- Email: [email protected]
High strain rate response of rabbit femur bones
Journal of Biomechanics 43 (2010) 3044–3050
Contents lists available at ScienceDirect
Journal of Biomechanics journal homepage: www.elsevier.com/locate/jbiomech www.JBiomech.com
High strain rate response of rabbit femur bones Vasanth Chakravarthy Shunmugasamy a, Nikhil Gupta a,n, Paulo G. Coelho b a
Composite Materials and Mechanics Laboratory, Mechanical and Aerospace Engineering Department, Polytechnic Institute of New York University, 6 MetroTech Center, Brooklyn NY 11201, USA Department of Biomaterials and Biomimetics, New York University, New York, NY 10010, USA
b
a r t i c l e in f o
a b s t r a c t
Article history: Accepted 30 June 2010
Strain rate dependence of the mechanical response of hard tissues has led to a keen interest in their dynamic properties. The current study attempts to understand the high strain rate characteristics of rabbit femur bones. The testing was conducted using a split-Hopkinson pressure bar equipped with a high speed imaging system to capture the fracture patterns. The bones were also characterized under quasi-static compression to enable comparison with the high strain rate results. The quasi-static compressive moduli of the epiphyseal and diaphyseal regions were measured to be in the range of 2–3 and 5–7 GPa, respectively. Under high strain rate loading conditions the modulus is observed to increase with strain rate and attains values as high as 15 GPa for epiphyseal and 30 GPa for diaphyseal regions of the femur. The strength at high strain rate was measured to be about twice the quasi-static strength value. A large number of small cracks initiated on the specimen surface close to the incident bar. Coalescence of crack branches leading to fewer large cracks resulted in specimen fragmentation. In comparison, the quasi-static failure was due to shear cracking. & 2010 Elsevier Ltd. All rights reserved.
Keywords: High strain rate Mechanical properties Fracture Rabbit femur
1. Introduction Bones have a complex and functionally graded microstructure, making it challenging to determine their structure–property relationships. The bone remodeling process depends on the stresses and the strains the bones are subjected to (Osbjorn and Daniel, 2004). The correlation between stresses and strains depends on mechanical properties of the bone. Dependence of mechanical properties on strain rates can significantly change this correlation and affect the bone remodeling process. The high strain rate loading on bones can be encountered in various situations including sport or recreational activities such as running and jumping, falls, and auto accidents, among others (Parenteau et al., 1996; Tencer et al., 2002). Characterizing the mechanical properties of bones under high strain rates can provide additional insight into how bones behave under these loading conditions. These studies can help in understanding the damage mechanisms, designing physical therapy routines, and providing key data for informed design rationale in developing medical devices or protective systems (Gupta and Cho, 2010). In the present study rabbit femur bones are evaluated for deformation and fracture characteristics at high strain rates using a split-Hopkinson pressure bar (SHPB) setup. Compared to quasi-static compression, where strain rates on the order of 10 3 10 2 s 1
n
Corresponding author. Tel.: +1 718 260 3080; fax: + 1 718 260 3532. E-mail address: [email protected] (N. Gupta).
0021-9290/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.jbiomech.2010.06.034
are exerted, strain rates as high as 103 s 1 are achieved in high strain rate testing. The SHPB setup is often utilized to characterize high strain rate properties of engineered and natural composites, such as particle reinforced polymer matrix composites (Song et al., 2007b), metal matrix composites (Tan et al., 2007) and bones (Ferreira et al., 2006; Katsamanis and Raftopoulos, 1990; Lewis and Goldsmith, 1975). SHPB has also been used for testing muscular tissues (Song et al., 2007a; Van Sligtenhorst et al., 2006). A previous study tested complete rabbit femur bones under flexural loading conditions at different deformation rates (Pal and Saha, 1984). The bending stiffness of bones was measured to be higher at higher deformation rates. In addition, the specimen failure mode was observed to be strain rate dependent, where fracture occurred in a single plane under quasi-static loading and crack branching occurred at high strain rates. The high strain rate tests conducted on human femoral cortical bones reported an average elastic modulus of 19.9 GPa at strain rate of 100 s 1 as compared to 16.2 GPa under quasi-static compression (Katsamanis and Raftopoulos, 1990). Studies related to failure mechanisms have observed that the prominent failure mode was shear cracking under quasi-static compression, while at high strain rates splintering of specimens dominated (McElhaney, 1966). The use of high speed imaging systems has been very illustrative in observing the failure pattern in the bovine muscle tissue specimens under high strain rate loading (Van Sligtenhorst et al., 2006). The aim of the present study was to characterize the mechanical response of rabbit femur bones under a wide range
V.C. Shunmugasamy et al. / Journal of Biomechanics 43 (2010) 3044–3050
of strain rates and analyze the fracture pattern by means of high speed imaging. The tested specimens were examined using a scanning electron microscope (SEM) to understand the failure mechanisms at different strain rates.
3045
sampling rate. The displacement of the test equipment crosshead was used to calculate the strain values. The cut specimens used for testing are shown in Fig. 1(b) and (c). Rigid compression supports of steel were used to minimize the compliance errors during the compression test. The compliance error in the present test protocol is expected to be negligible (Odgaard and Linde, 1991; Tony et al., 1997).
