369
A model for fatigue crack propagation and remodelling
in compact bone
D Taylor and P J Prendergast
Department of Mechanical Engineering, Trinity College, Dublin, Ireland
Abstract: The process of fatigue in bone is of interest for a number of reasons. Fatigue damage in
vivo can eventually lead to stress fracture, and may also act as a stimulus for bone remodelling and
adaptation. The aim of this paper is to develop a theoretical model which describes the growth of
fatigue cracks, especially of microcracks. The growth behaviour of microcracks is complicated by
their interactions with the surrounding microstructure. This problem has been identified by researchers
working on fatigue in engineering materials. Their work can be adapted to develop an equation in
which the growth rate of cracks is related to applied stress conditions and also to a microstructural
parameter, d, which is defined as the spacing of barriers to crack growth. The model can be used to
generate stress/life data for comparison with in vitro fatigue experiments. It can also be used to
investigate two hypotheses: that microcracking stimulates repair and that the level of fatigue damage
can act as a signal to initiate adaptation processes of deposition or resorption.
Keywords:
fatigue, compact bone, cracks, remodelling, fracture mechanics
NOTATION
crack length
rate constants used in the prediction of crack
growth
d
microstructural unit, equal to the spacing of
barriers to crack growth
da/dN fatigue crack growth rate, per cycle
F
constant dependent on crack geometry
m
exponent used in the prediction of crack growth
rate
n, n∞
exponents used in the prediction of crack
growth rate
N
number of cycles
N
number of cycles to failure
f
R
stress ratio, equal to the minimum stress in the
fatigue cycle divided by the maximum stress
a
C, C∞
DK
DK
Ds
1
th
range of cyclic stress intensity, K
threshold value of DK below which no crack
growth occurs
range of cyclic stress
INTRODUCTION
Compact bone is susceptible to failure when subjected
to cyclic loading. Fatigue failures, also termed stress
The MS was received on 17 July 1996 and was accepted for publication
on 8 April 1997.
H04996 © IMechE 1997
fractures, occur in vivo (1); fatigue cracking has been
positively linked to remodelling responses to repair
damage in osteonal bone (2) and has been proposed as
a mechanism for bone adaptation processes such as
deposition and resorption, which occur after changes in
stress levels in vivo. An understanding of fatigue, and an
ability to predict its effects, is clearly important. The aim
of the present paper is to develop a theoretical model of
fatigue in bone which can be applied to the prediction
of stress fractures, remodelling and adaptation. A critical
feature of this model is that it is based on the mechanism
of fatigue damage, which is the growth of cracks. Many
workers have measured fatigue in bone in vitro [e.g. references (3–5)] by subjecting samples to cyclic loading at
prescribed amplitudes of stress or strain and noting the
number of cycles to failure. These results show large
amounts of scatter and variation from one test to
another, some of which is due to differences in test protocol and some to real variations in bone’s properties.
Figure 1 shows the mean line taken through data of
Carter and Caler (3) which was recorded at a stress ratio
R (defined as the ratio between minimum stress and
maximum stress in the cycle) of −1. Such studies are
important, but they provide limited information because
they do not consider the mechanism of fatigue, which is
the initiation and propagation of cracks. Other studies
(6–8) have examined bone microscopically in order to
describe the mechanism of cracking. It is found that
cracks initiate very easily—in fact cracks approximately
50–100 mm long are already present in bone samples due
to normal in vivo loading (9, 10). Microcracks tend to
Proc Instn Mech Engrs Vol 211 Part H
370
D TAYLOR AND P J PRENDERGAST
Fig. 1 Predicted S/N curve for bone at R=0 and R=−1, and experimental data (mean line) at
R=−1 from Carter and Caler (3)
initiate in regions of weakness and stress concentration
in the microstructure, though currently there is little consensus on the details of their location. Martin and Burr
(11) proposed that cracks in Haversian bone originate
in extraosteonal material and grow around osteons in
cement lines, whereas Vashishth et al. (12) reported that
cracks can initiate within osteons. Cracking along
cement lines and between lamellae is, however, more
commonly observed (13).
