Juumul of Orthopaedic Rereclrclz 10:818-825 Raven Press. Ltd., Ncw York
0 1992 Orthopaedic Research Society
A Theory of Fatigue Damage Accumulation and Repair in Cortical Bone Bruce Martin Orthopedic Research Luhorntories, University of California, Davis,California. U.S.A.
Summary: An analysis is presented of the balance between the accumulation and repair of fatigue damage in osteonal bone. Fatigue damage is defined in terms of cracks seen histologically when precautions are taken to avoid preparation artifact. The rate of occurrence of such damage is assumed to be proportional to the product of applied peak-to-peak stress, raised to a power, and the loading frequency. The rate of damage repair is assumed to be proportional to the activation rate for osteonal remodeling, and to the mean crosssectional area of the resulting osteons. An additional factor is introduced to account for the possibility that damagc provokcs nearby remodeling. The theory is used to compare data from previous experiments of two types: fatigueto-failure, and studies in which histologically observable cracks are made more numerous by repetitive loading. The analysis shows that there is a measure of agreement between the results of the two kinds of experiments, but the current data are too limited, and the results are too dependent on the mode of loading, to adequately test the theory. However, the analysis provides a framework for designing experiments to more efficiently clarify the relationships between fatigue failure, cracks seen in histologic sections, and the rate at which such cracks are repaired by osteonal remodeling. Key Words: Fatigue-Remodeling-Damage-Cracks-Osteon.
Engineering analyses of fatigue damage and fatigue strength of bone are sparse in relationship to their importance in orthopedics. A fatigue S-N curve (applied stress or strain versus number of cycles to failure) f o r bone was first published by King and Evans (19). Subsequently, other investigators (notably Carter et al.) have established that bone behaves similarly to engineering composite materials in that its elastic modulus declines as fatigue progresses, the log of N is linearly proportional to the log of S, and cracks can be observed within fatigued bone that are apparently fatigue damage (5-9,16,27,28). Other investigators have studied histologic evidence of fatigue damage in normal bone
specimens and in those loaded short of fatigue failure (3,4,14,24). Many investigators have suggested that remodeling serves to repair fatigue damage and prevent fatigue fracture (9,10,11,13,23). A clear understanding of clinical fatigue fracture in bone requires knowledge of how early histologic damage is repaired by remodeling o r , if repair is not sufficiently effective, progresses to the damage associated with failure. The present paper proposes a theory that addresses three important questions: How does fatigue crack damage depend on S and N? What is its relationship to fatigue failure? What is the relationship between remodeling and damage repair?
Received July 26, 1991; accepted May 6 , 1992. Address correspondence and reprint requests to Dr. R. B. Martin at Orthopaedic Research Laboratory, TB-150, University of California, Davis, CA 95616, U.S.A.
THEORY
The theory is predicated on several limitations and simplifying assumptions. First, only osteonal 818
FATIGUE DAMAGE AND REPAIR IN CORTICAL BONE
remodeling of primary or secondary intracortical bone is treated. It is assumed that the word osteon refers to secondary osteons formed by basic multicellular units (BMU) (1 3 ) . The simplified intracortical geometry depicted in Fig. 1 is assumed. A volume 4 is occupied by an arbitrary mixture of osteons, osteon fragments, and primary bone. A continuum mechanics approach is assumed, so that & is appropriately siLed to serve as a control volume with respect to both osteons and fatigue damage. Osteons are assumed to be circular and parallel; Volkmann's canals are ignored. R is a typical histologic cross-section through &. The number of load cycles required for fatigue failure depends on the magnitude of the continuum level peak-to-peak (P-P) range of stress or strain imposed during each cycle. As fatigue damage accumulates, the elastic modulus declines, and the relationship between stress and strain changes. Because most experiments on bone have been performed under stress control, the theory will be formulated in terms of stress. However, other results have been reported in terms of either strain or stress normalized by elastic modulus (5). To make various experimental results more comparable, in this article stress will usually be normalized by an estimate of the bone's elastic modulus at the start of experimental loading. Bone fatigue is strain rate sensitive (22,28). In most fatigue experiments, each load cycle follows immediately after the last, and the strain rate increases with the loading frequency (load cycleshnit
Y +
FIG. 1. Diagram of the idealized model for osteonal bone.
819
time). In vivo, these variables may be coupled (as in running vs. walking), or they may not be (as in ballet dancing, where leaps of similar strain rate may occur at varying intervals). In this paper, strain rate and loading frequency are assumed to be independent variables, and both are assumed to be important. Fatigue damage is not a well-defined cntity (30). It may be defined simply as a parameter that varies from 0 in virgin structures to 1 in failed structures, or in terms of a reduction in elastic modulus (14). Crack-based definitions of damage have also been proposed, such as total crack length (12). In composite materials, microxopic cracks appear as the result of repetitive loading (17). These cracks tend to run along lamellar interfaces, extending fatigue life by diverting energy from the propagation of transverse cracks, which would cause more immediate failure (IS). In bone, similar cracks are normally present (4); in the human rib, about half appear in osteonal cement lines (14). The number of such cracks increases after repetitive loading (3,24). Therefore, as a starting point it is reasonable to define damage (D)as the total lengthlmm2 of cracks measured in histologic cross-sections by methods and criteria that exclude artifactual cracks (4). Note that by this definition, D = number of crackslmm' x mean crack length. Damage Formation Rate (DF) Regardless of how fatigue damage is defined, or what other variables are involved, it is reasonable to postulate that its rate of formation is a function of loading frequency VL) and the resulting mean P-P stress (u). It is useful to examine fatigue failure data to gain insight regarding the nature of this relationship for cortical bone. Figure 2 shows a log-log plot of stress versus the number of cycles to failure, NFAIL, for three experiments, each involving different uniaxial loading conditions. Each of the three plots may be fitted to the equation log cr
=
A
+ B log NFAIL
(1)
The intercept ( A ) and slope ( B ) of each experiment's regression line are shown in the upper portion of Table 1. For each of the three experiments, A = - 2 , but B is more variable. Equation 1 implies that the fatigue-to-failure results for bone may be expressed as
@ is a control volume and Cl is a typical cross-section
through it.
