D. Carnelli. D. Gastaldi V. Sassi R. Contro. C. Ortiz. P. Vena. 1 Introduction

Transcription

1 D. Carnelli Department of Structural Engineering, Laboratory of Biological Structure Mechanics (LaBS), Politecnico di Milano, Italy; Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA D. Gastaldi V. Sassi R. Contro Department of Structural Engineering, Laboratory of Biological Structure Mechanics (LaBS), Politecnico di Milano, Italy C. Ortiz Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA P. Vena Department of Structural Engineering, Laboratory of Biological Structure Mechanics (LaBS), Politecnico di Milano, Italy; IRCCS, Istituto Ortopedico Galeazzi, Milano, Italy A Finite Element Model for Direction-Dependent Mechanical Response to Nanoindentation of Cortical Bone Allowing for Anisotropic Post-Yield Behavior of the Tissue A finite element model was developed for numerical simulations of nanoindentation tests on cortical bone. The model allows for anisotropic elastic and post-yield behavior of the tissue. The material model for the post-yield behavior was obtained through a suitable linear transformation of the stress tensor components to define the properties of the real anisotropic material in terms of a fictitious isotropic solid. A tension-compression yield stress mismatch and a direction-dependent yield stress are allowed for. The constitutive parameters are determined on the basis of literature experimental data. Indentation experiments along the axial (the longitudinal direction of long bones) and transverse directions have been simulated with the purpose to calculate the indentation moduli and the tissue hardness in both the indentation directions. The results have shown that the transverse to axial mismatch of indentation moduli was correctly simulated regardless of the constitutive parameters used to describe the post-yield behavior. The axial to transverse hardness mismatch observed in experimental studies (see, for example, Rho et al. [1999, Elastic Properties of Microstructural Components of Human Bone Tissue as Measured by Nanoindentation, J. Biomed. Mater. Res., 45, pp ] for results on human tibial cortical bone) can be correctly simulated through an anisotropic yield constitutive model. Furthermore, previous experimental results have shown that cortical bone tissue subject to nanoindentation does not exhibit piling-up. The numerical model presented in this paper shows that the probe tip-tissue friction and the post-yield deformation modes play a relevant role in this respect; in particular, a small dilatation angle, ruling the volumetric inelastic strain, is required to approach the experimental findings. DOI: / Keywords: cortical bone, indentation, finite element, anisotropic yield function 1 Introduction The technique of nanoindentation allows for the mechanical characterization of bone tissue at small length scales, thereby providing a pathway to explore the relationship between mechanical properties and fundamental structural constituents 1 6. It also holds potential for use as a diagnostic tool 7. Traditional methods of estimating material properties from indentation data include the Oliver Pharr contact mechanical analytical formulation 8, which defines an indentation modulus that, in the case of an isotropic, elastic half-space continuum, can be explicitly related to the elastic constants. Since bone is a hierarchical multiscale material, homogenized responses are achieved at sufficiently large length scales while the mechanical properties of distinct microand nanostructural features can be probed at smaller length scales. In addition to possessing a hierarchical structure, bone is also anisotropic 9. Mineralized collagen fibrils, the fundamental building blocks of bone, are structurally anisotropic 10, and Contributed by the Bioengineering Division of ASME for publication in the JOUR- NAL OF BIOMECHANICAL ENGINEERING. Manuscript received July 20, 2009; final manuscript received February 18, 2010; accepted manuscript posted March 1, 2010; published online June 18, Assoc. Editor: Ellen M. Arruda. hence, parallel fibered bone exhibits a transverse normal to fiber axis /axial parallel to fiber axis elastic modulus ratio Osteonal bone is less anisotropic, exhibiting a transverse r =normal to Haversian canal long bone axis /axial z=parallel to Haversian canal long bone axis /elastic modulus ratio ,13. The latter observation can be attributed to the fact that in osteonal bone, mineralized collagen fibrils bundle into layers with varying directions to form a plywood-like structure unit, as well as rotate around their axes within these layers 10,14. Cortical bone is also known to exhibit a tension/compression yield asymmetry 15, which is associated to a pressure-dependent mechanical behavior Analysis of nanoindentation data using more refined constitutive laws and finite element analyses FEA can provide deeper insights into these unique structural and mechanical features of the bone tissue Previously reported literature in this area includes the use of elastic anisotropy, plastic pressure-independent anisotropy, and strain hardening 20 and, separately, a nanogranular pressure-dependent yield models, such as the Drucker Prager 17 or the Mohr Coulomb models 18, capturing the known tension/compression yield stress mismatch. Another study 21, utilized a viscoelastic-plastic, isotropic, constitutive model based Journal of Biomechanical Engineering Copyright 2010 by ASME AUGUST 2010, Vol. 132 /