2. Materials and methods 2.3. High-strain rate testing 2.1. Test materials Rabbit femur bones were selected for testing in this study. The rabbits were 120–180 days old and had a mean weight of 3 kg. The rabbits were euthanized and the femurs were obtained by sharp dissection. The tissues surrounding the femur were carefully removed and the bones were stored at 20 1C prior to testing (Sedlin and Hirsch, 1966). A total of 8 rabbit femur bones were used for quasistatic and high strain rate compression testing. The bones were cut into 10 mm long specimens (Fig. 1) using a low speed precision diamond blade saw (IsoMets; Buehler Ltd., Lake Placid, NY). Care was taken to ensure that the cut surfaces of the specimen were parallel to each other to minimize the errors in the results (Tony et al., 1997). The surfaces were maintained with less than 1% difference in the specimen length along the entire cross-section. The specimens were cut from the epiphyses–metaphyses (marked 1 and 5 in Fig. 1(a)) and the diaphyses regions (marked 2, 3 and 4 in Fig. 1(a)). The actual density (Sharp et al., 1990) of the bones was measured using the Archimedes’ principle. The average epiphyseal and diaphyseal bone densities were found to be 1622 7284 and 2048 7 454 kg/m3, respectively, which are in line with previously published data (Katsamanis and Raftopoulos, 1990; Rho et al., 1993).
Schematic representation of the SHPB setup is shown in Fig. 2. Prior to testing, the specimen was placed between two long slender bars, called incident and transmitter bars. A striker bar impacts the incident bar to generate an elastic wave (Kolsky, 1964). The incident wave undergoes reflection and transmission at the incident bar–specimen and specimen–transmitter bar interfaces depending on the relative impedance of the bar and specimen materials. Strain gages were bonded to the incident and the transmitter bars to record the strain response with respect to time. A digital oscilloscope was used to record the strain signals and transfer data to a computer for analysis. The incident, reflected, and transmitted wave signals were used to calculate stress, strain, and strain rate with respect to time (t). Stress was calculated from the time dependent strain signal of the transmitted wave, eT(t), by
sðtÞ ¼
Compression testing was carried out on an Instron 4469 universal test system equipped with a 50 kN load cell. The testing was conducted using displacement control mode and at the constant compression rate of 1 mm/min. Load–displacement data were obtained using Bluehill 2.0 software at 20 MHz
Fig. 1. (a) Rabbit femur bone showing markings along which the specimens were cut, (b) epiphyseal specimen from the epiphyses–metaphyses region and (c) diaphyseal specimen.
ð1Þ
where A0 and As are the cross-section areas of the bar and the specimen, respectively, and E is the modulus of the bar material. The strain rate was determined from the strain signal of the reflected wave, eR(t), by
e_ ðtÞ ¼
2.2. Quasi-static testing
A0 EeT ðtÞ As
2c0 eR ðtÞ ls
ð2Þ
where c0 and ls are the speed of sound in the bar and the specimen length, respectively. The strain rate was integrated with respect to time to
Fig. 3. Comparison of the incident stress wave with the summation of transmitted and reflected stress wave.
Fig. 2. Schematic representation of split-Hopkinson pressure bar apparatus used in high strain rate testing.
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obtain strain Z t eðtÞ ¼ e_ ðtÞdt
ð3Þ
0
Eqs. (1) and (3) can be used to develop stress–strain graphs. The maximum value of stress, called strength, and the corresponding strain were determined from these graphs. The values of mechanical properties of incident and transmitter bar material, Inconel, taken in the calculations include modulus, Poisson’s ratio, and density of 205 GPa, 0.307, and 8890 kg/m3, respectively. In addition, the speed of sound in Inconel alloy was taken as 4802 m/s. The SHPB setup was calibrated by a pilot test, where no specimen was used between the incident and the transmitter bars. The results of the test are shown in Fig. 3, where the incident stress wave is compared with the summation of reflected and transmitted stress waves. A close match between the two stress pulses shows that the dispersion effects are negligible.
Shear failure of diaphyses specimens subjected to quasi-static compression testing is shown in Fig. 8. The fracture pattern of epiphyses and diaphyses specimens tested at strain rates of 353, 225 and 497 s 1 are shown in Fig. 9. Significant fragmentation
2.4. Characterization of failure mode The high speed images were obtained using a Redlake V-3 camera at an average acquisition frequency of 6500 frames/s. A Hitachi S-3500N SEM was used to analyze damage caused due to the stress wave propagation in the high strain rate testing.
3. Results The microstructure of an as-cut rabbit femur bone (before mechanical testing) is shown in Fig. 4. Haversian canals and the osteocytic lacunae are observed in the cross-section of the sectioned rabbit femur specimens, Fig. 4(a). The surface on the endosteal in the longitudinal direction is shown in Fig. 4(b). This surface was not subjected to any specimen preparation procedure and is the natural appearance. The microstructures of failed specimens were compared with Fig. 4 to identify the failure features. Fig. 5 shows a representative set of stress–strain graphs for quasi-static compression. The modulus and the strength values calculated for two bones tested in this study are presented in Table 1. The strength is defined as the peak stress value in the stress–strain graphs. A representative set of incident and transmitted strain signals from the SHPB testing for epiphyses specimen is shown in Fig. 6(a). The stress on the front and the back face of the specimen are plotted in Fig. 6(b) to show that the specimen compression occurred under dynamic equilibrium. The incident and the transmitter bar signals are used to plot the stress–strain graphs for high strain rate testing using Eqs. (1) and (3). The stress–strain graphs for epiphyseal and diaphyseal regions tested at high strain rates are shown in Fig. 7(a) and (b), respectively. Tables 2 and 3 show the compressive modulus and strength values of epiphyses and diaphyses specimens tested at high strain rates.
