A Study of Fatigue Crack Tip Characteristics using Discrete Dislocation

Dynamics

MS Huanga,b, ZH Lia, J Tongb1

a Department of Mechanics, Huazhong University of Science and Technology, Wuhan, Hubei, 430074, China

b Mechanical Behaviour of Materials Group, School of Engineering, Anglesea Building, Anglesea Road, University of

Portsmouth, Portsmouth, PO1 3DJ, UK

Abstract 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

1Corresponding author: Tel: +44 (0)23 9284 2326; fax: +44 (0)23 9284 2351.

Email address: jie.tong@port.ac.uk

of the crack tip seems to be a physical phenomenon, which may be of particular significance

for developing a physical-based model of crack growth.

Keywords: Crack tip; Cyclic response; Discrete dislocation dynamics; Disclocation climb;

Grain boundary; Ratchetting; Slip band.

1. Introduction

Mechanistic understanding of fatigue crack deformation 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 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 also 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 (1991). Their results

also showed strain ratchetting occurring primarily in some of the localised slip bands. Using

the same material model (Meric, 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, Méric et al.,

1991; Nouailhas et al., 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 γ '

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 constitutive model to capture the non-Schmid response of a nickel alloy. Vattre and

Fedelich (2011) developed a micromechanical constitutive model with a pseudo-cubic slip

law which improved the estimation of the strain hardening anisotropy. Although dislocation-

microstructure interaction was incorporated, these constitutive models were formulated

within a continuum plasticity framework. 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 ρ(x ) as an independent local variable in the mesoscopic

constitutive models cannot be justified by a spatial averaging at the scale of channel width

(Vattre 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-center-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 axial 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 (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 m. 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 μm×58 μm,

and contains 150 grains with an average grain size of 5μm. As indicated in Fig.1, a strain-

controlled monotonic load was applied to the RC in the y-direction at a strain rate of ε=1/s.

In the x-direction, the following periodic boundary conditions were applied:

{ U xA−U x

B=U xC−U x

D

U yA−U y

C=U yB−U y

D=ε h (1)

where{U xA ,U x

B ,U xC ,U x

D} and {U yA ,U y

B ,U yC ,U y

D} are the displacements at representative

points A-D in x- and y-direction, respectively.

Following the 2D DDD framework by Van der Giessen (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:

~u=∑k=1

nd

uk ,~σ=∑k=1

nd

σk (2)

where nd is the total number of dislocations, and {uk , σ k } 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:

σ xxk +σ yy

k =2(ϕ ' ( z )+ϕ ' ( z ))

σ yyk −σxx

k +i 2σxyk =2(z ϕ ' ' ( z )+ψ ' ( z ))

uk ( z )=uxk+ iuy

k = 12 μ

[κϕ (z )−z ϕ' ( z )−ψ (z)] (3)

where σ xxk

and σ yyk are two normal stresses in the x and y direction, σ xy

k is the shear stress,

(z) and (z) are two potential functions, i is the pure imaginary, μ the shear modulus and

κ=3−4 ν for plain strain deformation. The two potential functions, (z) and (z), are

associated with dislocation via:

ϕ=ϕ0=γ ln (z−zd), ψ=ϕ0=γ ln ( z−zd )−γzd

z−zd

with γ=μ(bx+i by )4 π (1−v)

(4)

where zd=xd+i yd is the coordinate of the dislcoation, bx+ib y the Burgers vector of the

dislocation, v the Poisson's ratio. Inevitably, the dislocation fields (3) will introduce an

additional displacement and stress {~u ,~σ } at the RC boundary, which makes the boundary

condition (1) 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:

{T=T−~T=T−~σ .nt on St

U=U−~U on Su

(5)

where ~U∧~σ are the dislocation displacement and stress at the displacement (Su ¿ and

traction boundaries (S¿¿ t)¿, respectively, 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 (ˆ) fields:

u=~u+u , ε=~ε+ε , σ=~σ+ σ . (6)

For the kth dislocation, the in-plane component of the Peach–Koehler force controlling its

glide may be formulated as

f k=mk ( σ+∑~σ )bk (7)

where mk is the unit vector normal to its slip plane and bk 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 f fr=τ fr bk was introduced and the

effective Peach–Koehler force f eff may be written as:

f eff={fk−f fr if f k> f fr

f k+f fr if f k← f fr

0 if |( f k )|< f fr(8)

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

vk=f eff /B (9)

where B is the drag coefficient.

