The local symmetry criterion predicts that cracks under mixed-mode loading will deflect so that the local Mode II stress intensity factor, kII, is zero. An efficient approach is presented for its implementation in finite element simulations of crack propagation, based on the observed linearity of kII over a wide range of deflection angles. This linearity is explained by expressing kII, for the kinked crack, in terms of the global stress intensity factors, KI and KII, and parallel T-stress, Sigma T, for the original straight crack. Deflection angles and crack paths obtained agreed with those obtained using a more cautious search algorithm, and with other deflection criteria. This has been particularly applicable to crack-growth simulations in graded materials, where elastic asymmetry results in mixed-mode loading

Hoffman, M.

Tilbrook, M.

University of Queensland eSpace

SIF2004 Structural Integrity and Fracture. http://eprint.uq.edu.au/archive/00000836 
Implementation Of The Local Symmetry Criterion For Crack-Growth 
Simulations 
Matthew Tilbrook* and Mark Hoffman 
School of Materials Science & Engineering, University of New South Wales, Sydney, NSW 2052 
 
 
 
ABSTRACT: The local symmetry criterion predicts that cracks under mixed-mode loading will deflect so 
that the local Mode II stress intensity factor, kII, is zero. An efficient approach is presented for its 
implementation in finite element simulations of crack propagation, based on the observed linearity of kII over 
a wide range of deflection angles. This linearity is explained by expressing kII, for the kinked crack, in terms 
of the global stress intensity factors, KI and KII, and parallel T-stress, σT, for the original straight crack. 
Deflection angles and crack paths obtained agreed with those obtained using a more cautious search 
algorithm, and with other deflection criteria. This has been particularly applicable to crack-growth 
simulations in graded materials, where elastic asymmetry results in mixed-mode loading. 
1 INTRODUCTION 
Cracks frequently experience mixed-mode loading which causes them to deflect and curve. This 
influences crack propagation rate and, therefore, remaining life and structural integrity. Examples 
include cracks in complex-shaped components with highly asymmetric stress distribution, such as 
rotors and flanges [Miranda et al., 2003], and in heterogeneous material structures such as 
interfaces [Rice, 1988] or functionally graded materials (FGMs) [Gu and Asaro, 1997].  
Deflection criteria for cracks under mixed-mode loading have received considerable attention 
[Qian and Fatemi, 1996]. For example, the maximum tangential stress (MTS) criterion [Erdogan 
and Sih, 1963] predicts that cracks extend in the direction of maximum crack-tip tangential stress. 
This is calculated by expressing the crack-tip stress distribution in terms of the stress intensity 
factors (SIFs) for the original crack, KI and KII, so that the propagation direction, θM, may be 
determined directly from crack-tip mode-mixity, Ψ = tan-1(KII/KI): 
  Ψtan8
Ψtan8122
2
θtan
2
M +−=⎟⎠
⎞⎜⎝
⎛
                (1) 
Certain other criteria require a small test-kink to be extended from the crack-tip at angle, θ, 
which is varied to enable calculation of the optimum kink angle, θ = θK, in which, for example, the 
mechanical energy release rate, G, is maximised [Nuismer, 1975] or crack-tip loading is 
symmetrical [Goldstein and Salganik, 1974]. The latter, known as the local symmetry or kII = 0 
criterion, assumes that cracks will always propagate in the Mode I direction, so that kII, the Mode II 
SIF for the kink, is zero. This criterion has been shown to provide accurate predictions [Qian and 
Fatemi, 1996]. Implementation within a finite-element model involves the extension of a test-kink 
from the crack-tip, whose direction is varied to attain a minimum for kII. The successive deletion 
and remeshing that this necessitates extends computation time significantly [Becker et al., 2001, 
Tilbrook et al., 2004]. 
Investigating the local symmetry criterion for use in simulations of cracks in graded materials, it 
was observed that the variation in Mode II SIF, kII, with test-kink direction is approximately linear 
over a wide range of angles. This linearity was utilised in an improved algorithm for finding the 
optimum deflection angle. This paper discusses the linearity of the kII variation and its 
implementation for application to crack path predictions.  
SIF2004 Structural Integrity and Fracture. http://eprint.uq.edu.au/archive/00000836 
2 STRESS INTENSITY FACTORS FOR KINKED CRACKS 
For an infinitesimally small kink extending at an angle, θ, from the tip of an existing crack, as in 
Figure 1(b), the stress field acting on the kink may be assumed equivalent to that around the main 
crack in the absence of the kink, as in Figure 1(a) [Anderson, 1995]. Hence, SIFs at the kink-tip are: 
 cIIII rTAKCKCk 11211)( ++=θ  cIIIII rTAKCKCk 22221)( ++=θ   (2a,b) 
where Cij and Ai are angular functions given by Palaniswamy and Knauss [1978] and rc, related to 
process zone size [Smith et al., 2001], is equal to the kink-length. The full expression for kII(θ) is: 
   TrKKk cIIIII 2
)2sin(]
2
3cos3
2
[cos
4
1]
2
3sin
2
[sin
4
1)(
πθθθθθθ +⎟⎠
⎞⎜⎝
⎛+⎟⎠
⎞⎜⎝
⎛+⎟⎠
⎞⎜⎝
⎛+⎟⎠
⎞⎜⎝
⎛=          (3) 
 
