Perhaps the simplest way to approach this topic is to consider it in the context of stress concentrations. Consider the stress concentration on the edge of an elliptical hole in an infinite plate (Fig. 8.2a). The plate bears a tensile stress, s, normal to the long axis of the ellipse. The stress at each end of this axis is

where b and a are the minor and major radii of the ellipse. (For a circular hole, this reduces to 3 s.) However, this equation loses its validity if a»b and the ellipse becomes crack-like (Fig. 8.2b). In that case, the stresses at a point given by polar coordinates (r, Θ) relative to the end of the b-axis are


The stress in the y direction along the Θ = 0 line is
As one approaches the crack tip, r becomes very small relative to the crack length, a, and the stress concentration very large. At the crack tip itself, r = 0 and σ _y is singular (i.e., it is infinite). To avoid this difficulty, one may move the 2r term to the other side of the equation, obtaining σ _y (2r)^1/2, which is not singular at the crack tip but is still proportional to the “intensity” of the stress at the crack tip. One can then define:
That is, the stress intensity in the vicinity of the crack tip is proportional to the nominal stress, s, and the square root of the crack length, a. This stress intensity is what drives the crack forward.

Irwin (1957) used linear elasticity theory to obtain equations for the stresses near the tips of cracks with several different geometries and loading situations. He found that in general
where k is a coefficient peculiar to the particular problem. As one approaches the crack tip, the “other terms” became less important and can be ignored. Because of the π in these equations, a formal stress intensity factor, K, was defined so as to include π as well:
K has units of Pa m^1/2; compare Eq. (8.6) with Eq. (8.4). C is a dimensionless constant depending on the size and shape of the flaw and the size, shape, and mode of loading of the specimen, and s is a pertinent stress component in the specimen beyond the influence of the crack, for example, tensile stress perpendicular to the plane of the crack. K characterizes the state of stress near the crack tip; therefore it is known as a characterizing parameter. Its utility stems from the fact that if K exceeds a critical value, the crack will propagate; otherwise, it will not. There is no need to make calculations of the local stresses to gain this information.

To reiterate, the fundamental concept in linear elastic fracture mechanics is that K determines whether a crack will propagate or not, and if so, how fast. In practice, K bears a subscript denoting the type of cracking that is being considered.

Of the various ways in which a specimen may be loaded to extend cracks, three are commonly discussed in fracture mechanics (Fig. 8.3). Mode I cracks are propagated perpendicular to the line of action of a tensile load, as when a wedge splits a log lengthwise. Mode II cracks result when shearing forces are applied parallel to the plane of the crack and parallel to the direction of its propagation. Mode III cracks are a “tearing” phenomenon; in this case, the shearing is perpendicular to the crack propagation direction. Mode I is the most commonly discussed type of cracking.

As an example, consider the Mode I cracking that occurs when a plate of width b, containing a crack of length a on one edge, is loaded in tension. In this case, the C in Eq. (8.6) turns out to be:
where q = π a/2b. As shown in Eq. (8.6), when multiplied by (π a)^1/2 and the tensile stress in the plate, this C gives the stress intensity factor, K _I, for this situation. If this K _I is greater than a critical stress intensity factor, K _Ic, the crack propagates rapidly clear through the structure. K _Ic is also called the fracture toughness; it is a material property, like yield stress. Table 8.1 gives some typical values of K _Ic for metals, ceramics, polymers, and cortical bone (Pilkey 1994). By conducting many experiments and tabulating the results, the means for calculating K _I in various situations and comparing them to the critical values have been amassed for many conventional engineering applications. Figure 8.4 gives formulae for C in a few simple cases.

The measurement of K is beyond the scope of this book because it is complicated by the need to ensure that all the assumptions underlying its calculation are fulfilled. The basic process is simple, however, as illustrated in Fig. 8.5 for mode I cracking. A specimen with a standardized crack is manufactured, and the crack is propagated across the specimen by loading it in tension. The applied load and crack opening displacement are measured and plotted. A failure load is identified on the load-displacement curve using several rules. K _Ic is then calculated by multiplying the failure load by a formula derived from the length of the crack and the shape and dimensions of the specimen.