Eventually these cracks grow to such a length as to
cause failure; Wright and Hayes (14) have measured the
growth rates of macrocracks. But because initiation is
relatively easy and microstructural features provide barriers to growth, the majority of life is spent in the
microcrack stage. Martin and Burr (11) have drawn
parallels with composite materials (e.g. carbon fibre
reinforced polymers) and many workers have noted similar behaviour in metals (15, 16). There are always found
to be certain microstructural features which act as barriers to crack growth. The nature of these features
depends on the material; e.g. grain boundaries in metals,
fibres or lamellar boundaries in composites. In
Haversian bone the osteon boundary clearly impedes
cracks, whether they originate inside the osteon or not
(11, 12), so the cement line is an important barrier. Some
workers [for example, Martin (10) and Prendergast and
Taylor (17)] have developed theoretical models to
describe bone remodelling and adaptation based on the
cumulative damage approach, in which fatigue damage
is defined as some single-valued function which increases
during fatigue life. Damage parameters may be mechanistically based (e.g. crack number density, crack length
density) or not (e.g. remaining life, reduction in stiffness).
The present work follows current thinking in the analysis
of fatigue in engineering materials, using the methods of
fracture mechanics to describe the behaviour of individual cracks, with special emphasis on their interaction
with the local microstructure. The objectives of the work
Proc Instn Mech Engrs Vol 211 Part H
are: to develop a theoretical model which quantitatively
describes the rate of growth of a fatigue crack; to use
this model to predict existing stress/life data in vitro; and
to show how the model can be used to link microcracking with remodelling and adaptation phenomena.
2 METHODS
The aim is to develop an equation for the fatigue crack
growth rate, da/dN (in units of mm/cycle). Paris and
Erdogan (18) showed that da/dN could be related to the
range of stress intensity, DK, where
DK=FDs(pa)1/2
(1)
Here Ds is the range (maximum to minimum) of applied
stress, a is the crack length and F is a geometric constant
whose value depends on crack geometry and loading
mode. In what follows, the theory is developed assuming
a straight, surface crack, for which F=1.12. However,
the same approach applies for all crack orientations with
only the value of F changing. The Paris law, which has
now been shown to apply for a wide range of materials
(19, 20), is
da/dN=C(DK )n
(2)
Here C and n are constants; Wright and Hayes (14)
measured values for bone of C=1.3×10−5 mm/cycle
and n=4.5. This approach, in common with most
methods in fracture mechanics, is based on stress, rather
than strain. In fact the equations can easily be reformatted in terms of cyclic strain, and this would be an equally
valid way to proceed. But, assuming constant strain rate
and temperature, there is a one-to-one relationship
between stress and strain in bone, and in most bones
fatigue is caused by a stress-based input, usually related
to body weight. Equation (2) can be modified to include
H04996 © IMechE 1997
A MODEL FOR FATIGUE CRACK PROPAGATION
the presence of a threshold, DK
th
da/dN=C(DK−DK )n
(3)
th
The threshold has not been measured for bone; an
approximate value of 0.2 MPa(m)1/2 can be deduced by
a standard theoretical method (see Appendix). In the
present work this has only a very small effect on the final
result and so its precise value is not critical.
Unfortunately, equation (3) applies only to long
cracks. Microcracks, which can be defined as cracks
which are of the same order of size as the microstructure,
show faster crack growth, growth at DK<DK , and
th
growth rates which decrease as crack length increases.
These effects have been widely studied in other materials
(15, 16). They occur because cracks initiate in weak
regions of the microstructure, so that their early growth
is relatively easy. Growth becomes more difficult as the
crack encounters microstructural barriers. In this study,
d is defined as the average distance between such barriers, and a value of 100 mm is chosen. The exact value
of d will depend on the type of bone and also on the
location of cracks, however, the choice of d does not
strongly influence the predictions of the model.