Fig. 5. Representative quasi-static compressive stress–strain graphs of rabbit femur bones. The numbers correspond to the bone section marked in Fig. 1.
Table 1 Quasi-static compressive modulus and strength of the rabbit bones. Numbera Bone 1
1 2 3 4 5 a
Bone 2
Compressive modulus (GPa)
Strength (MPa)
Compressive modulus (GPa)
Strength (MPa)
2.74 5.71 6.90 6.36 2.66
56.3 92.4 103.9 96.5 42.0
2.90 7.93 9.11 – 1.93
53.7 72.3 111.1 – 39.2
Regions shown in Fig. 1.
Fig. 4. Micrographs of rabbit femur bone showing (a) cross-section of the diaphyses specimen revealing osteocytes in the lacunae (arrows) sectioning through the haversian system (arrowheads) and (b) surface texture at the endosteal surface of epiphyses specimen.
V.C. Shunmugasamy et al. / Journal of Biomechanics 43 (2010) 3044–3050
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Fig. 6. (a) Typical strain signals for an epiphyses specimen from the SHPB testing and (b) the stress on the front and back face of the specimen.
Fig. 7. Stress–strain graphs from the high strain rate testing of (a) the epiphyses and (b) the diaphyses specimens.
Table 2 Compressive modulus and strength of epiphyses specimens tested at high strain rates. Strain rate (s 1)
Compressive modulus (GPa)
Strength (MPa)
274 353 546 616 722
12.41 13.68 10.17 8.67 8.99
100.6 99.2 99.8 86.4 82.4
Table 3 Compressive modulus and strength of diaphyses specimens tested at high strain rates. Strain rate (s 1)
Compressive modulus (GPa)
Strength (MPa)
225 309 497 609 775
14.38 20.18 29.95 29.97 17.76
187.0 215.9 242.8 214.4 180.0
under multiple cracks is observed in these specimens. Figs. 10 and 11 show fracture patterns of a diaphyseal specimen tested at a strain rate of 225 s 1. Extensive crack initiation was observed at
Fig. 8. Failure pattern of a diaphyseal specimen under quasi-static compression testing. The big arrows denote the direction of compression. The small arrows point to the cracks in the specimen.
the specimen surface that was facing the incident bar, as shown in Fig. 10(a) and (b). The surface of bone facing the transmitter bar shows cracks propagating from the periosteal to the endosteal through the bone cortex as seen in Fig. 10(c). At high strain rate shearing of the bone cortex eventually leads to cracking and failure. The bone cortex that was cracked open due to the high strain rate compression is shown in Fig. 10(d). The failure also exposed some canaliculi present in the ossified bone as shown in Fig. 11.
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Fig. 9. (a) Sequence of images showing the fracture pattern in an epiphyseal specimen tested at a strain rate of 353 s 1. The total time period for the entire sequence is 873 ms. Sequence of images showing the fracture pattern in a diaphyseal specimen tested at a strain rate of (b) 225 s 1 and (c) 497 s 1. The total time period captured in (b) and (c) image sequences are 1024 and 986 ms, respectively. The letters ‘‘I’’ and ‘‘T’’ represent the bone surface proximal to the incident and the transmitted bars, respectively, and the arrows represent the direction of compression.
Fig. 10. Fracture pattern of a diaphyseal specimen tested at 225 s 1 strain rate showing in (a) and (b) crack branching at the surface that was in proximity of the incident bar and (c) crack passing from periosteal to endosteal, penetrating through the bone cortex at the surface which was in proximity to the transmitted bar (d) fracture surface features of a diaphyseal specimen tested at 225 s 1 strain rate. The arrows in (d) show the direction of compression.
V.C. Shunmugasamy et al. / Journal of Biomechanics 43 (2010) 3044–3050
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Fig. 11. Fracture surface of a diaphyseal specimen tested at 225 s 1 strain rate showing (a) shear fracture through lacunae and haversian system canals and (b) higher magnification of the same region depicting fracture through the lacunae and the haversian system canals.
