Experiments and analyses were performed to determine th e cause of a nonlinear force-deflection response observed in four-point flexural fatigue of beams ofcortical bone machined from the mid-diaphysis of the equine third metacarpus. Observable grooves which formed on the beam surface at supports and load noses were found to be the primary cause of the nonlinearity. An additional geometric nonlinearity at large deflections revealed by finite element modeling may be minimized by using the smallest diameter supports and load noses recommended in ASTM 790. However, frictional constraint of the beams at the load noses and supports can occur at low load levels and should be avoided by using roller-bearing supports and load noses. or some equivalent method

Gibeling, J. C.

Gibson, V. A.

Griffin, L. V.

Martin, R B

Stover, S. M.

English

DigitalCommons@CalPoly

ARTIFACTUAL NONLINEARITY DUE TO WEAR GROOVES AND F RICTION IN FOUR-POINT BENDING EXPERIMENTS OF CORTICAL BONE L. V. Griffin,* J. C. G ibeling,* V. A. Gibson,t R. B. Martint and S. M. Stoverf *Division of Materials Science and Engineering, College of Engineering; t Orthopaedic Research Laboratories, School ofMedicine; and t Veterinary Orthopedic Research Laboratory, School ofVeterinary Medicine, University of California at Davis, Davis, CA 95616, U.S.A. Abstract- Experiments and analyses were performed to determine th e cause of a nonlinear force-deflection response observed in four-point flexural fatigue of beams ofcortical bone machined from the mid-diaphysis of the equine third metacarpus. Observable grooves which formed on the beam surface at supports and load noses were found to be the primary cause of the nonlinearity. An additional geometric nonlinearity at large deflections revealed by finite element modeling may be minimized by using the smallest diameter supports and load noses recommended in ASTM 790. However, frictional constraint of the beams at the load noses and supports can occur at low load levels and should be avoided by using roller-bearing supports and load noses. or some equivalent method. INTRODilJCfiON Fatigue injuries such as suess fractures in military recruits, ath letes, and thoroughbred racehorses are a common occurrence. T o understand the mechanisms which cause these injmies, bone is subjected to experi­mentation under conditions similar to those experienced in vivo. Be­cause bone is a complex material subjected to multimode loadingstates, a linn base of reliable experimental material p roperties data obtained under simplified loading modes suc!h as tension, compression, bending, and torsion is needed. This facili,tates the development of reliable predictive models of whole bone behavior. In one such study of the flexural fatigue response of cortical bone taken from the equine third metacarpus (cannon bone), 24 beams (10x4 x IOOmm) were machined in conformity with ASTM 790 for flexural testing (Gibson et a/., 1995). These bone beams were cyclically loaded in four-point bending to a maximum flexural strain of 10,000 microstrain (J.u;J at 2Hz while immersed in a physiologic saline solution at 37•c. The supports and load noses (ASTM D790) were 9.5 mm diameter stainless steel cylinders. with support and load nose spacings of 64 and 32 mm, respectively. A tangent (elastic) modulus was defined as tbe slope of the stress-strain curve in the linear region. A secant modulus was defined as the stress range divided by the strain range for one load cycle. Early flexural fatigue tests produced unexpected results. While the tangent modulus tended to decrease with the number of cycles, as expected, the secant modulus began to increase near the end of the fatigue life, especially in specimens taken from the dorsal regions of the cannon bone (Gibson el at., 1995). The secant modulus increase was associated with the development of a nonlinear force-deflection curve. While nonlinear stiffening of cortical bone in bending had not been reported in the literature. a recent study indicated that previously reported nonlinear behavior of trabecular bone was an artifact caused by crushing and frictional effects at the loading platens (Keaveny eta/., 1994). For this reason. we were concerned that if the flexural testing method produced artifactual nonlinearity, the reliability of our data would similarly be diminished. It was observed that grooves approximately 0.25 mm in depth formed on the surface of the beam under the supports and load noses (Fig. I) ofmany specimens, including those that exhibited the increasing secant modulus. during the conduct of fatigue experiments (Gibson et a/., 1995). A review of the literature revealed a note in ASTM 0790-86 on flexural tests indicating that crushing of the material at the