A study of fatigue crack tip characteristics using discrete dislocation dynamics

International Journal of Plasticity 54 (2014) 229–246

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A study of fatigue crack tip characteristics using discrete dislocation dynamics Minsheng Huang a,b, Jie Tong b,⇑, Zhenhuan Li a a b

Department of Mechanics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China Mechanical Behaviour of Materials Group, School of Engineering, Anglesea Building, Anglesea Road, University of Portsmouth, Portsmouth PO1 3DJ, UK

a r t i c l e

i n f o

Article history: Received 15 February 2012 Received in final revised form 26 August 2013 Available online 8 September 2013 Keywords: Crack tip Cyclic response Discrete dislocation dynamics Disclocation climb Grain boundary

a b s t r a c t The near-tip deformation of a transgrannular crack under cyclic loading conditions has been modelled using discrete dislocation dynamics (DDD) with both dislocation climb and dislocation-grain boundary (GB) penetration considered. A representative cell was built to model the constitutive behaviour of the material, from which the DDD model parameters were fitted against the experimental data. The near-tip constitutive behaviour was simulated for a transgranular crack in a polycrystalline nickel-based superalloy. A phenomenon of cyclic creep or strain ratchetting was reproduced, similar to that obtained using viscoplastic and crystal-plastic models in continuum mechanics. Ratchetting has been found to be associated with dislocation accumulation, dislocation climb and dislocation-GB penetration, among which dislocation climb seems to be the dominant mechanism for the cases considered at elevated temperature. Ratchetting behaviour seems to have a distinctive discrete characteristic in that more pronounced ratchetting occurred within slip bands than elsewhere. Multiple slip systems were activated in grains surrounding the crack tip, as opposed to single active slip system in grains away from the crack tip. The present DDD results show that, the near-tip ratchetting strain ahead of the crack tip seems to be a physical phenomenon, which may be of particular significance for developing a physicalbased model of crack growth. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction Mechanical response and lifetime assessment of engineering materials under cyclic loading conditions has been a continuing driver for the development of new/improved constitutive material models, notably but not exclusively of those reported recently (Chaboche, 2008; Taleb and Cailletaud, 2011; Xiao et al., 2012; Yu et al., 2012; Pham et al., 2013; Chaboche et al., 2013), where aspects of cyclic response, such as cyclic hardening/softening and saturation (Pham et al., 2013), dynamic strain ageing (Yu et al., 2012; Chaboche et al., 2013) and strain ratchetting (Abdel-Karim, 2010; Taleb and Cailletaud, 2011), have been carefully studied. Although these studies have significantly enhanced our understanding of the overall stress–strain behaviour of engineering materials under cyclic loading conditions, the immediate impact of such a progress is often on the studies of fatigue crack initiation. Fatigue crack propagation, on the other hand, despite of being extremely important to damage tolerance design implemented virtually on all fracture critical components and structures, has not always benefited immediately from the most recent developments. Mechanistic understanding of fatigue crack growth may be traced back to Rice (1967) who provided a seminal analysis of stress and strain fields near an idealised stationary crack tip under tensile and anti-plane shear cyclic loadings. It was found ⇑ Corresponding author. Tel.: +44 (0) 23 9284 2326; fax: +44 (0) 23 9284 2351. E-mail address: [email protected] (J. Tong). 0749-6419/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijplas.2013.08.016

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that the crack-tip cyclic plastic deformation may be adequately determined by the variation in a stress intensity factor, and the reversed plastic-zone size due to load reversal is one quarter of the size of the maximum plastic zone. Considerable analytical research has since been carried out to study the controlling parameters of crack-tip deformation and crack propagation, notably including the well-known Hutchinson–Rice–Rosengren (HRR) field for power-law hardening materials; the RR (Riedel and Rice, 1980) and the HR (Hui and Riedel, 1981) fields for power-law creep materials. Numerous finite element (FE) analyses have been carried out to model the crack-tip deformation using cyclic plasticity and crystal plasticity constitutive models (e.g., Sehitoglu and Sun, 1991; Pommier and Bompard, 2000; Zhao et al., 2001; Tvergaard, 2004; Zhao and Tong, 2008). Keck et al. (1985) demonstrated the dependency of crack-tip stress–strain field and plastic-zone size on loading frequency and hold time, where low frequency and introduced hold time at maximum load led to increased crack-tip deformation and plastic-zone size. Characteristic strain ratchetting near a crack tip was found by Zhao et al. (2001) and Zhao and Tong (2008), where tensile strain normal to the crack plane was found to accumulate progressively. Flouriot et al. (2003) investigated the crack-tip strain field in a single crystal using the elasto(visco)-plastic model developed by Meric et al. (1991). Their results also showed strain ratchetting occurring primarily in some of the localised slip bands. Using the same material model (Meric et al., 1991), Marchal et al. (2006) found that ratchetting appears to be on octahedral slip systems and the amount of ratchetting depends on the distance from the crack tip. Dunne et al. (2007) used a simplified crystal plasticity model to study the low cycle fatigue crack nucleation. Their predicted locations of the persistent slip bands coincided well with the experimentally observed sites of crack nucleation. Using a crystal plasticity model (Busso et al., 2000), Lin et al. (2011) studied the near-tip deformation of a transgranular crack in a compact tension specimen for a polycrystalline nickel alloy. Ratchetting phenomenon was once again found near the crack tip, and the shear deformation on the slip planes was found to accumulate with the increase of the number of cycles. Nickel-based superalloys have been used for gas turbine discs applications, where fatigue and creep deformation is of primary concerns. Extensively studies (for example, Meric et al., 1991; Nouailhas and Cailletaud, 1995; Dalby and Tong, 2005; Zhan and Tong, 2007a, b; Lin et al. 2011; Tong et al., 2011) have been carried out to understand the material constitutive and crack growth behaviour at elevated temperature. It is well known that the interaction between dislocations and material microstructure, e.g., grain boundary (GB) and the second phase c0 precipitate, plays an important role in dictating the stress–strain response of the material. Modelling of dislocation-microstructure interaction has been attempted by formulating the constitutive laws. For instance, Fedelich (1999, 2002) introduced some microstructure parameters, including precipitate size, channel width and lattice mismatch, into his dislocation-based crystal plasticity constitutive law, and investigated the influence of microstructure parameters on the mechanical behaviour of a single crystal Ni-based superalloy. Busso et al. (2000) proposed a gradient- and rate-dependent crystallographic formulation for a single crystal Ni-based superalloy CMSX4, and investigated the effects of precipitate size and channel width on mechanical behaviour. Shenoy et al. (2008) formulated a rate-dependent, microstructure-sensitive crystal plasticity model for a polycrystalline Ni-base superalloy, which has the capability to capture first-order effects on the stress–strain response due to grain size, precipitate size distribution and precipitate volume fraction. Tinga et al. (2010) introduced the interaction of dislocations with the microstructure (such as the dislocations shear and climb over the precipitates) into a single crystal dislocation density-based constitutive model to capture the non-Schmid response of a nickel alloy. Vattré and Fedelich (2011) developed a micromechanical dislocation density-based constitutive model with a pseudo-cubic slip law which improved the estimation of the strain hardening anisotropy. Although both the material microstructure and dislocation density evolution were incorporated, these constitutive models were formulated only within a continuum plasticity framework. Since dislocation density-based constitutive continuum models can consider the interaction between dislocations and internal material microstructures, such as grain boundaries, secondary phases and dislocation cells, many researchers have recently focused their efforts on these approaches, leading to significant developments in this area (notably, Fan and Yang, 2011; Hamelin et al., 2011; Barlat et al., 2013; Bertin et al., 2013; Franz et al., 2013; Hansen et al.,2013; Li et al., 2013; Resende et al., 2013; Shanthraj and Zikry, 2013). However, as pointed out by Berdichevsky and Dimiduk (2005), the application of continuum plasticity is questionable at the scale of the dislocation structure. This issue becomes particularly crucial for typical microstructures of nickel alloys, since the use of dislocation density qðxÞ as an independent local variable in the mesoscopic constitutive models cannot be justified by a spatial averaging at the scale of channel width (Vattré and Fedelich, 2011). When a crack is concerned, dislocations tend to be organised into heterogeneous dislocation structures (such as slip bands) within an area of micron or sub-micron size ahead of the crack tip. Since continuum constitutive models only consider dislocation evolution phenomenally or statistically, they cannot accurately describe these local heterogeneous dislocation structures and capture the local non-homogeneous deformation field ahead of a crack tip. To consider the discreteness of dislocation structure ahead of a crack tip, Cleveringa et al. (2000) carried out a two dimensional (2D) discrete dislocation dynamics (DDD) analysis of crack-tip deformation field and crack growth in a FCC (face-centre-cubic) single crystal under mode I loading. It was found that the local stress concentration associated with discrete dislocation patterning ahead of the crack tip can lead to stress levels much higher than the yield stress, and indeed high enough to cause atomic separation. Van der Giessen et al. (2001) performed a 2D DDD simulation of the crack-tip deformation field for a stationary plane strain mode I crack. Their results showed that crack-tip deformation field and dislocation structure depend on slip system orientation; and the opening stress in the immediate vicinity of the crack tip is much larger than that predicted by continuum slip theory. Deshpande et al. (2003) modelled edge-cracked single crystal specimens of varying sizes subject to both monotonic and cyclic loading using 2D DDD simulation. It was found that the fatigue crack growth threshold decreases substantially with the crack size when it is below a critical value. Brinckmann and Van der Giessen