4. Discussion The average quasi-static compressive moduli of the epiphyses and diaphyses specimens of the first bone were 2.70 and 6.32 GPa, respectively. Such quasi-static values were substantially lower compared to their high strain rate counterparts, which ranged between 8 and 15 GPa for epiphyses specimens and 15–30 GPa for diaphyses specimens. The modulus increase appears at the expense of the ductility. A comparison of stress–strain graphs in Figs. 5 and 7 shows that at high strain rates the peak stress value is attained at lower strains. Studies on human femoral cortical bone similarly reported an average quasi-static modulus of 16.2 GPa and a 23% higher average dynamic modulus of 19.9 GPa at 100 s 1 strain rate (Katsamanis and Raftopoulos, 1990). Another study on human femoral bones supported a similar trend by finding the modulus values of 17.9 GPa under quasi-static and 40.7 GPa at 1500 s 1 strain rate (McElhaney and Byars, 1965). Since each tested specimen has a unique strain rate value in SHPB testing, standard deviations in mechanical properties cannot be calculated. It should be noted that the modulus values reported in the present study and in these cited studies are calculated as the slope of the stress–strain diagrams at different strain rates. In SHPB testing the strain rate is not constant throughout the test. The strain rate rises rapidly in the beginning of the test before becoming constant at a certain value. However, the initial part of compression where a steep rise in strain rate occurs is also where the modulus is calculated. Therefore, the reported modulus is not the modulus in a strict sense, but simply the slope of the stress– strain graph. The failure patterns at high strain rate are captured for an epiphyseal specimen in Fig. 9(a) and for diaphyseal specimens in Fig. 9(b) and (c). The time duration covered in the image sets shown in Fig. 9(a)–(c) are 873, 1024, and 986 ms, respectively. Epiphyseal specimens have lower strength and modulus leading to significant crushing of specimens even at 353 s 1 strain rate. At higher strain rates, the specimens were completely crushed, which is evident from stress drop in the stress–strain graphs. In comparison, the diaphyses specimens have higher modulus and strength. Their high strain rate failure pattern shows prominent fragmentation in Fig. 9(b). Fig. 9(c) shows the crack initiation on the specimen on the side that is proximal to the incident bar. The cracks propagate along the specimen’s length, leading to the fragmentation of the specimen. Compared to the quasi-static compressive failure shown in Fig. 8, where shear cracking of the specimen was the main failure mode, at high strain rates the specimens fractured primarily under the cracking in the direction of the applied load. In human and
bovine cortical femurs shear failures at low strain rates and splintering of specimens at high strain rates were observed (McElhaney and Byars, 1965), which is similar to the behavior observed in the present tests for diaphyses specimens. The fracture pattern of diaphyseal specimens is further analyzed using a SEM. The crack initiation in the specimen from the side proximal to incident bar, as seen in Fig. 9(c), can also be visualized in Fig. 10(a) and (b). These figures show that a large number of cracks initiate in the specimen due to a lack of enough time for deformation to take place under equilibrium conditions. These cracks tend to coalesce into a few large cracks that propagate towards the transmitter bar and constitute failure in the form of fragmentation as shown in Fig. 10(c). The effect of shearing can be observed in Fig. 10(d), as the fracture progresses under the passing of elastic waves through the specimen causing failure. The arrows included in this figure indicate the direction of compression. In Fig. 11, the cross-sections of lacunae, observed on the failure surface, have been considerably elongated, compared to Fig. 5. The specimen fragments also showed other microstructural features such as haversian canals, which were also severely deformed. 5. Conclusions The modulus values at high strain rate were found to be higher compared to the quasi-static compressive modulus. The shear cracks were prominent in quasi-static compressive failure. Shear cracking, along with specimen fragmentation, was prominent at high strain rates. The failure strain decreased but the specimen strength increased at high strain rates. These observations on failure modes are similar to the failure mechanisms observed in other types of bones tested in the published literature. The understanding of the bone fracture patterns at high strain rates can be valuable in designing armors and protective equipment. All of the experiments were conducted within the biological ethics protocol of NYU and NYU-Poly.
Conflict of interest statement There is no conflict of interest.
Acknowledgements This research is supported by the National Science Foundation through grant CMMI-0726723 and the NYU-Poly–NYU joint seed
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grant. MAE Department at NYU-Poly is acknowledged for the support and facilities provided. The authors thank Dung Dinh Luong and Dr. Nguyen Q. Nguyen for help in experimental work and data processing.
References ~ Ferreira, F., Vaz, M.A., Simoes, J.A., 2006. Mechanical properties of bovine cortical bone at high strain rate. Materials Characterization 57, 71–79. Gupta, N., Cho, K., 2010. Blast protection materials. Advanced Materials & Processes 168, 32–33. Katsamanis, F., Raftopoulos, D.D., 1990. Determination of mechanical properties of human femoral cortical bone by the Hopkinson bar stress technique. Journal of Biomechanics 23, 1173–1184. Kolsky, H., 1964. Stress waves in solids. Journal of Sound and Vibration 1, 88–110. Lewis, J.L., Goldsmith, W., 1975. The dynamic fracture and prefracture response of compact bone by split Hopkinson bar methods. Journal of Biomechanics 8, 27–40. McElhaney, J.H., 1966. Dynamic response of bone and muscle tissue. Journal of Applied Physiology 21, 1231–1236. McElhaney, J.H., Byars, E.F., 1965. Dynamic response of biological materials. ASME Publication 65-WA-HUF-9, pp. 1–8. Odgaard, A., Linde, F., 1991. The underestimation of Young’s modulus in compressive testing of cancellous bone specimens. Journal of Biomechanics 24, 691–698. Osbjorn, M.P., Daniel, E.L., 2004. The aging of Wolff’s law: ontogeny and responses to mechanical loading in cortical bone. American Journal of Physical Anthropology 125, 63–99.
