J. BIOMED. MATER. RES. SYMPOSIUM
No. 7, pp. 637-648 (1976)
The Fracture Mechanics of Fatigue Crack Propagation in Compact Bone T. M. WRIGHT and W. C. HAYES, Department of Materials Science and Department of Applied Mechanics, Stanford University, Stanford, California
Summary The purpose of this investigation was to apply the techniques of fracture mechanics to a study of fatigue crack propagation in compact bone. Small cracks parallel to the long axis of the bone were initiated in standardized specimens of bovine bone. Crack growth was achieved by cyclically loading these specimens. The rate of crack growth was determined from measurements of crack length versus cycles of loading. The stress intensity factor a t the tip of the crack wtts calculated from knowledge of the applied load, the crack length, and the specimen geometry. A strong correlation was found between the experimentally determined crack growth rate and the applied stress intensity. The relationship takes the form of a power law similar to that for other materials. Visual observation and scanning electron microscopy revealed that crack propagation occurred by initiation of subcritical cracks ahead of the main crack.
INTRODUCTION
Fatigue Fractures of Bone Fatigue fractures of bone have been noted clinically for more than a century. These fractures often occur during prolonged exercise involving repeated (or cyclic) bone loading. Athletes, military recruits, and ballet dancers are among those who suffer fatigue fractures (Fig. 1). I n many cases, the patient remains unaware of a developing fatigue crack until it propagates to the point of inducing pain or catastrophic failure. When a developing fatigue crack is observed clinically, there is presently no method of determining how near the bone is to complete fracture. Such a method would require two kinds of information: 1) the size t o which a fatigue crack must 637 @ 1976 by John Wiley & Sons, Inc.
638
WRIGHT AND HAYES
Fig. 1. X-Ray of tibia1 fatigue fracture.
grow to cause complete fracture, and 2) a quantitative description of the fatigue crack propagation process. The purpose of this investigation was to apply the techniques of fracture mechanics t o a study of fatigue crack propagation in compact bone. It is important t o note that the data in this study was collected in vitro and thus does not include the in vivo healing response. Such data will, however, serve as a n effective lower bound when used in predicting
MECHANICS OF FATIGUE CRACK PROPAGATION
639
the remaining life. This paper describes the results for propagation in a direction parallel to the long axis of the bone. Before reporting these results, a review of the basic concepts of fracture mechanics and its application to bone is provided.
Fracture Mechanics Linear elastic fracture mechanics considers the stress intensification around cracks in a body. Fracture criteria are established for structures containing cracks or crack-like defects. The approach is to relate the magnitude of the stresses a t the crack tip to the applied nominal stress, the material properties, and the size of the defect (Fig. 2). This magnitude can be expressed by a single Y
0
=-
(2nr)
112
cos
2 2
e sin -)38 (1 - s i n 2 2
Fig. 2. Schematic of the elastic stress distribution near the tip of a crack.
640
WRIGHT AND HAYES
parameter, the stress intensity factor. The relationship between this parameter and the other variables has been established for several geometric configurations.2 It takes the form :
K = f ( Y , u, 4 where K is the stress intensity factor, Y is a geometry parameter, u is the applied stress, and a is the crack size. The fracture criterion is that the crack will propagate to failure whenever the stress intensity at the crack tip exceeds a critical value called the critical stress intensity factor K,. The critical stressintensity factor is thus a measure of the material's resistance to unstable crack propagation. The critical stress-intensity factor is a function of structural configuration, crack size, and external loading. When the stress intensity factor is equal to or greater than the critical stress intensity, catastrophic fracture of the material will result. The application of fracture mechanics to anisotropic materials has been investigated both theoretically3 and experimentally. * While such studies showed that the fracture mechanics approach is indeed applicable, it was found that the distribution of stresses around the crack tip is dependent on the orientation of the crack. Stress intensity factors can therefore be defined only for specific crack orientations. In orthotropic materials, these orientations are parallel to the principal directions of elastic symmetry. For bone, these directions are parallel and normal to the long axis. The application of fracture mechanics to compact bone has received limited attention. Piekarski15Pope and