Uniaxial and Multiaxial Fatigue Life Prediction of the Trabecular Bone

Based on Physiological Loading: A Comparative Study

S. J. FATIHHI,1 M. N. HARUN,1,2 MOHAMMED RAFIQ ABDUL KADIR,3 JAAFAR ABDULLAH,4 T. KAMARUL,5

ANDREAS OCHSNER,6 and ARDIYANSYAH SYAHROM1,2

1Faculty of Mechanical Engineering, Universiti Teknologi Malaysia, Johor Baharu, Malaysia; 2Sport Innovation andTechnology Centre (SITC), Institute of Human Centered Engineering (IHCE), Universiti Teknologi Malaysia, Johor Baharu,Malaysia; 3Medical Implant Technology Group, Faculty of Biomedical Engineering and Health Science, Universiti TeknologiMalaysia, Johor Baharu, Malaysia; 4Centre for Computed Tomography and Industrial Imaging, Malaysia Nuclear Agency,

Bangi, Malaysia; 5Tissue Engineering Group, NOCERAL, Department of Orthopaedic Surgery, Faculty of Medicine, Universityof Malaya, 50603 Kuala Lumpur, Malaysia; and 6School of Engineering, Griffith University, Gold Coast Campus,

4222 Southport, Australia

(Received 18 December 2014; accepted 19 March 2015)

Associate Editor Eiji Tanaka oversaw the review of this article.

Abstract—Fatigue assessment of the trabecular bone hasbeen developed to give a better understanding of boneproperties. While most fatigue studies are relying on uniaxialcompressive load as the method of assessment, in variouscases details are missing, or the uniaxial results are not veryrealistic. In this paper, the effect of three different loadhistories from physiological loading applied on the tra-becular bone were studied in order to predict the first failuresurface and the fatigue lifetime. The fatigue behaviour of thetrabecular bone under uniaxial load was compared to that ofmultiaxial load using a finite element simulation. The plasticstrain was found localized at the trabecular structure undermultiaxial load. On average, applying multiaxial loadsreduced more than five times the fatigue life of the trabecularbone. The results provide evidence that multiaxial loading isdominated in the low cycle fatigue in contrast to the uniaxialone. Both bone volume fraction and structural model indexwere best predictors of failure (p < 0.05) in fatigue for bothtypes of loading, whilst uniaxial loading has indicated bettervalues in most cases.

Keywords—Trabecular bone, Fatigue, Multiaxial, Uniaxial,

Failure behaviour, Microarchitecture.

INTRODUCTION

The last few decades attest to increasing interests inthe fatigue behaviour of bone. Fatigue fractures in theelderly population are associated with bone changes

with increasing age.6,20 With the ongoing increase inlifespan it becomes more important to understandbone properties, thus increasing the importance of fa-tigue evaluation. As fatigue fracture can take place inboth the cortical and the trabecular bone during pro-longed exercise in healthy adults (known as stressfractures) and in elderly patients with osteoporosis, theaccumulation of microcracks, permanent strain andstrength reduction can gradually occur with increasingcycles of loading during ambulatory activities.6,17,45

Experimental and computational studies have soughtto characterize fatigue behaviour and failure of thetrabecular bone mostly under uniaxial compressiveloading. However, in regular daily activities, the wholebone is loaded with multiaxial stress states besidesmultidirectional functional and abnormal loadingfrom physiological trauma.18,36,37 A combination ofaxial compression-tension and torsion-shear imposedon the fatigue evaluation may produce a basis for bonefatigue due to post trabecular bone failure with manyconsequences during normal gait adaptation.

The uniaxial studies may provide loose estimationson their multiaxial fatigue life counterparts21 and giveinaccurate real life predictions. Bergmann et al.3 havereported the components of loading and the sig-nificance of torsional moment on hip implants duringrunning. These loading states may affect the trabecularstructure integrity from the femoral head joint over theneck to the intertrochantric region. Severe effects couldlead to proximal tibial and distal femur osteoarthritisduring gait,55 while the torsional moment is morepronounced in hip joints due to abnormal gait and

Address correspondence to Ardiyansyah Syahrom, Sport Inno-

vation and Technology Centre (SITC), Institute of Human Centered

Engineering (IHCE), Universiti Teknologi Malaysia, Johor Baharu,

Malaysia. Electronic mail: ardiyans@gmail.com

Annals of Biomedical Engineering (© 2015)DOI: 10.1007/s10439-015-1305-8

© 2015 Biomedical Engineering Society

traumatic pathology. However, the direct effects of thetrabecular bone under fatigue loading decreases theirstiffness and increases the number of microcracks.Whether these could lead to osteoarthritis due to in-sufficient repair of joints or damage from mechanicalstress5 remains unclear. The damage of mechanicalstress could develop under pathogenic causes such asexcessive body weight, loss of strength, or mechanicalinjuries which play a role in reducing bone strength.5

Nevertheless, the effects of trabecular damage andprolonged deterioration as consequences of the os-teoarthritis damage process are still unclear. Under-standing this situation may improve the insight oftrabecular bone implications towards osteoarthritistreatment and osteoporotic pathological fracture withimplantation treatment or bone grafting which iserected in trabecular sites.54

Cyclic loading has been found to create microfrac-tures in the bone in which the fatigue failure becomesmore severe under torsional load.8,43,44,58 However,bone is subjected in vivo to superimposed axial–tor-sional or multiaxial loading.7,9,50 Nevertheless, studiesfocusing on axial loading only24,57 have not providedmuch information on the age-related alterations in themechanisms of bone failure and remodelling.

Fatigue studies have been reported by several re-searchers through experimental13,27,41 and finite elementsimulation25,30,33 to predict and model the fatigue be-haviour of the trabecular bone. Finite element simula-tions are useful to prevent possible fracture initiation andto optimize the treatment of the trabecular bone whileexperimental methods could be cumbersome. Many ofthesemodels implemented idealizedmodels or 2Dmodelsto estimate the fatigue life of bone.As such, the predictionof the fatigue damage behaviour of the trabecular bone isnot always accomplished completely due to their complexmicroarchitectural and failure behavior.

To the best of our knowledge, no computationalstudies have been proposed to predict the fatigue life ofthe trabecular bone under multiaxial loading condi-tions. Thus, the complex multiaxial or triaxial loadingof such strenuous activities needs to be executed ex-perimentally. The aim of this study is to compare thefatigue life and failure behaviour of the trabecular boneunder uniaxial and multiaxial cyclic loading based onphysiological gait loading. Specifically, this studycompares the combination of lateral and vertical forcesfraction from gait loading that are believed to affectfatigue life and behavior of the trabecular bone. Thefatigue failure differs in terms of modulus reductionand plastic strain accumulation in comparison touniaxial and multiaxial loading, which were also ex-amined.

MATERIALS AND METHOD

Sample Preparation and µCT Imaging

Trabecular bone samples were extracted from thehind-limb of fresh bovine cadavers gathered from alocal slaughter house, then cut from the femoral ballusing a ±150 rpm diamond saw (Behringer GmbH,typeSLB 230 DG HA, Kirchardt) under copious lu-bricant irrigation to minimize heat generation andstrut breakage. Saline water was used as a lubricant toensure that the temperature did not exceed 46 °C toprotect the sample from heat-related damage.23 Aninfrared thermometer (Fluke 62 Mini Infrared Ther-mometer) was used in order to observe the temperatureof the sample based on the blade and coring bit.During the cutting and drilling procedure, the processwas stopped at several stages to measure the tem-perature in order to not to exceed the critical level.Cylinder samples with a total length of 15 mm and adiameter of 10 mm were then drilled using a 1.5 mmthick diamond-tip coring bit at 150–250 rpm.46 Anelectronic caliper was used to measure the length anddiameter of the specimens. The cylindrical bone sam-ples were ultrasonically cleaned (Cest ultrasonic, modelP1100SR, USA)46 with a chemical detergent (Pumi-cized, Gent-l-kleen, USA)32 for marrow removal. Ex-cessive marrow was further cleansed with water and airjets. The prepared samples were stored in plastic bagsand frozen at −20 °C.52,56 The scanning process wasdone 24 h after the freezing process.

