Trabecular bone fracture is closely related to the trabecular architecture, microdamage accumulation, and bone tissue properties. Micro-finite-element models have been used to investigate the elastic and yield properties of trabecular bone but have only seen limited application in modeling the microstructure dependent fracture of trabecular bone. In this research, dynamic fracture in two-dimensional (2D) micrographs of ovine (sheep) trabecular bone is modeled using the cohesive finite element method. For this purpose, the bone tissue is modeled as an orthotropic material with the cohesive parameters calculated from the experimental fracture properties of the human cortical bone. Crack propagation analyses are carried out in two different 2D orthogonal sections cut from a three-dimensional 8mm diameter cylindrical trabecular bone sample. The two sections differ in microstructural features such as area fraction (ratio of the 2D space occupied by bone tissue to the total 2D space), mean trabecula thickness, and connectivity. Analyses focus on understanding the effect of the rate of loading as well as on how the rate variation interacts with the microstructural features to cause anisotropy in microdamage accumulation and in the fracture resistance. Results are analyzed in terms of the dependence of fracture energy dissipation on the microstructural features as well as in terms of the changes in damage and stresses associated with the bone architecture variation. Besides the obvious dependence of the fracture behavior on the rate of loading, it is found that the microstructure strongly influences the fracture properties. The orthogonal section with lesser area fraction, low connectivity, and higher mean trabecula thickness is more resistant to fracture than the section with high area fraction, high connectivity, and lower mean trabecula thickness. In addition, it is found that the trabecular architecture leads to inhomogeneous distribution of damage, irrespective of the symmetry in the applied loading with the fracture of the entire bone section rapidly progressing to bone fragmentation once the accumulated damage in any trabeculae reaches a critical limit.

1.
Wang
,
X.
,
Liu
,
X.
, and
Niebur
,
G. L.
, 2004, “
Preparation of On-Axis Cylindrical Trabecular Bone Specimens Using Micro-CT Imaging
,”
ASME J. Biomech. Eng.
0148-0731,
126
(
1
), pp.
122
125
.
2.
Fyhrie
,
D. P.
, and
Schaffler
,
M. B.
, 1995, “
The Adaptation of Bone Apparent Density to Applied Load
,”
J. Biomech.
0021-9290,
28
(
2
), pp.
135
146
.
3.
Jacobs
,
C. R.
,
Simo
,
J. C.
,
Beaupre
,
G. S.
, and
Carter
,
D. R.
, 1997, “
Adaptive Bone Remodeling Incorporating Simultaneous Density and Anisotropy Considerations
,”
J. Biomech.
0021-9290,
30
(
6
), pp.
603
136
.
4.
Turner
,
C. H.
, 1998, “
Three Rules for Bone Adaptation to Mechanical Stimuli
,”
Bone (N.Y.)
8756-3282,
23
(
5
), pp.
399
407
.
5.
Goldstein
,
S. A.
,
Wilson
,
D. L.
,
Sonstegard
,
D. A.
, and
Matthews
,
L. S.
, 1983, “
The Mechanical Properties of Human Tibial Trabecular Bone as a Function of Metaphyseal Location
,”
J. Biomech.
0021-9290,
16
(
12
), pp.
965
969
.
6.
Fischer
,
K. J.
,
Jacobs
,
C. R.
, and
Carter
,
D. R.
, 1995, “
Computational Method for Determination of Bone and Joint Loads Using Bone Density Distributions
,”
J. Biomech.
0021-9290,
28
(
9
), pp.
1127
1135
.
7.
Lotz
,
J. C.
, and
Hayes
,
W. C.
, 1990, “
The Use of Quantitative Computed Tomography to Estimate Risk of Fracture of the Hip From Falls
,”
J. Bone Jt. Surg., Am. Vol.
0021-9355,
72
(
5
), pp.
689
700
.
8.
Bonfield
,
W.
, 1987, “
Advances in the Fracture Mechanics of Cortical Bone
,”
J. Biomech.
0021-9290,
20
, pp.
1071
1081
.
9.
Melvin
,
J. W.
, 1993, “
Fracture Mechanics of Bone
,”
ASME J. Biomech. Eng.
