FRACTURE

TOUGHNESS

OF COMPACT

BONE”

W. BONFIELD and P. K. DATTA~ Department

of Materials

Queen Mary College.

London,

England

Abstract-The tensile fracture stress (urTr) of longitudinal bovine tibia compact bone specimens was measured as a function of the length (c) and radius of curvature (r) of machined edge cracks. It was established that, for a given value of r, then

where A and I3 are constants. A fracture mechanics method was utilised to derive values of the fracture toughness (as defined by the critical stress intensity factor, Klc), the specific surface energy (7) and the “intrinsic” flaw size (ci,). The advantages and limitations of this approach are discussed.

INTRODUCTION

where

fracture characteristics of compact bone specimens with an “unnotched” surface (i.e. without artificially introduced surface notches or cracks) have been comprehensively investigated (e.g. Evans, 1973) but relatively little attention has been given to the fracture of precracked or “notched” bone specimens. It was demonstrated by Bonfield and Li (1966) that the presence of surface cracks significantly reduced the energy absorbed during fracture of bovine femur and tibia sections under impact conditions, for both longitudinal and transverse orientations. This result was confirmed and extended by Pope and Outwater (1972) in measurements of the fracture energy in bending of precracked specimens taken from a number of different bones. However, these experiments did not permit an evaluation of the fracture toughness of bone as dctined by the critical stress intensity factor for crack propagation t&j. In general for a semi-infinite sheet, containing a centre notch of length 2c, with an uniaxial stress (cr) applied perpendicular to the crack length, we have, following Andrews (1972), that the stress intensity factor (K) is given by:

The

K = d (nc)‘. Hence, the critical stress intensity tion (Klc) is given by:

(1) for crack propaga-

K,c = rJrr (7rc)2.

(2)

where 4, is the uniaxial stress required to fracture a specimen. For a single edge notch of length c in a specimen of width (IV), pin loaded in tension, a correction for the *‘finite width” of the specimen is made in the relations detailed by Brown and Srawley (1966) (for c/W < 0.6). with: K,,. = a,, c+ r;

(3)

* Rewired 4 hay 1975: ret&d 6 June 1975. t Now at Department of Chemistry and Metallurgy, Glasgow College of Technology. Glasgow. Scotland. 131

Y = I.99 - 0,41(c/W) + 18~7o(C/W)2 - 38.48(~/B’)~ + 53.85(c/B’J*. For an elastic solid. we also have (Griffith. that:

K,c = (~EY)~.

(4) 1920)

(5)

where E is the Young’s modulus and y is the specific surface energy (i.e. surface energy per unit area) for fracture. The applicability of the above approach to the fracture of compact bone has been investigated in a series of experiments, in which the tensile fracture stress of longitudinal bovine tibia compact bone specimens (CT,,)was measured as a function of crack length (c) and radius of curvature (r). These results demonstrate that, for a constant value of r. a,, is proportional to c-1. and are utilised to calculate values of the fracture toughness (K,,), the specific surface energy (y) and the intrinsic crack length (c,,) responsible for fracture in a “perfect” (un-notched) bone section.

EXPERtMENTAL

PROCEDURE

“Sheet” tensile specimens, with a gauge length of 25.4 mm, a width of 18.0 mm and a thickness of 2.0 mm. were prepared from 2 to 3-yr old bovine tibia sections. The bone was slit parallel to its long axis and the specimen shoulders (of width 21,Omm) and gauge length obtained by milling of the slit edges. A hole, of 5.On-m dia. was drilled at the mid-point of each shoulder. The specimen shoulders were supported on both sides by matching steel sheet (1 mm thick) which was bonded to the shoulders with “quick setting” epoxy resin. In addition, a slit perpendicular to the tensile axis of width 0.38 mm (with a range of lengths from I.25 to 14,Omm in different specimens), was drilled from the midpoint of the specimen edge. The tip of each slit or crack was carefully formed to give a constant radius of curvature of 0.38 mm. A second series of specimens was prepared

W. BONFIELDand P. K. DATTA

132

Table 1. Effect of crack length (c) on the fracture stress (orlr) of longitudinal bovine tibia compact bone specimens with a constant radius of curvature (r) = 0.38 mm Crack length (c) (mm) Fracture stress (q,) (MNm-‘)

1.25 2.5 60.3 40.0

with a constant crack length (6.0mm) and a range of radii of curvature from 0.19 to 1.25 mm. During most of this procedure, the bone was immersed in, or kept moist with, Ringer’s solution. The test specimen was mounted in an Instron testing machine by flexible pin and ball joint couplings, which allowed accurate alignment. Prior to testing, the specimen was equilibrated at room temperature. It was then deformed in tension at a constant rate of 3 x 10m4 set- ’ to fracture and the fracture stress measured. The crack was observed microscopically during tensile straining. A typical test specimen is shown in Fig. 1. The specimen has been deformed to fracture and the fracture path from the crack tip across the specimen can be seen. RESULTS

