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Micromechanics of bone strength and fracture
J. B,omechanr~s Prmled
Vol. 26. No. 4/5. pp 439
446, 1993
‘r
m Great Britarn
MICROMECHANICS
OF BONE STRENGTH
0021 9290193 S6.0+.00 1993 Pergamon Press Ltd
AND FRACTURE
JOSEPH F. MAMMONE* and SAMUEL M. HUDSON~ *Department of Radiology, Thomas Jefferson University Hospital. Philadelphia, PA 19107, U.S.A.; and tFiber and Polymer Science Program, Box 8301. Centennial Campus, North Carolina State University. Raleigh,
NC 27695-8301,
U.S.A.
Abstract-The mechanical properties of bone were modeled in the context of a filled polymeric composite containing a collagenous matrix and a hydroxyapatite filler. The longitudinal and transverse moduli of cortical bone as a composite with perfect alignment of filler particles were calculated to be 34.5 and 5.3 GPa, respectively. When considering that particle orientation is arranged within a distribution about the long axis. moduli close to the experimentally measured values are achieved. The calculated tensile strength of I.7 GPa is higher than the experimental values, which may be attributable to intrinsic sample flaws and biological heterogeneity. The mode of tensile failure in this model is particle-matrix debonding, which may explain fatigue or stress fractures. Overall, the filled composite model of bone helps explain the roles of mineralization fraction, particle shape and orientation, and other attributes of the constituent phases in understanding the tensile properties. The fundamentals of bone behavior in compression are less well understood. It is proposed that incorporation of an inorganic phase in bone was teleologically necessary for vertebrates to achieve adequate levels of compressive strength.
INTRODUCTION The mechanical properties of bone have significant clinical importance, especially in understanding fracture behavior as a function of mineralization. Osteoporosis and consequent fractures are a major cause of disability and morbidity in the elderly. If we can obtain an insight into the determinants of bone strength, then better methods to image, assess, and treat patients with abnormal bone can be identified. The composition of bone tissue is extremely complex compared to most engineering composites. The organization of bone within a Haversian system consists of a central canal surrounded by concentric lamella. Lamellae are observed at the level of a light microscope. A number of attempts have been made to describe the biomechanical properties of bone at this level (Ascenzi, 1988), but this description does not go beyond the histological level. Others have simply studied the forces in bone macroscopically as exemplified in models using finite element analysis (Manhanian and Piziali, 1988). If one considers the elements that comprise bone at the molecular level, namely, the collagenous matrix and hydroxyapatite crystals, then a more fundamental understanding may be achieved. The view that bone may be considered as a two-phase composite to explain its mechanical properties was first suggested by Currey (1964). This initial work applied the simple rule of mixtures where the modulus of the composite is calculated by averaging the proportional contributions of each phase. This rule is useful in estimating the modulus of unidirectional continuous fiber composites but is not considered a proper model for bone. Katz (1977,1980,
Accepted in jinal form 2 November 1992. 439
1981) developed a model of Haversian bone such that individual osteons are treated as circular coherent cylindrical entities packed in a hexagonal manner. The osteons are built up of concentric lamellae containing ground substance and hydroxyapatite. One of the objectives in this model was to account for the anisotropy in the modulus of bone. The smallest elements that are treated are those that are observed microscopically. There are, however, interactions at the molecular level that may be important in determining the mechanical properties of bone. These interactions would take place within each Iamella. Our approach in this investigation is to apply modern composite theory to the smallest constituent microstructural elements that make up bone and treat them as a filled polymeric composite. The structure function relationships in bone are addressed at essentially the level of the extracellular matrix. The mechanical behavior of a composite at this level is a function of the properties of the individual phases (matrix and filler), their interaction, and adhesion between the two. Other key parameters are crystal size, filler aspect ratio, and volume fraction of mineralization. Such information may be useful in describing the mechanism of fracture and fatigue phenomena. The complex nature of bone microstructure may necessitate that, ultimately, some combination of mechanical models will be needed. An understanding at the molecular level may also provide useful information for subsequent histologic-based models. While modern composite theories are useful in understanding the tensile properties of bone, our theoretical knowledge of compressive strength is rather elementary. Euler described the buckling of a thin strut under compression, but no adequate description has been developed for composites. However, compressive strength represents a crucial functional aspect
J. F. MAMMONEand S. M.
440
in bone. We propose that bone was teleologically designed to solve the problem of compressive strength. THEORETICAL
METHODS
An idealized composite stress-strain relationship under tension is depicted in Fig. 1. At low levels of tensile strain, stress is proportional to the strain and the constant of proportionality is known as Young’s modulus. Lower levels of strain represent the elastic or ‘spring-like’ region before any irreversible damage is induced. At higher levels of strain the yield point is reached, which represents the onset of plastic deformation. In bone, it occurs at a strain level of approximately 1% (Crowninshield and Pope, 1974). In the zone of plastic deformation, a composite is able to absorb energy via matrix-filler debonding before ultimate failure. For a filled polymeric composite, interfacial debonding represents the primary mode of failure (Trachte and Dibenedetto, 1971). This is most likely an important mechanism in bone, although failure can take place also between larger elements, such as those at the lamella level. The critical strain release rate or energy required to extend a crack exhibits a positive correlation with mineralization (Wright and Hayes, 1977), implying that increased levels of particle-matrix bonding results in increased fracture energy. In engineering composites, fracture toughness is increased by increasing the filler concentration or by improved adhesion (Trachte and Dibenedetto, 1971; Jancar and Kucera, 1990). The stiffness, toughness, and compressive strength of a polymeric solid such as collagen can be enhanced by incorporation of a second hard inorganic phase. For spherically symmetric inclusions, predictions of stiffness based on Kerner’s (1956) equations have been applicable over a wide range of volume fraction of the inorganic phase. In this theory, the properties of a matrix phase are changed by the presence of a particle with different properties. Direct application of Kerner’s equations to bone would be limited because
HUDSON
the hydroxyapatite inclusions are not spherical. For ellipsoidal fillers, Eshelby (1957) has calculated the elastic field for a dilute suspension. Chow (1977, 1978, 1980) has subsequently generalized Eshelby’s theory to estimate the five elastic constants of a filled composite at a finite volume concentration of filler particles. Chow’s general theory reduces to Kerner’s equations for spherical fillers. Particle shape is characterized by the aspect ratio, p, which is the ratio of the major to minor axes. For a composite model of bone the theory is very useful because it addresses the principal relevant parameters such as filler dimension, interfacial debonding, elastic constants of the constituent phases, and volume fraction of filler or mineralization. Following Chow, in a longitudinally oriented composite where the filling particles are firmly bonded to the matrix, Young’s modulus is given by
where E,, is the longitudinal modulus of the composite, E, is the matrix modulus, k is the bulk modulus, p is the shear modulus, and 4 is the volume fraction of filler. The subscripts f and m refer to the filler and matrix, respectively. & and Gj are functions of k, p, 0, and Poisson’s ratio of the matrix and are defined in the Appendix. The filling particles are aligned along the long axis but are randomly distributed and, thus, are transversely isotropic. The experimental data of Reilly and Burstein (1975) also support a transversely isotropic bone model. Because the composite strain is low with respect to the matrix phase, it does not carry much tensile load and transfers primarily shear stress to the high-modulus filler particles. Alternatively, this could be viewed as a function of the difference in modulus between the organic and inorganic phase. The transverse composite modulus, E9,,, in this model is given as
$=1+ m
(k,lk,-l)G,+(~,/~,-l)(G,C+K,I)~ 2KlG3+G1K3
(2) FailiJre Point
STRAIN
Fig. 1. Idealized stress-strain curve for bone.