load noses and beam supports may produce an artifactual 'toe' in the force-deflection curve (ASTM, 1992). Therefore, the grooves were hy­pothesized to be the primary cause of the nonlinearity. Experiments and analyses were conducted to determine: (i} if crushing. rather than friction a nd wear, could produce grooves similar to those observed in the fatigue experiments; (ii) the effect ofgrooves on the force- deflection response; (iii) the effect of friction at the supports-to-beam interfaces on the force-deflection response. METHODS AND MATERIALS To evaluate whether crushing could produce the observed grooves, a Hertzian mathematical model ofan elastic cylinder in contact with an elastic, semi-infinite medium in plane strain was used to test if direct contact would produce deformations as large as those observed (Joh­nson, 1985). The stainless steel cylinder support used in the experiments was modeled as 9.5 mm in diameter, with an elastic modulus of 195 GPa and a Poisson ratio of 0.305. The bone was considered to be isotropic with a modulus of 17 GPa and a Poisson ratio of 0.3. The contact load applied to the beam was 280 N, which corresponds to the maximum load at the supports and load noses during the flexural fatigue tests. No attempt was made to assess inelastic deformation of the bone ma·terial at the point of contact. An experiment using two Plexiglas beams was conducted to deter­mine if grooves at the supports and load noses would produce the nonlinear response that we observed in bone. Two 100 x 10 x 5 mm beams (approximately the same dimensions as the bone samples) were machined from a single piece ofPlexiglas. The same four-point bending test fixtures as the bone experiment were used. Grooves with a radius of 4.75 mm and a depth of0.3 mm were machined on one beam at the sites located below the supports and load noses; the other beam had no grooves. Both Plexiglas beams were tested for force--<:leflection response using an MTS 809.10 load frame with MTS TestStar 2 control elec­tronics. Both beams were loaded monotonically to failure at a constant actuator displacement rate of I mms - 1 . Force was measured using a 2200 N load cell and tbe mid-span displacement was measured using 1 mm Fig. I. A typical groove that appeared on fatigued bone specimens at the load pt)ints and supports. The groove depth is approximately 0.2 rom. an LVDT (model DC-E 125, Schaevitz Engineering). Stresses and strains were calculated from the force- deflection response using ele­mentary beam theory. Three two-dimensional. plane strain, finite element method (FEM) models were constructed in Patran• and solved using Abaqus0 . Due to lhc symmetry of four-point bending, one half of the total beam was modeled. Each FEM model included second-order interfacial elements on the beam surface in the vicinity of lhe contact points and on the surface ofthe loading noses and supports, allowing for the possibility of frictional contact. Each beam was modeled using 800 nine-node elements with an aspect ratio of I : I. The supports and load noses were modeled as semi-i:ircular with radius 4.5 mm, each containing 150sh·node triangle elements. T he solution was obtained using an incremental, iterative technique due to the presence of nonlinearities, such as friction. Large deformation theory was used in the analysis. The beam was modeled as isotropic with an elastic modulus of I 7 GPa and a Poisson ratio o f 0.3. The supports were isotropic with an elastic modulus of 207 GPa and a Poisson ratio of 0.305. The beam was loaded incrementally by a ramp to approximately I0,000 pEat the outer fiber of the beam, which was the strain level of the fatigue test. Three levels ofkinetic friction (ll•l were used in each model: 0.00, 0.25, and 0.50, which bracket experi· mentally obtained wet and dry kinetic friction coefficients (Griffin e1 nl.. !995). Friction was modeled as a step-function, assuming the static friction coefficient was equal to the coefficient of kinetic friction. The first FEM model was a beam with no grooves; the second, a beam with grooves the shape of a circular segment of radius 4.5 mm and depth 0.25 mm located d irectly under the supports and load noses (circular grooves); the third. a beam with grooves the shape ofa circular segment of radius 5.4 mm and depth 0.25 mm located directly under the supports and load noses (oblong grooves I, similar to those observed on fatigued bone samples (Fig. 1 ). RfSULTS The Hertzian model predicted highly localized compressive stresses ofapproximately 180 MPa and shear stressesofapproximately 55 MPa at the contact site. The grooved Plexiglas beam produced a nonlinear force-deflection curve similar to that observed in the bone(Fig. 2). To simplify compari­son, and account for the variability of the elastic modulus, the force-deflection results for each data set were normalized by the force and displacement which would produce 10,000 JU as calculated by elementary beam theory. The Plexiglas beaJn without grooves prod­uced a linear force-deflection response. T he FEM analysis of the ungrooved beam under the frictionless loading predicted a small amount ofnonlinearity in the force-<letlection Plexiglas Beams 1.0 CD (,) 0.8 ... 