Murphy16and Pope and Outwater7 measured the energy to cause fracture, but did not determine the critical stress-intensity factor K,. Melvin and Evanss and Margel-Robertsong measured critical stress-intensity factors and critical strain energy release rates. Fatigue Crack Growth Previous studies of the fatigue of bone have concentrated on the conventional endurance limit approach to fatigue, seeking to establish the cyclic life of bone as a function of the applied stress.'OJ1 Few investigations have concentrated on the mechanical factors leading to fatigue failure of bone. Fatigue fracture processes in anisotropic materials such as bone involve three steps: 1) the initiation of a microcrack, 2) stable growth of the microcrack under cyclic loading, and 3) unstable crack propagation at a critical crack size. Much of
MECHANICS OF FATIGUE CRACK PROPAGATION
64 1
the useful life of the structure involves the growth of a defect from some subcritical size t o the critical size. The rate a t which the fatigue crack grows depends on the range of cyclic stress intensity a t the crack tip. This relationship has been studied for a wide variety of materials and has been found t o generally obey a power law:L2 da/dN = CARn where a is the crack length, N is the number of cycles of loading, AK is the cyclic range of the stress intensity factor, and C and n are constants. T o establish the values of the constants in the growth-rate law, cracks must be initiated in standardized specimens for which the stress intensity expression is known. Crack growth can be achieved by cyclicalIy loading the specimens. The experimentally determined crack growth rate can then be related t o the applied range of stress intensity at the crack tip. Thus, the fracture mechanics approach t o fatigue incorporates all the pertinent variables (cyclic stress, geometry, and flaw size) into one parameter, the stress intensity factor. This allows fatigue crack propagation data collected from one loading configuration t o apply t o other geometries for which the stress intensity expressions are also known. The remaining life of a cracked structure subjected t o fatigue can be determined from the fatigue crack growth-rate law and the critical crack size for fracture. The rate of fatigue crack growth is dependent on the range of K . The critical crack size is defined by K,. The remaining life of the structure is then found by integrating the crack propagation data from the initial K t o K,.
TESTING METHODS For the present investigation, longitudinally oriented double cantilever beam specimensL3were machined from the mid-diaphysis of fresh, adult bovine femora and tibia. Test specimens were 7.6 em long and 2.3 cm wide (Fig. 3). Thicknesses ranged from 0.32-0.41 cm. Specimens were machined in a wet condition, equilibrated in Ringer’s solution overnight, and stored a t - 20°C until testing.I4 Testing was conducted on a n MTS closed-loop hydraulic test system. Four cyclic load amplitudes (from 5 f 5 to 15 j=15 kg) were used. Tests were run a t 1-5 Hz with a sinusoidal wave form. During testing, the specimen was immersed in a water bath maintained
WRIGHT AND HAYES
642
Fig. 3.
Double cantilever beam bovine test specimen.
a t 37°C. As the test progressed, periodic measurements of crack length were made by using a traveling microscope attached to the load frame of the test system. The number of elapsed cycles was automatically recorded on the counter panel of the test system. Microstructure studies of the fractured specimens were performed by cutting away small cross sections of bone near the fracture surface. These samples were ultrasonically degreased in trichloroethylene, embedded in methyl methacrylate, and viewed with a reflected light microscope. Fracture surfaces were ultrasonically degreased and coated with gold-palladium for examination in a scanning electron microscope.
ANALYSIS The data collected from each specimen was plotted as crack length versus the accompanying number of cycles of loading (Fig. 4a). These data were curve-fitted by a least-squares, cubic-spline program. The result is a smooth curve yielding accurate values of the slope or crack-tip velocity. The corresponding value of the stress intensity factor was calculated from the boundary collocation expression of Gross and Srawley:15 K = - ( Pa 3.46 2.38 BH3I2 a
+
where P is the applied load, H is the half-beam height, and B is the specimen thickness.
MECHANICS OF FATIGUE CRACK PROPAGATION
643
4.0
' I
r4
3 0
3.0
v
1.51 0
I
4
2
8
6
10
12
ELAPSED CYCLES (Thousands)
(a)
h
al
2.6 0
1.4
2.8
3.0
3.2
AK (m/m3/2) ( l o g scale)
(b) Fig. 4. (a) Crack length plotted against the elapsed number of cycles of load application. Curve shown is the result of the least-squares, cubic-spline program. (b) The resulting crack propagation rates plotted against the range of stress intensity with a linear regression curve.