Definitions of 5 and c are provided in the Appendix. The model can be further extended to predict the ultimate tensile strength (Chow, 1982). The ratio of the maximum stress within the composite to the applied stress is denoted by the stress concentration factor. If this local stress is sufficient, it may lead to evolution of a defect and subsequent failure of the composite. For aligned reinforcing particles, the stress concentration factor is highest at the ends of the particles, which is where an adhesion failure may take place. For a crack to form, the strain energy must exceed the energy of debonding (that required to create new surfaces of matrix and filler). Thus, WayA,
(3)
where y is the work of adhesion and A is the surface
Micromechanics of bone strength and fracture area created. From calculation of the strain energy required for crack formation, Chow (1982) derives a critical particle size, d,. Inclusions larger than d, will provide flaws, or stress concentrations greater than the inherent flaw size and reduce the ultimate strain. The critical size is given as d,= 12-JE,e:[E,],
(4)
where e, is the ultimate strain of the matrix, and [Eo] is the effective Young’s modulus of the composite: [E,,, =v.
m
The ultimate tensile strength at this point can then be calculated and is given by Chow (1982) as
The inverse square-root dependence of strength on particle size has been previously observed both theoretically (Nicholson, 1979) and experimentally (Leidner and Woodhams, 1974). Behavior under compression is a more complicated problem. No adequate theory has yet been developed to model compression in a filled composite. Before computation of Young’s modulus from equations (1) and (2) data on the constituent phases of bone need to be defined. Properties of hydroxyapatite crystals were reported by Katz and Ukraincik (1971). The bulk modulus is given as 89 GPa, the shear modulus is 44.5 GPa, and Poissons’ ratio is 0.22. The true modulus of the collagenous matrix is not known; however, Currey (1969) has estimated it to be approximately 1.47 GPa. This is a reasonable value, considering that the moduli of highly crosslinked aromatic epoxy resin systems range between 2.5 and 3.5 GPa. Poisson’s ratio for the organic matrix may be assumed to be 0.35, which is close to the value of many polymeric systems. For calculation purposes, the bulk modulus, k, and shear modulus, p, of collagen may be estimated from Young’s modulus, E, and Poissons ratio, v, using known identities for isotropic solids:
k=E 3(1-v)’ E p=2f2v. The filled composite model approximates cortical bone most closely, where there is a maximum concentration of hydroxyapatite and collagen matrix and a lack of fat and hematopoietic elements. In cortical bone, the volume fraction of filler or percent mineralization is approximately 0.65 (Currey, 1975). Paracrystalline aggregates of hydroxyapatite from bone have an aspect ratio of approximately 10, with a width of 4.0-4.5 nm and length of 40 nm (Ascenzi et al., 1978). For calculation of the tensile strength, an estimate of the work of adhesion is needed. In the case of
441
complete particle wetting, this should be close to the surface energy of hydroxyapatite. From Kelly (1973), the surface free energy of a solid, y, is estimated to be y = Ea,/lO,
(10)
where E is Young’s modulus and a, is the equilibrium separation between atomic planes. For hydroxyapatite, given that a,=0.94 nm and E= 114 GPa (Katz and Ukraincik, 1971), y is estimated to be lo4 ergcm-‘. The critical filler dimension in bone is estimated from equation (4). Equation (4) requires an estimate for the ultimate strain of the collagen matrix. While this is unknown, the calculation may be performed using a realistic range of 0.14.5 strain. Assuming a value of lo4 erg cm-’ for the work of adhesion, the critical diameter is approximately 2-5 pm for a matrix strain of 0.1-0.5. Thus, the effect of particle size is felt only when there are inclusions larger than this critical value. The measured sizes of hydroxyapatite are well below this value and should, therefore, not degrade bone tensile strength.
RESULTS
Longitudinal Young’s modulus may now be calculated from equation (1) as a function of mineralization. For particle aspect ratios of 1, 5, and 10, the results are plotted in Fig. 2. The transverse modulus is similarly calculated from equation (2) for particle aspect ratios of 1 and 10 and the results are plotted in Fig. 3. Both the longitudinal and transverse moduli are enhanced by the incorporation of an inorganic filler. The longitudinal modulus also increases with increasing aspect ratio of the filler particles. Note that, when the aspect ratio is one (i.e. for spherical fillers), the longitudinal and transverse moduli are equal. This is also the case described in the original Kerner formula. The calculated longitudinal and transverse moduli for cortical bone are 34.5 and 5.3 GPa, respectively. These results are obtained for an average mineralization of 0.65 and particle aspect ratio of 10, with alignment of all particles in the longitudinal direction. The ultimate tensile strength of bone is calculated from equation (6). For the same aspect ratio of 10 and percentage mineralization of 0.65 the tensile strength is 1.7 GPa.
DISCUSSION
The structural requirements of vertebrate bone include tensile and compressive strength, stiffness, and toughness with minimum weight of its components. With its incorporation of a mineral phase into a filled polymeric composite, bone is a unique material in nature. We have applied modern composite theories to understand bone mechanics at essentially the level of the extracellular matrix. The model provides an
J. F. MAMMONEand S. M. HUDSON
442
til I
0.8 Mineralization Fraction
mineralization and aspect ratio is clearly illustrated.
m 8 8-2- 7-.
1 10
l!
0
I 0.4
I 0.2
I 0.6
1 0.8
Mineralization Fraction Fig. 3. Calculated transverse moduli of cortical bone as a filled composite with all particles aligned with the long axis as a function of volume fraction of filler for particle aspect ratios of 1 and 10. Perpendicular to the strain direction, the reinforcing effect is not as efficient as in the parallel direction.
insight into the contributions of the properties of the constituent phases, mineralization, interfacial adhesion, and filler particle shape and orientation. The results for the longitudinal modulus of cortical bone can be viewed more realistically by considering that not all hydroxyapatite crystals are perfectly aligned and there is some distribution of directions about the long axis. Currey (1969) has introduced a misalignment correction, 0, such that the estimated modulus, Eg, varies as the crystals deviate from the long axis by an angle 19.This equation is given as F=cos40+s e
0
-
I
20
-
I
40
-
I
60
*
I
1
80
Misalignment Angle
Fig. 2. Calculated longitudinal moduli of cortical bone as a filled composite with all particles aligned with the long axis as a function of volume fraction of filler for particle aspect ratios of 1, 5, and 10. The reinforcing effect of increasing
t=
o!
sin40 Ego
where E. and Ego are the longitudinal and transverse modulus, respectively, given perfect orientation, p is the bone shear modulus, and v is Poisson’s ratio.
Fig. 4. Calculated longitudinal Young’s modulus for cortical bone as a filled composite with a fractional mineralization of 0.65 and hydroxyapatite ellipsoids with an aspect ratio of 10 as a function of the angle of orientation of the filler with the long axis. Most bone crystallites align within 30” of the longitudinal axis (Sasaki et al., 1989), yielding effective modulus values within the experimentally observed range.
