that the grain size itself strongly influences the splitting transformation. The normalized critical stress σ 0c /μ drops considerably with grain coarsening, especially for small grains. This signifies that the decrease in grain size in NCMs hinders the splitting transformation. Considering that a is the distance from the intersection point O2 of the blunt nanocrack tip and GBO2 A to the nearest TJ A, when the grain size is relatively small, a = 0.3d is approximately a few nanometers. In these circumstances, the dis-

Please cite this article as: Y. Zhao, L. Xu, Effect of blunt nanocracks on the splitting transformation of grain boundary dislocation piled up at triple junctions, International Journal of Solids and Structures (2018), https://doi.org/10.1016/j.ijsolstr.2018.02.025

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Fig. 9. Normalized critical stress σ 0c /μ versus nanocrack blunting ρ for different surface stresses τ s0 for α = 55°, ω = 30°, N = 5, d = 25 nm, p = 50 nm and ε = 0.5d.

Fig. 10. Normalized critical stress σ 0c /μ versus angle ω for different surface stresses τ s0 for α = 55°, ρ /p = 10−5 , d = 25 nm, N = 5, p = 50 nm and ε = 0.5d.

tance from GBDs piled up at TJ A to the blunt nanocrack tip is relatively short, it leads to the enormous shielding effect to the splitting transformation of the first head GBD produced by the interaction of GBDs piled up at TJ A with nanocrack, which is distinctly stronger as GBD pile-up approaches the blunt nanocrack tip. The stress fields of these GBDs compensate, in part, for the local stresses near the nanocrack tip for a certain grain size, thus playing the dominant role in the shielding effect (Bobylev et al., 2010). Accordingly, because of GBD shielding effect, the growing grain size relatively stimulates the splitting transformation. Li et al. (2014) developed a theoretical model for nanograin rotation that could be achieved through dislocation climb and showed that the occurrence of nanograin rotation and the coalescence of grains depend on the external stress level and grain size. Moreover, the critical shear stress decreases with the coarsening of grains. These results agree with those shown in Fig. 8. Moreover, the positive (negative) surface stress first decreases (increases) the critical stress and then facilitates (suppresses) the splitting transformation for the other given parameters. There exists a critical grain size dc for the barrierless splitting transformation. The positive (negative) surface stress sharply decreases (enhances) the critical grain size dc , i.e., dc ≈ 90 nm without the surface stress effect, dc ≈ 50 nm with a positive one, and dc ≈ 120 nm with a negative one. This implies that as the grain becomes finer, the splitting transformation may be energetically favorable, accounting for the effect of the positive surface stress. Furthermore, as the grain coarsens to a certain value, the influence of the surface stress on the nanocrack surface is relatively more obvious. We vary the normalized curvature radius ρ /p of nanocrack tip, which characterizes nanocrack blunting, to investigate its influence on the normalized critical stress σ 0c /μ near the nanocrack tip for different surface stresses. When the surface stress is ignored, as shown in Fig. 9, the critical stress σ 0c /μ slightly decreases first and then remarkably enhances with nanocrack blunting, leading to the critical curvature radius ρ c to make the splitting transformation easiest to occur for the other given parameters. Because the nanocrack blunting can significantly modify the stress field around the nanocrack tip and generally release the partial stress near the nanocrack tip, and then the splitting transformation of GBD piled up at TJ A can be hampered. The curves also indicate that there exists a threshold stress to make the splitting transformation energetically favorable, below which no energetically favorable splitting transformation occurs regardless of the curvature radius. As the nanocrack blunts, the critical stress σ 0c /μ reduces (rises) with

the positive (negative) surface stress, and the splitting transformation occurs more easily (difficultly). A negative stress slightly increases the critical curvature radius ρ c . Fig. 10 shows the effect of the angle ω, characterizing the geometry of the splitting between the Burgers vector bA of head GBD and the Burgers vector bT1 of the first resultant GBD, on the normalized critical stress σ 0c /μ for different surface stresses. Herein, for facilitating the analysis, we assume that η = ω. This figure demonstrates that the normalized critical stress σ 0c /μ increases markedly with the increase in the value of the angle ω. In other words, an increase in the angle ω of GB hinders the splitting transformation. This result is in fact the same as that found by Fedorov et al. (2003). Thus, the splitting transformation occurs effectively at TJ A with a small GB angle. Apparently, the effect of the surface stress gradually becomes more visible as the value of the angle ω increases. Figs. 11a and b show the normalized critical stress σ 0c /μ as a function of the angle η, characterizing the geometry of the splitting between the Burgers vector bA of head GBD and the Burgers vector bT2 of the second resultant GBD, for different angles ω and surface stresses τ s0 for α = 55°, ρ /p = 10−5 , d = 25 nm, N = 5, p = 50 nm and ε = 0.5d. The results reveal that the normalized critical stress σ 0c /μ markedly increases with an increase in the angle η. A positive (negative) surface stresses decreases (rises) the normalized critical stress σ 0c /μ and contributes to (holds back) the splitting transformation. The impact of surface stress is increasingly visible with the increase in the angle ω. Moreover, when the angle η reduces to a certain value ηc , the splitting transformation can occur in a non-barrier way. The positive (negative) surface stress slightly increases (decreases) ηc . As shown in Fig. 11a, the curves of the normalized critical stress σ 0c /μ intersect at a certain angle η0 , η0 ≈ 25° in this case. A larger angle ω induces a slight decrease in the normalized critical stress σ 0c /μ in the case of η < η0 , while a distinct increase in the case of η > η0 . This implies that the increasing angle η evidently enhances the effect of the angle ω on the normalized critical stress. Variation observed in the normalized critical stress σ 0c /μ with the number ε c1 of GBDs composing a pile-up in the initial stage at TJ A is given in Fig. 12 for different surface stresses. It is found that a further increase in the number N of GBDs causes a slight increase in the critical stress σ 0c /μ in an almost linear way. The results obtained by the work of Fedorov et al. (2003) show that the critical stress decreases with an increase in the number N of GBDs, which is contrary to our findings. This may be because an increase in the