Secondary osteons are typically 200 mm in diameter (11),
but if cracks initiate between osteons then osteon separation will be important, and this is probably smaller. The
size of growth units in laminar bone is about 120 mm (7)
and in plexiform bone is 125 mm (11). It has been shown
for many other materials that growth rates decrease to
a minimum as the barrier is approached (21, 22); the
crack will then either break through the barrier and,
under constant Ds, continue on to failure, or else it will
be arrested premanently at the barrier. In this study a
growth law is proposed for short cracks of the following
form:
da/dN =C∞(DK )n∞[(d−a)/d ]m
(4)
s
Here da/dN is the growth rate of short cracks, defined
s
by a<d. The equation is assumed to follow the same
form as equation (3) except for the addition of a term
which depends on the distance between the crack tip and
the barrier (d−a), normalized by the spacing of barriers,
d. If crack length is approximately equal to d, this is a
transition zone between short and long crack behaviour.
This can be represented by adding the two terms, giving
an expression for the total crack growth rate
da/dN=C(DK−DK )n+C∞(DK )n∞[(d−a)/d ]m
(5)
th
Each term is taken to be equal to zero if its value
becomes negative; the first term will dominate if crack
length is significantly greater than d, and the second term
if it is significantly less. C∞, n∞ and m are unknown constants. It is assumed that n∞=n, i.e. that the effect of
stress intensity is the same for both long and short
cracks. This means that a 10 per cent increase in stress
will have the same effect on short crack growth as on
long crack growth. This is by no means certain but it
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371
simplifies the equations and there is some evidence for
it in the literature (15). Values for C∞ and m were chosen
as 0.013 mm/cycle and 5 respectively, in order to provide
a best fit to stress/life data below.
3 RESULTS
The fatigue crack growth rate is independent of the stress
in the long crack region but not in the short crack region,
see Fig. 2 which shows values of da/dN as a function of
DK for various stress levels. Note the logarithmic scale—
growth rates change very rapidly especially in the region
of the short/long crack transition. Figure 3 shows da/dN
as a function of crack length, taking three different stress
levels; in this plot the effect of the barrier at a=d is
more clearly appreciated. At a sufficiently low stress
(10.1 MPa—not shown here) both terms in equation (5)
become 0 at a=d; the crack is predicted to arrest at the
barrier. The prediction therefore is as follows: below
approximately 10 MPa microcracks do grow, but only
to be arrested at the microstructural barrier. In fact, this
is a prediction of the fatigue limit, and it is significant
that this fatigue limit is determined not by crack
initiation, but by the inability of existing cracks to
propagate.
Figure 1 shows predictions of stress/life data for comparison with the results of Carter and Caler (3). The
number of cycles to failure can be predicted by integrating the growth rate in equation (5), starting from some
initial crack length. This was taken to be 50 mm based
on measurements of cracks in bone removed after
normal physiological use (6, 9). A further complication
here is the fact that equation (5) was developed for a
stress ratio R (the ratio of minimum stress to maximum
stress) of 0, which is approximately the case in vivo,
whereas the data of Carter and Caler were measured at
R=−1 (fully reversed loading). Taylor (15) has shown
that these two can be related in a wide range of materials
by changing DK by a factor of 1.6.
It is possible to use equation (5) to estimate the behaviour of cracks in vivo. Martin and Burr (11) proposed
that bone in humans and other vertebrates typically
experiences a strain range of 1 500 microstrains at a frequency of 1.5 million applications per year. Assuming
an elastic modulus of 20 GPa, which is typical for compact bone, equation (5) predicts that an initial crack of
length 50 mm will increase by 30 mm over a period of
2.2×104 cycles, which corresponds to 5.4 days. This is
of the same order of magnitude as the estimated rate of
osteon replacement of 30 mm/day (11). This adds support to a model of bone in which fatigue crack growth
is continually being matched by repair processes. If such
repair processes did not operate, the same crack would
grow to cause failure within 91 days. This emphasizes
the fact that bone is operating well above its fatigue
limit, relying on crack detection and repair to prevent
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D TAYLOR AND P J PRENDERGAST
Fig. 2 Predicted crack growth rate (da/dN ) for compact bone, as a function of cyclic stress
intensity range, DK, at three different values of the cycle stress range
Fig. 3 Predicted crack growth rate (da/dN ) for compact bone, as a function of crack length, at
three different values of the cycle stress range
failure. This conclusion contrasts with the assertion of
Martin and Burr (11) that bone operates below its
fatigue limit in vivo.