The prepared samples were then set up for the CTscan (Skyscan 1172, Kontich, Belgium). The micro-graph of the sample determined its alignment with theprincipal trabecular direction. A three-dimensionalmodel with a diameter of 5 mm and a length of 5 mm ofthe trabecular bone was extracted from the central partof the prepared sample and scanned with a highresolution of 20 µm. Despite the small size used, the 3Dmodel satisfied the continuum assumption for tra-becular bone which is at least to have three to five intertrabecular lengths in size.28 This is due to the limitationin computational power used in this study that could notpossibly construct the whole trabecular model. Quan-tification of the trabecular microarchitecture for bonevolume fraction (BV/TV), trabecular thickness (Tb.Th),trabecular separation (Tb.Sp), anisotropy (DA), bonesurface area (BS/BV), connectivity density (Conn.D)and structural model index (SMI) have been done usingImage J (Ver. 1.46r, National Institutes of Health,USA). The data is presented in mean, standard de-viation (SD) and range, as shown in Table 1. The datawas statistically analyzed using MINITAB® Release14.12.0 statistical software (MINITAB, USA) with

FATIHHI et al.

which we determined the regression coefficient (R2) andPearson’s correlation in between trabecular microar-chitecture and its fatigue properties.

Finite Element Analysis of Fatigue Loading

The finite element model was analysed using thecommercial COMSOL Multiphysics finite elementsoftware (COMSOLMultiphasic software, Burlington,USA, version 3.4). The cylindrical axis was assignedthe z-direction for axial fatigue loads. Multiaxial loadsfrom three axes contributed to the combined axial andtorsional moment. As the deformation behavior ofbovine bone has been found to be similar to that ofhuman bone,27 this study used trabecular bone of thebovine femoral hip as a material representation. Inhumans, this site experiences more multiaxial loadtransfer through body weight than bone or joints atother sites.38 The nonlinear-elastic/plastic materialbehavior of the trabecular bone was modelled with thefollowing values: 1000 MPa as the elastic tissue mod-ulus, a Poisson’s ratio of 0.3 and a linear hardeningmodulus of 0.05E2 for the finite deformation-basedplasticity model.19,26 This model is assigned to toleratenet section yielding with plasticity analysis assumingstress concentration in which the local stresses exceedthe yield limit of a material with a nominally elasticregion. Thus, in order to make accurate predictionsrelated to the trabecular behaviour in vivo, this plas-ticity based model is required. The validity and com-parison of the plasticity model was made with Garciaet al.19 Figure 1a shows a cyclic plasticity curve inwhich the result followed the typical curve pattern.Initial stress and strain at yield were determined fromprevious studies.51 All the constitutive parameters forfatigue were taken from previous studies.41 To performa mesh convergence study, the loading stress on thecylindrical trabecular sample caused by the prescribedtop displacement is the integral of the z-component of

the reaction force over the top surface of the modelgiven as

F ¼Z

rz2prdr ð1Þ

where the factor 2πr comes from the integration incylindrical coordinates of the trabecular 2D cut surfaceand σz is the total stress in the z-direction. This integralis computed by the integration operator, which appliesa summation method over nodes for calculating thereaction force in the vertical direction. The back-cal-culated modulus was consistent with a previousstudy.51 This study used tetrahedral elements with aminimum element size of 0.02 mm and a total numberof almost 400,000 elements, as shown in Fig. 2.

To mimic the real physiological loading duringnormal walking, the reconstructed gait loads in themodel were applied as a time dependent analysis alongits longitudinal axis in the form of r ¼ 1�cosð2pÞ

2 t (wherethe time t is within 0–1 s in a cycle). The equivalentmaximum stress from the gait cycle was converted intoa single load cycle. For the multiaxial load from thegait pattern during walking, the initial applied triaxialload (Fz = vertical direction force, Fy andFx = anterior-posterior and medial–lateral forces) onthe top surface of the trabecular bone was converted toa combined axial (Fz) and torsional moment[Txy = (Fx + Fy) * radius of the specimen]. A custommultigrid solver was used to solve this problem.

For the fatigue analysis, the finite element simula-tion was done using a fully elastoplastic analysis with acombined Basquin and Coffin–Manson equation inorder to estimate the fatigue life in the regions of lowand high cycle fatigue. The cycles to failure were pre-dicted in a variable load range from 0 up to the pre-determined stress level as the strain is used as thedamage variable for uniaxial and multiaxial loads. Thethree loadings give parabolic equations from3 in Fig. 3were extracted through Matlab (MATLAB® software,Math works, USA) and can be seen in the “Imple-mentation of gait loading” Appendix section. Thisfigure shows the plotted graph of the contact force onthe femoral hip during normal walking. For simplifi-cation, this total load from gait cycles was convertedinto 75% from the external load47 and a typicalbodyweight of 60 kg was assumed in order to applyonto the trabecular end. The stress applied under fa-tigue simulation was predetermined from 10 to 100%increments of the stress level under one gait cycle forboth uniaxial and multiaxial loading. Uniaxial loadswere derived from the total resultant force through hipcontact force only. The multiaxial load was the com-bination of axial and torsional load from the calcu-

TABLE 1. Microstructural parameter of the samples.

Parameter Mean SD Range

BV/TV 0.411 0.08 0.273–0.544

Tb.Th (mm) 0.243 0.027 0.199–0.312

Tb.Sp (mm) 0.458 0.115 0.277–0.692

BS/BV 0.201 0.022 0.179–0.251

DA 0.599 0.139 0.297–0.804

Conn.D (1/mm3) 4.588 2.053 0.837–8.798

SMI 0.631 0.652 −0.476 to 1.932

Porosity (%) 58.90 7.692 45.60–72.70

BV/TV bone volume fraction, Tb.Th trabecular separation, Tb.Sp

trabecular, BS/SV bone surface area, DA degree anisotropy, Conn.

D connectivity density, SMI structural model index.

Uniaxial and Multiaxial Fatigue Life Prediction

lated circumferential moment in the x- and y-directionwith respect to the cylindrical model radius (Fig. 3).

Implementation of Plasticity

Additive decomposition of strains is the most suit-able approach in the elastic region and thus imple-mented for small strains plasticity models. Therelationship between stress and strain is provided as;

r� r0 ¼ C : e� e0 � eth � ep � ecr� � ð2Þ

where, σ is the Cauchy stress tensor, ε is the total straintensor, σ0 is the initial stress, ε0 is the strain tensors, εth isthe thermal strain tensor, εp is plastic strain, εcr is creepstrain tensor, and C is fourth-order elasticity tensor.

Removing inelastic and initial strains from the totalstrain tensor gives out the elastic strain tensor as fol-lows;

eel ¼ e� e0 � eth � ep � ec ð3ÞThe elastic strain tensor is simplified when initialstrains equal zero with no creep or thermal expansionas;

eel ¼ e� ep ð4ÞThe Piola–Kirchhoff stress is used instead of theCauchy stress when the model includes geometricnonlinearity. Thus, the additive decomposition ofstrains is acknowledged as the Green–Lagrange strainssummation. Furthermore, strain is implied in the largeplastic strain decomposing the total deformation gra-dient tensor multiplicatively into the elastic deforma-tion gradient and plastic deformation gradient.

F ¼ FelFp ð5ÞEliminating the plastic deformation from the totaldeformation gradient,

Fel ¼ FF�1p ð6Þ

so the elastic Green–Lagrange strain tensor is computedfrom the elastic deformation gradient tensor Fel

eel ¼ 1=2 FTelFel � I

� � ð7Þand the plastic Green–Lagrange strain tensor is com-puted from the plastic deformation gradient tensor

ep ¼ 1=2 FTpFp � I

� �ð8Þ

As opposed to the small strain formulation describedin Eq. (4), the total, plastic and elastic Green–La-grange strain tensors are related as

eel ¼ FTp e� ep� �

F�1p ð9Þ

Under multiplicative decomposition, the elastic rightCauchy–Green tensor and the plastic right Cauchy–Green tensor are defined as

Cel ¼ FTelFel and Cp ¼ FT

pFp ð10Þ

Plastic Flow

The increment of the plastic strain tensor, έp and itsrelationship with the current state of stress, σ, when amaterial yields under further loading is defined by theflow rule. The direction of increment in plastic strainunder small plastic strain is defined by;

�ep ¼ k@Qp=@r ð11Þwhere λ is a positive multiplier (also known as consis-tency parameter or plastic multiplier). The current stateof stress and the load history are the factors whichcontribute to the positive multiplier. Qp is the plasticpotential which defines the surface perpendicular to thedirection of the plastic strain increment, έp.