 
 
 
 
 
 
 
 
 
Figure 1. Schematic of crack-tip (a) before and (b) after kinking has occurred. 
 
The case of r → 0, where the T-stress does not influence kII, is considered first. The effect of T-
stress on kII(θ) will be addressed subsequently. The trigonometric functions in Equation 3 may be 
expressed as Taylor series expansions, around θ = θM, the deflection angle given by the MTS 
criterion:  
ie. ...)(
2
cos
48
1)(
2
sin
8
1)(
2
cos
2
1
2
sin
2
sin 32 +−⎟⎠
⎞⎜⎝
⎛−−⎟⎠
⎞⎜⎝
⎛−−⎟⎠
⎞⎜⎝
⎛+⎟⎠
⎞⎜⎝
⎛=⎟⎠
⎞⎜⎝
⎛
M
M
M
M
M
MM θθθθθθθθθθθ (4) 
The value for θM may be determined from Equation 1, or from Equation 3 for kII(θ) = T = 0. Using 
such series expansions, kII(θ) may be expressed as a series of terms, kII(n)(θ), where n is the order in 
θ:  
...)()()()()()( )4()3()2()1()0( +++++= θθθθθθ IIIIIIIIIIII kkkkkk               (5) 
The initial, zeroth-order, term is equal to zero, due to the kII(θ) = 0 criterion. The first-order term is  
     )(
2
3sin9
2
sin
8
)(
2
3cos3
2
cos
8
)()1( M
MMII
M
MMI
II
KKk θθθθθθθθθ −⎟⎟⎠
⎞
⎜⎜⎝
⎛ ⎟⎠
⎞⎜⎝
⎛+⎟⎠
⎞⎜⎝
⎛−−⎟⎟⎠
⎞
⎜⎜⎝
⎛ ⎟⎠
⎞⎜⎝
⎛+⎟⎠
⎞⎜⎝
⎛=       (6) 
This is non-zero and is linear in (θ-θM). The second-order term is: 
2
M
MMII2
M
MMI)2(
II )(2
3cos9
2
cos
32
K)(
2
3sin9
2
sin
32
K)(k θθθθθθθθθ −⎟⎟⎠
⎞
⎜⎜⎝
⎛ ⎟⎠
⎞⎜⎝
⎛+⎟⎠
⎞⎜⎝
⎛−−⎟⎟⎠
⎞
⎜⎜⎝
⎛ ⎟⎠
⎞⎜⎝
⎛+⎟⎠
⎞⎜⎝
⎛−=     (7) 
As the derivatives of sine and cosine are cyclic, a dominant part of this expression is proportional to 
kII(θM), and hence approaches zero as θ approaches θM. Similarly the fourth-order term, and all 
subsequent even terms, will undergo partial cancellation, reducing the magnitude of the symmetric 
component of the kII(θM) variation. For angles within 30° (~0.5 rad) of θM, the increase in power of 
(θ - θM) leads to significant diminishing of the higher order terms. The third term, for example, has 
a dominant part which is proportional to, but much smaller than, the first term: 
Crack  
r 
x1
2v(x) 
2u(x) 
KII 
Kink kI 
KI 
KI 
kII 
KII kI 
θ θk kII 
(b) (a) 
∆ak 
Crack  
SIF2004 Structural Integrity and Fracture. http://eprint.uq.edu.au/archive/00000836 
 3MMMII
3
M
MMI)3(
II )(2
3
sin81
2
sin
196
K
)(
2
3
cos27
2
cos
196
K
)(k θθθθθθθθθ −⎟⎟⎠
⎞
⎜⎜⎝
⎛ ⎟⎠
⎞⎜⎝
⎛+⎟⎠
⎞⎜⎝
⎛+−⎟⎟⎠
⎞
⎜⎜⎝
⎛ ⎟⎠
⎞⎜⎝
⎛+⎟⎠
⎞⎜⎝
⎛−=    (8) 
The remaining part will also be small, and will partially cancel with the remainder from the 
second term. Consequently, the dominant linear behaviour may be explained by the partial 
