1. 8wnrhrrucsVd.
23. No. II.pp.
1173 IIR4. IWO
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TECHNICALNOTE
DETERMINATION OF MECHANICAL PROPERTIES OF HUMAN FEMORAL CORTICAL BONE BY THE HOPKINSON BAR STRESS TECHNIQUE FOTIOS KATSAMANIS*
and DEMETRIOS D. RAFroPOULOSt:
* Hellenic Naval Academy. 18501 Piraeous. Greece and t Dept. of Mechanical Engineering. The University of Toledo, Toledo, OH 43606. U.S.A. Abstract-The Hopkinson bar stress technique and a universal testing machine (Instron 1125) have been used to iqvestigate the dynamic and static mechanical properties of cortical bone taken from a human femur rcspectivqly.We found that the average dynamic Young’s modulus value (E, = 19.9CPa) to be 23 % higher than the ~averagestatic Young’s modulus value (E,= 16.2 GPa). Furthermore, the Poisson’s ratio did not exhibit aey significant variation for the two dilTerent types of loading. No dilTerencewas observed between the value of the dynamic Young’s modulus in tension and those found in compression. A comparison was made of the results of this study with those found by other researchen using dilTerent techniques, such as ultrasonics, and it was found that they agree well with most of the results of previous studies, Finally, the viscosity for cortical bone found in this study correlates with viscosity reported by Tennyson et al. [E.vpl Much. II, 502-507 (1972)] for ten days post mortem age specimens.
INTRODUCTION
The determination of mechanical bone properties has been an important topi/: ofstudy for many years. Reilly et of. (1974) tested human a4 bovine specimens but found no statistically significant ariation in the moduli of elasticity detcrmined by the two‘iIloading modes (tension and compression). Bargren et of. (1914) came lo the same conclusion when they tested hydrated r(nd air dried cortical bone human femur specimens altcm~tely in axial tension and compression. In contrast to the lbwtr values of the previously mentioned studies, the high+l values for Young’s modulus were obtained in static tc$rilc tests, using Tuckerman’s optical gage, on longitudinal tiovinc tibia sections, by Bonfield and Li (1966).In the last, two decades, other techniques have been introduced for d termining mechanical properties of biomaterials, such 1 the ultrasonic method. Lang (1970) was one of Ihe first lo b an ultrasonic technique to measure the elastic propertia Of bone. Since Lang’s initial study, several invatigalon havd made use of pulse ultrasonic technique to me&e the elastic properties bf bone. In particular, Loon and Katz (1976). Land et al. 119791.and Van Buskirk et al. (1981) uodd thii &nique. Ashman et al. (1984) used a continuous ullr nit wave technique. II is well know -R that bone is a viscoelastic material and that its me&an’ I prop&a are affected by its deformation rate. MEthaney =i (1966) investigated the response of cubes of cortical bone to pression loads at various strain rates, and found that tI?e Young’s modulus and ultimate stress increased and theIstrain to failure decreased with increasing strain rate. Sammjirco et al. (1971) tested whole human and canine bones in t rsion at two rates of deformation. Tennyson er al. (1972) the split Hopkinson bar technique to study the viscoe+ tic response of bovine femur bone subjected (0 dynamic~colnpreasive loading. A linear viscoelastic m&l daaibing Ithe mechanical behavior of bovine bone. including an estimate of the parameters. was obtained when Received in fit@ fom I7 April 1990. 1 Author to whbm correspondence should be addressed.
these bones were tested while still very fresh. Lewis and Goldsmith (1973. 1975) developed a biaxial split Hopkinson bar apparatus to measure dynamic material properties of compact bovine bone in compression. tension. torsion and combined torsion and compression. They concluded that in compression bone was viscoclastic and that fracture stress increased with strain rate. A major advantage in Ihe Lewis and Goldsmith technique was that strain measurements were obtained directly from the strain gagcr attached (0 the specimens. That is. these authors claim that their strain values were inherently more accurate than those obtained by Tennyson et a/. who deduckd strain by the normal split Hopkinson bar method using the magnitude of the incident reflected and trasmitted waves. Furthermore. Lewis and Goldsmith also pointed out that one penalty which must be paid for this accuracy was that the specimen had to be dried out for half-an-hour whik the strain gage was mounted. Pelker and Saha (1983) studied the traveling wave characteristics for a single compressive pulse in fresh and embalmed human long bones. The stress wave was generated by the longitudinal impact ofa steel ball on one end of the bone and was monitored by bonded semiconductor strain gages. The dynamic properties, namely wave velocity. attenualion coefficient, and dispersion were correlated with various parameters such as mineral density. porosity and cross-sectional area of the specimens. In the present study the Hopkinson bar technique was used to investigate the dynamic characteristics of cortical bone of a human femur. Stress waves were produced by the longitudinal impact of steel sphcrrr of 4.76 mm in diameter, fired from an air gun onto the end of long specimens. These specimens were cut along the shaft of the femur and machined to haw a uniform rectangular cross-section.Stress waves were detected by means of strain gages bonded in the longitudinal and cransversedirections at three stations. Since the cross-sectional dimensions of the specimens used were small relative lo pulse kngth. onedimensional wave theory in rods (Kolsky. 1963) was applicabk to this study. The aims of this investigation are: (1) Firstly, to measure the dynamic Young’s modulus and Poisson’s ratio at various locations on specimens prepared
1173
Technical Note
1174
I
I
-i----w---
_..___,
. ___
I (alRl@tFamur
I
1
(
b 1 Shan
( Mwhyals
)
Fig. I. The femur and the spccimcns used in this investigation.
from the same femur using the Hopkinson bar technique. and to comparc these values with the corresponding static values obtained from tensile cxpcrimcnts in a universal testing machine (Instron 1125). as well as with those reported by other investigators. (2) Secondly, to dctcrminc whether the modulus or clasticity of bone in tension is dilTcrrnt than that in compression as mportcd by some previous invcstigators(McEIhanty et al.. 1964; Simkin and Robin, 1973). (3) Thirdly, to measure bone’s viscoclastic characteristic prramctcrs, namely the attenuation cocllicicnt and viscosity, assuming that bone behaves as Voigt viscoelastic element and to compare the results with those reported by other rcscarchcrs. TEST SPECIMENS-EXPERIMENTAL
SET-UP
Tests were run on bone specimens from the shaft (diaphysis) of the right femur of an adult (45-55 yr old) male cadaver obtained from the Medical College of Ohio, at Toledo [Fig l(a)]. The lcmur was delivered to the laboratory fresh. However. when the specimens wcrc made. they were left for several days to dry out in order to mount the strain gages. This femur. therefore, should bc considered embalmed. dry and not wet. The first step, in preparing the specimens, was sectioning the diaphysis into three long pieces along its longitudinal axis. The numbers (I), (II). and (III) were assigned to the specimens as indicated in Fig. l(b). From each pica two strips were cut; these were milled to obtain long specimens (rods) of rectangular cross-section. One of them was cut into four short picas labeled (A), (B). (C), and (D). The dimensions of the specimen arc shown in Fig. I(c). The long specimens were used to measure dynamic mechanical properties at various locations the short specimens were used to measure the corresponding static properties. Furthcrmore. the longitudinal axis and the loading imposed on the spccimcnswere parallel to the long axis of the bone; and these specimens will hcncerorth be referred to as longitudinal specimens.
The static cxpcrimcnts were performed in a universal testing machine (lnstron 1125) and using very low speed.The short specimens were machined and shaped to appropriate dimensions. A hole at each end of the spccimcn was drilled to avoid slippage, see Fig. 1. Two strain gages were bonded on each specimen. One. EA-1312SBT-120, was in the longltudinal direction and the other. CEA-13-062VW-350, was bonded in the transverse direction. Strain gages were placed on both sides ol test spccimcns to eliminate the clTcctsof bending. Standard tcnsilc tests were conducted on these. specimenson the universal testing machine. Strain indicators were connected to the longitudinal and transverse strain gages, respectively; both the strains and tensile loads were recorded during testing. To dynamically load the bone specimens,a stresspulse was introduced by means of an air gun dcvelopcd for that purpose. The gun was designed to fire steel spheres of 12.7, 9.53, 6.35 and 4.26 mm diameter and it was powered by compressed air that was triggered by a quick action clcctromagnetic valve. The projectile velocities wcrc dctcrmincd From the signals recorded on a storage oscilloscope produced by the interruption ot two light beams a distance of 50 mm apart just ahead or the gun barrel in the path ol the projectile. The bone specimen was suspended in the horizontal position by means of two strings. Four strain gages ol 3.17 mm gage length (Micromcasurcntcnt Co.) were carefully bonded at three stations labeled 1.2, and 3 with M-Bond MO adhesive. The rcsistancc of these strain gages was 120 D and their gage (actor 211. Two pairs olstrain gages were bonded on each gage station; one pair in the longitudinal direction and the other pair in the transverse direction on opposite sides of the specimen. Each pair was connected in seriesso as to eliminate any antisymmetrical components 0r the slms pulse. In view ol the minimum pulse length, approximately 80 mm. the gage length of 3.17 mm was sufficiently short to accurately reflect the strain history at a given station. Each strain gage output was checked separately before being connected in series with the remaining strain gages. The gages were incorporated in potentiometric circuits (Dally and Riley, 1978) connected to dual trace digital storage
Technical Note
osdoscopu
(7ckronix 468) witb IO MHz useful storage battdwidtb. The oac&%sopns were triggered from tbc signal generated upon itperruption by the projectik ofa lucr beam focuacd on a pbotoodl just ahead of the impect point. To minimixc dcctrot$c noise c&c& light-wcigbt sbieldcd cabks that containad four cbannd potentiotnctric circuits wem used to connect ,strain gages with the gage box. For tbc connaXio0 baw+n the pip box and the digital storage oscilloscopa. co xial cables were used. A 24 V battery was conmftcd in P& Ikl to the four-channel potcntiornctric circuit in order toiobtaio an adapJote output signal from the strain gages.To avoid damage ofthe bonded strain gages due to local beating c&sed by the 50 mA electric current passing through them, t&24 V DC voltage was applied to the gage box only for a short time period (sS 8). The experimental arrangement used in this study is shown schematically in Fig. 2 A similar experimental arrangement was used by Goldsmith and katsamanis (1979) and Katsamanis and Goldsmith (1982) The specimens were carefully aligned to provide an axial impact. A polaroid camera was used to obtain the strain gage records from tbe~storage oscilloscopes. Many preliminary tests were conducted in order to select optimal arrangements for the reported (ests. For the specimen configuration cmployed. the small steel spheres of 4.76 mm diameter were fired from the air gun ivith a v&city of 22 m s- ‘. These spheres produced a short duration stress pulse (~25 cs) with an amplitude of I 156 w which provided an appropriate input. As noted. four ctumncls were available to store strain gage records; IWO channels were nccdcd for each gage station. one for longitudinal and one for the transverse strain. Tbercforc. data were taken from only two gage stations during each run. In order to obtain all the ncccssarydata the cxpcrimcnt was repeated with thc:samc conditions several times. The calibration factor of the four channel gage box was sckctcd at 92.2 p mv - ’ for all tests.
