Effect of nanograin boundary sliding on nanovoid growth by dislocation shear loop emission in nanocrystalline materials

European Journal of Mechanics A/Solids 49 (2015) 419e429

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Effect of nanograin boundary sliding on nanovoid growth by dislocation shear loop emission in nanocrystalline materials Yingxin Zhao a, b, Qihong Fang a, b, *, Youwen Liu a, b a b

State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha, 410082, PR China College of Mechanical and Vehicle Engineering, Hunan University, Changsha, 410082, PR China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 22 May 2013 Accepted 14 September 2014 Available online 22 September 2014

A theoretical model is suggested to describe the effect of nanograin boundary (NGB) sliding at triple junctions of grain boundaries on nanovoid growth by incipient emission of dislocation shear loop from its surface in mechanically loaded nanocrystalline materials. In the framework of the model, NGB sliding deformation represents generation of disclination dipoles at triple junctions of NGBs. The analytical expression of critical stress for the first dislocation emission is derived. The effects of NGB sliding deformation, the sizes of nanograin and nanovoid on critical stress and the corresponding most favorable slip plane for dislocation emission are evaluated quantitatively in the deformed nanocrystalline materials. The results indicate that NGB sliding deformation releases, in part, the high stresses near the nanovoid, thereby nanovoid growth is slowed down or even arrested with the increment of the strength of NGB sliding deformation in nanocrystalline solids. There exists a range of nanovoid size to make critical stress keep almost unchanged with the appearance of NGB sliding deformation. There is also a critical nanograin size to make critical stress reach the maximum value, at which dislocation emission and nanovoid growth occur most difficultly. © 2014 Elsevier Masson SAS. All rights reserved.

Keywords: Nanograin boundary sliding Nanovoid growth Critical stress

1. Introduction Ductile failure in nanocrystalline metals and ceramics normally takes place and the failure processes are the subject of rapidly growing research efforts motivated by the development of technologies exploiting the outstanding mechanical properties of these materials (Meyers et al., 2006; Tvergaard and Vadillo, 2007; Pastor and Kondo, 2013). The overall ductility of nanocrystalline materials is mainly controlled by the ability of the material to withstand the growth of nano- or micron-sized voids up to the point at which fracture is triggered by the sudden coalescence of neighbor voids into a crack (Tvergaard, 2001). These voids can span several orders of magnitude from the nanoscale range to about 100 microns (Niordson, 2008). Since Rice and Tracey (1969) first proposed the model of void ductile growth based on continuum plasticity and studied the influences of the remote tension and strain rate on microscopic void growth rate and the volume changes. A significant amount of experimental observations (Becker et al., 1989),

* Corresponding author. State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha, 410082, PR China. Tel.: þ86 731 89822841; fax: þ86 731 88822330. E-mail addresses: [email protected], [email protected] (Q. Fang). http://dx.doi.org/10.1016/j.euromechsol.2014.09.003 0997-7538/© 2014 Elsevier Masson SAS. All rights reserved.

computational simulations (Mathur et al., 1994; Tvergaard, 2012; Tang et al., 2012) and theoretical analyses (Mathur et al., 1996; Lubarda et al., 2004; Flandi and Leblond, 2005; Bilger et al., 2007; Tszeng, 2008; Meyers et al., 2009) have been carried out to characterize and assess the effects of different factors on nano or micron-sized void growth, including different stress states and strain rates, surface energies, lattice frictions as well as void shapes, sizes, spatial distributions and volume fractions, etc. These researches indicate that dislocation shear loop is nucleated at random site along the slip plane when the magnitude of the local resolved shear stress exceeds a critical value over a nucleation period of time. Once generated, dislocations glide along their slip planes, interact, possibly annihilate, pileup and exit at the free surface of the void. The emission and outward expansion of this dislocation loop by externally applied stresses carry away the material from the void and are responsible for the plastic deformation needed to accommodate significant void growth. However, concerning the continuous process of void growth and its synchronization with dislocation activity, some studies (Bringa et al., 2010; Bulatov et al., 2010) lately argued that the void growth proceeds by the emission of a special type of dislocation shear loop, which can expand as the partial or perfect dislocations, and then evolve into a prismatic loop through reaction, or develop into twin. Many recent MD simula€la € et al., 2004; Rudd et al., 2007) tions (Marian et al., 2004; Seppa

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Y. Zhao et al. / European Journal of Mechanics A/Solids 49 (2015) 419e429

