Determination of the transverse elastic coefficients of bone

J. t?iomechonrs.

1977, Vol

10. pp. 64-9.

Pergamon

Press.

Printed m Great Br~tam

DETERMINATION OF THE TRANSVERSE COEFFICIENTS OF BONE*

ELASTIC

NECIP BERME Bioengineering Unit, University of Strathclyde, Glasgow. Scotland

and YALCIN MENGI and ERK INGER Middle East Technical University, Ankara, Turkey

Abstract-A method for determining the elastic coepicients of bone as a transversely isotropic material is described. The constitutive equations relating the stresses to strains were solved simultaneously for tension and hydrostatic compression tests. Cylindrical specimens with symmetry axes oriented with the long axis of the fresh bovine Haversian femur were tested in tension, torsion and hydrostatic compression, and the results were used to calculate the five independent elastic coefficients.

INTRODUCTION The advances in the materials

technology and the relative success of the total hip and knee replacement joints have encouraged many orthopaedic surgeons and designers to undertake design and development of various other replacement joints. However, the long term success rate of some of the designs has not been very high, and failure of an implant within a few years after surgery has not been uncommon (Souter, 1973). A knowledge of the loads transmitted by the implants is necessary if design improvements are to be achieved. A critical part of the design information is the loading at the implant/bone interface. An understanding of the interface loading is not only necessary to design for strength, but also required if resorption of bone due to either excessive loading or lack of loading is to be avoided (Freidenberg and French, 1952; Miller et al., 1976). The magnitudes of both the normal and shear stresses at the interface critically depend on the mechanical properties of all the structures involved (Svensson, 1976). The implanted materials are isotropic, as most engineering materials are, and two elastic coefficients are sufficient to describe their behaviour for small deformations. However, bone is not only non-isotropic, but also non-homogenous; its properties vary depending on location. Evans (1957) summarised the literature on the properties of bone in his well known work. Later, Currey (1970), Yamada and Evans (1970), Evans (1973) and others gave accounts of further studies. Since then, there have been several active groups investigating bone properties. Many of the early investigators who studied the cortical long bones were concerned only with its properties in the direction of its long axis. Dempster and Liddicoat (1952) were among the first who investigated the ani* Received 25 August 1976. 643

sotropic behaviour of bone. Lang (1969) and later, Yoon and Katz (1973) treated animal and human bone respectively as transversely isotropic materials. and determined their elastic moduli at ultrasonic frequencies. The longitudinal and transverse behaviour of bone at lower frequencies has been determined by testing separate specimens oriented in these directions. Although the longitudinal specimens can be readily obtained the availability of the transverse specimens of desirable proportions is limited by the size and shape of the intact bone. For small deformations bone behaves as a linear material, and its compression and tension moduli are equal (Reilly et al., 1974). Therefore a simple method can be used to determine the modulus of elasticity of bone in the transverse direction by testing longitudinal specimens only. In this paper cortical bone is modelled as a transversely isotropic material, and a method of evaluating its linear elastic constants is outlined. The experiments involve measuring the axial and transverse strains when a specimen is subjected to hydrostatic compression in addition to the standard tension tests.

ANALYSIS

In this analysis cortical bone is treated as a uniform transversely isotropic material, with its long axis as the symmetry axis. This assumes that in the transverse plane bone is isotropic. As will be discussed later, this assumption in fact can be relaxed and the elastic moduli in three orthogonal directions can be calculated. In both cases it is essential for the long axis of the test specimen to be oriented with a principal direction, otherwise the shear strains cannot be eliminated from the solution for the normal stresseq and the number of equations would not be sufficient to calculate the elastic coefficients.

644

NECIP BERME, YALCINMENGI and ERK INGER

When a transversely isotropic specimen is referred to a cylindrical co-ordinate system (r, 6, z), where the z-axis coincides with the symmetry axis, then the constitutive equations which relate the stresses to strains become (Love, 1944): 7,

c 11

708 7 zs

=

Cl2

Cl3

0

0

0.-

0

0

0

c 12

Cl,

Cl3

Cl3

C 13

c33

0

0

0

C44

0

0 0

7ez

0

0

0

7 l,

0

0

0

0

G4

0

0

0

0

0

! 7rtl

Even if all these strain components were known the simple tension test is not sufficient to solve for the four unknown coefficients of the stiffness matrix. At least two more independent equations are necessary. The most convenient way of obtaining these equations using a longitudinal tensile test specimen appeared to be by subjecting the specimen to a hydrostatic compression. For a pressure p, let the corresponding strain components be given as:

l,,

; E,, = c;,. = c;, f. E&q= E&$

Now, it is possible to find the elastic coefficients from the equation:

CC.,

0

where, rij and fii are the stress and strain components, and Cij are the elastic coefficients. The stiffness matrix has only five independent coefficients, and Ce6 is given as: C 66 = &Cl1 -

G2).