For FCC crystal structure, three active slip systems were considered and distributed

randomly in each grain, as illustrated in Fig. 1(a), with an intersection angle of 56.40between

each two slip systems. The dislocation sources were randomly placed on the slip planes with

a given density ρ sou. 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=τnucb within a period of time, a dislocation dipole may be generated, and it

consists of two opposite dislocations with a distance Lnuc=Gb

2 π (1−v)τnuc. The dislocation

source strength has a Gaussian distribution with a mean strength of τ nuc and a standard

deviation of α τnuc. 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 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 and its

neighbouring subcells were 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 n can be

calculated efficiently. In addition, for a polycrystalline material, most dislocations pile up

against the grain boundaries with a zero 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:

{Σij=1A∫ σ ij dA=¿ 1

A∫ (~σ ij+ σ ij)dA ¿ Εij=1A∫ εijdA= 1

A∫ (~ε ij+ε ij )dA

(10)

where Σij and Eij are the average stress and strain for the RC model, A is the area of the

RC, and σ ij and ε ij 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 μm×58 μm), which contains a transgranular crack,

was built for DDD simulation of crack tip deformation, as shown in Fig.2. The average grain

size is 5m. Fine mesh (~ 0.6μm¿ 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, ∆ K=6 MPa√m, load

ratio R=0.1 and a load frequency f=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 (1:13000). This assumption can effectively consider the interaction between a crack

and a dislocation field. 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):

ϕ=ϕ0+ϕd, ψ=ψ0+ψd with

ϕd ( z )=γ ln ( z−zd )+2 γlnζ−γ ln (ζ− mζ d )−γ ln(ζ− 1

ζ d )+γζ d (1+mζ d

2 )−ζd (ζ d2+m)

ζ d ζ d (ζ d2−m )(ζ− 1

ζ d)

ψd ( z )=γ ln ( z−zd )−γzd

z−zd+2 γ lnζ−γ ln(ζ− m

ζ d )−γ ln(ζ− 1ζ d ) (11)

+γζ d (ζ d

2+m3 )−mζ d(ζ d2+m)

ζ d ζ d (ζ d2−m) (ζ−m

ζ d)

−ζ 1+mζ 2

ζ2−mϕd '

(ζ)

where,{ϕ0 , ψ0 } are the potential functions in Eq.(4), z=x+iy=R(ζ+mζ

), R=(a+b)/2 ,

m=(a−b)/(a+b), and {a ,b }are the short and the long axis of the elliptical crack.

Substituting (11) into the 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 Ω 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:

V c=2 πD 0

bln(R/b)exp (−∆ Esd

kBT )[exp( Fc∆V ¿ /bk BT )−1] (12)

where b is the magnitude of the Burgers vector, ∆ E sd is the vacancy self-diffusion energy,

∆V ¿ is the vacancy formation volume approximately equalling to b3, k B 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

F c=sk (σ+∑~σ )bk (13)

where sk is the unit vector in the slip direction. The model by Davoudi et al. (2012) seems to

be more accurate than the drag-type 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=1123K

and diffusion constant D0=1.27×10−4m2/s (Marzocca and Picasso, 1996). Since the vacancy

self-diffusion activation energy for Ni-based superalloys is in the range 257–283kJ/mol

(Heilmaier et al., 2009), a minimum value ∆ E sd=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.(3) 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 (x0 , y0) to a new location (x0 , y1):

b2π [−tan( y− y0

x−x0 )−tan( x−x0

y− y0 )+ tan( y− y1

x−x0 )+ tan( x−x0

y− y1 )]. (14)

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 realize. 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 (x0 , y0) to point B (x0 , y1) and further glides to point C (x1 , y1) 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 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:

b2π [−tan( y− yc

x−xc )−tan( x−xc

y− yc )] (15)

where (xc , yc) 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 the loading rate feasible for computational purposes, hence

allowance must be made in the interpretation of the results. Nevertheless, as it will be

shown later, that the observed ratcheting phenomenon will be more pronounced at lower

frequencies, hence 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.