also confirmed that the eventual formation of prismatic loop is a necessary and unavoidable stage of void growth. In the light of these investigations, dislocation mechanics have been verified to be critical to the initial stage of void growth in the ductile fracture of materials. Due to superior mechanical and physical properties, nanocrystalline metals and ceramics have represented the subject of intensive research efforts in recent years (Ovid'ko, 2007). Nanocrystalline materials exhibit either ductile or brittle fracture behavior, depending on their structural characteristics and the conditions of mechanical loading. Many experimental reports have shown that failure in ductile nanocrystalline is typically characterized by void and dimple and is viewed to occur through the void nucleation and coalescence mechanisms (e.g. Tvergaard, 1997; Li and Ebrahimi, 2005). From the experimental observations and theoretical analyses, nanovoids in nanocrystalline metals and ceramics normally appear within grain boundaries (GBs) and their triple junctions, and then their growth is driven by plastic deformation (Westwood et al., 2000, 2004; Querin et al., 2007; Ovid'ko et al., 2011). The plastic deformation of nanocrystalline solids is definitely controlled by their specific structural features such as nanoscopic sizes of grains, large amounts of GBs and triple junctions. A nanocrystalline material is a polycrystal with grain sizes of characteristic linear dimension less than 100 nm. GBs and their triple junctions due to high-density ensembles are characteristic constituents of nanocrystalline materials. They serve as obstacles for conventional lattice dislocation glide and crucially affect the outstanding mechanical behavior of nanocrystalline materials (see, e.g. Zhou et al., 2008a; Shi and Zikry, 2009), such as very large value of the flow stress and the strengthening effect under superplastic deformation. However, when the mean grain size is lower than some critical value, GBs in nanocrystalline solids can effectively carry plastic flow and operate the softening structural elements during plastic deformation (Borg, 2007; Ganghoffer et al., 2008; Wei and Gao, 2008). In these cases, the reduction in the grain size plays the dominant role in toughness enhancement. In any event, the mechanical characteristics of nanocrystalline metals and ceramics crucially depend on nanograin size and specific deformation mechanisms operating in these materials, and the ductility of nanocrystalline materials should be adequately described in terms of the deformation mechanisms. Recently, rapidly growing attention has been paid to GB sliding of the dominant plastic deformation mechanism in nanocrystalline materials, which represents a strengthening or softening micromechanism under different circumstances (see, e.g., Fleck et al., 1989; Zhou et al., 2008b), and a specific deformation mode (Gutkin et al., 2003; Zhao et al., 2013). Rudd and Belak (2002) used atomistic simulations to investigate the influence of pre-existing microstructure on void growth, such as dislocation activity, grain boundary sliding, twinning, and so on. They found that the growth of nanoscale void in polycrystalline copper depends strongly on the void location relative to grain boundaries and their junctions, and the interaction of the void with intersecting boundaries and dislocation structure. Computer simulations (Latapie and Farkas, 2004) showed that lattice dislocation emission and GB sliding deformation process contribute approximately 60% and 40%, respectively, to the blunting of a crack in nanocrystalline a-Fe with grain size d ¼ 9 nm. Stress concentration near pre-existent/growing blunt crack induces GB sliding which leads to formation of dislocations at triple junctions of GBs (Ovid'ko and Sheinerman, 2012). The superposition of the external stresses concentrated near the tips of blunt cracks and stresses created by these dislocations are capable of initiating generation and growth of nanocracks (Ovid'ko and Sheinerman, 2005; Zhou et al., 2006; Wu et al., 2007; Ovid'ko and Sheinerman, 2008). Wang et al. (2011) suggested a theoretical

model to describe the growth of nanovoid at triple junction of nanocrystalline metal under equal biaxial remote stress. Within their description, dislocations emitted from nanovoid surface are stopped at GBs and the stress field generated by arrested dislocations can prevent further dislocation emission. The critical stress is derived for dislocation emission by considering the effects of applied stress, image stress introduced by existing nanovoids and surface stress. However, the effects of NGBs and their deformation are ignored. With the role of nanovoids being crucial in fracture process in nanocrystalline materials, the importance of theoretically studying the influence of NGB deformation on the nanovoid growth to understand and control ductile fracture has been recognized. In addition, most initial voids in materials are usually of nanoscale or micron size, and thus their growth is also scaledependent, as directly or indirectly evidenced by some experiments and theoretical analyses (Tvergaard, 1996; Tvergaard and Niordson, 2004; Zhou et al., 2007; Lubarda, 2011). One of relatively simple ways to cope with this scale-dependence within a continuum description is the non-classical approach of surface/ interface elasticity of Gurtin-Murdoch (Gurtin and Murdoch, 1975, 1978), which is a common place in the theory of elasticity accounting for the surface/interface stresses. Since NGBs end at triple junctions, which, in the initial state of the defect configuration, are assumed to be geometrically balanced in the sense that misorientation angles of their adjacent grain boundaries are balanced. This indicates that there are no misorientation angle gaps at the triple junctions, and they do not create internal stresses. NGB sliding gives rise to movement of the triple junctions from their initial positions to new positions. This process of NGB sliding through triple junctions generally suffers from misorientation mismatch and is capable of producing the dipoles of wedge disclinations (rotational defects associated with crystal lattice orientation incompatibilities) at and near these junctions (Padmanabhan and Gleiter, 2004; Mohamed, 2007; Ovid'ko and Sheinerman, 2009; Luo and Liu, 2011). Because in the theory of defects in solids, a wedge disclination at a GB is normally defined as a line defect dividing GB fragments characterized by different tilt misorientation angles (Kleman and Friedel, 2008). The strength of wedge disclination represents the difference between the tilt misorientation angles of the interface fragments (see, Romanov and Kolesnikova, 2009; Bobylev et al., 2011). GBs misorientations also play a significant part in NGB sliding, since at low angle regime, higher misorientation corresponds to higher GB energy (Watanabe et al., 1999). At high angle regime, there is no direct relation between misorientation and GB energy (Monzen et al., 1990). Shi and Zikry (2009) showed that the maximum GB sliding for a random intermediate angle model is approximately 3 nm and occurs at a GB  with misorientation angle of 19 , and this is approximately 50% higher than the low angle GB aggregate. Wedge disclinations with the strength ±u produced by GB sliding in nanocrystalline materials frequently serve as highly powerful stress sources. Ovid’ko and Sheinerman (2007) demonstrated that in contrast to Ni for u ¼ p/4, nanocrack generation at disclination dipoles formed by GB sliding in nanocrystalline Cu, due to lower flow stress and shear modulus, is favorable only at very large values of u ¼ p/3 or more. Ovid'ko and Skiba (2012) calculated the dependence of the critical stress, the minimum external stress at which the process of the dislocation emission from GBs is energetically favorable, on the disclination strength in the cases of nanocrystalline silicon carbide and nanocrystalline aluminum. When the disclination strength u ranges    from 10 to 30 , the ratio tc(u ¼ 0 )/tc(u) ranges from 1.5 to 8.1 in the case of SiC and from 1.4 to 4.5 in the case of Al. Within these descriptions, the main aim of this paper is to suggest and theoretically describe the growth of nanovoid at triple junction by the emission of the incipient dislocation shear loops,