0148-0731,
115
, pp.
549
554
.
10.
Cody
,
D. D.
,
Gross
,
G. J.
,
Hou
,
F. J.
,
Spencer
,
H. J.
,
Goldstein
,
S. A.
, and
Fyhrie
,
D. P.
, 1999, “
Femoral Strength is Better Predicted by Finite Element Models Than QCT and Dxa
,”
J. Biomech.
0021-9290,
32
(
10
), pp.
1013
1020
.
11.
Crawford
,
R. P.
,
Cann
,
C. E.
, and
Keaveny
,
T. M.
, 2003, “
Finite Element Models Predict In Vitro Vertebral Body Compressive Strength Better Than Quantitative Computed Tomography
,”
Bone (N.Y.)
8756-3282,
33
(
4
), pp.
744
750
.
12.
Keyak
,
J. H.
, 2001, “
Improved Prediction of Proximal Femoral Fracture Load Using Nonlinear Finite Element Models
,”
Med. Eng. Phys.
1350-4533,
23
, pp.
165
173
.
13.
Odgaard
,
A.
,
Kabel
,
J.
,
van Rietbergen
,
B.
,
Dalstra
,
M.
, and
Huiskes
,
R.
, 1997, “
Fabric and Elastic Principal Directions of Cancellous Bone are Closely Related
,”
J. Biomech.
0021-9290,
30
(
5
), pp.
487
495
.
14.
van Rietbergen
,
B.
,
Huiskes
,
R.
,
Eckstein
,
F.
, and
Rüegsegger
,
P.
, 2003, “
Trabecular Bone Tissue Strains in the Healthy and Osteoporotic Human Femur
,”
J. Bone Miner. Res.
0884-0431,
18
(
10
), pp.
1781
1788
.
15.
van Rietbergen
,
B.
,
Weinans
,
H.
,
Huiskes
,
R.
, and
Odgaard
,
A.
, 1995, “
A New Method to Determine Trabecular Bone Elastic Properties and Loading Using Micromechanical Finite Element Models
,”
J. Biomech.
0021-9290,
28
(
1
), pp.
69
81
.
16.
Bayraktar
,
H. H.
,
Gupta
,
A.
,
Kwon
,
R. Y.
,
Papadopoulos
,
P.
, and
Keaveny
,
T. M.
, 2004, “
The Modified Super-Ellipsoid Yield Criterion for Human Trabecular Bone
,”
ASME J. Biomech. Eng.
0148-0731,
126
(
6
), pp.
677
684
.
17.
Bayraktar
,
H. H.
,
Morgan
,
E. F.
,
Niebur
,
G. L.
,
Morris
,
G. E.
,
Wong
,
E. K.
, and
Keaveny
,
T. M.
, 2004, “
Comparison of the Elastic and Yield Properties of Human Femoral Trabecular and Cortical Bone Tissue
,”
J. Biomech.
0021-9290,
37
(
1
), pp.
27
35
.
18.
Niebur
,
G. L.
,
Feldstein
,
M. J.
, and
Keaveny
,
T. M.
, 2002, “
Biaxial Failure Behavior of Bovine Tibial Trabecular Bone
,”
ASME J. Biomech. Eng.
0148-0731,
124
(
6
), pp.
699
705
.
19.
Niebur
,
G. L.
,
Feldstein
,
M. J.
,
Yuen
,
J. C.
,
Chen
,
T. J.
, and
Keaveny
,
T. M.
, 2000, “
High Resolution Finite Element Models With Tissue Strength Asymmetry Accurately Predict Failure of Trabecular Bone
,”
J. Biomech.
0021-9290,
33
(
12
), pp.
1575
1583
.
20.
Stolken
,
J. S.
, and
Kinney
,
J. H.
, 2003, “
On the Importance of Geometric Nonlinearity in Finite-Element Simulations of Trabecular Bone Failure
,”
Bone (N.Y.)
8756-3282,
33
(
4
), pp.
494
504
.
21.
Chen
,
J. Y.
,
Huang
,
Y.
, and
Ortiz
,
M.