2 on a spring-dashpot approach to viscoelasticity and on the Ramberg Osgood relationship for the isotropic post-yield behavior. While these works have provided significant new insights into bone indentation behavior, accurate predictions of directiondependent indentation moduli 23,1, direction-dependent hardness 1, and negligible pile-up 1,17 simultaneously still remain a challenge. In this paper, we approach this problem by formulating a FEA model which captures elastic and post-elastic anisotropy, as well as pressure-dependent yield tension/compression asymmetry of cortical bone tissue. The elastic behavior of the tissue is modeled through the transversely isotropic elastic tensor 24 and the postelastic behavior is introduced via a yield surface and a flow rule obtained by a suitable stress transformation of the Drucker Prager surface by means of a stress transformation rule to account for material anisotropy. A tension/compression yield stress mismatch, typical of the cortical bone tissue observed at the macroscopic scale 15,16, is accounted for. Parametric studies on the degree of tissue anisotropy, the friction between the indenter and tissue, and the amount of volumetric inelastic strain governed by the dilatation angle in a nonassociative plastic flow rule were carried out. Load versus penetration depth was predicted and the Oliver Pharr analysis was used to estimate indentation properties. The transverse/axial indentation moduli E T /E A, the transverse/axial hardness values H T /H A, as well as the pile-up, are predicted and compared with experimental data on cortical bone reported in literature 1, Methods 2.1 Elastic Anisotropic Model for Cortical Bone. Cortical bone is modeled as a homogeneous continuum. A transversely isotropic elastic model is used and obtained through a suitable transformation of equations for isotropic linear elasticity 24. Using indicial notation, the stress-strain relationship for a linear elastic, isotropic material in the small deformation regime is ij = kk ij +2 e ij 1 and the relevant stiffness tensor is C ijkl = ij kl +2 I ijkl 2 where ij is the Cauchy stress tensor, e ij is the deviatoric component of the deformation tensor, kk is the specific volumetric deformation summation over index k is assumed, and are the Lamè constants, ij is the Kronecker symbol, and I ijkl is the fourth-order unit tensor. I ijkl = 1 2 ik jl + il jk In order to transform the above equations into a set of equations for anisotropic materials, three perpendicular unit vectors g 1, g 2, and g 3 are introduced having the physical meaning of principal material directions; furthermore, the second-order tensor G ij is introduced as follows: G ij = p 1 g 1 i g 1 j + p 2 g 2 i g 2 j + p 3 g 3 3 i g j 4 where p 1, p 2, and p 3 represent the weights of the three directions in terms of elastic properties. We simulate an anisotropic material by substituting the second-order identity tensor ij with a new second-order tensor I ij = p 0 ij + G ij 5 so the anisotropic elastic constitutive law now reads C ijkl = I ij I kl + I ik I jl + I il I jk 6 In matrix notation it reads 3 C = 2 r r 1 r 1 r 2 r 1 r r 1 r 2 2 r r 2 r 2 r r 1 r 3 r 2 r 3 2 r r r 1 r r 1 r r 2 r 3 with r i = p 0 + p i 8 The constraint 3 i=0 p i =1 is also introduced in order to obtain a set of independent constitutive parameters. The value p 0 =1 has been assumed. For transversely isotropic simulations, p 1 = p A and p 2 = p 3 = p T, where the subscripts A and T refer to the axial and transverse directions, respectively. In the finite deformation regime, the above expounded elastic stress-strain relationships are used in incremental form ij =C ijhk hk with suitable rotations of the principal material directions g i during the deformation process. 2.2 Yield Locus for Anisotropic Tissue. The post-elastic mechanical behavior of the bone tissue was modeled using an elasticplastic constitutive law with anisotropic yield locus and a nonassociative plastic flow rule. Traditional procedures for deriving the constitutive equations for anisotropic elastoplastic materials are based on the description of appropriate yield and potential functions in terms of the characteristic material properties and principal directions. The satisfaction of invariance conditions in these cases can be difficult 25. A procedure to guarantee that these conditions are satisfied is to define the properties of the real anisotropic material in terms of a fictitious isotropic solid. This is achieved by relating the stress between the real and fictitious spaces using a linear tensor transformation 26. In this work, an extension of the Drucker Prager plasticity isotropic model 27 to the anisotropic case has been formulated by defining a suitable stress tensor projection into the anisotropic space allowing for material principal directions g. The proposed modified Drucker Prager model is able to account for the tension/ compression yield stress mismatch exhibited by the cortical bone tissue under macroscopic tension/compression mechanical loading 15,28. The plasticity domain for an isotropic material can be written in the general form as ij =0 The second-order tensor M 2 is introduced, the components of which depends on the principal material directions g as follows: / Vol. 132, AUGUST 2010 Transactions of the ASME