Given the average experimental values of the shear modulus and Poisson’s ratio to be 5.5 GPa and 0.35, respectively, the contribution to longitudinal Young’s modulus for cortical bone as a filled composite with a fractional mineralization of 0.65 and filler aspect ratio of 10 is plotted in Fig. 4 as a function of the angle of particle orientation. The calculated values for the moduli of cortical bone as a filled composite are compared with the experimentally measured values in Table 1. For perfect alignment, the calculated value for Young’s modulus is higher than the experimentally reported values. This is understandable because, in cortical bone, not all crystals are aligned along the long axis and there is a distribution of orientations. When the misalignment angle increases to 20”, values consistent with experiment are achieved. If an exact distribution function of hydroxyapatite orientation were known, integration with equations (1) (2), and (9) could provide a true estimate of the modulus of cortical bone in the context of a filled composite. Sasaki et al. (1989) have investigated crystal orientation by small-angle X-ray scattering and X-ray pole figure analysis. They reported that hydroxyapatite generally orients parallel to the long axis but there is a significant amount of orientation in other directions. In general, the number of crystals oriented in any particular direction decreases with increasing deflection from the long axis. The large majority of crystals are oriented within plus or minus 30” of the long axis. Chen and Gundjian (1974) have also reported measuring a distribution function with a width on the order of 30”. Given these degrees of misalignment of hydroxyapatite crystals, correlation with Fig. 4 and Table 1 shows that estimates of Young’s modulus are obtained in the range of experimentally measured values. The calculated transverse modulus of 5.3 GPa for a composite containing perfectly aligned filler particles is lower than that observed experimentally
Micromechanics
Table 1. Comparison
model*
and Raffopoulos
(1990)
Schaffler and Burr (1988) Vincentelli and Grigorov (1985) Ashman Lipson
et al. (1984) and Katz (1984)
Bonfield and Grynpas (1977) Currey (1975) Reilly and Burstein (1975)
*Mineralization
fraction
Table
of 0.65 and particle
aspect
17 15 9.5 11.5
Martin and Ishida (1989) Vincentelli and Grigorov (1985) Currey (1975) Reilly and Burstein (1975) Burstein et al. (1972)
(Table 1). The low value is due to the fact that, in the model, there are no particles aligned in the transverse dimension. In reality, however, there are a small number of hydroxyapatite crystals oriented perpendicular to the long axis (Sasaki et al., 1989), providing additional stiffness in this dimension. The calculated tensile strength of cortical bone with a particle aspect ratio of 10 and mineralization of 0.65 is 1.7 GPa. A comparison of the calculated value with the various experimentally measured values is given in Table 2. The predicted ultimate tensile strength based on a filled polymeric composite model is much higher than that observed experimentally for cortical bone. This is not entirely surprising as bone contains numerous mechanical flaws such as pores for nutrient vessels that would lower the measured ultimate strength. In general, experimental data in the literature for tensile strength have not been reproducible. This variability may be attributed to bone’s inherent anisotropy in properties, and biological heterogen-
bone as
Sample 0’ Misalignment 10; 20 30” Human femur
Human femur Canine femur Plexiform bovine femur Haversian bovine femur Bovine femur Bovine femur Human femur Human femur, Haversian
ratio of 10.
Ultimate tensile strength (MPa) model
of cortical
Bovine femur Human tibia
2. Comparison between the calculated tensile a filled composite and various experimentally
Study Present
5.3
34.5 31.2 23.9 17.1 16.2 (Static) 19.9 (Dynamic) 22.1 19.7 18.0 20.0 20.1 30 25 18.5 22.7 17.0 22.7
443
and transverse moduli determined values
Transverse modulus (GPa)
Longitudinal modulus (GPa)
Katsamanis
and fracture
between the calculated values of the longitudinal a filled composite and various experimentally
Study Present
of bone strength
1700 106 162 133 125 133 175
strength of cortical determined values
bone
as
Sample Aspect ratio of 10 Mineralization of 0.65 Bovine femur Human tibia Bovine femur Human femur Bovine femur
eity. The effects of sample preparation, introduction of flaws, and testing procedures also affect the fracture data. In an analysis of compact bone from a wide variety of species, ultimate tensile strength could not definitely be related to any combination of parameters (Currey, 1990). At the same time, other authors have been able to correlate tensile strength with certain parameters (Rice et al., 1988). Young’s modulus, which reflects the stiffness of bone, is enhanced by the addition of an inorganic filler. The filled polymeric composite model is successful in predicting the elastic moduli and accounting for their anisotropy (Table l), but overestimates the tensile strength (Table 2). This pattern is typical of calculations based on a molecular model. At low strain levels, Young’s modulus is nearly a direct function of interactions at the molecular level. The presence of flaws, for example, does not necessarily affect the measured values of the initial modulus. Thus, while the model is a simplification of true bone tissue, in
444
J. F. MAMMONE and S. M. HUDSON
that it does not consider superstructure or histology, reasonably satisfactory values of Young’s modulus are obtained. This is because the determinants of Young’s modulus in bone are at the microlevel. With other materials, such as fibers, or structural composites, similar success with molecular models is achievable in the calculation of Young’s modulus. The estimation of tensile strength is not as straightforward. For example, when the theoretical tensile strength of fibers is computed on a molecular basis by using the energy required to break covalent bonds, it is overestimated by at least an order of magnitude. This is especially true in ceramic fibers where the presence of various flaws can affect dramatically the strength but not the modulus. The overestimation of bone tensile strength still indicates that key determinants of bone strength will also include flaws and tissue structure. Tensile strength is enhanced by the addition of an inorganic filler, but not nearly as much as compressive strength. Ultimate tensile strength in bone is not as crucial, from a functional standpoint, as compressive strength. Tensile loads within the body can be borne by other structural elements such as tendons and ligaments. In fact, as Currey (1984) has pointed out, the skeletal system is designed to minimize tensile stresses. Still, bone tensile strength is important in several situations. One is at tendinous and ligamentous insertions. At these points bone responds by increasing its cross-sectional area in the form of tubercles and tuberosities, thereby reducing the overall tensile stress. Tensile failure at these locations results in avulsion fractures. Other loading situations where tension is important are in torsion and bending. When bone is subjected to a bending stress, the convex or outer surface experiences a tensile stress, while the concave or inner surface is loaded in compression. In bone, the side in tension fails first, producing a transverse fracture. In an organic solid or one at low levels of mineralization, tensile strength is greater than compressive strength. Increasing mineralization improves the compressive strength of bone such that it exceeds tensile strength (Kaplan et al., 1985). An organicbased composite would fail first on the compressive side. Fatigue or stress fractures may occur when bone is stressed beyond the yield point repeatedly but before ultimate failure. Thus, the fracture may not be visualized macroscopically. The energy absorbed by bone may result in a significant amount of particle-matrix debonding. While usually not apparent radiographitally, these fractures are positive on bone scanning with uptake of the radiotracer technetium methylene diphosphonate. This agent localizes in bone via interaction on the exposed hydroxyapatite crystal surface to produce insoluble technetium calcium phosphate complexes (Pendergrass et al., 1973). Although nonspecific, these observations are consistent with a filled polymeric composite model of bone where the primary mechanism of failure is interfacial debonding.
