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                                                                       D. Nowell et alii, Frattura ed Integrità Strutturale, 25 (2013) 1-6; DOI: 10.3221/IGF-ESIS.25.01 
 

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Special Issue: Characterization of Crack Tip Stress Field 
  
 
 
Characterisation of crack tip fields under  
non-uniform fatigue loading 
 
 
D. Nowell, M.E. Kartal  
University of Oxford, Department of Engineering Science, Parks Road, Oxford, OX1 3PJ, UK. 
 
P.F.P. de Matos 
U3Is, Unidade de Investigação e Internacionalização do ISVOUGA, Santa Maria da Feira, Portugal. 
 

 
ABSTRACT. The paper analyses previously reported work, which uses digital image correlation to measure 
fatigue crack closure.  As well as determining crack opening loads, the information on crack shape may be used 
to estimate the stress intensity factor, as well as other parameters in more complex models of crack tip fields.  A 
number of specimens were subjected to single overload cycles, which produced a significant retardation in crack 
growth rate.  The method previously applied to the analysis of constant amplitude loading is here used to 
analyse the single overload case.  The stress intensity factor history is found to be very different in the two cases 
and the consequences of this observation for analysis of fatigue crack propagation are discussed. 
 
KEYWORDS. Digital image correlation; Displacement measurement; Crack-tip fields; Elastic-plastic fatigue 
crack. 

 
 
 
INTRODUCTION  
 

he study of crack tip stress, strain, and displacement fields has a long history.  One hundred years ago, Inglis 
recognized that a sharp elastic crack would exhibit a stress (and strain) singularity [1]. Twelve years later 
Westergaard [2] produced a stress function solution for a crack in a biaxial stress field which formed the basis for 

the later concept of the stress intensity factor.  An excellent compilation of seminal papers in the field has been provided 
by Sanford [3,4].  The stress intensity factor has proved to be an extremely useful concept for the assessment of cracks 
and it has been widely used in industry.  Paul Paris’ extension of the concept to its use in fatigue crack propagation [5] is 
particularly worthy of mention.  However, it is clear that the stress intensity approach has many limitations.  In particular, 
all fatigue cracks must exhibit some irreversibility at or close to the crack tip, otherwise they would not propagate.  The 
stress intensity approach attempts to characterize this by a similitude argument, which depends on the assumption of 
small scale yielding (i.e. that the process zone of the crack tip is contained within a wider ‘K-dominant’ region).  Use of 
the stress intensity approach has particular limitations for the case of non-uniform loading, since load history effects are 
difficult to capture.  Hence, there has been increasing interest in improved characterization of crack tip fields.  In parallel, 
new experimental techniques, such as digital image or volume correlation (DIC/DVC); X-ray tomography; and large scale 
synchrotron facilities have provide tools which allow various models of crack tip behavior to be tested. 
A significant number of papers were presented at the First I.J. Fatigue & FFEMS Joint Workshop on the Characterisation 
of Crack Tip Stress Fields held in Forni di Sopra in 2011.  This resulted in the publication of two special issues of the 
journals [6, 7], but the workshop showcased only a small proportion of the work in the field.  On the experimental side, 
other recent papers of note include tomographic studies of cracked specimens under live loading [8] and in-situ SEM 

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D. Nowell et alii, Frattura ed Integrità Strutturale, 25 (2013) 1-6; DOI: 10.3221/IGF-ESIS.25.01                                                                            
 

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studies of deformations in the crack tip region.  The latter work has led to a suggested life prediction approach which 
obviates the need to characterize the number of cycles experienced by the crack [9].  Despite the existence of rainflow and 
similar approaches, cycle counting can be a difficulty in non-uniform loading situations.  The experimental work reported 
in the current paper springs from an investigation of crack closure originally carried out by de Matos [10].  However,  the 
data was later re-processed to compare with a two-parameter crack tip field model proposed by Pommier and co-workers 
[11].  This work was reported at the Forni di Sopra workshop [12] and in the subsequent special issue [13].  However, the 
majority of the work reported was concerned with constant amplitude loading.  A few experiments on simple overload 
cycles were carried out and the results of one of these experiments were presented in [13].  The displacement fields were 
converted into the two parameter description (K, ) and provided an interesting contrast to the results obtained from 
constant amplitude loading.  The current paper therefore sets out to explore these results in more detail and to analyse a 
wider range of variable amplitude experiments. 
 
