A multiscale analysis of the mechanical behavior of bovine Haversian cortical bone is presented in the frame-work of linear elasticity. Cortical bone displays a complex microstructure that includes four phases: Haversian canals, osteons, cement lines, and interstitial bone. Based on close experimental observations, a Monte Carlo algorithm is implemented to build the natural bone composite microstructure. To represent the hierarchical nature of bone, the algorithm incorporates macroscopic morphological components, such as its porosity and osteonal volume fraction, as well as microscopic parameters, such as the characterized distributions of the osteons diameters. Bone local mechanical properties are measured by nanoindentation and microextensometry. The three-dimensional microstructures are discretized by a finite element method in order to evaluate the representative volume element of bovine cortical bone. The numerical model calculates the macroscopic bulk and material Young’s moduli and describes the local stress and strain. How geometrical or mechanical factors affect bone failure is investigated through a comparison of the macroscopic anisotropy and local strain to experimental data.

References

References
1.
Marshall
,
D.
,
Johnell
,
O.
, and
Wedel
,
H.
, 1996, “
Meta-Analysis of How Well Measures of Bone Mineral Density Predict Occurrence of Osteoporotic Fractures
,”
Maturitas
,
25
(
2
), pp.
157
158
.
2.
Stone
,
K. L.
,
Seeley
,
D. G.
,
Lui
,
L. Y.
,
Cauley
,
J. A.
,
Ensrud
,
K.
, and
Browner
,
W. S.
, 2003, “
Osteoporotic Fractures Research Group. BMD at Multiple Sites and Risk of Fracture of Multiple Types: Long-Term Results From the Study of Osteoporosis Fractures
,”
Bone Miner. Res.
,
18
, pp.
1947
1954
.
3.
Gottesman
,
G.
, and
Hashin
,
Z.
, 1980, “
Analysis of Viscoelastic Behaviour of Bones on the Basis of Microstructure
,”
J. Biomech.
,
13
, pp.
89
96
.
4.
Aoubiza
,
B.
,
Crolet
,
J. M.
, and
Meunier
,
A.
, 1996, “
On the Mechanical Characterization of Compact Bone Structure Using the Homogenization Theory
,”
J. Biomech.
,
29
(
12
), pp.
1539
1547
.
5.
Carando
,
S.
,
Barbos
,
M. P.
,
Ascenzi
,
A.
, and
Boyde
,
A.
, 1989, “
Orientation of Collagen in Human Tibial and Fibular Shaft and Possible Correlation With Mechanical Properties
,”
Bone
,
10
(
2
), pp.
139
142
.
6.
Predoi-Racila
,
M.
, and
Crolet
,
J. M.
, 2008, “
Human Cortical Bone: The Sinupros Model. Part Idescription and Elastic Macroscopic Results
,”
Comput. Methods Biomech. Biomed. Eng.
,
11
(
2
), pp.
169
187
.
7.
Currey
,
J. D.
, 2003, “
The Many Adaptations of Bone
,”
J. Biomech.
,
36
, pp.
1485
1495
.
8.
Boivin
,
G.
, and
Meunier
,
P. J.
, 2002, “
The Degree of Mineralization of Bone Tissue Measured by Computerized Quantitative Contact Microradiography
,”
Calcified Tissue Int.
,
70
, p.
503
.
9.
Reilly
,
D. T.
, and
Burstein
,
A. H.
, 1975, “
The Elastic and Ultimate Properties of Compact Bone Tissue
,”
J. Biomech.
,
8
(
6
), pp.
393
405
.
10.
Cowin
,
S. C.
, 2001,
Bone Mechanics Handbook
,
CRC Press
,
Boca Raton, FL
.
11.
Hoc
,
T.
,
Henry
,
L.
,
Verdier
,
M.
,
Aubry
,
D.
,
Sedel
,
L.
, and
Meunier
,
A.
, 2006, “
Effect of Microstructure on the Mechanical Properties of Haversian Cortical Bone
,”
Bone
,
38
, pp.
466
474
.
12.
Wright
,
T. M.
, and
Hayes
,
W. C.
, 1977, “
Fracture Mechanics Parameters for Compact Bone—Effects of Density and Specimen Thickness
,”
J. Biomech.
,
10
, pp.
419
430
.
13.
Hui
,
S. L.
,
Slemenda
,
C. M.
, and
Johnston
,
C. C.
, 1988, “
Age and Bone Mass as Predictors of Fracture in a Prospective Study
,”
J. Clin. Invest.
,
81
, pp.
1804
1809
.
14.
Schaffler
,
M. B.
, and
Burr
,
D. B.
, 1988, “
Stiffness of Compact Bone. Effect of Porosity and Density
,”
J. Biomech.
,
21
, pp.
13
16
.
15.
Yeni
,
Y. N.
,
Brown
,
C. U.
,
Wang
,
Z.
, and
Norman
,
T. L.
, 1997, “
The Influence of Bone Morphology on Fracture Toughness of the Human Femur and Tibia
,”
Bone
,
21
(
5
), pp.
453
459
.