3 M 2 ij = q 0 ij + q 1 g 1 i g 1 j + q 2 g 2 i g 2 j + q 3 g 3 3 i g j 10 in which a proper weight is assigned to each material direction through the parameters q i. A new stress tensor ij projected into the anisotropic space is introduced ij = M 4 ijkl kl 11 in which the fourth-order tensor M 4 is defined as 4 M ijkl = 1 2 M 2 ikm 2 jl + M 2 il M 2 jk 12 The new yield locus for the anisotropic model is therefore expressed in a general form as ij =0 13 The yield locus of the modified Drucker Prager plasticity model for the anisotropic bone tissue is as follows: ij =3 3 p + eq =0 14 in which the projected stress components p and are defined as p = 1 3 ii; = 3 2 S ijs ij 15 and and eq are the material parameters; whereas S ij is the deviatoric component of the Cauchy stress tensor ij. The parameter rules the tension/compression yield stress ratio; whereas, the parameter eq depends on the tensile or compressive yield stress or elastic limit of the tissue. For sake of simplicity, a perfect elastic-plastic material model has been assumed; that is to say no strain hardening has been introduced. The flow rule, which determines the irreversible strain plastic strain within the tissue is as follows: d ij p = d Q pq pq ij 16 in which the plastic potential Q has been introduced. Q =3 3 def p + 17 where def is the dilatation angle. In the case in which def =, the associative plastic flow rule is obtained. In the above flow rule, d is a scalar parameter that can be obtained by solving the nonlinear set of equations Eq. 16, the stress-strain incremental relationship d =C d d p and the consistency condition d =0. These equations are solved by means of the iterative Newton Raphson procedure applied at each integration point and at each time increment during the finite element simulation. To this purpose, the standard computational procedures for elastic-plastic finite element models have been adopted in which consistent tangent stiffness matrix has been computed to solve for the displacement vector see, for example, Ref. 29. In the specific case, a transversely isotropic elastic-plastic behavior has been used in which the plane of isotropy of the tissue is perpendicular to the axial direction for long bones by making the following assumptions: q 1 = q A ; q 2 = q 3 = q T 18 Under these assumptions the stress components p and are p = q 0 + q A q 0 + q T 2 19 = 2 1 q 0 + q A q 0 + q T q in which 4 q 123 = q q A +2q T q q 2 A + q 2 T +4q A q T q q A q 2 T + q 2 A q T q 0 + q 2 2 A q T 21 The tensile and compressive yield stress along the axial direction t A, c A can be obtained by setting 2 = 3 =0. t eq 1 A = 1+ 3 q 0 + q A 2 ; A c eq 1 = 1+ 3 q 0 + q A 2 22 whereas the tensile and compressive yield stress along the transverse direction t T, c T can be obtained by setting 1 = 3 =0 t eq 1 T = 1+ 3 q 0 + q T 2 ; T c eq 1 = 1+ 3 q 0 + q T 2 23 The tensile to compressive yield stress ratio turns out to be independent from the material direction and ruled by the material parameter only. R tc = t c = and the axial to transverse yield stress ratio is as follows: R AT = A t t = c A c T = q 0 + q T 2 T q 0 + q A 2 25 Similar to the elastic formulation, the conditions q 0 =1; q 0 + q A +2q T =1 26 are adopted, thus only one of the q i material parameters results to be independent. 2.3 Finite Element Model of Indentation Tests. The virtual indenter employed was axisymmetric, conical with an angle of 70.3 deg and the same area-to-depth ratio as a three sided Berkovich pyramid. The indenter was modeled as perfectly rigid since it is composed of diamond, which is much stiffer than bone E diamond =1141 GPa; E bone 20 GPa. The indentations along the axial, as well as transverse directions, were simulated. Given the transversely isotropic constitutive behavior of cortical bone, an axisymmetric model was employed for indentation along the axial direction. For indentations along the transverse direction, a three-dimensional model was created. In the latter case, only a quarter of the bone tissue/indenter system was modeled by exploiting the symmetry of the material at 90 deg and the axisymmetry of the conical indenter. In both models the bone sample is represented as cylinder of 100 m in both height and radius Figs. 1 and 2. Suitable mesh refinement was applied under the indenter tip, where large strain gradients occur. The twodimensional mesh had 12,658 quadratic elements of both types CAX6 and CAX8. The characteristic element size in the refined region was 70 nm see Fig. 2. In order to simulate the effect of an infinite solid domain in the region far from the indenter tip, linear infinite elements were used 30. The three-dimensional finite element mesh had 57,600 eight-node, hexahedral elements. The el- Journal of Biomechanical Engineering AUGUST 2010, Vol. 132 /