 
EXPERIMENTAL WORK 
 

he fatigue experiments carried out by de Matos have been reported elsewhere, but a brief summary is given here 
for clarity.  Fig. 1 shows the main experimental set-up, which employs a CT specimen manufactured from 6082 
T6 aluminium alloy.  The specimen is loaded in fatigue and the area along the crack flanks is monitored using a 

Questar long range microscope and a digital camera (a low-cost USB webcam).  A second camera is used to measure the 
crack length on the opposite side of the specimen, but this does not provide sufficient resolution for displacement 
measurement.  The use of the microscope means that only a small area (approx. 600 x 400 m) close to the crack tip is 
imaged, but this permits high resolution and high accuracy evaluation of displacements.  It should be noted that, because 
the primary focus of the investigation was crack closure, most of the image is along the crack wake, and there is very little 
information collected from the region ahead of the crack tip. 
 

 
 

Figure 1: Experimental configuration: side and top views. 
 

Images were collected during fatigue loading at 30fps, whilst the specimen was loaded in a servo-hydraulic test machine at 
a frequency of 0.25 Hz.  A set of 360 images was collected over four loading cycles at a number of discrete points during 
crack propagation.  The loading frequency was increased between these measurement steps in order to propagate the 
crack in a reasonable time interval.  Each set of images was analysed using a public domain Matlab script [14].  Rather 
than analyse a full displacement field, the relative displacement of five pairs of points along the crack flank was 
determined (Fig. 2).  This information proved useful in investigating crack closure directly, but may also be employed to 
determine crack tip loading history.  The elastic model for a sharp crack predicts that the displacement along the crack 
faces is given by: 

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                                                                       D. Nowell et alii, Frattura ed Integrità Strutturale, 25 (2013) 1-6; DOI: 10.3221/IGF-ESIS.25.01 
 

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4
( )

2
I

i

K r
u r

E 
                (1) 

 
so that a plot of log(relative displacement) against log(r) may be used to estimate the stress intensity factor at any 
particular time during the crack loading history. 
 

 
 

Figure 2: Typical image showing the crack tip, together with the five pairs of measurement points. 
 
 
RESULTS 

 
hilst the main experimental programme concerned the investigation of behaviour under constant amplitude 
loading, a small number of specimens were subjected to a single overload in an otherwise constant amplitude 
load history.  Analysis of this situation represents a useful first step towards a full theory which can be applied 

in any general loading case.  Fig. 3a shows the schematic loading history, whereas Fig. 3b shows crack growth rate data 
(plotted against nominal K) for a number of 3mm thick specimens, including two  (CTF6 and CTF8) which were 
subjected to an overload.  The resulting retardation in crack growth rate can clearly be seen. 
 
 
  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Figure 3: (a) Schematic load history for single overload specimen; (b) Variation of crack growth rate with nominal elastic  K for 3mm 
thick specimens, including two with single overloads. 

W 

Load

Time

(a) (b) 

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D. Nowell et alii, Frattura ed Integrità Strutturale, 25 (2013) 1-6; DOI: 10.3221/IGF-ESIS.25.01                                                                            
 

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The experimental data is processed as described above and used to determine the stress intensity factor history for the 
crack.  In the case of specimen CTF6, the variation of K with load, just before the overload, is shown in Fig. 4(a).  It can 
be seen that the behaviour is quite similar to that predicted by an elastic analysis of the specimen geometry.  However, 
there is an offset between the theoretical and experimental lines, caused by crack closure.  Alternatively, this phenomenon 
may be thought of as a residual (negative) K caused by the residual stress (and displacement) field present.  The two lines 
have similar slope since, once the crack is fully open at a load of approximately 0.5 kN, the behaviour is predominantly 
elastic, with only a small zone of cyclic plasticity close to the crack tip. 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Figure 4:  Variation of measured stress intensity factor with load for specimen CTF6 (a) Before the overload and (b) Immediately after 
the overload. 