16.
Yeni
,
Y. N.
, and
Norman
,
T. L.
, 2000, “
Calculation of Porosity and Osteonal Cement Line Effects on the Effective Fracture Toughness of Cortical Bone in Longitudinal Crack Growth
,”
J. Biomed. Mater. Res.
,
51
(
3
), pp.
504
509
.
17.
Taylor
,
D.
, 1998, “
Microcrack Growth Parameters for Compact Bone Deduced From Stiffness Variations
,”
J. Biomech.
,
31
(
7
), pp.
587
592
.
18.
Nalla
,
R. K.
,
Kinney
,
J. H.
, and
Ritchie
,
R. O.
, 2003, “
Mechanistic Fracture Criteria for the Failure of Human Cortical Bone
,”
Nat. Mater.
,
2
, pp.
164
168
.
19.
O. Brien
,
F. J.
,
Taylor
,
D.
, and
Lee
,
T. C.
, 2005, “
The Effect of Bone Microstructure on the Initiation and Growth of Microcracks
,”
J. Orthopaedic Res.
,
23
, pp.
475
480
.
20.
Budyn
,
E.
, and
Hoc
,
T.
, 2007, “
Multiple Scale Modeling of Cortical Bone Fracture in Tension Using X-Fem
,”
Rev. Eur. Méc. Numér. (Eur. J. Comput. Mech.)
,
16
, pp.
213
236
.
21.
Budyn
,
E.
, and
Hoc
,
T.
, 2008, “
Fracture Strength Assessment and Aging Signs Detection in Human Cortical Bone Using an x-Fem Multiple Scale Approach
,”
Comput. Mech.
,
42
(
4
), pp.
579
591
.
22.
Philipson
,
B.
, 1965, “
Composition of Cement Line in Bone
,”
J. Histochem. Cytochem.
,
13
(
4
), pp.
270
281
.
23.
Burr
,
D. B.
,
Shaffler
,
M. B.
, and
Frederickson
,
R. G.
, 1988, “
Composition of the Cement Line and its Possible Mechanical Role as a Local Interface in Human Compact Bone
,”
J. Biomech.
,
21
(
11
), pp.
939
945
.
24.
Egerer
,
A. K.
,
Saha
,
S.
,
McMillan
,
P. J.
, and
Rivera
,
J.
, 1996, “
Morphology of the Cement Line in Human Bone and its Relationship to Bone Strength
,”
Biomedical Engineering Conference, Proceedings of the 1996 Fifteenth Southern
, pp.
7
10
.
25.
Torquato
,
S.
, 2000,
Random Heterogeneous Media—Microstructure and Macroscopic Properties—Interdisciplinary Applied Mathematics—Mechanics and Materials
,
Springer-Verlag
,
New York
.
26.
abaqus, 2004.
User’s Manual
, Hibbit, Version 6.3,
Karlsson & Sorensen
, Providence.
27.
Robertson
,
D. M.
,
Robertson
,
D.
, and
Barrett
,
C. R.
, 1978, “
Fracture Toughness, Critical Crack Length and Plastic Zone Size in Bone
,”
J. Biomech.
,
11
, pp.
359
364
.
28.
Oliver
,
W.
, and
Pharr
,
G. M.
, 1992, “
An Improved Technique for Determining Hardness and Elastic Modulus Using Load and Displacement Sensing Indentation Experiments
,”
J. Mater. Res.
,
7
(
6
), pp.
1564
1583
.
29.
Ascenzi
,
A.
, and
Bonucci
,
E.
, 1968, “
The Compressive Properties of Single Osteons
,”
Anat. Rec.
,
161
, pp.
377
392
.
30.
Ascenzi
,
A.
, and
Bonucci
,
E.
, 1968, “
The Tensile Properties of Single Osteons
,”
Anat. Rec.
,
158
, pp.
375
783
.
31.
Fan
,
Z.
,
Swadener
,
E.
,
Rho
,
J. Y.
,
Roy
,
M. E.
, and
Pharr
,
G. M.
, 2002, “
Anisotropic Properties of Human Tibial Cortical Bone as Measured by Nanoindentation
,”
J. Orthopaedic Res.
,
20
, pp.
806
810
.
32.
Swadener
,
J. G.
,
Rho
,
J. Y.
, and
Pharr
,
G. M.
, 2001, “
Effects of Anisotropy on Elastic Moduli Measured by Nanoindentation in Human Tibial Cortical Bone
,”
J. Biomed. Mat. Res.
,
57
, pp.
108
112
.
33.
Fan
,
Z.
,
Rho
,
J. Y.
, and
Swadener
,
E.
, 2004, “
Three-Dimensional Finite Element Analysis of the Effects of Anisotropy on Bone Mechanical Properties Measured by Nanoindentation
,”
J. Mater. Res.
,
19
(1), pp.
114
123
.
34.
Bonfield
,
W.
, and
Li
,
C. H.
, 1967, “
Anisotropy of Nonelastic Flow in Bone
,”
J. Appl. Phys.
,
38
(
6
), pp.
2450
2455
.