4 Fig. 3 Sketch of deformed surface under the indenter. The measure of the sink-in pile-up is shown. Fig. 1 View of the two-dimensional finite element mesh used for axial indentation simulations with detail of discretization under the conical indenter; the left side represents the axis of symmetry ement characteristic length in the refined region was 30 nm. The analyses were conducted using the commercial finite element code ABAQUS/STANDARD 30. Due to large displacement and displacement gradients that occur during an indentation test, large strains, and large rotations are expected, therefore the large deformation theory was used in the numerical models. The loading and unloading phases of the experiments are simulated by applying the displacement and repositioning the indenter to its original position in two separate steps. The frictional interaction between the indenter and bone tissue was modeled by means of a shear stress proportional to the contact pressure, which accounts for the Coulomb friction between the surfaces and by means of a hard contact algorithm based on Lagrange multipliers to account for the contact along the direction normal to the surfaces. The anisotropic elastoplastic constitutive model here presented is implemented into the FORTRAN subroutine UMAT and linked to the commercial finite element code ABAQUS Calculation of Indentation Moduli and Tissue Hardness From Numerical Simulations of the Indentation Tests. In order to compare with experimental data reported in literature, the Oliver Pharr method 8 was employed to calculate the indentation moduli E and hardness H in the axial and transverse directions from the FEA simulated unloading force P versus displacement h data. This method is based on the assumption that the initial part of the P h unloading curve is unaffected by inelastic phenomena. The method is applied by fitting the top 60% of the unloading curve using the following fitting equation: P = A h h r n 27 in which P is the load applied on the indenter, where h r, A, and n are the fitting parameters. The parameter h r has the physical meaning of residual depth. The indentation modulus E and indentation hardness H are therefore calculated as E = S 2 A p 28 H = P max 29 A p respectively, in which S is the contact stiffness dp/ dh calculated by differentiating Eq. 28 at P= P max. S = An h h r n 1 30 and A p h p is the projected contact area calculated at maximum indentation load P max, which is obtained from the finite element solution. For a conical indenter A p = 2 in which is the radial coordinate of the nodal point in contact with the rigid surface see Fig. 3. Calculations for reduced moduli and hardness were carried out for axial as well as transverse indentation simulations. 2.5 Numerical Examples. The parameter sets used in the numerical simulations have been chosen with the purpose to assess the role of anisotropy of yield locus, the deformation modes associated to inelastic phenomena nonassociative flow rule, and the friction between the indenter and the tissue surface. Furthermore, indenter-tissue friction was included in the FEA simulations because of its relevant role in predicting the absence or very limited residual pile-up observed experimentally after nanoindentation 17,20. The parameters used in the FEA simulations are summarized Table 1. Four sets of analyses have been carried out by changing one parameter at a time see Table 1 ; in particular, the analysis set R AT A refers to simulations of axial indentation in which the ratio Table 1 Values of the simulation parameters for the four sets of numerical analyses Analysis set def a c c T / A Fig. 2 View of the three-dimensional finite element mesh used for simulations of transverse indentations; only the detail under the indenter is shown. Axial direction is the z axis, transverse directions are: r indentation direction and c Fr A def A A AT R A AT R T / Vol. 132, AUGUST 2010 Transactions of the ASME

5 Table 2 Elastic constants employed in the numerical simulations GPa GPa p T p A p Fig. 4 Transverse to axial yield stress ratio versus constitutive parameter q A. Dashed line represents an isotropic yield function. c T / c A has been chosen in the interval 1 0.5, consistent with reported literature 15,28, where c T / c A =1 represents the case of isotropic yield locus. The constitutive parameter q A has been selected on the basis of Eq. 25 such as a prescribed value of the transverse to axial yield stress ratio is obtained see Fig. 4. The constitutive parameter eq was chosen such that axial compressive yield stress is c A = 182 MPa for all analyses, whereas the transverse compressive stress changes according to the value of R AT see Fig. 5. The constitutive parameter is set to =0.13, which corresponds to a fixed tension to compression yield stress ratio R tc =0.63 and to an axial tensile yield stress t A =115 MPa, as reported in Ref. 15 for cortical human femur; the dilatation angle def was set to 0.1. These tensile yield data are consistent with those reported in Refs. 31,32, which refers to cortical bovine femur data; a slightly lower value was reported for human cortical femur. Similarly, the analysis set R AT T refers to simulations of transverse indentation tests. All the above analyses are carried out for a =0. In the analysis set Fr A, the friction coefficient between the indenter and the tissue surface