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(2004) used DDD method to model fatigue crack initiation from a free surface. Their results revealed the evolution of dislocation structures which led to the accumulation of stresses. Déprés et al. (2004) carried out a three-dimensional (3D) DDD simulation to simulate the dynamic evolution of the dislocation microstructure and the topography of a free surface under cyclic loads. They deduced a mechanism for the formation of intense slip bands and the initiation of fatigue cracks. The advantage of the DDD method is that it models the plastic deformation directly through the evolution of discrete dislocations, hence it can capture the formation of dislocation structure at a microscopic scale. However, the existing DDD simulations for crack problems are limited to single crystal materials. To our knowledge, there is no published work for DDD simulation of cyclic crack-tip deformation in a polycrystalline material. The objective of this work is to carry out a DDD simulation of near-tip deformation for a transgranular crack in a polycrystalline Ni-based superalloy under cyclic model I loading condition. A representative cell (RC) was built to model the monotonic deformation of the material, from which the DDD model parameters, including slip plane friction stress, dislocation source strength and density, were fitted against the experimental data. Using the DDD method, crack tip deformation was then simulated for a compact tension specimen. The submodel contains a transgranular crack and 150 randomly oriented grains with an average grain size of 5 lm. A closed-form deformation field for dislocations near the crack tip was employed to account for the interaction between the dislocations and the crack. The displacement boundary condition for the DDD submodel was obtained from the FE analyses of the global CT specimen using a visco-plastic constitutive law. The primary interests of the study were the near-tip stress and strain responses and their evolution with cycles, as well as the associated evolution of the dislocation distribution ahead of the crack tip. 2. Methodology 2.1. The DDD framework A 2D representative cell (RC), as shown in Fig. 1, was built for DDD simulation of monotonic deformation of a polycrystalline nickel-based alloy. This RC has an area of 58 lm  58 lm, and contains 150 grains with an average grain size of 5 lm. As indicated in Fig. 1, a strain-controlled monotonic load was applied to the RC in the y-direction at a strain rate of e_ ¼ 1=s. In both x and y directions, the following periodic boundary conditions were applied:

(

U Ax  U Bx ¼ U Cx  U Dx U Ay  U Cy ¼ U By  U Dy ¼

R

e_ hdt

ð1Þ

where fU Ax ; U Bx ; U Cx ; U Dx g and fU Ay ; U By ; U Cy ; U Dy g are the displacements at representative points A–D in x- and y-direction, respectively. Following the 2D DDD framework by Van der Giessen and Needleman (1995), the above problem was solved by a superposition of the DDD part and the linear elastic part. In the DDD part, the plastic deformation was simulated directly by the evolution of discrete dislocations. The deformation field (~) in the material induced by these dislocations may be expressed as:

Fig. 1. The periodical polycrystalline material model (58 lm  58 lm) with 150 grains and an average grain size of 5 lm. Three slip systems with an intersection angle of 54.7° were randomly distributed. A triangular waveform was applied in the y direction along edges AB and CD with a displacement _ down ¼ e_ h, where h is the height of the model and e_ ¼ 1=s is the strain rate. A periodic boundary condition was applied on edge AC and BD as rate, U_ up y  Uy U Ax  U Bx ¼ U Cx  U Dx .