Y. Zhao et al. / European Journal of Mechanics A/Solids 49 (2015) 419e429

and then obtain critical stress for dislocation emission from nanovoid surface in deformed nanocrystalline materials. The impacts of the stresses produced by disclinations dipoles formed due to NGB sliding deformation, the sizes of nanograin and nanovoid on nanovoid growth are addressed quantitatively. This work highly identifies that NGB sliding deformation is capable of suppressing nanovoid growth at triple junction. With the results of our theoretical analyses achieved, it is natural to expect that it can be used as quantitative tool to understand material ductile failure by nanovoid evolution under the condition of GB deformation. This is an essential aspect of the design analyses of structures potentially targeted by explosive or projectile impacts. 2. Modeling and solution Within our model, NGB sliding occurs under the applied remote tensile stress or induced by stress concentration near pre-existent/ growing blunt cracks. In the present study, we consider the situation that due to NGB sliding occurrence, the angle gaps u and u appear at the nearest NGB triple junction p(zp ¼ reid) and the corresponding triple junction n(zn ¼ (zp þ qei2)eid), respectively, whose gaps gradually grow during the formation process (Wu et al., 2007; Ovid'ko et al., 2011). Often, NGB sliding occurs on (many) different boundaries at the same time. For simplicity, our theoretical analyses are only focused on the structural transformations (re-arrangements of NGB dislocation structure) of the NGB pn, while the neighboring NGBs are handled as those having unchanged structures of NGB. In this case, the role of the neighboring NGBs is in only their contribution to the misorientation balance at the triple junctions p and n. With this role, the triple junctions p and n with the angle gaps ±u due to the NGB pn sliding are considered as a dipole of wedge disclinations configuration with the strengths ±u and the arm q, perpendicular to the corresponding NGB planes adjacent to the triple junction (Romanov, 2003; Kleman and Friedel, 2008). In doing so, the wedge disclinations with the strengths ±u compensate for misorientations of NGB fragments pn in the initial state. They are shown schematically in Fig. 1b. The arm q is the distance by which a triple junction shifts due to NGB sliding and in fact characterizes local plastic strain carried by NGB sliding in the vicinity of the triple junction under consideration (Wu et al., 2007; Ovid'ko and Sheinerman, 2008). As NGB sliding proceeds, the distance q between the disclinations gradually increases. Here, we suggest that the characteristic distance q is in the range from several crystal lattice parameters until the NGB length. NGB sliding is characterized by the macroscopic plastic shear strain ε1 ¼ ε2q/d, where ε2(<1) is the fraction of NGBs that carry NGB sliding and d is a mean nanograin size (Ovid'ko and Sheinerman, 2009). Note that, according to this definition, the plastic tensile strain along the loading direction is equal to ε1 under consideration. When the effect of sliding on a GB is represented by a disclination dipole, there is complete continuity of tractions and displacements along the boundary. This is a limiting case, when NGB sliding deformation can be characterized by the continuous mathematical function. Normally, sliding boundary does not resist shear stresses, here normal stresses and displacements remain continuous while tangential displacements may have a discontinuity. In the present study, however, assume the NGB sliding has been completed. Furthermore, NGB sliding through triple junctions of GBs leads to misorientation mismatch and produces the dipoles of wedge disclinations at and near these junctions. Herein, we mainly evaluated quantitatively the effect of disclination dipole due to NGB sliding deformation on dislocation emission in the deformed nanocrystalline materials. For simplicity of mathematics, the nanovoid is assumed to have a cylindrical form, which can greatly facilitate practical application

421

of the model. We consider that a potential void nucleation or grow site is isolated from the other potential sites and therefore the effect of the interaction between neighboring voids is negligible. Based on the physical ideas for dislocation emission from nanovoid surface elucidated in Lubarda (2011), suppose it takes place by the emission of multiple perfect edge dislocation dipoles with opposite Burgers vectors along the same slip plane, during which one dislocation with positive Burgers vector b is triggered and emitted toward the interior of the crystal and the rest other with negative Burgers vector b toward the nanovoid surface. Once the dislocation emission occurs, the nanovoid grows one Burgers vector (Fig. 1b) (Stevens et al., 1972; Wang et al., 2011). With the emission of dislocation dipoles, nanovoid growth increases gradually and eventually tends to asymptotic value. In this case, the material associated with the creation of the dislocation shear loops by such dipole emission gives rise to the expansion of the nanovoid and the corresponding outward transport from its surface.

Fig. 1. Nanograin boundary (NGB) sliding near the nanovoid in a deformed nanocrystalline solid: (a) general schematic view and (b) the magnified inset highlights a wedge disclination dipole due to NGB sliding and dislocation emission from the nanovoid surface.

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The two-dimensional problem will be solved analytically under remote tensile stresses (Fig. 1). Consider an edge dislocation dipole near a cylindrical nanovoid of radius R under external biaxial tensile stresses in an infinitely extended isotropic elastic matrix. The NGB sliding has been completed and the wedge disclination dipole has been produced. In this case, the emitted edge dislocation subjects to three forces: the first is produced by a wedge disclination dipole in the deformed nanocrystalline with a nanovoid; the second is due to the emitted edge dislocation exerted by the free surface of nanovoid and the rest edge dislocation at nanovoid surface; the third is generated from the applied remote stresses. The micromechanical model to be analyzed here is a perfectly cylindrical nanovoid undergoing the remote tensile ∞ loading s∞ 11 and s22 in infinite nanocrystalline solid, schematically presented in Fig. 1b. For brevity, we restrict our consideration to a two-dimensional and elastically isotropic grain structure. Assume the emitted dislocation with Burgers vector b1 ( ¼ bx þ iby ¼ bzei(q þ 4)) is located at z1, the rest edge dislocation with Burgers vector b0( ¼ b1) is located at the surface of the nanovoid z0 ¼ Rei4. Consider the situation where the dislocation dipoles are of edge character and their Burgers vectors lie along the slip plane that makes an angle q with the x-axis on the nanovoid surface. For plane strain problem, stress fields are expressed in terms of Muskhelishvili's complex potentials (Muskhelishvili, 1975) F(z) and. J(z)

h i syy þ sxx ¼ 2 FðzÞ þ FðzÞ

(1)

h 0 i syy  sxx þ 2isxy ¼ 2 zF ðzÞ  JðzÞ

(2)

srr þ isrq ¼

 ts 2ms þ ls  ts m ðsqq þ srr Þ þ ðsqq  srr Þ þ lþm R 4Rm  vðsqq  srr Þ m vðsqq þ srr Þ i i vq vq lþm (5)

Now, the stress fields produced by a wedge disclination dipole in the deformed nanocrystalline solid with a nanovoid are calculated in the following. In view of Fang et al. (2006), two complex potentials Fw(z) and Jw(z) caused by a wedge disclination dipole whose disclination lines located at the points zp( ¼ reid) and zn[¼ (zp þ qei2)eid] in the infinite matrix can be expressed as follows

