Issue 10

D.Taylor et alii, Frattura ed Integrità Strutturale, 10 (2009) 12-20; DOI: 10.3221/IGF-ESIS.10.02 13 of short cracks. Following the success of this work, we then applied the approach to some problems of clinical significance. One example of this type of work is described: the surgical management of bone defects. T HE THEORY OF CRITICAL DISTANCES : A BRIEF INTRODUCTION hat follows is a very brief introduction to the TCD: further details are available in [1] and in many other recent publications. The TCD recognises the fact that, in order to predict failure arising from a stress concentration feature such as a notch, it is not sufficient to know the stress and strain at the notch surface, at the maximum stress point, often known as the “hot spot”. Rather, it is essential to have information about the stress field in the vicinity of the notch, because fracture processes that involve crack initiation and propagation are strongly influenced by aspects of the stress field in this region, such as the gradient of stress or, to put it another way, the absolute volume of material which is experiencing high stress. This recognises the fact that cracking-type failures require, in general, a solution of the type which is now generally referred to as a “non-local approach”, characterised by a physical mechanism of failure involving a process zone in the vicinity of the crack tip in which failure, deformation and damage processes occur. A variety of methods, more or less complex, have been devised to make predictions using stress and strain information in this critical region. In our most strict definition of the TCD, it consists of a group of methods which have the following two features in common. Firstly, the use of a linear, elastic material model for the stress analysis. Secondly, the use of a material parameter which has the units of length, known as the critical distance, L. The value of L cannot be known a priori ; it can only be found by processing data from samples containing stress concentrations, tested to failure in the particular failure mode of interest. It is taken to represent a critical dimension in the material over which relevant failure processes occur. For example, in many cases it is found to be related to critical microstructural parameters such as grain size, which are known to control the material’s strength and toughness: this relationship will be discussed further below. Having stated this strict definition, it is important to point out that exceptions do occur, in which the TCD is used in cases where these conditions are violated. For example it may be appropriate to use a non-linear material model, and we have indeed done so ourselves as will be discussed below. Also, some realisations of the theory make use of a value of L which is not a material constant [4, 5], though these will not be considered in the present paper. We can define two different types of TCD methods. In the first type, predictions are made using information about the stress field, specifically the stress as a function of distance from the hot spot, on a line (known as the focus path) along which crack growth is expected to occur. The simplest example of this approach is the so-called Point Method, which uses only the stress at a given point, located a distance L/2 from the hot spot. Failure is predicted to occur if the stress at this point exceeds a critical value. A variant of this approach is the Line Method, in which the stress parameter is the average stress along the line, over a distance from zero to 2L from the hot spot. Area and volume averages have also been used, though these more complex methods do not seem to confer any more accuracy than the simple point and line methods. The second type of TCD method involves a modification of fracture mechanics, whereby the critical distance appears as the length of an imaginary crack located at the notch, or, alternatively, as the magnitude of finite crack growth increments [6]. Once such a modification is accepted, normal linear-elastic fracture mechanics approaches can be used. In what follows we will use approaches of the first type, i.e. stress-based methods, making use of finite element analysis (FEA) to obtain the appropriate stress fields. I NITIAL VALIDATION : NOTCH FRACTURE DATA e obtained from the published literature three extensive sets of data on the effect of notches on brittle fracture in bone. All three involved tests in which monotonically-increasing loads were applied until failure occurred. One publication [7] was concerned with the effect of notch length for sharp notches machined in bone samples, whilst the other two [8, 9] reported the results of tests conducted on whole bones, loaded in bending and torsion respectively, containing circular holes of various sizes. We found that the TCD was able to predict all this data. Fig.1 shows an example: the effect of hole size on failure load for bones loaded in torsion. Further details can be found in a recent publication [10] . At this point it may be worth pointing out that the TCD can be used with any type of applied loading, including multiaxial load cases, though an appropriate multiaxial failure criterion should be used. In the present study our criterion was simply the maximum principal stress: we have reported the use of other multiaxial criteria to predict fatigue and fracture in various materials, in an extensive series of previous publications (e.g. [11-13] ). W W