The plastic multiplier λ is determined by comple-mentarily or Kuhn–Tucker conditions;

FIGURE 1. (a) Typical example of cyclic plasticity model with different tangent modulus (red: 0.01E0 and black: 0.05E0) in low andhigh load level of fully reversed loading cycles. (b) Evolution of plastic strain with time in the point evaluation with fine meshcontrolled.

FATIHHI et al.

k � 0;Fy � 0 and kFy ¼ 0 ð12Þwhere Fy is the yield function. Fy < 0 implies the yieldsurface encloses the elastic region. When Fy = 0, plasticflow occurs.

The flow rule is said to be associated when theplastic potential coincides with the yield surface(Qp = Fy). Thus, with the conditions in Eq. (12), therate in Eq. (13) is solved.

�ep ¼ k @Fy=@r ð13ÞThe flow rule is said to be non-associated when theplastic potential does not coincide the yield function. Inthis manner, the rate in Eq. (14) is solved together withthe conditions in Eq. (13) for the plastic potential Qp.

�ep ¼ k @Qp=@r ð14ÞWith either Eq. (13) or (14), the plastic strain tensor έpevolution in the plastic element is implemented atGauss points.

Qp or the plastic potential is expressed at most bythree invariants of Cauchy’s stress tensor for isotropicplasticity as;

Qp rð Þ ¼ Qp I1 rð Þ; J2 rð Þ; J3 rð Þð Þ ð15ÞIn which the invariants of the stress tensor are

I1 rð Þ ¼ trace rð ÞJ2 rð Þ ¼ 1=2dev rð Þ : dev rð ÞJ3 rð Þ ¼ det dev rð Þð Þ

where I1(σ) is the first invariant of the Cauchy’s stresstensor. The first invariant represents the effect of themean stress. J2(σ) and J3(σ) are the second, and thirddeviatoric stress invariants respectively in which themagnitude of shear stress is represented by J2, and thedirection of the shear stress which depends on theapplied load and geometry of the models is representedby J3.

Thus, the increase of the plastic strain tensor έp isbroken down into

�ep ¼ k @Qp=@r ¼ kð@Qp=@I1 � @I1=@rþ @Qp=@J2 � @J2=@rþ@Qp=@J3 � @J3=@rÞ

which includes both deviatoric and volumetric parts ina general case. Symmetric tensor έp can be presentedwith the following properties;

FIGURE 2. Mesh convergence analysis performed to obtainthe optimum number of elements. Mesh (b) was selectedbased on the computation time for the simulation and con-verged range. Mesh (c) and (d) arrived at the same resultsrange with about 1–3 days of computational times for a model.

FIGURE 3. (a) Hip contact forces during normal gait walking cycle. The time-dependent load was used as single cycle load caseby assuming a body weight of 60 kg in order to estimate the forces applied during gait cycles, and (b) trabecular bone 3D realmodel extracted from femoral hip of cylindrical specimen with the applied boundary condition.

Uniaxial and Multiaxial Fatigue Life Prediction

@I1=@r ¼ I

@J2=@r ¼ dev rð Þ@J3=@r ¼ dev rð Þdev rð Þ � 2=3J2I

ð16Þ

Effective plastic strain rate is a common measure ofinelastic deformation. It can be defined by;

�eepe ¼ p2=3�ep : �ep ð17ÞThe reliance of the plastic potential on the first in-variant I1(σ), is the only factor which effects thevolumetric plastic strain rate έpvol or the trace of theincremental plastic strain tensor, as ∂J2/∂σ and ∂J3/∂σare deviatoric tensors

�epvol ¼ traceð�epÞ ¼ k traceð@Qp=@rÞ ¼ 3kð@Qp=@I1ÞAs the invariant I1(σ) has no affect on the plastic po-tential, the volumetric plastic strain rate is always zero.This is true for general plasticity, also known as J2plasticity, in von Mises and Tresca criteria.

Elasto-plastic analysis is chosen as the plasticitymodelfor large strains. This model implements multiplicativedecomposition of deformation.29,42,48 The associatedplastic flow rule can be written as the Lie derivative of theelastic left Cauchy–Green deformation tensor Bel:

�1=2L Belð Þ ¼ kð@/=@sÞBel ð18ÞAs what has been done for infinitesimal strain plas-ticity, the plastic multiplier λ and the yield function ϕ(written in terms of the Kirchhoff stress tensor, τ)satisfy the Kuhn–Tucker condition;

k � 0; / � 0; and k/ ¼ 0

The plastic dissipation equals the conjugate energypair, τ and d, where d is the rate of strain tensor thusyield function ϕ in References29,48 was written in termsof Kirchhoff stress, τ and not Cauchy stress, σ.

The Lie derivative of Bel is then written in terms ofthe plastic right Cauchy–Green rate

LðBelÞ ¼ F _C�1p FT ð19Þ

By using Eqs. (18) and (19), the either associated or non-associated plastic flow rule for large strains is written as48

�1=2F _C�1p FT ¼ kð@Qp=@sÞBel ð20Þ

together with the Kuhn–Tucker conditions for theplastic multiplier λ and the yield function Fy

k � 0;Fy � 0; and kFy ¼ 0 ð21ÞPlastic potential and the yield surface coincide witheach other as Qp = Fy for the associated flow rule. Forthe non-associated flow rule on the other hand, the Qp

unequal Fy.

On the left-hand side, the Cauchy–Green tensor iswritten in terms of the deformation gradient and the rightCauchy–Green tensor. Thus,Bel=FCp

−1F.For the inverseof the plastic deformation gradient Fp

−1, the plastic flowrule is then solved at Gauss points in the plastic element.Variables in Eq. (19) are thus replaced by;

_C�1p ¼ _F�1

p F�Tp þ F�1

p_F�Tp and Bel ¼ FelF

Tel ¼ FF�1

p F�Tp FT

Thus the flow rule is written as;

�1=2ð _F�1p F�T

p þ F�1p

_F�Tp Þ ¼ kF�1ð@Qp=@sÞFF�1

p F�Tp

ð22ÞThe plastic Green–Lagrange strain tensor can becomputed from the plastic deformation tensor afterintegration of the flow rule in Eq. (22);

ep ¼ 1=2 FTpFp � I

� �ð23Þ

and the elastic Green–Lagrange strain tensor is com-puted from the elastic deformation gradient tensorFel = FFp

−1

eel ¼ 1=2 FTelFel � I

� � ð24ÞThe effective plastic strain variable is computed as thetrue effective plastic strain (also called Hencky orlogarithmic plastic strain).

Yield Function

The yield function should be smooth which meansthat it can be continuously differentiated with respectto the stress when applying an associated flow rule.The following formula is implemented in this case;

Fy ¼ / rð Þ � rys ð25Þwhere σys = yield stress.

As commonly employed in general plasticity, theeffective stress ϕ(σ) in this predefined formula is the vonMises stress;

rmises ¼ p3J2ðrÞ ¼ p

3=2devðrÞ : devðrÞ ð26Þ

Hardening Models

In elastic–plastic materials, three different kinds ofhardening models are implemented in the plasticitymodel. Thus, either the associated or non-associatedflow rule can be solved by the plasticity algorithm forthe plastic potential Qp;

�ep ¼ kð@Qp=@rÞwith the yield function

Fy ¼ / rð Þ � rys0

FATIHHI et al.

In this case, the hardening function σh(εepe) can bedefined linearly as:

rys eepe� � ¼ rys0 þ rh eepe

� �Either the associated or non-associated flow rule issolved by the algorithm for the plastic potential Qp

�ep ¼ kð@Qp=@rÞwith the yield function defined as

Fy ¼ / r� rshiftð Þ � rys0 and rshift ¼ fðepÞwhere, σys0 = initial yield stress, and ϕ(σ) = von Misesstress.