This concept is supported indirectly in the composite literature, where it is generally known that the adhesion between the matrix and the filler affects the strength of a filled composite greatly (Trachte and Dibenedetto, 1971; Han et al., 1978; Jancar and Kucera, 1990). Direct observations of debonding at the crack tip have been made in epoxy resin with glass filler particles (Owen, 1979). In vivo fatigue testing on bone results have yielded a failure mechanism compatible with diffuse microcracking and debonding (Forwood and Parker, 1989). Interfacial debonding may, therefore, be considered as an important mechanism of failure. It has also been suggested that partial debonding of hydroxyapatite from the collagen matrix may be responsible for the degradation of the mechanical properties in aging and certain disease states (Bundy, 1985). Compressive strength is a key in vivo requirement in bone. The most important determinant of compressive strength is the degree of mineralization. The qualitative loss of mineralization results in the wellknown increased risk of fracture, whereas increased mineral density is seen to provide increased compressive strength (Lotz et al., 1990). As with tensile strength, compressive fracture data are extremely variable. Compressive testing of composites is extremely sensitive to testing procedures, even more so than in tension (Mammone and Uy, 1984). In bone, significant differences have been noted by simply improving the test coupon end constraint (Linde and Hvid, 1989.) The mechanics of compression are extremely complicated and no adequate model for a filled polymeric composite has been formulated as yet. An understanding of the microstructural requirements for compressive strength provides an insight as to why nature was forced to develop such a unique material like bone. Aside from bone, organisms provide for all of their functions with organic compounds and polymers (proteins, carbohydrates, fats). The genetic code is programmed directly to synthesize proteins. No conceivable protein or solely organic structure, however, can provide the compressive strength required by vertebrates for their increased size and mobility. Man has fabricated advanced polymeric composites that have extremely high stiffness and tensile strength but they still lack adequate compressive strength. The failure of organic-based materials in compression is due to the fact that they buckle early at the molecular level when loaded in compression (Mammone and Uy, 1984). There is little intermolecular support as the predominant forces are relatively weak dipole-dipole and van der Waals interactions. A covalent, polar-covalent, or ionic network-like molecular lattice that exists in inorganic oxides like hydroxyapatite is required to SUStain compressive loads. It had been suggested that increased crosslink density may be the mechanism of property enhancement in bone (Lees and Davidson, 1977). Although increased crosslinking will increase
445
Micromechanics of bone strength and fracture the modulus, it has been shown to have little effect on advanced organic fibers and simply tends to make the material more brittle (Mammone and Uy, 1984). Currey (1984) has felt that the most important feature of bone is its stiffness. While it is clear that stiffness is an essential mechanical function of bone, incorporation of an inorganic phase may not be necessary to achieve it. There are many man-made organic fibers with high stiffness as well as fibers found in nature (e.g. spider silk). A composite of such fibers would exhibit very high stiffness. What it would lack is compressive strength. There is also the nontrivial problem of achieving adequate fiber-matrix adhesion which has been a thorny problem with most high-performance organic fibers. This dilemma is similar to the problem confronting aerospace engineers using lightweight high-strength, high-modulus organic fibers. The composites exhibit high tensile strength and stiffness but their major drawback is inadequate compressive strength. Adequate compressive and tensile properties generally represent absolute design prerequisites for composites in structural applications. They are easy to measure and understand. In practice, bone tissue is subjected to more complicated stresses than just tension and compression. Three other clinically important mechanisms involved in fracture production are tension, shear. and bending. In composite design, these three mechanisms are generally addressed macroscopically by adjusting the spatial relationships of the reinforcing materials after compressive and tensile properties are optimized. Improved adhesion has been used in composites to enhance the interlaminar shear properties. The advantage of organic polymers, including proteins, is that they are lightweight, which is a desirable design characteristic from a mechanical standpoint. Inorganic materials are heavy, but have higher compressive strengths. Thus, nature solved its structural problems by fabricating a composite. It added enough inorganic filler to provide the compressive strength required but kept bone relatively light by employing a two-phase composite concept. Man too has had to resort to hybridization with inorganics to improve compressive properties in his most advanced composite materials (Mammone, 1986). In summary, the properties of the microstructural elements in bone may be explained in the context of a filled polymeric composite. The theory affords satisfactory modeling of tensile properties and incorporates the influence of key variables. In this model, the mode of failure is particle-matrix debonding, which can also explain some fatigue or stress fractures. Our subsequent work will be to fabricate hydroxyapatitefilled composites, examine tensile and compressive properties as a function of mineralization and try to understand further the mechanisms of failure. We have also tried to show that bone is a uniquely designed material in nature with its use of a solid inorganic phase. The proposed teleological purpose is BM26:4/5-F
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Kelly, A. (1973) Strong Solids, 2nd Edn. Clarendon Press, Oxford. Kerner, E. H. (1956) Elastic and thermoelastic properties of composite media. Proc. Phys. Sot. (Lond.) 69B, 808-813. Lees, S. and Davidson, C. L. (1977) The role of collagen in the elastic properties of calcified tissues. J. Biomechanics 10,473-486. Leidner, J. and Woodhams, R. T. (1974) Strength of polymeric composites containing spherical fillers. J. appl. Polymer Sci. 18, 1639-1654. Linde, F. and Hvid, I. (1989) The effect of constraint on the mechanical behavior of rrabecular bone specimens.
in the anisotropic mechanical properties of bone-transverse anisotropy. J. Biomechanics 22, 157-164. Schaffler, M. B. and Burr, D. B. (1988) Stiffness of compact bone: effects of porosity and density, J. Biomechanics 21, 13-16.
Trachte, K. L. and Dibenedetto, A. T. (1971) Fracture properties of poly (phenylene oxide) composites. Int. J. Poly. Mater. 1, 75-94. Vincentelli, R. and Grigorov, M. (1985) The effect of Haversian remodeling on the tensile properties of human cortical bone. J. Biomechanics 1% 201-207. Wright, T. M. and Hayes, W. C. (1977) Fracture mechanics parameters for compact bone-effects of density and specimen thickness. J. Biomechanics 10, 419-430.
APPENDIX The pertinent parameters as described by Chow (1977, 1978, 1980) are listed below. The filler aspect ratio is given as p and the mineralization fraction as 4. The subscripts f and m refer to the filler and matrix, respectively. &=l
-&ai
+(k,/k,-l)(l
J. Biomechanics 22,485-490.
Lipson, S. F. and Katz, J. L. (1984) The relationship between elastic properties and the microstructure of bovine cortical bone. J. Biomechanics 17, 231-240. Lotz, J. C., Gerhart, T. N. and Hayes, W. C. (1990) Mechanical properties of trabecular bone from the proximal femur: a quantitative CT study. J. Comput. Assist. Totnogr. 14, 107-l 14. Mammone, J. F. (1986) Hybrid composite of poly (p-phenylene-transbisbenzothiazole) and ceramic fiber. U.S. Patent Number 4,571,411. Mammone, J. F. and Uy, W. C. (1984) Exploratory development of high strength, high modulus polybenzothiazole fibers. Part II. Air Force Technical Report AFWAL-TR82-4154, Part II. Manhanian, S. and Piziali, R. L. (1988) Finite element evaluation of the AIA shear specimen for bone. J. Biomechnnics
(i= 1,3), Gi= 1+(/4/p,-
l)(l-4%
a, =4nQ/3-2(2rr-I)R, a,=4nQ/3+4(1-z)R, b1
=(+-3)
&=
Q-4(1-2n)R,
$_(4ql;;p’ (
>
2S,z,z=+;
Q+(4n-I)R,
s)
Q+21R,
s ,,,,-s,,,,=(niiZ~)Q+(4n-I)R.
21, 347-356.
Martin, R. B. and Ishida, J. (1989) The relative effects of collagen fiber orientation, porosity, density, and mineralization on bone strength. J. Biomechanics 22, 419-426. Nicholson, D. W. (1979) On the detachment of a rigid inclusion from an elastic matrix. J. Adhesion 10, 2551260. Owen. A. B. (1979) Direct observations of debondinpc at cradk tips inglass’bead-filled epoxy. J. Mater. Sci. Leetry14,
where
Q=&, m
and ccos-’ p-p(1
2521-2523.
Pendergrass, H. P., Potsaid, M. S. and Castronovo, F. P. (1973) The clinical use of 99mTc-diphosphonate (HEDSPA). A new agent for skeletal imaging. Radiology 107, 557-562. Reilly, D. T. and Burstein, A. H. (1975) The elastic and ultimate properties of compact bone tissue. J. Bio-
-p*p*1
2np[p(p2-l)1’2-cosh-‘p]
for pl.