cancellation of the even terms, resulting from the condition that kII(0)(θ) is close to zero. 
Accordingly, the linear term (Equation 6) may be used as an approximation for the kII(θ). 
The variation of kII(T), the T-stress contribution to kII, with angle is not linear [Smith et al., 
2001]: 
  [ ]2MMMMMcc)T(II )θ-θ)(θ2sin(2)θ-θ)(θ2cos(2)θ2sin(2rTT2r)θ2sin()θ(k −+≈= ππ      (9) 
Over the relevant range of angles however, the variation in kII(T) is sufficiently small that the 
non-linear component does not skew results appreciably. Indeed, the series-expansion arguments 
above can be extended to demonstrate that the T-stress term does not alter the linearity of kII(θ) 
significantly. 
3 FINITE ELEMENT IMPLEMENTATION 
Crack deflection behaviour was investigated with finite element analysis, which was conducted 
using ANSYS (Version 6.1, ANSYS Inc, Canonsburg, Pennsylvania). The structures were modelled 
in two dimensions under plane strain conditions using free-meshed isoparametric quadrilateral 
elements, with quarter-point singularity elements at the crack-tip, as in Tilbrook et al. [2004]. Free 
meshing was utilised, with significant refinement around the crack-tip to ensure solution accuracy. 
For graded material simulations, the spatial elastic property variation was applied to nodes. SIFs 
were calculated from nodal crack-opening displacement values near the crack-tip, both before (Kα) 
and after (kα) kinking. A small test kink, ∆aK, was extended from the crack-tip at varying angles, 
and kII was calculated from nodal displacements.  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Figure 2. Schematic showing variation of Mode II SIF, kII, with kink angle, θ. Discrepancies, ∆1 and ∆2, between actual 
kII values (●) and linear approximation values, for angles θ1 and θ2, resulting in error, ∆θg, in predicted optimum kink 
angle. Specimen configurations examined: (a) asymmetrically notched homogeneous specimen in three-point bend 
loading with initially straight crack, and (b) notched graded specimen in four-point bend loading with initially straight 
crack. A finite element mesh representative of those used for simulations of both specimen types is shown in (c). 
Initial upper and lower bounds for the optimum kink angle, θup and θlo, were determined from a 
wide interval around θM, the angle predicted by the MTS criterion. The original algorithm was 
Kink angle, θ 
M
od
e 
II 
S
IF
, k
II, 
at
 k
in
k-
tip
 