was varied between In the last analysis set A def the nonassociative flow rule for plastic deformation has been used by exploring different values of the dilatation angle def in Eq. 17. In these analyses, a friction contact between the indenter and the tissue has been assumed a =0.2. The analysis parameters for each of the above expounded sets are provided in Table 1. The elastic constants were kept unchanged for all simulations; the parameters,, and p i, reported in Table 2, correspond to the following elastic tensor entries units in GPa : C 11 = 21.24; C 12 = 14.35; C 13 = C 22 = 30.91; C 44 = 5.63; C 55 = The above values have been obtained by fitting the model parameters,, p r, and p a to the elastic constants reported in Ref. 15 for human cortical bone. All analyses were carried out in a displacement control mode in which a full loading-unloading indentation experiment is simulated. The indenter, initially at contact with the tissue surface, was displaced downward up to a maximum displacement of 210 nm on loading and thereafter reversed and displaced back to its original position on unloading. The constitutive model does not account for time dependent phenomena, therefore loading rate was not an analysis parameter. 3 Results The fitting parameters A, h r, and n as well as the maximum indentation force are reported in Table 3 for all simulations. Table 3 Fitting parameters n, A, and h r see Eq. 28 of the unloading P h curves and maximum indentation force P max for all the performed simulations n A mn nm n h r nm P max mn Fig. 5 Compressive yield stress along axial and transverse directions versus constitutive parameter q A. The parameter eq has been suitably chosen so that an axial compressive yield stress of 182 MPa is obtained for all values of q A. R AT A c T / c A = R AT A c T / c A = R AT A c T / c A = R AT A c T / c A = R AT A c T / c A = R AT A c T / c A = R AT T c T / c A = R AT T c T / c A = R AT T c T / c A = R AT T c T / c A = R AT T c T / c A = R AT T c T / c A = Fr A a = Fr A a = Fr A a = Fr A a = Fr A a = A def A def = A def A def = A def A def = A def A def = Journal of Biomechanical Engineering AUGUST 2010, Vol. 132 /

6 Fig. 6 Indentation moduli for simulations along axial and transverse indentations versus transverse to axial yield stress ratio 3.1 Effect of the Transverse to Axial Yield Stress Ratio T c Õ A c =0.5\1. The axial E A and transverse E T indentation moduli for values of T c / A c ranging from 0.5 to 1 are reported in Fig. 6. Consistent with results reported in literature 1, a higher indentation modulus was obtained in the axial direction GPa compared with the transverse direction GPa. Limited sensitivity to the yield stress ratio was observed for both the directions. The axial and transverse hardness as a function of the ratio T c / A c are reported in Fig. 7. Similar to the indentation moduli, the axial hardness values were also found to be higher in the axial direction compared with the transverse indentation simulations. However, hardness exhibited a high sensitivity to the transverse to axial yield stress ratio. In particular, axial hardness was found to range from about 0.90 GPa for T c / A c =1 i.e., isotropic yield locus to 0.72 GPa for T c / A c =0.5; whereas, transverse hardness was found to range from 0.90 GPa for T c / A c =1 to 0.51 GPa for T c / A c =0.5. Furthermore, the transverse to axial hardness ratio H T /H A is also highly sensitive to the parameter T c / A c see Fig. 8. The solid line in Fig. 8 represents the case of H T /H A = T c / A c. The results show that for T c / A c 0.8 the hardness ratio is higher than the yield stress ratio an amplification effect ; whereas, for T c / A c 0.8, H T /H A = T c / A c. In particular, for an isotropic yield Fig. 7 Indentation hardness for simulations along axial and transverse indentations versus transverse to axial yield stress ratio Fig. 8 Transverse to axial indentation moduli and hardness ratios versus transverse to axial yield stress ratio locus, H T /H A =0.98 has been found, indicating that the elastic anisotropy is not sufficient to justify the direction-dependent indentation hardness observed in experiments. Figure 9 shows contour maps of the inelastic shear strain p rz under the indenter for three reference cases isotropic yield locus, c T / c A =0.8, and c T / c A =0.5. From inspecting the contour maps of p rz, it can be observed that for high yield stress transverse to axial mismatch, the inelastic shear strain magnitude and its spatial gradient along p the radial direction increase see also Fig. 10 for line plots of rz as a function of the distance from the indentation point. This leads to a higher contact depth with respect to the isotropic case, resulting in an increase in the contact area, which in turn decreases the hardness. It must be kept in mind that, in this parametric study, the axial yield stress is kept constant; whereas, the transverse yield stress is not; therefore transverse hardness is continuously decreasing with c T / c A consistently with the decrease in c T. For comparison purposes, Fig. 8 reports also the transverse to axial indentation moduli ratio; it can be observed that the indentation moduli ratio is largely unaffected by the yield