- Email: [email protected]
Young's modulus of compact bone
YOUNG’S
MODULUS
OF COMPACT
BONE*
W. BONFIELDand P. K. DATTA Department of Materials, Queen Mary College, London EI 4NS Abstract-The Young’s modulus of longitudinal bovine tibia compact bone specimens (both ‘wet’ and ‘dry’) has been determined with two different microstrain measuring techniques.
EXPER1ME~T.U
INTRODtiCFION
PROCEDURE
Tensile ‘specimens with a gauge length of 2.75 cm of the elastic deformation of compact bone with its collagen and hydroxapatite contents has and rectangular cross section of 045 x 0.20cm were been attempted by several workers (Currey. 1964; Bon- prepared from 2 to 3 yr old bovine tibia sections. The bone was slit parallel to its long axis and the specimen field and Li, 1968; Currey, 1969a; Katz, 1971; Bonlield and Clark. 1973). A major factor in this analysis is the shoulders and gauge length obtained by milling of the slit edges. At all stages in this procedure. the bone was particular value taken for the Young’s modulus, but immersed in or kept moist with Ringers solution. The there exists a significant variation in the absolute test specimen was mounted in an Instron testing values obtained experimentally in different investigations. The highest value obtained in ‘static’ tensile tests machine by flexible pin and ball joint couplings, which (i.e. at a ‘low’ strain rate (- 10-4sec-1) relative to allowed accurate alignment. Prior to testing, the specimen was equilibrated at room temperature. It was then ultrasonic measurements) was 265 GNm- ’ measured deformed at a constant strain rate of 3 x lo-‘set- I. (Bonfield and Li, 1966), on longitudinal bovine tibia sections. with a high strain sensitivity (2 x 10m6) The elastic modulus, the microscopic yield stress (the stress to produce a non elastic strain of 2 x IO-“) and Tuckerman optical gauge, a result which is in contrast to the values, ranging from 7-19 GNm-’ determined in the stress-strain relationship were determined, either with a capacitance gauge which allowed continuous several other studies (McElhaney. 1966; Dempster strain measurements on an X-Y recorder or with a and Liddicoat. 1952; Currey, 1969b). Recently, some Tuckerman optical gauge, using a load-unload techsupport for the ‘higher’ modulus value was established nique. in resistance strain gauge measurements (Bonfield and Clark. 1973).on I2 month rabbit longitudinal tibia secRESULTS AND DISCUSIOS tions deformed in tension at a strain rate of I x IO-“ set- I, which gave an average value of 27.6 GNm-‘. Figure 1 shows typical stress-total strain curves However. it was considered that for further confirmadetermined by the two microstrain techniques for a tion of the ‘higher’ modulus value, a redetermination of the elastic modulus of bovine tibia sections, with 60microstrain techniques, but under more rigorous exContinuous mcosuremerrt x Lead-unload measurement perimental conditions, would be of value. This report SOpresents the results of a series of experiments which “E g 40had the following objectives: (1) To continuously measure the total strain, protim duced in a constant strain rate test ofa bovine longitudiE IA 2Qnal tibia section. with a capacitance strain gauge (Bonfield et al., 1973) (of sensitivity 1.2 x 10w6) and hence IO determine the Young’s modulus. , zoo0 IS00 ICC0 MO 0 (2) To compare the continuous capacitance gauge Total strain xIOe6 strain measurement with the discontinuous. load-unload. Tuckerman optical gauge strain measurement. Fig. 1. Stress-total strain curves of a bovine tibia longitudi(3) To evaluate quantitatively the effect of drying on nal specimen deformed in tension at a strain rate of 3 x 10e4 set- ‘, measured with a capacitance gauge (-) the Young’s modulus. and a Tuckerman gauge ( x ). The elastic region is extended (----) to demonstrate the deviation from linear behaviour. * Rrceirrd 31 Jtrl~’ 1973. X correlation
1
147
,
W. BOWIELD and
148
specimen tested immediately after preparation (designated as ‘wet’). It can be seen that the results obtained by the two methods are in good agreement (generally within 5 per cent). The Young’s modulus was measured as the slope of the small linear portion of the stresstotal strain curve. The average modulus value obtained from tests on 9 specimens was 27.3 GNm- ? with a range from 22.5 to 30.0 GNm-’ and hence is in good agreement with the value of 26.5 GNm-’ measured previously (Bonfield and Li. 1966). The microscopic yield stress (the stress to produce a non elastic strain of 2 x LO-‘) varied from -5-7 MNm-‘. and above this stress level. the amount of non elastic strain progressively increased with an increase in the tensile stress (as shown by the deviation from the extended elastic slope in Fig. 1). It was found that the non elastic strain produced by stresses up to 70 IMNm- ’ was completely recoverable with time after unloading to zero stress (i.e. the non elastic strain was anelastic). as demonstrated in Fig. 3. Consequently. coincident stress-total strain curves. above the microscopic yield stress, for the continuous and load-unload technique were only obtained if the specimen was allowed to recover exactly to its original length between each load-unload cycle (a process which required 5-g min at zero stress). It should be noted that a less sensitive strain measuring technique would not detect the initial linear stresstotal strain region (with a maximum strain of 230 x 10e6) and, for comparison purposes. a secant ‘modulus’ to 2 x 10e3 total strain in Fig. 1 gives a value of 22.5 GNm-? (i.e. 20 per cent lower than the modulus derived from the initial slope). Some ‘wet’ specimens were allowed to dry in air at room temperature for periods of 2 weeks and 2 months. Testing these ‘dry’ specimens, under identical conditions to the ‘wet’ specimens, revealed no significant variation in the Young’s modulus between the
Wet bane o 70
t
Dry bcne A After X After
2 weeks iq air. 2 months m aw
P. K. DAITA
I First test A Second test o Third test
II 0
I!
