2495
The Journal of Experimental Biology 202, 2495–2503 (1999)
Printed in Great Britain © The Company of Biologists Limited 1999
JEB1959
WHAT DETERMINES THE BENDING STRENGTH OF COMPACT BONE?
JOHN D. CURREY*
Department of Biology, University of York, York YO10 5YW, UK
*e-mail: [email protected]
Accepted 7 June; published on WWW 25 August 1999
Summary
The bending strength of a wide variety of bony types is
material allows a greater bending moment to be exerted
shown to be nearly linearly proportional to Young’s
after the yield point has been reached, thereby increasing
modulus of elasticity/100. A somewhat closer and more
the strength as calculated from beam formulae. (2) Loading
satisfactory fit is obtained if account is taken of the
in bending results in a much smaller proportion of the
variation of yield strain with Young’s modulus. This
volume of the specimens being raised to high stresses than
finding strongly suggests that bending strength is
is the case in tension, and this reduces the likelihood of a
determined by the yield strain. The yield stress in tension,
weak part of the specimen being loaded to failure.
which might be expected to predict the bending
strength, underestimates the true bending strength by
Key words: bone, bending strength, Young’s modulus, Weibull
approximately 40 %. This may be explained by two
effect.
phenomena. (1) The post-yield deformation of the bone
Introduction
There have been two general approaches to ‘explaining’
the mechanical properties of bone. One approach is to
explain the mechanical properties by reference to the
microscopic and fine structure of the bone. Katz (1971) made
a pioneering attempt to explain elastic properties in this way.
Examples of later attempts are those by Wagner and Weiner
(1992) and Sasaki et al. (1991), who sought to explain some
elastic properties by considering the effect of the mineral
crystal size on the properties. However, explanations of
failure, as opposed to elastic, behaviour using microscopic
and fine structural information have effectively all been
qualitative.
The other approach is to relate statistically the mechanical
behaviour to other features of the bone, such as porosity, the
general orientation of the bone’s structure or the mineral
content. This is phenomenological: one can explain
mechanical properties in terms of other variables and use
such relationships for prediction. Keller (1994) related
various compressive properties to mineral content and
porosity. Martin and Ishida (1989) examined the relationship
between tensile strength and mineral content, porosity, and
the predominant orientation of the bone. Similarly, Currey
(1987, 1990) used a large data set to examine the physical
and chemical properties of compact bone, in particular its
mineral content and porosity, and their effect on bone’s
mechanical properties. The properties mainly examined were
Young’s modulus of elasticity and those associated with
tensile failure. In the present study, I discuss results, from an
enlarged data set, for bending strength.
Materials and methods
The data set
The total data set consists of records from 951 specimens
from 67 compact mineralised tissues from 32 species.
Estimates of porosity were made for 370 of these specimens.
The bending tests reported here were carried out on 351
specimens from 52 bones from 23 species (Table 1). The 86
bending specimens for which estimates of porosity were made
came from 35 bones from 18 species. Not all the specimens
are of bone sensu stricto, there are also some results from
elephant’s tusk and sarus crane ossified tendon. However, all
tissues will be referred to as ‘bone’.
Mechanical testing
Both bending and tensile specimens were prepared. The
general method of preparation and testing was as described by
Currey (1988). It involved rough preparation with a bandsaw,
and smoothing with increasingly fine grades of carborundum
paper. Tensile specimens were shaped into a dumbbell shape
by a milling head guided by a pre-machined pattern. All
specimens, which had been kept deep-frozen after the animals’
death, were thawed and were then prepared and tested wet, at
room temperature (approximately 16 °C).
Tensile specimens
The central section, of uniform cross-sectional shape, was
13 mm long, with a square cross section of 1.8 mm×1.8 mm.