Pal, S., Saha, S., 1984. Effect of deformation rate on the flexural fracture behaviour of long bones. Medical and Biological Engineering and Computing 22, 251–254. ¨ Parenteau, C.S., Viano, D.C., Lovsund, P., Tingvall, C., 1996. Foot-ankle injuries: influence of crash location, seating position and age. Accident Analysis & Prevention 28, 607–617. Rho, J.Y., Ashman, R.B., Turner, C.H., 1993. Young’s modulus of trabecular and cortical bone material: ultrasonic and microtensile measurements. Journal of Biomechanics 26, 111–119. Sedlin, E.D., Hirsch, C., 1966. Factors affecting the determination of the physical properties of femoral cortical bone. Acta Orthopaedica 37, 29–48. Sharp, D.J., Tanner, K.E., Bonfield, W., 1990. Measurement of the density of trabecular bone. Journal of Biomechanics 23, 853–857. Song, B., Chen, W., Ge, Y., Weerasooriya, T., 2007a. Dynamic and quasi-static compressive response of porcine muscle. Journal of Biomechanics 40, 2999–3005. Song, B., Chen, W.W., Lu, W.Y., 2007b. Mechanical characterization at intermediate strain rates for rate effects on an epoxy syntactic foam. International Journal of Mechanical Sciences 49, 1336–1343. Tan, Z.H., Pang, B.J., Gai, B.Z., Wu, G.H., Jia, B., 2007. The dynamic mechanical response of SiC particulate reinforced 2024 aluminum matrix composites. Materials Letters 61, 4606–4609. Tencer, A.F., Kaufman, R., Ryan, K., Grossman, D.C., Henley, M.B., Mann, F., Mock, C., Rivara, F., Wang, S., Augenstein, J., Hoyt, D., Eastman, B., 2002. Femur fractures in relatively low speed frontal crashes: the possible role of muscle forces. Accident Analysis & Prevention 34, 1–11. Tony, M.K., Tania, P.P., Crawford, R.P., David, L.K., Albert, L., 1997. Systematic and random errors in compression testing of trabecular bone. Journal of Orthopaedic Research 15, 101–110. Van Sligtenhorst, C., Cronin, D.S., Wayne Brodland, G., 2006. High strain rate compressive properties of bovine muscle tissue determined using a split Hopkinson bar apparatus. Journal of Biomechanics 39, 1852–1858.
Contents lists available at ScienceDirect
Journal of Biomechanics journal homepage: www.elsevier.com/locate/jbiomech www.JBiomech.com
High strain rate response of rabbit femur bones Vasanth Chakravarthy Shunmugasamy a, Nikhil Gupta a,n, Paulo G. Coelho b a
Composite Materials and Mechanics Laboratory, Mechanical and Aerospace Engineering Department, Polytechnic Institute of New York University, 6 MetroTech Center, Brooklyn NY 11201, USA Department of Biomaterials and Biomimetics, New York University, New York, NY 10010, USA
b
a r t i c l e in f o
a b s t r a c t
Article history: Accepted 30 June 2010
Strain rate dependence of the mechanical response of hard tissues has led to a keen interest in their dynamic properties. The current study attempts to understand the high strain rate characteristics of rabbit femur bones. The testing was conducted using a split-Hopkinson pressure bar equipped with a high speed imaging system to capture the fracture patterns. The bones were also characterized under quasi-static compression to enable comparison with the high strain rate results. The quasi-static compressive moduli of the epiphyseal and diaphyseal regions were measured to be in the range of 2–3 and 5–7 GPa, respectively. Under high strain rate loading conditions the modulus is observed to increase with strain rate and attains values as high as 15 GPa for epiphyseal and 30 GPa for diaphyseal regions of the femur. The strength at high strain rate was measured to be about twice the quasi-static strength value. A large number of small cracks initiated on the specimen surface close to the incident bar. Coalescence of crack branches leading to fewer large cracks resulted in specimen fragmentation. In comparison, the quasi-static failure was due to shear cracking. & 2010 Elsevier Ltd. All rights reserved.
Keywords: High strain rate Mechanical properties Fracture Rabbit femur
1. Introduction Bones have a complex and functionally graded microstructure, making it challenging to determine their structure–property relationships. The bone remodeling process depends on the stresses and the strains the bones are subjected to (Osbjorn and Daniel, 2004). The correlation between stresses and strains depends on mechanical properties of the bone. Dependence of mechanical properties on strain rates can significantly change this correlation and affect the bone remodeling process. The high strain rate loading on bones can be encountered in various situations including sport or recreational activities such as running and jumping, falls, and auto accidents, among others (Parenteau et al., 1996; Tencer et al., 2002). Characterizing the mechanical properties of bones under high strain rates can provide additional insight into how bones behave under these loading conditions. These studies can help in understanding the damage mechanisms, designing physical therapy routines, and providing key data for informed design rationale in developing medical devices or protective systems (Gupta and Cho, 2010). In the present study rabbit femur bones are evaluated for deformation and fracture characteristics at high strain rates using a split-Hopkinson pressure bar (SHPB) setup. Compared to quasi-static compression, where strain rates on the order of 10 3 10 2 s 1
n
Corresponding author. Tel.: +1 718 260 3080; fax: + 1 718 260 3532. E-mail address: [email protected] (N. Gupta).
