Crack propagation under monotonic and cyclic loading was investigated in alumina-epoxy composite specimens, produced via a multi-step infiltration technique. Under monotonic loading, an increase in effective fracture toughness was observed, which was attributed to the development of a bridging zone behind the crack-tip. A similar increase in crack propagation resistance was observed under cyclic loading, though this was more difficult to quantify. This was achieved approximately using an adjustment based on the Paris-law relation. This adjustment approach is described in this paper, and is also applied to fatigue results for graded composite specimens, in which the increase in crack-growth resistance with crack-extension is accompanied by a spatial variation in intrinsic and extrinsic crack-growth resistance

Hoffman, M.

Moon, R. J.

Rutgers, L.

Tilbrook, M. T.

University of Queensland eSpace

SIF2004 Structural Integrity and Fracture. http://eprint.uq.edu.au/archive/00000836 
Effective Crack-Propagation Resistance Under Monotonic And Cyclic Loading 
M T Tilbrook, L Rutgers, R J Moon & M Hoffman 
School of Materials Science and Engineering, University of New South Wales, NSW 2052 
 
 
 
ABSTRACT: Crack propagation under monotonic and cyclic loading was investigated in alumina-epoxy 
composite specimens, produced via a multi-step infiltration technique. Under monotonic loading, an increase 
in effective fracture toughness was observed, which was attributed to the development of a bridging zone 
behind the crack-tip. A similar increase in crack propagation resistance was observed under cyclic loading, 
though this was more difficult to quantify. This was achieved approximately using an adjustment based on 
the Paris-law relation. This adjustment approach is described in this paper, and is also applied to fatigue 
results for graded composite specimens, in which the increase in crack-growth resistance with crack-
extension is accompanied by a spatial variation in intrinsic and extrinsic crack-growth resistance. 
1 INTRODUCTION 
In many homogeneous materials under monotonic loading, the development of a bridging zone 
leads to crack-extension toughening. The value for stress intensity factor (SIF) at the crack-tip, Ktip, 
is less than the applied SIF, Kapp(a), due to crack-extension dependent shielding effects, Kbr(∆a). 
For fracture, Ktip must be equal to intrinsic fracture toughness, K0, so the required value of Kapp 
increases with crack propagation. This equates to an increase in effective fracture toughness: 
Kceff(∆a) = K0 + Kbr(∆a).             (1) 
Crack extension effects are also likely to influence resistance to crack propagation under cyclic 
loading [Ritchie, 1999]. This is more complicated than under monotonic loading, due to progressive 
fatigue degradation of the bridging zone, and introduces a dependence on propagation-rate as well 
as on crack-extension, ie. Kceff(∆a,da/dN), where da/dN is the crack  extension per cycle. The 
traditional approach for quantifying fatigue behaviour by presenting crack propagation rates as a 
function of SIF amplitude therefore becomes insufficient when R-curve effects are present. 
This issue is pertinent to functionally graded materials (FGMs). As FGMs are typically 
composites, extrinsic fracture phenomena, such as crack-bridging, are likely to influence the 
propagation of cracks within them, and the influence of such phenomena will vary across the graded 
region [Cai and Bao, 1998]. For crack propagation in a graded composite structure with property 
variation in the x-direction, the applied SIF, intrinsic fracture toughness and the crack-tip shielding 
will all depend on crack-tip position [Tilbrook et al., 2004a].  
In this paper, the results of crack propagation experiments on alumina-epoxy composites under 
monotonic and cyclic loading are presented and analysed. A novel approach is described for 
approximately analysing cyclic fatigue results using an adjustment based on the Paris law relation. 
This analysis is demonstrated for homogeneous composite specimens and is also applied to a 
previously investigated graded composite specimen [Tilbrook et al., 2004b]. 
2 METHODS & MODELLING 
The alumina-epoxy composite system was chosen for the large disparity in elastic properties (EAl = 
390 GPa, EEp = 3.4 GPa) and failure strains (εAl <1%, εEp > 3%). Specimens were prepared by 
infiltrating alumina slip into pieces of open-celled polyurethane foam compressed to required 
density [Neubrand et al., 2002]. After drying, the foam was burnt out and the alumina sintered. The 
resulting porous ceramic bodies were infiltrated with epoxy resin. After curing, specimens were 
machined to size (45x3x6 mm3) and polished with diamond paste. Microstructural characterisation 
was conducted with optical microscopy and elastic properties were measured using the impulse 
SIF2004 Structural Integrity and Fracture. http://eprint.uq.edu.au/archive/00000836 
excitation technique [Tilbrook et al., 2004c]. The scale and morphology of the composite 
microstructure reflected that of the original polyurethane foam. 
Stable crack propagation from notches (~2mm long) sharpened using a razor blade technique 
was investigated under monotonic and cyclic four-point bend loading, using an Instron 1185 in 
displacement-control for fracture and an MTS 810 in load-control for fatigue (frequency = 8Hz, 
load ratio, R = 0.1). Applied loading was varied throughout fatigue tests in such a manner as to 
ensure that crack growth rates varied non-systematically with crack extension. Crack growth was 
monitored via a microscope and digital camera. SIF values were calculated from applied load, 
specimen dimensions and crack length.  
  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Figure 1: Crack growth in homogeneous composites. (a) Optical micrograph showing crack propagation path.  Evidence 
of discontinuous crack growth and surface porosity may be observed. (b) Effect of composition on initiation toughness, 
KC, under monotonic and cyclic (KC = Kmax, R = 0.1) loading, and on Young’s modulus, E, measured via impulse 
excitation. 
3 RESULTS & ANALYSIS 
Crack propagation occurred in a discontinuous manner, as illustrated in Figure 1(a), which was 
attributed to the brittle nature of both phases and the relatively coarse composite structure. Figure 
1(b) shows that an increase in epoxy content caused a decrease in the maximum SIF required to 
initiate crack growth (ie. crack initiation toughness) under monotonic and cyclic loading, and a 
decrease in effective Young’s modulus [Tilbrook et al., 2004c]. This is attributed to the 
significantly lower stiffness and higher susceptibility to fatigue degradation of the epoxy phase. 
An increase in fracture toughness with crack extension, under monotonic loading, was observed, 
as shown in Figure 2(a). This was attributed to crack-bridging by intact epoxy ligaments, due to the 
significantly higher failure strain of the epoxy phase. Figure 2(b) shows that, under cyclic loading, 
crack propagation appeared to follow the Paris law, (da/dN) = A(∆K)m, with some scatter. During 
fatigue testing, the applied SIF amplitude was varied so that the correlation between crack growth 
rates and crack extension would be negligible, and that any transient effects would be non-
systematic and would contribute to noise in the data. This was achieved by varying the applied SIF 
amplitude, within the range resulting in crack growth, numerous times throughout the crack 
propagation. 
 