NFAIL= Clcr"
(2)
J Orthop R e , , Vol. 10, N o , 6 , 1992
B . MAR TIN
820
in all in vitro fatigue tests), and if stant, one has
2.6 2.4
D
cn 2.2
cn W
+
I/)
c3
2.0
1 .a
0 A
1.4
1:
-:-.I
YFFERTY & RAJU KING & EVANS GRAY & K O R B A C H E R
1.2 0
1
2
3
4
5
6
7
LOG N FIG. 2. S-N curves for three cortical bone fatigue experiments. Gray and Korbacher (16) tested cow femur in compression; King and Evans (19) tested embalmed human femur as reversed cantilevers; Lafferty and Raju (22) tested bovine femur as rotating cantilevers. Here, stress was defined in megapascals before taking the logarithm. The upper portion of Table 1 shows least squares regression results with stress normalized by elastic modulus.
where log C = -A/B and q = - B P I . Table 1 shows that q is relatively large and varies with the experimental conditions. In general, a few cycles of high stress are equivalent to many cycles of moderate stress, but changes in the conditions can greatly alter N,,,, for a given u. Carter et al. (5-8) have added several sets of data to those shown in Fig. 2, using different types of uniaxial loading and a lower loading frequency. Table 1 shows that A , B, and q for these experiments are similar to earlier values. That fatigue-to-failure results may be represented by Eq. 2 suggests letting the damage formation rate be
Here, k is a damage rate coefficient. As noted above, fL describes only the number of load cycles per unit time; k captures variability due to strain rate and other loading conditions, and bone structure. Equation 3 assumes that there is a single type of load, with one significant stress component, cr. (If multiple stress components and types of loading must be considered, one could extend the theory by assuming linear superposition of damage and writing Eq. 3 as a sum of similar terms.) If there is only one significant stress component (as in most fatigue tests), and no damage repair (as
J Orthop Rrs, VoE. 10, No. 6, 1992
=
JDF dt
=
CJ
k uqf L J dt
andf, are con=
k aYN
(4)
where N = f‘t is the number of accumulated load cycles at time t. At failure, one would have D = ~ F - A I I .and NI?AIL
(D,A,lJk)icrq
(5)
If k can be determined, Eq. 5 can provide an estimate of the damage associated with failure. k can be estimated by measuring the damage produced by a given amount of repetitive loading, or by considering the rate of repair of the damage under equilibrium conditions. Each of these methods will be explored. Damage Repair Rate (DR)
Assuming that fatigue damage can only be removed by osteonal remodeling, one may postulate that the rate of damage repair is proportional to the amount of bone resorbed per day. Consider the cros5-section Cl in Fig. 1. If the BMU activation frequency is J;, BMU/mm2/dayand the osteonal cement line radius is R,, then the fractional area of Cl resorbed per day is
X
=
janR,’
(6)
If Cl contains D damage per unit area, and resorption and damage are randomly distributed, the mean damage repair rate yhould be
DR
=
DX
=
D fa nRC2
(7)
That is, if X percent of the bone matrix is resorbed/ day, Xpercent of D will be resorbed with it. In adult rib,.f, = 0.003 BMU/mmZ/day,and the radius of a typical human osteon is R , = 0.100 mm (15). Therefore, in the rib, X = 0.009%per day or 3% per year. If, however, new BMU are activated by (and thus, in the vicinity of) cracks, the efficiency of repair ought to be greater. Martin and Burr (23) have postulated that cement line cracks can extend far enough along an osteon to reduce the strain on the surfaces of its Haversian canal, and that this initiates a new BMU from the canal wall that repairs the adjacent crack. Other mechanisms have been postulated for the initiation of remodeling to repair fatigue damage, including interruption of the osteocyte network (14). The present theory is not predicated on any particular mechanism; it simply as-
82 1
FATIGUE DAMAGE AND REPAIR IN CORTICAL BONE TABLE 1. Data from fatigue-to:failrcre experiments Type of loading
Frequency Hz
Intercept A
Slope B
Y
C
30 30 30
1.YO0 1.666 -2.011
0.1023 -0.1409 -0.0657
9.78 7.10 15.22
2.67 x 10-19 1.49 X lo-" 2.46 x
2 2 2 2 2 0.5-2
-2.049 -2.202 -2.031 -2.041 - 1.554 - 1.598
-0.0697 -- 0,0664 -0.0818 0,0849 -0.1525 0. I873
14.35
3.95 x 3.91 x 1.48 x 9.14 x 6.46 x 2.94 x
Early experiments Rotating cantileverU Reversed cantileverb Compression" Experiments of Carter et al. Tensiond Tension" Compressiond Partially reversed" Fully reversed" Mixed
-
-
-
-
-
15.05 12.22 1 1.78 6.56 5.34
10-3" lo-" 10 *' lo-'' 10
Note: All experiments were under 3tress control. and failure was defined as complete fracture except for mixed, which was strain controlled with failure defined as a 30% reduction in stress. Lafferty and Raju (22), bovine femur, stress normalized by E = 22.3 GPa (25). King and Evans (19), embalmed human femur. stress normalized by E = 17.5 GPa (8). ' Gray and Korbacher (16), bovine femur. stress normalized by E = 22.3 GPa (25). Caler and Carter ( 5 ) . human femur, stress normalized by initial elastic modulus. ' From Carter and Caler (6) [see also Carter and Caler (7)1; human femur. stress normalized by initial modulus. Carter et al. (8), mixture of partially and fully reversed fatigue, human femur.
'
sumes that fatigue cracks may cause an osteonal BMU to be activated nearby. To accommodate such an effect, a damage repair specificity factor, F,, is introduced in Eq. 7:
fiR
=
Df;,r R C 2F ,
(8)
If, for example, F , = 5 , BMU would be five times more likely to remove damage than if their location were independent of cracks. One has, then, for the rate of accretion of fatigue damage,
ri
=
DF -
DR
In the equilibrium state one has damage is
D,,
=
b,
ered here. When the damage cquals that at failure, one has the conventional S-N curve represented by the intersection of the damage surface with the plane D = D F A I L .Smaller amounts of damage would be associated with equidamage contours (e.g., A, B, and C in Fig. 3) representing the intersections of other constant damage planes with the damage surface.
(9) =
Dk,and the
( k UYfLY(T Rr% Fs)
(10)
EXPERIMENTAL VERIFICATION A useful theory should suggest experiments to test it. One such experiment is illustrated by extending the concept of an S-N graph to include damage as a third logarithmic axis (Fig. 3). One has from Eq. 4,
log D
=
log k
+ 4 log S + log N
(1 1)
where stress (or strain) is now represented by S. The sloping triangular surface in Fig. 3 represents the fatigue damage associated with N cycles of P-P strain S. The leftmost portion of this surface lies beyond the endurance limit (where S is so low that fatigue failure never occurs) and will not be consid-
FIG. 3. Schematic three-dimensional logarithmic graph of damage, N and S. The damage surface is shown as the stippled triangular region. For simplicity's sake, the surface is shown as a plane, and no attempt is made to define it for S below the endurance limit. Lines A, B, and C represent equidamage contours. The uppermost edge of the damage surface (D = D,,,,) corresponds to the conventional S-N plot for fatigue failure, as in Fig. 2.