Young’s modulus of elasticity in bending was determined
quasi-statically, in three-point loading, in an Instron 1122 table
testing machine, allowance being made for machine
2496 J. D. CURREY
Table 1. The provenance and numbers of the specimens used in the bending tests
Number
Species
Alligator
Axis deer
Mother
Foetus
Bovine
9 year
2 year
Crocodile
Dolphin
Donkey 34 year
Dugong
Elephant
Fallow deer
Fin whale
Flamingo
Galapagos tortoise
106 year
Bone
Number
BS
[Ca]
p
Species
Femur
Tibia
26
3
26
−
1
−
Tibia
Metacarpus
Tibia
Femur
Femur
Frontal
Prefrontal
Radius
Radius
Radius
Ulna
Scapula
Tusk
Tibia
Radius
Bulla
Tarso-metatarsus
Tibiotarsus
Femur
Fibula
Tibia
Humerus
8
9
11
10
20
2
2
2
7
2
2
12
3
9
3
9
5
3
4
5
2
7
−
9
11
10
−
2
1
2
7
2
2
12
3
9
3
9
5
3
4
5
2
7
−
−
1
3
−
1
1
2
2
2
2
7
−
2
−
9
5
3
−
2
−
1
Grey seal
Horse 7 year
Jackass penguin
King penguin
Muntjac deer
Red deer
Reindeer
Roe deer
Sarus crane
Swan
Wallaby
White-sided dolphin
Bone
BS
[Ca]
p
Tibia
Femur
Humerus
Ulna
Radius
Femur
Tibia
Humerus
Ulna
Antler
Antler
Antler
Femur
Tarsometatarsus
Ossified tendon
Tibiotarsus
Radius
Humerus
Femur
Humerus
Tarsal
Radius
Tibia
Ulna
Rib
5
9
1
2
2
2
2
9
2
7
25
6
7
13
1
9
6
4
19
2
10
4
24
4
8
5
9
1
2
2
2
2
9
2
7
25
6
7
4
1
−
−
−
14
2
7
2
13
2
8
2
2
1
2
1
1
2
−
−
−
2
2
−
4
1
−
−
−
8
2
7
1
7
2
4
Alligator, Alligator mississippiensis; axis deer, Axis axis; bovine, Bos taurus; crocodile, Crocodylus sp.; dolphin, Delphinus sp.; donkey,
Equus caballus; dugong, Dugong dugon; elephant, Loxodonta africana; fallow deer, Dama dama; fin whale, Balaenoptera physalus; flamingo,
Phoenicopterus ruber; Galapagos tortoise, Geochelone midas; grey seal, Halichoerus grypus; horse, Equus caballus; jackass penguin,
Spheniscus demersus; king penguin, Aptenodytes patagonica; muntjac deer, Muntiacus muntjak; red deer, Cervus elaphus; reindeer, Rangifer
tarandus; roe deer, Capreolus capreolus; sarus crane, Grus antigone; swan, Cygnus olor; wallaby, Protemnodon rufogrisea; white-sided
dolphin, Lagenorhyncus acutus.
BS, number of specimens for which values of bending strength and Young’s modulus of elasticity were determined; [Ca], number of these
specimens for which calcium content was also determined; p, number of these specimens for which porosity was also determined; −, no values
determined.
compliance. An extensometer was then attached to the uniform
part of the section of the specimen, which was then loaded in
tension at a strain rate of about 0.2 s−1. Data were reduced from
photographs of the screen of a storage oscilloscope. More
recently, the outputs from the load cell and the extensometer
were captured and analysed using DASYLAB software. The
following mechanical properties were determined for the
tensile specimens: Young’s modulus of elasticity (E, actually
determined in bending), yield stress (σy), yield strain (εy) and
strain at failure (εult). The yield point (Fig. 1A) was taken to
be the point where the curve had deviated by a strain of 0.002
from the straight line describing the initial part of the curve
(Currey, 1990).
Bending specimens
The specimens tested in bending initially all had the same
dimensions (gauge length 30 mm, depth 2 mm, breadth
3.5 mm). Young’s modulus was determined as for the tensile
specimens (Fig. 1B). The specimens were broken in threepoint bending, and Young’s modulus and bending strength
were calculated using standard beam formulae. The
head speed was set so that failure occurred in approximately
20–30 s.