WRIGHT AND HAYES
644
RESULTS AND DISCUSSION Fatigue crack propagation rates between 7.2 X lo-’ m/cycle and 3.4 x m/cycle were measured for the bovine compact bone specimens. A strong correlation was found between the fatigue crack propagation rate and the range of stress intensity a t the crack tip. The data, along with the curves from a linear regression analysis, are presented in Figure 5. The scatter in the AK values for the same growth rate is comparable to that found by Margel-Robertsong in measuring critical stress-intensity factors for compact bone. It is apparent from the slopes n that the longitudinal fatigue crack propagation behavior is fairly uniform for the specimens tested. For any given increase in AK, due either to an increase in crack length or to applied load, the same relative increase in fatigue crack +Tibia.
n
OTibia, n
OFemur. n AFemUr. n OFemur.
@Femur, n
.Femur,
I
2
n
-
5.1, 1
-
2 . 9 , 1 Hz
-
I
-
/dy
3 3 , 1 Hz
=
HI
/,
2 3 . 83 ., 1 Hz
4.4. 5 Hz
0’ 1.2.
I 3
I Hz
A/
I
I
4 5 AK, MNm-3’2
I 6
..
I 7
I 8
I
Fig. 5 . Fatigue crack propagation rate plotted against the range of stress intensity for bovine specimens tested with cracks oriented parallel to the long axis of the bone.
MECHANICS O F FATIGUE CRACK PROPAGATION
645
growth rate occurs. The fact that the specimens tested a t 5 Hz had higher n values than those tested a t 1 Hz suggests that frequency influences fatigue crack propagation. This has been shown to be true for metalsI6 and polymer^.'^ However, insufficient data exists to determine what effect frequency had in this study. Figure 6 compares the fatigue crack propagation behavior of bone with that for other materials. Metals exhibit comparable fatigue growth rates for a given AK when normalized with respect to the elastic modulus. This suggests that the main factor in fatigue crack propagation in metals is the elastic strain concentration a t the tip of the crack, and not the individual flow properties of the metals.l8
4
(ml'*)
(log scale)
Fig. 6. Comparative fatigue crack propagation behavior in rnetals,'s polymers,*g and compact bone.
646
WRIGHT AND HAYES
Fig. 7. Scanning electron microscopy photomicrograph (200 X ) of fatigue fracture surface. Crack was propagating left to right.
Polymers do not exhibit such insensitivity to structural material properties. l9 Bone also does not appear to comply with the common behavior found in metals. Microstructure studies revealed a laminated bone structure consisting of alternating layers of elongated primary osteons and woven bone. I n other materials with a layered, fibrous structure (such as balsa wood), crack propagation is confined mostly to the weak interfaces between layers. Propagation occurs by initiation and growth of subcritical flaws ahead of the main crack.4 The main crack propagates by joining with these flaws. I n the bone specimens tested, this means of propagation was confirmed visually and with scanning electron microscopy. During
MECHANICS O F FATIGUE CRACK PROPAGATION
647
the test, small cracks were observed initiating ahead of the main crack tip. These smaller cracks did not always occur on the same plane as the main crack. When the main crack propagated by joining with the smaller cracks, the fracture surfaces assumed a rough appearance. The scanning electron microscopy photograph in Figure 7 shows the protruding areas originally between the crack tip and a smaller crack initiated ahead of the tip. Figure 7 also verifies that the crack propagation is not confined to a single interface, but rather skips across laminae.
CONCLUSIONS Fatigue tests of bovine compact bone specimens resulted in a strong correlation between the longitudinal fatigue crack propagation rate and the applied stress intensity a t the tip of the crack. This correlation takes the form of a power law such as that found for other materials. Longitudinal fatigue crack propagation occurred by initiation and growth of subcritical cracks ahead of the main crack. Fatigue crack length was increased when the main crack joined with these smaller flaws. This research was supported by the National Science Foundation through the Center for Materials Research a t Stanford University.
