Computational Modeling of Trabecular Bone in
ARCHNES
Lower Extremity Injuries due to Impac MASSACHUSETTSNl
OF TECHNOLOGY
by
JUN 2 5 2013
Wendy Pino
B.S., Aeronautics and Astronautics
-Le9B/\I.
Massachusetts Institute of Technology, Cambridge (2011)
Submitted to the Department of Mechanical Engineering
in partial fulfillment of the requirements for the degree of
Master of Science in Mechanical Engineering
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2013
© Massachusetts Institute of Technology 2013. All rights reserved.
...
Author .............................
..
........
Department of Mechanical Engineering
May ,. 2013
Certified by................
Rad*l A. Rafovifzky
Professor of Aeronautics and Astronautics
Thesis Supervisor
Certified by...............................
Lorna J. Gibson
Professor of Mechanical and Materials Science Engineering
4 .
Accepted by ...........................
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is1
der
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Cavid E. Hardt
Chairman, Department Committee on Graduate Theses
Computational Modeling of Trabecular Bone in Lower
Extremity Injuries due to Impact
by
Wendy Pino
Submitted to the Department of Mechanical Engineering
on May 10, 2013, in partial fulfillment of the
requirements for the degree of
Master of Science in Mechanical Engineering
Abstract
Lower extremity injuries resulting from improvised explosive devices (IEDs) pose a
serious threat to the safety of military troops. Reports from Operation Iraqi Freedom
and Operation Enduring Freedom identify IEDs as the cause for a substantial number
of lower extremity bone fractures. The Army Research Laboratory (ARL) concentrates part of its research efforts into better understanding impact related injuries.
Despite a significant number of articles on bone mechanical behavior, only a few consider high strain rates. In collaboration with ARL, we propose in this thesis to capture
via an in silico approach the dynamic and quasi-static responses of the trabecular
bone. To achieve this goal, we conducted large scale parallel finite element simulations on biofidelic and biomimetic morphological models of trabecular bone. The
biofidelic model was developed using an image-based tetrahedral meshing approach
on pCT images, courteously provided by Niebur's group at Notre Dame University,
of a human femural sample. The biomimetic model was developed from an analytical
model proposed by Wang and Cutifno [50] for a periodic unit cell. For the solid part of
the trabecular, a visco-elastic visco-plastic constitutive model developed by Socrate's
group [15] for cortical bone was applied. For the dynamic simulations, the effect of
strain rate on the response of the bone microstructure was investigated and compared to published experimental results. We observed a structural softening on the
stress-strain curve which takes its origins from the buckling that appears within the
spongious trabeculae structure. Finally we included fracture in our dynamic simulations using the discontinuous Galerkin method developed by Radovitzky's group [37]
to observe initiation and propagation of cracks within the trabecular and to capture
the resulting material softening in the post-yield stress-strain response.
Thesis Supervisor: Radil A. Radovitzky
Title: Professor of Aeronautics and Astronautics
3
4
Acknowledgments
I would like to express the deepest gratitude to my research advisor, Prof. Radl
Radovitzky, for giving me the opportunity to take part in the exciting and meaningful
research that his group is conducting.
I am extremely grateful for his invaluable
guidance, support and understanding.
I would like to thank him for sharing his
wealth of knowledge while helping me work through the tough problems that arose
and for sharing his genuine enjoyment of research. Radl, gracias por su interes en mi
bienestar y por hacerme sentir como en casa.
My sincere thanks go to Dr. Aurelie Jean for her constant encouragement and motivation. I would like to thank her for caring about my success and investing so much
time in teaching me new skills. Her insight has been extremely valuable to me. I
would also like to thank her for helping me make improvements to this thesis. She is
someone I aspire to be more like in my career and in life. I know that she will make
a wonderful professor.
I would also like to thank the members of the RR group, including Abiy Tasissa,
Adrian Rosolen, Andrew Seagraves, Brandon Talamini, Gauthier Becker, Lei Qiao,
Martin Hautefeuille, Michael Tupek, and Michelle Nyein, for their friendship, support,
and interest in my academic progress.
Additionally, I would like to express my gratitude towards Dr. Simona Socrate for
providing the constitutive model implemented in this study and for her hands-on
approach to investigating problems with me. I would also like to thank Prof. Glen
Niebur from the University of Notre Dame for generously providing images of trabecular bone which were used to create our model. For their contributions, I am
extremely appreciative.
