Issue 10

D. Taylor et alii, Frattura ed Integrità Strutturale, 10 (2009) 12-20; DOI: 10.3221/IGF-ESIS.10.02 16 Figure 3 : Threshold stress intensity range for fatigue crack growth in bone, defined at a crack growth rate of 3-6 x 10 -8 m/cycle; for details see [17]. As can be seen from the figures, the predictions are very satisfactory for both types of failure, even including data for very small crack lengths obtained using nanoindentation experiments [18] . It should be remarked that currently there is considerable controversy in the literature about the validity of measuring toughness using indentation, a technique which has been used for brittle material for many years but which is now being seriously questioned. The interested reader may wish to refer to recent letters in the Journal of Biomechanics arising from the publication by Mullins et al [18]. Currently, indentation is one of the few options available for estimating the toughness of materials at very small length scales, a subject which is of increasing interest given the advent of micro and nano scale materials and devices. The data in Figs 2 and 3 h ere all refer to crack growth in the transverse direction: bone is highly anisotropic so further work is needed to explore fracture properties in different directions. In making these predictions we used the same value for L as previously obtained from the predictions of notch fracture behaviour. This implies that L takes the same value in fatigue as in brittle fracture in this material, at least for cracking in the transverse direction. We have previously found significant differences between L values for fatigue and brittle fracture in metallic materials, but similar values for a polymer, PMMA. In the present case the fatigue data available are relatively sparse, so further validation is needed before this conclusion can be stated with confidence. It is perhaps worth considering at this stage why bone has this particular value of L. As noted above, L values for many materials are often related to the size of microstructural features which control fracture behaviour. Bone has a hierarchical structure, displaying features at a range of size scales, especially nanometres (the thickness of reinforcing crystals of hydroxyapatite), microns (the thickness of lamellae consisting of crystals and collagen fibres in a composite structure) and hundreds of microns (the size and spacing of structural units known as osteons). A number of mechanisms operating at the hundred-micron scale have been identified, notably uncracked ligaments bridging the crack faces [19] and the role of the osteon boundary in crack arrest (O'Brien et al. , 2005), in a manner similar to the grain boundary in metals. Figs 4 and 5 show examples of these mechanisms. Ritchie and co-workers have investigated these mechanisms in some detail and have laid particular emphasis on the role of uncracked ligaments. They showed a definite relationship between the rising R- curve for a given crack and the increasing number of uncracked ligaments observed as the crack extended [20]. In our studies on high-cycle fatigue in bone we have placed emphasis on the role of the osteon boundary, showing that the great majority of fatigue cracks become non-propagating when they reach the first boundary and developing relationships between crack length, growth rate and the proximity of this boundary [21, 22]. All of these various observations imply that L takes a value equal to a few hundred microns because this corresponds to the size scale on which important toughening mechanisms operate in this material. In fact, this turns out to be the case for many different materials. Fig.6 s hows the value of L for various different classes of materials, plotted against the relevant structural parameter d. In some cases there are very clear and demonstrable relationships between L and d: for example we showed that L takes values very close to d in steels failing by brittle cleavage fracture at low temperatures [23] . In other cases the relationship is less clear but for most materials it seems that L falls between d and 10d in magnitude. There are, however, some important exceptions: for example amorphous polymers such as PMMA have no microstructure as such, and yet have L values of the order of 0.1mm. This coincides with the typical size of crazes in the material. 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 1.5 2 2.5 3 Crack Length (mm) Stress Intensity Range (MPa.m^1/2) TCD Prediction Kruzic