Photo-Cella-
1175
RESULTS Figure 3 presents typical strain gage records as they were obtained using a polaroid camera from a digital storage oscilloscope. These records correspond to specimen (II) and show the propagating compressive stress pulse at stations I and 2. Figure 3(a) presents strain in the longitudinal direction (L,, Lx) and Fig 3(b) in the transverse direction (T,. TJ. In these records, the vertical setting was 2 mVdiv- ’ which corresponds to I85 pdiv-’ (by mistake L, was set at a little over 2mVdiv-r) and the grapb was taken inverted for convenience. Tbe horizontal scak was 10 pdiv- *. Figure 4 also presents typical strain gage records of stations 2 and 3 for the same specimen (If), The horizontal scak was taken as 20 p div- ’ and, therefore, two wave refkctions arc shown in addition to the initial compressive pulse (1st tensile and 2nd compressive). In Fig. 4(a). which corresponds to longitudinal strain (I!.,, tI), the vertical setting was 5 mVdiv_’ (460 p div- ‘) and in Fig 4(b). which corresponds to transverse strain (T,, 7,). the vertical setting was ZmVdiv-’ (185 pdiv-‘). These settings on the digital storage oscilloscope were appropriate for collecting accurate data. Photographs of the oscilloscope traces were enlarged by projection to increase measurement sensitivity. The velocity of wave propagation was determined from the time elapsed between tbc arrival of the pulse peak bctwccn the two gage stations. Pulse propagating velocities for the compressive and the reflected tensile pulses were obtained for the specimen regions B and C. No statistically significant difference was observed between the two pulse wave velocitics. Furthermore. since there is no dilkrcncc bctwccn the compressive and tensile speeds, the pulse propagating vclocities for the other two regions (A and D) of the specimen wcrc obtained by measuring the time bctwccn the arrival of the two peaks (compressive and reflected tcnsik pulse) at gage stations I and 3. mpcctivcly.
I 12-24V
8
On.011
Fig. 2. Schematic diagram of the experimental set-up for the determination of cortical bone dynamic mechanical properties.
1176
Technical Note
Since the nominal dimensions of the specimens used (I. II and 111) were 22Or : 10.5 x 4 mm and the minimum pulse kngth &as 80 mm, the wave velocity for the rod is given by the one-dimensional approximation:
(where E, is the dynamic Young’s modulus and p the mass density of the bone) and quite accurately represents the pulse propagation velocity. Knowing the density p of the bone at each region of the specimen and using quation (I). the dynamic Young’s moduli, for the compression (I?$) and tension (E;) pulse were obtained. Archimedes’ principle was the basis of the technique
employed to determine the mass density p of the bone, which was measured for each particular specimen region. The density measumncnts took place just before gage installation. That is. there was not significant drying between the time of density measurement and testing to signilkantly a&t the density measured. This technique also has been used by Ashman et af. (1984). The dynamic Poisson’s ratio v, was calculated at three locations for each specimen (gage stations 1. 2, and 3) by dividing the amplitude of transverse strain s, by the comaponding amplitude of the longitudinal strain st at the same gage station. Tables I. 2 and 3 summar& the rcsultr for each spcimen (1). (II). and (III). In these tables d is the corresponding distana that stress pulse propagates during time At’ or At’
Table 1. Experimental results for Young’s modulus (E,), specimen (I)
ala
Region
Al’(w)
At’(w)
Compression
Tension
17.70 Exp. 26
A+ 17.80
Exp. 27
B A C’
18.30
Exp. 28 Exp. 30
Dt A’ D B+C
Exp. 31 Exp. 33
D B+C C
55.4
18.70
59.7 x 2 55.4
3.11
17.00
59.7 x 2 55.5
3.04
38.30 30.40 38.70 30.00 35.70
35.70 39.10 30.80
37.30 18.50
Distance C: (km s-t) C: (km s- i)
18.2 39.10
A
E;+
Ed==-
d(m)
34.4 17.60
48.2 x 2 59.7 x 2 48.2 x 2 110.9 59.7 x 2 48.2 x 2 110.9 55.5
3.13
3.05 3.05
E’,(GPa)
2090
20.02
19.04
19.53
2080
19.04 18.89
2.96
2090
3.27
2080 2000
19.88 18.13 20.98
19.88 19.56
1910 2080 1910 2050 2080 1910 2050 Moo
18.81 19.42 19.35 19.46 19.46 19.02 18.35 17.80 20.95 17.58 19.60
18.81 19.42 19.35 19.46 19.02 18.35 19.37 18.59
3.17 3.09 3.21 3.11 3.05 3.13 2.98 2.99
G (GPa)
19.04 19.80 17.99
3.12
3.11
Density
2 @Pa)
3.23 3.16
E;
l The pulse propagating velocities for regions B and C wcrc obtained by measuring the time the tensile pulse took to travel from one station to the next station. t The pulse propagating velocities for regions A and D were obtained by measuring the time between the arrival of the two peaks (compressive and tensile pulse).
Table 2. Experimental results for Young’s modulus, specimen (It)
Region Exp. 10 Exp. 11 Exp. 12
Dt C’ D C B’
AW4
AI’(P)
Compression
Tension
33.50 17.40 3220 17.70 17.65
Exp. 18 Exp. I5 Exp. 13 Exp. 16
B A B A A B+C B A A D
17.70 17.65 37.6
At Exp. 14
17.40
18.4
16.8 37.2
18.20
17.50 36.80 37.10
35.6 16.8
36.0 17.6 36.8 36.8 34.40
G+E:
E,=-
d(mm) Distance C: (kms- ‘) C: (kms- t) p (kgm-‘) 103.8 55.8 103.8 55.8 55.7 119.4 55.7 119.4 55.7 119.4 119.4 111.5 55.7 119.4 119.4 103.8
3.10 3.21
3.21 3.22
3.15 3.16
3.15 3.16 3.18
3.32
3.03 3.21 3.96
3.19 3.24 3.22 3.10 3.17
3.13 3.32 3.25 3.25 3.02
2090 2160 2050 2160 2050 2160 2050 2050 2120 2160 2050 2050
E; (GPa) E, (GPa) 19.64 21.16 21.16 21.27 20.36 20.36 21.10 21.10 20.28 19.41 23.29 20.72 21.54 19.85 21.15 20.86 20.40 19.95 23.29 21.22 21.17 21.17 18.68
(GP: 19.64 21.16 21.27 20.36 21.10 20.28 21.35 20.72 20.69 21.15 20.86 20.18 22.26 21.17 21.17 18.68
l The pulse propagating velocities for regions B and C were obtained by measuring the time the tensile pulse took to travel from one station to the next station. t The pulse propagating velocities for regions A and D were obtained by measuring the time between the arrival of the two peaks (compressive and tensile pulse).
2mV(185p&)
Ll
(a)
-Tl 2mV(185p&)
-T2
(b) Fig. 3. Strain gage records for spccimcn (II) showing the propagating comprcsive stress pulse at stations I and 2. (a) Stress pulse in longitudinal direction (L,. L2) (for convenience L, was taken inverted); (h) stress pulse in transverse direction (T,, T, 1.
II77
L3 5mV(460y&)
(4
T2 2mV(185p&)
T3
(b) Fig. 4. Strain gage records for specimen (II) at stations 2 and 3. Two wave rcktions (1st tensile and 2nd compressive) are shown in addition IO compreGve pulse. (a) Stress pulse in the longitudinal direction (L,, L,);(b) stress pulse in the transverse direction (T>. T,).