3.0 30.8

5.5 21.0

60 20.7

10.0 12.5

14.0 5.3

The fracture stress (or,) is also plotted in Fig. 2 as a function of Cf. It can be seen that the results satisfy an equation of the form: crrr= AC-’ - B

(6)

where A and B are constants (= 2.60 MNrnm312 and 135 MNm-*, respectively). With a constant edge crack length (c) of 6 mm, the fracture stress (c+~)was also measured for a series of values of the tip radius of curvature, ranging from @19 to 1.25 mm. The results, shown in Table 2 indicate that the measured fracture stress was approximately independent of the crack tip radius of curvature. The data given above is for individual specimens, but a measure of the experimental scatter for a similar condition is given by a comparison of the three fracture stress values in Tables 1 and 2 for c = 5-5 mm, r = 038 mm (21*0MNm-*) and c = 6.0mm, r = 0.38 mm (20.7 and 25.5 MNm-*) respectively. This data, together with the small range of fracture stress values (24627.3 MNm-2) for the six specimens with c = 6.0mm in Table 2, suggests that the fracture stress range from 60.3 to 5.3 MNm- 2, for a variation in c from 1.25 to 14mm, does represent a significant experimental difference.

At the strain rate of 3 x 10e4 set- ‘, all the notched specimens exhibited a linear stress-strain curve to fracture. In addition microscopic examination of the machined crack during deformation revealed no observable crack extension prior to fracture for any of the specimens tested. With a constant radius of curvature (r) at the crack tip of 0.38 mm, as the edge crack length (c) was increased from 1.25 to 14mm, the fracture stress (urfr) DISCUSSION decreased from 60.3 to 53 MNm-‘, as shown tabulated in Table 1 and plotted in Fig. 2. For compariAs no crack extension was detected prior to fracson, an un-notched specimen, tested under identical ture, the measured fracture stress (a,,) may be taken conditions, had a mean fracture stress of as the stress to propagate the corresponding edge - 120MNm-‘. crack of length (c). The experimental equation (6), relating the fracture stress rrrrrfor compact bone to the edge crack length c, is similar to that generally found for polymeric solids, both with respect to the general form of the equation (Andrews, 1972) and to the failure to pass through the origin (as pointed out recently by Reed and Squires, 1974). The experimentally derived equation (6) between crfrand c-i, may be reasonably extrapolated to the particular value of a,, obtained for an un-notched specimen and a measure of the “intrinsic crack or flaw length”, designated as ce, obtained. For the present experimental situation, a,, for an unnotched specimen was - 120 MNm- * and hence the associated cc, value is - 370 pm (if an edge crack) and - 740 pm (if an internal flaw). The “intrinsic crack” in the present studies may actually be an “artificial crack” introduced by the specimen surface preparation. However this approach could clearly be apFig. 2. The variation of fracture stress (ur,) with edge crack plied to a specimen with a non-machined surface and length (c) (shown as X) and with c-t (shown as o) for provide a valuable insight into the role of lacunar longitudinal bovine tibia compact bone specimens deand vascular spaces in the initiation of fracture. With formed to fracture at a strain rate of 3 x 10m4set-r.

^_-:

__’

.;’

_

Fig. 1. A “sheet” tensile longitudinal bovine tibia compact bone specimen, with an edge ct ack (c 3.0 mm, Y = 0.38 mm) deformed to fracture at a strain rate of 3 -: lo-“ set-‘.

Fracture toughness of compact bone

133

Table 2. Effect of crack radius of curvature (v) on the fracture stress (ur,) of longitudinal bovine tibia compact bone specimens. with a constant crack length

(c) = 6.0 mm Crack radius of curvature (v) (mm) Fracture stress (urrr) (MNm-‘)

0.19 25.4

this prospect. it is intcrcsting to note that the magnitude of cg approaches the scale of the vascular spaces (30 100 ktm) rather than that of the lacunar spaces l-3 ALrn).(Bonfield and Clark. 1973). In addition the value of c,, is significantly less than that [c,, = I.3 mm (edge)] derived by Piekarski (1970) for a “fast propagating” fracture. in bending, of a bovine femur scction. If the Brown and Srawley “finite width” correction factor (13 is applied to the measured fracture values, this factor increases relatively the magnitude of the fracture stress for the larger crack lengths, but only results in a small reduction in the slope ,4 (to 2~20MNm~3’ ) and the intercept B (to 6,0MNm-‘) of equation (6). Extrapolation of the “corrected” slope gives c,, = 340 Itrn (edge) or 680 Arm (central). Equations (3) and (4) can also be utilised to calculate the critical stress intensity factor for crack propagation. K,c. for each particular value of err, and c. As all the stress-strain curves were linear to fracture. and no crack extension was detected prior to fracture. it is reasonable to assume that the measured values of g,r and c relate to crack propagation. This procedure gives values from 2.2 to 4.6 MNm-3 ‘. It should be emphasised that this result for K,, refers to a particular crack within a range of radii of curvature which is normally designated as “blunt”. The apparent independence of fracture strength on radius of curvature found in the present results is not predicted by the Inglis theory (Andrews. 1972). which suggests a relationship of the form: grn = (T(1 + &, C./r)