I am also thankful for the financial support provided by the Army Research Laboratory (ARL) and the Institute for Soldier Nanotechnologies (ISN). From ARL, I would
like to thank Dr. Reuben Kraft for his efforts during our collaboration.
5
I would like to thank Samuel Dyar for his help and encouragement every step of the
way. For his love and advice, I am deeply grateful. Lastly, and above all, I would like
to thank my parents for taking the risks and making the sacrifices that have made
this opportunity a reality. This would not be possible without their immense love
and support.
6
Contents
1
2
19
Introduction
19
. . . . . . . . . . . . . . . . . . . . . . .
1.1
M otivation.
1.2
Advanced Computational Mechanics Capabilities for Bone Fracture
A pplications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
1.3
O bjectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
1.4
O utline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
Trabecular Bone Properties
2.1
2.2
23
Biological Hierarchical Structure of Trabecular Bone . . . . . . . . . .
23
2.1.1
Observation of Microstructure . . . . . . . . . . . . . . . . . .
23
2.1.2
Characterization of Microstructure
. . . . . . . . . . . . . . .
24
Mechanical Behavior of Trabecular Bone . . . . . . . . . . . . . . . .
25
2.2.1
Most Relevant Characteristics . . . . . . . . . . . . . . . . . .
25
2.2.2
Relative Density and Anisotropy
. . . . . . . . . . . . . . . .
26
2.2.3
Strain Rate Effect on Mechanical Behavior . . . . . . . . . . .
28
2.2.4
Fracture Mechanism
. . . . . . . . . . . . . . . . . . . . . . .
29
2.2.5
Quasi-static Experimental Results . . . . . . . . . . . . . . . .
30
3 Morphological Modeling of Trabecular Bone Microstructure
33
. . . . . . . . . . . . . . .
33
3.1.1
State of A rt . . . . . . . . . . . . . . . . . . . . . . . . .
33
3.1.2
Geometrical Structure of a Single Unit Cell . . . . . . . .
34
3.1.3
Geometrical Structure of a Tessellation Unit Cell
. . . .
36
3.1
Periodic Biomimetic Microstructures
7
3.2
4
5
6
Geometrical Structure of the Tessellation . . . . . . . . . . . .
36
3.1.5
Trabecular Dimensions . . . . . . . . . . . . . . . . . . . . . .
38
Image-based Biofidelic Microstructure . . . . . . . . . . . . . . . . . .
40
3.2.1
41
M eshing Procedure . . . . . . . . . . . . . . . . . . . . . . . .
Material Behavior Modeling of Trabecular Bone
49
4.1
Previous Models in Literature . . . . . . . . . . . . . . . . . . . . . .
49
4.2
Constitutive Equations . . . . . . . . . . . . . . . . . . . . . . . . . .
50
4.3
Implementation in Summit . . . . . . . . . . . . . . . . . . . . . . . .
52
Modeling Framework
53
5.1
Computational Framework . . . . . . . . . . . . . . . . . . . . . . . .
53
5.2
Finite Deformation Numerical Formulation . . . . . . . . . . . . . . .
53
5.3
Discontinuous Galerkin . . . . . . . . . . . . . . . . . . . . . . . . . .
55
5.4
Solver: Newmark Scheme . . . . . . . . . . . . . . . . . . . . . . . . .
55
Numerical Simulation of Quasi-Static Test
6.1
7
3.1.4
Numerical Method: Dynamic Relaxation Solver
57
. . . . . . . . . . . .
57
6.1.1
Implementation in Summit . . . . . . . . . . . . . . . . . . . .
57
6.1.2
Example: Convergence of Prismatic Bar
. . . . . . . . . . . .
60
6.2
Quasi-Static Simulation of Biomimetic Trabecular Bone . . . . . . . .
63
6.3
Quasi-Static Simulation of Biofidelic Trabecular Bone . . . . . . . . .
65
6.4
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68
Numerical Simulation of Dynamic Test
69
7.1
Numerical Method: Parallel Design for Periodic Structures in Summit
69
7.2
Dynamic Simulation of Biomimetic Trabecular Bone . . . . . . . . . .
71
7.3
Dynamic Simulation of Biofidelic Trabecular Bone . . . . . . . . . . .
75
7.3.1
Trabecular Bone Sample A . . . . . . . . . . . . . . . . . . . .
75
7.3.2
Trabecular Bone Sample B . . . . . . . . . . . . . . . . . . . .
