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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 Printed in Great Britain © The Company of Biologists Limited 1999

JEB1959

The bending strength of a wide variety of bony types is shown to be nearly linearly proportional to Young’s modulus of elasticity/100. A somewhat closer and more satisfactory fit is obtained if account is taken of the variation of yield strain with Young’s modulus. This finding strongly suggests that bending strength is determined by the yield strain. The yield stress in tension, which might be expected to predict the bending strength, underestimates the true bending strength by approximately 40 %. This may be explained by two phenomena. (1) The post-yield deformation of the bone

material allows a greater bending moment to be exerted after the yield point has been reached, thereby increasing the strength as calculated from beam formulae. (2) Loading in bending results in a much smaller proportion of the volume of the specimens being raised to high stresses than is the case in tension, and this reduces the likelihood of a weak part of the specimen being loaded to failure.

Key words: bone, bending strength, Young’s modulus, Weibull effect.

Summary

Introduction

WHAT DETERMINES THE BENDING STRENGTH OF COMPACT BONE?

JOHN D. CURREY*

Department of Biology, University of York, York YO10 5YW, UK *e-mail: jdc1@york.ac.uk

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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 three-point 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).

Number

Species Bone BS [Ca] p

Alligator Femur 26 26 1

Axis deer Tibia 3 − −

Mother

Foetus Tibia 8 − −

Bovine Metacarpus 9 9 −

Tibia 11 11 1

9 year Femur 10 10 3

2 year Femur 20 − −

Crocodile Frontal 2 2 1

Prefrontal 2 1 1

Dolphin Radius 2 2 2

Donkey 34 year Radius 7 7 2

Dugong Radius 2 2 2

Ulna 2 2 2

Scapula 12 12 7

Elephant Tusk 3 3 −

Fallow deer Tibia 9 9 2

Radius 3 3 −

Fin whale Bulla 9 9 9

Flamingo Tarso-metatarsus 5 5 5

Tibiotarsus 3 3 3

Galapagos tortoise Femur 4 4 −

106 year Fibula 5 5 2

Tibia 2 2 −

Humerus 7 7 1

Number

Species Bone BS [Ca] p

Grey seal Tibia 5 5 2

Horse 7 year Femur 9 9 2

Jackass penguin Humerus 1 1 1

Ulna 2 2 2

Radius 2 2 1

Femur 2 2 1

Tibia 2 2 2

King penguin Humerus 9 9 −

Ulna 2 2 −

Muntjac deer Antler 7 7 −

Red deer Antler 25 25 2

Reindeer Antler 6 6 2

Roe deer Femur 7 7 −

Sarus crane Tarsometatarsus 13 4 4

Ossified tendon 1 1 1

Tibiotarsus 9 − −

Radius 6 − −

Swan Humerus 4 − −

Wallaby Femur 19 14 8

Humerus 2 2 2

Tarsal 10 7 7

Radius 4 2 1

Tibia 24 13 7

Ulna 4 2 2

White-sided dolphin Rib 8 8 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

[image:2.609.45.563.86.421.2]

specimens for which calcium content was also determined; p, number of these specimens for which porosity was also determined; −, no values determined.

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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.

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

Tension Yield region

Deformation

Load

A

Bending

Deformation

Load

B

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.

[Calcium] (mg g-1)

150 200 250 300 350

Young’s modulus (GPa)

0 10 20 30 40

A

150 200 250 300 350

Young’s modulus (GPa)

0 10 20 30 40

[Calcium] (mg g-1)

B

[image:3.609.60.293.68.534.2] [image:3.609.320.563.71.570.2]
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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−1bone, 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

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 log-transformed values, respectively. These coefficients are both highly significant (P<0.001), though not strong.

Bending strength

[image:4.609.49.560.100.166.2]

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−1bone bending strength may be high or low. High-porosity specimens tend to have low bending strengths Table 2. Relationship between Young’s modulus E (GPa), calcium content [Ca] (mg g1) and porosity (%) determined by linear

regression

Equation N t r2 P

logE=−8.58+4.05log[Ca] 855 30.35 0.52 <0.001

logE=−9.53+4.28log[Ca] 370 16.45 0.42 <0.001

logE=1.67–0.63logp 370 −18.56 0.48 <0.001

logE=5.06+2.75log[Ca]–0.451logp 370 11.48, −13.75 0.62 <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, r2is calculated allowing for the number of degrees of freedom.

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 %.

150 200 250 300 350

Bending strength (MPa)

0 50 100 150 200 250 300 350

A

[Calcium] (mg g-1)

A

150 200 250 300 350

Bending strength (MPa)

0 50 100 150 200 250 300 350

[image:4.609.48.551.444.693.2]
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(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.