When p= 1, I = 4n/3,
mechanics 8, 393-405.
Rice, J. C., Cowin, S. C. and Bowman, J. A. (1988) On the dependence of the elasticity and strength of cancellous bone on apparent density. J. Biomechanics 21, 155-168. Sasaki, N., Matsushima, N., Ikawa, T., Yamamura, H. and Fukuda, A. (1989) Orientation of bone mineral and its role
RJ_ 14Vm~ 8x l-v,,,’
5=
i=
K,
1+2(~,/~,-1)(1-~)s1212’ l+(P~/P~-l)(l-~)(s11*1-s33Il) 1+2(~f/~,-1)(1-~)s1212
Vol. 26. No. 4/5. pp 439
446, 1993
‘r
m Great Britarn
MICROMECHANICS
OF BONE STRENGTH
0021 9290193 S6.0+.00 1993 Pergamon Press Ltd
AND FRACTURE
JOSEPH F. MAMMONE* and SAMUEL M. HUDSON~ *Department of Radiology, Thomas Jefferson University Hospital. Philadelphia, PA 19107, U.S.A.; and tFiber and Polymer Science Program, Box 8301. Centennial Campus, North Carolina State University. Raleigh,
NC 27695-8301,
U.S.A.
Abstract-The mechanical properties of bone were modeled in the context of a filled polymeric composite containing a collagenous matrix and a hydroxyapatite filler. The longitudinal and transverse moduli of cortical bone as a composite with perfect alignment of filler particles were calculated to be 34.5 and 5.3 GPa, respectively. When considering that particle orientation is arranged within a distribution about the long axis. moduli close to the experimentally measured values are achieved. The calculated tensile strength of I.7 GPa is higher than the experimental values, which may be attributable to intrinsic sample flaws and biological heterogeneity. The mode of tensile failure in this model is particle-matrix debonding, which may explain fatigue or stress fractures. Overall, the filled composite model of bone helps explain the roles of mineralization fraction, particle shape and orientation, and other attributes of the constituent phases in understanding the tensile properties. The fundamentals of bone behavior in compression are less well understood. It is proposed that incorporation of an inorganic phase in bone was teleologically necessary for vertebrates to achieve adequate levels of compressive strength.
INTRODUCTION The mechanical properties of bone have significant clinical importance, especially in understanding fracture behavior as a function of mineralization. Osteoporosis and consequent fractures are a major cause of disability and morbidity in the elderly. If we can obtain an insight into the determinants of bone strength, then better methods to image, assess, and treat patients with abnormal bone can be identified. The composition of bone tissue is extremely complex compared to most engineering composites. The organization of bone within a Haversian system consists of a central canal surrounded by concentric lamella. Lamellae are observed at the level of a light microscope. A number of attempts have been made to describe the biomechanical properties of bone at this level (Ascenzi, 1988), but this description does not go beyond the histological level. Others have simply studied the forces in bone macroscopically as exemplified in models using finite element analysis (Manhanian and Piziali, 1988). If one considers the elements that comprise bone at the molecular level, namely, the collagenous matrix and hydroxyapatite crystals, then a more fundamental understanding may be achieved. The view that bone may be considered as a two-phase composite to explain its mechanical properties was first suggested by Currey (1964). This initial work applied the simple rule of mixtures where the modulus of the composite is calculated by averaging the proportional contributions of each phase. This rule is useful in estimating the modulus of unidirectional continuous fiber composites but is not considered a proper model for bone. Katz (1977,1980,
Accepted in jinal form 2 November 1992. 439
1981) developed a model of Haversian bone such that individual osteons are treated as circular coherent cylindrical entities packed in a hexagonal manner. The osteons are built up of concentric lamellae containing ground substance and hydroxyapatite. One of the objectives in this model was to account for the anisotropy in the modulus of bone. The smallest elements that are treated are those that are observed microscopically. There are, however, interactions at the molecular level that may be important in determining the mechanical properties of bone. These interactions would take place within each Iamella. Our approach in this investigation is to apply modern composite theory to the smallest constituent microstructural elements that make up bone and treat them as a filled polymeric composite. The structure function relationships in bone are addressed at essentially the level of the extracellular matrix. The mechanical behavior of a composite at this level is a function of the properties of the individual phases (matrix and filler), their interaction, and adhesion between the two. Other key parameters are crystal size, filler aspect ratio, and volume fraction of mineralization. Such information may be useful in describing the mechanism of fracture and fatigue phenomena. The complex nature of bone microstructure may necessitate that, ultimately, some combination of mechanical models will be needed. An understanding at the molecular level may also provide useful information for subsequent histologic-based models. While modern composite theories are useful in understanding the tensile properties of bone, our theoretical knowledge of compressive strength is rather elementary. Euler described the buckling of a thin strut under compression, but no adequate description has been developed for composites. However, compressive strength represents a crucial functional aspect
J. F. MAMMONEand S. M.
440
in bone. We propose that bone was teleologically designed to solve the problem of compressive strength. THEORETICAL
METHODS
An idealized composite stress-strain relationship under tension is depicted in Fig. 1. At low levels of tensile strain, stress is proportional to the strain and the constant of proportionality is known as Young’s modulus. Lower levels of strain represent the elastic or ‘spring-like’ region before any irreversible damage is induced. At higher levels of strain the yield point is reached, which represents the onset of plastic deformation. In bone, it occurs at a strain level of approximately 1% (Crowninshield and Pope, 1974). In the zone of plastic deformation, a composite is able to absorb energy via matrix-filler debonding before ultimate failure. For a filled polymeric composite, interfacial debonding represents the primary mode of failure (Trachte and Dibenedetto, 1971). This is most likely an important mechanism in bone, although failure can take place also between larger elements, such as those at the lamella level. The critical strain release rate or energy required to extend a crack exhibits a positive correlation with mineralization (Wright and Hayes, 1977), implying that increased levels of particle-matrix bonding results in increased fracture energy. In engineering composites, fracture toughness is increased by increasing the filler concentration or by improved adhesion (Trachte and Dibenedetto, 1971; Jancar and Kucera, 1990). The stiffness, toughness, and compressive strength of a polymeric solid such as collagen can be enhanced by incorporation of a second hard inorganic phase. For spherically symmetric inclusions, predictions of stiffness based on Kerner’s (1956) equations have been applicable over a wide range of volume fraction of the inorganic phase. In this theory, the properties of a matrix phase are changed by the presence of a particle with different properties. Direct application of Kerner’s equations to bone would be limited because
HUDSON
the hydroxyapatite inclusions are not spherical. For ellipsoidal fillers, Eshelby (1957) has calculated the elastic field for a dilute suspension. Chow (1977, 1978, 1980) has subsequently generalized Eshelby’s theory to estimate the five elastic constants of a filled composite at a finite volume concentration of filler particles. Chow’s general theory reduces to Kerner’s equations for spherical fillers. Particle shape is characterized by the aspect ratio, p, which is the ratio of the major to minor axes. For a composite model of bone the theory is very useful because it addresses the principal relevant parameters such as filler dimension, interfacial debonding, elastic constants of the constituent phases, and volume fraction of filler or mineralization. Following Chow, in a longitudinally oriented composite where the filling particles are firmly bonded to the matrix, Young’s modulus is given by
where E,, is the longitudinal modulus of the composite, E, is the matrix modulus, k is the bulk modulus, p is the shear modulus, and 4 is the volume fraction of filler. The subscripts f and m refer to the filler and matrix, respectively. & and Gj are functions of k, p, 0, and Poisson’s ratio of the matrix and are defined in the Appendix. The filling particles are aligned along the long axis but are randomly distributed and, thus, are transversely isotropic. The experimental data of Reilly and Burstein (1975) also support a transversely isotropic bone model. Because the composite strain is low with respect to the matrix phase, it does not carry much tensile load and transfers primarily shear stress to the high-modulus filler particles. Alternatively, this could be viewed as a function of the difference in modulus between the organic and inorganic phase. The transverse composite modulus, E9,,, in this model is given as
$=1+ m
(k,lk,-l)G,+(~,/~,-l)(G,C+K,I)~ 2KlG3+G1K3
(2) FailiJre Point
STRAIN
Fig. 1. Idealized stress-strain curve for bone.