References 1. J. M. Morris and L. D. Blickenstaff, Fatigue Fractures, Charles C Thomas, Springfield, Illinois, 1967. 2 . W. F. Brown, Jr. and J. E. Srawley, Plane Strain Crack Toughness Testing of High Strength Metallic Materials, American Society for Testing and Materials, Philadelphia, Pennsylvania, 1966. 3. D. D. Ang and W. L. Williams, J . A p p l . Mech., 28, 372 (1961). 4. E. M. Wu, J . A p p l . Mech., 34, 967 (1967). 5. K. Piekarski, J . A p p l . Phys., 41, 215 (1970). 6. M. H. Pope and M. C. Murphy, Med. B i d . Eng., 11, 763 (1974). 7. M. H. Pope and J. 0. Outwater, J . Biomech., 5, 457 (1972). 8. J. W. Melvin and F. G. Evans, paper presented a t Biomaterials Symposium, ASME, Detroit, Michigan, Nov., 1973. 9. D. R. Margel-Robertson, Ph.D. Thesis, Stanford University, Stanford, California, 1974. 10. S. A. V. Swanson, M. A. R. Freeman, and W. H. Day, Med. Biol.Eng., 9, 23 (1971).
WRIGHT AND HAYES
648
11. D. It. Carter and W. C . Hayes, J. Biochem., 9, 27 (1976).
12. 13. 14. 15.
16. 17. 18. 19.
P. C. Paris, Ph.D. Thesis, Lehigh University, 1962. S. Mostovoy, P. B. Crosley, and E. J. Ripling, J . Mater., 2, 661 (1967). E. I). Sedlin and C. Hirsch, Acta Orth. Scand., 37, 29 (1966). B. Gross and J. E. Srawley, Stress Intensity Factors by Boundary Collocation for Single-Edge-Notch Specimens Subject to Splitting Forces, NASA Tech. Note D-3295, 1966. R. P. Wei, Eng. Frac. Mech., I, 633 (1970). R. W. Hertzberg, Closed Loop Mag., 3, 12 (1973). S. Pearson, Nature, 211, 1077 (1966). R. W. Hertzberg, H. Nordberg, and J. A. Manson, J . Mater. Sci., 5 , 521 (1970).
No. 7, pp. 637-648 (1976)
The Fracture Mechanics of Fatigue Crack Propagation in Compact Bone T. M. WRIGHT and W. C. HAYES, Department of Materials Science and Department of Applied Mechanics, Stanford University, Stanford, California
Summary The purpose of this investigation was to apply the techniques of fracture mechanics to a study of fatigue crack propagation in compact bone. Small cracks parallel to the long axis of the bone were initiated in standardized specimens of bovine bone. Crack growth was achieved by cyclically loading these specimens. The rate of crack growth was determined from measurements of crack length versus cycles of loading. The stress intensity factor a t the tip of the crack wtts calculated from knowledge of the applied load, the crack length, and the specimen geometry. A strong correlation was found between the experimentally determined crack growth rate and the applied stress intensity. The relationship takes the form of a power law similar to that for other materials. Visual observation and scanning electron microscopy revealed that crack propagation occurred by initiation of subcritical cracks ahead of the main crack.
INTRODUCTION
Fatigue Fractures of Bone Fatigue fractures of bone have been noted clinically for more than a century. These fractures often occur during prolonged exercise involving repeated (or cyclic) bone loading. Athletes, military recruits, and ballet dancers are among those who suffer fatigue fractures (Fig. 1). I n many cases, the patient remains unaware of a developing fatigue crack until it propagates to the point of inducing pain or catastrophic failure. When a developing fatigue crack is observed clinically, there is presently no method of determining how near the bone is to complete fracture. Such a method would require two kinds of information: 1) the size t o which a fatigue crack must 637 @ 1976 by John Wiley & Sons, Inc.
638
WRIGHT AND HAYES
Fig. 1. X-Ray of tibia1 fatigue fracture.
grow to cause complete fracture, and 2) a quantitative description of the fatigue crack propagation process. The purpose of this investigation was to apply the techniques of fracture mechanics t o a study of fatigue crack propagation in compact bone. It is important t o note that the data in this study was collected in vitro and thus does not include the in vivo healing response. Such data will, however, serve as a n effective lower bound when used in predicting
MECHANICS OF FATIGUE CRACK PROPAGATION
639
the remaining life. This paper describes the results for propagation in a direction parallel to the long axis of the bone. Before reporting these results, a review of the basic concepts of fracture mechanics and its application to bone is provided.