Several investigators have measured K _Ic for cortical bone (Table 8.2). Values for bone are an order of magnitude less than those for structural metals but similar to a tough ceramic like Al₂O₃. In making these comparisons, it needs to be emphasized that the values for bone are for crack propagation parallel to the long axis of the bone, which is to say “with the grain.” If one tries to propagate a crack transversely, or perpendicular to the primary and secondary lamellar structure, the crack immediately turns and goes “with the grain.” Thus, the fracture toughness of bone is decidedly anisotropic, and the values shown are for the direction offering the least resistance to cracking.

By rearranging Eq. (8.6) and setting the stress intensity factor equal to the critical value, one may solve for the flaw size that will precipitate unstable crack propagation. For human tibial compact bone, Norman et al. (1995) found that K _Ic ≈ 4 MPa m^1/2. For a semicircular crack of radius a at the surface of a semi-infinite body loaded in tension (e.g., on the tension side of a large bone loaded in bending), the stress intensity factor is Pilkey (1994).
Here, θ is as defined in Fig. 8.2 and s is the stress elsewhere in the body, far from the crack. Setting θ = 0, the maximum tolerable crack radius would be
Assuming that E = 20 GPa and the strain (ε) is a physiologic 2000 micro-strain, one has s = E ε = (20 × 10⁹) (2 × 10⁻³) = 40 MPa and
which is a rather large flaw. Remember also that the K _Ic used here is probably considerably smaller than the one for a crack propagating across a bone.

Before Irwin and others developed the concept of a stress intensity factor, Griffith (1920) formulated the problem of crack propagation in brittle solids in terms of energy. He reasoned that energy was required to create the surfaces of the extended crack because all surfaces have “surface energy” associated with them. He conducted many experiments with glass rods to develop his idea into a theory.

Suppose it takes energy or work, dW _c, to propagate a crack for a distance, da. This energy usually comes from the work done in deforming the specimen via the strain energy, U, stored throughout its interior. Conservation of energy requires that
The right side of this equation is called the strain energy release rate and is denoted by G, after Griffith, who showed that for mode I propagation of a thin ellipsoidal crack the strain energy release rate is
where E is the elastic modulus. By setting up an experiment in which a crack can be slowly propagated in a material of modulus E, and its length (a) measured as a function of the applied stress (s), the strain energy release rate can be determined from Eq. (8.12).

The left side of Eq. (8.11) is called the crack growth resistance, R:
For a crack to grow, strain energy must be produced and released at least as fast as it is consumed by breaking molecular bonds in the material to extend the crack. For the crack to grow precipitously, G must be greater than a critical value, which is R; that is, G ≥ R.

By comparing Eqs. (8.6) and (8.12), it is apparent that G is proportional to K ². For mode I cracking in a plane stress situation,
It is important to understand that a propagating crack is “nourished” by the elastic energy of the mass under stress. Finkel (1985) pointed out that although cracks are driven by the strain energy stored in the mass under stress, they are themselves without inertia, and thus capable of turning instantaneously in any direction that promises more elastic energy.

In most materials, and certainly in bone, linear elasticity theory does not hold when stresses become high, and that includes the situation at the tip of a crack. There is a zone of plastic deformation around the crack tip that is generally shaped as depicted in Fig. 8.6. Note the “Mickey Mouse ears” shape of the plastic zone. Finkel has called the plastic zone the “ears” of the crack, and refers to computations showing that these ears may “wiggle” as the crack propagates. The plastic zone at the leading edge of a propagating crack has an ear-like characteristic shape.

For mode I cracking, the distance directly ahead of the crack that is plastically deformed is given by
where σ _y is the yield stress; k = 2 for plane stress or 6 for plane strain.¹ One way to accommodate the presence of a plastic zone in linear elastic fracture mechanics theory is to add this r _p (or some portion of it) to the crack length in the equations for K and G. Equivalently, when using the energy approach, another term must be added to Eq. (8.14): the rate at which energy is consumed by plastic deformation per unit length of crack extension.