35.
Cezayirlioglu
,
H.
,
Bahniuk
,
E.
,
Davy
,
D. T.
, and
Heiple
,
K. G.
, 1985, “
Anisotropic Yield Behavior of Bone Under Combined Axial Force and Torque
,”
J. Biomech.
,
18
(
1
), pp.
61
69
.
36.
Sabelman
,
E. E.
,
Koran
,
P.
,
Diep
,
N.
, and
Lineaweaver
,
W. C.
, 1997, “
Collagen/Hyaluronic Acid Matrices for Connective Tissue Repair
,”
First Smith & Nephew International Symposium: Advances in Tissues Engineering and Biomaterials
, July 20–23.
37.
Rho
,
J. Y.
,
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.
,
35
, pp.
189
198
.
38.
Katz
,
J. L.
, and
Ukraincik
,
K.
, 1971, “
On the Anisotropic Elastic Properties of Hydroxyapatite
,”
J. Biomech.
,
29
(
4
), pp.
221
227
.
39.
Skedros
,
J. G.
,
Holmes
,
J. L.
,
Vajda
,
E. G.
, and
Bloebaum
,
R. D.
, 2005, “
Cement Lines of Secondary Osteons in Human Bone are Not Mineral-Deficient: New Data in a Historical Perspective
,”
Anatom. Rec.
,
286A
(
1
), pp.
781
803
.doi
40.
Lakes
,
R.
, and
Saha
,
S.
, 1979, “
Cement Line Motion in Bone
,”
Nature
,
204
, pp.
501
503
.doi
41.
Katz
,
J. L.
,
Yoon
,
H. S.
,
Lipson
,
S.
,
Maharidge
,
R. L.
,
Meunier
,
A.
, and
Christel
,
P.
, 1984, “
The Effects of Remodeling on the Elastic Properties of Bone
,”
Calcified Tissue Int.
,
36
(suppl. 1), pp.
S31
36
.
42.
Henry
,
L.
, 2006, “
Approche Multiéchelle du Comportement Mécanique d’un Tissu Osseux Cortical
,” Ph.D thesis.
43.
Temizer
,
I.
, and
Wriggers
,
P.
, 2008, “
On a Mass Conservation Criterion in Micro-to-Ma cro Transitions
,”
J. Appl. Mech.
,
75
, pp.
054503
1–054503
4
.
44.
Rho
,
J. Y.
, 1996, “
An Ultrasonic Method for Measuring the Elastic Properties of Human Tibial Cortical and Cancellous Bone
,”
Ultrasonic
,
34
, pp.
777
783
.
45.
Mohsin
,
S.
,
O’Brien
,
F. J.
, and
Lee
,
T. C.
, 2006, “
Osteonal Crack Barriers in Ovine Compact Bone
,”
J. Anatomy
,
208
, pp.
81
89
.
46.
Hill
,
R.
, 1964, “
Theory of Mechanical Properties of Fibre-Strengthened Materials: I. Elastic Behaviour
,”
J. Mech. Phys. Solids
,
12
, pp.
199
212
.
47.
Jones
,
A. C.
,
Sheppard
,
A. P.
,
Sok
,
R. M.
,
Arns
,
C. H.
,
Limaye
,
A.
,
Averdunk
,
H.
,
Brandwood
,
A.
,
Sakellariou
,
A.
,
Senden
,
T. J.
,
Milthorpe
,
B. K.
, and
Knackstedt
,
M. A.
, 2004, “
Three-Dimensional Analysis of Cortical Bone Structure Using X-Ray Micro-Computed Tomography
,”
Physica A
,
339
, pp.
125
130
.
48.
Qiu
,
S.
,
Rao
,
D. S.
,
Fyhrie
,
D. P.
,
Palnitkar
,
S.
, and
Parfitt
,
A. M.
, 2005,“
The Morphological Association Between Microcracks and Osteocyte Lacunae in Human Cortical Bone
,”
Bone
,
37
, pp.
10
15
.
49.
Prendergast
,
P. J.
, and
Huiskes
,
R.
, 1996, “
Microdamage and Osteocyte-Lacuna Strain in Bone: A Microstructural Finite Element Analysis
,”
J. Biomech. Eng.Trans. ASME
,
118
, pp.
240
246
.
50.
Yeni
,
Y. N.
, and
Fyhrie
,
D. P.
, 2002, “
Fatigue Damage-Fracture Mechanics Interaction in Cortical Bone
,”
Bone
,
30
(
3
), pp.
509
514
.
51.
Vashishth
,
D.
,
Tanner
,
K. E.
, and
Bonfield
,
W.
, 2000, “
Contribution, Development and Morphology of Microcracking in Cortical Bone During Crack Propagation
,”
J. Biomech.
,
33
, pp.
1169
1174
.
52.
Guo
,
X. E.
,
Liang
,
L. C.
, and
Goldstein
,
S. A.
, 1998, “
Micromechanics of Osteonal Cortical Bone Fracture
,”
J. Biomed. Eng. Trans. ASME
,
120
, pp.
303
304
.
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