=

(‘11-

c12)(c33cll GlC33

L

=

c33

-

%3 +

Cl,

+

c3Jc12

-

c:3,

-

2c:3)

v*e = *

-

c33cl2

-

CT3

Cl,&3

-

G3

.(

c13tc12 c:,

-

-

cll)

CllC33

.

ZJ

[i].

C 11

Xl =

-

1 - x:

C12 =

X4Cll

Cl3

&-

=

(9)

=

x3

+

G3 -.

C 11

Here it should be noted that although the number of equations corresponding to uniaxial tension and hydrostatic compression tests is six, equation (Q due to elimination of unknown radial strains, constitutes only four equations. The shear modulus C44 can simply be obtained by subjecting the specimen to torsion along the long axis of the specimen. For a shear strain cBr corresponding to a shear stress of 7oz the shear modulus is given as: =

7 0s~

or.

(10)

EXPERIMENTS

(6)

In uniaxial tension (r5,, = u) let the strains be: G, = l:, ; $#J= E;@;E,,= r;,.

1

(4)

In the Poisson’s ratio expressions the first and second indices indicate the directions of the applied uniaxial stress and orthogonal strain respectively. If the specimen is subjected to normal stresses only, then, equation (1) reduces to:

ii;

1+e$

where,

c44

v,z =

[;]=!I

-p

c33

The Poisson’s ratios relate the strain in the direction of a uniaxial stress to a corresponding orthogonal strain value. Here it should be noted that v,, would be equal to v,@ since in the transverse plane the material is assumed to be isotropic. The three Poisson’s ratios take the form:

v

-P

4

C12’

.

u

(2)

It is not uncommon to describe the stress strain relationships in terms of two Young’s moduli, a shear modulus and three Poisson’s ratios. The Young’s moduli E,, and E,, are the ratios of the stress to strain in the radial ,and axial directions respectively. In terms of the elastic coefficients, they are expressed as: E,

(7)

(6)

The tests were performed to assess the applicability of the method and thus the number of specimens tested were limited; and fresh bovine femur, a material that has been investigated by other researchers, was used. Two rectangular segments were tirst extracted from the bone by hand sawing. Then, the specimens were turned on a lathe to a final uniform diameter of 6 mm. During turning a high cutting speed (157 rad/sec) with small feeds were employed and water was used as coolant. The finished length of the specimens was 50 mm.

635

Fig. 1. The pressure chamber and a specimen. The two sections of the chamber are connected contain the specimen. and the assembly is attached to the dead-weight tester.

to

647

Transverse elastic coefficients of bone Three electric resistance strain gauges with epoxy backings and pre-attached lead wires having gauge lengths of 0.79 mm were bonded around the periphery of each specimen at its mid-length. First the surfaces were wiped dry and then the areas where gauges were

2 t 0

P 4 5$

%s

\

to be applied were degreased using ether. The gauges were positioned making angles of 0,45 and 90 degrees with the long axes of the specimens and held in place using cellophane tape. One end of each piece of tape

was lifted, lifting the gauge assembly and Ethicon Bucrylate Cement (Isobutyl 2-Cyano Acrylate Monomer) was applied onto the bone surface. The tapes were repositioned and pressure was applied on the gauges pressing with the fingers for two minutes. T’he tapes were removed and three half bridge circuits were formed each consisting of a gauge of one specimen and the corresponding gauge of the other. This assured proper temperature compensation for the gauges. Finally, the gauges and the lead wires were coated with a water resistant gauge wat Each specimen was subjected to three different tests: tension, hydrostatic compression and torsion. During the tension and hydrostatic compression tests the strain reading of the axial and transverse strain gauges were monitored, while during the torsion test the output of the gauge oriented at 45 degrees with the long axis of the specimen being recorded. Strains were kept within the elastic range, therefore not only was it possible to employ each specimen for all three tests but also to check repeatability by performing the same series of tests for a second time. In all three tests a load-unload technique was used. Dead weights were employed to apply the loads. Immediately after the application of a load it was released and a higher load was applied when the strain values returned to their initial zero readings. Special close fitting collets were manufactured to apply the tensile loads. The clamps also incorporated universal joints to facilitate the concentric application of the forces. The system was suspended from one end and the load applied to the other. The maximum tension applied in all cases was 44.5 N, resulting in an axial stress of 1.58 MPa. A small cylindrical chamber was built for the hydrostatic pressure tests (Fig. 1). The chamber was