Table 1. The fitting DDD parameters

Young's

modulus

E

Poisson'

s ratio v

Dislocation source

density ρ sou

Dislocation mean

strength τ nuc

Dislocation strength

standard deviation

Slip friction

stress τ fr

185.3GPa 0.285 50 μm−2 250 MPa 0.1 τnuc 326 MPa

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

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 ∆K = 6 MPam, load ratio R=0.1 and loading frequency f=1000 Hz. A

high loading frequency was employed to allow DDD calculations to be completed within a

reasonable period of time. 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:

{Σ yy=1A1

∫σ yy dA=¿ 1A1

∫(~σ yy+σ yy)dA=∑i=1

n

∑k=1

4

ωk (~σ yy+σ yy)|J|¿Εyy=1A1

∫ ε yy dA= 1A1

∫(~ε yy+ ε yy)dA=∑i=1

n

∑k=1

4

ωk (~ε yy+ ε yy)|J|

(16)

where A1 is the area of the aggregate 1, n is the number of elements, k the number of

Gaussian point, and |J| 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 ∆1≈0.6 μm ahead of the crack

tip, while the fine mesh model in Fig.5 (b) with an average element size ∆2=13∆1

. The

maximum strains (ε cmax¿ registered at each loading cycle for aggregate 1 are plotted as a

function of loading cycles in Fig.5 (c) for the two meshes used (Fig.5 (a) and (b)). Although

the values of ε cmax differ for the two meshes, the behaviour depicted appears to be largely

identical. 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. The

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 a result of the deformation fields {~σ yy ,~ε yy } of discrete dislocations, which differ from

the FE results from continuum mechanics.

In the present DDD simulation, the dislocation sources were randomly distributed in the

material at a given density ρ sou=50 μm−2, 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, 2012) and the growth of microvoid

(Huang et al. 2007). Hence the effect of dislocation source distribution on the simulated

cyclic maximum strain ε cmax vs. number of cycles for 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 ε cmax, indicating that the discrete character of the

DDD method and the inherent stochastic properties affect significantly the strain evolution

over time. Nevertheless ratchetting response ahead of the crack tip is clearly captured in all

three cases. The maximum strain ε cmax might not always increase with cycles monotonically

(Fig 6). For instance, the maximum strain ε cmax of cycle 4 is lower than that of cycle 3 for

realization 3, again reflective of the discrete nature at microscopic scales, 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

expand 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. Dislocation climb was considered in the present

simulation and the maximum strain ε cmax is plotted 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-dependent

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 the

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 two loading frequencies,

f=1000 Hz and f=2000 Hz. It is evident that ratchetting response at f=1000 Hz is greater

than that at f=2000 Hz. 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 to larger strains. For the same reasons, the ratchetting rate d εcmax

dN is

also higher for lower loading frequency. As shown in Fig.9, the average ratchetting rate

equals approximately to 0.00425 for f=1000 Hz, while it is only about 0.00366 for

f=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 localized in slip

bands. This localized deformation may influence the ratchetting response ahead of the crack

tip. In this work, we considered three neighbouring element aggregates 1, 2 and 3, as

defined in Fig.10 (a). The centres of these three aggregates are within a distance of less than

0.5μm 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 localized 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 only be truly captured by DDD simulation, an

advantage over the continuum approach.

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 localized

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 is defined following Balint et al. (2006) as:

Γ=|γ 1|+|γ2|+|γ3|withγ i=msi~us ,t+~ut , s

2n ti (i=1 3) (17)

where γi is the quantity of slip for slip system i, mi and ni are the tangential and normal

vectors of the ith slip system, respectively, and ~u the displacement induced by all

dislocations. This accumulated plastic slip is plotted in Fig.12 at the maximum strain state

of cycle 5. It can be seen that is only significant in two slip bands near the crack tip,

although more slip bands can be observed in 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 . 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 . Away from the slip bands with high , 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 localized ratchetting response.

The normal stress σ yy 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 σ yy around the crack tip may be divided into four distinct regions

by four slip bands (indicated by the black solid lines). For the region right ahead of the crack

tip, the normal stress level is the highest. Although σ yy 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 σ yy is shielded by the dislocation stress field 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 quantity of plastic slip Γ and

high normal stress σ yy 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 -800MPa 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

Under cyclic loading conditions, a relatively small stress intensity factor range

ΔK=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 intensity, a

monotonic loading was applied up to K=18MPa√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 et al.,

1983; De Koning et al., 2003). TEM analysis by Sangid et al. (2011 a; b) 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) was introduced into the

present DDD program. Assuming that a dislocation penetration produces a GB extrusion

with a width b, the critical shear stress τ pass for dislocation penetration may be obtained

basing on the energy criterion:

τ passb∙b≥ Egbb+αμ Δb2 (18)

where Δ b=|b1−b2| is the magnitude of the difference between the Burgers vectors of

incoming b1 and outgoing b2 dislocations, α the material constant. In addition, the GB

energy density Egb may be expressed approximately as:

Egb={k ∆θ /θ1 at ∆θ≤θ1

k at θ1≤∆θ≤θ2

k ( π2 −∆θ)( π2 −θ2)

at ∆θ≥θ2

(19)

where ∆θ is the misorientation angle between two neighbouring grains. Values of the rest

parameters are taken to be θ1≈200, θ2≈700 and k ≈1000mJ /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) was not considered.