Increases in stress are known to lead to adaptation
processes such as deposition (23, 24). Under increased
loading, fatigue cracks will grow very much faster,
especially once they break through the microstructural
barrier and move up the rising part of the curve of Fig. 3.
In order to prevent stress fracture, action must be taken
to reduce the loading very quickly. The authors propose
that the passage of the crack through the barrier acts as
the signal for the adaptation process. Figure 4 shows the
time needed for a crack of 50 mm to reach the barrier at
100 mm. At above-normal stresses such as 50 MPa this
signal will go out within a few days, allowing remedial
action to be taken before total failure occurs. It should
be noted that this prediction is made without allowing
Proc Instn Mech Engrs Vol 211 Part H
for the normal repair process, which would tend to
retard crack growth, though the effect of repair would
diminish rapidly as stress increases. At the other end of
the scale, stress shielding or disuse may reduce stresses,
and thus crack growth rates, almost to zero. Under these
conditions microcracks of length 0.05–0.1 mm will be
removed by repair processes very quickly (within a few
days at the above figure of 30 mm/day). It is not clear
how this lack of microcracks is being detected, but
assuming that it is detected, the signal for remodelling
would be sent out within a few days.
4 DISCUSSION
This paper presents for the first time a theoretical model
of fatigue crack propagation in bone which includes both
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A MODEL FOR FATIGUE CRACK PROPAGATION
Fig. 4 Prediction of time for crack to reach the microstructural barrier, as a function of stress level
microcrack and macrocrack behaviour in a unified form.
Using equation (5), the crack growth rate can be estimated for any crack, as a function of the applied stress,
crack length and a microstructural parameter which
describes the spacing of barriers to crack propagation.
The equation is purely empirical in nature, its form has
been chosen to reflect the general characteristics of
fatigue growth as measured in many different materials,
for which the parameter (d−a) has been shown to control behaviour of microcracks. Its advantage over other
approaches, such as damage mechanics, is that it is firmly
rooted in the mechanism of the process and thus gives
researchers the potential to explain experimental observations in terms of microstructural processes.
At the present time it is not possible to verify the
equation by direct observation, except that part of it
which refers to macrocracks. The relationship between
theory and experiment is a synergistic one; the existence
of this model can act as a stimulus to the design of
experiments which will test its predictions, such as direct
measurements of microcrack growth; these have been
373
carried out for many other materials, see for example
Taylor et al. (25).
It has been shown in this paper that, by choice of the
relevant constants in the expression for microcrack
growth, experimental stress/life data can be predicted.
Variations in these data occur in part due to variations
in the type and quality of bone and also due to loading
factors such as R. These can be described by changes in
the values of the various constants. For example, a bone
which has a relatively fine microstructure will have a low
value of d, which will improve its resistance to microcracking, but it may, at the same time, have higher
values of the rate constants C and C∞. In this example
the material may have good fatigue resistance but, in
order to avoid stress fractures, it may be necessary to
detect and repair cracks when they are very small, in
comparison to a material with a coarse microstructure.
Microcrack interaction will also play a role; if a number
of cracks initiate close together they may coalesce, producing a larger crack, or they may hinder each other’s
growth.
The model is able to explain the fact that bone, in
normal use, contains damage. A number of workers (2,
6, 9) have observed diffuse microcracking with crack
lengths of the order of 50 mm. Figure 5 shows predictions, based on a stress of 18 MPa, of the number of
cycles to failure as a function of initial crack length.