, 1998, “
Fracture Analysis of Cellular Materials: A Strain Gradient Model
,”
J. Mech. Phys. Solids
0022-5096,
46
(
5
), pp.
789
828
.
22.
Camacho
,
G. T.
, and
Ortiz
,
M.
, 1996, “
Computational Modelling of Impact Damage in Brittle Materials
,”
Int. J. Solids Struct.
0020-7683,
33
(
20–22
), pp.
2899
2938
.
23.
Tomar
,
V.
,
Zhai
,
J.
, and
Zhou
,
M.
, 2004, “
Bounds for Element Size in a Variable Stiffness Cohesive Finite Element Model
,”
Int. J. Numer. Methods Eng.
0029-5981,
61
, pp.
1894
1920
.
24.
Tomar
,
V.
, and
Zhou
,
M.
, 2005, “
Deterministic and Stochastic Analyses of Dynamic Fracture in Two-phase Ceramic Microstructures With Random Material Properties
,”
Eng. Fract. Mech.
0013-7944,
72
, pp.
1920
1941
.
25.
Xu
,
X. P.
, and
Needleman
,
A.
, 1994, “
Numerical Simulations of Fast Crack Growth in Brittle Solids
,”
J. Mech. Phys. Solids
0022-5096,
42
, pp.
1397
1434
.
26.
Volokh
,
K. Y.
, 2004, “
Nonlinear Elasticity for Modeling Fracture of Isotropic Brittle Solids
,”
Trans. ASME, J. Appl. Mech.
0021-8936,
71
(
1
), pp.
141
143
.
27.
Xu
,
X. P.
, and
Needleman
,
A.
, 1994, “
Numerical Simulations of Fast Crack Growth in Brittle Solids
,”
J. Mech. Phys. Solids
0022-5096,
42
(
9
), pp.
1397
1434
.
28.
Ural
,
A.
, and
Vashishth
,
D.
, 2006, “
Cohesive Finite Element Modeling of Age-Related Toughness Loss in Human Cortical Bone
,”
J. Biomech.
0021-9290,
39
(
16
), pp.
2974
2982
.
29.
Diab
,
T.
, and
Vashishth
,
D.
, 2005, “
Effects of Fatigue Microdamage on Local Bone Tissue Properties
,”
Summer Bioengineering Conference
,
ASME
, Vail, CO.
30.
Mota
,
A.
,
Klug
,
W. S.
,
Ortiz
,
M.
, and
Pandolfie
,
A.
, 2003, “
Finite-Element Simulation of Firearm Injury to the Human Cranium
,”
Comput. Mech.
0178-7675,
31
, pp.
115
121
.
31.
Parkkari
,
J.
,
Kannus
,
P.
,
Palvanen
,
M.
,
Natri
,
A.
,
Vainio
,
J.
,
Aho
,
H.
,
Vuori
,
I.
, and
Järvinen
,
M.
, 1999, “
Majority of Hip Fractures Occur as a Result of a Fall and Impact on the Greater Trochanter of the Femur: A Prospective Controlled Hip Fracture Study With 206 Consecutive Patients
,”
Calcif. Tissue Int.
0171-967X,
65
, pp.
183
–187.
32.
Pandolfi
,
A.
,
Guduru
,
P. R.
,
Ortiz
,
M.
, and
Rosakis
,
A. J.
, 2000, “
Three Dimensional Cohesive-Element Analysis and Experiments of Dynamic Fracture in C300 Steel
,”
Int. J. Solids Struct.
0020-7683,
37
(
27
), pp.
3733
3760
.
33.
Pandolfi
,
A.
, and
Ortiz
,
M.
, 2002, “
An Efficient Adaptive Procedure for Three-Dimensional Fragmentation Simulations
,”
Eng. Comput.
0177-0667,
18
, pp.
148
159
.
34.
Yu
,
C.
,
Pandolfi
,
A.
,
Ortiz
,
M.
,
Coker
,
D.
, and
Rosakis
,
A. J.
, 2002, “
Three-Dimensional Modeling of Intersonic Shear-Crack Growth in Asymmetrically Loaded Unidirectional Composite Plates
,”
Int. J. Solids Struct.