.I Orthop Rec, Vol. 10, IVO.6, 19Y2
822
B . MAR TIN
A better understanding of the relationship between histologically apparent damage and fatigue could be gained by systematically varying S and N until the entire damage surface is revealed for various types of bone and modes of loading. (This approach could be extended to other measures of damage, such as decrements of elastic properties.) Estimation of q and k from Fatigue Crack Data Consider the experiments of Burr et al. (3,24), in which fatigue crack damage was measured in bones loaded in three-point bending at two different values of S for N = 10,000 cycles. These experiments involve movement on the damage surface along a line defined by its intersection with plane B, erected at log N = 4 (Fig. 4). We will call this line the damage experiment line (DEL). The initial damage in the specimens is represented by an equidamage contour, I. Loading of the specimens moved them upward along the DEL from point B, (the intersection of contour 1 with the DEL) to points B , or B2, depending on the load magnitude. Burr et al. (3,24) showed that there are normally about 0.018 cracksimm2 in the diaphysis of the adult dog radius. After 10,000 cycles of bending for which fL = 1 hz and the strain (or normalized stress) on the surface of the radius was u = 0.0015, the crack density (&) increased to 0.024 mm '. After 10,000 cycles of bending at u = 0.0025, K , increased to 0.056 cracksimm'. Mean crack length ( L , ) was not measured, but data from normal human rib (noted earlier), fdtigue-loas2ded cow bone ( 2 8 ) , and race
horse cannon bone (author's unpublished data) indicate that L, is 0.80-4.90 mm among different species and methods of loading. Therefore, one may assume that L, was not greatly affected by u, and write €or the incremental damage at ur = 0.0015, 6D'
=
K',L,
ka"N
=
(124
where K', = 0.024 - 0.018 = 0.006 is the number of additional cracks/mm2 produced by the experimental loading. At u" = 0.0025,
6D" = K" dL c
=
where K " d = 0.056 - 0.018 bining these equations, q
=
kurrqN
=
(12b)
0.038 rnmp2. Com-
[In(K",/Kr,)]/[In(ur'/u')]
=
3.6
(13)
Because it is based on only two data points, this estimate of q is not likely to be as accurate as those obtained from S-N curves. Either through inaccuracy or becauqe it represents different experimental circumstances, the value is somewhat smaller than those obtained for fatigue-to-failure of machined bone specimens. However, it is comparable with the failure-calculated values listed in Table 1 . To adequately test the theory using this kind of data, the experiments must be repeated using more values of u, and q should be calculated by regression rather than by Eq. 13. One can also imagine experiments in which specimens are examined for damage after various numbers of load cycles when the applied stress is held constant. These experiments would move specimens along DELs formed by the intersections of the damage surface with constant S planes. The repair side of the theory could be tested in several ways. One of these will be sketched using data from the literature. Estimation of k from Equilibrium Damage Equation 10 can be rearranged to give k
Y
FIG. 4. Sketch illustrating the damage experiments of Burr et al. in the context of the damage surface. See text for details.
J Orthop R e s , Vol. 10, No. 6 , 1992
= (7 R,2@'~D,Y(~qf')
(14)
Do may be estimated for human rib. Burr and Stafford (4) counted 0.14 fatigue cracks/mm2 in a sixth rib from a man in his 60s. Subsequently, the present author measured the mean length of these cracks to be 0.088 ? 0.038 mm. Assuming that fatigue damage was being resorbed as fast as it was being produced in this rib, D,, = 0.14 x 0.088 = 0.012 mm-'. To estimate the value off, to use in the calculation, an approach suggested by Beaupre et al. ( I )
FATIGUE DAMAGE A N D REPAIR IN CORTICAL BONE
may be used. By considering data from experiments in which the strain magnitude and number of load cycles per day necessary to prevent bone loss were determined, they found by regression that for this kind of equilibrium, fL =
(151
*rrl
where cr is P-P applied stress, JI is applied cycles/ day, $ = 0.00263 day-", and m ;= 4. It will be assumed that, normally, equilibrium simultaneously exists for both bone mass and fatigue damage repair. One may use Eq. 15 to substitute for,f, in Eq. 14, obtaining k
= (T
R,' f a Fs D,)/(cr"-" $'")
(16)
Because only a single CT value is being considered and larger values of u contribute much more to fatigue than moderate values, cr will be set equal to 0.002, consistent with in vivo strain measurements for vigorous activity (26). There being no reliable data on F,, the value 5 will arbitrarily be used. R,, til,and rn will be set equal to the previously mentioned typical values (R, = 0.100 mm, fu = 1.1 BMU/mm'/yr = 0.0030 BMU/mm'/day, m = 4). Finally, a value must be assigned to q. Given the sensitivity of q to the type of loading, a good estimate of k will depend on how representative q is of the type of loading experienced by the human rib. One would expect this to be some kind of bending. Table 2 shows the values of k and DFAII, = kC calculated using the values of q obtained from the fatigue experiments in Table 1, which involved bending or reversed tension-compression. The estimates of DFAILvary from 0.04 to 108 mm-'. It would be useful to compare the predicted D,,, values with the extreme case in which cracks occur between all the lamellae in every osteon. An osteon typically contains 16 lamellar interfaces (13). If the Haversian canal and cement line radii are R , = 0.020 mm and R , = 0.100 mm, respectively, and the lamellae are of uniform thickness. the total inTABLE 2. Damage rute coeflicient and damage-at-failure e5fimates bused on equilibrium in human rib Type of loading Rotating cantilever Reversed cantilever Partially reversed Fully reversed Mixed
k mm-l 3.8 x 2.7 x 1.2 x 0.9 x 4.9 x
1015 1013
DFAIT rnm-'
10*6 10"
9.9 0.04 108 62
IOU
1 .s
823
terlamellar length seen in a cross-section can be shown to be 6.28 mm. The cross-sectional area of the osteon would be TR,' = 0.0314 mm', so the total potential interlamellar crack damage is 200 mm-I. As expected, this maximum possible damage estimate is greater than the estimates of DFAIL. When one attempts to calculate k from the Burr experiments (using k = D/uqN), one finds that the result ("9 x 10' mm-') is much smaller than the values in Table 2. This could be because the calculation is very sensitive to the value of q, which may actually be >3.6. Another possibility is that the damage surface is curved rather than flat. In that case the linear formulation presented here could be modified to fit the surface by making k and/or q suitable functions of D . DISCUSSION
The theory developed here provides preliminary answers to the three question\ posed in the introduction. The relationships between fatigue crack damage and S, N , and fatigue failure are embodied in Eq. 3 and Fig. 3. For cortical bone, the relationship between fatigue damage repair and the variables of remodeling is given in Eq. 8, and Eq. 10 predicts equilibrium damage as a function of loading and remodeling conditions. This model was used to compare the results of three kinds of experiments found in the literature: fatigue-to-failure, fatigue to an increased level of damage, and equilibrium damage. There is a measure of agreement between the theory and such experiments, but the existing data are too sparse and inappropriately controlled to provide a good test. Figure 3 provides a paradigm for more rigorous testing of the damage formation side of the theory. Experiments to test the repair side of the theory would involve creating predictable amounts of damage in living animals and studying its diminishment as a function of remodeling using standard histomorphometric techniques. As the theory is tested, many other factors must be addressed. How are fatigue damage and failure affected by histologic type (e.g., plexiform, circumferential lamellar, osteonal) and histocomposition (e.g., porosity, mineralization, collagen fiber orientation). There are very limited data on the effects of strain rate on bone fatigue (21,22,28). How does strain rate affect f i r and D,,,'? Can the effects of strain rate be separated from those off; ? It may not be possible to simply use k to account for strain
J Orrhop Res, Vol. 10. No.6. 1992
824
B . MARTIN
log s
log N FIG. 5. Sketch relating the theory presented here to Palmgren-Miner analyses of fatigue damage in engineering materials. There is debate about whether equidamage contours (e.g., A, B, C) are parallel (as shown) or converge at the endurance limit.
