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. 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.

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. Biomed. Mater. Res.
0021-9304,
45
, pp.
48
54
.
2.
Ebenstein
,
D.
, and
Pruitt
,
L.
, 2006, “
Nanoindentation of Biological Materials
,”
Nanotoday
1748-0132,
1
, pp.
26
33
.
3.
Lewis
,
G.
, and
Nyman
,
J.
, 2008, “
Review, The Use of Nanoindentation for Characterizing the Properties of Mineralized Hard Tissues: State-of-the Art Review
,”
J. Biomed. Mater. Res., Part B: Appl. Biomater.
1552-4973,
87B
, pp.
286
301
.
4.
Rho
,
J.
,
Tsui
,
T. Y.
, and
Pharr
,
G. M.
, 1997, “
Elastic properties of Human Cortical and Trabecular Lamellar Bone Measured by Nanoindentation
,”
Biomaterials
0142-9612,
18
, pp.
1325
1330
.
5.
Rho
,
J.
,
Zioupos
,
P.
,
Currey
,
J. D.
, and
Pharr
,
G. M.
, 2002, “
Microstructural Elasticity and Regional Heterogeneity in Human Femoral Bone of Various Ages Examined by Nano-Indentation
,”
J. Biomech.
0021-9290,
35
, pp.
189
198
.
6.
Wang
,
X.
,
Chen
,
X.
,
Hodgson
,
P.
, and
Wen
,
C.
, 2006, “
Elastic Modulus and Hardness of Cortical and Trabecular Bovine Bone Measured by Nanoindentation
,”
Trans. Nonferrous Met. Soc. China
1003-6326,
16
, pp.
s744
s748
.
7.
Hansma
,
P.
,
Turner
,
P.
,
Fantner
,
G.
, 2006, “
Bone Diagnostic Instrument
,”
Rev. Sci. Instrum.
0034-6748,
77
, p.
075105
.
8.
Oliver
,
W.
, and
Pharr
,
G.
, 1992, “
An Improved Technique for Determining Hardness and Elastic Modulus Using Load and Displacement Sensing Indentation Experiments
,”
J. Mater. Res.
0884-2914,
7
, pp.
1564
1583
.
9.
Currey
,
J.
, 2002,
Bones: Structure and Mechanics
,
Princeton University Press
,
NJ
.
10.
Weiner
,
S.
,
Addadi
,
L.
, and
Wagner
,
H.
, 2000, “
Materials Design in Biology
,”
Mater. Sci. Eng., C
0928-4931,
11
, pp.
1
8
.
11.
Lees
,
S.
, and
Page
,
E.
, 1992, “
A Study of Some Properties of Mineralized Turkey Leg Tendon
,”
Connect. Tissue Res.
0300-8207,
28
, pp.
263
287
.
12.
Liu
,
D.
,
Wagner
,
H.
, and
Weiner
,
S.
, 2000, “
Bending and Fracture of Compact Circumferential and Osteonal Lamellar Bone of the Baboon Tibia
,”
J. Mater. Sci.: Mater. Med.
0957-4530,
11
, pp.
49
60
.
13.
Liu
,
D.
,
Wagner
,
H.
, and
Weiner
,
S.
, 1999, “
Anisotropic Mechanical Properties of Lamellar Bone Using Miniature Cantilever Bending Specimens
,”
J. Biomech.
0021-9290,
32
, pp.
647
654
.
14.
Fratzl
,
P.
, and
Weinkamer
,
R.
, 2007, “
Nature’s Hierarchical Materials
,”
Prog. Mater. Sci.
0079-6425,
52
(
8
), pp.
1263
1334
.
15.
Cowin
,
S.
, 1989,
Bone Mechanics
,
CRC
,
Boca Raton, FL
.
16.
Mercer
,
C.
,
He
,
R.
,
Wang
,
M. Y.
, and
Evans
,
A.
, 2006, “
Mechanisms Governing the Inelastic Deformation of Cortical Bone and Application to Trabecular Bone
,”
Acta Biomater.