Given the general multiaxial setting, the plasticstrain έp is known for the second-order tensor. In thissetting, the one-dimensional hardening law is imple-mented, i.e., σshift = cεp where the back stress σshiftshifted the stress tensor used in the yield function.Thus the hardening constant, c, can be computed by;

1=c ¼ 3=2 1=ETkin�1=Eð Þwhere, ETkin = the kinematic tangent modulus,E = Young’s modulus taken from the linear elastic ma-terial model

Thecurrentplastic strain functionand itshistorycanbedescribedby thebackstress. In the caseof linear kinematichardening, f(εp), a scalar measure is introduced, thus theamount of changes in the material microstructure can beevaluated. Here, expansion can be made on the linearfunction of the plastic strain tensor εp, or Prager’s hard-ening rule. It canbe represented simply as the plastic straintensor, ‖έp‖= √έp:έp. As the plastic strain rate can have anydirection, this variable is however, not always increasingthroughout the plastic flow. Thus, the reduction of εpmayoccur even though the plastic flow continues. By additionof kinematic hardening, plastic potentials as well as theyield surfaces are calculated with effective invariants.Here, the invariants of the tensor can be pictured as thestress without the back-stress. The effective invariant ofdeviatoric tensor is implemented when a von Mises plas-ticity is used together with kinematic hardening.

Numerical Solution of the Elastic–Plastic Conditions

The plastic flow rule used a backward Euler dis-cretization of the pseudo-time derivative. For smallplastic strains, this gives

ep � ep;old ¼ K@Qp=@r

where ‘old’ denotes the previous time step and Λ = λΔt,where Δt is the pseudo-time step length.

For large plastic strains, Eq. (22) gives

�1=2F 2MMT�MoldMT�MMT

old

� �¼Kð@Qp=@sÞFMMT

where M = Fp−1

The plastic state variables (εp and M, respectively)and the plastic multiplier, Λ, can be calculated bysolving this time-discretized flow rule together with thecomplementarily conditions at each Gauss point;

K � 0;Fy � 0 and KFy ¼ 0

This is done as follows:29

Elastic-predictor: Try the elastic solution εp = εp,old(orM=Mold) andΛ= 0. If this satisfies Fy ≤ 0 it is done.

Plastic-corrector: If Fy > 0, thus, the elastic solutiondoes not work. One has to solve the nonlinear systemconsisting of the flow rule and the equation Fy = 0using a damped Newton method.

Fatigue Failure Under Combination of Basquinand Coffin–Manson

Combined damage of low cycle fatigue (LCF) andhigh cycle fatigue (HCF) is considered as a damageindicator due to the strain amplitude of loading. It canbe hypothesized that if the physical state of damage isthe same then the fatigue life is depended on the loadingcondition. Suppose a component is subjected to a strainof amplitude, εa of N number of cycles at load level n,and Nf is the number of cycles to failure as defined bythe fatigue life corresponding to εepe. Therefore, thereduced life at the load level n is obtained as (Nf − N) isdefined as the nth level damage equivalent strain.Hence, the new damage indicator Dn is stated as,

Dn ¼ eepe � ea=ep � ea ð27Þwhere

ep ¼ fecwhen ea � e0f or eb when ea<e0f

The appropriate value of εp is selected based on thefatigue regime (LCF or HCF) in which the strainamplitude εa lies. Thus, εc and εb belong to low cycleand high cycle fatigue regions respectively.

The transformation of the damage Dn to the nextloading cycles n + 1 has come to an assumption inwhich damage D has occurred at the end of nth loadinglevel, due to the effect of εa+1 loading cycles as follows;

Dn ¼ eaþ1epe � eaþ1

� �= ep � eaþ1

� � ð28Þwhere;

ep ¼ fec when eaþ1 � e0f and eb when eaþ1<e0f

Then, the damage equivalent strain at loading leveln + 1, εa+1epe is calculated as,

e0fnþ1epe ¼ Dnðep � enþ1Þ þ enþ1 ð29ÞThe strain-life curve as shown in Fig. 11 provides thecorresponding equivalent number of cycles to failure

Uniaxial and Multiaxial Fatigue Life Prediction

N0fnþ1: The εn+1 is the strain at the level n + 1. The

corresponding residual life at load level n + 1, Nfn+1 iscalculated as follows supposing that it is subjected toNn+1 number of cycles;

Nfnþ1 ¼ N0fnþ1 �Nnþ1 ð30Þ

Therefore, strain (ε)n+1epe, which corresponds to εn+1epe at load level n + 1, is obtained from the strain-life curve as shown in Fig. 11. At the end of load leveln + 1, the cumulative damage is defined as;

Dnþ1 ¼ ðenþ1epe � enþ1Þ=ðep � enþ1Þwhere,

ep ¼ f ecwhen enþ1 � e0f and ebwhen enþ1< e

0f

For any given strain history, the damage indicator, Dn

can be calculated in similar manner. The equivalentstrain εnepe during the first cycle is equal to εn. Thus, thecorresponding damage indicator becomes Dn = 0. Thedamage indicator becomes Dn = 1 in similar mannerduring the last cycles, when εnepe is equal to εp.

Therefore, in fatigue failure of the trabecular, thenormalized damage indicator (Dn = 1) is used withimplementation of the above procedure. Failure isdefined as the first initiation of failure at the surface ofmaximum stress of the boundaries. The minimumnumber of cycles is the number of cycles after whichthe first failure is observed on the surface. Dividing thestress range by the strain range of each cycle En yieldsthe Young’s modulus. These Young’s moduli valuesare in opposite of tangent moduli, ETkin, known assecant moduli. The fractional loss in the modulus fromthe initial secant modulus, E0 to the current secantmodulus, En equals to the material damage in the nthcycle;

En=E0 ¼ 1�D ð31Þin which D is a theoretical value of continuum damageranging from zero to one and increases consistently.This value represents the degradation of mechanicalproperties of solid materials.35

It is often possible to represent the central part ofthe S–N curve which is in the high cycle fatigue (HCF)regime, by a straight line in a log–log diagram. Thisrelation is called the Basquin equation,

ra ¼ rf 2Nð Þb ð32ÞHere, N is the number of load reversals, so that 2 Nf isthe number of full cycles to failure at a stress amplitudewhereby σa, σf, and b are material parameters.

The low cycle fatigue (LCF) analogy to the Basquinequation is the Coffin–Manson equation

Dep=2 ¼ ef 2Nð Þc ð33Þ

where Δεp/2 is the plastic strain amplitude, εf is thefatigue ductility coefficient, 2 N is the number of re-versals to failure and c is the fatigue ductility exponent.A validation study in uniaxial fatigue has been per-formed using an equivalent 0.1–3% strain amplitudeand displacement were applied on the top of the spe-cimen model to predict the number of cycles to failurein each prescribed displacement amplitude. The Cof-fin–Manson relation is a low-cycle fatigue strain clas-sification relating fatigue strain size to the number ofcycles where Δεp is the plastic strain range, and εf(fatigue ductility coefficient) and c (fatigue ductilityexponent) are material parameters. The implication isthat the plastic strain range is a straight line whenplotted in a log–log S–N-type diagram against thenumber of cycles.

It is also possible to combine the Basquin andCoffin–Manson equations into a single expression,covering the entire range of LCF and HCF:

Dea=2 ¼ rf=E 2Nfð Þbþef 2Nfð Þc ð34ÞThe first term in Eq. (34) represents the elastic strain,the second the plastic strain, and Δε is the total strainrange or strain amplitude.

Using the elasto-plastic law (Eq. (35)), the transientboundary value problem is solved in high cycle fatigueas

r x; tð Þ ¼ Cep x; tð Þ�e x; tð Þ ð35Þto retrieve the real elasto-plastic stress and strain.

To determine the elastic stress and strain in high-cycle fatigue, a linear elastic boundary problem with aconstitutive law of42

r x; tð Þ ¼ C xð Þe x; tð Þ ð36Þis considered.

The combined low and high cycle fatigue behaviourwould be considered as the relative combination of theBasquin and Coffin–Manson equation as an assump-tion of the trabecular bone could have to be isotropichomogenous continuum materials and transverselyisotropic materials that have an axis of symmetryalong the axis of the cylindrical specimen. Thus, inorder to estimate the fatigue life, Nf, and plastic strainunder the consistent condition, Eq. (37)

Dr ¼ 0:0035 Nfð Þ�0:096þDe ¼ 0:352 Nfð Þ�0:981 ð37Þwas used in this simulation under high and low cyclefatigue where Δεp is the plastic strain and Δσ is theamplitude of the applied stress. Details on materialparameters used in this study are tabulated in Table 2and “Appendix”.

FATIHHI et al.