Propagation of a crack in bone is slowed (or accelerated!) by changes in the material ahead of or behind the tip of the propagating crack (Fig. 8.7). Crack toughening mechanisms ahead of the crack tip are called “Intrinsic mechanisms” and those that occur behind the crack tip are called “Extrinsic mechanisms” (Ritchie 2011). Intrinsic mechanisms involve changes in the material properties—for example, in a metal the material properties change from elastic to plastic in front of the crack tip. Extrinsic mechanisms act at the location of crack propagation (not in front of the crack) and, in bone, are strongly related to bridging of the crack by “ligaments” of unbroken tissue. For relatively large cracks (the kind in a fracture toughness specimen) a great deal of toughening comes from unbroken “ligaments” of bone tissue that bridge across the faces of the propagating crack. The ligaments hold the faces of the crack together creating a “closure stress”. Examination of the crack faces after failure of a piece of bone finds the remains of numerous ligaments that had to be broken in order to finally fracture the tissue.

There continues to be some debate about the role of microcrack formation in the toughening of bone tissue, but it is clear that younger/tougher bone forms much greater numbers of microcracks during failure than does older more brittle bone (Green et al. 2011). This is likely due to a more functional collagen matrix, a very important contributor to bone fracture toughness (Barth et al. 2011; Russell et al. 2012; Nyman et al. 2005; Nalla et al. 2003). The fracture toughness of bone tissue with the collagen matrix removed is extremely low and almost no ligaments are observed in the wake of the crack tip. Ligament bridging between the faces of microcracks by collagen fibers allows the microcrack to open, changing the shape of the surrounding tissue which consumes energy and thereby blunts propagation. One effect is that arrays of collagen bridged cracks at the micro level provide enough controlled deformation to enable tissue bridging across larger cracks at the macro level. This is an example of a hierarchical dependency in bone tissue where a macroscopic toughening mechanism (i.e. ligament bridging of large macroscopic cracks) depends on a more microscopic toughening mechanism (collagen matrix bridging of microcracks).

As a crack is propagated in a fracture specimen, the stress intensity to propagate the crack could decrease, stay the same or increase. Recall for linear elastic fracture mechanics that: . Experimentally, the process is to determine “s” and “a” as the crack propagates and then calculate the stress intensity factor “K.” Clearly, the best situation is for the stress intensity factor to increase with crack length so that there is a tendency for the crack to stop propagating. Healthy bone has the property that K increases with crack length resulting in what is called a “rising R-curve” where “R” is for “Resistance” (Fig. 8.8). The name “R-curve” is probably the most obscure in all of bone mechanics. Fortunately the concept is simple: An R-curve plots the changing fracture propagation toughness of the tissue as a crack elongates.

Both age and sequential degradation of the bone matrix using radiation correlate with a reduction of the propagation resistance (Fig. 8.9). The radiation experiment strongly shows the importance of the collagen matrix to fracture toughness. The implication of the age related decrease in the propagation resistance is that the collagen matrix is degrading with age in humans. There is a great deal of research currently being done to characterize how age and other conditions in humans affect both the bone matrix and fracture toughness. It is clear at the time of this writing that a significant portion of age related bone fragility in humans is caused by changes in the collagen matrix. What remains to be determined is what is changing in the matrix and whether it can be prevented or treated in vivo.

Paris (1964) proposed that the number of cycles required for fatigue failure of a specimen with a crack could be related to the crack’s stress intensity factor. His idea was that the crack length increase per load cycle, da/dN, would be proportional to the range of K values experienced during the loading: Δ K = K _max − K _min. A commonly suggested form for such a relationship is
where ξ and ζ are constants. Unfortunately, the Paris equation has not proven to be a consistently valid relationship. It describes the results at intermediate crack growth rates better than at low or high values of da/dN. Within that range, if crack propagation data are presented as a log-log plot of da/dN vs. Δ K, the slope and intercept should yield the coefficients ζ and ξ respectively.