Definitions of 5 and c are provided in the Appendix. The model can be further extended to predict the ultimate tensile strength (Chow, 1982). The ratio of the maximum stress within the composite to the applied stress is denoted by the stress concentration factor. If this local stress is sufficient, it may lead to evolution of a defect and subsequent failure of the composite. For aligned reinforcing particles, the stress concentration factor is highest at the ends of the particles, which is where an adhesion failure may take place. For a crack to form, the strain energy must exceed the energy of debonding (that required to create new surfaces of matrix and filler). Thus, WayA,
(3)
where y is the work of adhesion and A is the surface
Micromechanics of bone strength and fracture area created. From calculation of the strain energy required for crack formation, Chow (1982) derives a critical particle size, d,. Inclusions larger than d, will provide flaws, or stress concentrations greater than the inherent flaw size and reduce the ultimate strain. The critical size is given as d,= 12-JE,e:[E,],
(4)
where e, is the ultimate strain of the matrix, and [Eo] is the effective Young’s modulus of the composite: [E,,, =v.
m
The ultimate tensile strength at this point can then be calculated and is given by Chow (1982) as
The inverse square-root dependence of strength on particle size has been previously observed both theoretically (Nicholson, 1979) and experimentally (Leidner and Woodhams, 1974). Behavior under compression is a more complicated problem. No adequate theory has yet been developed to model compression in a filled composite. Before computation of Young’s modulus from equations (1) and (2) data on the constituent phases of bone need to be defined. Properties of hydroxyapatite crystals were reported by Katz and Ukraincik (1971). The bulk modulus is given as 89 GPa, the shear modulus is 44.5 GPa, and Poissons’ ratio is 0.22. The true modulus of the collagenous matrix is not known; however, Currey (1969) has estimated it to be approximately 1.47 GPa. This is a reasonable value, considering that the moduli of highly crosslinked aromatic epoxy resin systems range between 2.5 and 3.5 GPa. Poisson’s ratio for the organic matrix may be assumed to be 0.35, which is close to the value of many polymeric systems. For calculation purposes, the bulk modulus, k, and shear modulus, p, of collagen may be estimated from Young’s modulus, E, and Poissons ratio, v, using known identities for isotropic solids:
k=E 3(1-v)’ E p=2f2v. The filled composite model approximates cortical bone most closely, where there is a maximum concentration of hydroxyapatite and collagen matrix and a lack of fat and hematopoietic elements. In cortical bone, the volume fraction of filler or percent mineralization is approximately 0.65 (Currey, 1975). Paracrystalline aggregates of hydroxyapatite from bone have an aspect ratio of approximately 10, with a width of 4.0-4.5 nm and length of 40 nm (Ascenzi et al., 1978). For calculation of the tensile strength, an estimate of the work of adhesion is needed. In the case of
441
complete particle wetting, this should be close to the surface energy of hydroxyapatite. From Kelly (1973), the surface free energy of a solid, y, is estimated to be y = Ea,/lO,
(10)
where E is Young’s modulus and a, is the equilibrium separation between atomic planes. For hydroxyapatite, given that a,=0.94 nm and E= 114 GPa (Katz and Ukraincik, 1971), y is estimated to be lo4 ergcm-‘. The critical filler dimension in bone is estimated from equation (4). Equation (4) requires an estimate for the ultimate strain of the collagen matrix. While this is unknown, the calculation may be performed using a realistic range of 0.14.5 strain. Assuming a value of lo4 erg cm-’ for the work of adhesion, the critical diameter is approximately 2-5 pm for a matrix strain of 0.1-0.5. Thus, the effect of particle size is felt only when there are inclusions larger than this critical value. The measured sizes of hydroxyapatite are well below this value and should, therefore, not degrade bone tensile strength.
RESULTS
Longitudinal Young’s modulus may now be calculated from equation (1) as a function of mineralization. For particle aspect ratios of 1, 5, and 10, the results are plotted in Fig. 2. The transverse modulus is similarly calculated from equation (2) for particle aspect ratios of 1 and 10 and the results are plotted in Fig. 3. Both the longitudinal and transverse moduli are enhanced by the incorporation of an inorganic filler. The longitudinal modulus also increases with increasing aspect ratio of the filler particles. Note that, when the aspect ratio is one (i.e. for spherical fillers), the longitudinal and transverse moduli are equal. This is also the case described in the original Kerner formula. The calculated longitudinal and transverse moduli for cortical bone are 34.5 and 5.3 GPa, respectively. These results are obtained for an average mineralization of 0.65 and particle aspect ratio of 10, with alignment of all particles in the longitudinal direction. The ultimate tensile strength of bone is calculated from equation (6). For the same aspect ratio of 10 and percentage mineralization of 0.65 the tensile strength is 1.7 GPa.
DISCUSSION
The structural requirements of vertebrate bone include tensile and compressive strength, stiffness, and toughness with minimum weight of its components. With its incorporation of a mineral phase into a filled polymeric composite, bone is a unique material in nature. We have applied modern composite theories to understand bone mechanics at essentially the level of the extracellular matrix. The model provides an
J. F. MAMMONEand S. M. HUDSON
442
til I
0.8 Mineralization Fraction
mineralization and aspect ratio is clearly illustrated.
m 8 8-2- 7-.
1 10
l!
0
I 0.4
I 0.2
I 0.6
1 0.8
Mineralization Fraction Fig. 3. Calculated transverse moduli of cortical bone as a filled composite with all particles aligned with the long axis as a function of volume fraction of filler for particle aspect ratios of 1 and 10. Perpendicular to the strain direction, the reinforcing effect is not as efficient as in the parallel direction.
insight into the contributions of the properties of the constituent phases, mineralization, interfacial adhesion, and filler particle shape and orientation. The results for the longitudinal modulus of cortical bone can be viewed more realistically by considering that not all hydroxyapatite crystals are perfectly aligned and there is some distribution of directions about the long axis. Currey (1969) has introduced a misalignment correction, 0, such that the estimated modulus, Eg, varies as the crystals deviate from the long axis by an angle 19.This equation is given as F=cos40+s e
0
-
I
20
-
I
40
-
I
60
*
I
1
80
Misalignment Angle
Fig. 2. Calculated longitudinal moduli of cortical bone as a filled composite with all particles aligned with the long axis as a function of volume fraction of filler for particle aspect ratios of 1, 5, and 10. The reinforcing effect of increasing
t=
o!
sin40 Ego
where E. and Ego are the longitudinal and transverse modulus, respectively, given perfect orientation, p is the bone shear modulus, and v is Poisson’s ratio.
Fig. 4. Calculated longitudinal Young’s modulus for cortical bone as a filled composite with a fractional mineralization of 0.65 and hydroxyapatite ellipsoids with an aspect ratio of 10 as a function of the angle of orientation of the filler with the long axis. Most bone crystallites align within 30” of the longitudinal axis (Sasaki et al., 1989), yielding effective modulus values within the experimentally observed range.