117x
5mV(460pc)
L2
Ll 1 OmV(920pE)
L2
(b) Fig. 5. Strain page records for specimen (II) at stations I and Z in longitudinal direction (I.,. f_,).(a) Stress pulse was pencrafed by impacting slecl spheres or 4.27 mm diameter on the end of the specimen (impact velocity !2 m s- ’ 1. The corrcspondinp attenuation coellicicnt is a=O.Ol cm-‘. (b) Stress pulse was generated by impacting slwl spheres ol 9.5 mm diameter on the end of the specimen (impact velocity 10 m s _ ’ ). The corresponding attenuation coellicient is a = 0.006 cm - ’
1179
Technical Note
1181
Table 3. Experimental results for Young’s modulus, specimen (III)
t;+ E, &,=Region
Compression
Tension
19.3
16.70
54.1
18.40
51.1 X2 54.1
3.01
17.10
51.1 X 2 54.3
3.18
17.5
59.7 x 2 54.3
3.26
B* Exp. 20 Exp. 21
Exp. 22 Exp. 23 Exp. 24
Exp. 25
A+ B
18.0
A C’
17.10
D+ C
16.70
32.5 32.70 37.5
D B+C A D C
39.20 34.60
35.90 33.20 37.70
18.60
D
16.50 39.10
Distance C:(kms-‘)C:(kms-‘) 2.80
p(kgm-‘)
16.49
22.16
19.33
2.95
2080 2140
20.18 18.96 18.21
20.18 18.59
2080 2100
19.99
3.18
20.82
20.82
19.99 20.82
3.10
2060 2IM)
20.48 21.86 19.84
20.48 20.85
18.78 19.36 20.24 17.60 22.42
18.78 19.72 19.36 20.24 20.0 I
18.85
18.85
3.18 3.05
2060 2120 2080 2060 2100
3.02 3.08 3.17
2.92
59.7 x 2
2 (GPa)
2140
3.13
3.14
E:(GPa)
3.25 3.15
59.7 x 2 108.4 51.1 x 2 59.7 X 2 54.3
E;(GPa)
3.30 3.05
20.47
2060
18.95
l The pulse propagating velocities for regions B and C were obtained by measuring the time the tensile pulse took to travel from one station to the next station. t The pulse propagating velocities for regions A and D were obtained by measuring the time between the arrival of the two peaks (compressive and tensile pulse).
Table 4. Avcragc values of dynamic Young’s modulus E,. Specimen
RegiSn A
Region B
Region C
Region D
Ed(C)Pa)
E, (GPn) 19.22 21.35 18.96 19.84
E,(GPa) 19.08 20.76 20.59 20.14
E,(GPa) 18.84 19.87 19.59 (9.43
Table 5. Avrragc valucs(from four dilfcrent tcsts)ofdynamic Poisson’s ratio Poisson’s Ratio v(~!= I x IO’ s- ‘)
Spccimcn I II III Mean value
19.35 20.90 19.85 20.03
E, = IV.9 (GPa).
Station I II III Mean value
I
Station 2
St;ltion 3
0.360 0.354 0.365 0.360
0.350 0.360 0.340 0.350
0.370 0.382 0.357 0.370
Table 6. Static mechanical propertics [Young’s modulus (E,) and Poisson’s ratio (v.)] Region A Specimen I II Ill Mean value
E,GPa 15.2 16.3 16.8 16.1
v, 0.362 0.374 0.357 0.364
Region B E,GPa 16.0 17.1 16.8 16.6
Y, 0.348 0.345 0.356 0.350
Region C E,GPa 15.1 17.6 17.2 16.1
Region D v,
0.345 0.353 0.365 0.354
E,GPa 14.7 15.5 16.6 15.6
v, 0.355 0.345 0.360 0.353
I?,.- 16.2 GPa; F,=O.36.
where At’ and At’ represent the time it took the compressive and the reflective tensile pulse to travel from one station to the adjacent station respectively. The average val@ of the dynamic Young’s modulus for each sp&men at tt+ four tegions A B. C. and D as presented in Tables I. 2 and 31are presented in Table 4. When one takes the average values ~ofthe dynamic Young’s moduli in each region A. 8. and C., it is found that these values differ by less than 4 %. It can beconc)uded. therefore, that the shalt of the femur, when it is loaded dynamically along its axis. has a mean dynamic mo6ulu.s of elasticity of E, - 19.9 x lop Pa.
Table 5 prcscnts the dynamic Poisson’s ratios v, at gage stations I. 2, and 3 of the three specimens which wcrc calculated using the amplitudes of the longitudinal and transverse strain gages records at each station. No significant variation was observed. The avcragc value for the dynamic Poisson’s ratio obtained was v, = 0.36. Table 6 presents the static values of Young’s modulus and Poisson’s ratio at the region A. B. C. and D for Sections (I), (II), and(III) which were obtained by performing on the corresponding small specimens in a universal testing
experiments
Technical Note
1182
Table 7. Young’s modulus (E) and Poisson’s ratio (v) of cortical bone (human femur) obtained by different investigators* Authors’ names
E(GPa)
Ko (1953) 8edIin (1965) McElhaney (1966) Abendschein and Hyatt (1970) Bargren er al. (1974) Reilly and Burstein (1975) Lappi et al. (1979) Ashman et al. (1984) Present study
l
17.3 15.8 15.1 29.5 24.5 15.8 18.7 17.0 5.5 20.0 16.2 19.9
v
0.46 0.39 0.36 0.36 0.36
Comments Very low strain rate (wet bone) Bending unknown strain (wet bone) Low strain rate (1 x IO-’ 9-l) (wet bone) High strain rate (3 x 10s s-‘) (wet bone) Ultrasonics (wet bone) Hydrated (5.2 Hz) Air dried (5.2 Hz) Los strain rate (wet bone) Ultrasonics (wet bone) Ultrasonics (wet bone) Low strain rate (2 x lo-’ s-‘) (dry bone) High strain rate (I x IO* s- ’ ) (dry bone)
Loading direction parallel to bone axis.
(Instron 1125) machine and at the same level of strain (- 1200 p) with the dynamic experiments of this study. The static experiments (lnstron 1125) were performed with strain rate 1-2 x IO-ss-’ while the dynamicexperiments(air gun) with d- I x IO* s-‘. Table 6 gives the values of Young’s modulus (E,) obtained for the specimens under static load; it can be observed that these values vary in a similar manner to those found using dynamic loading for the tthree sectionsand for the four regions of the femur. No significant variations were observed in the static values of Poisson’s ratio between specimens or in the digerent regions of the same specimen. The average values for the Young’s modulus and Poisson’s ratio for static loading (t-2x lo-‘s-‘) and for all regions were found to be E,= 16.2 x 10’ Pa;
?,-0.36.
respectively. Due to the viscoelastic character of bone, stress waves attenuate as they propagate. This effect is shown clearly in Fig. S(a) which presents the longitudinal strain gage records at stations I and 2 of specimen (II). The horizontal scale was set at 5O/‘sdiv’* in order to obtain many reflections of the propagating stress pulse from the free ends of the specimen, while the vertical scale was 5 mV div- ’ which corresponds to a strain level 46Opdiv-‘. The attenuation coefficient o. of the cortical bone was calculated from the relation (Kolsky, 1963)
(2) where c, and et are the peak strains measured at two stations and d the distance between them. Figure 5(b) presents typical strain gage records of longitudinal strain at stations I and 2 of specimen II for these experimental conditions. The horizontal scale was 50 pdiv- ’ and the vertical scale was 10 mVdiv-’ which correspond to strain level 920pdiv-‘. The pulse width, measured at half maximum produced by these conditions, was approximately 50 JAS. DISCUSSION
The general purpose of this investigation was to evaluate the mechanical properties of human cortical bone during two diRerent time rates of loading (static and dynamic). The results herein. however. were based on a study of the mechanical properties of one femur only. Certainly, mechanical properties of bone vary with bone age and bone physical conditions during testing; such information is not provided by this study.