θ1 θ2 
∆2
∆1 
∆θg 
Linear 
Approximations 
θg 
θk θM 
Including T-stress  
Neglecting 
T-stress 
Interpolation 
between  
kII(1) & kII(2) 
(b) 
(c) 
(d) 
(a) 
SIF2004 Structural Integrity and Fracture. http://eprint.uq.edu.au/archive/00000836 
based on successive 30% reductions of the angular range, until a sufficiently small range was 
attained.  
The linearity of the angular kII variation was exploited in a simple algorithm for finding the 
optimum kink angle more efficiently. The initial angular bounds, θ1 = θlo and θ2 = θup, were 
estimated from θM as in the previous method. The optimum angle was estimated directly, from 
these angles and their corresponding kII values, by interpolation:  
  )1(
II
)2(
II
)1(
II
121g kk
k)( −−+= θθθθ                     (10) 
The estimate may be refined by calculating kII for angular bounds closer to the estimated 
optimum, ie. θ = θg ± χ with χ around 4°, then repeating the interpolation. With this approach, the 
optimum angle is estimated within ± 0.1° in 4 iterations, compared with 12 or more iterations 
required for an equivalent estimate with the original approach. 
To assess the accuracy of the approach, the error in estimated optimum kink angle was estimated 
from the intrinsic errors associated with the assumption of linearity. The values of kII for the initial 
angular bounds will be near the linear values, as in Figure 2(a), with each differing by a small error, 
∆x: 
  1k1
)1(
II )(mk ∆+−= θθ   2k2)2(II )(mk ∆+−= θθ            (11) 
where m is the linear gradient from (θ1,kII(1)) to (θ2,kII(2)). The estimated kink angle, from (10) is: 
)1(
II
)2(
II
)1(
II
)2(
II
12
21
g kk
kk)(
2 −
+−−+= θθθθθ                  (12) 
From (11) and (12), the error in θg, the difference between estimated and actual optimum kink 
angle, is: 
 )1(
II
)2(
II
)1(
II
)2(
II1221
kgg kk
kk
m2m2 −
+∆−∆+∆+∆−=−=∆ θθθ            (13) 
If the angular bounds were distributed fairly symmetrically about the optimum angle, then kII(2) 
and kII(1) would be similar in magnitude though opposite in sign, as would be the errors ∆1 and ∆2., 
due to the dominant odd-terms. Accordingly, both terms in (13) will be small, especially so in the 
refinement step, when the angular bounds are closer to, and more symmetrical about, the optimum 
angle. 
4 CRACK GROWTH SIMULATIONS 
The application of the improved algorithm to crack-growth simulations is demonstrated here for 
two cracked specimen configurations: an asymmetrically-notched homogeneous beam under three-
point bend loading, as in Figure 2(b); and an inhomogeneous beam, with longitudinal compositional 
variation, containing a straight crack under four-point bend loading, as in Figure 2(c). The geometry 
and meshing for each configuration was very similar. A representative mesh, comprised of 
approximately 3500 elements, is shown in Figure 2(d). 
Values of SIFs and T-stress for the initially unkinked cracks were calculated from nodal stress 
and displacement values near the crack tip. For the homogeneous specimen, KI = 1 MPa√m; KII = 
0.089 MPa√m; σT = 11 MPa; whilst for the graded specimen, KI = 2.45 MPa√m; KII = 1.15 
MPa√m; σT = 39 MPa. The results for these are plotted in Figure 4, which compares the FEM 
results with the predicted variation (Equation 3) and the linear approximation (Equation 6), with 
and without the inclusion. The high degree of linearity of the kII-θ relationship is apparent. Good 
agreement may be observed for both specimen configurations.  
 