stress ratio. The indentation simulations allowed the calculation of the pile-up at a maximum penetration depth of 210 nm Fig. 11. According to the sketch provided in Fig. 3, the pile-up here is measured as the vertical displacement of the external boundary of the contact area with respect to the original surface level. Positive values of pile-up indicate that the material rises with respect to the undeformed surface typical for ductile materials ; whereas, negative values indicate that the material sinks-in with respect to the undeformed surface. This latter case is typical of ceramic brittle materials with a low elasticity to yield stress ratio. The set of analyses R AT A shows an increasing pile-up for decreasing transverse to axial yield stress ratio. A maximum pile-up of 40 nm has been found for c T / c A =0.5 Fig. 11 curve labeled a = Effects of the Inelastic Deformation Mode and of Friction. The effect of friction between indenter and tissue surface was investigated in the analyses set Fr A see Table 1 in which axial indentations were simulated. It was determined that friction affects both the pile-up see Fig. 11 and the indentation hardness see Fig. 12. The increase in friction coefficient substantially decreases the pile-up for all values of T c / A c Fig. 11. For friction coefficients a 0.1 and for T c / A c 0.75, sink-in is observed. Simultaneously, increases in tissue hardness are observed up to a =0.2 see Fig. 12 ; for larger values of the a no further hardness increases take place data not shown. Notably, friction does not affect the axial indentation modulus see Fig. 12. Furthermore, the contribution of the inelastic deformation mode is required to avoid piling-up of the material for T c / A c =0.5. The / Vol. 132, AUGUST 2010 Transactions of the ASME

7 Fig. 9 Contour maps of the inelastic strain rz component p rz, where z and r are the directions parallel and perpendicular to the long bone axis direction, respectively in an axial indentation simulation for: c T / c A =1 left, c T / c A =0.8 center, and c T / c A =0.5 right set of analyses A A def have shown that the inelastic deformation mode, governed by the parameter def, has a moderate effect on the axial hardness and indentation moduli Fig. 13 ; whereas, it strongly affects the pile-up Fig. 14. Indeed, as shown in Fig. 14, the pile-up decreases markedly with decreasing def. For def 0.09, sink-in effect is observed. In this case, both directiondependent hardness Fig. 8 and absence of piling-up of the material Fig. 14 are simultaneously observed. 4 Discussions and Conclusions 4.1 Origins of Direction-Dependent Response. The transversely isotropic elasticity assumption is a primary determinant of the observed indentation moduli anisotropy ratio Fig. 8 compared with Eq. 31. It was confirmed that the degree of plastic anisotropy including the isotropic yield locus had little effect on the direction-dependent elastic moduli Fig. 6, which is attributed to the fact that little plasticity takes place during the initial stage of the unloading P h curve. The slight dependence of the indentation moduli on the transverse to axial yield stress ratio is attributed to differences in the computed contact area at maximum load. It was also shown that anisotropic post-elastic properties, i.e., anisotropic yield locus and plastic potential, are required in order Fig. 10 Plots of the inelastic strain p rz component as a function of the distance from the indentation point on the surface for: isotropic yield locus model, c T / c A =0.8, and c T / c A =0.5 Fig. 12 Axial hardness versus friction coefficient left vertical axis for def =0.1; axial hardness versus friction coefficient right vertical axis for def =0.1 Fig. 11 Pile-up versus transverse to axial yield stress ratio for three different values of tissue-probe friction coefficient Fig. 13 Axial hardness left vertical axis versus def ; axial indentation modulus right vertical axis versus def Journal of Biomechanical Engineering AUGUST 2010, Vol. 132 /

8 Fig. 14 Axial indentations: pile-up versus def for a =0.2 Fig. 15 Indentation modulus and indentation hardness from FEM simulations in the axial direction symbols ; shaded area represent the values mean±sd from experiments 1 to obtain a direction-dependent tissue hardness see Fig. 7. Indeed, isotropic yield surface predicts the values of the ratio of transverse to axial hardness very close the yield anisotropy ratio Fig. 8. However, interestingly an amplification effect is observed, whereby hardness anisotropy ratios exceed yield stress anisotropy ratios for T / A 0.8. This effect is owed to the decreasing of transverse yield stress, which produces a decreasing axial hardness for T / A Comparison With Previously Reported Experimental Data. The results are highly consistent with experimental data of literature 1,4 6. The indentation moduli values reported in Fig. 6 are comparable to those obtained by Rho et al GPa and 16.6 GPa in the axial and transverse directions, respectively, GPa, axial direction, and GPa, axial direction for human tibial cortical tissue. Regarding hardness Fig. 7, 1 reports a transverse to axial hardness ratio of 0.91 for human tibial cortical tissue. This anisotropy ratio corresponds to the one obtained