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~¼ u

nd X uk ;

r~ ¼

k¼1

nd X

rk

ð2Þ

k¼1

where nd is the total number of dislocations, and fuk ; rk g are the displacement and stress fields induced by the kth dislocation in a homogeneous infinite solid. Following the Muskhelishvili method, the stress and displacement fields of the kth dislocation are expressed as:

8 k r þ rkyy ¼ 2ð/0 ðzÞ þ /0 ðzÞÞ > > < xx rkyy  rkxx þ i2rkxy ¼ 2ðz/00 ðzÞ þ w0 ðzÞÞ > > : uk ðzÞ ¼ uk þ iuk ¼ 1 ½j/ðzÞ  z/0 ðzÞ  wðzÞ x

y

ð3Þ

2l

where r and r are two normal stresses in the x and y directions, respectively, rkxy is the shear stress, /ðzÞ and wðzÞ are two potential functions, i is the pure imaginary, l the shear modulus and j ¼ 3  4m for plain strain deformation. The two potential functions, /ðzÞ and wðzÞ, are associated with dislocation via: k xx

k yy

/ ¼ /0 ¼ c lnðz  zd Þ;

w ¼ /0 ¼ c lnðz  zd Þ  c

zd z  zd

with c ¼

lðbx þ iby Þ 4pð1  v Þ

ð4Þ

where zd ¼ xd þ iyd is the coordinate of the dislcoation, bx þ iby the Burgers vector of the dislocation, v the Poisson’s ratio. ~; r ~ g at the RC boundary, which Inevitably, the dislocation fields (6) will introduce an additional displacement and stress fu makes the boundary condition (2) unsatisfied. To correct this, a complementary field (^) was introduced and solved by the linear finite element (FE) method. The new boundary conditions may be written as:

(

~  nt T^ ¼ T  T~ ¼ T  r ^ ¼UU ~ on Su U

on St

ð5Þ

~ and r ~ are the dislocation displacement and stress at the displacement (Su ) and traction boundaries (St ), respectively, where U nt is the unit vector normal to the boundary. The actual fields in the material may be obtained by the superposition of the dislocation (~) and complementary (^) fields:

^þu ~; u¼u

e ¼ ^e þ ~e; r ¼ r^ þ r~ :

ð6Þ

For the kth dislocation, the in-plane component of the Peach–Koehler force controlling its glide may be formulated as k

^ þr ~ Þb f k ¼ mk ðr

ð7Þ k

k

where m is the unit vector normal to its slip plane and b is the Burgers vector. To consider the obstacle effects by the solution atoms in matrix and the Kear–Wilsdorf (KW) locking in the precipitates of Ni-based superalloys, a friction force k f fr ¼ sfr b was introduced and the effective Peach–Koehler force f eff may be written as:

f eff

8 k fr if f k > f fr > : 0 if absðf k Þ < f fr

ð8Þ

This effective Peach–Koehler force drives the kth dislocation to glide at a velocity:

v k ¼ f eff =B

ð9Þ

where B is the drag coefficient. For FCC crystals, three active slip systems were considered and distributed randomly in each grain, as illustrated in Fig. 1, with an intersection angle of 56.4° between each two slip systems. The dislocation sources were randomly placed on the slip planes with a given density qsou . According to Frank-Read dislocation nucleation mechanism, once the effective Peach–Koehler force f eff at these dislocation sources exceeds the dislocation source strength f nuc ¼ snuc b within a period of time, a dislocation dipole may be generated, and it consists of two opposite dislocations with a distance Lnuc ¼ 2pð1Gbv Þsnuc . The dislocation source strength has a Gaussian distribution with a mean strength of snuc and a standard deviation of asnuc . The dislocation annihilation occurs if the two opposite dislocations approach each other within a material-dependent critical distance Le = 6b. Both rigid and dislocation-penetrable grain boundaries (GBs) were considered to investigate the mechanisms of the cyclic response ahead of the crack tip, although rigid GB was the default configuration unless otherwise specified. To improve the computing efficiency, the RC was further divided into 40  40 subcells. For a dislocation i in subcell m, its interaction with the dislocation j in the same subcell m or its neighbouring subcells was calculated directly, while its interaction with the dislocations in the remote subcell k was computed using a so-called superdislocation method (Zbib et al., 1998). A superdislocation is the sum of all dislocations in remote subcell k and was assumed to be located at the centre of the remote subcell k. It was further assumed that dislocation i is also located at the centre of its subcell m. Following this algorithm, the interaction between the dislocation i in subcell m and the superdislocation in remote subcell k can be calculated efficiently. In addition, for a polycrystalline material, most dislocations pile up against the grain boundaries with a zero

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glide velocity. In our simulation, if the velocities for both dislocations i and j equal to zero in one load step, their interaction forces will be saved and used directly in the next step as if they remain immobile. Furthermore, the DDD program was programmed in parallel computing using the OpenMP interface, which makes the simulation of the cyclic response much more efficient. To obtain the global stress–strain response of the polycrystalline material, homogenisation based on averaging theorem over the RC area was adopted via the following area integral:

(

Rij ¼ A1 Eij ¼ A1

R R

R

rij dA ¼ A1 ðr~ ij þ r^ ij ÞdA R eij dA ¼ A1 ð~eij þ ^eij ÞdA

ð10Þ

where Rij and Eij are the average stress and strain for the RC model, A is the area of the RC, and rij and eij are the local stress and the strain field of the Gaussian points within the RC model. 2.2. DDD modelling of crack tip deformation A 150-grain finite element submodel (58 lm  58 lm), which contains a transgranular crack, was built for DDD simulation of crack tip deformation, as shown in Fig. 2. The average grain size is 5 lm. Fine mesh (0.6 lm) was used in grain 1 (G1), which contains the crack tip, as indicated in the inset. The displacement conditions applied on the outer boundary of the submodel were obtained by a FE simulation of the global CT model described by a viscoplastic constitutive law (Chaboche, 1989; Zhan and Tong, 2007a,b). Unless otherwise specified, the applied cyclic load has a stress intensity factor range pffiffiffiffiffi DK ¼ 6 MPa m, load ratio R ¼ 0:1 and a load frequency f ¼ 1000 Hz, a loading regime chosen mainly for computational efficiency. The 2D DDD scheme is similar to that described in the above section, but needs to consider the interaction between the crack and the dislocations. In the presence of a crack, the dislocation field must satisfy the traction free condition on the crack surface, which was considered by assuming a sharp crack as an elliptical hole with an extremely large aspect ratio (1:13,000). This assumption can effectively consider the interaction between a crack and a dislocation. For a dislocation near an elliptical void, its deformation field can be obtained theoretically by superposing a complementary term to the potential in Eq. (4) (Fischer and Beltz, 2001): 8   > /d ðzÞ ¼ c lnðz  zd Þ þ 2c lnf  c ln f  fm > > d > > >   > f ð1þmfd 2 Þfd ðf2d þmÞ > >   c ln f  f1 þ c d > > > d < fd fd ðfd 2 mÞ f 1 fd / ¼ /0 þ /d ; w ¼ w0 þ wd with     ð11Þ zd > d > ðzÞ ¼ c lnðz  z Þ  c þ 2 c lnf  c ln f  fmd  c ln f  f1 w > d zzd > d > > > > 0 2 þmÞ > þc fd ðf2d þm3 Þmfd ðfd 2  > /d ðfÞ  f 1þmf > f2 m > : fd fd ðf2d mÞ ffm d

Fig. 2. The cracked polycrystal model (58 lm  58 lm) with 150 grains and an average grain size of 5 lm. The crack tip is located in the grain 1 (G1), as illustrated in the inset. The displacement field U applied on the boundary was obtained from a visco-plastic FE submodel. The average mesh size in G1 is D1 ¼ 0:6 lm.