An examination of Tables 1, 2, and 3 reveals that the dynamic Young’s modulus was the same in tension and compression. In other expcrimenfs these two values were scattered. but their mean values were very close to those obtained in the experiments where no difference among them was observed. No statistically significant difference between the values of Young’s modulus in compression and those in tension was observed. This supports the results of Bargrcn et al. (1974) as well as those of Reilly er a/. (1974) who came to the same conclusion using dilferent techniques. Therefore, the results given by McElhaney et al. (19641. who stated that Young’s modulus in compression is higher (35 %) than that in tension. are dillirent from those observed in this study. From Table 4 it can be observed that the values of dynamic Young’s modulus calculated in Section (II) were 613% higher, in all regions, than the corresponding values of the other two sections,while the variation of Young’s modulus at various regions of the same section was between 3 and 7 %. This means that the spatial variations of dynamic Young’s modulus around the circumfercna of each region (A, B. C, and D) was about two times that along the same section. It is also noted that the lower values of the dynamic Young’s modulus were in region D which corresponds to the part of diaphysis next to the distal part of the femur of all (three) sections. To statistically evaluate the experimental results of Young’s modulus from the specimens of different sections or regions an analysis has been carried out by using the statistical software package, ‘Minitab’. To compare the digcrencesof results between regions, we employed the data of the same region from each section (13 data from region A. 16 from B. 14 from C and 10 from D). The differences among them were shown to be insignificant with a significance level of 5 %. The mean values for the Young’s Modulus in regions A. D. C and D were 20.172, 20.217, 20.192 and 19.445 GPa respectively. But when wecompared the diflcrences between sections (12 data points from Section 1,14 from 11and 12 from III), we found that the Young’s modulus in Section II was slightly larger than the values in the other two scctions: the mean values for E, in 8ections I. II and III were 19.234,21.014 and 19.883 GPa, respectively. An analysis of variance comparing the three sections rejected the hypothesis that Young’s modulus was the same at a 5 % significance kvel. A further analysis employing orthogonal contrasts and a significana level of 5 % showed that the Young’s modulus for Sections I and III are not statistically different while 8ection 11 is significantly different from both I and HI. In order to examine how much the mechanical properties, viz Young’s modulus and PoissonS ratio, of human cortical bone were atTectedby the strain rate, a comparison should be made between the values of Tables 4 and 5 with the corresponding ones of Table 6. From this comparison it can be
Technical Note obrcrvcd that the Young’s modulus is 23% higher during dynamic loading, while Poisson’s ratio was not affected by the two diRerent type d loadings It should be pointed out that the comparison of the mechanical properties of bone between static and dynamic loading was made in sp$cimcm taken from the same femur. Furthermore. all spc&mns were prepared (machined, etc.) carefully using the same proccdurcs. Experiments were alsj~ performed on Pkxiglas specimens of the same geometry the bone specimens of this study under similar loading E$S nditions (static-dynamic). No variation was observed in Poisson’s ratio but a much higher increase. of order 64.7 9, in Young’s modulus was observed between the corrcspot+ting values of static and dynamic loading. large variatibns in Young’s modulus for high polymers for intcmted(ate speed (E=4x lo-‘s-t) loading were found by Raftopbulos et of. (1976) and for dyanmic loading by Goldsmith and Katsamanis (1979) also. Table 7 presents the mechanical properties of bone (human femur) obtained from this investigation and those reported by other investigators. From this table it can be observed that the static value of Young’s modulus reported in the present study correlates well with those published for low strain rates such as those reported by Scdlin (1965) and Bargren et 01. (1974) for hydrated bone. The dynamic loading results have almost the same value ‘with those obtained by Ashman er al. (1984) using ultrasonic, technique. The values of the modulus of elasticity given by ~Lappi et af. (1979) using ultrasonic technique E- 5.5 x lop N m - I, and by McElhancy (1966). E* 29.5 x 10’ N m- ‘, arc too low and too high (as compared with the values reported by most other studies), respectively, and arc outside of tha range of this study. It has been demonstrated. in many investigations. that bone is anisotropic or probably orthotropic. If bone is considered orthotropfc, there arc six Poisson*s ratios, three of which arc independent and measurable. In the present study, however, because the cross-sectional dimensions of the specimens used arc ~small. only one Poisson’s ratio was measured. The Poisson’s ratio values tabulated in Table 7 show that the results of this study arc close to those of Lappi et elf. (1979) and arc within the range of those given by Ashman er al. (1984) ‘and Reilly and Buntcin (1975). The lateral dimensions (especially the thickness) of the rectangular cross-see/ion of the specimens used in this study arc considered sm/ll compared with the wavelengths (approx. 8 cm). It is assumed therefore. that the pulses were not influenced by dispersions due to variable phase velocities. That is. the attenuation is attributed to the viscoclastic cffccts only, and not to the Ichange of the pulse due to the variable phase velocity which should have appcarcd as attenuation also. Utilixing cquatton (2) the attenuation coefficients were computed from data such as in Fig Z(a).The calculated average value is a = 0.01 cm i ‘, which is smaller than those reported by Pclkcr and Saha (1983) for fresh (a=O.O23 cm-‘), cmbalmul (a=O.OSScm-‘), wet (a=O.O83cm-‘) and dry ((I ==0.069cm-‘) human long bones. It can be noted that the average pulse width (measured at half maximum) was ap proximately SOps ih the previous studies, whereas in this experiment it was only 9~13 ps. and, since the attenuation cocfIicicnt for visco&stic material increases with increasing frequency (Ko)sky,j 1956). it is not unrcasonabk that the values given by Pclkcr and Saha (1983) could be much higher than for the short ;duration stress pulse conditions of the present study. Thi$ is supported by the experiments pcrformed on the spccfmcn (I), (II), and (III) by impacting steel spheres of 4.76 mm diameter fired from an air gun with a velocity _ 22 m s- r. From Fig. S(b), the calculated average value of the attenuation coefficient is found to be o =0.006 cm-‘. In dur opinion. this diffcrcna was primarily due to the fact thkt the attenuation coefficients given by Pclkcr and Saha (1983) were obtained from experiments performed in a whole long bone (human femur) and the measured values correspond to the response of a structure
1183
which can be simulated by a tube (cortical bone) containing different material (cancctlous bone) inside. The values given in the present study corresponded to the material property of cortical bone only. Tennyson et al. (1972) assumed that bones behave as a Voigt viscoclastic clement and they measured the viscous coctBcicnt (viscosity) n as a function of post mortem age utiliing the split Hopkinson bar technique. The visoosity obtained in this study (nw3.7 x IO’ Nsm”) using a similar model to that used by Tennyson et al. correlates well with the viscosity (n = 2.1 x IU’ Ns m-‘) found by them for ten days post mortem age specimens. The aforesaid values were much lower than the value II= 32.1 x 10‘ Nsms2 given by Lewis and Goldsmith (1975) who assumed a different viscoclastic mc&l. Bargrcn et uf. (1974) measured this cocRiFicut on human femur specimens using sinusoidal strain rates of 5.2 and 73 Hz They also assumed a Voigt visooclastic model but their results were surprisingly high compared to the previous ones. The average values for viscosity reported by Bargrcn CI af.wcrcn=3.59x10’Nsm-2andn=3.99xt0’Nsm-2for hydrated and air dried human cortical bone. respectively. These values of course correspond to low frequency rcsponscs (5.2 and 7.5 Hz) while those reported by the abovcmentioned researchers and these authors correspond to high frequency responses (high strain rate). Nevertheless, the diffcrcna between these two groups of experimental results cannot be explained by these authors. Acknowledgemcnrx-The support of NATO Scientific AtTairs Division. B-1 110 Brussels. Belgium, Grant No. tOl/g7. and the Mechanical Engineering Department of The University of Toledo arc gratefully acknowledged. REFERENCES
Abcndschcin, W. and Hyatt, G. W. (1970) Ultrasonics and selected physical properties of bone. Cfin. Orthap. 69. 294-301. Ashman, R. B.. Cowin. S. C.. Van Buskirk. W. C. and Ria. J. C. (1984) A continuous wave technique for the mcasurcmcnt of the elastic properties of cortical bone. J. Biomt&xnics 17. 349-361. Bargrcn. J. H.. Andrew. C.. Bassett. L. and Gjclsvik. A. (1974) Mechanical properties of hydrated cortical bone. 1. Bionuckonics 7.239-245. Bonfield. W. and Li. C. H. (1966) Deformation and fracture of bone. J. appl. Phys. 37.869-875. Dally, J. W: and Riley. W. F. (1978) Experimental Stress Analysis (2nd Edn). McGraw-Hill, New York. Goldsmith, W. and Katsamanis, F. (1979) Fracture of no:chcd and perforated polymeric bars produced by longitudinal impact. Inr. 1. M&h. Set. 21.85-108. Katsamanis. F. and Goldsmith. W. (1982) Transverse impact on fluid-fiikd cylindrical tub&. I..appl.*Nech. 49,lS~l56. Ko, R. (1953) The tension test upon the compact substana of the long bones of human extremities. 1. Kyoro PreJ Med. Univ. 53, 50>525. Kolsky. H. (1956) The propagation of stress pulses in viscocl&ii solids. &frif. Mae. II, 693-710. Kolskv. H. (I%31 Stress Waves in Solids. Dover, New York. Lang, b. 8. ‘(1976)~Ultrasonic method for measuring elastic cocfficimts of bone and results on fresh and dried bovine bones /EEE Tronr. Biomed. Engng BME 17. IOI-10s. Lappi, V. G.. King, M. S. and May. I. L (1979) Determination of elastic constants for human femurs. 1. biomech. Engng 101. 193-197. Lewis. J. Land Goldsmith, W. (1973) A biaxial split Hopkinson bar for simultaneous torsion and compression. Rev. Sci. fntrrum. 44, 81 l-813. Lewis. J. L. and Goldsmith, W. (1975) The dynamic fracture and prcfracturc response of compact bone by split Hopkinson bar methods. J. Biomechanics 8. 27-40.
II&1
Technical Note
McElhaney. J. H. (1966) Dynamic response of bone and muscle tissue. 1. appl. Physiol. 21, 1231-1236. McElhaney. J. H.. Fogle. J.. Byars. E. and Weaver. G. (1964) Etiects of embalming on the mechanical properties of beef bone. 1. appl. Physiol. 19, 1234-1236. Pelkn. R. R. and Saha. S. (1983) Stress wave propagation in bone. J. Biornechanics 16.481489. Raftopoulos. D. D.. Karapanos, D. and Tbeocharis, P. S. (1976) A static and dynamic mechanical and optical behavior of hiah wlvmers. 1. Phvs. D. ad. Phvs. 9.869-877. Reilly. D. T. %.I &stein. A.*H. (1675)I& elastic and ultimate properties of compact bone tissue. J. Biomechanits 8, 393-405. Reilly, D. T., Burstein. A. H. and Frankel, V. H. (1974) The elastic modulus for bone. J. Biomechanics 7, 271-275. Sammarco. G. J.. Burstein. A. H.. Davis W. L. and Frankel. V. H. (1971) The biomechanics of torsional fractures: the
effect of loading on ultimate properties. 1. Biomechanics 4. 113-117. Sedlin. E. D. (1965) A rheological model for cortical bone: a study of the physical properties of human femoral samples. Acta &top. xand. Suppl 83. pp. l-77. Muaksgaard Copcnhapn. Simkin. A. and Robin, G. (1973) The mechanical testing of bone in bending. 1. Biomdanics 6.31-39. Tennyson, R. C.. Ewert. R. and Niranjan, V. (1972) Dynamic viscoelastic response of bone. &xnl Mech. 12.502-507. Van Buskirk. W.‘C.. Cowin. S. C. and Ward. R. N. (1981) Ultrasonic measurement of orthotropic elastic constants of bovine femoral bone. J. biomech. Engng 103, 67-72. Yoon. H. S. and Katz J. L. (1976) Ultrasonic wave propaption in human cortical bone--II. Measurements of elastic properties and microhardness. 1. Biomechanics9,459464.