Fracture Mechanics Linear elastic fracture mechanics considers the stress intensification around cracks in a body. Fracture criteria are established for structures containing cracks or crack-like defects. The approach is to relate the magnitude of the stresses a t the crack tip to the applied nominal stress, the material properties, and the size of the defect (Fig. 2). This magnitude can be expressed by a single Y
0
=-
(2nr)
112
cos
2 2
e sin -)38 (1 - s i n 2 2
Fig. 2. Schematic of the elastic stress distribution near the tip of a crack.
640
WRIGHT AND HAYES
parameter, the stress intensity factor. The relationship between this parameter and the other variables has been established for several geometric configurations.2 It takes the form :
K = f ( Y , u, 4 where K is the stress intensity factor, Y is a geometry parameter, u is the applied stress, and a is the crack size. The fracture criterion is that the crack will propagate to failure whenever the stress intensity at the crack tip exceeds a critical value called the critical stress intensity factor K,. The critical stressintensity factor is thus a measure of the material's resistance to unstable crack propagation. The critical stress-intensity factor is a function of structural configuration, crack size, and external loading. When the stress intensity factor is equal to or greater than the critical stress intensity, catastrophic fracture of the material will result. The application of fracture mechanics to anisotropic materials has been investigated both theoretically3 and experimentally. * While such studies showed that the fracture mechanics approach is indeed applicable, it was found that the distribution of stresses around the crack tip is dependent on the orientation of the crack. Stress intensity factors can therefore be defined only for specific crack orientations. In orthotropic materials, these orientations are parallel to the principal directions of elastic symmetry. For bone, these directions are parallel and normal to the long axis. The application of fracture mechanics to compact bone has received limited attention. Piekarski15Pope and Murphy16and Pope and Outwater7 measured the energy to cause fracture, but did not determine the critical stress-intensity factor K,. Melvin and Evanss and Margel-Robertsong measured critical stress-intensity factors and critical strain energy release rates. Fatigue Crack Growth Previous studies of the fatigue of bone have concentrated on the conventional endurance limit approach to fatigue, seeking to establish the cyclic life of bone as a function of the applied stress.'OJ1 Few investigations have concentrated on the mechanical factors leading to fatigue failure of bone. Fatigue fracture processes in anisotropic materials such as bone involve three steps: 1) the initiation of a microcrack, 2) stable growth of the microcrack under cyclic loading, and 3) unstable crack propagation at a critical crack size. Much of
MECHANICS OF FATIGUE CRACK PROPAGATION
64 1
the useful life of the structure involves the growth of a defect from some subcritical size t o the critical size. The rate a t which the fatigue crack grows depends on the range of cyclic stress intensity a t the crack tip. This relationship has been studied for a wide variety of materials and has been found t o generally obey a power law:L2 da/dN = CARn where a is the crack length, N is the number of cycles of loading, AK is the cyclic range of the stress intensity factor, and C and n are constants. T o establish the values of the constants in the growth-rate law, cracks must be initiated in standardized specimens for which the stress intensity expression is known. Crack growth can be achieved by cyclicalIy loading the specimens. The experimentally determined crack growth rate can then be related t o the applied range of stress intensity at the crack tip. Thus, the fracture mechanics approach t o fatigue incorporates all the pertinent variables (cyclic stress, geometry, and flaw size) into one parameter, the stress intensity factor. This allows fatigue crack propagation data collected from one loading configuration t o apply t o other geometries for which the stress intensity expressions are also known. The remaining life of a cracked structure subjected t o fatigue can be determined from the fatigue crack growth-rate law and the critical crack size for fracture. The rate of fatigue crack growth is dependent on the range of K . The critical crack size is defined by K,. The remaining life of the structure is then found by integrating the crack propagation data from the initial K t o K,.