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  where, f/0 ; w0 g are the potential functions in Eq. (4), z ¼ x þ iy ¼ R f þ mf , R ¼ ða þ bÞ=2, m ¼ ða  bÞ=ða þ bÞ, and fa; bg are the short and the long axis of the elliptical crack. Substituting (11) into Eq. (3), new stress and displacement fields for dislocations may be obtained and the crack surface traction free boundary condition will be satisfied automatically. 2.3. Incorporation of dislocation climb into the DDD model Since nickel-based alloys are mostly used for applications at elevated temperature, diffusion of point defects and its resultant dislocation climb cannot be neglected as in most previous DDD studies. Dislocation climb can reduce the back stress induced by dislocation pile-ups and renders more dislocations to be emitted from the dislocation sources. Recently, Keralavarma et al. (2012), Davoudi et al. (2012) and Danas and Deshpande (2013) introduced the climb of edge dislocations into a 2D DDD framework by different schemes independently. By combining the DDD modelling and the vacancy diffusion FE simulation, Keralavarma et al. (2012) coupled the dislocation climb, the stress field and the vacancy distribution field explicitly. Although this scheme has an advantage to solve the constant-load creep problem with a large time scale, it is not suitable to model the constant loading rate boundary problems. Danas and Deshpande (2013) incorporated the dislocation climb by a drag-type relation, in which the temperature T, the equilibrium concentration of vacancies c0 , average dislocation spacing l and the vacancy volume X were taken into account in the climb drag coefficient Bc . As opposed to the linear drag-type relations, Davoudi et al. (2012) derived the climb velocity from the steady-state solution of the diffusion equation as:

Vc ¼

     2pD0 DEsd F c DV  =b exp 1 exp  kB T b lnðR=bÞ kB T

ð12Þ

where b is the magnitude of the Burgers vector, DEsd is the vacancy self-diffusion energy, DV is the vacancy formation vol3 ume approximately equalling to b , kB the Boltzmann constant, T the absolute temperature, and D0 the pre-exponential diffusion constant. The climb force F c for the kth dislocation may be formulated as

 X  k ^þ F c ¼ sk r r~ b

ð13Þ

where sk is the unit vector in the slip direction. The model by Davoudi et al. (2012) seems to be more accurate than the dragtype models and more convenient to implement than the vacancy diffusion coupled model (Keralavarma et al., 2012), it was therefore employed in the present simulations with the following parameters selected as: Temperature T ¼ 1123 K and diffusion constant D0 ¼ 1:27  104 m2 =s (Marzocca and Picasso, 1996). Since the vacancy self-diffusion activation energy for Ni-based superalloys is in the range 257–283 kJ/mol (Heilmaier et al., 2009), a minimum value DEsd ¼ 257 kJ=mol was chosen to favour the process of dislocation climb. When the dislocation climb is taken into account, the displacement field of dislocation calculated by Eq. (6) needs to be amended to ensure the correct discontinuity between dislocations on the glide planes. Considering this, Davoudi et al. (2012) added the following extra terms to the displacement in the slip direction when a dislocation climbs from its location in the glide plane ðx0 ; y0 Þ to a new location ðx0 ; y1 Þ:

         b y  y0 x  x0 y  y1 x  x0  tan þ tan þ tan  tan 2p x  x0 y  y0 x  x0 y  y1

ð14Þ

These extra terms should be retained in the following loading steps. With continuous climbing of one dislocation, more and more extra terms should be introduced to correct its displacement field which makes this process very complex. In addition, Danas and Deshpande (2013) calculated the displacement by dislocation glide and climb separately with an incremental method. Since the time step for pure dislocation climb must be determined, this method is also complex and not convenient to realise. In the present paper, a simpler method was developed to calculate the displacement field considering dislocation climb, based on the model of Davoudi et al. (2012). When the kth dislocation climbs from point A ðx0 ; y0 Þ to point B ðx0 ; y1 Þ and further glides to point C ðx1 ; y1 Þ in one time step, as shown in Fig. 3, two additional fake dislocations are introduced into the simulation at the points A and B, respectively. The fake dislocation located at point A has a similar character to

t + Δt

t ( x0 , y1 )

( x0 , y0 )

Fig. 3. A schematic of the treatment of dislocation climb in the present DDD model. To ensure the correct discontinuity at the two slip planes after a climb, two red fake dislocations were placed at points A and B at the time t þ Dt.

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that of the kth dislocation, but opposite to that of fake dislocation at point B. Since the fake dislocations are introduced only for the correction of the displacement field with dislocation climb, they have no contribution to the stress field and remain immobile during all the time steps. The displacement of the fake dislocations may be written as:

     b y  yc x  xc  tan  tan 2p x  xc y  yc

ð15Þ

where ðxc ; yc Þ and b are the coordinates and the Burgers vector of the fake dislocation, respectively. Although the number of fake dislocations increases with continuous dislocation climb, the computational scale will not increase significantly as no stress calculation needs to be performed. In addition, since the spacing between two neighbouring potential slip planes is taken as b, a dislocation is not allowed to climb if the dislocation-climb distance is smaller than b in any given time step. Similar to the model of Davoudi et al. (2012), the time step for climb was taken as 100 times larger than that for glide. 3. Results and discussion 3.1. Determinaiton of material model parameters The material studied is a polycrystalline nickel-based superalloy for turbine discs. The DDD model was calibrated by fitting the stress–strain response of a strain-controlled test at a strain rate of 0.05%/s (Zhan and Tong, 2007a). It should be mentioned that the DDD model shown in Fig. 1 is under a uniaxial tensile loading rate 1/s, which is much greater than the experimental strain rate 0.05%/s. Due to the limitation in the time scale of the DDD method, 1/s is the lowest loading rate feasible for computational purposes, hence allowance must be made in the interpretation of the results. The parameter fitting in the DDD simulation was carried out based on monotonic experimental results to reduce the computational costs, hence a quantitative comparison between the DDD and the experimental results was abandoned. Nevertheless, as it will be shown later, that the observed ratchetting phenomenon is more pronounced at lower frequencies, hence the DDD parameters fitting and the results presented are not without practical significance. The final parameter values for the DDD model determined by a fitting process are listed in Table 1. The simulated stress–strain response is shown in Fig. 4, together with the experimental data. The linear and the initial yielding behaviour was well captured by the DDD-based simulation, whilst at higher strains the simulated strain hardening rate from DDD deviated from the experimental result after 0.8% strain. The DDD parameters were considered to be

