Fracture growth is considered as the competition between cleavage and dislocation self-organization in elastic-plastic solids. A self-consistent model is proposed to bridge the responses at relevant length scales, an elastic enclave in the immediate vicinity of crack tip, an array of disclination dipoles and macroscopic plastic deformation. The directional dependence of crack growth is studied. In the continuum limit, the flow stress is expressed by a spatial coupling in terms of a second-order gradient of the rotation strength of disclination dipoles. An estimate of the core size and the crack-tip shielding ratio is given by identification of the macroscopic plastic fields, the elastic field and the constitutive flow stress from the micromechanics consideration, on the boundary of elastic core. Strong dependence of apparent fracture toughness on the intrinsic surface energy and the ductile-to-brittle transition are examined

Jeffrey, R.

Mai, Y. W.

Zhang, X.

University of Queensland eSpace

 1
A Self-Consistent Model For Directional Dependence Of Crack Growth 
 
Xi Zhang1,   Rob Jeffrey1   and   Yiu-Wing Mai2 
1CSIRO Petroleum, PO Box 3000, Glen Waverley, Victoria 3150, Australia 
2Centre for Advanced Materials Technology (CAMT), School of Aerospace, Mechanical and Mechatronic 
Engineering J07, University of Sydney, Sydney, NSW 2006, Australia 
 
 
 
ABSTRACT 
Fracture growth is considered as the competition between cleavage and dislocation self-organization in 
elastic-plastic solids. A self-consistent model is proposed to bridge the responses at relevant length scales, an 
elastic enclave in the immediate vicinity of crack tip, an array of disclination dipoles and macroscopic plastic 
deformation. The directional dependence of crack growth is studied. In the continuum limit, the flow stress is 
expressed by a spatial coupling in terms of a second-order gradient of the rotation strength of disclination 
dipoles. An estimate of the core size and the crack-tip shielding ratio is given by identification of the 
macroscopic plastic fields, the elastic field and the constitutive flow stress from the micromechanics 
consideration, on the boundary of elastic core. Strong dependence of apparent fracture toughness on the 
intrinsic surface energy and the ductile-to-brittle transition are examined.  
Key words: Crack growth, direction, disclination dipole, kink, core size, shielding ratio 
 
INTRODUCTION 
Crack growth operates on different length scales due to high stress and strain gradients and 
continuum models allowing for finite scale effects are needed to account for multiscale 
mechanisms. Most studies on ductile versus brittle fracture behaviours focused on the competition 
of dislocation emission from the crack tip and cleavage fracture [Kelly et al., 1967; Rice and 
Thomson, 1974; Zhang, 1990; Fischer and Beltz, 2001]. However, other energy dissipation 
mechanisms such as preexisting mobile dislocations may play an important role. Creation and 
motion of dislocations near the crack tip may give rise to collective response in increasing the 
ductility of the materials [Orowan, 1945; Irwin, 1948]. Usually, plastic deformation on different 
length scales is interconnected and exceeds the free energy of the newly created surface by orders of 
magnitude. Fracture mechanics theories proceed from the idea of the existence of a close relation 
between length scales and the crack size so that some distributions and types of flaws may shield, 
and others anti-shield, the crack tip in terms of changing critical stress intensity factors [Zhang, 
1990] or considering energy dissipation mechanisms [Kysar, 2003].  
There exists a microdefect-free ligament directly ahead of the crack tip at the atomic scale level. 
A dislocation-free zone ahead of the crack tip has been detected in single crystals of stainless steels, 
copper and aluminium subjected to tension and cyclic loading [Ohr et al., 1982]. Outside this zone, 
a massive dislocation activity can take place on the mesoscopic level by stress-assisted cooperative 
dislocation movement. These cooperative effects manifest qualitatively new properties by forming a 
system of disclinations when their total number exceeds a certain value. There is a need to examine 
the effect of plastic dissipation on material fracture. 
Several proposals have been made to define the characteristic length for size effects. These 
include (i) the Burgers vector [Lipkin et al, 1996], (ii) the spacing of barrier to dislocation motion 
[Beltz et al., 1996], (iii) the spacing of Frank-Read sources [Kysar, 2003], 2/1−= ρcL  in which ρ  is 
the dislocation density, and (iv) a plastic zone with many low-mobility dislocations at a radius of 
the order 4310 − b has predicted [Suo et al., 1993]. The cooperative effect of high-density dislocations 
may behave as a disclination dipole or a superdislocation [Romanov and Vladimirov, 1992]. 
Especially, plastic deformation can be localized into a kink or misorientational band emitted from 
 2
the crack tip. The disclination dipoles distribute uniformly inside the misorientation band, but with 
different rotation strength corresponding to stress and strain gradients. The dipole arm proportional 
to the width of the kink band, is of the order of thousands of the Burger's vector, in the order of 0.1-
1 µm. These disclination dipoles also interact with each other so as to minimize the potential 
energy, and a continuum limit exists. In this paper, we will focus on the understanding of the effect 
of disclination plasticity on the crack growth in some preferred directions.  
 