23. No. II.pp.
1173 IIR4. IWO
a)zr Y?*)90 13w + uo C IWOPerpmon Pram pk
PrImed In GROI Bntun
TECHNICALNOTE
DETERMINATION OF MECHANICAL PROPERTIES OF HUMAN FEMORAL CORTICAL BONE BY THE HOPKINSON BAR STRESS TECHNIQUE FOTIOS KATSAMANIS*
and DEMETRIOS D. RAFroPOULOSt:
* Hellenic Naval Academy. 18501 Piraeous. Greece and t Dept. of Mechanical Engineering. The University of Toledo, Toledo, OH 43606. U.S.A. Abstract-The Hopkinson bar stress technique and a universal testing machine (Instron 1125) have been used to iqvestigate the dynamic and static mechanical properties of cortical bone taken from a human femur rcspectivqly.We found that the average dynamic Young’s modulus value (E, = 19.9CPa) to be 23 % higher than the ~averagestatic Young’s modulus value (E,= 16.2 GPa). Furthermore, the Poisson’s ratio did not exhibit aey significant variation for the two dilTerent types of loading. No dilTerencewas observed between the value of the dynamic Young’s modulus in tension and those found in compression. A comparison was made of the results of this study with those found by other researchen using dilTerent techniques, such as ultrasonics, and it was found that they agree well with most of the results of previous studies, Finally, the viscosity for cortical bone found in this study correlates with viscosity reported by Tennyson et al. [E.vpl Much. II, 502-507 (1972)] for ten days post mortem age specimens.
INTRODUCTION
The determination of mechanical bone properties has been an important topi/: ofstudy for many years. Reilly et of. (1974) tested human a4 bovine specimens but found no statistically significant ariation in the moduli of elasticity detcrmined by the two‘iIloading modes (tension and compression). Bargren et of. (1914) came lo the same conclusion when they tested hydrated r(nd air dried cortical bone human femur specimens altcm~tely in axial tension and compression. In contrast to the lbwtr values of the previously mentioned studies, the high+l values for Young’s modulus were obtained in static tc$rilc tests, using Tuckerman’s optical gage, on longitudinal tiovinc tibia sections, by Bonfield and Li (1966).In the last, two decades, other techniques have been introduced for d termining mechanical properties of biomaterials, such 1 the ultrasonic method. Lang (1970) was one of Ihe first lo b an ultrasonic technique to measure the elastic propertia Of bone. Since Lang’s initial study, several invatigalon havd made use of pulse ultrasonic technique to me&e the elastic properties bf bone. In particular, Loon and Katz (1976). Land et al. 119791.and Van Buskirk et al. (1981) uodd thii &nique. Ashman et al. (1984) used a continuous ullr nit wave technique. II is well know -R that bone is a viscoelastic material and that its me&an’ I prop&a are affected by its deformation rate. MEthaney =i (1966) investigated the response of cubes of cortical bone to pression loads at various strain rates, and found that tI?e Young’s modulus and ultimate stress increased and theIstrain to failure decreased with increasing strain rate. Sammjirco et al. (1971) tested whole human and canine bones in t rsion at two rates of deformation. Tennyson er al. (1972) the split Hopkinson bar technique to study the viscoe+ tic response of bovine femur bone subjected (0 dynamic~colnpreasive loading. A linear viscoelastic m&l daaibing Ithe mechanical behavior of bovine bone. including an estimate of the parameters. was obtained when Received in fit@ fom I7 April 1990. 1 Author to whbm correspondence should be addressed.
these bones were tested while still very fresh. Lewis and Goldsmith (1973. 1975) developed a biaxial split Hopkinson bar apparatus to measure dynamic material properties of compact bovine bone in compression. tension. torsion and combined torsion and compression. They concluded that in compression bone was viscoclastic and that fracture stress increased with strain rate. A major advantage in Ihe Lewis and Goldsmith technique was that strain measurements were obtained directly from the strain gagcr attached (0 the specimens. That is. these authors claim that their strain values were inherently more accurate than those obtained by Tennyson et a/. who deduckd strain by the normal split Hopkinson bar method using the magnitude of the incident reflected and trasmitted waves. Furthermore. Lewis and Goldsmith also pointed out that one penalty which must be paid for this accuracy was that the specimen had to be dried out for half-an-hour whik the strain gage was mounted. Pelker and Saha (1983) studied the traveling wave characteristics for a single compressive pulse in fresh and embalmed human long bones. The stress wave was generated by the longitudinal impact ofa steel ball on one end of the bone and was monitored by bonded semiconductor strain gages. The dynamic properties, namely wave velocity. attenualion coefficient, and dispersion were correlated with various parameters such as mineral density. porosity and cross-sectional area of the specimens. In the present study the Hopkinson bar technique was used to investigate the dynamic characteristics of cortical bone of a human femur. Stress waves were produced by the longitudinal impact of steel sphcrrr of 4.76 mm in diameter, fired from an air gun onto the end of long specimens. These specimens were cut along the shaft of the femur and machined to haw a uniform rectangular cross-section.Stress waves were detected by means of strain gages bonded in the longitudinal and cransversedirections at three stations. Since the cross-sectional dimensions of the specimens used were small relative lo pulse kngth. onedimensional wave theory in rods (Kolsky. 1963) was applicabk to this study. The aims of this investigation are: (1) Firstly, to measure the dynamic Young’s modulus and Poisson’s ratio at various locations on specimens prepared
1173
Technical Note
1174
I
I
-i----w---
_..___,
. ___
I (alRl@tFamur
I
1
(
b 1 Shan
( Mwhyals
)
Fig. I. The femur and the spccimcns used in this investigation.
from the same femur using the Hopkinson bar technique. and to comparc these values with the corresponding static values obtained from tensile cxpcrimcnts in a universal testing machine (Instron 1125). as well as with those reported by other investigators. (2) Secondly, to dctcrminc whether the modulus or clasticity of bone in tension is dilTcrrnt than that in compression as mportcd by some previous invcstigators(McEIhanty et al.. 1964; Simkin and Robin, 1973). (3) Thirdly, to measure bone’s viscoclastic characteristic prramctcrs, namely the attenuation cocllicicnt and viscosity, assuming that bone behaves as Voigt viscoelastic element and to compare the results with those reported by other rcscarchcrs. TEST SPECIMENS-EXPERIMENTAL
SET-UP
Tests were run on bone specimens from the shaft (diaphysis) of the right femur of an adult (45-55 yr old) male cadaver obtained from the Medical College of Ohio, at Toledo [Fig l(a)]. The lcmur was delivered to the laboratory fresh. However. when the specimens wcrc made. they were left for several days to dry out in order to mount the strain gages. This femur. therefore, should bc considered embalmed. dry and not wet. The first step, in preparing the specimens, was sectioning the diaphysis into three long pieces along its longitudinal axis. The numbers (I), (II). and (III) were assigned to the specimens as indicated in Fig. l(b). From each pica two strips were cut; these were milled to obtain long specimens (rods) of rectangular cross-section. One of them was cut into four short picas labeled (A), (B). (C), and (D). The dimensions of the specimen arc shown in Fig. I(c). The long specimens were used to measure dynamic mechanical properties at various locations the short specimens were used to measure the corresponding static properties. Furthcrmore. the longitudinal axis and the loading imposed on the spccimcnswere parallel to the long axis of the bone; and these specimens will hcncerorth be referred to as longitudinal specimens.
The static cxpcrimcnts were performed in a universal testing machine (lnstron 1125) and using very low speed.The short specimens were machined and shaped to appropriate dimensions. A hole at each end of the spccimcn was drilled to avoid slippage, see Fig. 1. Two strain gages were bonded on each specimen. One. EA-1312SBT-120, was in the longltudinal direction and the other. CEA-13-062VW-350, was bonded in the transverse direction. Strain gages were placed on both sides ol test spccimcns to eliminate the clTcctsof bending. Standard tcnsilc tests were conducted on these. specimenson the universal testing machine. Strain indicators were connected to the longitudinal and transverse strain gages, respectively; both the strains and tensile loads were recorded during testing. To dynamically load the bone specimens,a stresspulse was introduced by means of an air gun dcvelopcd for that purpose. The gun was designed to fire steel spheres of 12.7, 9.53, 6.35 and 4.26 mm diameter and it was powered by compressed air that was triggered by a quick action clcctromagnetic valve. The projectile velocities wcrc dctcrmincd From the signals recorded on a storage oscilloscope produced by the interruption ot two light beams a distance of 50 mm apart just ahead or the gun barrel in the path ol the projectile. The bone specimen was suspended in the horizontal position by means of two strings. Four strain gages ol 3.17 mm gage length (Micromcasurcntcnt Co.) were carefully bonded at three stations labeled 1.2, and 3 with M-Bond MO adhesive. The rcsistancc of these strain gages was 120 D and their gage (actor 211. Two pairs olstrain gages were bonded on each gage station; one pair in the longitudinal direction and the other pair in the transverse direction on opposite sides of the specimen. Each pair was connected in seriesso as to eliminate any antisymmetrical components 0r the slms pulse. In view ol the minimum pulse length, approximately 80 mm. the gage length of 3.17 mm was sufficiently short to accurately reflect the strain history at a given station. Each strain gage output was checked separately before being connected in series with the remaining strain gages. The gages were incorporated in potentiometric circuits (Dally and Riley, 1978) connected to dual trace digital storage
Technical Note
osdoscopu
(7ckronix 468) witb IO MHz useful storage battdwidtb. The oac&%sopns were triggered from tbc signal generated upon itperruption by the projectik ofa lucr beam focuacd on a pbotoodl just ahead of the impect point. To minimixc dcctrot$c noise c&c& light-wcigbt sbieldcd cabks that containad four cbannd potentiotnctric circuits wem used to connect ,strain gages with the gage box. For tbc connaXio0 baw+n the pip box and the digital storage oscilloscopa. co xial cables were used. A 24 V battery was conmftcd in P& Ikl to the four-channel potcntiornctric circuit in order toiobtaio an adapJote output signal from the strain gages.To avoid damage ofthe bonded strain gages due to local beating c&sed by the 50 mA electric current passing through them, t&24 V DC voltage was applied to the gage box only for a short time period (sS 8). The experimental arrangement used in this study is shown schematically in Fig. 2 A similar experimental arrangement was used by Goldsmith and katsamanis (1979) and Katsamanis and Goldsmith (1982) The specimens were carefully aligned to provide an axial impact. A polaroid camera was used to obtain the strain gage records from tbe~storage oscilloscopes. Many preliminary tests were conducted in order to select optimal arrangements for the reported (ests. For the specimen configuration cmployed. the small steel spheres of 4.76 mm diameter were fired from the air gun ivith a v&city of 22 m s- ‘. These spheres produced a short duration stress pulse (~25 cs) with an amplitude of I 156 w which provided an appropriate input. As noted. four ctumncls were available to store strain gage records; IWO channels were nccdcd for each gage station. one for longitudinal and one for the transverse strain. Tbercforc. data were taken from only two gage stations during each run. In order to obtain all the ncccssarydata the cxpcrimcnt was repeated with thc:samc conditions several times. The calibration factor of the four channel gage box was sckctcd at 92.2 p mv - ’ for all tests.