The cyclic stress-strain responses of aggregate 1 are plotted in Fig.15 for a dislocation-

penetrable GB, together with that for an impenetrable GB under ∆ K=6 MPa√m,

R=0.1∧f=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 τ pass 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

from 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 (Shanthraj and Zikry, 2013). In addition,

it can be seen from Eqs. (18) and (19) that the critical dislocation-GB penetration stress τ pass

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=1123K , 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 localized 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 to be

associated with dislocation climb and dislocation-GB 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).

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Figures

Fig. 1. The periodical polycrystalline material model (58 μm×58 μm) with 150 grains and an average

grain size of 5 μm. Three slip systems with an intersection angle of 54.70 were randomly distributed.

A triangular waveform was applied in the y direction along edges AB and CD with a displacement

rate, U yup−U y

down=ε h, where h is the height of the model and ε=1/s is the strain rate. A periodic

boundary condition was applied on edge AC and BD as U xA−U x

B=U xC−U x

D.

U yupwithU y

up−U ydown= ε h

U ydown

B

U xA−U x

B=U xC−U x

D

A

DC

h

54.70

Slip systems

Fig. 2. The cracked polycrystal model (58 μm×58 μm) with 150 grains and an average grain size of

5 μm. 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 G

1 is

∆1=0.6 μm

.

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+∆ t .

C (x1 , y1)

Solid: dislocation at time t+∆ t

Dashed: dislocation at time t

Fake dislocation 1

Fake dislocation 2

B (x0 , y1)

Slip plane

glide

climb

A (x0 , y0)

Grain 3

U

U

U

U

U

G1

Crack tip

Crack

Fig.4. A comparison of uniaxial tensile stress-strain response between the experiment data (ε=¿

0.005/s) and the DDD modelling (ε=¿1/s). Parameters in the DDD model were set as: Friction stress

σ f=326MPa, dislocation source strength = 250MPa, dislocation distribution Gaussian error = 0.1

and dislocation source density = 50μm−2 .

Experimental data

DDD simulation

(a) (b)

(c) (d)

Fig.5. The cyclic response of element aggregate (1) under ∆ K=6 MPa√m, R=0.1∧f=1000 Hz.

(a) The coarse mesh model of aggregate (1) ahead of the crack tip with an average element size

∆1≈0.6 μm. (b) The fine mesh model with an average element size ∆2=13∆1

. (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).

1 1

For mesh (a)

Fig.6. The effect of dislocation sources on the maximum strain of element aggregate 1 under cyclic

loading condition: ∆ K=6 MPa√m, R=0.1∧f=1000 Hz.

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 ∆ K=6 MPa√m, R=0.1∧f=1000 Hz.

Fig.8. The maximum strain evolution of element aggregate 1 under cyclic loading condition (

∆ K=6 MPa√m, R=0.1∧f=1000 Hz ), with and without considering dislocation climb.

Fig.9. The maximum strain of element aggregate 1 under cyclic loading (∆ K=6 MPa√m, R=0.1¿

at loading rates1000Hz and2000 Hz.

f=1000 Hz

f=2000 Hz

(a)

(b)

Fig.10. The cyclic stress-strain responses (b) under cyclic loading (∆ K=6 MPa√m,

R=0.1∧f=1000 Hz ¿ for the three neighbouring element aggregates 1-3, as shown in (a).

12 3Crack

(a) (b)

Fig.11. The slip trace of dislocations under cyclic loading (∆ K=6 MPa√m, R=0.1∧f=1000 Hz )

after (a) 1st and (b) 5th loading cycle.

Slip bands

GBGB

Slip bands

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.

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

Pa Pa

(a) (b)

(c)

Fig. 14. The crack tip deformation under a monotonic loading K=18MPa√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.

Blunting

K=18MPa√m

Fig.15. The cyclic stress-strain responses under ∆ K=6 MPa√m, R=0.1∧f=1000 Hz for rigid GB

and dislocation-penetrable GB models.