Other measures
The porosity of the cross section approximately 2 mm
behind the fracture line was determined for 370 specimens, of
which 86 were bending specimens. Porosity was determined
by a point-counting method. The mineral content was
determined by a colorimetric method and is expressed as
milligrams calcium per gram dry defatted bone. Details of the
methods are given in Currey (1988).
Bone bending strength 2497
40
A
A
Young’s modulus (GPa)
30
Load
Yield region
20
10
0
150
Tension
200
250
300
350
250
300
[Calcium] (mg g-1)
350
[Calcium] (mg g-1)
Deformation
40
B
B
Load
Young’s modulus (GPa)
30
20
10
Bending
Deformation
Fig. 1. (A) Idealised load versus deformation curve of a specimen
loaded in tension. There is a quite sharply defined yield region.
(B) Idealised load versus deformation curve of a specimen loaded in
bending. The Young’s modulus of elasticity is calculated from the
initial, straight part of the curve using standard beam formulae. The
bending strength is calculated from the highest load.
Statistical analyses
Most of the analyses were least-squares linear regressions
on log-transformed data. The data were transformed to reduce
the heteroscedasticity and curvilinearity of the plots, and also
to make various statistical comparisons more straightforward.
However, the raw untransformed data are generally presented
here to make the particular form of the relationships (if any)
more apparent.
0
150
200
Fig. 2. (A) Young’s modulus of elasticity in bending versus calcium
content for all specimens. (B) Young’s modulus of elasticity in
bending versus calcium content for those specimens whose porosity
was measured. Open circles, porosity <8 %; filled circles, porosity
>8 %. The value of 8 % was chosen arbitrarily.
Results
Young’s modulus
There is clearly a strong relationship calcium content and
Young’s modulus (Fig. 2A).
The relationship appears as if it might be sigmoidal, but
the use of using quadratic and cubic values of calcium content
2498 J. D. CURREY
Table 2. Relationship between Young’s modulus E (GPa), calcium content [Ca] (mg g−1) and porosity (%) determined by linear
regression
Equation
logE=−8.58+4.05log[Ca]
logE=−9.53+4.28log[Ca]
logE=1.67–0.63logp
logE=−5.06+2.75log[Ca]–0.451logp
N
t
r2
P
855
370
370
370
30.35
16.45
−18.56
11.48, −13.75
0.52
0.42
0.48
0.62
<0.001
<0.001
<0.001
<0.001, <0.001
There are two equations for Young’s modulus as a function of calcium content alone. The first is from the complete data set (see Fig. 2A),
the second is from the subset including only those specimens for which values for porosity (p) were measured (see Fig. 2B). In all tables
describing the results of statistical analysis, r2 is calculated allowing for the number of degrees of freedom.
as explanatory variables barely improves the statistical fit
over the use of linear regression. The highest values of
calcium content are associated with high moduli, and low
values with very low moduli. However, particularly in the
region between 230 and 280 mg calcium g−1 bone, there is a
very large range of values for Young’s modulus. Fig. 2B
shows the subset of specimens for which values of porosity
were also obtained. The great majority of specimens with low
calcium content (<220 mg g−1) have a high porosity (>8 %,
filled circles), and those that have a high calcium content
(>280 mg g−1) have a low porosity (open circles). In the
middle region, in general, lower values of Young’s modulus
are associated with porosities of 8 % or greater. This is borne
out by the statistical analysis in which porosity was
considered as a continuous variable (Table 2), which shows
that 40 % or more of the variance in Young’s modulus can
be explained, statistically, by calcium content, and nearly
Bending strength
Fig. 3A shows the relationship between calcium content
and bending strength (BS). Excluding group A (see below),
there is a positive relationship between calcium content and
bending strength. Low calcium content is always associated
with low bending strength, whereas above approximately
230 mg calcium g−1 bone bending strength may be high or low.
High-porosity specimens tend to have low bending strengths
350
A
B
300
300
250
250
Bending strength (MPa)
Bending strength (MPa)
350
50 % by porosity. If these two explanatory variables are
combined, over 60 % is explained. However, this means that
nearly 40 % of the variance is still unexplained by these two
variables. It is clear from these equations that porosity and
calcium content are themselves related: their correlation
coefficients are −0.48 and −0.44 for the raw and logtransformed values, respectively. These coefficients are both
highly significant (P<0.001), though not strong.
