Issue 10

D. Taylor et alii, Frattura ed Integrità Strutturale, 10 (2009) 12-20; DOI: 10.3221/IGF-ESIS.10.02 14 Note in particular from Fig. 1 that the TCD was able to predict the fact that small holes (hole diameter = 0.1 x bone diameter) have no effect on strength, a very useful finding for clinicians and one that was not predicted by other approaches. In the TCD approach this finding arises because if a notch is very small, the critical distance (being constant) becomes effectively much larger than the hole, so the stress at the critical point is similar to the nominal applied stress: effectively the hole has become “invisible” as far as this approach is concerned. A particularly encouraging aspect of this validation exercise was the fact that the appropriate value of the critical distance was found to be almost constant across the three sets of data. It is well known that other mechanical properties of bone, such as stiffness and strength, vary considerably, so we had expected that L would also vary, but this seems not to be the case: a value of 0.32-0.38mm was able to give good predictions throughout. Figure 1 : The effect of hole diameter (normalized by bone diameter) on fracture torque (normalized by fracture torque for bones containing no hole), for whole bones tested in torsion, containing single transcortical holes of various diameters. Predictions using the TCD and two other theories. The same situation arises in fibre composite materials, which are also known to have only a small range of L values [14] . In those materials, strength and toughness are roughly proportional to each other over quite a wide range of values, so that an increase in strength (for example by increasing the proportion of fibres) confers a similar proportional increase in toughness. An equation can be derived which links three material constants used in the TCD: L, K c and the critical stress for failure  o , as follows: 2 1        o c K L   (1) Since L is related to the ratio of strength to toughness, it stays constant if these two properties change in a proportionate manner. The critical stress parameter defining failure in bone,  o , was found to be slightly larger than the material’s tensile strength,  u as measured from tests conducted on plain, unnotched samples. We found that accurate predictions could be made using a critical stress of T  u where T had a constant of value 1.33. This finding is in line with our investigations of other materials, in which the value of T has been found to take values close to 1.0 for brittle ceramics and composites [15], in the range 1.4-3 for polymer s [1] and values typically greater than 3 for metals [16]. A precise interpretation of the meaning of the T parameter is still unclear. In considering the significance of this value it is worth noting the link between the three parameters of toughness, strength and L, as shown in equation 1 above. If two of these constants are known, the third can be calculated, which implies that only two of these three constants are of fundamental significance. In my personal opinion, the two fundamental parameters are L and K c . The value of  o differs from that of  u , in my view, because of two assumptions which we make in this analysis. Firstly, we assume that the material is linear and elastic, which of course it is not. It is significant that values of T become larger in materials and fracture processes involving more plasticity. Secondly, we assume that the mechanism of failure in a plain specimen is the same as that in a notched specimen. This is clearly not the case in some materials: plain specimens may fail differently due to, for example, plastic instability (necking) in ductile materials or the presence of pre-existing defects in brittle materials. It is interesting to note that, when we