TESTING METHODS For the present investigation, longitudinally oriented double cantilever beam specimensL3were machined from the mid-diaphysis of fresh, adult bovine femora and tibia. Test specimens were 7.6 em long and 2.3 cm wide (Fig. 3). Thicknesses ranged from 0.32-0.41 cm. Specimens were machined in a wet condition, equilibrated in Ringer’s solution overnight, and stored a t - 20°C until testing.I4 Testing was conducted on a n MTS closed-loop hydraulic test system. Four cyclic load amplitudes (from 5 f 5 to 15 j=15 kg) were used. Tests were run a t 1-5 Hz with a sinusoidal wave form. During testing, the specimen was immersed in a water bath maintained
WRIGHT AND HAYES
642
Fig. 3.
Double cantilever beam bovine test specimen.
a t 37°C. As the test progressed, periodic measurements of crack length were made by using a traveling microscope attached to the load frame of the test system. The number of elapsed cycles was automatically recorded on the counter panel of the test system. Microstructure studies of the fractured specimens were performed by cutting away small cross sections of bone near the fracture surface. These samples were ultrasonically degreased in trichloroethylene, embedded in methyl methacrylate, and viewed with a reflected light microscope. Fracture surfaces were ultrasonically degreased and coated with gold-palladium for examination in a scanning electron microscope.
ANALYSIS The data collected from each specimen was plotted as crack length versus the accompanying number of cycles of loading (Fig. 4a). These data were curve-fitted by a least-squares, cubic-spline program. The result is a smooth curve yielding accurate values of the slope or crack-tip velocity. The corresponding value of the stress intensity factor was calculated from the boundary collocation expression of Gross and Srawley:15 K = - ( Pa 3.46 2.38 BH3I2 a
+
where P is the applied load, H is the half-beam height, and B is the specimen thickness.
MECHANICS OF FATIGUE CRACK PROPAGATION
643
4.0
' I
r4
3 0
3.0
v
1.51 0
I
4
2
8
6
10
12
ELAPSED CYCLES (Thousands)
(a)
h
al
2.6 0
1.4
2.8
3.0
3.2
AK (m/m3/2) ( l o g scale)
(b) Fig. 4. (a) Crack length plotted against the elapsed number of cycles of load application. Curve shown is the result of the least-squares, cubic-spline program. (b) The resulting crack propagation rates plotted against the range of stress intensity with a linear regression curve.
WRIGHT AND HAYES
644
RESULTS AND DISCUSSION Fatigue crack propagation rates between 7.2 X lo-’ m/cycle and 3.4 x m/cycle were measured for the bovine compact bone specimens. A strong correlation was found between the fatigue crack propagation rate and the range of stress intensity a t the crack tip. The data, along with the curves from a linear regression analysis, are presented in Figure 5. The scatter in the AK values for the same growth rate is comparable to that found by Margel-Robertsong in measuring critical stress-intensity factors for compact bone. It is apparent from the slopes n that the longitudinal fatigue crack propagation behavior is fairly uniform for the specimens tested. For any given increase in AK, due either to an increase in crack length or to applied load, the same relative increase in fatigue crack +Tibia.
n
OTibia, n
OFemur. n AFemUr. n OFemur.
@Femur, n
.Femur,
I
2
n
-
5.1, 1
-
2 . 9 , 1 Hz
-
I
-
/dy
3 3 , 1 Hz
=
HI
/,
2 3 . 83 ., 1 Hz
4.4. 5 Hz
0’ 1.2.
I 3
I Hz
A/
I
I
4 5 AK, MNm-3’2
I 6
..
I 7
I 8
I
Fig. 5 . Fatigue crack propagation rate plotted against the range of stress intensity for bovine specimens tested with cracks oriented parallel to the long axis of the bone.
MECHANICS O F FATIGUE CRACK PROPAGATION
645
growth rate occurs. The fact that the specimens tested a t 5 Hz had higher n values than those tested a t 1 Hz suggests that frequency influences fatigue crack propagation. This has been shown to be true for metalsI6 and polymer^.'^ However, insufficient data exists to determine what effect frequency had in this study. Figure 6 compares the fatigue crack propagation behavior of bone with that for other materials. Metals exhibit comparable fatigue growth rates for a given AK when normalized with respect to the elastic modulus. This suggests that the main factor in fatigue crack propagation in metals is the elastic strain concentration a t the tip of the crack, and not the individual flow properties of the metals.l8
4
(ml'*)
(log scale)
Fig. 6. Comparative fatigue crack propagation behavior in rnetals,'s polymers,*g and compact bone.