200
150
A
100
50
200
150
100
50
0
0
150
200
250
300
[Calcium] (mg g-1)
350
150
200
250
300
350
[Calcium] (mg g-1)
Fig. 3. (A) Bending strength versus calcium content for all specimens loaded to failure in bending. The sample size is smaller than in Fig. 2A
because many specimens were not tested to failure in bending. The arrows indicate the subgroup A, consisting of highly mineralised bone from
whales, which was excluded from the statistical analysis. (B) Bending strength versus calcium content for the specimens whose porosity was
measured. Open circles, porosity <8 %; filled circles, porosity >8 %. Subgroup A is not shown, the porosity of these specimens was <8 %.
Bone bending strength 2499
Table 3. Regression relationships between bending strength BS (MPa), calcium content [Ca] (mg g−1) and porosity (%)
Equation
logBS=−4.25+2.70log[Ca]
logBS=−5.76+3.32log[Ca]
logBS=−0.39+1.25log[Ca]–0.491logp
N
t
r2
P
262
86
86
10.30
4.92
2.05, −7.06
0.29
0.22
0.50
<0.001
<0.001
<0.05, <0.001
There are two equations for BS as a function of [Ca]; the first is from the complete data set (Fig. 3A), the second is from the subset including
only those specimens for which porosity (p) was measured (Fig. 3B).
Table 4. Regression relationships between bending strength BS (MPa), porosity (%) and Young’s modulus E (GPa)
Equation
logBS=1.25+0.851logE
logBS=1.18+0.890logE
logBS=1.28+0.847E–0.053logp
N
t
r2
P
343
86
86
55.46
31.63
21.91, −1.60
0.90
0.92
0.90
<0.001
<0.001
<0.001, NS
There are two equations for BS as a function of E: the first is from the complete data set (Fig. 4A); the second is from the subset including
only those specimens for which porosity (p) was measured (Fig. 4B).
NS, not significant.
(Fig. 3B). The relationships, though highly significant
(Table 3) are not close ones (r2=0.22, 0.29).
Fig. 4A shows the relationship between Young’s modulus
and bending strength. Excluding group A, there is an extremely
tight, almost linear relationship. Fig. 4B shows that, although
specimens with high porosity predominate at the lower end of
the distribution, the few high-porosity specimens elsewhere are
part of the main distribution. The statistical analysis (Table 4)
shows that adding porosity as an explanatory variable had
virtually no effect on the strength of the relationship between
logBS and logE.
The fact that bending strength (excluding group A) has such
a close and nearly proportional relationship with Young’s
modulus, with the effect of porosity being very small,
immediately suggests that bending strength is determined
by some characteristic strain. The initial part of the
load–deformation curve of a bone specimen loaded in bending
is essentially straight (Fig. 1B). Since, in these circumstances,
350
350
B
300
300
250
250
Bending strength (MPa)
Bending strength (MPa)
A
200
150
A
100
50
200
150
100
50
0
0
0
10
20
Young’s modulus (GPa)
30
40
0
10
20
30
40
Young’s modulus (GPa)
Fig. 4. (A) Bending strength versus Young’s modulus of elasticity for all specimens. (B) Bending strength versus Young’s modulus of
elasticity for those specimens whose porosity was measured. Open circles, porosity <8 %; filled circles, porosity >8 %. Subgroup A is not
shown, the porosity of these specimens was <8 %.
2500 J. D. CURREY
Young’s modulus is equal to stress divided by strain (E=σ/ε),
if all bones were to fail in bending at the same strain in the
outermost fibre of the specimen (failure strain εult=k), then
bending strength would be exactly proportional to Young’s
modulus (σ=kE). In fact, the equations in Table 4 show that
bending strength is not directly proportional to Young’s
modulus: for the complete data set, the equation in Table 4 can
be rewritten as: BS=15.1E0.85.