RESULTS

The predicted fatigue failure and plastic strains ofthe trabecular bone model fall within the range of ex-perimental measurements from previous studies. Thesurface failure of the model was compared to the axialand multiaxial load of the hip contact force duringnormal gait walking with increasing cycles. The modelwas found consistent with previous experimentalfindings (see Fig. 4). A close agreement is presentedbetween the experimental data and the FE predictionsunder uniaxial compressive fatigue with a standarderror of 0.038. Due to the fact that under load control(which is used for the gait loading condition), the stressremained constant, stiffness reduced due to residualstress between load cycle 1 and next cycles in which thedisplacement control was prescribed on top of themodel surface. The displacement mean is less criticalwith respect to fatigue life or failure cycle. Underboundary conditions, both models were not restrainedagainst the transverse against displacement.

Figure 5 shows the uniaxial load under the influenceof a variable stress level applied to the effective plasticstrain in comparison to its multiaxial counterparts.The evolution of the plastic strain with multiaxial loadsproduces a higher range of increased strain with in-creasing stress than those with uniaxial loads. Thestrain increment was recorded to be on average be-tween 89.3 and 92.5% of effective plastic strain underminimum cycle to failure on multiaxial loads. At thispoint of the stress–strain curve the stiffness of thetrabecular model decreases with the increasing of stressas well as increasing number of cycles to failure.

Figure 6 shows differences between the total strain-life curve under uniaxial and multiaxial loads on thesame trabecular model. Obviously, the effective plasticstrain in both simulations decreases notably in the lowcycle fatigue region at the beginning of the fatiguescycles and reaches a gradual decrease at the maximumstress level. 10% increase of stress for both uniaxialand multiaxial loads would alter the fatigue behaviourat different regions of low and high cycle fatigue.Under the multiaxial load case, the structure is moreimminent to have low cycle numbers compared to the

TABLE 2. Paramater used in fatigue modelling.

Property Parameter Value Property group

Fatigue strength coefficient rf 0E corresponding to 7 MPa 0.0035 Basquin

Fatigue strength exponent b −0.096 Basquin

Fatigue ductility coefficient e0f 0.352 Coffin–Manson

Fatigue ductility exponent c −0.98 Coffin–Manson

Q Critical plane evaluation 3 NA

Initial yield stress σys0 8.4 (MPa) NA

Kinematic tangent modulus ETkin 0.05E0 NA

FIGURE 4. Comparison of finite element prediction with ex-perimental data from31 showing the relationship of appliedstrain on the cycles to failure. The effective plastic strains inthe finite element simulation were iteratively solved with in-creasing cycles from 1st to cycle to failure on the first surfaceelement. Gait load is taken from only resultant component offorces during normal walking. The small arrows show thestandard error calculated value between experiment and finiteelement simulation. The deviation of the simulated data fromthe actual experiment is approximately 4.5%.

FIGURE 5. Comparison of uniaxial loading and multiaxialloading relationship of applied stress to effective strain cor-responding to the vertical strain (z-direction). Mean shear/lateral strain were excluded.

Uniaxial and Multiaxial Fatigue Life Prediction

uniaxial load case which is dominant in the high cyclenumber of fatigue.

Table 3 shows the contour plot of cycles to failureand effective plastic strain with increasing cycles underuniaxial and multiaxial loading. The magnitude of

cycles to failure increased by about 40% under uni-axial loading compared to the multiaxial loadingcondition. Multiaxial load introduced higher stressconcentrations corresponding to the increase in plasticstrain under low load levels when compared to uniaxialload in which the model was intact even with higherfatigue cycles. Additionally, the failure behaviour withincreasing effective plastic strain under uniaxial andmultiaxial loading is notably different. Based on ourextensive literature study, we are not aware of anysimulation or experimental studies done on bulk tra-becular bone under multiaxial fatigue lifetime predic-tion. Thus, we placed our results in comparison toprevious studies41 for uniaxial ones.

In Fig. 7, the predicted Young’s moduli were com-pared for the cases of uniaxial and multiaxial loading.The applied stress level has a significant influence onthe degradation of the modulus which occurred uponfailure of the trabeculae. More reduction can be seenwith increasing apparent stress levels during fatiguelife. A considerable reduction of the Young’s modulusoccurs before the final catastrophic failure regardlessof the cycle level. Different trends in modulus reduc-tion can be observed between uniaxial and multiaxialloads at higher cycle numbers: the modulus reduction

FIGURE 6. Comparison of uniaxial and multiaxial loadingrelationship of applied stress to effective strain correspond tovertical direction changes of strain. The high-cycle fatigueregion (HCF) starts from N > 100 whereas the low-cycle fatigueregion (LCF) is in the range of 0 < N < 100.

TABLE 3. Contour of effective plastic strain predicted by FE simulations under uniaxial and multiaxial loading (final fatigueloading cycles taken from gait loading).

Load per gait loading (%)

Uniaxial Loading Multiaxial Loading

10

Nf = 37 Nf = 530

Nf = 171 Nf = 4550

Nf = 8943 Nf = 23090

Nf = 26464 Nf =6862

FATIHHI et al.

under multiaxial load is about ten times higher than inthe case of uniaxial loading.

Figure 8 shows the typical linear relationship ofbone volume fraction and SMI with cycles to failure inboth types of loading. A strong correlation for bothbone volume fraction and SMI (p < 0.05) exists. In-

crease of cycles to failure is associated with high vol-ume fraction and low SMI. Regression analysis forother microarchitectures was tabulated in Tables 4 and5 with a determination of coefficients and p-values forcycles to failure and effective plastic strain respectively.

Tables 4 and 5 list the coefficient values of microar-chitecture parameters and their relative differences in cy-cles to failure and effective plastic strain respectively. Allsamples have a strong correlation in cycles to failure withBV/TV and SMI regardless of the type of loading. Inmostmicroarchitectural parameters such as BV/TV, SMI, Tb.Th, and Conn.D, uniaxial loading showed better correla-tion in cycles to failure when compared to multiaxialloading. In contrast, Tb.Sp andDAhave good correlation(R2 > 0.50; p < 0.05) in the multiaxial loading case. Aninverse correlation is observed in the relation ofTb.Th andSMI with the effective plastic strain (p< 0.05).

DISCUSSION

The predicted values for the model were firstvalidated with previous experimental data for thestrain accumulation during each cycle loading andwere controlled with applied displacements between0.005 up to 0.15 mm (equivalent 0.1–3% strain) inorder to obtain reliable results for the 3D trabecularmodel. This corresponds to low strain to high strainwhich is consistent with the previous study.41 In com-parison to uniaxial loading, the model under multiaxialloading is deviated (approximately 16.18% standarderror from experimental data) the predictive valuemore than that of the uniaxial due to different pre-scribed boundary conditions which affected the strainresults. However, the uniaxial condition the boundarycondition had less effect on strain results. The re-strained model may affect the prediction of stress andstrain, and thus the failure state would be underesti-mated due to reduction of shielding effect on tra-becular struts at the site of failure. Under thiscondition of fatigue, the use of displacement is lesscritical than load controlled with respect to failurebehavior. Therefore, applied physiological gait loadingin this present study to assure a more reliable and saferprediction of fatigue lifetimes. In contrast, the use ofuniaxial load alone might be even dangerous due tounderestimated risk of fracture and over-estimatedfatigue lifetime.

In Fig. 5, the effect of multiaxial load is notablyunder large changes of total plastic strain with in-creasing cycles to failure when compared to uniaxialload. The linear curve of the uniaxial load reflects aslight decrease in axial stiffness compared to that of themultiaxial load with about 50% reduction with in-creasing cycles to failure.

FIGURE 7. Predicted normalized modulus with increment ofapplied stress corresponding to the number of cycles to fail-ure. Measurement of modulus based on equivalent stress andstrain tensor.

FIGURE 8. Linear relationship between BV/TV and cycles tofailure, Nf, (a) between SMI and cycles to failure, Nf, (b) num-ber of cycles to failure increased with BV/TV but decreasedwith SMI in both uniaxial and multiaxial loading.