Fw ðzÞ ¼

  Du   ln z  zp  lnðz  zn Þ þ Fw0 ðzÞ 2

zp Du zn  Jw ðzÞ ¼ 2 z  zn z  zp

'

srr ðtÞ þ isrq ðtÞ ¼

  vss ðtÞ 1 s sqq ðtÞ  i qq R vq

jtj ¼ R

ssqq ¼ ts þ ð2ms þ ls  ts Þεsqq

(4)

where ssqq and εsqq denote surface stress and strain, ms and ls are surface Lame constants, ts is the residual surface tension. For a coherent surface, surface strain εsqq is equal to the associated tangential strain in the abutting bulk material. Referring to the work of Fang and Liu (2006), and considering the additional constitutive equation for surface region in Eq. (3), the nonclassical stress boundary condition on the entire circular surface can be written as

! þ Jw0 ðzÞ

jzj > R

(7)



R2 z

þ

  2 R2 0 R2 R2 R þ 2 Jw Fw z z z z

jzj < R

(8)

The substitution of Eqs. (6) and (7) into Eq. (8) yields

F*w ðzÞ

*

Du z  z*n zp  zp zn  z*n zn  zp ln ¼ þ  þ 2 z z  z*p z  z*p z  z*n þ F*w0 ðzÞ

(3)

According to Povstenko (1993), the surface region is given as

(6)

where D ¼ m/[2p(1  y)], m is the shear modulus, y is Poisson's ratio of the infinite matrix. Fw0(z) and Jw0(z) are holomorphic in the matrix. To easily treat the boundary condition on the surface and by using the RiemanneSchwarz’s symmetry principle, it is convenient for the mathematical calculation to assume and introduce the following new analytic function:

F*w ðzÞ ¼ Fw where F (z) ¼ d[F(z)]/dz, the overbar represents the complex conjugate. Following the work of Gurtin and Murdoch (1975), and Sharma et al. (2003), considering the surface effect for the current problem, the boundary condition at the nanovoid surface can be expressed as

jzj > R

jzj < R

!

(9)

where z*p ¼ R2 =zp ,z*n ¼ R2 =zn ,F*w0 ðzÞ is holomorphic complex function in the nanovoid region jzj
Jw ðzÞ ¼

  2  0 R2 * R  zF F ðzÞ þ F ðzÞ w w w z z2

jzj > R

(10)

Substituting Eqs. (1) and (2) into Eq. (5), and considering Eq. (8), the following equation can be obtained:

  2  þ   2 0 0 R R2 0 R2 R  ða þ bÞ Fw ð1 þ aÞF*w ðtÞ þ atF*w ðtÞ þ ða þ bÞFw ¼ ð1  a  bÞFw ðtÞ  ða þ bÞtFw ðtÞ  aF*w t t t t jtj ¼ R   R2 *0 R2 ts  þ a Fw t t R

(11)

Y. Zhao et al. / European Journal of Mechanics A/Solids 49 (2015) 419e429

where a¼(2ms þ ls  ts)/(4Rm),b¼(2ms þ ls  ts)/[2R(l þ m)]. The superscripts þ and  denote the boundary values of a physical quantity as z approaches the circular surface from the hole and matrix region, respectively. The substitution of Eqs. (6) and (9) into Eq. (11) yields

423

where Eq. (13) in the region jzjR is the solution to Eq. (12) inside the matrix. They are no sense in actual physical meaning.

  2  2 þ   2 0 0 R R2 0 R R ð1 þ aÞF*w0 ðtÞ þ atF*w0 ðtÞ þ ða þ bÞFw0  ða þ bÞ Fw0  ð1  a  bÞFw0 ðtÞ  ða þ bÞtFw0 ðtÞ  aF*w0 t t t t þa

  R2 *0 R2 ts  Fw0 ¼ gðtÞ jtj ¼ R t t R

(12)

The above two first-order differential equations with variable coefficients can be solved by the power series method. Note that

where

. .  h i z h i z

  p n gðtÞ ¼½ð1  bÞDu=2ln t  zp =ðt  zn Þ þ Du=2 azp zp R2  a  b  Du=2 azn zn R2  a  b þ aDu=2 zp zp  zn zn t  zp t  zn       .

.  2 2 *2 * R2 t þ aDu=2 þ aDu zp  zn t R2  ða þ bÞDu=2lnzp t  zp z*2 z  z ðt  z Þ z R2  aDu=2 zp  z*p R2 t n n p n n .   i.



h

 t  z*p þ Du zn  aDu=2lnzp zn  ts =R þ ½ð1  bÞDu=2ln t  z*p = t  z*n  Du=2 ð1 þ aÞzp  ð1 þ a  bÞz*p 

2

        . 2 t  z*p  aDu=2 zn  z*n t t  z*n t  z*n  Du=2 zn  zp t þ aDu=2 zp  z*p t =2 ð1 þ aÞzn  ð1 þ a  bÞz*n

By the use of Plemeji formulation (Muskhelishvili, 1975), the solutions of Eq. (12) can be derived explicitly as

0

ð1 þ aÞF*w0 ðzÞ þ azF*w0 ðzÞ þ ða þ bÞFw0



R2 z

 ða þ bÞ

the complex potentials Fw0(z) and F*w0 ðzÞ can be taken from the following series expansions

 2 h   R2 0 R ¼½ð1  bÞDu=2ln z  zp =ðz  zn Þ þ Du=2 azp zp Fw0 z z . . i z h i z p n R2  a  b  Du=2 azn zn R2  a  b z  zp z  zn  .

R2  aDu=2 zp  z*p R2 z þ aDu=2 zp zp  zn zn       2 2 *2 * R2 z þ aDu=2 z  z ðz  z Þ z z  zp z*2 n n p n n  . .