(7)

where (T, is the maximum stress at the crack tip. oriented perpendicular to the tensile stress. However. although no cracks were observed around the machined crack tip, the presence of sub-microscopic cracks of constant radius of curvature is not precluded. Clearly. it is necessary to extend the experimental range towards the condition of a “sharp” and ultimately. an “atomically sharp” crack tip, at which I’ = a. where LI is the lattice parameter. For an elastic solid. a knowledge of values for K,, and the Young’s modulus E allows a calculation of the specific surface energy 7. from equation (5). However. it has been demonstrated (e.g. Currey. 1964: Bonfield and Li. 1966. 1968) that the deformation of compact bone is not completely elastic to fracture and contains significant anelastic and plastic contributions. But. as the total non-elastic deformation is small relative to the elastic deformation. it is reasonable. for a first approximation, to utilise equation (5) to obtain an estimate of the specific surface energy.

0.38 25.5

0.54 24.8

0.75 21.3

0.94 24.6

1.25 26.5

As discussed recently by Bonfield and Datta (1974), there is not a generally agreed value for the Young’s modulus (E) of longitudinal compact bone sections (at a strain rate comparable to that used in the present tests). However. if we take the upper and lower limits of quoted values of E, i.e. 27.3 and together with 19,0MNm-’ respectively, in K,,- = 4.6 MNm- 3’2. we obtain. by substitution equation (5). that ;’ = 3.9 x 10”~56 x 10’ J rn- ‘_ The presence of non-elastic deformation means that the derived values of 7 represent an over-estimate of the actual specific surface energy (7). A correction for the energy losses associated with non-elastic deformation will be developed in a later publication.

CONCLI’SIONS

(I ) For a given radius of curvature (r), the fracture stress (grr) of longitudinal bovine tibia compact bone sections. containing an edge crack of length (c) is given by: u , r = 2.60 cm i -

13.5 (uncorrected)

OK

or, = 2.20 c i - 6-O (with “finite width” correction). (2) For a constant crack length (c), a variation in the radius of curvature (r) of a “blunt” crack does not affect the fracture stress (ar,). (3) The fracture toughness (K,,). specific surface energy b) and “intrinsic crack length” (c,,) are 3.X4.6 MNm- 3 ?~ 3.9 x 10’~~5~6x lO’Jm_ and __ 340 Ltm. respectively. .4cl;r1ow/rt/y~rt1rnrc The provision of a Research Assistantship for P.K.D. by the Science Research Council and useful discussions with Professor acknowledged.

E. H. Andrews

are gratefully

REFERENCES

Andrews. E. H. (1972) Fr~tur~‘. Po/J,ru~ S&we by Jenkins, A. D.). pp. 579-619. North-Holland.

dam. Bonfield. W. and Li. C. H. (1966) Deformation

(Edited

Amster-

and fracture of bone. J uppl. Phps. 37. 8699875. Bonfield. W. and Li, C. H. (1968) The temperature dependence of the deformation of bone. J. Biomrchanics 1. 3233329. Bonfield. W. and Clark. E. A. (1973) Elastic deformation of compact bone. J. mar. Sci. 8. 159@95. Bonfield. -W. and Datta. P. K. (1974) Young’s modulus of compact bone. J. Bio~rtrchanics 7. 1477149. Brown, W. and Srawley, J. (1966) Plane strain crack testing of high strength metallic materials. Spec. Publ. No. 410. ASI‘M. Philadelphia

134

W. BONFIELD and P. K. DAI-~A

Currey, J. D. (1964) Three analogies to explain the mechanical properties of bone. Biorheology 2, l-10. Evans, F. G. (1973) Mechanical Properties of Bone Thomas, Springfield, IL. Griffith, A. (1920) The phenomena of rupture and flow in solids. Phil. Trans. R. Sot. London A, 221, 163-168.

Pope, M. H. and Outwater. J. 0. (1972) The fracture

characteristics of bone substance. J. Biomechanics 5. 451-465. Piekarski, K. (1970) Fracture of bone. J. appl. Phys. 41. 215-223.

Reed, P. E. and Squires, H. A. (1974) Application of fracture mechanics to plastics deformed at high strain rates. J. Mat. Sci. 9. 129.

Fracture toughness of compact bone.

FRACTURE TOUGHNESS OF COMPACT BONE” W. BONFIELD and P. K. DATTA~ Department of Materials Queen Mary College. London, England Abstract-The ten...
935KB Sizes 0 Downloads 0 Views