DISCLINATION-BASED CRACK-TIP PLASTICITY 
For a geometrical description of disclinations, we start with a uniform hollow cylinder having a 
radial cut, the typical wedge disclination of Somigliana type. By turning the faces of the cut and 
subsequently sticking them together, one can find a disclination related to the rotation of non-
deformed faces of the cuts by an angle ω , the Frank vector, about the fixed axis. 
dh
ω
dh
ω
 
Figure 1: Dislocation description of wedge disclinations and their dipoles 
There is a relationship between disclinations and dislocations. A straight wedge disclination can 
be represented in terms of an array of dislocations, as shown in Fig. 1. An disclination with the 
power ω  is identical to a semi-infinite wall of edge dislocation having linear density ρ :  
 dbdbdh //)2/tan(2/1 ωωρ ===        (1) 
in which dh  is the spacing of dislocations. 
ω
a
x 
y 2a V 
 
Figure 2: Disclination hardening mechanism. A steady moving disclination dipole motion is related to 
redistribution of dislocations in front of the dipole. 
The semi-infinite dislocation wall seems to be impossible in crystalline solids because it needs 
adequate energy. But, most disclinations with opposite sign can exist in the form of a wall cut off 
from two sides, as a disclination dipole illustrated in Fig. 1. It can be found in Romanov and 
Vladimirov (1992) that the energy for the formation of disclination dipoles is nearly the same as a 
single edge dislocation. To calculate the flow stress in the presence of disclinations, let us consider 
the motion of a disclination dipole that interacts with gliding dislocation loops in front of the 
moving disclination dipole at a steady speed V, as illustrated in Fig. 2. The central plane of the 
 3
microrotation band suffers the maximum shear stress that is equal to, [Romanov and Vladimirov, 
1992]:  
exy ygD τωσ += )(           (2) 
in which )1(2/ νπ −= GD ; )/()( 22 ayyayg += ; eτ  refers to the critical shear stress above which the 
disclination dipole nucleation occurs and is of the form )/1(* ce TT−= στ , in which T and cT  are the 
temperature and the reference temperature respectively, and *σ  is the reference stress for a special 
material. 
It follows from (2) that the maximum flow stress is:  
ωτσ Deflow 5.0+=            (3) 
 
 
cθcθ
 
Figure 3: Generation of a micro-orientational band at an angle cθ  to the crack line. The arrays 
of dislocations can be represented by disclination dipole motion at cθ . 
Materials ahead of a crack tip experience a non-uniform distribution of stresses. An example of a 
relaxation process, in this case, twinning, is shown in Fig. 3 with a periodic array of disclination 
dipoles to form a kink band. The kink band has been modelled [Flouriot et al., 2003] in a single 
crystal CT specimen using finite element method. It can be simplified by the disclination dipoles, as 
shown in Fig. 3, which move quasi-statically in the direction of cθ  to the crack line. The dipole 
motion reduces the stress levels directly in front of the crack tip and create a local overstress at a 
certain distance. The asymmetric crack geometry is expected in the presence of a preferred direction 
of mobile dislocations, as shown in Fig. 3.  
We now express the coupling term inside the band by a discrete form that takes into account the 
interaction of a dipole with its two nearest neighbours. It is assumed that the disclination power 
varies along the thickness of the kind band and the effective disclination arm is also equal to 2a. So, 
the flow stress in elastic medium is [Romanov and Vladimirov, 1992]:  
 ωτσ 222 ∇−=− Daeflow         (4) 
 