þ aDu zp  zn z R2  ða þ bÞDu=2lnzp zn  aDu=2lnzp zn  ts =R

jzj < R (13)

0

ð1  a  bÞFw0 ðzÞ  ða þ bÞzFw0 ðzÞ  aF*w0



R2 z

þa

  h  

R2 *0 R2 ¼½ð1  bÞDu=2ln z  z*n = z  z*p þ Du=2 ð1 þ aÞzp Fw0 z z i.

   z  z*p  Du=2 ð1 þ aÞzn  ð1 þ a  bÞz*n  ð1 þ a  bÞz*p 

2

  

 z  z*p z  z*n þ Du=2 zn  zp z  aDu=2 zp  z*p z   . 2 þ aDu=2 zn  z*n z z  z*n jzj > R (14)

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Y. Zhao et al. / European Journal of Mechanics A/Solids 49 (2015) 419e429

Fw0 ðzÞ ¼

∞ X

Ak zk

jzj > R

(15)

k¼0

F*w0 ðzÞ ¼

∞ X

Bk zk

jzj < R

(16)

k¼0

   fw ¼fwx  ifwy ¼ swxy ðz1 Þbx þ swyy ðz1 Þby þ i swxx ðz1 Þbx "  mb2z Fw ðz1 Þ þ Fw ðz1 Þ þ swxy ðz1 Þby ¼ Q pð1 þ kÞ # 0 z F ðz Þ þ Jw ðz1 Þ þ 1 w 1 Q (19)

Substituting Eqs. (15) and (16) into Eqs. (13) and (14), and comparing the coefficients of the same power terms, the unknown coefficients in the Eqs. (15) and (16) can be obtained:

A0 ¼ B0 ¼

.

ð1  a  bÞDu=2lnzp zn  ða þ bÞDu=2lnzp zn  ts =R a þ b þ ð1 þ aÞð1  a  bÞ=a .

ð1  a  bÞDu=2lnzp zn  ða þ bÞDu=2lnzp zn  ts =R 1 þ a þ aða þ bÞ=ð1  a  bÞ

where fwx and fwy are the force in the x and y directions, respectively. swxx,swyy,swxy are the components of the stress fields produced by the NGB sliding, and Q ¼ m(by  ibx)/[4p(1  y)].

;

;

A1 ¼ 0;

B1 ¼

 Du=2 ½1 þ ðk þ 1ÞaSk2  aðk  1ÞR2k Sk1 1 ð2a þ 2b  1Þ z1 ; Ak ¼ ; p  zn 1 þ 2a ½1 þ ðk  1Þða þ bÞ½1 þ ðk þ 1Þa  ðk þ 1Þðk  1Þaða þ bÞ

Bk ¼

½1 þ ðk  1Þða þ bÞSk1  ðk þ 1Þða þ bÞR2k Sk2 ; ðk  2Þ: ½1 þ ðk  1Þða þ bÞ½1 þ ðk þ 1Þa  ðk þ 1Þðk  1Þaða þ bÞ

k where Sk1 ¼ Du=2fðzk p  zn Þ½aðk þ 1Þ þ b  ð1  bÞ=k þ aðk þ 1Þ

=R2 ðzn zkþ1 n



zp zkþ1 Þg; p

 n

*k z*k n  zp ½ðb  1Þ=k þ 1 þ a  b  ak o

:  zn z*k1 þ ð1 þ a  akÞ zp z*k1 p n

Sk2 ¼ Du=2

fd ¼fdx  ifdy

From Eqs. (6) and (15), we obtain the complex potential Fw(z) in the matrix

Fw ðzÞ ¼

∞  X  Du   Ak zk ln z  zp  lnðz  zn Þ þ 2

jzj > R

Then, according to the Peach-Koehler formula, both the force fd produced by the interaction of edge dislocation with Burgers vector b1 and nanovoid as well as the rest edge dislocation with Burgers vector b0, and the force fs due to the remote external tensions acting on the edge dislocation can be calculated as follows:

(17)

k¼0

Substituting Eqs. (9), (15)e(17) into Eq. (10), this leads to

3 2   cd ðz1 Þ þ F cd ðz1 Þ5 ¼ by þ ibx 4 F

# "   c0 ðz Þ þ J c ðz Þ þ by  ibx z1 F d 1 d 1

fs ¼fsx  ifsy

(20)

3 2   cs ðz1 Þ þ F cs ðz1 Þ5 ¼ by þ ibx 4 F

# "   0 c c þ by  ibx z1 Fs ðz1 Þ þ Js ðz1 Þ

   9   z z > > z  zp z*p   zp  z*p z > ∞ h =

 . X R2 Du z  zn z  zp ðk þ 1ÞAk Jw ðzÞ ¼ 2 þ A0 þ B0 R2 z2 þ     z 2 >

> > > k¼1 > : þ zn  z*n z ; ðz  zn Þz*n þ z*n  z*p z R2 þ lnzp zn > i þ Bk R2k R2 zk2 jzj > R

(21)

8 > > > <

Substituting Eqs. (17) and (18) into Eqs. (1) and (2), the stress fields due to the NGB sliding can be obtained. For the edge dislocation with Burgers vector b1( ¼ bx þ iby ¼ bzei(q þ 4)) located at z1¼(R þ reiq)ei4, the force fw produced by the NGB sliding can be written as (Zhang and Li, 1991)

(18)

cd ðz1 Þ and J cd ðz1 Þ are the perturbation complex potentials where F cs ðz1 Þ are cs ðz1 Þ and J due to edge dislocation dipole in the matrix, F the complex potentials due to the remote external forces in the matrix.