1’

6 % Specimen

,/

I-

C,, GW Ct2 GPa) Cl3 @PaI CH @Pa) CM GPa) G6 @Pa) E,, (GPa) E,, (GPa) “z B= “rr “,0 1’C

92

p/’

I

r z

P

1’

P v

I -40

1’ I 40

0

Axial and circumferential

strains.

I

80 w

Fig. 2. Axial and circumferential strains vs tensile stress.

attached to a dead-weight tester through which pressures up to 3.45 MPa, were applied. The lead wires of the gauges were taken out of the chamber through a small opening which was subsequently sealed with epoxy resin. Normal saline was used as the compression medium. Pressures of such small magnitudes do not affect the strain gauge performance. therefore the readings obtained did not require modification (Kular, 1972). A Carl Schenk (model PWO) testing machine was used to perform the torsion tests. The maximum shear stress due to the applied torque was limited to 18 MPa. RESULTS

AND DISCUSSION

Figures 2 and 3 show the variation of the axial and circumferential strains in the tension and hydrostatic tests respectively. Within the range the stressstrain relationships were linear and repeatable. Regression lines were calculated for each specimen applying a least squares fit. For both specimens the scatter of the data was small, the maximum deviation

Table 1. Elastic coefficients and technical constants Elastic coefficients & technical constants

/ 2I

Specimen 1

Specimen 2

19.0 10.6 11.5 30.1 5.10 4.2 21.2 12.0 0.388 0.428 0.218

20.2 9.4 11.4 30.1 5.98 5.4 21.3 14.2 0.387 0.317 0.259

1 + 2 combined 19.5 10.0 11.5 30.1 5.48 4.15 21.2 13.2 0.388 0.374 0.239

NECIP BERME,YALCM MENGI and ERK INGER

648

4 *ZZ

488

t

i

\ ?

Specimen

I

$ Specimen

2

6

\

\ -120

1

‘4,

-

t

0

40

-80

Axiol cinumkrmliol

-’

slroins,

pc

Fig. 3. Axial and circumferential strains vs hydrostatic pressure.

for any measured strain from its regression line being 3~. The slopes of the regression lines were used to calculate the elastic coefficients, and the technical constants. The results obtained are given in Table 1. The elastic constants were also evaluated by combining the raw data corresponding to both specimens and listed in the last column of the Table. The longitudinal modulus of elasticity E,,, and the Poisson’s ratio v,, values of the two specimens were close to each other, and they agreed with those calculated from the tension tests only. The other elastic constants, which depend on both tests, showed a much greater variation. Such a scatter would be expected in tests of this nature due to the second and higher order terms involved in the calculation of the results. The two other transversely isotropic models of bone relevant to this study are due to Lang (1969), and Reilly and Burstein (1975). The former gives results for bovine phalanx obtained for ultrasonic frequencies, whereas the latter has tested specimens oriented in various directions from bovine Haversian femoral compact bone and gives results for both tension and compression tests. They are listed in Table The elastic moduli obtained both in the longitudinal and radial directions compare favourably with the results given in Table 2. Although the scatter in the