0 u. "C 0.6 Q) ~ 0.4 Ill E... 0 0.2 .... - -Ungroovedz 0.0 0.0 0.2 0.4 0.6 0.8 1.0 (a) Nonnallzed Displacement Bone Samples 1.0 Q) .­e 0.8 0 u. 1 0.6 . ii !:! 0.4 E0.2 - - --RcCier Suppoo-1$0 z · · • • • . BeamTheory 0.0 0.0 0.2 0.4 0.6 0.8 1.0 (b) Nonnalized Displacement Fig. 2. (a)The force-deflection curves for the Plexiglas beams with, and without, grooves. (b) The force-<leflection response of three representa­tive bone beams fatigued with fixed supports compared to that of a bone beam using roller suppons and beam theory. Only beams with fixed supports developed grooves. curve associated with large deflections [Fig. 3(a)]. As the friction coef­ficient increased, the amount of nonlinearity increased. The degree of nonlinearity depended on the geometry of the grooves and the value of the kinetic friction coefficient [Fig. 3(b) and (c)]. As the friction coeffic­ient increased. the final displacement of the beam also became signifi­cantly less than that predicted by beam theory. whio::h is manifest as an increasin.g secant modulus. The case of oblong grooves produced a re­sponse very similar to that seen in the actual test of the bone beam [compare Fig. 2(b) with Fig. 3(b)]. Examination of the stress state in the FEM model indicated that an additional axial tension is present between the load noses as evidenced by a shift in the neutral axis toward the compressive edge (Fig. 4). Essentially, the maximum value of the tensile stress is greater than the maximum compressive stress. Furthermore, there were no shear stres­ses between the inner supports. DISCUSSION Crushing does not appear to be the primary cause of the grooves observed in the fatigue experiments. The compressive stresses at the contact points were estimated to be 180 MPa. The longitudinal com­pressive yield strength ofcortical bone tissue of the equine third meta­carpus is 150-250 MPa (Les er a/., 1994). The compressive yield in the transverse direction should be less than these reported values, and suggests that some crushing could occur. However, the model predicts lower stresses if the modulus input is lower. The Poisson ratio may be varied up to 0.5 (due to isotropic linear elasticity restrictions) and results in slightly higher stress predictions. Strictly speaking, the Hertzian model is not valid due to the anisot­ropy of bone. which was modeled as isotropic. However, bone beams No Grooves Oblong Grooves Cl) u ~ 0.8 ~ 0 u. "0 .§ii e 0 z Cl) u.. 0.8 0 u. 0.6"0 CD .t! 0.4ii e 0.20 z 0.0 0.2 0.4 0.6 0.8 1.0 Nonnalized Displacement (a) Circular Grooves 0 u. 0.6 .! 0.4ii e 0.20 z 0.0 ._-+-----il----+---' 0.0 0.2 0.4 0.6 0.8 1.0 Nonnalized Displacement (b) ---- Beam Theory _.__ 1-\ =0.00 --6- 1-\ =0.25 --..- 1-\ = 0.50 0.0 ...-t----+--+--+-----1 0.0 0.2 0.4 0.6 0.8 1.0 Nonnalized Displacement (c) Fig. 3. FEM and beam theory force-deflection curves for various beam models: (a) no grooves; (b) oblong grooves; (c) circular grooves. Note the similarity of the force-deflection behavior of the beam with oblong grooves to that of actual bone shown in Fig. 2(b). Beam Midspan 14.2 MPaGeometric centerline of Tensionthe beam Compression Actual position of the neutral axis Fig. 4. The effect of grooves and friction in the mid-span flexural stress state of the beam as determined by the FEM. Notice the definite shift of the neutral axis of bending from the geometric center of the beam toward the compressive side of the beam predicted at small deflections. cycled to failure at 10,000 Jl£ in flexure using roller bearing supports have produced no residual indentations on the beam surface, nor have they ex.hibited any nonlinear stiffening behavior (Gibson er a/., 1995). Therefore, crushing does not seem to be the cause of the grooves or the nonUnearity. We postulate the primary cause of the grooves to be predominately wear. The cause of the observed nonlinearity appears to be a combination of grooves and friction at the supports and load noses. The effect of grooves on the force-deflection behavior i~ clearly demonstrated in the Plexiglas beam experiment. The Plexiglas beams were cut from the same piece of material and loaded at the same rate, so the material behaviorof the Plexiglas is common to both. Since the only difference is that one beam had grooves while the other did