Uniaxial and Multiaxial Fatigue Life Prediction

Figure 6 shows the total plastic strain-life curvebetween the experimental data from Moore et al.41

with close agreement to the predicted cycles to failure(N) in low cycle fatigue from this study with a slightover-estimation of cycles to failure after 1000 cycles inthe high cycle fatigue region. However, the same slopeis obtained with a separated case for high cycle fatigue.The material properties of the finite element modelwere assumed to be isotropic and homogenous and notanisotropic. Thus, the differences in the high cycle fa-tigue region could be eliminated by a different finiteelement model which considers the heterogeneity ofmaterial properties existing in the experimental sam-ples. Cyclic hardening observed in both load typescontributes to the decrease of plastic strain.15 Totalplastic strain obtained through the simulation could bemore lower than one to see in experimental study oncortical bone within the range of 2000–6000 με orsomewhere higher under multiaxial loading could bedependent on initial tissue properties applied on themodel. This error should be minimized with the con-sideration of inhomogeneity in initial value applied inthe models. Multiaxial loading gives 38% higher av-erage difference in comparison to that of uniaxial(p < 0.01). The multiaxial load further increases thetotal plastic strain when compared to uniaxial loading(see Table 3). This condition is attributed to the cyclichardening at an early stage of the cycle load and isprolonged to the high-cycle fatigue region under uni-axial load. In contrast, ‘cut-off’ of cycles to failure inlow-cycle fatigue regions takes place in the case ofmultiaxial loading. This off-axis load application wasreported to decrease the fatigue life in both human andbovine bone13,20 and also may increase the suscepti-bility to microdamage.59 Under the same situation, themultiaxial stress state contributes to the reduction offatigue life with the presence of shear effects undercircumferential x- and y-components of force super-imposed with axial loading. Through this study, stiff-ness reduction decreased about tenfold on averagefrom uniaxial to multiaxial loads. Shear stress has beenreported to reduce more than half of the maximumaxial stress in other experimental studies with corticalbone.58 The local shear stress within the trabecularstructure under multiaxial loading may lower and ac-celerate the cycles to failure and thus shorten the fa-tigue life. Even at a small magnitude, the combinationof x- and y-force components during gait analysis (seeFig. 2) results in a highly reduced load bearing capa-bility of the trabecular bone structure. This clearlydetermined the small effect of shear stress which mayfrequently deteriorate the trabecular architecture thusreducing stiffness in off-axis direction with respect touniaxial loading. The substantial heterogeneity of thetrabecular architecture may distort yielding and the

flow stress under different loading paths. This is re-sulting in oblique failure under both loading condi-tions and manifests a typical failure profile found inanisotropic materials.58 In fact, multiaxial loading ismore significant in the divertion of the failure pathduring traumatic fracture of long bones.44 However,no other similar study has reported the failure of thetrabecular bone under combination loading.

The modulus reduction under multiaxial loadingconditions is more significant to fatigue failure whencompared to uniaxial loads (see Fig. 7). Consistentreduction has been described elsewhere.40 This com-bination of axial load and torsional moment in mul-tiaxial loading altered the strain path and graduallyaffected the torsion with increasing of applied stress. Asignificant reduction of stiffness in every fatigue cycleleads to final microcracks.4,25 Ten percent apparentmodulus reduction under uniaxial loading in whichrapid failure occurred has been reported in only 1–2%of trabeculae failures.39 Under axial compressive fa-tigue, strain concentration has been observed to con-tribute to the deviation of apparent behavior in whichit appeared to be linear.12 In such, mixed mode stressand strain would deviate to more complex behaviourthan in uniaxial loading. Under multiaxial loading,rapid modulus reduction may first initiate microcrackin the trabecular structure than uniaxial loading, thusinteraction of mineral in fibrils is affected by a singleload upon yielding. This phenomenon is depicted bythe increase of strain in multiaxial loading which ishigher than that of the uniaxial loading.

Consolidation ofmineralizationdegrees in the samplesmodel may affect results of failure analysis as alterationsin mineral content give complex implications on bonemechanics,10 though major lack of understanding is laidout on its interaction. However, Currey11 has concludedthat the collagen content in trabecular bone is not worthyof note in terms of mechanical properties which wassupported by studies of Bailey et al.1 and Hommingaet al.31 Thus, more focus should be posed to evaluatepossible consequences of trabecular structural changes onthe mechanical properties of the trabecular bone. In thisstudy, the uniaxial and multiaxial with specific microar-chitectural parameters difference was used to predict thefailure and effective plastic strain. Traditionally, to di-agnose vertebral osteoporosis, bone mineral density(BMD) is measured. However, this method fails to ac-count the changes in the trabecular bone and quantifyhow this changes affect the quality of the trabecular.34,49

Age-related changes of the trabecular bone included adecrease in BV/TV and Conn.D, an increase in Tb.Sp, ashift from plate-like trabecula to a rod-like struc-ture.16,49,53

An estimation of plate or rod characteristic of thetrabeculae can be explained by measuring SMI. SMI is

FATIHHI et al.

highly dependent on the trabecular type (plate- to rod-like structure) thus the mechanical behaviour can bepredicted under both uniaxial and multiaxial loading.Change in the type of trabecular structure is well re-lated to age.14,22 This study shows a significant effect ofthis parameter to fatigue life which contributes to thestrength of the trabecular structure. In both loadingconditions, low-SMI demonstrated high endurancetowards fatigue loading. On the contrary, fatigue lifecycles decreased significantly in samples with high-SMI. However, increase in effective plastic strain isalso observed in some cases with low cycle fatigue. Asthe sample’s structure is dominated by rod-like tra-beculae, the ability of the sample to sustain prolongedloading is reduced.

BV/TV and Conn.D values affect the fatigue life ofthe trabecular bone under multiaxial loading sig-nificantly. Severe shear effect could be associated to thereduction of fatigue life in which alteration on tra-becular orientation can be seen. In comparison to thesamples under uniaxial loading, the horizontal tra-becular struts are less affected in which most of theload was bore on vertical struts. Thus, the samplefailed by bending. This condition is closely associated

with the thinning of the trabecular and an increase inanisotropy with increasing age in osteoporotic bone.However, under multiaxial loading, the trabecularstructure was loaded in multiple directions in whichresulting in failure of the struts irrespective to its ori-entation. High bone volume fraction demonstrated loweffective plastic strain corresponding to strain at fail-ure. The effective plastic strain is not dependent onbone volume fraction exclusively due to variations onthe degree of anisotropy, thus other relevant factors inmicroarchitectural changes should be accounted.

CONCLUSION

The intention of the fatigue analyses done in thisstudy was to compare the effects of multiaxial loadingunder normal walking conditions. Significant reductionin fatigue life under multiaxial loading is greater than60–80% when compared to the uniaxial one. Decreasedin cycles to failure of lower BV/TV, Tb.Th, and Conn.Din both loading conditions, while decreases in cycles tofailure of higher SMI is demonstrated. Cycles to failureand effective plastic strain do not depend on the degreeof anisotropy in both loading conditions, while weakcorrelation of cycles to failure and effective plastic strainin Conn.D is found for multiaxial loading. On average,applying multiaxial loads reduced the fatigue life morethan five times. Computational simulations from thisstudy reveal the importance to consider multiaxial loadin fatigue analyses in order to improve fatigue under-standing based on realistic load cases which were notstudied previously.

APPENDIX

Implementation of Gait Loading

A set of polynomial functions derived from Matlabrelative to gait loading during normal walking wereused to formulate the time dependent behaviour oftrabecular bone under fatigue analysis.

Fx ¼ �1E6 �0:3434t7 þ 1:1756t6 � 1:5667t5�

þ1:0239t4 � 0:3369t3 þ 0:0490t2 � 0:0013t

þ0:0002ÞFy ¼ �1E5 �1:1068t7 þ 3:8818t6 � 4:8999t5

�þ 2:4244t4 � 0:0797t3� 0:2734t2

þ 0:0542t� 0:0010ÞFz ¼ �1E5 � 2:9006t7 þ 7:0557t6� 3:5732t5

�� 3:5934t4 þ 4:4087t3� 1:6199t2

þ 0:2244t þ 0:0048Þ

TABLE 5. Determination of coefficient of microstructuralparameter of trabecular bone samples and p value relative toeffective plastic strain in both uniaxial and multiaxial loading

condition.

Parameter

Uniaxial Multiaxial

R2 p value R2 p value

BV/TV 0.409 0.082 0.361 0.129

Tb.Th (mm) 0.016 0.948 0.593 7.45e−3*Tb.Sp (mm) 0.095 0.424 0.175 0.474

BS/BV 0.467 0.438 0.435 0.063

DA 0.154 0.529 0.195 0.424

Conn.D (1/mm3) 0.093 0.705 0.019 0.938

SMI 0.509 0.026* 0.491 0.033*

* Significant difference with p value <0.05.