THE MODEL 
We shall build our qualitative model on the framework previously proposed for cleavage in the 
presence of mobile dislocations[Suo et al., 1993; Beltz et al., 1993; and Lipkin et al., 1996]. Fig. 4 
shows schematically the physical basis for our analyses of initiation of cleavage fracture under 
small scale yielding condition. The crack tip is located in a single grain and the material 
surrounding the crack is allowed to plastically deform with strain hardening prior to cleavage. There 
exists a core region with radius cR  in which the material behaves as an elastic medium. The 
continuum assumption of disclination dipoles can be justified, as the radius is greater than cR .  
In the region outside the core, a power-law elastic-plastic constitutive law is valid. The 
asymptotic stress field for homogeneous deformation at small distance, r, from the crack tip can be 
taken as the modified singular HRR field [O'Dowd and Shih, 1991]. The effective stress σ  in this 
so-called J-Q field has the form: 
 4
 0
)1/(1
2
0
2
0 )( σσσβ
ξσσ Q
r
K
I n
nI
n
+= +∞        (5) 
in which Q represents the crack-tip constraint; ∞IK  is the applied stress intensity factor that 
characterizes the elastic field well beyond the plastic zone, and the measured energy release rate 
EKG I /
2∞∞ = ξ . 0σ  is the yield strength. The parameter ξ  represents the state of loading mode, 1 for 
plane stress and 21 ν−  for plane strain. n and β  are material constants in power-law plasticity. nσ  is 
a weak function of θ . nI  is a wake function of the work-hardening exponents. 
 
Rc 
Plastic 
R 
Core 
Elastic σflow
r 
Rc
R
LEFM 
J-Q Field 
Core LEFM
Disclination-based 
Plasticity Solution 
 
Figure 4: Physical sources for cleavage fracture in the presence of plastic flow (a) microrotational 
bands, plastic zone and elastic core; (b) idealized stress distribution at cθ  ahead of the tip. 
In the core region, the stress field inside the core region takes the form of an elastic singularity 
described by a crack-tip stress intensity factor ξ/tiptip EGK = . The effective stress inside it can be 
approximately evaluated from the standard solutions in LEFM: 
 
r
Ktipλσ =           (6) 
in which the angular factor )2/(cos)cos3588(4 222 θθννπλ −+−=  for plane strain Mode I cracks. 
Furthermore, the second gradient of the rotation ω  is expressed as follows: 
22,2112,2212,1111,12
2 εεεεω −+−=∇        (7) 
Combining (4) and (7), we can obtain the expression for the flow stress as: 
),(~)()(2 )1/(2
0
2
2
0 θσβ
ξεβτσ n
r
K
Ir
aD nnI
n
eflow ∆−= +∞      (8) 
It is expected that the maximum radius of the core occurs along the angle cθ  where the value of 
∆~  reach the maximum. Enforcing continuity of effective stress on the core boundary along the fixed 
direction, cθθ = , we can equate the equation (5), (6) and (8) to calculate the radius of the elastic 
core R  and the ratio tipGG /∞ . Thus we can obtain: 
 25.01
0
5.0 )(])([)( −−− −+=
a
RQ
a
Ra
a
R ne
ξ
δλ
σ
τ
ξ
δλ      (9) 
 15.0 }/])(){[( +− −= nnn Qa
R
a
RI σξ
δλβχ       (10) 
in which 
a
EGtip
2
0σ
δ = ; 
a
EG
2
0σ
χ ∞=  and nnE
Da )/(~21 σβ ∆= . 
 5
 