rate; for example, both k and q may be functions of strain rate, as well as other loading conditions. And of course, the theory must be extended to cancellous bone. When the damage surface is viewed from above (Fig. 5 ) , equidamage contours appear as prefailure damage curves, such as those used in the PalmgrenMiner analysis of residual fatigue life and its more recent descendants (2). This connection is significant because it relates the theory presented here to engineering analyses of damage and residual fatigue life in other materials. There is debate in the engineering literature about the slope of the equidamage contours relative to the failure line, and their relationship to the endurance limit (2,20,29). Knowledge of the shapes of damage surfaces for bone and other materials would substantially resolve these questions. APPENDIX
Mathematical Symbols S N
R lJ
fL NFAIL
A B C
Stress or strain Number of load cycles Region of cortical bone Cross-section through @ Peak-to-peak stress, usually normalized by elastic modulus Loading frequency Number of cycles at failure Intercept of the log-log S-N curve Slope of the log-log S-N curve Ratio. -A/B
J Orthop Rer. Vol. 10, N o . 6 , 1992
-B-l Damage (6D = an increment of damage) Equil ibrium damage Damage at failure Damage formation rate Damage repair rate Damage rate coefficient Time Fractional part of il resorbed per day Radii of cement line and Haversian canal, respectively BMU activation frequency Damage repair specificity factor Empirical constants Fatigue crack density (cracks per unit area of cross-section) Mean length of fatigue cracks Acknowledgment: I t h an k the individuals who reviewed this p ap er for their extremely helpful suggestions, which significantly improved t h e work.
REFERENCES 1. Beaupre GS, Orr TE. Carter DK: An approach to time-
dependent bone modeling and remodeling-application: a preliminary remodeling simulation. J Orthop Res 8:662470. 1990 2. Ben-Amoz M: A cumulative damage theory for fatigue life prediction. Eng Frwture Mech 37:341-347, 1990 3. Burr DB, Martin RB. Schaffler ME, Radin EL: Bone remodeling in response to in vivo fatigue microdamage. J Riomech 18:18%200, 1985 4. Burr DB, Stafford T: Validity of the bulk-staining technique to separate artifactual from in vivo bone microdamage. Clin Orrhop Re1 Res 260305-308, 1990 5. Caler WE, Carter DR: Bone creep-fatiguc damage accumulation. I Biomech 22:625-6,35, 1989 6. Carter DR, Caler WE: A cumulative damage model for bone fracture. J Orthop Res 3:8&90. 1985 7. Carter DR; Caler WE: Cycle-dependent and time-dependent bone fracture with repeated loading. J Bionzech Eng 105: 166170, 1983 8. Carter D, Caler WE, Spengler DM. Frankel VH: Fatigue behavior of adult cortical bone: The influence of mean strain and strain runge. Arta Orthop Scand 52:481490, 1981 9. Carter DR, Hayes WC: Compact bone fatigue damage: a microscopic examination. Clin Orthop Re1 Rex 127:265-274, 1977 10. Chamey A, Tschantz P: Mechanical influences in bone remodeling. Experimental research on Wolff's law. J Biomech 5:173-180, 1972 11. Currey JD: Stress concentrations in bone. Q J Microsc Sci 103:111-1 33, 1962 12. Fong JT: What i s fatigue damage? In: Damage in composite materials: basic mechanisms, accumulation, tolerance, and characterization, ed by K L Reifsnider. Philadelphia, American Society for Testing and Materials, 1980 13. Frost HM: Bone remodelling dynamics. Springfield, IL, Charles c. Thomas, 1963 14. Frost HM: Presence of microscopic cracks in vivo in bone. Henry Ford Hosp Med Bull 8:27-35, 1960
FATIGUE DAMAGE A N D REPAIR IN CORTICAL BONE 15. Frost HM: Tetracycline-based histological analysis of bone remodeling. c'ulcif Tissue Res 3:211-237, 1969 16. Gray RJ, Korbacher GK: Compressive fatigue behavior of bovine compact bone. J Biomech 7287-292, 1974 17. Jamison RD, Schulte K , Reifsnider KL, Stinchcomb WW: Characterization and analysis of damage mechanisms in tension-tension fatigue of graphite/epoxy laminates. In: Effects sf defects in composite materials, Philadelphia, American Society for Testing and Materials, 1984 18. Kelly A, Davies GJ: The principles of the fibre reinforcement of metals. MetnllurX Rev 10:l-77. 196.5 19. King AT, Evans FG: Analysis of fatigue strength of human compact bone by the Weibull method. In: Digest of the 7th international conferences an medictrl and 6iologic.al engineering, ed by B Jacobson, Stockholm, Royal Academy of Engineering Sciences, 1967 20. Kujawski D, Ellyin F: A cumulative damage theory for fatigue crack initiation and propagation. int J Fatigrie 6:83-88, 1984 21. Lafferty JF: Analytical model of the fatigue characteristics of bone. Aviat Space Environ Med 49:17&174, 1978 22. Lafferty JF, Raju PVV: The influence of stress frequency on the fatigue strength of cortical bone. J Biomech Eng 101: 112-113, 1979
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23. Martin RB, Burr DB: A hypothetical mschanism for the stimulation of osteonal remodeling by fatigue damage. J Biomech 15: 137-139. 1982 24. Mori S . Burr DB: Increased intracortical remodeling following fatigue microdamage. Transactions, combined meeting of the Orthopaedic Research Societies of the USA. Japan, and Canada. Banff. Canada, October 21-23, 1991 25. Reiily DT, Burstein AH, Frunkel VH: The elastic modulus for bone. J Biomech 7:211-275, 1974 26. Rubin CT: Skeletal strain and the functional significance of bone architecture. CalciJ'Tissue Int 36s: 11-18, 1984 27. Schaffler MB, Radin EL, Burr DB: Long-term fatiguc behavior of compact bone at low strain magnitude and rate. BoPie 111321-326, 1990 28. Schaffler MB. Radin EL, Burr DB: Mechanical and morphological effccts of strain rate on fatigue of compact bone. Bone 10207-214, 1989 29. Subramanyan S: A cumulative damage rule based on the knee point of the S-N curve. J Eng ,Materials Techno/ 98: 316-321, 1Y76 30. Talreja R: Fnfigue of composite materials, Lancaster, PA, Technomic, 1987
J Ortho!) Res, Vol. 10, No. 6 , 1992
0 1992 Orthopaedic Research Society
A Theory of Fatigue Damage Accumulation and Repair in Cortical Bone Bruce Martin Orthopedic Research Luhorntories, University of California, Davis,California. U.S.A.