 
 
 
0
1
2
3
4
5
6
7
8
0
50
100
150
200
250
300
350
400
0 10 20 30 40 50 60
K
C
 cyclic loading
K
C
 exponential fit
K
C
 monotonic loading
E expt
E EMA fit
Volume Fraction of Epoxy [%]
In
iti
at
io
n 
To
ug
hn
es
s,
 K
C
 [M
P
a 
m
0.
5 ]
Young's M
odulus, E
 [G
Pa]
Discontinuous 
crack growth 
Porosity 
Branching 
Notch-tip 
Tortuous 
crack path
(b) 
(a) 
500 µm 
SIF2004 Structural Integrity and Fracture. http://eprint.uq.edu.au/archive/00000836 
  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Figure 2: Crack propagation in homogeneous composites (a) R-curves for homogeneous specimens under monotonic 
loading. (b) Fatigue crack growth rates for several different compositions, showing Paris law behaviour. 
 
It was observed that the SIF amplitude required for crack growth generally increased with crack 
extension. It is believed this, and some of the scatter in Figure 2(b), are due to the effect of crack-
extension toughening on fatigue crack growth rate [Tilbrook et al., 2004b]. Assuming this is the 
case, and that these effects may be extracted from the fatigue data, the following analysis is 
proposed.  
The underlying assumption is that, for a given composition, the mechanisms of crack advance at 
the crack-tip, and accordingly the value of the Paris-law exponent, do not vary with crack-
extension. The Paris law may be rearranged to form an expression equal to the Paris law coefficient, 
A, which is assumed constant for a given specimen: 
  ( ) ⎟⎠
⎞⎜⎝
⎛∆= −
dN
daKA m              (2) 
This suggests that the measured values, ∆Kmeas and (da/dN)(meas), for SIF amplitude and crack 
growth rate could be related to adjusted values, ∆Kadj and (da/dN)adj: 
  ( ) ( ) (adj)m(adj)(meas)mmeas
dN
daK
dN
daK ⎟⎠
⎞⎜⎝
⎛∆=⎟⎠
⎞⎜⎝
⎛∆ −−           (3) 
Measured ∆K values that resulted in differing crack-growth rates may be compared, by 
estimating the adjusted values that correspond to a standard adjusted crack growth rate, (da/dN)adj = 
(da/dN)std: 
( )
( )m measdNda
std
dN
da
measadj KK )(
)(
∆=∆            (4) 
In this work, the value for (da/dN)std was arbitrarily chosen as 10-7 m/cycle. Essentially, this 
adjustment represents an estimate, assuming that the Paris law is applicable, of the SIF amplitude 
value which would have resulted in crack extension at the standard rate. The adjusted SIF was 
interpreted as a measure of fatigue crack propagation resistance, KCF, ie. ∆Kadj = KCF. This 
adjustment method does not provide absolute results, rather it provides an approximate way of 
quantifying the relative variation in fatigue crack-propagation resistance. The variation in adjusted 
SIF with crack extension is shown in Figure 3(a) for several alumina-epoxy composite specimens. 
A systematic increase in fatigue resistance with crack extension is observed, comparable with the 
10-10
10-9
10-8
10-7