in this study for a transverse to axial yield stress ratio c T / c A =0.9 Fig. 8. For bovine bone, the transverse to axial hardness ratio H T /H A =0.76 reported in Ref. 6 as well as the magnitude of the hardness values H A =811 MPa and H T =647 MPa are all consistent with a transverse to axial yield stress ratio c c T / A 0.6 in this model. Compressive and tensile axial yield stresses used in the above analyses 180 MPa and 115 MPa, respectively correspond to a material cohesion eq =141 MPa, as defined in the Drucker Prager strength criterion 14. This material parameter is consistent with that found in Ref. 17 eq =122 MPa, which refers to microscale indentation experiments and it is lower than that found in Ref. 16 eq =171 MPa, which refers to macroscale experiments. These differences may suggest that a size dependence exists with the cohesive strength increasing with increasing size scale as expected when investigating tissues exhibiting a hierarchical structure. In order to carry out a quantitative comparison of the FEA predictions to the magnitude of E and H reported in Ref. 1, axial indentation simulations were run to different penetration depths h max up to 500 nm which is consistent with that used in Ref. 1. Since the FEA simulations assume a homogeneous continuum rather than taking into account specific microstructural features, as in Ref. 33, the estimated mechanical properties, such as indentation modulus and hardness, are, by definition, insensitive with respect to length scale maximum force and depth. Figure 15 verified that the axial indentation modulus and hardness were indeed approximately independent from penetration depth; in the same figure the shaded area bounds the experimental findings mean sd for both E and H from Ref. 1. In the transverse direction, the computed indentation modulus and hardness were E T =16.39 GPa and H T =0.58 GPa, respectively; whereas, experiments in Ref. 1 reported E T = GPa and H T = GPa, respectively. Therefore, good agreement was obtained for indentation moduli as well as for indentation hardness between predicted and experimentally reported values. Another key result of this study is that pile-up of the material is affected by: i the tissue-indenter friction and ii the inelastic deformation mode in this model represented by the dilatation angle def. In particular, the pile-up is small or even becomes negative i.e., sink-in when the dilatation angle is decreased and when the probe-tissue friction is accounted for. This finding has a great relevance since experiments have shown that bone tissue indentation produce a negligible pile-up 17,20. High values of the dilatation angle def with the upper limit def =, i.e., associative flow rule implies high volumetric dilatation for all values of the hydrostatic stress component. It is therefore justifiable to assume small values of def to reduce the volume increase during the inelastic deformation process. 4.3 Model Limitations. In the following we provide a summary of the limitations of the current model. The elastic-plastic constitutive law does not represent the most general anisotropic transversely isotropic stress-strain relationship. In particular, the elastic tensor 6 in which r 2 =r 3 =r T is assumed, depends on three independent parameters instead of five, as required by transverse isotropy; this issue is discussed in Ref. 24. Furthermore, the post-yield constitutive law is characterized by a compressive to tensile yield stress ratio independent of material direction; which does not match exactly with experimental finding see, for example, Ref. 15. This model has the advantage to be dependent on fewer parameters, as opposite to more general models, and simpler sensitivity analysis. The results have shown that the above mentioned limitations do not affect the main conclusions described in this research. A further limitation concerns the assumed conical geometry for the indenter. This is a typical assumption for finite element simulations of nanoindentation and the equivalence between simulations carried out with spherical indenters and those with actual pyramidal geometry has been well documented 34,35. It should be also underlined that the effect of pore water has been disregarded in this study. The water acts as a plasticizer for the collagenous phase filling the 50% volume fraction of interfibrillar and the 20% of the extrafibrillar space and may have a significant effect on tissue anisotropy properties, as shown in Refs , in which hydrated and dehydrated bone tissues, where subject to nanoindentation. In the above mentioned papers, the effect of hydration on the transverse to axial ratio of the indentation response was investigated by means of spherical indentation with tip radius of 20 m. In particular, in Ref. 36, itwas found that hydrated bone samples did not show a statistically significant transverse to axial indentation moduli ratio, in contrast to / Vol. 132, AUGUST 2010 Transactions of the ASME