 
SIF2004 Structural Integrity and Fracture. http://eprint.uq.edu.au/archive/00000836 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Figure 3. Variation of Mode II stress intensity factor, kII, with kink angle, θ, for initially straight crack in (a) an 
asymmetrically notched homogeneous specimen under three-point bend loading, and (b) in a notched graded specimen 
under four-point bend loading. Comparison of predictions, from exact expression and linear approximation, with results 
from finite element model.  
  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Figure 4. (a) Non-linear component of kII variation. Plot showing discrepancy between linear approximation and exact 
expression, along with second and third order terms from series expansion, for crack in graded specimen. (b) Predicted 
propagation paths for cracks in homogeneous and graded specimens. Predictions obtained using the improved algorithm 
are compared with those obtained using the more rigorous algorithm, with the (kII)min and Gmax criteria. 
The non-linear component of the kII(θ) variation for the graded specimen is illustrated in Figure 
4(a). The relative discrepancy between the linear approximation and the exact expression is plotted 
against θ, and compared with the second and third order terms (Equations 7 and 8), and the T-stress 
contribution. It may be observed that the relative discrepancy is small, and the higher order terms 
(b) 
(a) 
(b) (a) 
-6 105
-4 105
-2 105
0
2 105
4 105
6 105
0.4 0.5 0.6 0.7 0.8 0.9 1 1.1
FE Results
Exact Expression
Linear Approximation
M
od
e 
II 
SI
F,
 K
II [
M
P
a 
m
1/
2 ]
Test-kink angle, θ [rad]
Neglecting T-stress
Including T-stress
-2 105
-1 105
0
1 105
2 105
3 105
4 105
0 0.2 0.4 0.6 0.8 1
FE Results
Exact Expression KII
Linear Approximation
M
od
e 
II 
S
IF
, k
II [
M
P
a 
m
1/
2 ]
Test-kink angle, θ [rad]
Neglecting T-stress
Including T-stress
-1 105
-5 104
0
5 104
1 105
0.4 0.5 0.6 0.7 0.8 0.9 1 1.1
M
od
e 
II 
S
IF
, K
II [
M
P
a 
m
1/
2 ]
Test-kink angle, θ [rad]
Difference between exact expression
and linear approximation
Difference between FE results 
and linear approximation
Second-order term K
II
(2)
Third-order term K
II
(3)
T-stress term K
II
(T)
0
2
4
6
8
10
12
14
0 2 4 6 8
MTS
(k
II
)
min
G
max
Impr. (k
II
)
min
y 
[m
m
]
x [mm]
Homogeneous
Specimen
Graded
Specimen
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have a very minor influence, in the case investigated. This indicates that the use of the linear 
approximation in estimating optimum deflection angle is valid, though the use of refinement step is 
recommended. 
Crack propagation paths were simulated for both specimen configurations, utilising the improved 
algorithm. For comparison, simulations were also conducted using the original algorithm with the 
(kII)min criterion and the Gmax criterion, and using the MTS criterion which does not require a test-
kink. Predicted crack paths are shown in Figure 4(b) for both specimen configurations. Clearly there 
is excellent agreement between results from all approaches, with the only exception being a 
divergence of the MTS predictions for both specimens later in the crack propagation, due to T-
stresses which influence crack path but are not taken into account in MTS predictions [Tilbrook et 
al., 2004].  
5 CONCLUSIONS 
1. The variation of Mode II stress intensity factor, kII, at the tip of a kinked crack with the 
deflection angle of the kink is close to linear, to a very close approximation, over a wide range 
of deflection angles, and is predominantly antisymmetric about kII = 0.  
2. This was explained in terms of the expression for Mode II SIF for a small kink, which can be 
expanded as a series. The linear term, and subsequent odd terms, dominated due to partial 
cancellation of the even terms. 
3. This has been utilised in an algorithm for prediction of optimum deflection angles, which 
increased the efficiency of crack path simulations significantly without compromising accuracy.   
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materials, Int. J. Sol. Str., 38, 5545 (2001) 
Erdogan F and Sih G C. On the crack extension in plates under plane loading and transverse shear, J. Basic. 
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Goldstein, R V and Salganik, R L, Brittle fracture of solids with arbitrary cracks, Int. J. Fract., 10, 507 
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Gu, P and Asaro, R J, Crack Deflection in Functionally Graded Materials. Int. J. Sol. Struct. 34, 3085 (1997). 
Kim, J-H and Paulino G, Finite element evaluation of mixed mode stress intensity factors in functionally 
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Implementation Of The Local Symmetry Criterion For Crack-Growth Simulations

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Implementation Of The Local Symmetry Criterion For Crack-Growth Simulations

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