The basic means of presenting fatigue data is an S-N curve: a plot of the applied stress or strain vs. the number of cycles required for failure at that load level. A fatigue S-N curve was first measured for bone by King and Evans (1967). Subsequently, in the 1970s and 1980s, several other investigators (notably Carter and co-workers) reported similar experiments, establishing that bone behaves similar to engineering materials in fatigue. That is, its elastic modulus declines as fatigue progresses, the log of N is linearly proportional to the log of S, and cracks can be seen within fatigued bone which is apparently fatigue damage. The concept that the fatigue behavior of bone is similar to that of composite materials was also established in this period.

Fatigue life, which we now represent by N _F, is related to the applied stress in a very nonlinear way. When one regresses log N _F vs. log S for bone specimens fatigued in various modes of loading, one finds that
where c is a coefficient and the exponent q is a number between 5 and 15, depending on the type of bone and the mode of loading. Clearly, small changes in load magnitude (S), or the kind of bone and mode of loading (q), can result in very large changes in fatigue life. Caler and Carter (1989) and Pattin et al. (1996) have shown that the fatigue life of human femoral cortex is longer in compression than in tension. Testing in uniaxial tensile and compressive fatigue at 2 Hz, Pattin et al. produced data that, when fitted to Eq. (8.17), had the following coefficients:

where the stress range, S, is from zero to a peak stress normalized by the initial elastic modulus. (Therefore, S is also the initial strain, and has units of microstrain.) These data were obtained at superphysiologic strains (2600–6600 με). If one may extrapolate the results to a physiologic peak strain of 2000 με, the fatigue life would be 4.1 million cycles in tension and 9.3 million cycles in compression.

Although we typically think of microdamage in bone as linear cracks, damage caused by cyclic loading of bone actually comes in different flavors. Three distinct forms of damage can be identified in bone, distinguished by their morphology, their mechanical and physiological roles, and by the manner in which they are repaired. Linear microcracks (Fig. 8.10a) are typically about 40–100 μm “long” in cross-section, but may run for hundreds of microns or more than a millimeter longitudinally in bone (Burr and Martin 1993; Taylor and Lee 1998). They are less than 2 μm “wide” in cross-section and are typically repaired by normal bone remodeling processes in which the crack is resorbed by osteoclasts, and new bone is laid down in its place. If many linear microcracks accumulate, either because of high and repeated stresses or because remodeling is suppressed through disease or treatments for disease (e.g. bisphosphonates used to mitigate osteoporosis), they can degrade the elastic modulus of the bone to the extent that a stress fracture (as in athletes and soldiers), or a complete fracture (as in osteoporotic people who sustain atypical femoral fractures) can occur. A second form of microdamage, diffuse damage (Fig. 8.10b), is composed of a collection of very small cracks, most less than 10 μm long in cross-section. The cracks themselves can be difficult to detect with standard light microscopy, even on stained sections, but can be identified as individual cracks using confocal microscopy (Fazzalari et al. 1998; Vashishth et al. 2000). These cracks probably do not cause a significant degradation of the mechanical properties of bone, but, like linear microcracks, may be important in stress relief when bone is repeatedly loaded. They do not create the signals necessary to stimulate whole bone remodeling (Herman et al. 2010), so why do they not accumulate? Recent observational and experimental evidence suggests that they may heal through an intrinsic matrix repair process in which the defect is eliminated by filling with mineral (Boyde 2003), or in which non-collagenous proteins (perhaps osteopontin) act like a glue to bring the surfaces of bone back into contact (Seref-Ferlengez et al. 2012). Microfractures are a third type of microdamage in bone. These are not cracks at all, but full fractures of trabecular bone, which heal through callus formation and subsequent callus remodeling, just as a full fracture of a long bone would heal. Microfractures have a deleterious effect on the mechanical properties of whole bone as they disrupt the connectivity of the trabecular lattice.