AN EXAMPLE 
Consider the semi-infinite crack problems in an infinite medium. Table 1 gives the material 
constants required for solving Eqs. (9) and (10). Figure 5 shows the variations of the core size and 
the shielding ratio in the case of Q=0 and *σ =0. It can be seen that both the core size and the 
shielding ratio depend much on the work-hardening exponents and the intrinsic fracture toughness 
tipG . For the normalized quantity δ  up to 410 , bGtip 0/σ <300 [Beltz et al., 1996] because 
0/σE ~1000 and a/b=20. An increase in n and tipG  gives rise to an increase in the core size and ∞G , 
although a reverse trend is found for n=2.5 and δ  is less than 500. A slight variation in tipG  leads to 
a dramatic change in Rc/a and tipGG /∞ , especially for low hardening materials. The larger the value 
of n is, the more important the shielding ratio. This trend is in good agreement with existing results 
[Suo et al., 1993; and Beltz et al., 1996]. The shielding ratio can reach 410  for n=10 and 310  for n=5, 
the same order as those using FEM technique [Suo et al., 1993; Beltz et al., 1996]. In addition, the 
core size ranges from several to 100 times of the dipole arm. As a ranges from several to 50b, the 
core region would have the order of several hundreds of the lattice constants, that is, on the order of 
0.1-1µm, in good agreement with the predictions [Lipton et al., 1996; Beltz et al., 1996].  
 
Table 1: Selected values of HRR parameters under plane strain condition 
n cθ  nσ  nI  ∆~  
10.0 0.648 0.78 4.5 -1.9385 
5.0 0.517 0.61 5.0 -1.6542 
2.5 0.247 0.20 5.7 -1.5533 
 
δ
0 2000 4000 6000 8000 10000
R c
/a
0
20
40
60
80
100
120
n=10
5
2.5
δ
0 2000 4000 6000 8000 10000
K
oo
/K
tip
1
10
100
1000
10000
n=10
5
2.5
 
Figure 5: (a) Core size versus the intrinsic toughness and (b) Crack-tip shielding ratio versus the 
intrinsic toughness for various n at Q=0 and cTT = . 
The Ductile-Brittle transition is reported in Fig. 6 through the dependence of fracture toughness 
on temperature, under the conditions 0* /σσ =1, 1.5, 2 and δ=500. The sharp drop of tipGG /∞  at low 
temperature implies the occurrence of transition. It is interesting to note that the dislocation-free 
zone increase with decreasing temperature. Therefore, the plastic deformation is retarded by the 
expansion of the dislocation-free zone. Lower fracture toughness is expected in this case. 
Figure 7 displays the dependence of aRc /  and tipGG /∞  on the crack-tip constraint. It can be seen 
that the fracture toughness is enhanced by the loss of crack-tip constraint. This feature is more 
pronounced when n=10. It reveals that the constraint effect is more important for low work-
hardening materials. 
 6
Moreover, the kink band is located at an angle cθ  to the crack line, rather than directly ahead of 
the crack tip in the previous model [Lipkin et al., 1996]. The strain localization in these kink bands 
can yield a shielding effect on the crack tip field. Sometimes, it leads to the blunting of the crack 
tip. This toughening mechanism is well known [Romanov and Vladimirov, 1992]. 
T/Tc
0.0 0.2 0.4 0.6 0.8 1.0
R c
/a
0
5
10
15
20
σ∗/σ0=2.01.51.0
 T/Tc
0.0 0.2 0.4 0.6 0.8 1.0
K
oo
/K
tip
0
50
100
150
200
250
σ∗/σ0=2.01.5
1.0
 
Figure 6: Dependence of (a) the core size and (b) the shielding ratio on temperature for varying 
values of the ratio 0* /σσ  at n=10, Q=0 and δ=500. 
Q
-1.0 -0.5 0.0 0.5 1.0 1.5 2.0
R c
/a
0
20
40
60
80
n=10
5
2.5
Q
-1.0 -0.5 0.0 0.5 1.0 1.5 2.0
K
oo
/K
tip
10
100
1000
10000
n=10
5
2.5
 
Figure 7: Dependence of (a) core size and (b) shielding ratio on crack-tip constraint for various 
values of n at cTT =  and δ=2500. 
 