1742-7061,
2
, pp.
59
68
.
17.
Mullins
,
L.
,
Bruzzi
,
M.
, and
McHugh
,
P.
, 2009, “
Calibration of a Constitutive Model for the Post-Yield Behaviour of Cortical Bone
,”
J. Mech. Behav. Biomed. Mater.
1751-6161,
2
, pp.
460
470
.
18.
Tai
,
K.
,
Ulm
,
F.
, and
Ortiz
,
C.
, 2006, “
Nanogranular Origins of the Strength of Bone
,”
Nano Lett.
1530-6984,
6
, pp.
2520
2525
.
19.
Tai
,
K.
,
Dao
,
M.
,
Suresh
,
S.
,
Palazoglu
,
A.
, and
Ortiz
,
C.
, 2007, “
Nanoscale Heterogeneity Promotes Energy Dissipation in Bone
,”
Nature Mater.
1476-1122,
6
, pp.
454
462
.
20.
Fan
,
Z.
,
Rho
,
J.
, and
Swadener
,
J.
, 2004, “
Three-Dimensional Finite Element Analysis of the Effects of Anisotropy on Bone Mechanical Properties Measured by Nanoindentation
,”
J. Mater. Res.
0884-2914,
19
, pp.
114
123
.
21.
Zhang
,
J.
,
Niebur
,
G.
, and
Ovaert
,
T.
, 2008, “
Mechanical Property Determination of Bone Through Nano- and Micro-Indentation Testing and Finite Element Simulation
,”
J. Biomech.
0021-9290,
41
, pp.
267
275
.
22.
Swadener
,
J.
, and
Pharr
,
G.
, 2001, “
Indentation of Elastically Anisotropic Half-Spaces by Cones and Parabolas of Revolution
,”
Philos. Mag. A
0141-8610,
81
, pp.
447
466
.
23.
Swadener
,
J.
,
Rho
,
J.
, and
Pharr
,
G.
, 2001, “
Effects of Anisotropy on Elastic Moduli Measured by Nanoindentation in Human Tibial Cortical Bone
,”
J. Biomed. Mater. Res. Part A
1549-3296,
57
, pp.
108
112
.
24.
Zysset
,
P.
, and
Curnier
,
A.
, 1995, “
An alternative model for Anisotropic Elasticity Based on Fabric Tensors
,”
Mech. Mater.
0167-6636,
21
, pp.
243
250
.
25.
Malvern
,
L.
, 1969,
Introduction to the Mechanics of a Continous Medium
,
Prentice-Hall
,
Englewood Cliffs, NJ
.
26.
Car
,
E.
,
Oller
,
S.
, and
Oñate
,
E.
, 2000, “
An Anisotropic Elastoplastic Constitutive Model for Large Strain Analysis of Fiber Reinforced Composite Materials
,”
Comput. Methods Appl. Mech. Eng.
0045-7825,
185
, pp.
245
277
.
27.
Drucker
,
D.
, and
Prager
,
W.
, 1952, “
Soil Mechanics and Plastic Analysis or Limit Design
,”
Q. Appl. Math.
0033-569X,
10
, pp.
157
165
.
28.
Meyers
,
M.
,
Chen
,
P.
,
Yu-Min Lin
,
A.
, and
Seki
,
Y.
, 2008, “
Biological Materials: Structure and Mechanical Properties
,”
Prog. Mater. Sci.
0079-6425,
53
, pp.
1
206
.
29.
Zienkiewicz
,
O.
, and
Taylor
,
R.
, 1991,
The Finite Element Method
,
4th ed.
,
McGraw-Hill
,
England
, Vol.
2
.
30.
ABAQUS
, 2009, Documentation Manual, ABAQUS Simulia.
31.
Bayraktar
,
H.
,
Morgan
,
E.
,
Niebur
,
G.
,
Morris
,
G.
,
Wong
,
E.
, and
Keaveny
,
T.
, 2004, “
Comparison of the Elastic and Yield Properties of Human Femoral Trabecular and Cortical Bone Tissue
,”
J. Biomech.
0021-9290,
37
, pp.