There is no improvement in life to be expected by reducing crack length below 50 mm, which is half the size of
the microstructural unit, d. It would be uneconomic for
bone to repair these cracks; however, cracks greater than
2d cause a rapid reduction in life, thus any repair process
will concentrate on detecting and eliminating cracks in
this range, which is the same size as that of secondary
osteons. It was shown earlier that the rate of replacement
of osteons is of the same order as the rate of growth of
these microcracks at physiological stress levels.
It has been postulated that damage accumulation
could act as a signal to initiate remodelling or adaptation
processes such as bone deposition (11, 17). In the present
Fig. 5 Predicted fatigue life (cycles to failure) at a stress of 18 MPa as a function of initial
crack length
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D TAYLOR AND P J PRENDERGAST
model this can be linked directly to the relationship
between crack length and microstructure. For both
increased and decreased stress conditions, the monitoring of microcrack length can be effective in indicating
the need for action within a relatively short time. The
mechanism by which this monitoring is achieved is
beyond the scope of this paper; but it is obvious that
cracks which disrupt the microstructure in this way will
have a significant effect on many measurable parameters.
For example, Prendergast and Huiskes (26) have predicted that microcracks of the order of the osteon have
a significant effect on bone cells: microcracks unload the
strain from the cells that may carry out adaptation, i.e.
the osteocytes in the lacunae and the bone lining cells
on the Haversian canal wall. Zioupos et al. (27) have
measured significant levels of microdamage in compact
bone long before failure indicating that microdamage
could serve as a stimulus for remodelling without undue
danger of immediate bone fracture.
In conclusion, it should be noted that this theoretical
model has a long way to go, in particular it needs to be
verified by direct measurement. However, it provides a
basis for the consideration of fatigue cracking in bone
and for the discussion of how fatigue cracks influence
remodelling and adaptation processes. In particular, it
introduces for the first time a microstructural parameter,
d, which makes it possible to discuss how bone’s microstructure affects its fatigue and other properties.
9
10
11
12
13
14
15
16
17
18
19
ACKNOWLEDGEMENTS
20
Financial support was provided by Forbairt, the
Industrial Development Authority in Ireland.
21
22
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A MODEL FOR FATIGUE CRACK PROPAGATION
APPENDIX
In the absence of a measured value for DK , it is possible
th
to make an estimate using a method which has been
shown to be valid for many other materials (15). This
is based on the assumption that the threshold is characterized by a plastic zone ahead of the crack whose size
is equal to the size of the smallest microstructural units
in the material capable of inhibiting plasticity. The plastic zone size, r , is calculated using a formula from Knott
y
(28) which assumes the material to be a homogeneous
continuum with a yield strength s
y
r =0.04 (DK/s )2
(6)
y
y
If the plastic zone size is less than the size of the microstructural unit, s, the material can no longer be treated
as a homogeneous continuum, and crack advance
becomes much more difficult. Taylor (15) showed that
the condition r =s corresponds to a value of DK at
y
which the rate of crack growth, da/dN is equal to the
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375
spacing of atoms in the material; this relationship was
demonstrated for a range of materials in which s varied
from 1 mm to 100 mm. Crack growth rates less than this
value must necessarily be achieved by a discontinuous
process of crack advance.
For compact bone the following parameter values
were used: s =120 MPa; s=1 mm; interatomic spacy
ing=0.24 nm. The value of s was chosen as a typical
interlamellar distance and the atomic spacing is that of
the Ca–O bond in hydroxyapatite. The resulting prediction is that a stress intensity range of 0.61 MPa (m)1/2
will correspond to a growth rate of 2.4×10−7 mm/cycle.
This is achieved in equation (3) by using a value of DK
th
equal to 0.2 MPa (m)1/2.
This result can only be regarded as an approximate
estimate, the accuracy of which depends on the choice
of parameter values. Ideally the threshold value should
be measured experimentally. However, in the present
case the value of DK has only a minor effect on the
th
predictions made using equation (5), so this estimate is
satisfactory for the purposes of this study.
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