 

Fig. 4b shows equivalent data just after the overload.  A cursory inspection suggests that the behaviour appears as 
expected.  The slope of the curve remains approximately parallel to the elastic line, and the offset has increased (perhaps 
as a result of the increased magnitude of residual stress resulting from the overload).  However, closer inspection reveals 
some unexpected features.  The analysis yields negative stress intensity factors, which we would not expect to be 
physically admissible on their own.  This must mean that the relative displacement between the crack flanks is varying as -
r, where  is a positive constant.  Further, this would imply that the stress field at the tip of the crack is compressive 
and reasonably well described by the singular Westergaard solution.  One possible explanation for these observations is 
that the crack is being held open by the additional crack opening displacement caused by the overload.  Investigation of 
the crack profile broadly confirms this view.  As the crack propagates after the overload, the behaviour gradually returns 
to that which exists previously.  Fig. 5 shows a similar plot of load against K taken 23,500 cycles after the overload.  The 
negative offset is substantially reduced, although negative stress intensity factor values still exist.  Moreover it is apparent 
that the variation is no longer linear over the range of load plotted.  The deviation from linearity at lower load values is 
almost certainly caused by the onset of closure, which was absent immediately after the overload. 
The discussion above suggests that a single parameter, K, may be insufficient to fully characterize the crack tip 
environment, particularly under non-uniform loading conditions.  An improvement can almost certainly be obtained by 
explicitly acknowledging the role of crack tip plasticity.  A simple model has been proposed by Pommier and Hamam [11] 
which relies on partitioning the displacement field into elastic and plastic components.  The elastic component, uel, is the 
usual field associated with K, whereas the plastic field, upl, is associated with crack tip opening.  To a first approximation it 
may be taken as that given by a unit dislocation located at the crack tip.  Hence 
 

I el plu K u u              (2) 

As far as the relative displacements between the crack faces are concerned, this leads to 

8
( )

2
I

y

K r
u r

E



             (3) 

(a) (b) 

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                                                                       D. Nowell et alii, Frattura ed Integrità Strutturale, 25 (2013) 1-6; DOI: 10.3221/IGF-ESIS.25.01 
 

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Hence, a plot of uy against √r should give a straight line with a gradient of 8KI/(E√(2)) and an intercept of ,  and values 
of K and  may be obtained from the experimental results in a similar manner to the purely elastic case. Fig. 6 shows the 
variation of K with  for the cycle soon after the overload (i.e. corresponding to the elastic analysis shown in Fig. 5b).  
This shows evidence of the hysteresis in of K/ space, which Pommier and Hamam suggest is associated with fatigue 
crack propagation.  According to their hypothesis, absence of a cyclic variation in  corresponds to loading beneath the 
threshold value of K.  

 
 

Figure 5:  Variation of measured stress intensity factor with load for specimen CTF6, 23,500 cycles after the overload. 
 

 
 

Figure 6:  Variation of K with  when the elastic/plastic approach is used to analyse a cycle after the overload. 
 
 
CONCLUSIONS 
 

he results presented here have demonstrated that displacement information collected in the crack wake can be 
analysed using digital image correlation in order to give reliable values of the stress intensity factor.  For the case 
of constant amplitude loading, the results are broadly as expected and show a linear relationship of K with load, 

suggesting that the behaviour is predominantly elastic.  There is, however, an offset to the experimental curve, compared 
with the linear elastic model, which can be attributed to the existence of crack closure.  The situation when a non-uniform 
load history is examined appears more complex.  Here, a simple case of a single overload in an otherwise constant 
amplitude loading has been examined.  Even this relatively simple load history gives rise to some surprising results.  For 
some time after the overload, negative stress intensity factors are measured when using a purely elastic model.  Normally it 
is only possible to obtain positive values, as the crack closes and allows transmission of compressive forces across the 

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D. Nowell et alii, Frattura ed Integrità Strutturale, 25 (2013) 1-6; DOI: 10.3221/IGF-ESIS.25.01                                                                            
 

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crack faces.  The values measured here can only be possible when a significant crack tip opening displacement is also 
present.  This suggests that a more sophisticated description of crack tip fields is necessary, perhaps including elastic and 
plastic terms as suggested by Pommier and Hammam [11].  Future work will include the comparison of the measured 
results with strip yield and finite element models.  Further experiments are also planned, which will look at the region 
ahead of the crack tip as well as along the crack flanks.  This should allow correlation of the results obtained in the crack 
wake with conditions in and around the process zone itself. 
 