TABLE 4. Microstructural parameter of trabecular bonesamples with determination of coefficients (R2) and p value inrelation to cycles to failure, Nf in both uniaxial and multiaxial

loading condition.

Parameter

Uniaxial Multiaxial

R2 p value R2 p value

BV/TV 0.702 8.07e−4* 0.660 2.11e−3*Tb.Th (mm) 0.689 1.11e−3* 0.621 4.55e−3*Tb.Sp (mm) 0.404 0.086 0.671 0.002*

BS/BV 0.231 0.341 0.145 0.554

DA 0.148 0.545 0.382 0.107

Conn.D (1/mm3) 0.584 0.048* 0.348 0.144

SMI 0.737 3.19e−4* 0.673 1.59e−3*

* Significant difference with p value <0.05.

Uniaxial and Multiaxial Fatigue Life Prediction

The initial gait loading reduce to the trabecular loadingas the following;

Normal force, Fz = −1E5 (−2.1755t7 + 5.2918t6

− 2.6799t5 − 2.6951t4 + 3.3065t3 − 1.2149t2 + 0.1683t+ 0.0036)

Torsional moment, T = 1E3 (1.1352t7 − 3.9095t6

+ 5.1417t5 − 3.1659t4 + 0.8622t3 − 0.0542t2 − 0.0168t− 0.0003)

Thus, Txy = 1E3 (0.8514t7 − 2.9321t6 + 3.8563t5

− 2.3744t4 + 0.6467t3 − 0.0407t2 − 0.0126t − 0.0002)

Influencing Parameter in Fatigue Life Prediction

Figure 9 shows increasing cycles to failure withfunction of fatigue strength coefficient while otherparameters are constant. This figure also shows a smalldeviation (lower than 5%) of different Q value andpredetermined fatigue strength coefficient in all cases.Q values determines the number of evaluation pointsused in the search for the critical plane and thus itcontrols the computational time. A smooth transitionin the results indicates that the specified searchresolution for the critical plane is sufficient to correctlycapture the fatigue response. A mesh convergence alsohas been done on Fig. 10 to estimate accurate fatiguelife in low cycle fatigue and effective plastic strain andthis sensitivity analysis is consistent under static initialmodulus and fatigue properties. Details on the defini-tion of fatigue parameter are shown in Fig. 11. Fatiguestrength exponent, b and fatigue strength coefficient, σfnormally cover the entire range of high cycle fatiguethat can be explained in logarithmic increase of fatiguecycles. Both increase of these two parameter will in-crease fatigue failure prediction. In contrast to high

cycle fatigue, effect of fatigue ductility coefficient, ε ´f onlow cycle fatigue prediction is insignificant to plasticstrain. However the prediction of number of cycle tofailure is affected. Increasing value of fatigue ductilitycoefficient could result in higher prediction of fatiguecycle in which the same effect can be obtained by re-ducing the value of fatigue ductility exponent, c. In thisstudy, consistent initial fatigue parameter and coeffi-cients derived from the bovine bone and consider validin all type of analysis regarding to fatigue life predic-tion.

ACKNOWLEDGMENTS

This project was sponsored by the KementerianPendidikan Malaysia (KPM) through Grant scheme(R.J130000.7809.4F355). The authors would also liketo thank the Research Management Centre, UniversitiTeknologi Malaysia, for managing the project.

FIGURE 10. Mesh convergence analysis performed to obtainthe optimum number of elements in prediction of cycle tofailure and effective plastic strain.

FIGURE 9. Effect of search resolution, Q, on critical planesetting evaluation in fatigue analysis. The accuracy of thealgorithm is determined by the spacing of the points, whichcan be selected by the critical point evaluation as the searchresolution setting Q = (N + 1).

FIGURE 11. Typical fatigue life curve under combinationBasquin and Coffin–Manson.

FATIHHI et al.

REFERENCES

1Bailey, A. J., T. J. Sims, E. N. Ebbesen, J. P. Mansell, J.S. Thomsen, and L. Mosekilde. Age-related changes in thebiochemical properties of human cancellous bone collagen:relationship to bone strength. Calcif. Tissue Int. 65:203–210,1999.2Bayraktar, H. H., A. Gupta, R. Y. Kwon, P. Pa-padopoulos, and T. M. Keaveny. The modified super-el-lipsoid yield criterion for human trabecular bone. J.Biomech. Eng. 126:677–684, 2004.3Bergmann, G., F. Graichen, A. Rohlmann, A. Bender, B.Heinlein, G. N. Duda, M. O. Heller, and M. M. Morlock.Realistic loads for testing hip implants. Biomed. Mater. Eng.20:65–75, 2010.4Bowman, S. M., X. E. Guo, D. W. Cheng, T. M. Keaveny,L. J. Gibson, W. C. Hayes, and T. A. McMahon. Creepcontributes to the fatigue behavior of bovine trabecularbone. J. Biomech. Eng. 120:647–654, 1998.5Brandt, K. D., P. Dieppe, and E. L. Radin. Etiopatho-genesis of osteoarthritis. Rheum. Dis. Clin. N. Am. 34:531–559, 2008.6Burr, D. B., M. R. Forwood, D. P. Fyhrie, R. B. Martin,M. B. Schaffler, and C. H. Turner. Bone microdamage andskeletal fragility in osteoporotic and stress fractures. J.Bone Miner. Res. 12:6–15, 1997.7Burr, D. B., C. Milgrom, D. Fyhrie, M. Forwood, M.Nyska, A. Finestone, S. Hoshaw, E. Saiag, and A. Simkin.In vivo measurement of human tibial strains during vig-orous activity. Bone 18:405–410, 1996.8Carter, D. R., W. E. Caler, D. M. Spengler, and V.H. Frankel. Fatigue behavior of adult cortical bone: theinfluence of mean strain and strain range. Acta Orthop.52:481–490, 1981.9Carter, D. R., and D. M. Spengler. Mechanical propertiesand composition of cortical bone. Clin. Orthop. Relat. Res.135:192–217, 1978.

10Chen, P. Y., and J. McKittrick. Compressive mechanicalproperties of demineralized and deproteinized cancellousbone. J. Mech. Behav. Biomed. Mater. 4:961–973, 2011.

11Currey, J. D. Role of collagen and other organics in themechanical properties of bone. Osteoporos. Int. 14(Suppl 5):S29–S36, 2003.

12Dendorfer, S., H. J. Maier, and J. Hammer. Fatiguedamage in cancellous bone: an experimental approach fromcontinuum to micro scale. J. Mech. Behav. Biomed. Mater.2:113–119, 2009.

13Dendorfer, S., H. J. Maier, D. Taylor, and J. Hammer.Anisotropy of the fatigue behaviour of cancellous bone. J.Biomech. 41:636–641, 2008.

14Ding, M., A. Odgaard, F. Linde, and I. Hvid. Age-relatedvariations in the microstructure of human tibial cancellousbone. J. Orthop. Res. 20:615–621, 2002.

15Dutta, K., S. Sivaprasad, S. Tarafder, and K. K. Ray.Influence of asymmetric cyclic loading on substructureformation and ratcheting fatigue behaviour of AISI 304LNstainless steel. Mater. Sci. Eng. A 527:7571–7579, 2010.

16Eckstein, F., M. Fischbeck, V. Kuhn, T. M. Link, M.Priemel, and E.-M. Lochmuller. Determinants andheterogeneity of mechanical competence throughout thethoracolumbar spine of elderly women and men. Bone35:364–374, 2004.

17Freeman, M. A., W. H. Day, and S. A. Swanson. Fatiguefracture in the subchondral bone of the human cadaverfemoral head. Med. Biol. Eng. 9:619–629, 1971.

18Fyhrie, D. P., and D. R. Carter. Femoral head apparentdensity distribution predicted from bone stresses. J. Bio-mech. 23:1–10, 1990.

19Garcia, D., P. Zysset, M. Charlebois, and A. Curnier. Athree-dimensional elastic plastic damage constitutive lawfor bone tissue. Biomech. Model. Mechanobiol. 8:149–165,2009.

20George, W. T., and D. Vashishth. Damage mechanismsand failure modes of cortical bone under components ofphysiological loading. J. Orthop. Res. 23:1047–1053, 2005.