I
2
3
I
I
4
5 Time,
11 6
7
11 8
9
I to
!
II
min
Fig. 3. The recovery of nonelastic strain for a bovine tibia longitudinal specimen with time at zero stress, after unloading from a stress of 45 MN/m’, measured in three successive tests. specimens. However, the amount of non elastic strain produced by a given stress decreased with the time of drying, as shown in Fig. 2, although the non elastic strain remained completely anelastic. Hence a secant ‘modulus’ to 2 x 10e3 total strain is slightly increased. It should be noted that the stress-non elastic strain curve measured originally (Bonfield and Li, 1966), agrees closely with the curve for ‘wet’ bone in the present results. It is concluded that the linear elastic deformation of bovine tibia compact sections is limited to a maximum strain of 230 x 10s6. The average Young’s modulus measured in this region by a continuous measuring microstrain technique is 27.3 GNm- ‘, which is in good agreement with the ‘higher’ value (26.5 GNm-‘) reported (Bonfield and Li. 1966) previously for the load-unload technique. A ‘modulus’ measured for a larger total strain (as for example with a measurement technique of lower strain sensitivity) contains a non elastic strain contribution and gives a significantly lower value. It is suggested that this distinction is of importance in considering the results of in vioo measurements of strain in bone (Lanyon. 1973). which give values of s 200 x IO-” and are hence comparable to the strain limit of elastic behaviour in the present tests. Acknowkfgemenrs-This research forms part of a programme on the deformation and fracture of bone supported by the Science Research Council.
REFERESCES
1
0
1
/
10
20
I
;o
Nonelastic
40
I
50 strain
x
I
I
60
70
Bonfield
9;
10e6
Fig. 1. The effect of drying on the stress-nonelastic curve of bovine tibia longitudinal specimens.
strain
W. and
Li. C. H. (1966)Deformation and fracture
of bone. J. appl. Phys. 37, 869-875. Bonfield. W. and Li. C. H. (1968) The temperature dependence of the deformation of bone. J. Biomechanics1. 323329
Young’s modulus of compact bone Bonfield W. and Clark. E. A. (1973) EIastic deformation of compact bone. J. Mater. Sci. 8, 1590-1594. Bonfield. W.. Datta. P. K.. Edwards. B. C. and Plane. D. C. (1973) A capacitance gauge for microstrain measurement in tension. J. Mnrer. Sci. 8, 1832-1834. Currey. J. D. (1964) Three analogies to explain the mechanical properties of bone. Biorheolog_v2, l-10. Currey, J. D. (1969a) The relationship between the stiffness and the mineral content of bone. J. Biomechnnics 2, 477480. Currey, J. D. (1969b) The mechanical consequences of variation in the mineral content of bone. J. Eiomechanics 2. I-
119
Dempster. W. T. and Liddicoat. R. T. (19521Compact bone as a non-isotropic material. &I. J. Am. 91, 33 1-342. Katz. J. L. (1971) Hard tissue as a composite material-t Bounds on the elastic behaviour. J. Biomtichuntcs -1, 455173. Lanyon. L. H. (1973) Analysis of surface bone strain in the calcaneus of sheep during normal locomotion. J. Biomrchanics 6. 41-49. McElhane). J. H. (1966) Dynamic rrsponje ol’ bons and muscle tissue. J. nppl. Ph,vsioL 21, 1231-1336.