0020-7683,
39
, pp.
6135
6157
.
35.
Kabel
,
J.
,
van Rietbergen
,
B.
,
Dalstra
,
M.
,
Odgaard
,
A.
, and
Huiskes
,
R.
, 1999 “
The Role of an Elective Isotropic Tissue Modulus in the Elastic Properties of Cancellous Bone
,”
J. Biomech.
0021-9290,
32
, pp.
673
680
.
36.
Bayraktar
,
H. H.
,
Morgan
,
E. F.
,
Niebur
,
G. L.
,
Morris
,
G. E.
,
Wong
,
E. K.
, and
Keaveny
,
T. M.
, 2004, “
Comparison of the Elastic and Yield Properties of Human Femoral Trabecular and Cortical Bone Tissue
,”
J. Biomech.
0021-9290,
37
, pp.
27
35
.
37.
Kowalczyk
,
P.
, 2006, “
Orthotropic Properties of Cancellous Bone Modelled as Parameterized Cellular Material
,”
Comput. Methods Biomech. Biomed. Eng.
1025-5842,
9
(
3
), pp.
135
147
.
38.
Knets
,
I. V.
, 1978, “
Mechanics of Biological Tissues. A Review
,”
Mech. Compos. Mater.
0191-5665,
13
, pp.
434
440
.
39.
Anderson
,
T. L.
, 1994,
Fracture Mechanics: Fundamentals and Applications
,
CRC
,
Boca Raton, FL
.
40.
Shet
,
C.
, and
Chandra
,
N.
, 2002, “
Analysis of Energy Balance When Using Cohesive Zone Models to Simulate Fracture Processes
,”
ASME J. Eng. Mater. Technol.
0094-4289,
124
, pp.
440
450
.
41.
Tvergaard
,
V.
, and
Hutchinson
,
J. W.
, 1992, “
The Relation Between Crack Growth and Fracture Process Parameters in Elastic-Plastic Solids
,”
J. Mech. Phys. Solids
0022-5096,
40
, pp.
1377
1397
.
42.
Minnaar
,
K.
, 2002, “
Experimental and Numerical Analysis of Damage in Laminate Composites Under Low Velocity Impact Loading
,” Ph.D. thesis, Georgia Institute of Technology, Atlanta, GA.
43.
Ortiz
,
M.
, and
Pandolfi
,
A.
, 1999, “
Finite Deformation Irreversible Cohesive Elements for Three Dimensional Crack-Propagation Analysis
Int. J. Numer. Methods Eng.
0029-5981,
44
(
9
), pp.
1267
1282
.
44.
Cornec
,
A.
,
Scheider
,
I.
, and
Schwalbe
,
K.-H.
, 2003, “
On the Practical Application of the Cohesive Zone Model
,”
Eng. Fract. Mech.
0013-7944,
70
, pp.
1963
1987
.
45.
Sorensen
,
B. F.
, and
Jacobsen
,
T. K.
, 2003, “
Determination of Cohesive Laws by the J Integral Approach
,”
Eng. Fract. Mech.
0013-7944,
70
, pp.
1841
1858
.
46.
Espinosa
,
H. D.
,
Dwivedi
,
S.
, and
Lu
,
H.-C.
, 2000, “
Modeling Impact Induced Delamination of Woven Fiber Reinforced Composites With Contact∕Cohesive Laws
,”
Comput. Methods Biomech. Biomed. Eng.
1025-5842,
183
, pp.
259
290
.
47.
Reilly
,
D. T.
, and
Burstein
,
A. H.
, 1975, “
The Elastic and Ultimate Properties of Compact Bone Tissue
,”
J. Biomech.
0021-9290,
8
, pp.
393
405
.
48.
Zhai
,
J.
,
Tomar
,
V.
, and
Zhou
,
M.
, 2004, “
Micromechanical Modeling of Dynamic Fracture Using the Cohesive Finite Element Method
,”
ASME J. Eng. Mater. Technol.
0094-4289,
126
, pp.
179
–191.
49.
Krieg
,
R. D.
, and
Key
,
S. W.
, 1973, “
Transient Shell Response by Numerical Integration
,”
Int. J. Numer. Methods Eng.