0021-9290/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.jbiomech.2010.06.034
are exerted, strain rates as high as 103 s 1 are achieved in high strain rate testing. The SHPB setup is often utilized to characterize high strain rate properties of engineered and natural composites, such as particle reinforced polymer matrix composites (Song et al., 2007b), metal matrix composites (Tan et al., 2007) and bones (Ferreira et al., 2006; Katsamanis and Raftopoulos, 1990; Lewis and Goldsmith, 1975). SHPB has also been used for testing muscular tissues (Song et al., 2007a; Van Sligtenhorst et al., 2006). A previous study tested complete rabbit femur bones under flexural loading conditions at different deformation rates (Pal and Saha, 1984). The bending stiffness of bones was measured to be higher at higher deformation rates. In addition, the specimen failure mode was observed to be strain rate dependent, where fracture occurred in a single plane under quasi-static loading and crack branching occurred at high strain rates. The high strain rate tests conducted on human femoral cortical bones reported an average elastic modulus of 19.9 GPa at strain rate of 100 s 1 as compared to 16.2 GPa under quasi-static compression (Katsamanis and Raftopoulos, 1990). Studies related to failure mechanisms have observed that the prominent failure mode was shear cracking under quasi-static compression, while at high strain rates splintering of specimens dominated (McElhaney, 1966). The use of high speed imaging systems has been very illustrative in observing the failure pattern in the bovine muscle tissue specimens under high strain rate loading (Van Sligtenhorst et al., 2006). The aim of the present study was to characterize the mechanical response of rabbit femur bones under a wide range
V.C. Shunmugasamy et al. / Journal of Biomechanics 43 (2010) 3044–3050
of strain rates and analyze the fracture pattern by means of high speed imaging. The tested specimens were examined using a scanning electron microscope (SEM) to understand the failure mechanisms at different strain rates.
3045
sampling rate. The displacement of the test equipment crosshead was used to calculate the strain values. The cut specimens used for testing are shown in Fig. 1(b) and (c). Rigid compression supports of steel were used to minimize the compliance errors during the compression test. The compliance error in the present test protocol is expected to be negligible (Odgaard and Linde, 1991; Tony et al., 1997).
2. Materials and methods 2.3. High-strain rate testing 2.1. Test materials Rabbit femur bones were selected for testing in this study. The rabbits were 120–180 days old and had a mean weight of 3 kg. The rabbits were euthanized and the femurs were obtained by sharp dissection. The tissues surrounding the femur were carefully removed and the bones were stored at 20 1C prior to testing (Sedlin and Hirsch, 1966). A total of 8 rabbit femur bones were used for quasistatic and high strain rate compression testing. The bones were cut into 10 mm long specimens (Fig. 1) using a low speed precision diamond blade saw (IsoMets; Buehler Ltd., Lake Placid, NY). Care was taken to ensure that the cut surfaces of the specimen were parallel to each other to minimize the errors in the results (Tony et al., 1997). The surfaces were maintained with less than 1% difference in the specimen length along the entire cross-section. The specimens were cut from the epiphyses–metaphyses (marked 1 and 5 in Fig. 1(a)) and the diaphyses regions (marked 2, 3 and 4 in Fig. 1(a)). The actual density (Sharp et al., 1990) of the bones was measured using the Archimedes’ principle. The average epiphyseal and diaphyseal bone densities were found to be 1622 7284 and 2048 7 454 kg/m3, respectively, which are in line with previously published data (Katsamanis and Raftopoulos, 1990; Rho et al., 1993).
Schematic representation of the SHPB setup is shown in Fig. 2. Prior to testing, the specimen was placed between two long slender bars, called incident and transmitter bars. A striker bar impacts the incident bar to generate an elastic wave (Kolsky, 1964). The incident wave undergoes reflection and transmission at the incident bar–specimen and specimen–transmitter bar interfaces depending on the relative impedance of the bar and specimen materials. Strain gages were bonded to the incident and the transmitter bars to record the strain response with respect to time. A digital oscilloscope was used to record the strain signals and transfer data to a computer for analysis. The incident, reflected, and transmitted wave signals were used to calculate stress, strain, and strain rate with respect to time (t). Stress was calculated from the time dependent strain signal of the transmitted wave, eT(t), by
sðtÞ ¼
Compression testing was carried out on an Instron 4469 universal test system equipped with a 50 kN load cell. The testing was conducted using displacement control mode and at the constant compression rate of 1 mm/min. Load–displacement data were obtained using Bluehill 2.0 software at 20 MHz
Fig. 1. (a) Rabbit femur bone showing markings along which the specimens were cut, (b) epiphyseal specimen from the epiphyses–metaphyses region and (c) diaphyseal specimen.
ð1Þ
where A0 and As are the cross-section areas of the bar and the specimen, respectively, and E is the modulus of the bar material. The strain rate was determined from the strain signal of the reflected wave, eR(t), by
e_ ðtÞ ¼
2.2. Quasi-static testing
A0 EeT ðtÞ As
2c0 eR ðtÞ ls
ð2Þ
where c0 and ls are the speed of sound in the bar and the specimen length, respectively. The strain rate was integrated with respect to time to
Fig. 3. Comparison of the incident stress wave with the summation of transmitted and reflected stress wave.
Fig. 2. Schematic representation of split-Hopkinson pressure bar apparatus used in high strain rate testing.