Thus, fatigue damage is not a narrowly defined quantity in either bone or other materials. In the engineering literature, it has usually been defined theoretically as a parameter that varies from zero in virgin structures to unity in failed structures, based on changes in mechanical properties, but without any specific reference to cracks or other evidence of physical damage, which may be difficult to detect. For example, damage is often defined as the fractional reduction in elastic modulus. Alternatively, when cracks are apparent, as in certain kinds of composite materials, damage may be defined in terms of the numbers or lengths of microscopic cracks that appear after loading.

Cracks are relatively easy to find in normal bone. Frost (1960) first described such damage, and linked it to fatigue. Burr and Stafford (1990) counted 0.14 such cracks/mm² in the ribs of a 60-year-old man. The number of such cracks increases with age in the human femoral diaphysis and neck cortex (Schaffler et al. 1995), and in the trabecular bone of the femoral head (Mori et al. 1997). Microcracks accumulate exponentially after age 40 (Schaffler et al. 1995). Although the data suggest increased skeletal fragility caused by microcrack damage in aging women, they do not show why some older women fracture and others do not. Mori et al. (1997) found that older women have significantly more microcracks than younger women, but older women who fracture did not have more microcracks than older women who did not fracture.

Microdamage burdens may be higher in trabecular than cortical bone; Wenzel et al. (1996) found approximately 5 microcracks/mm² in trabecular bone from the spine, without a significant increase with age. In the spine, disk pathology may contribute to microdamage (Hasegawa et al. 1995), either because loads are improperly attenuated or because stress concentrations are created in the vertebral body.

If linear microcracks and diffuse damage both perform the function of stress relief, then why are two different kinds of damage created? This probably has to do with the nature of the stresses, and with the quality of the bone matrix. Linear microcracks are preferentially found in areas of higher compressive stress (Diab and Vashishth 2005), are associated with greater matrix deformation before the cracking occurs (Diab et al. 2006) and may be associated with shear strains/stresses in the matrix caused by the compression (Yeni et al. 2003). About 80–90 % of all linear microcracks in cortical bone are found in the interstitial matrix between osteons where mineralization is high (Schaffler et al. 1995; Norman and Wang 1997; Wasserman et al. 2005). Also, because bone from older individuals tends to be more highly cross-linked and may be more fully mineralized, linear microcracks are found more commonly in older people than younger people, and tend to be on average three times longer (136 μm in older people vs. 63 μm in younger folks) (Diab et al. 2006). This reduces the fatigue life of bone by about 75 % between the ages of 40 and 80 years old. Diffuse damage, on the other hand, is more likely to be found in areas of high tensile stress, and more often seen than linear microcracks in bone from younger people (Diab and Vashishth 2005; Diab et al. 2006). Although stress is dissipated by linear microcracks that are attracted to and trapped by lamellar interfaces, the stress reduction associated with larger areas of smaller cracks initially increases the toughness of bone (Parsamian and Norman 2001), and is likely to be more effective in resisting a catastrophic fracture (Green et al. 2011). When linear cracks form, on the contrary, residual toughness declines. Although some have suggested that linear cracks are simply the natural sequelae to diffuse damage (Parsamian and Norman 2001; Vashishth et al. 1997), the very specific age- location- and stress-associated patterns of these two types of damage (Boyce et al. 1998) make it more likely that they are very distinct forms of damage that are not related by any temporal sequence.

The resistance of any material to fatigue failure is a function of its resistance to both the initiation and propagation of cracks. These two factors are in turn highly dependent on the microstructural characteristics of the material. Materials that resist the initiation of microcracks often cannot effectively resist their propagation; on the other hand, those materials in which cracks are easily started are often difficult to propagate a crack through. Because the fatigue life of a material is a function of both crack nucleation and growth, a material in which cracks are easily started, but find it difficult to grow, can often provide greater resistance to fatigue failure than one in which cracks are difficult to initiate. In fact, many highly fatigue-resistant materials derive this property primarily from their resistance to crack growth rather than initiation. Bone is such a material.