What determines the bending strength of compact bone?

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.

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’.

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).

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 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 (σ_y), yield strain (ε_y) and strain at failure (ε_ult). 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).

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.

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).

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.

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 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 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.

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 230mgcalcium g⁻¹ bone bending strength may be high or low. High-porosity specimens tend to have low bending strengths (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, 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 ε_ult=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=15.1E^0.85.

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 ε_y and Young’s modulus E (Fig. 5). The equation is:

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 Eε_y is the yield stress σ_y and should be equivalent to the bending strength. This yield stress σ_y, 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_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.

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 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 ε_ult to yield strain ε_y, or ‘strain ratio’ ε_ult/ε_y, is shown in Fig. 8 plotted against Young’s modulus for all the tension specimens. The equation for the relationship is:

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 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.