21George, W. T., and D. Vashishth. Susceptibility of aginghuman bone to mixed-mode fracture increases bonefragility. Bone 38:105–111, 2006.

22Gong, H., M. Zhang, L. Qin, K. K. Lee, X. Guo, and S.Q. Shi. Regional variations in microstructural properties ofvertebral trabeculae with structural groups. Spine 31:24–32,2006.

23Gronkiewicz, K., P. Majewski, G. Wisniewska, M. Pihut,B. W. Loster, and S. Majewski. Experimental research onthe possibilities of maintaining thermal conditions withinthe limits of the physiological conditions during intraoralpreparation of dental implants. J. Physiol. Pharmacol. 60(Suppl 8):123–127, 2009.

24Guillen, T., A. Ohrndorf, G. Tozzi, J. Tong, and H.J. Christ. Compressive fatigue behavior of bovine cancel-lous bone and bone analogous materials under multi-steploading conditions. Adv. Eng. Mater. 14:B199–B207, 2012.

25Guo, X.-D. E., T. A. McMahon, T. M. Keaveny, W.C. Hayes, and L. J. Gibson. Finite element modeling ofdamage accumulation in trabecular bone under cyclicloading. J. Biomech. 27:145–155, 1994.

26Gupta, A., H. Bayraktar, J. Fox, T. Keaveny, and P. Pa-padopoulos. Constitutive modeling and algorithmic im-plementation of a plasticity-like model for trabecular bonestructures. Comput. Mech. 40:61–72, 2007.

27Haddock, S. M., O. C. Yeh, P. V. Mummaneni, W.S. Rosenberg, and T. M. Keaveny. Similarity in the fatiguebehavior of trabecular bone across site and species. J.Biomech. 37:181–187, 2004.

28Harrigan, T. P., M. Jasty, R. W. Mann, and W. H. Harris.Limitations of the continuum assumption in cancellousbone. J. Biomech. 21:269–275, 1988.

29Hill, R. A theory of the yielding and plastic flow ofanisotropic metals. Proc. R. Soc. Lond. A Math. Phys. Sci.193:281–297, 1948.

30Homminga, J., B. R. McCreadie, T. E. Ciarelli, H. Wei-nans, S. A. Goldstein, and R. Huiskes. Cancellous bonemechanical properties from normals and patients with hipfractures differ on the structure level, not on the bone hardtissue level. Bone 30:759–764, 2002.

31Homminga, J., B. R. McCreadie, T. E. Ciarelli, H. Wei-nans, S. A. Goldstein, and R. Huiskes. Cancellous bonemechanical properties from normals and patients with hipfractures differ on the structure level, not on the bone hardtissue level. Bone 30:759–764, 2002.

32Kohles, S. S., J. B. Roberts, M. L. Upton, C. G. Wilson, L.J. Bonassar, and A. L. Schlichting. Direct perfusion mea-surements of cancellous bone anisotropic permeability. J.Biomech. 34:1197–1202, 2001.

33Kosmopoulos, V., C. Schizas, and T. S. Keller. Modelingthe onset and propagation of trabecular bone microdamageduring low-cycle fatigue. J. Biomech. 41:515–522, 2008.

34Legrand, E., D. Chappard, C. Pascaretti, M. Duquenne, S.Krebs, V. Rohmer, M.-F. Basle, and M. Audran. Trabe-cular bone microarchitecture, bone mineral density, and

Uniaxial and Multiaxial Fatigue Life Prediction

vertebral fractures in male osteoporosis. J. Bone Miner. Res.15:13–19, 2000.

35Lemaitre, J., and J. L. Chaboche. Mechanics of SolidMaterials. Cambridge: Cambridge University Press, 1990.

36Lotz, J. C., E. J. Cheal, and W. C. Hayes. Fracture pre-diction for the proximal femur using finite element models:part I-linear analysis. J. Biomech. Eng. 113:353–360, 1991.

37Lotz, J. C., E. J. Cheal, and W. C. Hayes. Fracture predictionfor the proximal femur using finite element models: part II-nonlinear analysis. J. Biomech. Eng. 113:361–365, 1991.

38Lotz, J. C., E. J. Cheal, and W. C. Hayes. Stress distribu-tions within the proximal femur during gait and falls: im-plications for osteoporotic fracture. Osteoporos. Int. 5:252–261, 1995.

39Makiyama, A. M., S. Vajjhala, and L. J. Gibson. Analysisof crack growth in a 3D Voronoi structure: a model forfatigue in low density trabecular bone. J. Biomech. Eng.124:512–520, 2002.

40Michel, M. C., X. D. Guo, L. J. Gibson, T. A. McMahon,and W. C. Hayes. Compressive fatigue behavior of bovinetrabecular bone. J. Biomech. 26:453–463, 1993.

41Moore, T. L., and L. J. Gibson. Fatigue of bovine tra-becular bone. J. Biomech. Eng. 125:761–768, 2003.

42Ottosen, N. S., and M. Ristinmaa. The Mechanics ofConstitutive Modeling. London: Elsevier Science, 2005.

43Pattin, C. A., W. E. Caler, and D. R. Carter. Cyclic me-chanical property degradation during fatigue loading ofcortical bone. J. Biomech. 29:69–79, 1996.

44Peterson, D. L., J. S. Skraba, J. M. Moran, and A.S. Greenwald. Fracture of long bones: rate effects undersingular and combined loading states. J. Orthop. Res. 1:244–250, 1984.

45Schaffler, M. B., K. Choi, and C. Milgrom. Aging andmatrix microdamage accumulation in human compactbone. Bone 17:521–525, 1995.

46Shim, V. P. W., L. M. Yang, J. F. Liu, and V. S. Lee.Characterisation of the dynamic compressive mechanicalproperties of cancellous bone from the human cervicalspine. Int. J. Impact Eng. 32:525–540, 2005.

47Silva, M. J., T. M. Keaveny, and W. C. Hayes. Loadsharing between the shell and centrum in the lumbar ver-tebral body. Spine 22:140–150, 1997.

48Simo, J. C. Algorithms for static and dynamic multiplica-tive plasticity that preserve the classical return mappingschemes of the infinitesimal theory. Comput. Methods Appl.Mech. Eng. 99:61–112, 1992.

49Stauber, M., and R. Muller. Age-related changes in tra-becular bone microstructures: global and local mor-phometry. Osteoporos. Int. 17:616–626, 2006.

50Stone, J. L., G. S. Beaupre, and W. C. Hayes. Multiaxialstrength characteristics of trabecular bone. J. Biomech.16:743–752, 1983.

51Syahrom, A., M. Abdul Kadir, J. Abdullah, and A. Och-sner. Mechanical and microarchitectural analyses of can-cellous bone through experiment and computer simulation.Med. Biol. Eng. Comput. 49:1393–1403, 2011.

52Teo, J. C. M., K. M. Si-Hoe, J. E. L. Keh, and S. H. Teoh.Correlation of cancellous bone microarchitectural pa-rameters from microCT to CT number and bone me-chanical properties. Mater. Sci. Eng. C 27:333–339, 2007.

53Thomsen, J. S., E. N. Ebbesen, and L. I. Mosekilde. Age-related differences between thinning of horizontal andvertical trabeculae in human lumbar bone as assessed by anew computerized method. Bone 31:136–142, 2002.

54Tomar, V. Insights into the effects of tensile and com-pressive loadings on microstructure dependent fracture oftrabecular bone. Eng. Fract. Mech. 76:884–897, 2009.

55Turner, M. S. The association between tibial torsion andknee joint pathology. Clin. Orthop. Relat. Res. 302:47–51,1994.

56van Lenthe, G. H., M. Stauber, and R. Muller. Specimen-specific beam models for fast and accurate prediction ofhuman trabecular bone mechanical properties. Bone39:1182–1189, 2006.

57Varvani-Farahani, A., and H. Najmi. A damage assess-ment model for cadaveric cortical bone subjected to fatiguecycles. Int. J. Fatigue 32:420–427, 2010.

58Vashishth, D., K. E. Tanner, and W. Bonfield. Fatigue ofcortical bone under combined axial-torsional loading. J.Orthop. Res. 19:414–420, 2001.

59Wang, X., J. Guyette, X. Liu, R. K. Roeder, and G.L. Niebur. Axial-shear interaction effects on microdamagein bovine tibial trabecular bone. Eur. J. Morphol. 42:61–70,2005.

FATIHHI et al.