Photo-Cella-
1175
RESULTS Figure 3 presents typical strain gage records as they were obtained using a polaroid camera from a digital storage oscilloscope. These records correspond to specimen (II) and show the propagating compressive stress pulse at stations I and 2. Figure 3(a) presents strain in the longitudinal direction (L,, Lx) and Fig 3(b) in the transverse direction (T,. TJ. In these records, the vertical setting was 2 mVdiv- ’ which corresponds to I85 pdiv-’ (by mistake L, was set at a little over 2mVdiv-r) and the grapb was taken inverted for convenience. Tbe horizontal scak was 10 pdiv- *. Figure 4 also presents typical strain gage records of stations 2 and 3 for the same specimen (If), The horizontal scak was taken as 20 p div- ’ and, therefore, two wave refkctions arc shown in addition to the initial compressive pulse (1st tensile and 2nd compressive). In Fig. 4(a). which corresponds to longitudinal strain (I!.,, tI), the vertical setting was 5 mVdiv_’ (460 p div- ‘) and in Fig 4(b). which corresponds to transverse strain (T,, 7,). the vertical setting was ZmVdiv-’ (185 pdiv-‘). These settings on the digital storage oscilloscope were appropriate for collecting accurate data. Photographs of the oscilloscope traces were enlarged by projection to increase measurement sensitivity. The velocity of wave propagation was determined from the time elapsed between tbc arrival of the pulse peak bctwccn the two gage stations. Pulse propagating velocities for the compressive and the reflected tensile pulses were obtained for the specimen regions B and C. No statistically significant difference was observed between the two pulse wave velocitics. Furthermore. since there is no dilkrcncc bctwccn the compressive and tensile speeds, the pulse propagating vclocities for the other two regions (A and D) of the specimen wcrc obtained by measuring the time bctwccn the arrival of the two peaks (compressive and reflected tcnsik pulse) at gage stations I and 3. mpcctivcly.
I 12-24V
8
On.011
Fig. 2. Schematic diagram of the experimental set-up for the determination of cortical bone dynamic mechanical properties.
1176
Technical Note
Since the nominal dimensions of the specimens used (I. II and 111) were 22Or : 10.5 x 4 mm and the minimum pulse kngth &as 80 mm, the wave velocity for the rod is given by the one-dimensional approximation:
(where E, is the dynamic Young’s modulus and p the mass density of the bone) and quite accurately represents the pulse propagation velocity. Knowing the density p of the bone at each region of the specimen and using quation (I). the dynamic Young’s moduli, for the compression (I?$) and tension (E;) pulse were obtained. Archimedes’ principle was the basis of the technique
employed to determine the mass density p of the bone, which was measured for each particular specimen region. The density measumncnts took place just before gage installation. That is. there was not significant drying between the time of density measurement and testing to signilkantly a&t the density measured. This technique also has been used by Ashman et af. (1984). The dynamic Poisson’s ratio v, was calculated at three locations for each specimen (gage stations 1. 2, and 3) by dividing the amplitude of transverse strain s, by the comaponding amplitude of the longitudinal strain st at the same gage station. Tables I. 2 and 3 summar& the rcsultr for each spcimen (1). (II). and (III). In these tables d is the corresponding distana that stress pulse propagates during time At’ or At’
Table 1. Experimental results for Young’s modulus (E,), specimen (I)
ala
Region
Al’(w)
At’(w)
Compression
Tension
17.70 Exp. 26
A+ 17.80
Exp. 27
B A C’
18.30
Exp. 28 Exp. 30
Dt A’ D B+C
Exp. 31 Exp. 33
D B+C C
55.4
18.70
59.7 x 2 55.4
3.11
17.00
59.7 x 2 55.5
3.04
38.30 30.40 38.70 30.00 35.70
35.70 39.10 30.80
37.30 18.50
Distance C: (km s-t) C: (km s- i)
18.2 39.10
A
E;+
Ed==-
d(m)
34.4 17.60
48.2 x 2 59.7 x 2 48.2 x 2 110.9 59.7 x 2 48.2 x 2 110.9 55.5
3.13
3.05 3.05
E’,(GPa)
2090
20.02
19.04
19.53
2080
19.04 18.89
2.96
2090
3.27
2080 2000
19.88 18.13 20.98
19.88 19.56
1910 2080 1910 2050 2080 1910 2050 Moo
18.81 19.42 19.35 19.46 19.46 19.02 18.35 17.80 20.95 17.58 19.60
18.81 19.42 19.35 19.46 19.02 18.35 19.37 18.59
3.17 3.09 3.21 3.11 3.05 3.13 2.98 2.99
G (GPa)
19.04 19.80 17.99
3.12
3.11
Density
2 @Pa)
3.23 3.16
E;
l The pulse propagating velocities for regions B and C wcrc obtained by measuring the time the tensile pulse took to travel from one station to the next station. t The pulse propagating velocities for regions A and D were obtained by measuring the time between the arrival of the two peaks (compressive and tensile pulse).
Table 2. Experimental results for Young’s modulus, specimen (It)
Region Exp. 10 Exp. 11 Exp. 12
Dt C’ D C B’
AW4
AI’(P)
Compression
Tension
33.50 17.40 3220 17.70 17.65
Exp. 18 Exp. I5 Exp. 13 Exp. 16
B A B A A B+C B A A D
17.70 17.65 37.6
At Exp. 14
17.40
18.4
16.8 37.2
18.20
17.50 36.80 37.10
35.6 16.8
36.0 17.6 36.8 36.8 34.40
G+E:
E,=-
d(mm) Distance C: (kms- ‘) C: (kms- t) p (kgm-‘) 103.8 55.8 103.8 55.8 55.7 119.4 55.7 119.4 55.7 119.4 119.4 111.5 55.7 119.4 119.4 103.8
3.10 3.21
3.21 3.22
3.15 3.16
3.15 3.16 3.18
3.32
3.03 3.21 3.96
3.19 3.24 3.22 3.10 3.17
3.13 3.32 3.25 3.25 3.02
2090 2160 2050 2160 2050 2160 2050 2050 2120 2160 2050 2050
E; (GPa) E, (GPa) 19.64 21.16 21.16 21.27 20.36 20.36 21.10 21.10 20.28 19.41 23.29 20.72 21.54 19.85 21.15 20.86 20.40 19.95 23.29 21.22 21.17 21.17 18.68
(GP: 19.64 21.16 21.27 20.36 21.10 20.28 21.35 20.72 20.69 21.15 20.86 20.18 22.26 21.17 21.17 18.68
l The pulse propagating velocities for regions B and C were obtained by measuring the time the tensile pulse took to travel from one station to the next station. t The pulse propagating velocities for regions A and D were obtained by measuring the time between the arrival of the two peaks (compressive and tensile pulse).
2mV(185p&)
Ll
(a)
-Tl 2mV(185p&)
-T2
(b) Fig. 3. Strain gage records for spccimcn (II) showing the propagating comprcsive stress pulse at stations I and 2. (a) Stress pulse in longitudinal direction (L,. L2) (for convenience L, was taken inverted); (h) stress pulse in transverse direction (T,, T, 1.
II77
L3 5mV(460y&)
(4
T2 2mV(185p&)
T3
(b) Fig. 4. Strain gage records for specimen (II) at stations 2 and 3. Two wave rcktions (1st tensile and 2nd compressive) are shown in addition IO compreGve pulse. (a) Stress pulse in the longitudinal direction (L,, L,);(b) stress pulse in the transverse direction (T>. T,).