MODULUS
OF COMPACT
BONE*
W. BONFIELDand P. K. DATTA Department of Materials, Queen Mary College, London EI 4NS Abstract-The Young’s modulus of longitudinal bovine tibia compact bone specimens (both ‘wet’ and ‘dry’) has been determined with two different microstrain measuring techniques.
EXPER1ME~T.U
INTRODtiCFION
PROCEDURE
Tensile ‘specimens with a gauge length of 2.75 cm of the elastic deformation of compact bone with its collagen and hydroxapatite contents has and rectangular cross section of 045 x 0.20cm were been attempted by several workers (Currey. 1964; Bon- prepared from 2 to 3 yr old bovine tibia sections. The bone was slit parallel to its long axis and the specimen field and Li, 1968; Currey, 1969a; Katz, 1971; Bonlield and Clark. 1973). A major factor in this analysis is the shoulders and gauge length obtained by milling of the slit edges. At all stages in this procedure. the bone was particular value taken for the Young’s modulus, but immersed in or kept moist with Ringers solution. The there exists a significant variation in the absolute test specimen was mounted in an Instron testing values obtained experimentally in different investigations. The highest value obtained in ‘static’ tensile tests machine by flexible pin and ball joint couplings, which (i.e. at a ‘low’ strain rate (- 10-4sec-1) relative to allowed accurate alignment. Prior to testing, the specimen was equilibrated at room temperature. It was then ultrasonic measurements) was 265 GNm- ’ measured deformed at a constant strain rate of 3 x lo-‘set- I. (Bonfield and Li, 1966), on longitudinal bovine tibia sections. with a high strain sensitivity (2 x 10m6) The elastic modulus, the microscopic yield stress (the stress to produce a non elastic strain of 2 x IO-“) and Tuckerman optical gauge, a result which is in contrast to the values, ranging from 7-19 GNm-’ determined in the stress-strain relationship were determined, either with a capacitance gauge which allowed continuous several other studies (McElhaney. 1966; Dempster strain measurements on an X-Y recorder or with a and Liddicoat. 1952; Currey, 1969b). Recently, some Tuckerman optical gauge, using a load-unload techsupport for the ‘higher’ modulus value was established nique. in resistance strain gauge measurements (Bonfield and Clark. 1973).on I2 month rabbit longitudinal tibia secRESULTS AND DISCUSIOS tions deformed in tension at a strain rate of I x IO-“ set- I, which gave an average value of 27.6 GNm-‘. Figure 1 shows typical stress-total strain curves However. it was considered that for further confirmadetermined by the two microstrain techniques for a tion of the ‘higher’ modulus value, a redetermination of the elastic modulus of bovine tibia sections, with 60microstrain techniques, but under more rigorous exContinuous mcosuremerrt x Lead-unload measurement perimental conditions, would be of value. This report SOpresents the results of a series of experiments which “E g 40had the following objectives: (1) To continuously measure the total strain, protim duced in a constant strain rate test ofa bovine longitudiE IA 2Qnal tibia section. with a capacitance strain gauge (Bonfield et al., 1973) (of sensitivity 1.2 x 10w6) and hence IO determine the Young’s modulus. , zoo0 IS00 ICC0 MO 0 (2) To compare the continuous capacitance gauge Total strain xIOe6 strain measurement with the discontinuous. load-unload. Tuckerman optical gauge strain measurement. Fig. 1. Stress-total strain curves of a bovine tibia longitudi(3) To evaluate quantitatively the effect of drying on nal specimen deformed in tension at a strain rate of 3 x 10e4 set- ‘, measured with a capacitance gauge (-) the Young’s modulus. and a Tuckerman gauge ( x ). The elastic region is extended (----) to demonstrate the deviation from linear behaviour. * Rrceirrd 31 Jtrl~’ 1973. X correlation
1
147
,
W. BOWIELD and
148
specimen tested immediately after preparation (designated as ‘wet’). It can be seen that the results obtained by the two methods are in good agreement (generally within 5 per cent). The Young’s modulus was measured as the slope of the small linear portion of the stresstotal strain curve. The average modulus value obtained from tests on 9 specimens was 27.3 GNm- ? with a range from 22.5 to 30.0 GNm-’ and hence is in good agreement with the value of 26.5 GNm-’ measured previously (Bonfield and Li. 1966). The microscopic yield stress (the stress to produce a non elastic strain of 2 x LO-‘) varied from -5-7 MNm-‘. and above this stress level. the amount of non elastic strain progressively increased with an increase in the tensile stress (as shown by the deviation from the extended elastic slope in Fig. 1). It was found that the non elastic strain produced by stresses up to 70 IMNm- ’ was completely recoverable with time after unloading to zero stress (i.e. the non elastic strain was anelastic). as demonstrated in Fig. 3. Consequently. coincident stress-total strain curves. above the microscopic yield stress, for the continuous and load-unload technique were only obtained if the specimen was allowed to recover exactly to its original length between each load-unload cycle (a process which required 5-g min at zero stress). It should be noted that a less sensitive strain measuring technique would not detect the initial linear stresstotal strain region (with a maximum strain of 230 x 10e6) and, for comparison purposes. a secant ‘modulus’ to 2 x 10e3 total strain in Fig. 1 gives a value of 22.5 GNm-? (i.e. 20 per cent lower than the modulus derived from the initial slope). Some ‘wet’ specimens were allowed to dry in air at room temperature for periods of 2 weeks and 2 months. Testing these ‘dry’ specimens, under identical conditions to the ‘wet’ specimens, revealed no significant variation in the Young’s modulus between the
Wet bane o 70
t
Dry bcne A After X After
2 weeks iq air. 2 months m aw
P. K. DAITA
I First test A Second test o Third test
II 0
I!