The fact that the exponent is less than unity suggests that, if
bending specimens do fail at some characteristic strain, then
that strain is not quite constant, but decreases as a function of
Young’s modulus. Unfortunately, it is not possible to
determine the yield strain of bending specimens, the onset of
yield being too gentle (Fig. 1B). However, the yield strain of
tensile specimens can be measured. Such measurements were
taken from broadly the same bones as those used to produce
the bending data set. The only two tensile specimens from the
whale’s bulla, which had very low strains at yield (0.0017 and
0.0018), were excluded from the statistical analyses. There is
a weak, although highly significant, negative correlation
between yield strain εy and Young’s modulus E (Fig. 5). The
equation is:
εy = 0.0086 − 0.000089E ,
(1)
where E is measured in GPa (P<0.001, r2=0.20).
If we assume that this equation applies to the bending
specimens as well as to the tensile specimens, we can, given
the Young’s modulus of the bending specimens, use it to
predict their yield strain. It was argued above that, if strain at
yield (and therefore failure) is invariant, bending strength
should simply be proportional to Young’s modulus. However,
Fig. 5 shows that strain at yield is to some extent a negative
function of E. Assuming, for the moment, that bone is elasticperfectly brittle and fails at yield, then Eεy is the yield stress
σy and should be equivalent to the bending strength. This yield
stress σy, expressed in units of MPa, is used below rather than
the constant strain implied by the use of Young’s modulus.
Fig. 6 shows the relationship between bending strength and
yield stress; results from the statistical analysis are given in
Table 5. Using yield stress produces only a slightly better fit,
in terms of r2, than using Young’s modulus (r2=0.90 versus
r2=0.88, respectively). The former equation (BS=1.19+1.66σy)
will give nearly zero bending strength when Young’s modulus
is zero, which is satisfactory, whereas the latter equation
(BS=32.7+9.55E) gives a large, significant positive value for
bending strength, which must be wrong (Fig. 4; Table 4).
Examination of the residuals for the two equations shows the
differences. The standardised residuals (expressed in terms of
the standard deviation) of yield stress, despite possibly being
somewhat heteroscedastic, are much better behaved than the
residuals for Young’s modulus, their general distribution being
horizontal (Fig. 7B), whereas the residuals for Young’s
modulus appear to increase and then decrease as a function of
the fitted values (Fig. 7A). This was confirmed by regression
analysis. Bending strength was fitted by linear equations and
also by quadratic equations (Table 5). The quadratic equation
involving Young’s modulus improved the fit slightly, reducing
the unexplained variance by 13 %, and also made the curve
pass very close to the origin, which is a further improvement.
The quadratic equation involving yield stress did not improve
the fit at all, indicating that the yield stress model is a better
one than the Young’s modulus model.
350
0.015
Bending strength (MPa)
300
Yield strain
0.010
0.005
250
200
150
100
50
0
0
0
0
10
20
30
50
100
150
200
Yield stress (MPa)
Young’s modulus (GPa)
Fig. 5. Strain at yield versus Young’s modulus of elasticity for all
tensile specimens.
Fig. 6. Bending strength versus yield stress for all specimens.
Subgroup A is not shown in this diagram. The broken line is the line
of equality.
Bone bending strength 2501
Table 5. Regression relationships between bending strength BS (MPa), Young’s modulus E (GPa) and yield stress σy (MPa),
using quadratic terms
Equation
BS=32.7+9.55E
BS=−2.17+14.8 E–0.165E2
BS=1.19+1.66σy
BS=−4.41+1.79σy–0.0006σy2
N
t
r2
P
343
343
343
343
50.84
19.50, 7.12
54.92
11.31, −0.82
0.88
0.90
0.90
0.90
<0.001
<0.001, <0.001
<0.001
<0.001, NS
Three of the values of R2 in this table are the same. This is correct, and not caused by typos.
NS, not significant.
Discussion
The bending strength of compact bone can best be explained
as a function of yield strain; yield strain is similar for all bone,
although it declines slightly as Young’s modulus increases.