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obtain strain Z t eðtÞ ¼ e_ ðtÞdt
ð3Þ
0
Eqs. (1) and (3) can be used to develop stress–strain graphs. The maximum value of stress, called strength, and the corresponding strain were determined from these graphs. The values of mechanical properties of incident and transmitter bar material, Inconel, taken in the calculations include modulus, Poisson’s ratio, and density of 205 GPa, 0.307, and 8890 kg/m3, respectively. In addition, the speed of sound in Inconel alloy was taken as 4802 m/s. The SHPB setup was calibrated by a pilot test, where no specimen was used between the incident and the transmitter bars. The results of the test are shown in Fig. 3, where the incident stress wave is compared with the summation of reflected and transmitted stress waves. A close match between the two stress pulses shows that the dispersion effects are negligible.
Shear failure of diaphyses specimens subjected to quasi-static compression testing is shown in Fig. 8. The fracture pattern of epiphyses and diaphyses specimens tested at strain rates of 353, 225 and 497 s 1 are shown in Fig. 9. Significant fragmentation
2.4. Characterization of failure mode The high speed images were obtained using a Redlake V-3 camera at an average acquisition frequency of 6500 frames/s. A Hitachi S-3500N SEM was used to analyze damage caused due to the stress wave propagation in the high strain rate testing.
3. Results The microstructure of an as-cut rabbit femur bone (before mechanical testing) is shown in Fig. 4. Haversian canals and the osteocytic lacunae are observed in the cross-section of the sectioned rabbit femur specimens, Fig. 4(a). The surface on the endosteal in the longitudinal direction is shown in Fig. 4(b). This surface was not subjected to any specimen preparation procedure and is the natural appearance. The microstructures of failed specimens were compared with Fig. 4 to identify the failure features. Fig. 5 shows a representative set of stress–strain graphs for quasi-static compression. The modulus and the strength values calculated for two bones tested in this study are presented in Table 1. The strength is defined as the peak stress value in the stress–strain graphs. A representative set of incident and transmitted strain signals from the SHPB testing for epiphyses specimen is shown in Fig. 6(a). The stress on the front and the back face of the specimen are plotted in Fig. 6(b) to show that the specimen compression occurred under dynamic equilibrium. The incident and the transmitter bar signals are used to plot the stress–strain graphs for high strain rate testing using Eqs. (1) and (3). The stress–strain graphs for epiphyseal and diaphyseal regions tested at high strain rates are shown in Fig. 7(a) and (b), respectively. Tables 2 and 3 show the compressive modulus and strength values of epiphyses and diaphyses specimens tested at high strain rates.
Fig. 5. Representative quasi-static compressive stress–strain graphs of rabbit femur bones. The numbers correspond to the bone section marked in Fig. 1.
Table 1 Quasi-static compressive modulus and strength of the rabbit bones. Numbera Bone 1
1 2 3 4 5 a
Bone 2
Compressive modulus (GPa)
Strength (MPa)
Compressive modulus (GPa)
Strength (MPa)
2.74 5.71 6.90 6.36 2.66
56.3 92.4 103.9 96.5 42.0
2.90 7.93 9.11 – 1.93
53.7 72.3 111.1 – 39.2
Regions shown in Fig. 1.
Fig. 4. Micrographs of rabbit femur bone showing (a) cross-section of the diaphyses specimen revealing osteocytes in the lacunae (arrows) sectioning through the haversian system (arrowheads) and (b) surface texture at the endosteal surface of epiphyses specimen.
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Fig. 6. (a) Typical strain signals for an epiphyses specimen from the SHPB testing and (b) the stress on the front and back face of the specimen.
Fig. 7. Stress–strain graphs from the high strain rate testing of (a) the epiphyses and (b) the diaphyses specimens.
Table 2 Compressive modulus and strength of epiphyses specimens tested at high strain rates. Strain rate (s 1)
Compressive modulus (GPa)
Strength (MPa)
274 353 546 616 722
12.41 13.68 10.17 8.67 8.99
100.6 99.2 99.8 86.4 82.4
Table 3 Compressive modulus and strength of diaphyses specimens tested at high strain rates. Strain rate (s 1)
Compressive modulus (GPa)
Strength (MPa)
225 309 497 609 775
14.38 20.18 29.95 29.97 17.76
187.0 215.9 242.8 214.4 180.0
under multiple cracks is observed in these specimens. Figs. 10 and 11 show fracture patterns of a diaphyseal specimen tested at a strain rate of 225 s 1. Extensive crack initiation was observed at
Fig. 8. Failure pattern of a diaphyseal specimen under quasi-static compression testing. The big arrows denote the direction of compression. The small arrows point to the cracks in the specimen.
the specimen surface that was facing the incident bar, as shown in Fig. 10(a) and (b). The surface of bone facing the transmitter bar shows cracks propagating from the periosteal to the endosteal through the bone cortex as seen in Fig. 10(c). At high strain rate shearing of the bone cortex eventually leads to cracking and failure. The bone cortex that was cracked open due to the high strain rate compression is shown in Fig. 10(d). The failure also exposed some canaliculi present in the ossified bone as shown in Fig. 11.
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Fig. 9. (a) Sequence of images showing the fracture pattern in an epiphyseal specimen tested at a strain rate of 353 s 1. The total time period for the entire sequence is 873 ms. Sequence of images showing the fracture pattern in a diaphyseal specimen tested at a strain rate of (b) 225 s 1 and (c) 497 s 1. The total time period captured in (b) and (c) image sequences are 1024 and 986 ms, respectively. The letters ‘‘I’’ and ‘‘T’’ represent the bone surface proximal to the incident and the transmitted bars, respectively, and the arrows represent the direction of compression.