646
WRIGHT AND HAYES
Fig. 7. Scanning electron microscopy photomicrograph (200 X ) of fatigue fracture surface. Crack was propagating left to right.
Polymers do not exhibit such insensitivity to structural material properties. l9 Bone also does not appear to comply with the common behavior found in metals. Microstructure studies revealed a laminated bone structure consisting of alternating layers of elongated primary osteons and woven bone. I n other materials with a layered, fibrous structure (such as balsa wood), crack propagation is confined mostly to the weak interfaces between layers. Propagation occurs by initiation and growth of subcritical flaws ahead of the main crack.4 The main crack propagates by joining with these flaws. I n the bone specimens tested, this means of propagation was confirmed visually and with scanning electron microscopy. During
MECHANICS O F FATIGUE CRACK PROPAGATION
647
the test, small cracks were observed initiating ahead of the main crack tip. These smaller cracks did not always occur on the same plane as the main crack. When the main crack propagated by joining with the smaller cracks, the fracture surfaces assumed a rough appearance. The scanning electron microscopy photograph in Figure 7 shows the protruding areas originally between the crack tip and a smaller crack initiated ahead of the tip. Figure 7 also verifies that the crack propagation is not confined to a single interface, but rather skips across laminae.
CONCLUSIONS Fatigue tests of bovine compact bone specimens resulted in a strong correlation between the longitudinal fatigue crack propagation rate and the applied stress intensity a t the tip of the crack. This correlation takes the form of a power law such as that found for other materials. Longitudinal fatigue crack propagation occurred by initiation and growth of subcritical cracks ahead of the main crack. Fatigue crack length was increased when the main crack joined with these smaller flaws. This research was supported by the National Science Foundation through the Center for Materials Research a t Stanford University.
References 1. J. M. Morris and L. D. Blickenstaff, Fatigue Fractures, Charles C Thomas, Springfield, Illinois, 1967. 2 . W. F. Brown, Jr. and J. E. Srawley, Plane Strain Crack Toughness Testing of High Strength Metallic Materials, American Society for Testing and Materials, Philadelphia, Pennsylvania, 1966. 3. D. D. Ang and W. L. Williams, J . A p p l . Mech., 28, 372 (1961). 4. E. M. Wu, J . A p p l . Mech., 34, 967 (1967). 5. K. Piekarski, J . A p p l . Phys., 41, 215 (1970). 6. M. H. Pope and M. C. Murphy, Med. B i d . Eng., 11, 763 (1974). 7. M. H. Pope and J. 0. Outwater, J . Biomech., 5, 457 (1972). 8. J. W. Melvin and F. G. Evans, paper presented a t Biomaterials Symposium, ASME, Detroit, Michigan, Nov., 1973. 9. D. R. Margel-Robertson, Ph.D. Thesis, Stanford University, Stanford, California, 1974. 10. S. A. V. Swanson, M. A. R. Freeman, and W. H. Day, Med. Biol.Eng., 9, 23 (1971).
WRIGHT AND HAYES
648
11. D. It. Carter and W. C . Hayes, J. Biochem., 9, 27 (1976).
12. 13. 14. 15.
16. 17. 18. 19.
P. C. Paris, Ph.D. Thesis, Lehigh University, 1962. S. Mostovoy, P. B. Crosley, and E. J. Ripling, J . Mater., 2, 661 (1967). E. I). Sedlin and C. Hirsch, Acta Orth. Scand., 37, 29 (1966). B. Gross and J. E. Srawley, Stress Intensity Factors by Boundary Collocation for Single-Edge-Notch Specimens Subject to Splitting Forces, NASA Tech. Note D-3295, 1966. R. P. Wei, Eng. Frac. Mech., I, 633 (1970). R. W. Hertzberg, Closed Loop Mag., 3, 12 (1973). S. Pearson, Nature, 211, 1077 (1966). R. W. Hertzberg, H. Nordberg, and J. A. Manson, J . Mater. Sci., 5 , 521 (1970).