Please cite this article as: Y. Zhao, L. Xu, Effect of blunt nanocracks on the splitting transformation of grain boundary dislocation piled up at triple junctions, International Journal of Solids and Structures (2018), https://doi.org/10.1016/j.ijsolstr.2018.02.025

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Fig. 11. Normalized critical stress σ 0c /μ versus angle η for different angles ω and surface stresses τ s0 for α = 55°, ρ /p = 10−5 , d = 25 nm, N = 5, p = 50 nm and ε = 0.5d.

Fig. 12. Normalized critical stress σ 0c /μ versus number N for different surface stresses τ s0 for α = 55°, ω = 30°, ρ /p = 10−5 , d = 25 nm, p = 50 nm and ε = 0.5d.

number N of GBDs can induce a more intensive shielding to the stress intensity near the vicinity of nanocrack tip, and impede the splitting transformation. The positive (negative) surface stress decreases (enhances) the critical stress σ 0c /μ and makes the splitting transformation more easy (difficult) for the other given parameters.

5. Conclusion We used a theoretical model to describe in detail the splitting transformation of GBD piled up at TJ A near a blunt nanocrack in plastically deformed and mechanically loaded NCM. The influence of various parameters of the blunt pre-nanocrack, the characteristics of grain and GBs, and the surface stress on critical conditions for the splitting transformation of the first head GBD within pileup at TJ A is determined clearly in theory. The main results of our work are summarized as follows: (1) The positive (negative) surface stress decreases (increases) the critical stress, significantly decreases (rises) the energy of splitting transformation, and then facilitates (suppresses) the very beginning of the GB deformation. (2) The increase in distance ε rapidly decreases the critical stress, and promotes the splitting transformation. A critical distance ε c exists, at which splitting transformation can occur in a non-barrier way with positive surface stress, for example, ɛc /d ≈ 0.87 in this condition.

(3) A critical angle α c exists, which makes splitting transformation most difficult to occur. Regardless of whether the surface stress is considered or not, all the curves of normalized critical stress intersect at a point α 0 , at which the effect of surface stress on critical stress can be negligible. When α < α 0 , the critical stress decreases (rises) with positive (negative) surface stress. However, when α > α 0 , the contrary effect of surface stress is presented. When negative surface stress appears, only the external load is applied, and for a certain value of stress, the splitting transformation may occur in a certain range of the angle α , in this case 32° < α < 77°. The splitting transformation occurs more easily for a short nanocrack, and the effect of nanocrack length on critical stress is strongly dependent on the angle α . (4) Critical stress slightly reduces and then remarkably enhances with nanocrack blunting, thus leading to a critical curvature radius to make splitting transformation easiest to occur. There exists a threshold stress to make splitting transformation energetically favorable, below which no energetically favorable splitting transformation occurs regardless of the curvature radius. (5) Critical stress decreases considerably with coarsening of grain, and with decrease in the number N of GBDs within pile-up, and the values of the angles ω and η. There exist critical grain size dc and angle ηc for the barrierless splitting transformation. The positive (negative) surface stress sharply reduces (rises) the critical grain sizedc and slightly increases (decreases) the critical angle ηc . The effect of surface stress gradually becomes more visible as the angle ω increases. Increase in the angle η evidently enhances the effect of the angle ω on critical stress.

Acknowledgments The authors would like to deeply appreciate the support from the NNSFC (11602170) and China Postdoctoral Science Foundation (2016M590199).

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Please cite this article as: Y. Zhao, L. Xu, Effect of blunt nanocracks on the splitting transformation of grain boundary dislocation piled up at triple junctions, International Journal of Solids and Structures (2018), https://doi.org/10.1016/j.ijsolstr.2018.02.025