Fig. 10. Fracture pattern of a diaphyseal specimen tested at 225 s 1 strain rate showing in (a) and (b) crack branching at the surface that was in proximity of the incident bar and (c) crack passing from periosteal to endosteal, penetrating through the bone cortex at the surface which was in proximity to the transmitted bar (d) fracture surface features of a diaphyseal specimen tested at 225 s 1 strain rate. The arrows in (d) show the direction of compression.
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Fig. 11. Fracture surface of a diaphyseal specimen tested at 225 s 1 strain rate showing (a) shear fracture through lacunae and haversian system canals and (b) higher magnification of the same region depicting fracture through the lacunae and the haversian system canals.
4. Discussion The average quasi-static compressive moduli of the epiphyses and diaphyses specimens of the first bone were 2.70 and 6.32 GPa, respectively. Such quasi-static values were substantially lower compared to their high strain rate counterparts, which ranged between 8 and 15 GPa for epiphyses specimens and 15–30 GPa for diaphyses specimens. The modulus increase appears at the expense of the ductility. A comparison of stress–strain graphs in Figs. 5 and 7 shows that at high strain rates the peak stress value is attained at lower strains. Studies on human femoral cortical bone similarly reported an average quasi-static modulus of 16.2 GPa and a 23% higher average dynamic modulus of 19.9 GPa at 100 s 1 strain rate (Katsamanis and Raftopoulos, 1990). Another study on human femoral bones supported a similar trend by finding the modulus values of 17.9 GPa under quasi-static and 40.7 GPa at 1500 s 1 strain rate (McElhaney and Byars, 1965). Since each tested specimen has a unique strain rate value in SHPB testing, standard deviations in mechanical properties cannot be calculated. It should be noted that the modulus values reported in the present study and in these cited studies are calculated as the slope of the stress–strain diagrams at different strain rates. In SHPB testing the strain rate is not constant throughout the test. The strain rate rises rapidly in the beginning of the test before becoming constant at a certain value. However, the initial part of compression where a steep rise in strain rate occurs is also where the modulus is calculated. Therefore, the reported modulus is not the modulus in a strict sense, but simply the slope of the stress– strain graph. The failure patterns at high strain rate are captured for an epiphyseal specimen in Fig. 9(a) and for diaphyseal specimens in Fig. 9(b) and (c). The time duration covered in the image sets shown in Fig. 9(a)–(c) are 873, 1024, and 986 ms, respectively. Epiphyseal specimens have lower strength and modulus leading to significant crushing of specimens even at 353 s 1 strain rate. At higher strain rates, the specimens were completely crushed, which is evident from stress drop in the stress–strain graphs. In comparison, the diaphyses specimens have higher modulus and strength. Their high strain rate failure pattern shows prominent fragmentation in Fig. 9(b). Fig. 9(c) shows the crack initiation on the specimen on the side that is proximal to the incident bar. The cracks propagate along the specimen’s length, leading to the fragmentation of the specimen. Compared to the quasi-static compressive failure shown in Fig. 8, where shear cracking of the specimen was the main failure mode, at high strain rates the specimens fractured primarily under the cracking in the direction of the applied load. In human and
bovine cortical femurs shear failures at low strain rates and splintering of specimens at high strain rates were observed (McElhaney and Byars, 1965), which is similar to the behavior observed in the present tests for diaphyses specimens. The fracture pattern of diaphyseal specimens is further analyzed using a SEM. The crack initiation in the specimen from the side proximal to incident bar, as seen in Fig. 9(c), can also be visualized in Fig. 10(a) and (b). These figures show that a large number of cracks initiate in the specimen due to a lack of enough time for deformation to take place under equilibrium conditions. These cracks tend to coalesce into a few large cracks that propagate towards the transmitter bar and constitute failure in the form of fragmentation as shown in Fig. 10(c). The effect of shearing can be observed in Fig. 10(d), as the fracture progresses under the passing of elastic waves through the specimen causing failure. The arrows included in this figure indicate the direction of compression. In Fig. 11, the cross-sections of lacunae, observed on the failure surface, have been considerably elongated, compared to Fig. 5. The specimen fragments also showed other microstructural features such as haversian canals, which were also severely deformed. 5. Conclusions The modulus values at high strain rate were found to be higher compared to the quasi-static compressive modulus. The shear cracks were prominent in quasi-static compressive failure. Shear cracking, along with specimen fragmentation, was prominent at high strain rates. The failure strain decreased but the specimen strength increased at high strain rates. These observations on failure modes are similar to the failure mechanisms observed in other types of bones tested in the published literature. The understanding of the bone fracture patterns at high strain rates can be valuable in designing armors and protective equipment. All of the experiments were conducted within the biological ethics protocol of NYU and NYU-Poly.
Conflict of interest statement There is no conflict of interest.
Acknowledgements This research is supported by the National Science Foundation through grant CMMI-0726723 and the NYU-Poly–NYU joint seed
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grant. MAE Department at NYU-Poly is acknowledged for the support and facilities provided. The authors thank Dung Dinh Luong and Dr. Nguyen Q. Nguyen for help in experimental work and data processing.
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