CONCLUSIONS 
In this paper, an effort was made to examine the directional dependence of crack growth in the 
presence of intense plastic deformation. The mobile dislocations can evolve into a kink band to 
shield the crack tip. A continuum model based on the disclination mechanism is proposed to 
account for relaxation of stress levels ahead of the crack tip. However, stress relaxation may be 
hindered by the cleavage process. A self-consistent model is presented in this paper to account for 
the competition between energy dissipation in terms of disclinations and the cleavage fracture in 
homogeneous materials. The elastic core size and the crack-tip shielding ratio are obtained for crack 
growth orientated in one direction with maximum absolute value of the second-order gradient of 
rotation. For low hardening materials, the core size is larger in order than the dislocation spacing. In 
addition, strong dependence of apparent fracture toughness on the intrinsic surface energy and the 
ductile-to-brittle transition caused by both thermal effect and crack–tip constraint are found. 
 
REFFERENCES 
Beltz, G. E., Rice, J. R., Shih, C. F. and Xia, L. A self-consistent model for cleavage in the presence of 
plastic flow, Acta Mater. 44, 3943(1996). 
Fischer, L.L. and Beltz, G. E. The effect of crack blunting on the competition between dislocation nucleation 
 7
and cleavage, J. Mech. Phys. Solids 49, 635(2001). 
Flouriot, S, Forest, S., Cailletaud, G., Koster, A., Remy, L., Burgardt, B., Gros, V., Mosset, S. and Delautre, 
J. Strain localization at the crack tip in single crystal CT specimens under monotonous loading: 3D Finite 
Element analyses and application to nickel-base superalloys, Int. J. Fracture 124, 43(2003). 
Irwin, G. R. Fracture mechanics, In: Fracturing of Metals, Eds. Jonassen, F. et al., American Society of 
Metals, Cleveland, 147(1948). 
Kelly, A., Tyson W., and Cottrell, A. Ductile and brittle crystals, Phil Mag 15 567(1967). 
Kysar, J. W. Energy dissipation mechanisms in ductile fracture, J. Mech. Phys. Solids 51, 795(2003) 
Lipkin, D. M., Clark, D. R. and Beltz, G. E. A strain-gradient model of cleavage fracture in plastically 
deforming materials, Acta Mater. 44, 4051(1996). 
O'Dowd, N. P., and Shih, C. F. Family of crack-tip fields characterized by a triaxility parameter, I Structure 
of fields, J. Mech. Phys. Solids 39, 989(1991). 
Ohr S. M., Horton, J. A., and Chung, S. J. (1982) Direct observation of crack tip dislocation behavior during 
tensile and cyclic deformation, Defects, Fracture and Fatigue, G. C. Sih and J. W. Provan, eds., Martinus 
Nijhoff Publishers, the Netherlands, 3-15. 
Orowan, E. (1945) Notch brittleness and the strength of metals, Proceedings of the Society of Engineers and 
Shipbuilders, Paper No. 1063, Scotland. 
Rice, J. R. and Thomson, R. Ductile versus Brittle behavior of crystals, Phil. Mag. 29, 73(1974). 
Romanov, A. E. and Vladimirov, V. I. (1992) Disclinations in crystalline solids, chapter 47 in Dislocations in 
Solids, No. 9, F. R. N. Nabarro ed., Elsvise Science Publishers, 191. 
Suo, Z., Shih, C. F. and Beltz, G. E. A theory for cleavage cracking in the presence of plastic flow, Acta 
Metall. Mater. 41,1551(1993). 
Zhang, T.-Y. Computer Simulation of Semibrittle Fracture, Zeitschrift für Metallkunde, 81 63(1990). 
 


A Self-Consistent Model For Directional Dependence Of Crack Growth

http://espace.library.uq.edu.au/view/UQ:10005/XiZhang_sif04.pdf

A Self-Consistent Model For Directional Dependence Of Crack Growth

Abstract

Similar works

Full text

Available Versions

University of Queensland eSpace