 
ACKNOWLEDGEMENT 
 

.F.P. de Matos would like to gratefully acknowledge the support of the Portuguese Fundação para a Ciência e a 
Tecnologia (FCT) for providing a D.Phil. scholarship (reference SFRH/BD/12989/2003, financed by POSI). 
 

 
 
REFERENCES  
 
[1] Inglis, C.E., Stresses in a plate due to the presence of cracks and sharp corners, Trans of the Institution of Naval 

Architects, 55 (1913) 219-230. 
[2] Westergaard, H.M., Bearing pressures and cracks, Jnl Appl. Mech., 6 (1939) 49-53. 
[3] Sanford, R.J., Selected Papers on Foundations of Linear Elastic Fracture Mechanics, SEM/SPIE, USA (1997). 
[4] Sanford, R.J., Selected Papers on Crack Tip Stress Fields, SEM/SPIE, USA (1997). 
[5] Paris, P., Erdogan, F., A critical analysis of crack propagation laws, Jnl Basic Engineering, 85 (1963) 528-534. 
[6] James, M.N., Yates, J.R., Susmel, L., Iacoviello, F., Nowell, D., Lazzarin, P., Patterson, E.A., Carpinteri, A., 

Characterisation of Crack Tip Stress Fields, Proceedings of the 1st Joint International Journal of Fatigue/Fatigue & 
Fracture of Engineering Materials & Structures conference, held in Forni di Sopra, UD, ITaly, 7-9 March, 2011, Int. 
Jnl Fatigue, 46 (2013) 1-1.  

[7] James, M.N., Yates, J.R., Susmel, L., Iacoviello, F., Guest Editorial: Special Issue on Characterisation of Crack Tip 
Stress Fields, Fatigue Fract. Engng Mater. Struct., 36 (2013) 1-2. 

[8] Limodin, N., Rethore, J., Buffiere, J.Y., Influence of closure on the 3D propagation of fatigue cracks in a nodular cast 
iron investigated by X-ray tomography and 3D volume correlation, Acta Materiala, 58 (2010) 2957-2967. 

[9] Zhang, W., Liu, Y., Investigation of incremental fatigue crack growth mechanisms using in situ SEM testing, Int. Jnl 
Fatigue, 42 (2012) 14-23. 

[10] de Matos, P.F.P., Nowell, D., Experimental and numerical investigation of thickness effects in plasticity-induced 
fatigue crack closure, Int Jnl Fatigue, 31 (2009) 1795-1804. 

[11] Pommier, S., Hamam, R., Incremental model for fatigue crack growth based on a displacement partitioning 
hypothesis of mode I elastic-plastic displacement fields, Fatigue Fract. Engng Mater. Struct., 30 (2006) 582-598. 

[12] Nowell, D., Kartal, M.E., de Matos, P.F.P., Measurement and modelling of near-tip displacement fields for fatigue 
cracks in 6082 T6 aluminium, Proc. First I.J. Fatigue & FFEMS Joint Workshop, Forni di Sopra, Italy, March 7-9, 
2011, Gruppo Italiano Frattura, 2011. 

[13] Nowell, D., Kartal, M.E., de Matos, P.F.P., Digital image correlation measurement of near-tip fatigue crack 
displacement fields: constant amplitude loading and load history effects, Fatigue Fract. Engng Mater. Struct., 36 (2013) 
3-13. 

[14] Eberl, C. Thompson, R., Gianola, R., Digital image correlation and tracking with Matlab, Matlab Central file exchange 
(2006) http://www.mathworks.co.uk/matlabcentral/fileexchange/12413-digital-image-correlation-and-tracking. 

 
 
 

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