117x
5mV(460pc)
L2
Ll 1 OmV(920pE)
L2
(b) Fig. 5. Strain page records for specimen (II) at stations I and Z in longitudinal direction (I.,. f_,).(a) Stress pulse was pencrafed by impacting slecl spheres or 4.27 mm diameter on the end of the specimen (impact velocity !2 m s- ’ 1. The corrcspondinp attenuation coellicicnt is a=O.Ol cm-‘. (b) Stress pulse was generated by impacting slwl spheres ol 9.5 mm diameter on the end of the specimen (impact velocity 10 m s _ ’ ). The corresponding attenuation coellicient is a = 0.006 cm - ’
1179
Technical Note
1181
Table 3. Experimental results for Young’s modulus, specimen (III)
t;+ E, &,=Region
Compression
Tension
19.3
16.70
54.1
18.40
51.1 X2 54.1
3.01
17.10
51.1 X 2 54.3
3.18
17.5
59.7 x 2 54.3
3.26
B* Exp. 20 Exp. 21
Exp. 22 Exp. 23 Exp. 24
Exp. 25
A+ B
18.0
A C’
17.10
D+ C
16.70
32.5 32.70 37.5
D B+C A D C
39.20 34.60
35.90 33.20 37.70
18.60
D
16.50 39.10
Distance C:(kms-‘)C:(kms-‘) 2.80
p(kgm-‘)
16.49
22.16
19.33
2.95
2080 2140
20.18 18.96 18.21
20.18 18.59
2080 2100
19.99
3.18
20.82
20.82
19.99 20.82
3.10
2060 2IM)
20.48 21.86 19.84
20.48 20.85
18.78 19.36 20.24 17.60 22.42
18.78 19.72 19.36 20.24 20.0 I
18.85
18.85
3.18 3.05
2060 2120 2080 2060 2100
3.02 3.08 3.17
2.92
59.7 x 2
2 (GPa)
2140
3.13
3.14
E:(GPa)
3.25 3.15
59.7 x 2 108.4 51.1 x 2 59.7 X 2 54.3
E;(GPa)
3.30 3.05
20.47
2060
18.95
l The pulse propagating velocities for regions B and C were obtained by measuring the time the tensile pulse took to travel from one station to the next station. t The pulse propagating velocities for regions A and D were obtained by measuring the time between the arrival of the two peaks (compressive and tensile pulse).
Table 4. Avcragc values of dynamic Young’s modulus E,. Specimen
RegiSn A
Region B
Region C
Region D
Ed(C)Pa)
E, (GPn) 19.22 21.35 18.96 19.84
E,(GPa) 19.08 20.76 20.59 20.14
E,(GPa) 18.84 19.87 19.59 (9.43
Table 5. Avrragc valucs(from four dilfcrent tcsts)ofdynamic Poisson’s ratio Poisson’s Ratio v(~!= I x IO’ s- ‘)
Spccimcn I II III Mean value
19.35 20.90 19.85 20.03
E, = IV.9 (GPa).
Station I II III Mean value
I
Station 2
St;ltion 3
0.360 0.354 0.365 0.360
0.350 0.360 0.340 0.350
0.370 0.382 0.357 0.370
Table 6. Static mechanical propertics [Young’s modulus (E,) and Poisson’s ratio (v.)] Region A Specimen I II Ill Mean value
E,GPa 15.2 16.3 16.8 16.1
v, 0.362 0.374 0.357 0.364
Region B E,GPa 16.0 17.1 16.8 16.6
Y, 0.348 0.345 0.356 0.350
Region C E,GPa 15.1 17.6 17.2 16.1
Region D v,
0.345 0.353 0.365 0.354
E,GPa 14.7 15.5 16.6 15.6
v, 0.355 0.345 0.360 0.353
I?,.- 16.2 GPa; F,=O.36.
where At’ and At’ represent the time it took the compressive and the reflective tensile pulse to travel from one station to the adjacent station respectively. The average val@ of the dynamic Young’s modulus for each sp&men at tt+ four tegions A B. C. and D as presented in Tables I. 2 and 31are presented in Table 4. When one takes the average values ~ofthe dynamic Young’s moduli in each region A. 8. and C., it is found that these values differ by less than 4 %. It can beconc)uded. therefore, that the shalt of the femur, when it is loaded dynamically along its axis. has a mean dynamic mo6ulu.s of elasticity of E, - 19.9 x lop Pa.
Table 5 prcscnts the dynamic Poisson’s ratios v, at gage stations I. 2, and 3 of the three specimens which wcrc calculated using the amplitudes of the longitudinal and transverse strain gages records at each station. No significant variation was observed. The avcragc value for the dynamic Poisson’s ratio obtained was v, = 0.36. Table 6 presents the static values of Young’s modulus and Poisson’s ratio at the region A. B. C. and D for Sections (I), (II), and(III) which were obtained by performing on the corresponding small specimens in a universal testing
experiments
Technical Note
1182
Table 7. Young’s modulus (E) and Poisson’s ratio (v) of cortical bone (human femur) obtained by different investigators* Authors’ names
E(GPa)
Ko (1953) 8edIin (1965) McElhaney (1966) Abendschein and Hyatt (1970) Bargren er al. (1974) Reilly and Burstein (1975) Lappi et al. (1979) Ashman et al. (1984) Present study
l
17.3 15.8 15.1 29.5 24.5 15.8 18.7 17.0 5.5 20.0 16.2 19.9
v
0.46 0.39 0.36 0.36 0.36
Comments Very low strain rate (wet bone) Bending unknown strain (wet bone) Low strain rate (1 x IO-’ 9-l) (wet bone) High strain rate (3 x 10s s-‘) (wet bone) Ultrasonics (wet bone) Hydrated (5.2 Hz) Air dried (5.2 Hz) Los strain rate (wet bone) Ultrasonics (wet bone) Ultrasonics (wet bone) Low strain rate (2 x lo-’ s-‘) (dry bone) High strain rate (I x IO* s- ’ ) (dry bone)
Loading direction parallel to bone axis.
(Instron 1125) machine and at the same level of strain (- 1200 p) with the dynamic experiments of this study. The static experiments (lnstron 1125) were performed with strain rate 1-2 x IO-ss-’ while the dynamicexperiments(air gun) with d- I x IO* s-‘. Table 6 gives the values of Young’s modulus (E,) obtained for the specimens under static load; it can be observed that these values vary in a similar manner to those found using dynamic loading for the tthree sectionsand for the four regions of the femur. No significant variations were observed in the static values of Poisson’s ratio between specimens or in the digerent regions of the same specimen. The average values for the Young’s modulus and Poisson’s ratio for static loading (t-2x lo-‘s-‘) and for all regions were found to be E,= 16.2 x 10’ Pa;
?,-0.36.
respectively. Due to the viscoelastic character of bone, stress waves attenuate as they propagate. This effect is shown clearly in Fig. S(a) which presents the longitudinal strain gage records at stations I and 2 of specimen (II). The horizontal scale was set at 5O/‘sdiv’* in order to obtain many reflections of the propagating stress pulse from the free ends of the specimen, while the vertical scale was 5 mV div- ’ which corresponds to a strain level 46Opdiv-‘. The attenuation coefficient o. of the cortical bone was calculated from the relation (Kolsky, 1963)
(2) where c, and et are the peak strains measured at two stations and d the distance between them. Figure 5(b) presents typical strain gage records of longitudinal strain at stations I and 2 of specimen II for these experimental conditions. The horizontal scale was 50 pdiv- ’ and the vertical scale was 10 mVdiv-’ which correspond to strain level 920pdiv-‘. The pulse width, measured at half maximum produced by these conditions, was approximately 50 JAS. DISCUSSION
The general purpose of this investigation was to evaluate the mechanical properties of human cortical bone during two diRerent time rates of loading (static and dynamic). The results herein. however. were based on a study of the mechanical properties of one femur only. Certainly, mechanical properties of bone vary with bone age and bone physical conditions during testing; such information is not provided by this study.