Nevertheless, the bending strength of bone cannot be explained
simply by assuming that the bone yields at some strain that is
a negative function of Young’s modulus and that the bending
moment at failure is predicted by that strain. Fig. 6 shows that
although yield stress is directly linearly proportional to
bending strength, if bending strength is assumed to equal yield
stress, the predicted values of bending strength are too low by
a factor of approximately 40 % (Fig. 6, broken line).
Bending strength is simple to measure, but this simplicity is
misleading. When a specimen is bent, one side is loaded in
compression, the other in tension, and strain varies
continuously, and in theory linearly, with distance from the
neutral axis. If a material is homogeneous and behaves linearly
elastically, the stress also varies linearly. Bending strength
formulae make this assumption. In an extremely brittle
specimen, the beam formula may be adequate. However, if the
outermost fibres of a specimen yield, and show some post-yield
deformation, the proportionality of stress and strain ceases. The
specimen may undergo a greater and greater bending moment,
with a resulting larger and larger calculated maximum bending
stress, which may be spurious. This matter was analysed
theoretically and experimentally for bone by Burstein et al.
(1972).
Because the stress at a particular strain is nearly proportional
to Young’s modulus, for a similar yield strain, a high Young’s
modulus will be associated with a high yield stress and
therefore a high bending moment when the specimen yields.
If, furthermore, the material shows a reasonable amount of
post-yield strain, then the bending moment will continue to
increase for a while, and the apparent bending strength will be
higher. If the bone has a low modulus, it will yield at a rather
low stress, and even the large amount of post-yield
deformation characteristically shown by low-modulus
specimens will not increase the apparent bending strength
4
4
B
3
3
2
2
Residual of yield stress
Residual of Young's modulus
A
1
0
-1
1
0
-1
-2
-2
-3
-3
-4
-4
0
100
200
Fitted value
300
400
0
100
200
300
400
Fitted value
Fig. 7. (A) Standardised residuals (expressed in terms of their standard deviations) versus fitted values in the equation BS=32.7+9.55E, where
BS is bending strength and E is Young’s modulus of elasticity. (B) Standardised residuals versus fitted values in the equation BS=1.19+1.66σy,
where σy is yield stress.
2502 J. D. CURREY
350
300
Bending strength (MPa)
Strain ratio
10
5
2
250
200
150
100
A
50
1
0
0
5
10
15
20
25
30
35
Young’s modulus (GPa)
50
100
150
200
250
300
350
Predicted bending strength (MPa)
Fig. 8. Strain ratio (ultimate strain/yield strain) versus Young’s
modulus of elasticity for tensile specimens. Note that the ordinate is
on a logarithmic scale.
sufficiently to make up for the low stress at yield. Burstein et
al. (1972) showed that, for a rectangular cross section, as was
used in the present tests, if the post-yield strain is equal to the
elastic strain, then the bending moment will be raised by a
factor of approximately 1.5 compared with a completely brittle
material. If the post-yield strain is five times the elastic strain,
then the factor is 1.7. It requires, therefore, little post-yield
strain to reap most of the benefits of not being completely
brittle. This post-yield behaviour of bone may partly
explain why the predictions of yield stress in the present study,
which assume that bone is linearly elastic–totally brittle, are
too low.
The ratio of ultimate strain εult to yield strain εy, or ‘strain
ratio’ εult/εy, is shown in Fig. 8 plotted against Young’s
modulus for all the tension specimens. The equation for the
relationship is:
log(εult/εy) = 0.996 − 0.0289logE
0
(2)
(P<0.001, r2=0.37). The ordinate is plotted on a logarithmic
scale; strain ratios near unity indicate brittle or near-brittle
behaviour. Many of the more compliant specimens have a
strain ratio of the order of 10–15. According to the calculations
of Burstein et al. (1972), such specimens would be expected to
have a bending strength approximately twice their yield
strength. In the present specimens, therefore, there is a
variation of approximately a factor of two in the difference
between the yield strength and the bending strength; the bones
with a lower Young’s modulus tend to have higher factors (Fig.