Summary: An analysis is presented of the balance between the accumulation and repair of fatigue damage in osteonal bone. Fatigue damage is defined in terms of cracks seen histologically when precautions are taken to avoid preparation artifact. The rate of occurrence of such damage is assumed to be proportional to the product of applied peak-to-peak stress, raised to a power, and the loading frequency. The rate of damage repair is assumed to be proportional to the activation rate for osteonal remodeling, and to the mean crosssectional area of the resulting osteons. An additional factor is introduced to account for the possibility that damagc provokcs nearby remodeling. The theory is used to compare data from previous experiments of two types: fatigueto-failure, and studies in which histologically observable cracks are made more numerous by repetitive loading. The analysis shows that there is a measure of agreement between the results of the two kinds of experiments, but the current data are too limited, and the results are too dependent on the mode of loading, to adequately test the theory. However, the analysis provides a framework for designing experiments to more efficiently clarify the relationships between fatigue failure, cracks seen in histologic sections, and the rate at which such cracks are repaired by osteonal remodeling. Key Words: Fatigue-Remodeling-Damage-Cracks-Osteon.
Engineering analyses of fatigue damage and fatigue strength of bone are sparse in relationship to their importance in orthopedics. A fatigue S-N curve (applied stress or strain versus number of cycles to failure) f o r bone was first published by King and Evans (19). Subsequently, other investigators (notably Carter et al.) have established that bone behaves similarly to engineering composite materials in that its elastic modulus declines as fatigue progresses, the log of N is linearly proportional to the log of S, and cracks can be observed within fatigued bone that are apparently fatigue damage (5-9,16,27,28). Other investigators have studied histologic evidence of fatigue damage in normal bone
specimens and in those loaded short of fatigue failure (3,4,14,24). Many investigators have suggested that remodeling serves to repair fatigue damage and prevent fatigue fracture (9,10,11,13,23). A clear understanding of clinical fatigue fracture in bone requires knowledge of how early histologic damage is repaired by remodeling o r , if repair is not sufficiently effective, progresses to the damage associated with failure. The present paper proposes a theory that addresses three important questions: How does fatigue crack damage depend on S and N? What is its relationship to fatigue failure? What is the relationship between remodeling and damage repair?
Received July 26, 1991; accepted May 6 , 1992. Address correspondence and reprint requests to Dr. R. B. Martin at Orthopaedic Research Laboratory, TB-150, University of California, Davis, CA 95616, U.S.A.
THEORY
The theory is predicated on several limitations and simplifying assumptions. First, only osteonal 818
FATIGUE DAMAGE AND REPAIR IN CORTICAL BONE
remodeling of primary or secondary intracortical bone is treated. It is assumed that the word osteon refers to secondary osteons formed by basic multicellular units (BMU) (1 3 ) . The simplified intracortical geometry depicted in Fig. 1 is assumed. A volume 4 is occupied by an arbitrary mixture of osteons, osteon fragments, and primary bone. A continuum mechanics approach is assumed, so that & is appropriately siLed to serve as a control volume with respect to both osteons and fatigue damage. Osteons are assumed to be circular and parallel; Volkmann's canals are ignored. R is a typical histologic cross-section through &. The number of load cycles required for fatigue failure depends on the magnitude of the continuum level peak-to-peak (P-P) range of stress or strain imposed during each cycle. As fatigue damage accumulates, the elastic modulus declines, and the relationship between stress and strain changes. Because most experiments on bone have been performed under stress control, the theory will be formulated in terms of stress. However, other results have been reported in terms of either strain or stress normalized by elastic modulus (5). To make various experimental results more comparable, in this article stress will usually be normalized by an estimate of the bone's elastic modulus at the start of experimental loading. Bone fatigue is strain rate sensitive (22,28). In most fatigue experiments, each load cycle follows immediately after the last, and the strain rate increases with the loading frequency (load cycleshnit
Y +
FIG. 1. Diagram of the idealized model for osteonal bone.
819
time). In vivo, these variables may be coupled (as in running vs. walking), or they may not be (as in ballet dancing, where leaps of similar strain rate may occur at varying intervals). In this paper, strain rate and loading frequency are assumed to be independent variables, and both are assumed to be important. Fatigue damage is not a well-defined cntity (30). It may be defined simply as a parameter that varies from 0 in virgin structures to 1 in failed structures, or in terms of a reduction in elastic modulus (14). Crack-based definitions of damage have also been proposed, such as total crack length (12). In composite materials, microxopic cracks appear as the result of repetitive loading (17). These cracks tend to run along lamellar interfaces, extending fatigue life by diverting energy from the propagation of transverse cracks, which would cause more immediate failure (IS). In bone, similar cracks are normally present (4); in the human rib, about half appear in osteonal cement lines (14). The number of such cracks increases after repetitive loading (3,24). Therefore, as a starting point it is reasonable to define damage (D)as the total lengthlmm2 of cracks measured in histologic cross-sections by methods and criteria that exclude artifactual cracks (4). Note that by this definition, D = number of crackslmm' x mean crack length. Damage Formation Rate (DF) Regardless of how fatigue damage is defined, or what other variables are involved, it is reasonable to postulate that its rate of formation is a function of loading frequency VL) and the resulting mean P-P stress (u). It is useful to examine fatigue failure data to gain insight regarding the nature of this relationship for cortical bone. Figure 2 shows a log-log plot of stress versus the number of cycles to failure, NFAIL, for three experiments, each involving different uniaxial loading conditions. Each of the three plots may be fitted to the equation log cr
=
A
+ B log NFAIL
(1)
The intercept ( A ) and slope ( B ) of each experiment's regression line are shown in the upper portion of Table 1. For each of the three experiments, A = - 2 , but B is more variable. Equation 1 implies that the fatigue-to-failure results for bone may be expressed as
@ is a control volume and Cl is a typical cross-section
through it.