values of the Poisson’s rations determined by the different techniques appear to be large the results are approximately within one standard deviation of those given by Reilly and Burstein (1975). The shear modulus Cb4 was determined independent of the remaining four elastic mod&i and is 5.48 MPa for the average of the two specimens. This value agrees with the one given by Lang (1969). The small compression chamber used for the hydrostatic tests restricted the strain measurement method to the use of electric resistance strain gauges. To cause minimum interference with the material behaviour the spreading of the adhesive beyond the gauge area was avoided. Similarly, protective coating for the gauges was restricted to their surfaces only. The very close agreement of the longitudinal modulus of elasticity calculated here with the values cited above is an indication of the acceptability of the use of electric resistance strain gauges. However, the experimental technique described is not limited by this strain measurement method. Other means of measurement can also be utilised provided that a suitable hydrostatic compression chamber is designed. To the authors’ knowledge, there is no information in the literature on the behaviour of bone in hydrostatic compression. The compression medium used did not appear to have an adverse effect on the repeatability of the tests. However, the effect of still higher pressures and different media is not yet known. Here, bone is treated as a transversely isotropic material: its properties in the radial and circumferential directions are assumed to be the same. No effort was made to identify these directions, and the circumferential gauges were arbitrarily sited. However, identical strain readings were obtained for the two specimens. A study of the two orthogonal elastic moduli in the transverse plane could be carried out using specimens with square cross sections where the faces are aligned with radial and tangential directions of the intact bone. The transverse strains on the two adjacent faces and the longitudinal strain measured during the tensile and the hydrostatic tests produce enough information to calculate the three orthogonal elastic moduli. This assumes that the symmetry planes of any orthotropic behaviour are normal to the radial and circumferential directions. In conclusion, the method described is a useful technique for testing bone. It reduces the number of specimens required to obtain directional variations in

Table 2. Elastic properties: mean values (SD.) Technical constants E, @Pa) E,, @PaI CA, @Pa) v*, V,, v,z

Lang (1969)

Reilly & Burstein (1975) Compression Tension

11.3 22.0 5.4 0.487 0.397 0.204

10.4(1.64) 10.1(l.78) 23.1(3.18) 22.3(4.61) 3.57 (0.25) 0.29 (0.079) 0.40 (0.209) 0.Sl(0.236) 0.51 (0.115) -

649

Transverse elastic coefficients of bone bone properties. of the transverse

and also permits the determination prqperties of smaller long bones.

REFERENCES

Currey. J. D. (1970) The mechanical properties of bone. C/in. Orthop. 73. 210-231. Dempster. W. T. and Liddicoat, R. T. (1952) Compact bone as a non-isotropic material. Am. J. Anat. 91. 331-362. Evans, F. G. (1957) Srress and Strain in Bone. Thomas, Springfield, IL. Evans. F. G. (1973) Mechanical Properties of Bone. Thomas, Springfield, IL. Freidenbern. Z. B. and French. G. (1952) The effect of known c&npressive forces on fracture healing. Surg. Gyn. Ohstet. 94, 743-748. Kular, G. S. (1972) Use of foil strain gauge at high hydrostatic pressure. Exp. Mech. 12(7), 311-316. Lang, S. B. (1969) Elastic coefficients of animal bone. Science 165, 287-288. Love. A. E. H. (1944) A Treatise on the Mathematical Theor)) of Efasticify. Dover, New York, pp. 159-161. Miller. J., Burke, D. L.. Stachiewicz, J. W. and Kelebay, L. C. (1976) The fixation of major load-bearing metal prostheses to bone. An experimental study comparing smooth to porous surfaces in a weight bearing mode. Digest of Papers-11th Int. Conf. on Medical and Biological Engineering, Ottowa, Canada pp 528-529.

Reilly. D. T.. Burstein. A. H. and Frankel, V. H. (1974) The elastic modulus of bone. J. Biomechanics 7, 271.-275. Reilly. D. T. and Burstein. A. H. (1975) The elastic and ultimate properties of compact bone. J. Biomechanics 8. 393-405. Souter, W. A. (1973) Arthroplasty of the elbow. Orrhop. Clin. N. Am. 4(2), 395413. Svensson. N. L., Valliappan. S. and Wood. R. D. (1976) Stress analysis of human femur with implanted Charnley prosthesis. Report No. 1976/AMil. University of New South Wales, Australia. Yamada H. (1970) Strength of Biological Materials. (Edited by Evans, F. G.) Williams & Wilkins, Baltimore, pp. 19-79. Yoon, S. H. and Katz, J. L. (1973) The elastic properties and microstructure of human cortical bone: experimental. Presented at the 51st General Session of the International Association for Dental Research. Washington. DC. NOMENCLATURE cij

E,, Ezz P

r. 0. 2 xi Eij,

“ij

a Tij

r;,

Gi

elastic coefficient moduli of elasticity pressure cylindrical co-ordinates a variable strain Poisson’s ratio tensile stress stress.