Table 1 The fitting DDD parameters. Young’s modulus E

Poisson’s ratio m

185.3 GPa

0.285

Dislocation source density

Dislocation mean strength

snuc

Dislocation strength standard deviation

Slip friction stress

qsou

250 MPa

0:1snuc

326 MPa

2

50 lm

sfr

Fig. 4. A comparison of uniaxial tensile stress–strain response between the experiment data (e_ ¼ 0:005=s) and the DDD modelling (e_ ¼ 1=s). Parameters in the DDD model were set as: Friction stress rf ¼ 326 MPa, dislocation source strength snuc ¼ 250 MPa, dislocation distribution Gaussian error = 0.1 and dislocation source density qsou ¼ 50 lm2 .

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reasonable in describing the material behaviour for relatively small strains, but should only be regarded as approximate at large strains. 3.2. Near-tip stress–strain response The DDD simulations of a transgranular crack (Fig 2) were carried out for 5 cycles at a stress intensity range of pffiffiffiffiffi DK ¼ 6 MPa m, load ratio R ¼ 0:1 and loading frequency f ¼ 1000 Hz. A high loading frequency was employed to allow DDD calculations to be completed within a reasonable period of time. Quantitatively, the cyclic loading conditions pffiffiffiffiffi DK ¼ 6 MPa m and f ¼ 1000 Hz result in location-dependent variable strain rates in the range of 0.02–43.66/s in this cracked model. Since the tension loading rate 1/s is roughly somewhere in the middle of this range, the fitted DDD parameters by the tension simulation at 1/s seems to be reasonable as a compromise. To study the normal stress–strain response ahead of the crack tip, element aggregate 1 was defined in Fig. 5(a) and (b). Based on the superposition scheme employed in the DDD model, the normal stress and strain for aggregate 1 are calculated as follows: 8 n X 4 X R R > 1 1 > ^ yy þ r ~ yy ÞdA ¼ > ð R r r xk ðr^ yy þ r~ yy ÞjJj yy ¼ A yy dA ¼ A > 1 1 < i¼1 k¼1 ð16Þ n X 4 > X R R > > 1 1 > Eyy ¼ eyy dA ¼ A1 ð^eyy þ ~eyy ÞdA ¼ xk ð^eyy þ ~eyy ÞjJj : A1 i¼1 k¼1

where A1 is the area of the aggregate 1, n is the number of elements, k the number of Gaussian point, and jJj the Jacobian determinant. To investigate the mesh sensitivity on the stress–strain response, two meshes were considered in Fig. 5(a) and (b), respectively. The coarse mesh model in Fig. 5(a) has an average element size D1  0:6 lm ahead of the crack tip, while the fine mesh model in Fig. 5(b) with an average element size D2 ¼ D1 =3. The maximum strains (emax ) registered at c each loading cycle for aggregate 1 are plotted as a function of loading cycles in Fig. 5(c) for the two meshes used

pffiffiffiffiffi Fig. 5. The cyclic response of element aggregate (1) under DK ¼ 6 MPa m, R ¼ 0:1 and f ¼ 1000 Hz. (a) The coarse mesh model of aggregate (1) ahead of the crack tip with an average element size D1 ¼ 0:6 lm. (b) The fine mesh model with an average element size D2 ¼ 13 D1 . (c) The response of maximum strain vs. number of cycle of aggregate (1) from the two models. (d) The cyclic stress–strain response of aggregate (1) from model (a).

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(Fig. 5(a) and (b)). Although the values of emax differ for the two meshes, the behaviour depicted appears to be largely idenc tical. Considering the computational costs, the following results were calculated using the coarse mesh (Fig. 5(a)). The normal cyclic stress–strain response is presented in Fig. 5(d) for aggregate 1, where the stress–strain loops exhibit a progressive shift in the direction of increasing tensile strain, a phenomenon known as ratchetting, where the plastic deformation during the loading portion is not balanced by an equal amount of yielding in the reverse loading direction. This local microscopic ratchetting response may be of particular significance for crack growth, as it may eventually lead to material separation near the crack tip. Ratchetting has already been recognised as a fatigue failure mechanism for metallic materials and alloys under asymmetric cyclic stressing (Yaguchi and Takahashi, 2005; Kang et al., 2006). Ratchetting strain near a crack tip also has been identified in a series studies carried out in our group, using time-independent and time-dependent cyclic plasticity (Zhao et al., 2004; Zhao and Tong, 2008; Cornet et al., 2009), crystal plasticity and simple power-law hardening material models (Tong et al., 2011). Furthermore, the concept of ratchetting has been used to predict crack growth (Zhao and Tong, 2008; Cornet et al., 2009) and the predictions compared reasonably well with some preliminary experimental results (Tong et al., 2011). It should be mentioned that ratchetting response can only be reproduced by standard FE modelling if a constitutive law with kinematic hardening is employed. For the present DDD simulations, no phenomenological hardening laws were introduced, hence it represents a physical approach to model crack tip field, from which ratchetting strain ahead of the crack tip has been identified for the first time. The ‘‘jerky’’ stress–strain responses of aggregate 1 shown in Fig. 5(d) are ~ yy ; ~eyy g of discrete dislocations, which differ from the FE results from continuum a result of the deformation fields fr mechanics. In the present DDD simulation, the dislocation sources were randomly distributed in the material at a given density qsou ¼ 50 lm2 , which may have an influence on the simulated mechanical responses, as demonstrated in the DDD simulations of tensile and compressive behaviour of micro-single crystals (Deshpande et al., 2005; Akarapu et al., 2010), the polycrystalline plasticity in thin films (Zhou and LeSar, 2011; Hou et al. 2009) and the growth of microvoid (Huang et al. 2007). Hence the effect of dislocation source distribution on the simulated cyclic maximum strain emax vs. number of cycles for c aggregate 1 was examined by using three different dislocation source distributions with the same density, and the results are plotted in Fig 6. It seems that dislocation source distribution does have a significant influence on the cyclic maximum strain emax , indicating that the discrete character of the DDD method and the inherent stochastic properties affect signific cantly the strain evolution over time. Nevertheless ratchetting response ahead of the crack tip is clearly captured in all three cases. The maximum strain emax might not always increase with cycles monotonically (Fig 6). For instance, the maximum c strain emax of cycle 4 is lower than that of cycle 3 for realisation 3, again reflective of the discrete nature at microscopic scales, c although overall ratchetting behaviour is clearly established. To understand the physical mechanisms of the near-tip ratchetting behaviour, dislocation density evolution with cycle is plotted against the tensile strain for grain 1 (G1) in Fig. 7. It seems that, although the dislocation density decreases to a certain extent during the unloading stage, it increases significantly during the loading stage, leading to an overall increase of dislocation density as well as plastic and total strains with cycle. This suggests that ratchetting may be a result of extra dislocation slip (or glide) quantity induced by the accumulation of dislocations. In Ni-based superalloys, since the friction stress introduced by the solid solution atoms and second phase precipitates is very high, not all the dislocation dipoles can recover to their original states and annihilate with each other during the unloading stage. Thus dislocation density does not decrease significantly at this stage. Consequently, in the subsequent loading stage, the sustained dislocation dipoles continue to ex-