NFAIL= Clcr"
(2)
J Orthop R e , , Vol. 10, N o , 6 , 1992
B . MAR TIN
820
in all in vitro fatigue tests), and if stant, one has
2.6 2.4
D
cn 2.2
cn W
+
I/)
c3
2.0
1 .a
0 A
1.4
1:
-:-.I
YFFERTY & RAJU KING & EVANS GRAY & K O R B A C H E R
1.2 0
1
2
3
4
5
6
7
LOG N FIG. 2. S-N curves for three cortical bone fatigue experiments. Gray and Korbacher (16) tested cow femur in compression; King and Evans (19) tested embalmed human femur as reversed cantilevers; Lafferty and Raju (22) tested bovine femur as rotating cantilevers. Here, stress was defined in megapascals before taking the logarithm. The upper portion of Table 1 shows least squares regression results with stress normalized by elastic modulus.
where log C = -A/B and q = - B P I . Table 1 shows that q is relatively large and varies with the experimental conditions. In general, a few cycles of high stress are equivalent to many cycles of moderate stress, but changes in the conditions can greatly alter N,,,, for a given u. Carter et al. (5-8) have added several sets of data to those shown in Fig. 2, using different types of uniaxial loading and a lower loading frequency. Table 1 shows that A , B, and q for these experiments are similar to earlier values. That fatigue-to-failure results may be represented by Eq. 2 suggests letting the damage formation rate be
Here, k is a damage rate coefficient. As noted above, fL describes only the number of load cycles per unit time; k captures variability due to strain rate and other loading conditions, and bone structure. Equation 3 assumes that there is a single type of load, with one significant stress component, cr. (If multiple stress components and types of loading must be considered, one could extend the theory by assuming linear superposition of damage and writing Eq. 3 as a sum of similar terms.) If there is only one significant stress component (as in most fatigue tests), and no damage repair (as
J Orthop Rrs, VoE. 10, No. 6, 1992
=
JDF dt
=
CJ
k uqf L J dt
andf, are con=
k aYN
(4)
where N = f‘t is the number of accumulated load cycles at time t. At failure, one would have D = ~ F - A I I .and NI?AIL
(D,A,lJk)icrq
(5)
If k can be determined, Eq. 5 can provide an estimate of the damage associated with failure. k can be estimated by measuring the damage produced by a given amount of repetitive loading, or by considering the rate of repair of the damage under equilibrium conditions. Each of these methods will be explored. Damage Repair Rate (DR)
Assuming that fatigue damage can only be removed by osteonal remodeling, one may postulate that the rate of damage repair is proportional to the amount of bone resorbed per day. Consider the cros5-section Cl in Fig. 1. If the BMU activation frequency is J;, BMU/mm2/dayand the osteonal cement line radius is R,, then the fractional area of Cl resorbed per day is
X
=
janR,’
(6)
If Cl contains D damage per unit area, and resorption and damage are randomly distributed, the mean damage repair rate yhould be
DR
=
DX
=
D fa nRC2
(7)
That is, if X percent of the bone matrix is resorbed/ day, Xpercent of D will be resorbed with it. In adult rib,.f, = 0.003 BMU/mmZ/day,and the radius of a typical human osteon is R , = 0.100 mm (15). Therefore, in the rib, X = 0.009%per day or 3% per year. If, however, new BMU are activated by (and thus, in the vicinity of) cracks, the efficiency of repair ought to be greater. Martin and Burr (23) have postulated that cement line cracks can extend far enough along an osteon to reduce the strain on the surfaces of its Haversian canal, and that this initiates a new BMU from the canal wall that repairs the adjacent crack. Other mechanisms have been postulated for the initiation of remodeling to repair fatigue damage, including interruption of the osteocyte network (14). The present theory is not predicated on any particular mechanism; it simply as-
82 1
FATIGUE DAMAGE AND REPAIR IN CORTICAL BONE TABLE 1. Data from fatigue-to:failrcre experiments Type of loading
Frequency Hz
Intercept A
Slope B
Y
C
30 30 30
1.YO0 1.666 -2.011
0.1023 -0.1409 -0.0657
9.78 7.10 15.22
2.67 x 10-19 1.49 X lo-" 2.46 x
2 2 2 2 2 0.5-2
-2.049 -2.202 -2.031 -2.041 - 1.554 - 1.598
-0.0697 -- 0,0664 -0.0818 0,0849 -0.1525 0. I873
14.35
3.95 x 3.91 x 1.48 x 9.14 x 6.46 x 2.94 x
Early experiments Rotating cantileverU Reversed cantileverb Compression" Experiments of Carter et al. Tensiond Tension" Compressiond Partially reversed" Fully reversed" Mixed
-
-
-
-
-
15.05 12.22 1 1.78 6.56 5.34
10-3" lo-" 10 *' lo-'' 10
Note: All experiments were under 3tress control. and failure was defined as complete fracture except for mixed, which was strain controlled with failure defined as a 30% reduction in stress. Lafferty and Raju (22), bovine femur, stress normalized by E = 22.3 GPa (25). King and Evans (19), embalmed human femur. stress normalized by E = 17.5 GPa (8). ' Gray and Korbacher (16), bovine femur. stress normalized by E = 22.3 GPa (25). Caler and Carter ( 5 ) . human femur, stress normalized by initial elastic modulus. ' From Carter and Caler (6) [see also Carter and Caler (7)1; human femur. stress normalized by initial modulus. Carter et al. (8), mixture of partially and fully reversed fatigue, human femur.
'
sumes that fatigue cracks may cause an osteonal BMU to be activated nearby. To accommodate such an effect, a damage repair specificity factor, F,, is introduced in Eq. 7:
fiR
=
Df;,r R C 2F ,
(8)
If, for example, F , = 5 , BMU would be five times more likely to remove damage than if their location were independent of cracks. One has, then, for the rate of accretion of fatigue damage,
ri
=
DF -
DR
In the equilibrium state one has damage is
D,,
=
b,
ered here. When the damage cquals that at failure, one has the conventional S-N curve represented by the intersection of the damage surface with the plane D = D F A I L .Smaller amounts of damage would be associated with equidamage contours (e.g., A, B, and C in Fig. 3) representing the intersections of other constant damage planes with the damage surface.
(9) =
Dk,and the
( k UYfLY(T Rr% Fs)
(10)
EXPERIMENTAL VERIFICATION A useful theory should suggest experiments to test it. One such experiment is illustrated by extending the concept of an S-N graph to include damage as a third logarithmic axis (Fig. 3). One has from Eq. 4,
log D
=
log k
+ 4 log S + log N
(1 1)
where stress (or strain) is now represented by S. The sloping triangular surface in Fig. 3 represents the fatigue damage associated with N cycles of P-P strain S. The leftmost portion of this surface lies beyond the endurance limit (where S is so low that fatigue failure never occurs) and will not be consid-
FIG. 3. Schematic three-dimensional logarithmic graph of damage, N and S. The damage surface is shown as the stippled triangular region. For simplicity's sake, the surface is shown as a plane, and no attempt is made to define it for S below the endurance limit. Lines A, B, and C represent equidamage contours. The uppermost edge of the damage surface (D = D,,,,) corresponds to the conventional S-N plot for fatigue failure, as in Fig. 2.
