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Research Papers

A Finite Element Model for Direction-Dependent Mechanical Response to Nanoindentation of Cortical Bone Allowing for Anisotropic Post-Yield Behavior of the Tissue

[+] Author and Article Information
D. Carnelli

Department of Structural Engineering, Laboratory of Biological Structure Mechanics (LaBS), Politecnico di Milano, 20133 Italy; Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139

D. Gastaldi, V. Sassi, R. Contro

Department of Structural Engineering, Laboratory of Biological Structure Mechanics (LaBS), Politecnico di Milano, 20133 Italy

C. Ortiz

Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139

P. Vena

Department of Structural Engineering, Laboratory of Biological Structure Mechanics (LaBS), Politecnico di Milano, 20133 Italy; IRCCS, Istituto Ortopedico Galeazzi, Milano, 20161 Italyvena@stru.polimi.it

J Biomech Eng 132(8), 081008 (Jun 18, 2010) (10 pages) doi:10.1115/1.4001358 History: Received July 20, 2009; Revised February 18, 2010; Posted March 01, 2010; Published June 18, 2010; Online June 18, 2010

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 [1999, “Elastic Properties of Microstructural Components of Human Bone Tissue as Measured by Nanoindentation,” J. Biomed. Mater. Res., 45, pp. 48–54] 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.

Copyright © 2010 by American Society of Mechanical Engineers
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References

Figures

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Figure 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

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Figure 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.

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Figure 3

Sketch of deformed surface under the indenter. The measure of the sink-in (pile-up) is shown.

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Figure 4

Transverse to axial yield stress ratio versus constitutive parameter qA. Dashed line represents an isotropic yield function.

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Figure 5

Compressive yield stress along axial and transverse directions versus constitutive parameter qA. The parameter σ¯eq has been suitably chosen so that an axial compressive yield stress of 182 MPa is obtained for all values of qA.

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Figure 6

Indentation moduli for simulations along axial and transverse indentations versus transverse to axial yield stress ratio

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Figure 7

Indentation hardness for simulations along axial and transverse indentations versus transverse to axial yield stress ratio

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Figure 8

Transverse to axial indentation moduli and hardness ratios versus transverse to axial yield stress ratio

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Figure 9

Contour maps of the inelastic strain rz component (ϵrzp, where z and r are the directions parallel and perpendicular to the long bone axis direction, respectively) in an axial indentation simulation for: σTc/σAc=1 (left), σTc/σAc=0.8 (center), and σTc/σAc=0.5 (right)

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Figure 10

Plots of the inelastic strain ϵrzp component as a function of the distance from the indentation point on the surface for: isotropic yield locus model, σTc/σAc=0.8, and σTc/σAc=0.5

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Figure 11

Pile-up versus transverse to axial yield stress ratio for three different values of tissue-probe friction coefficient

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Figure 12

Axial hardness versus friction coefficient (left vertical axis) for αdef=0.1; axial hardness versus friction coefficient (right vertical axis) for αdef=0.1

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Figure 13

Axial hardness (left vertical axis) versus αdef; axial indentation modulus (right vertical axis) versus αdef

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Figure 14

Axial indentations: pile-up versus αdef for μa=0.2

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Figure 15

Indentation modulus and indentation hardness from FEM simulations in the axial direction (symbols); shaded area represent the values (mean±sd) from experiments (1)

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