An examination of Tables 1, 2, and 3 reveals that the dynamic Young’s modulus was the same in tension and compression. In other expcrimenfs these two values were scattered. but their mean values were very close to those obtained in the experiments where no difference among them was observed. No statistically significant difference between the values of Young’s modulus in compression and those in tension was observed. This supports the results of Bargrcn et al. (1974) as well as those of Reilly er a/. (1974) who came to the same conclusion using dilferent techniques. Therefore, the results given by McElhaney et al. (19641. who stated that Young’s modulus in compression is higher (35 %) than that in tension. are dillirent from those observed in this study. From Table 4 it can be observed that the values of dynamic Young’s modulus calculated in Section (II) were 613% higher, in all regions, than the corresponding values of the other two sections,while the variation of Young’s modulus at various regions of the same section was between 3 and 7 %. This means that the spatial variations of dynamic Young’s modulus around the circumfercna of each region (A, B. C, and D) was about two times that along the same section. It is also noted that the lower values of the dynamic Young’s modulus were in region D which corresponds to the part of diaphysis next to the distal part of the femur of all (three) sections. To statistically evaluate the experimental results of Young’s modulus from the specimens of different sections or regions an analysis has been carried out by using the statistical software package, ‘Minitab’. To compare the digcrencesof results between regions, we employed the data of the same region from each section (13 data from region A. 16 from B. 14 from C and 10 from D). The differences among them were shown to be insignificant with a significance level of 5 %. The mean values for the Young’s Modulus in regions A. D. C and D were 20.172, 20.217, 20.192 and 19.445 GPa respectively. But when wecompared the diflcrences between sections (12 data points from Section 1,14 from 11and 12 from III), we found that the Young’s modulus in Section II was slightly larger than the values in the other two scctions: the mean values for E, in 8ections I. II and III were 19.234,21.014 and 19.883 GPa, respectively. An analysis of variance comparing the three sections rejected the hypothesis that Young’s modulus was the same at a 5 % significance kvel. A further analysis employing orthogonal contrasts and a significana level of 5 % showed that the Young’s modulus for Sections I and III are not statistically different while 8ection 11 is significantly different from both I and HI. In order to examine how much the mechanical properties, viz Young’s modulus and PoissonS ratio, of human cortical bone were atTectedby the strain rate, a comparison should be made between the values of Tables 4 and 5 with the corresponding ones of Table 6. From this comparison it can be
Technical Note obrcrvcd that the Young’s modulus is 23% higher during dynamic loading, while Poisson’s ratio was not affected by the two diRerent type d loadings It should be pointed out that the comparison of the mechanical properties of bone between static and dynamic loading was made in sp$cimcm taken from the same femur. Furthermore. all spc&mns were prepared (machined, etc.) carefully using the same proccdurcs. Experiments were alsj~ performed on Pkxiglas specimens of the same geometry the bone specimens of this study under similar loading E$S nditions (static-dynamic). No variation was observed in Poisson’s ratio but a much higher increase. of order 64.7 9, in Young’s modulus was observed between the corrcspot+ting values of static and dynamic loading. large variatibns in Young’s modulus for high polymers for intcmted(ate speed (E=4x lo-‘s-t) loading were found by Raftopbulos et of. (1976) and for dyanmic loading by Goldsmith and Katsamanis (1979) also. Table 7 presents the mechanical properties of bone (human femur) obtained from this investigation and those reported by other investigators. From this table it can be observed that the static value of Young’s modulus reported in the present study correlates well with those published for low strain rates such as those reported by Scdlin (1965) and Bargren et 01. (1974) for hydrated bone. The dynamic loading results have almost the same value ‘with those obtained by Ashman er al. (1984) using ultrasonic, technique. The values of the modulus of elasticity given by ~Lappi et af. (1979) using ultrasonic technique E- 5.5 x lop N m - I, and by McElhancy (1966). E* 29.5 x 10’ N m- ‘, arc too low and too high (as compared with the values reported by most other studies), respectively, and arc outside of tha range of this study. It has been demonstrated. in many investigations. that bone is anisotropic or probably orthotropic. If bone is considered orthotropfc, there arc six Poisson*s ratios, three of which arc independent and measurable. In the present study, however, because the cross-sectional dimensions of the specimens used arc ~small. only one Poisson’s ratio was measured. The Poisson’s ratio values tabulated in Table 7 show that the results of this study arc close to those of Lappi et elf. (1979) and arc within the range of those given by Ashman er al. (1984) ‘and Reilly and Buntcin (1975). The lateral dimensions (especially the thickness) of the rectangular cross-see/ion of the specimens used in this study arc considered sm/ll compared with the wavelengths (approx. 8 cm). It is assumed therefore. that the pulses were not influenced by dispersions due to variable phase velocities. That is. the attenuation is attributed to the viscoclastic cffccts only, and not to the Ichange of the pulse due to the variable phase velocity which should have appcarcd as attenuation also. Utilixing cquatton (2) the attenuation coefficients were computed from data such as in Fig Z(a).The calculated average value is a = 0.01 cm i ‘, which is smaller than those reported by Pclkcr and Saha (1983) for fresh (a=O.O23 cm-‘), cmbalmul (a=O.OSScm-‘), wet (a=O.O83cm-‘) and dry ((I ==0.069cm-‘) human long bones. It can be noted that the average pulse width (measured at half maximum) was ap proximately SOps ih the previous studies, whereas in this experiment it was only 9~13 ps. and, since the attenuation cocfIicicnt for visco&stic material increases with increasing frequency (Ko)sky,j 1956). it is not unrcasonabk that the values given by Pclkcr and Saha (1983) could be much higher than for the short ;duration stress pulse conditions of the present study. Thi$ is supported by the experiments pcrformed on the spccfmcn (I), (II), and (III) by impacting steel spheres of 4.76 mm diameter fired from an air gun with a velocity _ 22 m s- r. From Fig. S(b), the calculated average value of the attenuation coefficient is found to be o =0.006 cm-‘. In dur opinion. this diffcrcna was primarily due to the fact thkt the attenuation coefficients given by Pclkcr and Saha (1983) were obtained from experiments performed in a whole long bone (human femur) and the measured values correspond to the response of a structure
1183
which can be simulated by a tube (cortical bone) containing different material (cancctlous bone) inside. The values given in the present study corresponded to the material property of cortical bone only. Tennyson et al. (1972) assumed that bones behave as a Voigt viscoclastic clement and they measured the viscous coctBcicnt (viscosity) n as a function of post mortem age utiliing the split Hopkinson bar technique. The visoosity obtained in this study (nw3.7 x IO’ Nsm”) using a similar model to that used by Tennyson et al. correlates well with the viscosity (n = 2.1 x IU’ Ns m-‘) found by them for ten days post mortem age specimens. The aforesaid values were much lower than the value II= 32.1 x 10‘ Nsms2 given by Lewis and Goldsmith (1975) who assumed a different viscoclastic mc&l. Bargrcn et uf. (1974) measured this cocRiFicut on human femur specimens using sinusoidal strain rates of 5.2 and 73 Hz They also assumed a Voigt visooclastic model but their results were surprisingly high compared to the previous ones. The average values for viscosity reported by Bargrcn CI af.wcrcn=3.59x10’Nsm-2andn=3.99xt0’Nsm-2for hydrated and air dried human cortical bone. respectively. These values of course correspond to low frequency rcsponscs (5.2 and 7.5 Hz) while those reported by the abovcmentioned researchers and these authors correspond to high frequency responses (high strain rate). Nevertheless, the diffcrcna between these two groups of experimental results cannot be explained by these authors. Acknowledgemcnrx-The support of NATO Scientific AtTairs Division. B-1 110 Brussels. Belgium, Grant No. tOl/g7. and the Mechanical Engineering Department of The University of Toledo arc gratefully acknowledged. REFERENCES
Abcndschcin, W. and Hyatt, G. W. (1970) Ultrasonics and selected physical properties of bone. Cfin. Orthap. 69. 294-301. Ashman, R. B.. Cowin. S. C.. Van Buskirk. W. C. and Ria. J. C. (1984) A continuous wave technique for the mcasurcmcnt of the elastic properties of cortical bone. J. Biomt&xnics 17. 349-361. Bargrcn. J. H.. Andrew. C.. Bassett. L. and Gjclsvik. A. (1974) Mechanical properties of hydrated cortical bone. 1. Bionuckonics 7.239-245. Bonfield. W. and Li. C. H. (1966) Deformation and fracture of bone. J. appl. Phys. 37.869-875. Dally, J. W: and Riley. W. F. (1978) Experimental Stress Analysis (2nd Edn). McGraw-Hill, New York. Goldsmith, W. and Katsamanis, F. (1979) Fracture of no:chcd and perforated polymeric bars produced by longitudinal impact. Inr. 1. M&h. Set. 21.85-108. Katsamanis. F. and Goldsmith. W. (1982) Transverse impact on fluid-fiikd cylindrical tub&. I..appl.*Nech. 49,lS~l56. Ko, R. (1953) The tension test upon the compact substana of the long bones of human extremities. 1. Kyoro PreJ Med. Univ. 53, 50>525. Kolsky. H. (1956) The propagation of stress pulses in viscocl&ii solids. &frif. Mae. II, 693-710. Kolskv. H. (I%31 Stress Waves in Solids. Dover, New York. Lang, b. 8. ‘(1976)~Ultrasonic method for measuring elastic cocfficimts of bone and results on fresh and dried bovine bones /EEE Tronr. Biomed. Engng BME 17. IOI-10s. Lappi, V. G.. King, M. S. and May. I. L (1979) Determination of elastic constants for human femurs. 1. biomech. Engng 101. 193-197. Lewis. J. Land Goldsmith, W. (1973) A biaxial split Hopkinson bar for simultaneous torsion and compression. Rev. Sci. fntrrum. 44, 81 l-813. Lewis. J. L. and Goldsmith, W. (1975) The dynamic fracture and prcfracturc response of compact bone by split Hopkinson bar methods. J. Biomechanics 8. 27-40.
II&1
Technical Note
McElhaney. J. H. (1966) Dynamic response of bone and muscle tissue. 1. appl. Physiol. 21, 1231-1236. McElhaney. J. H.. Fogle. J.. Byars. E. and Weaver. G. (1964) Etiects of embalming on the mechanical properties of beef bone. 1. appl. Physiol. 19, 1234-1236. Pelkn. R. R. and Saha. S. (1983) Stress wave propagation in bone. J. Biornechanics 16.481489. Raftopoulos. D. D.. Karapanos, D. and Tbeocharis, P. S. (1976) A static and dynamic mechanical and optical behavior of hiah wlvmers. 1. Phvs. D. ad. Phvs. 9.869-877. Reilly. D. T. %.I &stein. A.*H. (1675)I& elastic and ultimate properties of compact bone tissue. J. Biomechanits 8, 393-405. Reilly, D. T., Burstein. A. H. and Frankel, V. H. (1974) The elastic modulus for bone. J. Biomechanics 7, 271-275. Sammarco. G. J.. Burstein. A. H.. Davis W. L. and Frankel. V. H. (1971) The biomechanics of torsional fractures: the
effect of loading on ultimate properties. 1. Biomechanics 4. 113-117. Sedlin. E. D. (1965) A rheological model for cortical bone: a study of the physical properties of human femoral samples. Acta &top. xand. Suppl 83. pp. l-77. Muaksgaard Copcnhapn. Simkin. A. and Robin, G. (1973) The mechanical testing of bone in bending. 1. Biomdanics 6.31-39. Tennyson, R. C.. Ewert. R. and Niranjan, V. (1972) Dynamic viscoelastic response of bone. &xnl Mech. 12.502-507. Van Buskirk. W.‘C.. Cowin. S. C. and Ward. R. N. (1981) Ultrasonic measurement of orthotropic elastic constants of bovine femoral bone. J. biomech. Engng 103, 67-72. Yoon. H. S. and Katz J. L. (1976) Ultrasonic wave propaption in human cortical bone--II. Measurements of elastic properties and microhardness. 1. Biomechanics9,459464.