I
2
3
I
I
4
5 Time,
11 6
7
11 8
9
I to
!
II
min
Fig. 3. The recovery of nonelastic strain for a bovine tibia longitudinal specimen with time at zero stress, after unloading from a stress of 45 MN/m’, measured in three successive tests. specimens. However, the amount of non elastic strain produced by a given stress decreased with the time of drying, as shown in Fig. 2, although the non elastic strain remained completely anelastic. Hence a secant ‘modulus’ to 2 x 10e3 total strain is slightly increased. It should be noted that the stress-non elastic strain curve measured originally (Bonfield and Li, 1966), agrees closely with the curve for ‘wet’ bone in the present results. It is concluded that the linear elastic deformation of bovine tibia compact sections is limited to a maximum strain of 230 x 10s6. The average Young’s modulus measured in this region by a continuous measuring microstrain technique is 27.3 GNm- ‘, which is in good agreement with the ‘higher’ value (26.5 GNm-‘) reported (Bonfield and Li. 1966) previously for the load-unload technique. A ‘modulus’ measured for a larger total strain (as for example with a measurement technique of lower strain sensitivity) contains a non elastic strain contribution and gives a significantly lower value. It is suggested that this distinction is of importance in considering the results of in vioo measurements of strain in bone (Lanyon. 1973). which give values of s 200 x IO-” and are hence comparable to the strain limit of elastic behaviour in the present tests. Acknowkfgemenrs-This research forms part of a programme on the deformation and fracture of bone supported by the Science Research Council.
REFERESCES
1
0
1
/
10
20
I
;o
Nonelastic
40
I
50 strain
x
I
I
60
70
Bonfield
9;
10e6
Fig. 1. The effect of drying on the stress-nonelastic curve of bovine tibia longitudinal specimens.
strain
W. and
Li. C. H. (1966)Deformation and fracture
of bone. J. appl. Phys. 37, 869-875. Bonfield. W. and Li. C. H. (1968) The temperature dependence of the deformation of bone. J. Biomechanics1. 323329
Young’s modulus of compact bone Bonfield W. and Clark. E. A. (1973) EIastic deformation of compact bone. J. Mater. Sci. 8, 1590-1594. Bonfield. W.. Datta. P. K.. Edwards. B. C. and Plane. D. C. (1973) A capacitance gauge for microstrain measurement in tension. J. Mnrer. Sci. 8, 1832-1834. Currey. J. D. (1964) Three analogies to explain the mechanical properties of bone. Biorheolog_v2, l-10. Currey, J. D. (1969a) The relationship between the stiffness and the mineral content of bone. J. Biomechnnics 2, 477480. Currey, J. D. (1969b) The mechanical consequences of variation in the mineral content of bone. J. Eiomechanics 2. I-
119
Dempster. W. T. and Liddicoat. R. T. (19521Compact bone as a non-isotropic material. &I. J. Am. 91, 33 1-342. Katz. J. L. (1971) Hard tissue as a composite material-t Bounds on the elastic behaviour. J. Biomtichuntcs -1, 455173. Lanyon. L. H. (1973) Analysis of surface bone strain in the calcaneus of sheep during normal locomotion. J. Biomrchanics 6. 41-49. McElhane). J. H. (1966) Dynamic rrsponje ol’ bons and muscle tissue. J. nppl. Ph,vsioL 21, 1231-1336.