10-6
106 107
Fa
tig
ue
 C
ra
ck
 G
ro
w
th
 R
at
e,
 (d
a/
dN
) [
m
/c
yc
]
Applied Stress Intensity Factor Range, ∆K [Pa m1/2]
45% Epoxy
   m = 25
30% Epoxy
   m = 22 6% Epoxy
   m = 27
11% Epoxy
   m = 24
0
1
2
3
4
5
6
0 0.5 1 1.5 2 2.5
E
ffe
ct
iv
e 
Fr
ac
tu
re
 T
ou
gh
ne
ss
, ∆
K
 [M
P
a 
m
1/
2 ]
Crack Extension, ∆a [mm]
30% Epoxy
60% Epoxy
5% Epoxy
(b) (a) 
SIF2004 Structural Integrity and Fracture. http://eprint.uq.edu.au/archive/00000836 
R-curves in Figure 2(a). It appears that the crack-extension effects are less significant under cyclic 
than monotonic loading, which is consistent with the susceptibility to fatigue degradation of the 
epoxy phase. Furthermore, an increase in epoxy volume fraction results in a decrease in the 
bridging effect, both absolutely and relatively, which may also be explained in terms of the inferior 
toughness and fatigue resistance of the epoxy phase. 
   
                        
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Figure 3: (a) Variation of effective fatigue crack propagation resistance, ie. adjusted SIF as in Equation (4), with crack 
length for a range of compositions. (b) Fatigue crack growth rate results from Figure 2(b) corrected to remove crack 
extension effect, as in Eq. (7). 
 
Applying a fit to the fatigue resistance curve, the effective fatigue resistance, KCF(∆a), may be 
predicted at a given crack extension, ∆a. Rearranging Equation (3) and making the substitution 
∆Kadj = KCF(∆a) yields: 
  ( )
(meas)m
CF
meas(std)
dN
da
aK
K
dN
da ⎟⎠
⎞⎜⎝
⎛=⎟⎟⎠
⎞
⎜⎜⎝
⎛
∆
∆⎟⎠
⎞⎜⎝
⎛            (5) 
A form of the Paris law similar to this, in which the applied SIF amplitude is normalised by 
fracture toughness, has been used to quantify fatigue behaviour in grain-bridging ceramics [Gilbert 
et al., 1997]. Expanding the logarithms of each side in Equation (5), and writing (da/dN)std as a 
constant, A’: 
( ) ( ) ( )( )∆aKlogmKlogmA'log
dN
dalog CF
meas
(meas)
−∆+=⎟⎠
⎞⎜⎝
⎛         (6) 
This implies that each measurement of fatigue crack growth rate may be corrected, removing the 
dependence on crack-extension:  
( ) ( ) ( )measCF(meas)(corr) KlogmA'logKlogmdNdalogdNdalog ∆+=+⎟⎠⎞⎜⎝⎛=⎟⎠⎞⎜⎝⎛        (7) 
10-8
10-7
10-6
10-5
10-4
10-3
10-2
106 107
C
or
re
ct
ed
 F
at
ig
ue
 C
ra
ck
 G
ro
w
th
 R
at
e,
 (d
a/
dN
)c
or
r  [
m
/c
yc
]
Applied Stress Intensity Factor Range, ∆K [Pa m1/2]
45% Epoxy
30% Epoxy
6% Epoxy
11% Epoxy0
1
2
3
4
5
6
7
0 0.5 1 1.5
Fa
tig
ue
 R
es
is
ta
nc
e,
 ∆K
ad
j  [
M
P
a 
m
1/
2 ]
Crack Extension, ∆a [mm]
6% Epoxy
11% Epoxy
30% Epoxy
45% Epoxy
(b) 
(a) 
SIF2004 Structural Integrity and Fracture. http://eprint.uq.edu.au/archive/00000836 
Figure 3(b) demonstrates that applying this correction to the data improves the linearity of the 
data significantly, which confirms that much of scatter was attributable to R-curve effects.  
 