9 what was found on dehydrated samples. The model presented in this paper applies for dehydrated tissue samples as in Refs. 1,39 ; however, it is still to be investigated whether the independence of indentation moduli on test directions found for hydrated samples in Ref. 36 is owed to the specific indenter geometry. Further studies are needed in order to extend these findings to pyramidal indentations. The choice of a plasticity framework for the numerical simulation of indentation of the bone tissue is widely accepted one of the most recent papers which deals with this issue is Ref. 17. Although it is well known that plastic deformation, as defined for metals, are not physical in bone tissue, it has been shown by several experimental studies on the nanoscale deformation modes of bone that a stress-strain curve of tissue exhibits a nonlinear relationship with the achievement of a yield stress, which may be correlated with the onset progressive disruption of the sacrificial bonds during which the tangent to the stress-strain curve progressively decreases This behavior can be phenomenologically reproduced by the elastic-plastic constitutive law. The disruption of sacrificial bonds, however, generates a decrease in elastic modulus of the tissue in a damage-like mechanism. An appropriate physically consistent post-yield model should define a flow rule, i.e. the law ruling the development of inelastic strain, accounting for deformation energy, which has been shown to be dissipated by shearing of a thin glue layer between mineralreinforced collagen fibrils 43. Further developments of the presented model accounting for the most relevant above specified limitations will improve our knowledge of the post-elastic behavior of the cortical bone tissue and its relationship with the tissue constitution. In summary, the technique of indentation is increasingly being used for bone mechanics characterization and has potential for use as a diagnostic tool. Our study provides a rigorous scientific linkage between refined constitutive models representing the tissue s fundamental material properties and the complex multiaxial stress field generated during a penetration. This work provides interesting results on the relationship between bone indentation anisotropy and anisotropy in uniaxial stress fields. More specifically, the model presented is able to predict direction-dependent indentation moduli, direction-dependent hardness, and negligible pile-up simultaneously. In the future, this model may be used to assess the effect of variations of constitutive parameters due to age, injury, and/or disease on bone mechanical performance in multiaxial stress fields. Furthermore, microstructurally based FEA models could be coupled to the present continuum-based approach to assess the role of localized deformation mechanisms and specific microstructural features and to determine the limitations and applicability of the continuum approach. Acknowledgment The authors would like to thank the MIT-Italy Program and the Progetto Rocca for supporting Davide Carnelli s one year scholarship at the MIT. We also gratefully acknowledge the U.S. Army through the MIT Institute for Soldier Nanotechnologies Contract No. DAAD D0002, the National Security Science and Engineering Faculty Fellowship NSSEFF Program, the National Science Foundation MIT Center for Materials Science and Engineering Contract No. DMR , and the NSF under Grant No. CMMI Furthermore, Riccardo Lucchini and Matteo Ponzoni are gratefully acknowledged for their contribution in performing the numerical simulations. Nomenclature A parameter for fitting P h curve A p indentation contact area C ij components of elasticity tensor matrix notation C ijkl fourth-order elasticity tensor E indentation modulus E A indentation modulus along axial direction E T indentation modulus along transverse direction e ij deviatoric strain components g i k ;i,k=1..3 components of unit vectors along principal material directions H hardness H A hardness along axial direction H T hardness along transverse direction h r residual indentation depth I ijkl fourth-order unit tensor M 2 ij second-order constitutive tensor 4 M ijkl fourth-order constitutive tensor n parameter for fitting P h curve P indentation load P max maximum indentation load p i ; i=0.3 or i=a,t weight parameters for elastic anisotropy p hydrostatic pressure in fictitious anisotropic space q i ; i=0.3 or i=a,t weight parameters for inelastic anisotropy R AT axial/transverse yield strength ratio R tc tension/compression yield strength ratio S slope of P h curve S ij deviatoric stress components in fictitious anisotropic space Drucker Prager constitutive parameter def dilatation angle ij Kronecker delta function ij total strain components p rz inelastic strain components elastic constitutive parameter elastic constitutive parameter a friction coefficient compressive strength along the axial direction tensile strength along the axial direction compressive strength along the transverse direction t T tensile strength along the transverse direction ij components of Cauchy stress tensor eq material cohesion parameter ij stress components in fictitious anisotropic space c A t A c T References Von Mises equivalent stress in fictitious anisotropic space Drucker Prager yield locus 1 Rho, J., Roy, M., Tsui, T., and Pharr, G., 1999, Elastic Properties of Microstructural Components of Human Bone Tissue as Measured by Nanoindentation, J. 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