pffiffiffiffiffi Fig. 6. The effect of dislocation sources on the maximum strain of element aggregate 1 under cyclic loading condition: DK ¼ 6 MPa m, R ¼ 0:1 and f ¼ 1000 Hz.

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pffiffiffiffiffi Fig. 7. The dislocation density in grain 1 ahead of the crack tip plotted as a function of the normal strain of aggregate (1) under DK ¼ 6 MPa m, R ¼ 0:1 and f ¼ 1000 Hz.

pand which introduce further plastic deformation. In addition, new dislocation sources can be nucleated since the shielding effect on dislocation sources by pile-ups may be released by the dislocation climb, which leads to further increase of dislocation density and to plastic strain accumulation. 3.3. Influence of climb and other factors on ratchetting response Dislocation climb is an important deformation mechanism for nickel-based superalloys at elevated temperature. It seems plausible that dislocation climb may influence the ratchetting response ahead of a crack tip. The maximum strain emax is plotc ted in Fig. 8 as a function of loading cycles for both with and without consideration of dislocation climb. It can be seen that, when dislocation climb is taken into account, a pronounced increase in maximum strain with cycles can be found. However, if dislocation climb is neglected, the maximum strain seems oscillate with the cycle and no significant ratchetting response is captured. Thus, dislocation climb would seem to be an important mechanism responsible for ratchetting ahead of a crack tip. This may be further explained as follows: During plastic deformation, dislocations pile up against the grain boundaries. These pile-ups can introduce strong back stresses on the dislocation sources, making further dislocation glide and nucleation more difficult on the same slip plane (Davoudi et al., 2012; Nicola et al., 2006). Work hardening rate may be enhanced by the pile-ups. Recovery processes, on the other hand, weaken the hardening effect by rearrangement and annihilation of dislocations. In the present DDD simulations, recovery occurs through climb. Dislocation climb is a temperature- and time-depen-

pffiffiffiffiffi Fig. 8. The maximum strain evolution of element aggregate 1 under cyclic loading condition (DK ¼ 6 MPa m, R ¼ 0:1 and f ¼ 1000 Hz), with and without considering dislocation climb.

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pffiffiffiffiffi Fig. 9. The maximum strain of element aggregate 1 under cyclic loading DK ¼ 6 MPa m and R ¼ 0:1 at loading rates 1000 Hz, 2000 Hz and 10,000 Hz.

dent process, which can disperse dislocations on different slip planes, reduce the number of dislocations in each pile-up and decrease the back stress. With increasing loading cycles (loading time), the number of dislocation climb events also increases gradually. As a result, the back stress is enhanced by the pile-ups and then released by dislocation climb alternately, leading to an increase in dislocation emission and slip quantity with cycles. The enhanced ratchetting response shown in Fig. 8 is clearly associated with dislocation climb. When no dislocation climb is considered, recovery mechanism is not possible hence ratchetting is not well-defined, if anything definitive. It should be mentioned that, since the stress concentration ahead of the crack tip can promote dislocation climb indicated by Eq. (12), ratchetting may be more pronounced near the crack tip than that away from the crack tip. Since recovery can also be achieved by dislocation cross slip not considered in the present DDD simulations, they might act as an alternative mechanism for ratchetting response. In addition, dislocation-GB penetration may also be one of the recovery mechanisms, its influence on the ratchetting responses will be examined in the following section. To investigate the effect of loading rate on the ratchetting response ahead of the crack tip, the cyclic maximum strain of aggregate 1 was shown in Fig. 9 for three loading frequencies, f ¼ 1000 Hz, f ¼ 2000 Hz and f = 10,000 Hz. It is evident that the maximum ratchetting strain at any given loading cycle decreases monotonically with increasing the loading frequency, which shows a significant loading rate effect. This indicates that lower loading frequency may result in higher microscopic strain ahead of the crack tip. When loading frequency is lower, the time taken to complete a cycle is longer, hence more dislocations have sufficient time to climb from its original slip plane to a new position. Consequentially, more dislocations can be further nucleated and more back stress relaxed, introducing more slip quantity. As a result, lower loading frequency leads demax

c to larger strains. For the same reasons, the ratchetting rate dN is also higher at lower loading frequencies. As shown in Fig. 9, the average ratchetting rate equals approximately to 0.00425 at f ¼ 1000 Hz, about 0.00366 for f ¼ 2000 Hz, while it is only

demax

c 0.00168 for f = 10,000 Hz. Further, after the second load cycle, the ratchetting rate dN for f = 10,000 Hz decreases to a very low value about 0.0009, which seems to indicate a saturation state and very different from the situations for f ¼ 1000 Hz and f ¼ 2000 Hz. In summary, both the strain field and the ratchetting rate ahead of the crack tip are closely related to the loading rate. It should be noted that these high frequencies were used to reduce the computational costs. Much lower frequencies are usually experienced in this type of alloys, hence it is not inconceivable that much more pronounced ratchetting response may be found in applications. It is well known that the plastic deformation in crystalline materials tends to localised in slip bands. This localised deformation may influence the ratchetting response ahead of the crack tip. In this work, we considered three neighbouring element aggregates 1, 2 and 3 in the same grain G1 containing the crack tip, as defined in Fig. 10(a). The centres of these three aggregates are within a distance of less than 0.5 lm to each other. The cyclic stress–strain responses for the three aggregates are presented in Fig. 10(b). It seems that, although these three aggregates are close to each other, their cyclic stress–strain responses are quite different. The most pronounced ratchetting response is found in aggregate 1; whilst the ratchetting strain and its rate are much weaker in aggregate 2, even though it is right front of the crack tip. Furthermore, the ratchetting response for aggregate 3 is very weak and almost negligible. These results clearly show that ratchetting is a highly localised event. Flouriot et al. (2003) investigated the strain localisation ahead of a crack tip in a single crystal material. Their experimental results showed that ratchetting is more significant in a localised slip band, similar to that shown by Marchal et al. (2006). This microstructure characteristic of plastic deformation and ratchetting response near the crack tip can be truly captured by DDD simulation, as shown also by the following Figs. 11 and 12.