Given the average experimental values of the shear modulus and Poisson’s ratio to be 5.5 GPa and 0.35, respectively, the contribution to longitudinal Young’s modulus for cortical bone as a filled composite with a fractional mineralization of 0.65 and filler aspect ratio of 10 is plotted in Fig. 4 as a function of the angle of particle orientation. The calculated values for the moduli of cortical bone as a filled composite are compared with the experimentally measured values in Table 1. For perfect alignment, the calculated value for Young’s modulus is higher than the experimentally reported values. This is understandable because, in cortical bone, not all crystals are aligned along the long axis and there is a distribution of orientations. When the misalignment angle increases to 20”, values consistent with experiment are achieved. If an exact distribution function of hydroxyapatite orientation were known, integration with equations (1) (2), and (9) could provide a true estimate of the modulus of cortical bone in the context of a filled composite. Sasaki et al. (1989) have investigated crystal orientation by small-angle X-ray scattering and X-ray pole figure analysis. They reported that hydroxyapatite generally orients parallel to the long axis but there is a significant amount of orientation in other directions. In general, the number of crystals oriented in any particular direction decreases with increasing deflection from the long axis. The large majority of crystals are oriented within plus or minus 30” of the long axis. Chen and Gundjian (1974) have also reported measuring a distribution function with a width on the order of 30”. Given these degrees of misalignment of hydroxyapatite crystals, correlation with Fig. 4 and Table 1 shows that estimates of Young’s modulus are obtained in the range of experimentally measured values. The calculated transverse modulus of 5.3 GPa for a composite containing perfectly aligned filler particles is lower than that observed experimentally
Micromechanics
Table 1. Comparison
model*
and Raffopoulos
(1990)
Schaffler and Burr (1988) Vincentelli and Grigorov (1985) Ashman Lipson
et al. (1984) and Katz (1984)
Bonfield and Grynpas (1977) Currey (1975) Reilly and Burstein (1975)
*Mineralization
fraction
Table
of 0.65 and particle
aspect
17 15 9.5 11.5
Martin and Ishida (1989) Vincentelli and Grigorov (1985) Currey (1975) Reilly and Burstein (1975) Burstein et al. (1972)
(Table 1). The low value is due to the fact that, in the model, there are no particles aligned in the transverse dimension. In reality, however, there are a small number of hydroxyapatite crystals oriented perpendicular to the long axis (Sasaki et al., 1989), providing additional stiffness in this dimension. The calculated tensile strength of cortical bone with a particle aspect ratio of 10 and mineralization of 0.65 is 1.7 GPa. A comparison of the calculated value with the various experimentally measured values is given in Table 2. The predicted ultimate tensile strength based on a filled polymeric composite model is much higher than that observed experimentally for cortical bone. This is not entirely surprising as bone contains numerous mechanical flaws such as pores for nutrient vessels that would lower the measured ultimate strength. In general, experimental data in the literature for tensile strength have not been reproducible. This variability may be attributed to bone’s inherent anisotropy in properties, and biological heterogen-
bone as
Sample 0’ Misalignment 10; 20 30” Human femur
Human femur Canine femur Plexiform bovine femur Haversian bovine femur Bovine femur Bovine femur Human femur Human femur, Haversian
ratio of 10.
Ultimate tensile strength (MPa) model
of cortical
Bovine femur Human tibia
2. Comparison between the calculated tensile a filled composite and various experimentally
Study Present
5.3
34.5 31.2 23.9 17.1 16.2 (Static) 19.9 (Dynamic) 22.1 19.7 18.0 20.0 20.1 30 25 18.5 22.7 17.0 22.7
443
and transverse moduli determined values
Transverse modulus (GPa)
Longitudinal modulus (GPa)
Katsamanis
and fracture
between the calculated values of the longitudinal a filled composite and various experimentally
Study Present
of bone strength
1700 106 162 133 125 133 175
strength of cortical determined values
bone
as
Sample Aspect ratio of 10 Mineralization of 0.65 Bovine femur Human tibia Bovine femur Human femur Bovine femur
eity. The effects of sample preparation, introduction of flaws, and testing procedures also affect the fracture data. In an analysis of compact bone from a wide variety of species, ultimate tensile strength could not definitely be related to any combination of parameters (Currey, 1990). At the same time, other authors have been able to correlate tensile strength with certain parameters (Rice et al., 1988). Young’s modulus, which reflects the stiffness of bone, is enhanced by the addition of an inorganic filler. The filled polymeric composite model is successful in predicting the elastic moduli and accounting for their anisotropy (Table l), but overestimates the tensile strength (Table 2). This pattern is typical of calculations based on a molecular model. At low strain levels, Young’s modulus is nearly a direct function of interactions at the molecular level. The presence of flaws, for example, does not necessarily affect the measured values of the initial modulus. Thus, while the model is a simplification of true bone tissue, in
444
J. F. MAMMONE and S. M. HUDSON
that it does not consider superstructure or histology, reasonably satisfactory values of Young’s modulus are obtained. This is because the determinants of Young’s modulus in bone are at the microlevel. With other materials, such as fibers, or structural composites, similar success with molecular models is achievable in the calculation of Young’s modulus. The estimation of tensile strength is not as straightforward. For example, when the theoretical tensile strength of fibers is computed on a molecular basis by using the energy required to break covalent bonds, it is overestimated by at least an order of magnitude. This is especially true in ceramic fibers where the presence of various flaws can affect dramatically the strength but not the modulus. The overestimation of bone tensile strength still indicates that key determinants of bone strength will also include flaws and tissue structure. Tensile strength is enhanced by the addition of an inorganic filler, but not nearly as much as compressive strength. Ultimate tensile strength in bone is not as crucial, from a functional standpoint, as compressive strength. Tensile loads within the body can be borne by other structural elements such as tendons and ligaments. In fact, as Currey (1984) has pointed out, the skeletal system is designed to minimize tensile stresses. Still, bone tensile strength is important in several situations. One is at tendinous and ligamentous insertions. At these points bone responds by increasing its cross-sectional area in the form of tubercles and tuberosities, thereby reducing the overall tensile stress. Tensile failure at these locations results in avulsion fractures. Other loading situations where tension is important are in torsion and bending. When bone is subjected to a bending stress, the convex or outer surface experiences a tensile stress, while the concave or inner surface is loaded in compression. In bone, the side in tension fails first, producing a transverse fracture. In an organic solid or one at low levels of mineralization, tensile strength is greater than compressive strength. Increasing mineralization improves the compressive strength of bone such that it exceeds tensile strength (Kaplan et al., 1985). An organicbased composite would fail first on the compressive side. Fatigue or stress fractures may occur when bone is stressed beyond the yield point repeatedly but before ultimate failure. Thus, the fracture may not be visualized macroscopically. The energy absorbed by bone may result in a significant amount of particle-matrix debonding. While usually not apparent radiographitally, these fractures are positive on bone scanning with uptake of the radiotracer technetium methylene diphosphonate. This agent localizes in bone via interaction on the exposed hydroxyapatite crystal surface to produce insoluble technetium calcium phosphate complexes (Pendergrass et al., 1973). Although nonspecific, these observations are consistent with a filled polymeric composite model of bone where the primary mechanism of failure is interfacial debonding.