27
35
.
32.
Reilly
,
D.
, and
Burstein
,
A.
, 1975, “
The Elastic and Ultimate Properties of Compact Bone Tissue
,”
J. Biomech.
0021-9290,
8
, pp.
393
405
.
33.
Wang
,
L.
,
Song
,
J.
,
Ortiz
,
C.
, and
Boyce
,
M.
, 2009, “
Anisotropic Design of a Multilayered Biological Exoskeleton
,”
J. Mater. Res.
0884-2914,
24
, pp.
3477
3494
.
34.
Dao
,
M.
,
Chollacoop
,
N.
,
Van Vliet
,
K.
,
Venkatesh
,
T.
, and
Suresh
,
S.
, 2001, “
Computational Modeling of the Forward and Reverse Problems in Instrumented Sharp Indentation
,”
Acta Mater.
1359-6454,
49
, pp.
3899
3918
.
35.
Qin
,
J.
,
Huang
,
Y.
,
Xiao
,
J.
, and
Hwang
,
K.
, 2009, “
The Equivalence of Axisymmetric Indentation Model for Three-Dimensional Indentation Hardness
,”
J. Mater. Res.
0884-2914,
24
, pp.
776
783
.
36.
Bembey
,
A.
,
Koonjul
,
V.
,
Bushby
,
A.
,
Ferguson
,
V.
, and
Boyde
,
A.
, 2005, “
Contribution of Collagen, Mineral and Water Phases to the Nanomechanical Properties of Bone
,”
Material Research Society, Symposium Proceedings
, Vol.
844
.
37.
Oyen
,
M.
,
Ko
,
C.
,
Bembey
,
A.
,
Bushby
,
A.
, and
Boyde
,
A.
, 2005, “
Nanoindentation and Finite Element Analysis of Resin-Embedded Bone Samples as a Three-Phase Composite Material
,”
Material Research Society Symposium
, Vol.
874
.
38.
Akhtar
,
R.
,
Morse
,
S.
, and
Mummery
,
P.
, 2005, “
Nanoindentation of Bone in a Physiological Environment
,”
Materials Research Society Symposium Proceedings
, Vol.
84
, pp.
87
92
.
39.
Fan
,
Z.
,
Swadener
,
J.
,
Rho
,
J.
,
Roy
,
M.
,
Pharr
,
G.
, 2002, “
Anisotropic Properties of Human Tibial Cortical Bone as Measured by Nanoindentation
,”
J. Orthop. Res.
0736-0266,
20
, pp.
806
810
.
40.
Gupta
,
H.
,
Wagermaier
,
W.
,
Zickler
,
G.
,
Raz-Ben Aroush
,
D.
,
Funari
,
S.
,
Roscheger
,
P.
,
Wagner
,
H.
, and
Fratzl
,
P.
, 2005, “
Nanoscale Deformation Mechanisms in Bone
,”
Nano Lett.
1530-6984,
5
, pp.
2108
2111
.
41.
Gupta
,
H.
,
Seto
,
J.
,
Wagermaler
,
W.
,
Zaslansky
,
P.
,
Boesecke
,
P.
, and
Fratzl
,
P.
, 2006, “
Cooperative Deformation of Mineral and Collagen in Bone at the Nanoscale
,”
Proc. Natl. Acad. Sci. U.S.A.
0027-8424,
103
, pp.
17741
17746
.
42.
Gupta
,
H.
,
Roschger
,
P.
,
Wagner
,
H.
, and
Fratzl
,
P.
, 2006, “
Mechanical Modulation at the Lamellar Level in Osteonal Bone
,”
J. Mater. Res.
0884-2914,
21
, pp.
1913
1921
.
43.
Peterlik
,
H.
,
Roschger
,
P.
,
Klaushofer
,
K.
, and
Fratzl
,
P.
, 2006, “
From Brittle to Ductile Fracture of Bone
,”
Nature Mater.
1476-1122,
5
, pp.
52
55
.
You do not currently have access to this content.