Y. Zhao et al. / European Journal of Mechanics A/Solids 49 (2015) 419e429

Using the above same method, in light of the work of Zeng et al. c0 ðz Þ, J cd ðz1 Þ, F cs ðz1 Þ, cd ðz1 Þ, F (2012), the six complex potentials F d 1 0 c c Fs ðz1 Þ and Js ðz1 Þ can be expressed as follows:

cd ðz1 Þ ¼ F

g0 z1  Re

c0 ðz Þ ¼  F  d 1

2

cd ðz1 Þ ¼ R J z21

i4

þ

g0 z1  Re "

∞ X

∞ X

Ck zk 1

2 Dk ¼

(22)

k¼0

 i4 2



Ck zk 1 þ

k¼0

∞ X

kCk zk1 1

(23)

k¼1

∞ X

kCk zk 1 þ

k¼0

∞ X

Dk R2k zk 1 þ

k¼0 i4

g0 z 1 g0 z1 Re g0 z 1 g0 þ þ 2   þ 2 i4 i4 2 z1 R R  Re z z1  Re 1

#

g0 i4

z1  Re

425

3

c 5 4 g1 g0 5 þ a þ ak þ c6 þ c4 zkþ1 Rei4 kþ1 c4 1 2

k1

3 i4 *k1 Re g z þ g  d g þ g 1 0 1 0 1k 1 6 7 6 7 6 7 2k R 6 7 6 7  6 7 *k2 z1 4 5 * * g1 z1 z1  z1 ðk  1Þ z1 R2k

ðk  1Þ;

z*1 ¼ R2 =z1 , d1k is the Kronecker delta. Based on the work of Stagni (1993) and xfemit of the force along the Burgers vector direction acting on the emitted edge dislocation can be written as

femit ¼fx cosðq þ 4Þ þ fy sinðq þ 4Þ ¼ Re½fw þ fd þ fs cosðq þ 4Þ  Im½fw þ fd þ fs sinðq þ 4Þ (28) where fw, fd and fs can be obtained from Eqs. (19)e(21).

(24) 3. Critical stress for dislocation emission 2 cs ðz1 Þ ¼ ð1 þ 2aÞA  a  1 þ c3 R B F c1 z21

(25)

2 c0 ðz Þ ¼ a  1 þ c3 2R B F s 1 3 c1 z1

(26)

" # 2 2 2 cs ðz1 Þ ¼ B þ R 2ð1  bÞA  3ða  1 þ c3 Þ R B  3a þ c6 R B J c1 c4 z21 z21 z21 (27) ∞ where gj ¼  imbj/[p(1 þ k)](j ¼ 0,1),A ¼ ðs∞ 11 þ s22 Þ=4, ∞ Þ=2, c ¼ 1 þ (a þ b)(k  1) þ [a(a þ b)(k þ 1)(1  k)]/  s B ¼ ðs∞ 1 22 11 (1 þ a þ ak), c2 ¼ a(1  k)[(a þ b)(k þ 1)  1]/(1 þ a þ ak), c3 ¼ a2(1  k)(k þ 1)/(1 þ a þ ak), c4 ¼ 1 þ a(k þ 1) þ a(a þ b)(1  k)(k þ 1)/[1 þ (a þ b)(k  1)], c5 ¼ (a þ b)(k þ 1)  1 þ (a þ b)2(k þ 1)(k  1)/[(a þ b)(1  k)  1], c6 ¼ (a þ b)(1 þ k)(1 þ a  ak)/[1 þ (a þ b)(1  k)],

. i. h

. ðb  1Þ  ats =½Rð1  bÞ; C0 ¼ ð2a þ bÞ g1 z1 þ g0 Rei4

.

i. h

Rei4 ð1  bÞ D0 ¼ ð2a þ 2b  1Þ g1 =z1 þ g0  ð1  a  bÞts =½ð1  bÞR; 2 3 ½ða þ bÞðk  1Þ þ c2 R2k 4 g1 g0 þ Ck ¼ kþ1 5 c1 z1 kþ1 Rei4 "

k1 ka  a  1 þ c3 þ  g1 z*k1 þ g0 Rei4 1 c1 #   g1 z*1 z1  z*1 ðk  1Þz*k2 1  d1k ðg1 þ g0 Þ  ; z1

Referencing to the criterion from Lubarda et al. (2004) and Lubarda (2011), assume dislocation will be emitted from the nanovoid surface if its equilibrium distance r from the surface is equal to the dislocation core cut-off radius r0 (one half of edge dislocation width, which represents the extent of edge dislocation core spreading). For simplicity, neglecting the influence of the different axial tensions, herein we consider the case of the equal ∞ biaxial loading at infinity. Thus we suppose that s∞ 11 ¼ s22 ¼ s, it yields A ¼ s/2, B ¼ 0. Substituting them into Eq. (28) and letting femit ¼ 0, the applied critical stress scr for the first edge dislocation emission from nanovoid surface can be expressed as follows:

scr ¼

Re½fw þ fd cosðq þ 4Þ  Im½fw þ fd sinðq þ 4Þ Im½Tsinðq þ 4Þ  Re½Tcosðq þ 4Þ

(29)

where T ¼ ðby þ ibx Þð1 þ 2aÞ þ ðby  ibx Þð1  bÞR2 =z20 . 4. Numerical results and discussions Using the above obtained analytic expression of critical stress for dislocation emission, we examine critical stress of nanovoid growth in the deformed nanocrystalline materials. In doing so, we will use the typical parameters of the nanocrystalline material Al: m ¼ 27 GPa and y ¼ 0.34 (Hirth and Lothe, 1982); the nanocrystalline material ceramic 3C-SiC: m ¼ 217 GPa and y ¼ 0.23 (Ding et al., 2004). In addition, suppose the normalized critical stress for edge dislocation emission from nanovoid surface by the axial loading scr0 ¼ scr/s, the magnitude of Burgers vector of the emitted edge dislocation b1¼0.25 nm, the intrinsic lengths of nanovoid surface a ¼ ms/m,b ¼ ls/m and c ¼ ts/m. According to the former work (Miller and Shenoy, 2000), the newly introduced intrinsic lengths can be either positive or negative, depending on the material type and surface crystallographic orientation. They have the dimension of length and the realistic magnitude is nearly 1A . Moreover, in the   following calculation, letd ¼ 10 ,2 ¼ 30 , r0 ¼ bz,ε1 ¼ 0.01 and ε2 ¼ 0.3, if they are not specifically given. Fig. 2 reveals that the normalized critical stress scr0 as a function of emission angle q is shown considering the effect of the different NGB sliding strengths in deformed nanocrystalline metal Al and ceramic 3C-SiC. The results reveal that critical stress for dislocation emission in deformed metal Al is obviously smaller than that in

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Fig. 2. Dependence of critical stress scr0 on emission angle q with the different nanocrystalline materials and disclination strengths for a ¼ b ¼ c ¼ 0, R ¼ 5 nm and d ¼ 15 nm.