not, it is concluded that the cause of the nonlinearity in the force-deflection curves for the Plexiglas beams is the grooves. The FEM models of the grooved beams demonstrate a stiffening nonlinear force-displacement response [Fig. 3(b) and (c)]. As friction at the supports-to-beam interfaces was increased, the predicted force­displacement nonlinearity increased considerably -even in the beam with no grooves [Fig. 3(a)]. Apparently friction serves to constrain the beam at the suppons and load noses. Since the material modeled in the FEM beams is strictly linear elastic, the observed nonlinearity must be caused by the test configuration. The shift of the neutral axis toward the compressive edge of the beam can appear at low loads (Fig. 4). Be.:ause neutral axis shift appears even in the ungrooved FEM beam, the cause is apparently constraint of the beam at the supports and load noses by friction and/or grooves. The FEM model of a beam with no grooves and no friction prod\tced a small amounU of nonlinearity [Fig. 3(a)]. which is also observed experimentally in bone with roller-bearing supports and load noses [Fig. 2(b)]. This nonlinearity is a geometric effect, caused by the circular supports and load points combined with tbe large deflections in reach­ing 10,000 JtC. By using the large diameter supports, the test configura­lion behaves as a nonlinear stiffening spring (Shigley and Mischke. 1989). The geometric nonlinear stiffening may be minimized by using the smallest diameter support and load nose recommended by ASTM 790. or conducting studies at small deflection levels. It should be mentioned that while the Plexiglas beam demonstration of the nonlinear stiffening is not a powerful test, on its own, the result is strengthened by considering that the only cortical bone beams that have exhibited the nonlinear stiffening behavior had grooves and were loaded using fixed supports that can produce friction at the load noses and supports. Furthermore, the FEM results agree with the Ple"'iglas beam experiment and the observation tbat grooves produce a force­displacement curve exhibiting nonlinear stiffening in bone beams. Thus, friction and grooves produce nonlinearity in the force-deflection curve as well as altering the expected stress state between the load noses. In a controlled test environment, all sources of linearity. or nonlinearity, need to be understood to avoid mistaking experimental artifact for material behavior. We conclude that application of the flexural mode of testing to bone requires minimization of nonlinear stiffening behavior associated with friction and wear groove formation by the use of roller­bearing supports and load noses. or some equivalent method. ;lcknowledgemems-- This work was supported by NIH gra nt AR41644. We also acknowledge the assistance of the Hearst foundation. Mr and Mrs Amory J. Cooke, the California Horse Racing Board Postmortem program, and the California Center for Equine Hc:al th and Perfor­mance, University ofCalifornia. Davis. with funds provid<!d by the Oak Tree Racing Association. the State of California satellite wagering fund. and contributions by private donors. ASTM Staudard D790M-86, Stam.Janl tc:st mc:thous for flc:xural prop­erties of unreinforced plastics and electrical insulating materials [Metric]. 1992 Annual Book of ASTM Standards. Vol. 3.01, pp. 279- 287. Carter, D. R. and Hayes, W. C. (1977) The compressive behavior of bone as a two-phase porous structure. J. Bone Jt Surg. 59, 954-962. Gibson, V. A., Stover, S.M., Martin, R. B., Gibeling, J. C. eta/. ( 1995) Fatigue behavior of the equine third metacarpus: mechanical prop­erty analysis. J. orthop. Res. 13, 861-868. Griffin, L. V., Gibeling, J. C. and Martin, R. B. (1995) Unpublished data. Johnson, K. L. (1985) Contact Mechanics. pp. 129-134. Cambridge University Pr-ess, London. Keaveny, T. M., Guo, E .. Wachtel, E. F .. McMahon, T. A. and Hayes. W. C. (1994) Trabecular bone exhibits fully linear elastic behavior and yields at low strains. J. Biomechanics 27, 1127- 1136. Les, C. M .. Keyak, J . H ., Stover, S. M., Taylor, K. T. and Kaneps, A. J. (1994) Estimation of material properties in the equine metacarpus with use of quantitative computed tomography. J. orthop. R f!'S. 12, 822-833. Shigley, J. E. and Mischke, C. R. (1989) Mechanical Engineering Desig11 (5th Edn). McGraw-Hill, New York. 

Artifactual nonlinearity due to wear grooves an friction in four-point bending experiments of cortical bone

https://digitalcommons.calpoly.edu/cgi/viewcontent.cgi?article=1068&amp;context=bmed_fac

Artifactual nonlinearity due to wear grooves an friction in four-point bending experiments of cortical bone

Abstract

Similar works

Full text

Available Versions

DigitalCommons@CalPoly