[image:5.609.45.568.87.141.2]

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, Table 3. Regression relationships between bending strength BS (MPa), calcium content [Ca] (mg g1) and porosity (%)

Equation N t r2 P

logBS=−4.25+2.70log[Ca] 262 10.30 0.29 <0.001

logBS=−5.76+3.32log[Ca] 86 4.92 0.22 <0.001

logBS=0.39+1.25log[Ca]–0.491logp 86 2.05, −7.06 0.50 <0.05, <0.001

[image:5.609.47.568.209.264.2]

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 N t r2 P

logBS=1.25+0.851logE 343 55.46 0.90 <0.001

logBS=1.18+0.890logE 86 31.63 0.92 <0.001

logBS=1.28+0.847E–0.053logp 86 21.91, −1.60 0.90 <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. 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 %.

Young’s modulus (GPa)

0 10 20 30 40

Bending strength (MPa)

0 50 100 150 200 250 300 350

A

A

Young’s modulus (GPa)

Bending strength (MPa)

0 50 100 150 200 250 300 350

0 10 20 30 40

[image:5.609.56.560.458.700.2]
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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 εyand 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 elastic-perfectly brittle and fails at yield, then Eεyis the yield stress

σyand 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+1.66σ

[image:6.609.308.557.459.701.2]

y) 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.

Fig. 5. Strain at yield versus Young’s modulus of elasticity for all tensile specimens.

Young’s modulus (GPa)

Yield strain

0 0.005 0.010 0.015

0 10 20 30

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.

Yield stress (MPa)

0 50 100 150 200

Bending strength (MPa)

[image:6.609.43.290.469.711.2]
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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).

[image:7.609.52.569.100.168.2]

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 Table 5. Regression relationships between bending strength BS (MPa), Young’s modulus E (GPa) and yield stress σy (MPa),

using quadratic terms

Equation N t r2 P

BS=32.7+9.55E 343 50.84 0.88 <0.001

BS=2.17+14.8 E–0.165E2 343 19.50, 7.12 0.90 <0.001, <0.001

BS=1.19+1.66σy 343 54.92 0.90 <0.001

BS=−4.41+1.79σy–0.0006σy2 343 11.31, −0.82 0.90 <0.001, NS

Three of the values of R2in this table are the same. This is correct, and not caused by typos. NS, not significant.

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 σyis yield stress.

Fitted value

0 100 200 300 400

Residual of Young's modulus

-4 -3 -2 -1 0 1 2 3 4

A

Fitted value

0 100 200 300 400

Residual of yield stress

-4 -3 -2 -1 0 1 2 3 4

[image:7.609.57.559.457.698.2]
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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 εultto 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 (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

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(2C)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

Young’s modulus (GPa)

0 5 10 15 20 25 30 35

Strain ratio

[image:8.609.307.558.72.313.2]

1 2 5 10

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.

Predicted bending strength (MPa)

0 50 100 150 200 250 300 350

Bending strength (MPa)

0 50 100 150 200 250 300 350

[image:8.609.47.286.72.310.2]

A

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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.

References

Burstein, A. H., Currey, J. D., Frankel, V. H. and Reilly, D. T.

(1972). The ultimate properties of bone tissue: the effects of yielding. J. Biomech. 5, 34–44.

Currey, J. D. (1979). Mechanical properties of bone with greatly

differing functions. J. Biomech. 12, 313–319.

Currey, J. D. (1987). The evolution of the mechanical properties of

amniote bone. J. Biomech. 20, 1035–1044.

Currey, J. D. (1988). The effect of porosity and mineral content on

the Young’s modulus of elasticity of compact bone. J. Biomech.

21, 131–139.

Currey, J. D. (1990). Physical characteristics affecting the tensile

failure properties of compact bone. J. Biomech. 23, 837–844.

Katz, J. L. (1971). Hard tissue as a composite material. I. Bounds on

the elastic behavior. J. Biomech. 4, 455–473.

Keller, T. S. (1994). Predicting the compressive mechanical behavior

of bone. J. Biomech. 27, 1159–1168.

Martin, R. B. and Ishida, J. (1989). The relative effects of collagen

fiber orientation, porosity, density and mineralization on bone strength. J. Biomech. 22, 419–426.

Sasaki, N., Ikawa, T. and Fukuda, A. (1991). Orientation of mineral

in bovine bone and the anisotropic mechanical properties of plexiform bone. J. Biomech. 24, 57–61.

Wagner, H. D. and Weiner, S. (1992). On the relationship between

the microstructure of bone and its mechanical stiffness. J. Biomech.

25, 1311–1320.

Weibull, W. (1951). A statistical distribution function of wide

Figure

Table 1. The provenance and numbers of the specimens used in the bending tests
Fig. 1. (A) Idealised load versusbending. The Young’s modulus of elasticity is calculated from theinitial, straight part of the curve using standard beam formulae
Table 2. Relationship between Young’s modulus E (GPa), calcium content [Ca] (mgg−1) and porosity (%) determined by linearregression
Table 3. Regression relationships between bending strength BS (MPa), calcium content [Ca] (mgg−1) and porosity (%)
+4

References

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