deformed ceramic 3C-SiC, which suggests that the deformed metal Al is easier than the ceramic 3C-SiC to make dislocation emission from nanovoid surface under the other same conditions. The minimum value of critical stress for dislocation emission increases with rising disclination strength u. It also indicates that NGB sliding deformation releases, in part, the high stresses near the nanovoid, thereby enhances critical stress for dislocation emission and makes nanovoid growth more difficultly. The greater the strength of NGB sliding is, the more difficultly nanovoid grows by edge dislocation emission from nanovoid surface. As is seen in Fig. 3, when emission angle q is fixed, critical stress scr0 varies with respect to nanovoid size R with the different nanocrystalline materials and disclination strengths. It is found that critical stress is distinctly dependent on nanovoid size R. For both deformed nanocrystalline metal Al and ceramic 3C-SiC, critical stress increases with the decrement of nanovoid size. Especially when nanovoid size is below2 nm, critical stress significantly increases as nanovoid size decreases, and the results become meaningless since the applied stress is too high in our present model analyses. However, when NGB sliding deformation appears, there may be a range of nanovoid size to make critical stress keep almost  unchanged, such as 1.7 nm
Fig. 3. Dependence of critical stress scr0 on nanovoid size R with the different nano crystalline materials and disclination strengths fora ¼ b ¼ c ¼ 0,q ¼ 55 and d ¼ 20 nm.

Fig. 4. Dependence of critical stress scr0 on nanograin size d with the different dis clination strengths and the surface elasticity forc ¼ 0,q ¼ 55 and R ¼ 10 nm.

stress. In other words, dislocation emission is more difficult and thereby nanovoid growth is slowed down or even arrested with the stronger NGB sliding deformation in nanocrystalline solids. That is, a nanocrystalline material tends to further depress the ductile failure when NGB sliding deformation enhances. The results reveal that the release of the stresses near the nanovoid caused by NGB sliding deformation is very effective and significant for stopping further nanovoid growth, which are of crucial importance for the ductility of nanocrystalline materials. Fig. 4 plots critical stress scr0 versus nanograin size d with the different disclination strengths and the surface elasticity. It indicates that nanograin size will have negligible effect on critical stress when NGB sliding deformation vanishes, while nanograin size yields an evident influence on critical stress as NGB sliding progresses. Critical stress first increases sharply and then decreases slowly with further nanograin coarsening. NGB sliding suppresses dislocation emission from nanovoid surface in nanocrystalline materials, and the suppression significantly depends on nanograin

Fig. 5. Dependence of critical stress scr0 on emission angle q with the different disclination strengths and surface residual stresses fora ¼ b ¼ 0,R ¼ 4 nm and d ¼ 10 nm.

Y. Zhao et al. / European Journal of Mechanics A/Solids 49 (2015) 419e429

Fig. 6. Dependence of critical stress scr0 on disclination strength u with the different  nanovoid sizes and the surface elasticity forc ¼ 0,q ¼ 55 and d ¼ 25 nm.

size. There is a critical nanograin size to make critical stress reach the maximum value and dislocation emission occur most difficultly.  In the situation with u ¼ 10 and a ¼ b ¼ 0, a critical nanograin size has its highest values of around 34 nm. When nanograin size is below the critical value, critical stress appears to dramatically decrease with further nanograin refinement. Decreasing nanograin size below the critical value results in softening rather than hardening, and this gives rise to ductile failure of the deformed nanocrystalline materials. In addition, the critical value of nanograin size increases with growing the strength of NGB sliding. These results indicate that NGB sliding deformation effectively hinders dislocation emission from nanovoid surface and suppresses nanovoid growth, and therefore impedes the ductility failure of nanocrystalline materials. We also see that the surface stress characterized by positive (negative) surface elasticity increases (decreases) critical stress. That is, the emission of dislocation from nanovoid surface becomes more difficult (easily) for nanovoid characterized by positive (negative) surface elasticity. The positive

Fig. 7. Dependence of critical stress scr0 on disclination strength u with the different widths of dislocation core and the surface residual stresses for  a¼ b ¼ 0,q ¼ 55 ,R ¼ 8 nm and d ¼ 25 nm.

427

(negative) surface elasticity makes the critical nanograin size increase (reduce). It is clearly shown that material hardening or softening of nanovoid surface due to surface stresses characterized by surface elasticity has a pronounced effect on critical stress for dislocation emission and nanovoid growth. Critical stress scr0 versus emission angle q with the different disclination strengths and the surface residual stresses are depicted in Fig. 5. We can see that, critical stress and the relative most probable critical angle for dislocation emission increase with rising the strength of NGB sliding deformation u. Surface stress characterized by positive (negative) surface residual stress decreases (increases) the minimum critical stress and the relative most probable emission angle. This implies clearly that dislocation is more easily (difficultly) emitted from nanovoid surface due to the local softening (hardening) on the surface induced by surface residual stresses. Fig. 6 shows the influence of disclination strength u on critical stress scr0 with the different nanovoid sizes and the surface elasticity. The curves show that critical stress scr0 increases with an increase of disclination strength u. Critical stress scr0 falls as nanovoid size increases, and makes nanovoid growth more easily occur. Increasing disclination strength enhances the effect of nanovoid size on critical stress. It is also shown that surface effect yields an evident influence on critical stress, and its influence is almost not affected by changing the disclination strength. Fig. 7 shows the influence of disclination strength u on critical stress scr0 with the different widths of dislocation core and the surface residual stresses. Increasing the emitted dislocation core leads to the decrease in critical stress, indicating that dislocation with a wider core is more easily emitted than dislocation with a narrower core. As the core of the emitted dislocation rises, the influence of the surface residual stresses visibly weakens. 5. Conclusions In summary, the main contribution of this paper is the presentation of a theoretical model to describe the effect of NGB sliding on nanovoid growth by the emission of the incipient dislocation shear loops from a cylindrical nanovoid surface in deformed nanocrystalline materials. Special emphasis is to elaborate the dependence of threshold condition of dislocation emission upon the sizes of nanograin and nanovoid as well as the features of disclinations dipole due to NGB sliding. Major conclusions from the study are as follows: 1) The value of critical stress and the relative most probable critical angle for dislocation emission increase with rising the strength of NGB sliding deformation, which indicates NGB sliding deformation releases, in part, the high stresses near the nanovoid, thereby nanovoid growth is slowed down or even arrested with the stronger NGB sliding deformation in nanocrystalline solids. Surface stress characterized by positive (negative) surface elasticity increases (decreases) critical stress. Surface stress characterized by positive (negative) surface residual stress decreases (increases) critical stress and the relative most probable emission angle. Surface effect yields an evident influence on critical stress, while its influence is almost not affected by changing the disclination strength. 2) Critical stress increases with the decrement of nanovoid size. When NGB sliding deformation appears, there exists a range of nanovoid size to make critical stress keep almost unchanged. Increasing disclination strength enhances the effect of nanovoid size on critical stress. 3) Nanograin size has no effect on critical stress without NGB sliding, while nanograin size yields an evident influence on