.I Orthop Rec, Vol. 10, IVO.6, 19Y2
822
B . MAR TIN
A better understanding of the relationship between histologically apparent damage and fatigue could be gained by systematically varying S and N until the entire damage surface is revealed for various types of bone and modes of loading. (This approach could be extended to other measures of damage, such as decrements of elastic properties.) Estimation of q and k from Fatigue Crack Data Consider the experiments of Burr et al. (3,24), in which fatigue crack damage was measured in bones loaded in three-point bending at two different values of S for N = 10,000 cycles. These experiments involve movement on the damage surface along a line defined by its intersection with plane B, erected at log N = 4 (Fig. 4). We will call this line the damage experiment line (DEL). The initial damage in the specimens is represented by an equidamage contour, I. Loading of the specimens moved them upward along the DEL from point B, (the intersection of contour 1 with the DEL) to points B , or B2, depending on the load magnitude. Burr et al. (3,24) showed that there are normally about 0.018 cracksimm2 in the diaphysis of the adult dog radius. After 10,000 cycles of bending for which fL = 1 hz and the strain (or normalized stress) on the surface of the radius was u = 0.0015, the crack density (&) increased to 0.024 mm '. After 10,000 cycles of bending at u = 0.0025, K , increased to 0.056 cracksimm'. Mean crack length ( L , ) was not measured, but data from normal human rib (noted earlier), fdtigue-loas2ded cow bone ( 2 8 ) , and race
horse cannon bone (author's unpublished data) indicate that L, is 0.80-4.90 mm among different species and methods of loading. Therefore, one may assume that L, was not greatly affected by u, and write €or the incremental damage at ur = 0.0015, 6D'
=
K',L,
ka"N
=
(124
where K', = 0.024 - 0.018 = 0.006 is the number of additional cracks/mm2 produced by the experimental loading. At u" = 0.0025,
6D" = K" dL c
=
where K " d = 0.056 - 0.018 bining these equations, q
=
kurrqN
=
(12b)
0.038 rnmp2. Com-
[In(K",/Kr,)]/[In(ur'/u')]
=
3.6
(13)
Because it is based on only two data points, this estimate of q is not likely to be as accurate as those obtained from S-N curves. Either through inaccuracy or becauqe it represents different experimental circumstances, the value is somewhat smaller than those obtained for fatigue-to-failure of machined bone specimens. However, it is comparable with the failure-calculated values listed in Table 1 . To adequately test the theory using this kind of data, the experiments must be repeated using more values of u, and q should be calculated by regression rather than by Eq. 13. One can also imagine experiments in which specimens are examined for damage after various numbers of load cycles when the applied stress is held constant. These experiments would move specimens along DELs formed by the intersections of the damage surface with constant S planes. The repair side of the theory could be tested in several ways. One of these will be sketched using data from the literature. Estimation of k from Equilibrium Damage Equation 10 can be rearranged to give k
Y
FIG. 4. Sketch illustrating the damage experiments of Burr et al. in the context of the damage surface. See text for details.
J Orthop R e s , Vol. 10, No. 6 , 1992
= (7 R,2@'~D,Y(~qf')
(14)
Do may be estimated for human rib. Burr and Stafford (4) counted 0.14 fatigue cracks/mm2 in a sixth rib from a man in his 60s. Subsequently, the present author measured the mean length of these cracks to be 0.088 ? 0.038 mm. Assuming that fatigue damage was being resorbed as fast as it was being produced in this rib, D,, = 0.14 x 0.088 = 0.012 mm-'. To estimate the value off, to use in the calculation, an approach suggested by Beaupre et al. ( I )
FATIGUE DAMAGE A N D REPAIR IN CORTICAL BONE
may be used. By considering data from experiments in which the strain magnitude and number of load cycles per day necessary to prevent bone loss were determined, they found by regression that for this kind of equilibrium, fL =
(151
*rrl
where cr is P-P applied stress, JI is applied cycles/ day, $ = 0.00263 day-", and m ;= 4. It will be assumed that, normally, equilibrium simultaneously exists for both bone mass and fatigue damage repair. One may use Eq. 15 to substitute for,f, in Eq. 14, obtaining k
= (T
R,' f a Fs D,)/(cr"-" $'")
(16)
Because only a single CT value is being considered and larger values of u contribute much more to fatigue than moderate values, cr will be set equal to 0.002, consistent with in vivo strain measurements for vigorous activity (26). There being no reliable data on F,, the value 5 will arbitrarily be used. R,, til,and rn will be set equal to the previously mentioned typical values (R, = 0.100 mm, fu = 1.1 BMU/mm'/yr = 0.0030 BMU/mm'/day, m = 4). Finally, a value must be assigned to q. Given the sensitivity of q to the type of loading, a good estimate of k will depend on how representative q is of the type of loading experienced by the human rib. One would expect this to be some kind of bending. Table 2 shows the values of k and DFAII, = kC calculated using the values of q obtained from the fatigue experiments in Table 1, which involved bending or reversed tension-compression. The estimates of DFAILvary from 0.04 to 108 mm-'. It would be useful to compare the predicted D,,, values with the extreme case in which cracks occur between all the lamellae in every osteon. An osteon typically contains 16 lamellar interfaces (13). If the Haversian canal and cement line radii are R , = 0.020 mm and R , = 0.100 mm, respectively, and the lamellae are of uniform thickness. the total inTABLE 2. Damage rute coeflicient and damage-at-failure e5fimates bused on equilibrium in human rib Type of loading Rotating cantilever Reversed cantilever Partially reversed Fully reversed Mixed
k mm-l 3.8 x 2.7 x 1.2 x 0.9 x 4.9 x
1015 1013
DFAIT rnm-'
10*6 10"
9.9 0.04 108 62
IOU
1 .s
823
terlamellar length seen in a cross-section can be shown to be 6.28 mm. The cross-sectional area of the osteon would be TR,' = 0.0314 mm', so the total potential interlamellar crack damage is 200 mm-I. As expected, this maximum possible damage estimate is greater than the estimates of DFAIL. When one attempts to calculate k from the Burr experiments (using k = D/uqN), one finds that the result ("9 x 10' mm-') is much smaller than the values in Table 2. This could be because the calculation is very sensitive to the value of q, which may actually be >3.6. Another possibility is that the damage surface is curved rather than flat. In that case the linear formulation presented here could be modified to fit the surface by making k and/or q suitable functions of D . DISCUSSION
The theory developed here provides preliminary answers to the three question\ posed in the introduction. The relationships between fatigue crack damage and S, N , and fatigue failure are embodied in Eq. 3 and Fig. 3. For cortical bone, the relationship between fatigue damage repair and the variables of remodeling is given in Eq. 8, and Eq. 10 predicts equilibrium damage as a function of loading and remodeling conditions. This model was used to compare the results of three kinds of experiments found in the literature: fatigue-to-failure, fatigue to an increased level of damage, and equilibrium damage. There is a measure of agreement between the theory and such experiments, but the existing data are too sparse and inappropriately controlled to provide a good test. Figure 3 provides a paradigm for more rigorous testing of the damage formation side of the theory. Experiments to test the repair side of the theory would involve creating predictable amounts of damage in living animals and studying its diminishment as a function of remodeling using standard histomorphometric techniques. As the theory is tested, many other factors must be addressed. How are fatigue damage and failure affected by histologic type (e.g., plexiform, circumferential lamellar, osteonal) and histocomposition (e.g., porosity, mineralization, collagen fiber orientation). There are very limited data on the effects of strain rate on bone fatigue (21,22,28). How does strain rate affect f i r and D,,,'? Can the effects of strain rate be separated from those off; ? It may not be possible to simply use k to account for strain
J Orrhop Res, Vol. 10. No.6. 1992
824
B . MARTIN
log s
log N FIG. 5. Sketch relating the theory presented here to Palmgren-Miner analyses of fatigue damage in engineering materials. There is debate about whether equidamage contours (e.g., A, B, C) are parallel (as shown) or converge at the endurance limit.