0029-5981,
7
, pp.
273
–286.
50.
Belytschko
,
T.
,
Chiapetta
,
R. L.
, and
Bartel
,
H. D.
, 1976, “
Efficient Large Scale Non-Linear Transient Analysis by Finite Elements
,”
Int. J. Numer. Methods Eng.
0029-5981,
10
, pp.
579
596
.
51.
Zhai
,
J.
, and
Zhou
,
M.
, 2000, “
Finite Element Analysis of Micromechanical Failure Modes in Heterogeneous Brittle Solids
,”
Int. J. Fract.
0376-9429,
101
, pp.
161
180
, Special Issue on Failure Mode Transition in Solids.
52.
Thomsen
,
J. S.
,
Ebbesen
,
E. N.
, and
Mosekilde
,
L.
, 1998, “
Relationships Between Static Histomorphometry and Bone Strength Measurements in Human Iliac Crest Bone Biopsies
Bone (N.Y.)
8756-3282,
22
(
2
), pp.
153
163
.
53.
Tijssens
,
M.
, 2000, “
On the Cohesive Surface Methodology for Fracture of Brittle Heterogeneous Solids
,” Technical University Delft.
54.
Scheider
,
I.
, and
Brocks
,
W.
, 2003, “
Simulation of Cup-Cone Fracture Using The Cohesive Model
Eng. Fract. Mech.
0013-7944,
70
, pp.
1943
1961
.
55.
Falk
,
M. A.
,
Needleman
,
A.
, and
Rice
,
J. R.
, 2001, “
A Critical Evaluation of Dynamic Fracture Simulations Using Cohesive Surfaces
,”
J. Phys. IV
1155-4339,
11
, pp.
43
52
.
56.
Cameron
,
J. R.
,
Skofronick
,
J. G.
, and
Grant
,
R. M.
, 1999,
Physics of the Body
,
2nd ed.
,
Medical Physics
,
Madison, WI
, p.
96
.
57.
Niebur
,
G. L.
,
Yuen
,
J. C.
,
Hsia
,
A. C.
, and
Keaveny
,
T. M.
, 1999, “
Convergence Behavior of High-Resolution Finite Element Models of Trabecular Bone
,”
ASME J. Biomech. Eng.
0148-0731,
121
(
6
), pp.
629
635
.
58.
Ravi-Chandar
,
K.
, and
Knauss
,
W. G.
, 1984, “
An Experimental Investigation into Dynamic Fracture-I. Crack Initiation and Arrest
,”
Int. J. Fract.
0376-9429,
25
, pp.
247
262
.
59.
Ravi-Chandar
,
K.
, and
Knauss
,
W. G.
, 1984, “
An Experimental Investigation Into Dynamic Fracture II. Microstructural Aspects
,”
Int. J. Fract.
0376-9429,
26
, pp.
65
80
.
60.
Ravi-Chandar
,
K.
, and
Knauss
,
W. G.
, 1984, “
An Experimental Investigation Into Dynamic Fracture III. Steady-State Crack Propagation and Crack Branching
,”
Int. J. Fract.
0376-9429,
26
, pp.
141
154
.
61.
Ravi-Chandar
,
K.
, and
Knauss
,
W. G.
, 1984, “
An Experimental Investigation Into Dynamic Fracture IV. On the Interaction of Stress Waves With Propagating Cracks
,”
Int. J. Fract.
0376-9429,
26
, pp.
192
203
.
62.
Vanleene
,
M.
,
Mazeran
,
P.-E.
, and
Ho Ba Thoa
,
M.-C.
, 2006, “
Influence of Strain Rate on the Mechanical Behavior of Cortical Bone Interstitial Lamellae at the Micrometer Scale
J. Mater. Res.
0884-2914,
21
(
8
), pp.
2093
2097
.
63.
Adharapurapu
,
R. R.
,
Jiang
,
F.
, and
Vecchio
,
K. S.
, 2006, “
Dynamic Fracture of Bovine Bone
,”
Mater. Sci. Eng., C
0928-4931,
26
, pp.
1325
1332
.
You do not currently have access to this content.