This concept is supported indirectly in the composite literature, where it is generally known that the adhesion between the matrix and the filler affects the strength of a filled composite greatly (Trachte and Dibenedetto, 1971; Han et al., 1978; Jancar and Kucera, 1990). Direct observations of debonding at the crack tip have been made in epoxy resin with glass filler particles (Owen, 1979). In vivo fatigue testing on bone results have yielded a failure mechanism compatible with diffuse microcracking and debonding (Forwood and Parker, 1989). Interfacial debonding may, therefore, be considered as an important mechanism of failure. It has also been suggested that partial debonding of hydroxyapatite from the collagen matrix may be responsible for the degradation of the mechanical properties in aging and certain disease states (Bundy, 1985). Compressive strength is a key in vivo requirement in bone. The most important determinant of compressive strength is the degree of mineralization. The qualitative loss of mineralization results in the wellknown increased risk of fracture, whereas increased mineral density is seen to provide increased compressive strength (Lotz et al., 1990). As with tensile strength, compressive fracture data are extremely variable. Compressive testing of composites is extremely sensitive to testing procedures, even more so than in tension (Mammone and Uy, 1984). In bone, significant differences have been noted by simply improving the test coupon end constraint (Linde and Hvid, 1989.) The mechanics of compression are extremely complicated and no adequate model for a filled polymeric composite has been formulated as yet. An understanding of the microstructural requirements for compressive strength provides an insight as to why nature was forced to develop such a unique material like bone. Aside from bone, organisms provide for all of their functions with organic compounds and polymers (proteins, carbohydrates, fats). The genetic code is programmed directly to synthesize proteins. No conceivable protein or solely organic structure, however, can provide the compressive strength required by vertebrates for their increased size and mobility. Man has fabricated advanced polymeric composites that have extremely high stiffness and tensile strength but they still lack adequate compressive strength. The failure of organic-based materials in compression is due to the fact that they buckle early at the molecular level when loaded in compression (Mammone and Uy, 1984). There is little intermolecular support as the predominant forces are relatively weak dipole-dipole and van der Waals interactions. A covalent, polar-covalent, or ionic network-like molecular lattice that exists in inorganic oxides like hydroxyapatite is required to SUStain compressive loads. It had been suggested that increased crosslink density may be the mechanism of property enhancement in bone (Lees and Davidson, 1977). Although increased crosslinking will increase
445
Micromechanics of bone strength and fracture the modulus, it has been shown to have little effect on advanced organic fibers and simply tends to make the material more brittle (Mammone and Uy, 1984). Currey (1984) has felt that the most important feature of bone is its stiffness. While it is clear that stiffness is an essential mechanical function of bone, incorporation of an inorganic phase may not be necessary to achieve it. There are many man-made organic fibers with high stiffness as well as fibers found in nature (e.g. spider silk). A composite of such fibers would exhibit very high stiffness. What it would lack is compressive strength. There is also the nontrivial problem of achieving adequate fiber-matrix adhesion which has been a thorny problem with most high-performance organic fibers. This dilemma is similar to the problem confronting aerospace engineers using lightweight high-strength, high-modulus organic fibers. The composites exhibit high tensile strength and stiffness but their major drawback is inadequate compressive strength. Adequate compressive and tensile properties generally represent absolute design prerequisites for composites in structural applications. They are easy to measure and understand. In practice, bone tissue is subjected to more complicated stresses than just tension and compression. Three other clinically important mechanisms involved in fracture production are tension, shear. and bending. In composite design, these three mechanisms are generally addressed macroscopically by adjusting the spatial relationships of the reinforcing materials after compressive and tensile properties are optimized. Improved adhesion has been used in composites to enhance the interlaminar shear properties. The advantage of organic polymers, including proteins, is that they are lightweight, which is a desirable design characteristic from a mechanical standpoint. Inorganic materials are heavy, but have higher compressive strengths. Thus, nature solved its structural problems by fabricating a composite. It added enough inorganic filler to provide the compressive strength required but kept bone relatively light by employing a two-phase composite concept. Man too has had to resort to hybridization with inorganics to improve compressive properties in his most advanced composite materials (Mammone, 1986). In summary, the properties of the microstructural elements in bone may be explained in the context of a filled polymeric composite. The theory affords satisfactory modeling of tensile properties and incorporates the influence of key variables. In this model, the mode of failure is particle-matrix debonding, which can also explain some fatigue or stress fractures. Our subsequent work will be to fabricate hydroxyapatitefilled composites, examine tensile and compressive properties as a function of mineralization and try to understand further the mechanisms of failure. We have also tried to show that bone is a uniquely designed material in nature with its use of a solid inorganic phase. The proposed teleological purpose is BM26:4/5-F
a need to improve its compressive strength, which is simply unachievable using organic-based materials alone. REFERENCES
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APPENDIX The pertinent parameters as described by Chow (1977, 1978, 1980) are listed below. The filler aspect ratio is given as p and the mineralization fraction as 4. The subscripts f and m refer to the filler and matrix, respectively. &=l
-&ai
+(k,/k,-l)(l
J. Biomechanics 22,485-490.
Lipson, S. F. and Katz, J. L. (1984) The relationship between elastic properties and the microstructure of bovine cortical bone. J. Biomechanics 17, 231-240. Lotz, J. C., Gerhart, T. N. and Hayes, W. C. (1990) Mechanical properties of trabecular bone from the proximal femur: a quantitative CT study. J. Comput. Assist. Totnogr. 14, 107-l 14. Mammone, J. F. (1986) Hybrid composite of poly (p-phenylene-transbisbenzothiazole) and ceramic fiber. U.S. Patent Number 4,571,411. Mammone, J. F. and Uy, W. C. (1984) Exploratory development of high strength, high modulus polybenzothiazole fibers. Part II. Air Force Technical Report AFWAL-TR82-4154, Part II. Manhanian, S. and Piziali, R. L. (1988) Finite element evaluation of the AIA shear specimen for bone. J. Biomechnnics
(i= 1,3), Gi= 1+(/4/p,-
l)(l-4%
a, =4nQ/3-2(2rr-I)R, a,=4nQ/3+4(1-z)R, b1
=(+-3)
&=
Q-4(1-2n)R,
$_(4ql;;p’ (
>
2S,z,z=+;
Q+(4n-I)R,
s)
Q+21R,
s ,,,,-s,,,,=(niiZ~)Q+(4n-I)R.
21, 347-356.
Martin, R. B. and Ishida, J. (1989) The relative effects of collagen fiber orientation, porosity, density, and mineralization on bone strength. J. Biomechanics 22, 419-426. Nicholson, D. W. (1979) On the detachment of a rigid inclusion from an elastic matrix. J. Adhesion 10, 2551260. Owen. A. B. (1979) Direct observations of debondinpc at cradk tips inglass’bead-filled epoxy. J. Mater. Sci. Leetry14,
where
Q=&, m
and ccos-’ p-p(1
2521-2523.
Pendergrass, H. P., Potsaid, M. S. and Castronovo, F. P. (1973) The clinical use of 99mTc-diphosphonate (HEDSPA). A new agent for skeletal imaging. Radiology 107, 557-562. Reilly, D. T. and Burstein, A. H. (1975) The elastic and ultimate properties of compact bone tissue. J. Bio-
-p*p*1
2np[p(p2-l)1’2-cosh-‘p]
for p
When p= 1, I = 4n/3,
mechanics 8, 393-405.
Rice, J. C., Cowin, S. C. and Bowman, J. A. (1988) On the dependence of the elasticity and strength of cancellous bone on apparent density. J. Biomechanics 21, 155-168. Sasaki, N., Matsushima, N., Ikawa, T., Yamamura, H. and Fukuda, A. (1989) Orientation of bone mineral and its role
RJ_ 14Vm~ 8x l-v,,,’
5=
i=
K,
1+2(~,/~,-1)(1-~)s1212’ l+(P~/P~-l)(l-~)(s11*1-s33Il) 1+2(~f/~,-1)(1-~)s1212