8). This phenomenon of post-yield resistance to bending
would, therefore, seem to be of the right magnitude to explain
why the bending strengths of the specimens are greater than
Fig. 9. Relationship between observed bending strength and that
predicted from the theory of Burstein et al. (1972) and regressions
from Figs 5 and 8. The isolated whale bulla specimens labelled A do
not form part of the general pattern. The broken line is the line of
equality.
those predicted by yield stress. It is surprising, however, that
some of the more brittle bones do not show a lower bending
strength, closer to their presumed yield stress.
It may be taking theory too far to apply the equations of
Burstein et al. (1972) to the present data because to do so
requires the building of prediction on prediction. It requires the
combination of two regressions: yield strain as a function of
Young’s modulus and strain ratio as a function of Young’s
modulus. Both these relationships are rather loose.
The calculations of Burstein et al. (1972) use γ, defined as
the ratio of pre-yield strain to ultimate strain (the inverse of the
strain ratio), where C=2{1−√[1−(γ−1)2]/(γ−1)2}. The ratio of
calculated bending stress to stress at yield in tension is then
given by 0.25{C(3−γ2)+2(2−C)3/γC}.
These calculations assume that the specimens have a
rectangular cross section (which is the case in this
investigation) and that the specimens do not yield in
compression before failure. If the specimens do yield in
compression, the effect will be negligible down to a value of
γ of approximately 0.5. For lower values of γ, the calculated
value of bending strength starts to deviate less from the actual
yield stress than predicted by the above equations.
The value of γ as a function of Young’s modulus was
calculated from the regression equation for the data shown in
Fig. 8, and the strain at yield as a function of Young’s modulus
was calculated from the regression equation for the data shown
in Fig. 5.
The results of these calculations are shown in Fig. 9. Except
for the group of eight outliers that form subgroup A (see
Bone bending strength 2503
below), the predicted values and the actual values are closely
similar until a predicted value of bending strength of
approximately 180 MPa. Thereafter, the actual values become
greater than the predicted values.
Another factor that may be important for the present results
was first analysed by Weibull (1951). A tensile specimen
undergoes roughly the same stress all through its volume,
while a bending specimen undergoes high tensile stresses
within a rather small volume close to one surface opposite the
central loading point. Any real material is not quite uniform
and has a distribution of strengths throughout its volume. It is
likely, therefore, that a tensile specimen will yield at a lower
calculated stress than a bending specimen simply because the
stress in a tensile specimen has a larger volume over which to
‘seek out’ a weak part of the specimen. Therefore, the stresses
in the outermost fibres of the bending specimens before they
yield will on average be higher than the overall yield stress
reached if the same specimens had been loaded in tension. The
values used to calculate yield stress here were from tensile
specimens, which may explain why the bending moments of
the bending specimens tended to be higher than predicted by
the yield stress model. The specimens that deviated most were
the stronger specimens, which had the highest Young’s
modulus and were, in general, more brittle. The Weibull effect
is more pronounced in brittle materials; however, its magnitude
cannot be quantified from the present data.
The specimens that comprised subgroup A (Figs 3, 4, 9)
were from very highly mineralised tympanic bullae from the
fin whale Balaenoptera physalus (Currey, 1979) with a high
Young’s modulus. They had a low bending strength (Figs 3,
4), a low strength in tension (approximately 25 MPa) and a
very low strain at yield and fracture. These specimens were
completely brittle, showing no post-yield strain in tension.
Also, in tension, they broke at a lower strain than all the other
bones. Therefore, in bending, they cannot make use of their
high Young’s modulus, both because they break at a lower
strain than other bones and because they undergo no post-yield
deformation, so the bending moment cannot increase after
yield. In this very highly mineralised bone, the few cracks that
appear in the pre-yield region presumably find it so easy to
travel that they become fatal, giving a strain at yield
(approximately 0.002) considerably less than that reached by
other bones (>0.004), and the material fails in a brittle manner
at a low bending moment.
Professor Mike Ashby, Miss Debra Balderson and Drs
Terry Crawford and Justin Molloy gave me helpful advice. I
thank an anonymous referee who encouraged me actually to
apply the predictions of Burstein et al. (1972) to the data. The
results were surprising and interesting.
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