rate; for example, both k and q may be functions of strain rate, as well as other loading conditions. And of course, the theory must be extended to cancellous bone. When the damage surface is viewed from above (Fig. 5 ) , equidamage contours appear as prefailure damage curves, such as those used in the PalmgrenMiner analysis of residual fatigue life and its more recent descendants (2). This connection is significant because it relates the theory presented here to engineering analyses of damage and residual fatigue life in other materials. There is debate in the engineering literature about the slope of the equidamage contours relative to the failure line, and their relationship to the endurance limit (2,20,29). Knowledge of the shapes of damage surfaces for bone and other materials would substantially resolve these questions. APPENDIX
Mathematical Symbols S N
R lJ
fL NFAIL
A B C
Stress or strain Number of load cycles Region of cortical bone Cross-section through @ Peak-to-peak stress, usually normalized by elastic modulus Loading frequency Number of cycles at failure Intercept of the log-log S-N curve Slope of the log-log S-N curve Ratio. -A/B
J Orthop Rer. Vol. 10, N o . 6 , 1992
-B-l Damage (6D = an increment of damage) Equil ibrium damage Damage at failure Damage formation rate Damage repair rate Damage rate coefficient Time Fractional part of il resorbed per day Radii of cement line and Haversian canal, respectively BMU activation frequency Damage repair specificity factor Empirical constants Fatigue crack density (cracks per unit area of cross-section) Mean length of fatigue cracks Acknowledgment: I t h an k the individuals who reviewed this p ap er for their extremely helpful suggestions, which significantly improved t h e work.
REFERENCES 1. Beaupre GS, Orr TE. Carter DK: An approach to time-
dependent bone modeling and remodeling-application: a preliminary remodeling simulation. J Orthop Res 8:662470. 1990 2. Ben-Amoz M: A cumulative damage theory for fatigue life prediction. Eng Frwture Mech 37:341-347, 1990 3. Burr DB, Martin RB. Schaffler ME, Radin EL: Bone remodeling in response to in vivo fatigue microdamage. J Riomech 18:18%200, 1985 4. Burr DB, Stafford T: Validity of the bulk-staining technique to separate artifactual from in vivo bone microdamage. Clin Orrhop Re1 Res 260305-308, 1990 5. Caler WE, Carter DR: Bone creep-fatiguc damage accumulation. I Biomech 22:625-6,35, 1989 6. Carter DR, Caler WE: A cumulative damage model for bone fracture. J Orthop Res 3:8&90. 1985 7. Carter DR; Caler WE: Cycle-dependent and time-dependent bone fracture with repeated loading. J Bionzech Eng 105: 166170, 1983 8. Carter D, Caler WE, Spengler DM. Frankel VH: Fatigue behavior of adult cortical bone: The influence of mean strain and strain runge. Arta Orthop Scand 52:481490, 1981 9. Carter DR, Hayes WC: Compact bone fatigue damage: a microscopic examination. Clin Orthop Re1 Rex 127:265-274, 1977 10. Chamey A, Tschantz P: Mechanical influences in bone remodeling. Experimental research on Wolff's law. J Biomech 5:173-180, 1972 11. Currey JD: Stress concentrations in bone. Q J Microsc Sci 103:111-1 33, 1962 12. Fong JT: What i s fatigue damage? In: Damage in composite materials: basic mechanisms, accumulation, tolerance, and characterization, ed by K L Reifsnider. Philadelphia, American Society for Testing and Materials, 1980 13. Frost HM: Bone remodelling dynamics. Springfield, IL, Charles c. Thomas, 1963 14. Frost HM: Presence of microscopic cracks in vivo in bone. Henry Ford Hosp Med Bull 8:27-35, 1960
FATIGUE DAMAGE A N D REPAIR IN CORTICAL BONE 15. Frost HM: Tetracycline-based histological analysis of bone remodeling. c'ulcif Tissue Res 3:211-237, 1969 16. Gray RJ, Korbacher GK: Compressive fatigue behavior of bovine compact bone. J Biomech 7287-292, 1974 17. Jamison RD, Schulte K , Reifsnider KL, Stinchcomb WW: Characterization and analysis of damage mechanisms in tension-tension fatigue of graphite/epoxy laminates. In: Effects sf defects in composite materials, Philadelphia, American Society for Testing and Materials, 1984 18. Kelly A, Davies GJ: The principles of the fibre reinforcement of metals. MetnllurX Rev 10:l-77. 196.5 19. King AT, Evans FG: Analysis of fatigue strength of human compact bone by the Weibull method. In: Digest of the 7th international conferences an medictrl and 6iologic.al engineering, ed by B Jacobson, Stockholm, Royal Academy of Engineering Sciences, 1967 20. Kujawski D, Ellyin F: A cumulative damage theory for fatigue crack initiation and propagation. int J Fatigrie 6:83-88, 1984 21. Lafferty JF: Analytical model of the fatigue characteristics of bone. Aviat Space Environ Med 49:17&174, 1978 22. Lafferty JF, Raju PVV: The influence of stress frequency on the fatigue strength of cortical bone. J Biomech Eng 101: 112-113, 1979
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23. Martin RB, Burr DB: A hypothetical mschanism for the stimulation of osteonal remodeling by fatigue damage. J Biomech 15: 137-139. 1982 24. Mori S . Burr DB: Increased intracortical remodeling following fatigue microdamage. Transactions, combined meeting of the Orthopaedic Research Societies of the USA. Japan, and Canada. Banff. Canada, October 21-23, 1991 25. Reiily DT, Burstein AH, Frunkel VH: The elastic modulus for bone. J Biomech 7:211-275, 1974 26. Rubin CT: Skeletal strain and the functional significance of bone architecture. CalciJ'Tissue Int 36s: 11-18, 1984 27. Schaffler MB, Radin EL, Burr DB: Long-term fatiguc behavior of compact bone at low strain magnitude and rate. BoPie 111321-326, 1990 28. Schaffler MB. Radin EL, Burr DB: Mechanical and morphological effccts of strain rate on fatigue of compact bone. Bone 10207-214, 1989 29. Subramanyan S: A cumulative damage rule based on the knee point of the S-N curve. J Eng ,Materials Techno/ 98: 316-321, 1Y76 30. Talreja R: Fnfigue of composite materials, Lancaster, PA, Technomic, 1987
J Ortho!) Res, Vol. 10, No. 6 , 1992