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pffiffiffiffiffi Fig. 10. The cyclic stress–strain responses (b) under cyclic loading (DK ¼ 6 MPa m, R ¼ 0:1 and f ¼ 1000 Hz) for the three neighbouring element aggregates 1–3, as shown in (a).

pffiffiffiffiffi Fig. 11. The slip trace of dislocations under cyclic loading (DK ¼ 6 MPa m, R ¼ 0:1 and f ¼ 1000 Hz) after (a) 1st and (b) 5th loading cycle.

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Fig. 12. The contour plot of slip quantity at the maximum strain of loading cycle 5. Element aggregates 1 and 2 are on or mostly on the major slip bands, whilst aggregate 3 is away from the slip bands.

3.4. Crack-tip dislocation slip trace and deformation fields Since plastic deformation and ratchetting behaviour are direct results of dislocation evolution, dislocation slip traces defined as the trajectories of dislocation motion are plotted in Fig. 11 at the maximum strain state for cycle 1 (a) and cycle 5 (b), respectively. It can be seen that all the three slip systems in G1 are activated. Some dislocation slips are localised in areas with a narrow width to form slip bands. For all three slip systems in G1 containing the crack tip, the slip bands can be clearly identified. Further, both the intensity and the width of the slip bands at cycle 5 are much enhanced than those at cycle 1, indicating the accumulative nature of discrete dislocations with cycle. The wider slip bands and thus the larger ratchetting strain at cycle 5 are facilitated by climb and stress concentration, as a result of the formation of dislocation structures during loading. To describe the intensity of the dislocation slip, a parameter named as accumulated plastic slip C is defined following Balint et al. (2006) as:

C ¼ jc1 j þ jc2 j þ jc3 j with ci ¼ mis

~ s;t þ u ~ t;s i u nt ði ¼ 1—3Þ 2

ð17Þ

where ci is the quantity of slip for slip system i, mis and nit are the tangential and normal vectors of the ith slip system, respec~ the displacement induced by all dislocations. This accumulated plastic slip C is plotted in Fig. 12 at the maxtively, and u imum strain state of cycle 5. It can be seen that C is only significant in two slip bands near the crack tip, although more slip bands can be observed in the grain G1 from Fig. 11. One of the two major slip bands is directed approximately 45° ahead of the crack tip, indicating a more plausible crack growth path. Aggregate 1 is right on this major slip band, aggregate 2 is on the edge of this slip band while aggregate 3 is away from the area with high C. Considering the ratchetting responses of these three aggregates shown in Fig. 10, a conclusion may be easily reached that the ratchetting response ahead of the crack tip is associated with slip bands with high plastic slip C. Away from the slip bands with high C, the slip of dislocations is weak and dislocation climb more difficult. As a result, recovery process is less likely to occur hence ratchetting response relatively weak. In short, the heterogeneous dislocation motion seems to be responsible for the localised ratchetting response. The normal stress ryy fields at the maximum and minimum strain state for cycle 5 are presented in Fig. 13(a) and (b), respectively. From Fig. 13(a) for the maximum strain state, the field of normal stress ryy around the crack tip may be divided into four distinct sectors by four slip bands (indicated by the black solid lines). This is consistent with Rice’s (1987) asymptotic solution that the near-tip stress state is uniform (independent of h) within finite angular sectors at the crack tip, and the stress state jumps discontinuously at boundaries between sectors. The present DDD simulations show that this stress discontinuity is induced by the strong dislocation generation and motion in the intensive slip bands, as shown in Figs. 11 and 12. For the region right ahead of the crack tip, the normal stress level is the highest. Although ryy at the crack tip may be strongly shielded by the nucleated dislocations, it can also be enhanced by the dislocation structures ahead of the crack tip (O’day and Curtin, 2005). In the regions upper and below the crack tip, the normal stress ryy is shielded to relatively low values. Further, for the region behind the crack tip, a compressive stress field can be found even at the maximum strain state. The distribution of stress field seems to be discontinuous between two sides of a slip bands. Slip bands with both large

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Fig. 13. The stress contours of loading cycle 5 at (a) the maximum strain and (b) the minimum strain.

quantity of plastic slip C and high normal stress ryy may be indicative of potential crack growth paths. On the other hand, no such distinctive regions can be observed at the minimum stress state. Instead, high compressive stresses up to 800 MPa are found around the crack tip due to the back stresses from the residual dislocations. Because of the high compressive stresses, the singularity of the crack tip may be eliminated and the stress–strain response ahead of the crack tip shows a Bauschinger effect. 3.5. Monotonic response ahead of the crack tip under high stress intensity pffiffiffiffiffi Under cyclic loading conditions, a relatively small stress intensity factor range DK ¼ 6 MPa m has been used in the above DDD modelling to reduce the computational costs. To evaluate the deformation ahead of the crack tip at a higher stress pffiffiffiffiffi intensity, a monotonic loading was applied up to DK ¼ 18 MPa m at the same loading rate as that for cyclic cases. The dislocation slip trace near the crack tip is plotted in Fig. 14(a). It can be seen that, for the grains in the vicinity of the crack tip, multiple slip systems are activated, whilst only single slip systems are active for grains away from the crack tip. More slip systems are necessary to accommodate the complex stress state near a crack tip. Also, the area near the crack tip with a high dislocation density has a similar shape to that of plastic zone predicted by continuum mechanics. In addition, blunting of the crack tip can be clearly observed in Fig. 14(b). The detail of the crack tip in Fig. 14(b) also shows that most of the blunting comes from the upper surface of crack. The total strain of aggregate 1 is as high as 0.16 due to its position on the major slip band, as opposed to 0.05 for aggregate 3 away from the slip band. It is clear that plastic strain localization will be more significant when the external applied load is increased. In addition, the hardening rate of aggregate 1 is also increased at larger strains due to stronger back stresses introduced by the intensive dislocation pile-ups. 3.6. The effects of dislocation-GB penetration It is well known that grain boundaries (GBs) play a key role in dislocation evolutions and subsequent deformation response (such as the Hall–Petch effect) in polycrystals. The interactions between dislocations and GBs include dislocation absorption, reflection, emission and transmission (Shen et al., 1986, 1988), among which slip transmission (penetration) has been frequently observed (Carrington and Mclean, 1965; Mughrabi, 1983; De Koning et al., 2003). TEM analysis by Sangid et al. (2011a, 2011b) has found penetration of GBs is inherent in a polycrystalline Ni-based superalloy U720. Since dislocation-GB penetration can relax dislocation pile-ups, it may be one of mechanisms for recovery in addition to dislocation climb. A dislocation-penetrable GB model was hence introduced in the present analysis. To examine the effect of GB penetration on ratchetting response, dislocation climb is neglected in the following analysis for simplicity. The dislocation-penetrable GB model developed by Li et al. (2009) and Hou et al. (2009) was introduced into the present DDD program. Assuming that a dislocation penetration produces a GB extrusion with a width b, the critical shear stress spass for dislocation penetration may be obtained basing on the energy criterion:

spass b  b P Egb b þ alDb2

ð18Þ

where Db ¼ jb1  b2 j is the magnitude of the difference between the Burgers vectors of incoming b1 and outgoing b2 dislocations, a the material constant. In addition, the GB energy density Egb may be expressed approximately as:

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pffiffiffiffiffi Fig. 14. The crack tip deformation under a monotonic loading K ¼ 18 MPa m at a loading rate similar to that of the cyclic loading. (a) The slip trace of dislocations; (b) the blunting of crack tip by the dislocations and (c) the stress–strain response of element aggregate 1.

Egb

8 at Dh 6 h1 > < kDh=h1 at h1 6 Dh 6 h2 ¼ k >



: p k 2  Dh = p2  h2 at Dh P h2

ð19Þ

where Dh is the misorientation angle between two neighbouring grains. Values of the rest parameters are taken to be h1  20 , h2  70 and k  1000 mJ=m2 (Sangid et al., 2011b). For simplicity, the dislocation debris produced in the process of GB transmission was assumed to be totally absorbed by the GB. As a result, dislocation emission from the dislocation debris suggested in Li et al. (2009) and Fan et al. (2011) was not considered. The cyclic stress–strain responses of aggregate 1 are plotted in Fig. 15 for a dislocation-penetrable GB, together with that pffiffiffiffiffi for an impenetrable GB under DK ¼ 6 MPa m, R ¼ 0:1 and f ¼ 1000 Hz. It can be seen that, when dislocation penetration across GB is forbidden, no ratchetting response can be observed for aggregate 1, even if it is right on the slip band. However, when dislocation penetration is introduced, clear ratchetting behaviour of aggregate 1 can be captured although it may not be as pronounced as that in Fig. 4(c) when dislocation climb was considered. Nevertheless dislocation-GB penetration seems to be one of important mechanisms for the cyclic ratchetting response ahead of the crack tip. This is because that, with increasing number of dislocations in the pile-up, the Peach–Kohler force on the leading dislocation of the pile-up will exceed the critical stress spass and force the leading dislocation to enter into the neighbouring grains. As a result, the number of dislocations in the pile-up decreases, which also leads to the decrease in the back-stress on the dislocation source. New plastic deformation may be produced by the newly-nucleated dislocations from the relaxed dislocation sources by dislocation penetration in each loading cycle, which leads to recovery and ratchetting response ahead of the crack tip. This may be the reason that dislocation-GB penetration has been proposed to be incorporated into a crystal plasticity theory recently (Ma et al.,

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pffiffiffiffiffi Fig. 15. The cyclic stress–strain responses under DK ¼ 6 MPa m, R ¼ 0:1 and f ¼ 1000 Hz for rigid GB and dislocation-penetrable GB models.

2006; Shanthraj and Zikry, 2013). In addition, it can be seen from Eqs. (18) and (19) that the critical dislocation-GB penetration stress spass is on the order of several GPas. For the loading cases considered, only about 300 hundred dislocations successfully pass the GB during the DDD simulations, which leads to a relatively weak ratchetting response. At an elevated temperature T ¼ 1123 K, our DDD simulations seem to suggest that dislocation climb may be a more dominant mechanism for ratchetting than dislocation-GB penetration, although both dislocation-GB penetration and dislocation climb clearly contribute to the ratchetting response, an insight perhaps useful for further mechanistic modelling of crack growth in this type of polycrystalline alloys. 4. Conclusions Crack tip deformation has been studied in a polycrystalline Ni-based superalloy under cyclic model I loading condition using DDD simulations. Both dislocation climb and dislocation-GB penetration were considered in the DDD model. Strain ratchetting near a crack tip was observed for selected element aggregates in the vicinity and ahead of the crack tip, consistent in trend with the results from our crystal plastic and viscoplastic FE analyses. The dislocation density ahead of the crack tip was found to increase with the number of cycles. Multiple slip systems were activated for grains surrounding the crack tip as opposed to single active slip system found in remote grains. Ratchetting responses from the DDD simulation appear to be highly localised in that more significant ratchetting occurred within the slip bands than elsewhere. Although both dislocation climb and dislocation-GB penetration contribute to the local microscopic ratchetting response ahead of the crack tip, dislocation climb seems to be the dominant mechanism at the elevated temperature considered. This is the first time that ratchetting strain ahead of the crack tip is shown and validated to be associated with dislocation climb and dislocationGB penetration. Acknowledgments The authors wish to express their thanks for the financial supports from NSFC (11272128 and 11072081) and the Fundamental Research Funds for the Central Universities (2012QN024). References Abdel-Karim, M., 2010. An evaluation for several kinematic hardening rules on prediction of multiaxial stress-controlled ratcheting. Int. J. Plast. 26, 711– 730. Akarapu, S., Zbib, H.M., Bahr, D.F., 2010. Analysis of heterogeneous deformation and dislocation dynamics in single crystal micropillars under compression. Int. J. Plast. 26, 239–257. Balint, D.S., Deshpande, V.S., Needleman, A., Van der Giessen, E., 2006. Discrete dislocation plasticity analysis of the wedge indentation of films. J. Mech. Phys. Solids 54, 2281–2303. Barlat, F., Ha, J., Grácio, J.J., Lee, M.-G., Rauch, E.F., Vincze, G., 2013. Extension of homogeneous anisotropic hardening model to cross-loading with latent effects. Int. J. Plast. 46, 130–142. Bertin, N., Capolungo, L., Beyerlein, I.J., 2013. Hybrid dislocation dynamics based strain hardening constitutive model. Int. J. Plast. 49, 119–144. Berdichevsky, V., Dimiduk, D., 2005. On failure of continuum plasticity theories on small scales. Scr. Mater. 52, 1017–1019. Brinckmann, S., Van der Giessen, E., 2004. A discrete dislocation dynamics study aiming at understanding fatigue crack initiation. Mater. Sci. Eng. A 387– 389, 461–464. Busso, E.P., Meissonnier, F.T., O’Dowd, N.P., 2000. Gradient-dependent deformation of two-phase single crystals. J. Mech. Phys. Solids 48, 2333–2361.

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