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critical stress as NGB sliding progresses. Critical stress first increases sharply and then decreases slowly with further nanograin coarsening. The suppression of NGB sliding significantly depends on nanograin size. There is a critical nanograin size to make critical stress reach the maximum value, and then dislocation emission and nanovoid growth occur most difficultly. The critical nanograin size increases with growing the strength of NGB sliding. The positive (negative) surface elasticity makes critical nanograin size increase (reduce). Acknowledgments The authors would like to deeply appreciate the support from the NNSFC (11172094, 11372103 and 11172095), the New Century Excellent Talents in University (NCET-11-0122), Hunan Provincial Natural Science Foundation for Creative Research Groups of China (12JJ7001), the Fok Ying-Tong Education Foundation, China (141005) and Interdisciplinary Research Project of Hunan University. References Becker, R., Needleman, A., Suresh, S., Tvergaard, V., Vasudevan, A.K., 1989. An analysis of ductile failure by grain boundary void growth. Acta Metall. 37, 99e120. Bilger, N., Auslender, F., Bornert, M., Moulinec, H., Zaoui, A., 2007. Bounds and estimates for the effective yield surface of porous media with a uniform or a nonuniform distribution of voids. Eur. J. Mech. A. Solids 26, 810e836. Bobyle, S.V., Ishizaki, T., Kuramoto, S., Ovid’ko, I.A., 2011. Formation of nanocrystals due to giant-fault deformation in Gum. Metals. Scr. Mater. 65, 668e671. Borg, U., 2007. A strain gradient crystal plasticity analysis of grain size effects in polycrystals. Eur. J. Mech. A. Solids 26, 313e324. Bringa, E.M., Lubarda, V.A., Meyers, M.A., 2010. Response to “Shear impossibilityComments on ‘Void growth by dislocation emission’ and ‘Void growth in metals’”. Scr. Mater. 63, 148e150. Bulatov, V.V., Wolfer, W.G., Kumar, M., 2010. Shear impossibility: comments on “Void growth by dislocation emission” and “Void growth in metals: atomistic calculations”. Scr. Mater. 63, 144e147. Ding, Z., Zhou, S., Zhao, Y., 2004. Hardness and fracture toughness of brittle materials: a density functional theory study. Phys. Rev. B 70, 184117. Fang, Q.H., Liu, Y.W., 2006. Size-dependent interaction between an edge dislocation and a nanoscale inhomogeneity with interface effects. Acta Mater. 54, 4213e4220. Fang, Q.H., Liu, Y.W., Jiang, C.P., Li, B., 2006. Interaction of a wedge disclination dipole with interfacial cracks. Eng. Fract. Mech. 73, 1235e1248. Flandi, L., Leblond, J.B., 2005. A new model for porous nonlinear viscous solids incorporating void shape effects e I: theory. Eur. J. Mech. A. Solids 24, 537e551. Fleck, N.A., Hutchinson, J.W., Tvergaard, V., 1989. Softening by void nucleation and growth in tension and shear. J. Mech. Phys. Solids 37, 515e540. Ganghoffer, J.F., Brekelmans, W.A.M., Geers, M.G.D., 2008. Distribution based model for the grain boundaries in polycrystalline plasticity. Eur. J. Mech. A. Solids 27, 737e763. Gurtin, M.E., Murdoch, A.I., 1975. A continuum theory of elastic material surfaces. Arch. Rat. Mech. Anal. 57, 291e323. Gurtin, M.E., Murdoch, A.I., 1978. Surface stress in solids. Int. J. Solids Struct. 14, 431e440. Gutkin, M.Yu, Ovid’ko, I.A., Skiba, N.V., 2003. Crossover from grain boundary sliding to rotational deformation in nanocrystalline materials. Acta Mater. 51, 4059e4071. Hirth, J.P., Lothe, J., 1982. Theory of Dislocations. Wiley, New York. Kleman, M., Friedel, J., 2008. Disclinations, dislocations, and continuous defects: a reappraisal. Rev. Mod. Phys. 80, 61e115. Li, H., Ebrahimi, F., 2005. Ductile-to-brittle transition in nanocrystalline metals. Adv. Mater. 17, 1969e1972. Latapie, A., Farkas, D., 2004. Molecular dynamics investigation of the fracture behavior of nanocrystalline a-Fe. Phys. Rev. B 69, 134110. Lubarda, V.A., 2011. Emission of dislocations from nanovoids under combined loading. Int. J. Plast. 27, 181e200. Lubarda, V.A., Schneider, M.S., Kalantar, D.H., Remington, B.A., Meyers, M.A., 2004. Void growth by dislocation emission. Acta Mater. 52, 1397e1408. Luo, J., Liu, F., 2011. Stress analysis of a wedge disclination dipole interacting with a circular nanoinhomogeneity. Eur. J. Mech. A. Solids 30, 22e32. Marian, J., Knap, J., Ortiz, M., 2004. Nanovoid cavitation by dislocation emission in aluminum. Phys. Rev. Lett. 93, 165503. Mathur, K.K., Needleman, A., Tvergaard, V., 1994. Ductile failure analyses on massively parallel computers. Comput. Methods Appl. Mech. Eng. 119, 283e309.

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