4 APPLICATION TO GRADED SPECIMENS 
The fatigue data analysis method has also proved useful in application to step-graded alumina-
epoxy composite specimens. The analysis is demonstrated here for a specimen investigated 
previously, with the relevant experimental and computational simulation procedures given 
elsewhere [Tilbrook et al., 2004b, 2004d]. A step-graded specimen, produced by a similar method 
to the homogeneous composites described above, is illustrated in Figure 4(a). Significant crack 
deflection is observed, which was attributed to elastic property asymmetry around the notch. 
Accordingly, cracks traversed the gradient region, thereby experiencing a variation in composition 
and properties at the crack-tip.  
Figure 4(b) shows the effective crack-propagation resistance profile for the crack in a stepped 
gradient, in which stable propagation occurred across two compositional steps. A steady increase in 
effective crack propagation resistance was observed and attributed to the development of a bridging 
zone behind the crack-tip. As the crack moved from one step to the next, the effective fatigue 
resistance, ∆Kadj, and relative stress intensity, K/P, both decreased suddenly. The decrease in 
resistance was more dominant, so the crack advanced significantly and a reduction in applied 
loading was required in order to avoid unstable crack growth. After crossing the interface, the crack 
propagation resistance began to increase again, corresponding to further development of the 
bridging zone.  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Figure 4: Graded alumina-epoxy specimens, investigated by Tilbrook et al. [2004b]. (a) Graded specimen showing 
dimensions and crack rpopagation path. (b) Load profiles for fatigue crack propagation (a) across an interface between 
compositional steps. 
 
It is apparent that the variation in effective crack-propagation resistance in graded materials 
results from a competition between (1) crack-extension effects, which tend to increase resistance, 
and (2) spatial variation in intrinsic and extrinsic contributions to crack-propagation resistance. The 
adjustment method developed in this work for homogeneous composites provides a useful approach 
for analysing this phenomenon in graded composites. 
 
Notch 
∼12 mm 
Length ∼130 mm ∼30 mm ∼40 mm 
95% Alumina 
5% Epoxy 
95% Alumina  
5% Epoxy 
100%  
Epoxy 
Crack 
Path 
Graded 
Regions
(a) 
2
3
4
5
4
5
6
7
8
9
10
0 1 2 3 4 5 6 7
Crack Extension, ∆a [mm]
Adjusted 
Experimental 
Results
K/P
Fa
tig
ue
 R
es
is
ta
nc
e,
 ∆K
ad
j  [
M
P
a 
m
1/
2 ] N
orm
alised S
IF, K
/P [x10
3 m
-3/2] 
Step
Interface
∆Kadj
SIF2004 Structural Integrity and Fracture. http://eprint.uq.edu.au/archive/00000836 
5 CONCLUSIONS 
Crack propagation in homogeneous and graded alumina-epoxy composites, under monotonic and 
cyclic loading, was investigated via experiments and FE simulations, and several conclusions were 
reached: 
1. Crack-extension toughening effects in homogeneous specimens were observed under monotonic 
loading and also, by adjusting SIF amplitude values, detected under cyclic loading. 
2. The adjustment of SIF amplitudes provides a useful approach for quantifying the relative increase 
in crack propagation resistance with crack-extension, under cyclic loading, for homogeneous and 
graded materials. 
3. Compositional variation experienced by cracks propagating in graded specimens led to a decrease 
in crack propagation resistance, which occurred suddenly at step-interfaces in the specimens 
studied.  
REFERENCES 
Cai, H and Bao, G, Crack Bridging in Functionally Graded Coatings, Int. J. Sol. Struct., 35, 701 (1998) 
Gilbert, C J, Dauskardt, R H and Ritchie, R O, Microstructural Mechanisms of Cyclic Fatigue-Crack 
Propagation in Grain-Bridging Ceramics, Ceramics International, 23, 413 (1997) 
Neubrand, A, Chung, T J, Rödel, J, Steffler, E D and Fett, T, Residual stresses in functionally graded plates, 
J. Mater. Res., 17, 2912 (2002) 
Ritchie R O, Mechanisms of fatigue crack propagation in ductile and brittle solids, Int. J. Fract., 100, 55 
(1999) 
Tilbrook, M T, Moon, R J and Hoffman, M. Crack Propagation in Graded Composites, Comp. Sci. Tech., in 
publication (2004a) 
Tilbrook M T, Rutgers, L, Moon, R J and Hoffman, M, “Fatigue Crack Propagation in Graded Composites,” 
Proc. Australasian Conf. Comp. Mat. (2004b) 
Tilbrook M T, Moon, R J and Hoffman, M, “On the Mechanical Properties of Alumina-Epoxy Composites 
with an Interpenetrating Network Structure,” Mat. Sci. Engng. A, accepted (2004c) 
Tilbrook, M T, and Hoffman, M., Implementation of the local symmetry criterion for crack-growth 
simulations, Proc. Structural Integrity and Fracture 2004, these proceedings (2004d) 


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Effective Crack-Propagation Resistance Under Monotonic And Cyclic Loading

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