75
7.4
Dynamic Fracture Simulations using Discontinuous Galerkin
7.4.1
. . . . .
81
Fracture Simulation of Biomimetic Unit Cell . . . . . . . . . .
81
8
Fracture Simulation of Biofidelic Trabecular Bone . . . . . . .
81
D iscussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
85
7.4.2
7.5
8
87
Conclusions
8.1
D iscussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
87
8.2
Recommendations for Future Work
. . . . . . . . . . . . . . . . . . .
89
8.2.1
Quasi-static Fracture Simulation . . . . . . . . . . . . . . . . .
89
8.2.2
Large Scale Bone Modeling
. . . . . . . . . . . . . . . . . . .
89
A Neighbor Numbers for Communication Map
9
95
10
List of Figures
2-1
Structure of cortical bone (a) 3D Sketch of cortical bone, (b) Haversian system cut, (c) Haversian system photomicrograph, Fridez P.
Mod lisation de l'adaptation osseuse externe. PhD thesis, In Physics
Department, EPFL, Lausanne, 1996 . . . . . . . . . . . . . . . . . . .
2-2
25
Scanning electron micrograph of low density (a) rod-like trabecular
bone and high density (b) plate-like trabecular bone taken from the
femoral head of a human specimen [9].
2-3
. . . . . . . . . . . . . . . . .
26
Plot of compressive strength vs density of trabecular bone as presented
by [10]. The plots show earlier (a) and later (b) studies conducted human bone specimens, unless specified otherwise. Relative compressive
strengths are normalized by oy, = 182 MPa and relative densities are
normalized by ps = 1800 kg/m.
2-4
. . . . . . . . . . . . . . . . . . . . .
28
Stress results found in literature for quasi-static experimental tests on
trabecular bone samples. . . . . . . . . . . . . . . . . . . . . . . . . .
30
3-1
Bottom part of Cuitifio unit . . . . . . . . . . . . . . . . . . . . . . .
35
3-2
Single Cuitifio unit . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
3-3
Two Cuitifno unit cells joined together. . . . . . . . . . . . . . . . . .
36
3-4
Tessellation unit cell containing 73,121 tetrahedral elements and 12,634
nodes, generated using gmsh.
3-5
. . . . . . . . . . . . . . . . . . . . . .
37
Tessellation based on the Cuitifio model consisting of 564,104 tetrahedral elements. Each color represents a different processor. . . . . . . .
11
38
3-6
Slice of pCT viewed in ImageJ [42] consisting of 384 x 384 pixels and
representing an 8mm x 8mm cross-sectional area.
3-7
. . . . . . . . . . . .
41
Triangulated cubes based on label combinations taken from each vertex
of the cube, as shown by Lorensen and Cline [22]. . . . . . . . . . . .
3-9
38
Marching cube used for mesh triangulation placed between two slices
of images, as shown by Lorensen and Cline [22].
3-8
. . . . . . . . . . .
42
Layered stacks of segmented binary images of trabecular bone taken
from pCT and imported in AMIRA. Each slice consists of 384x384x1
pixels. There are 371 slices in the z direction.
. . . . . . . . . . . . .
43
3-10 Trabecular bone mesh after processing pCT images in Amira and then
meshing in Ansys to create a mesh consisting of 2, 379, 141 linear tetrahedral elements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-11 Trabecular bone meshes for sample A and sample B.
. . . . . . . . .
45
46
3-12 Zoomed in image of part of the trabecular bone mesh in the same
vicinity for meshes created using different processes. The mesh that
results from AMIRA and Ansys is a higher quality mesh. . . . . . . .
4-1
Stress-strain solid curve fits from the Johnson constitutive model compared with data points from the McElhaney experiment. [15] . . . . .
4-2
50
Constitutive model schematic representing visco-elastic, visco-plastic
bone from [15] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6-1
47
51
Spatial distribution of the displacement field along the z direction after convergence of DR solver, for a displacement prescribed at both
extremities of the prismatic bar along the z direction. . . . . . . . . .
6-2
60
Evolution of the pressure as a function of the position along the prismatic bar at different iterations of the DR solver: the response reaches
static equilibrium as the number of DR iterations increases.
6-3
. . . . .
61
Evolution of the displacement as a function of the position along the
prismatic bar at different iterations of the DR solver: the response
reaches static equilibrium as the number of DR iterations increases.
12
.
62
6-4
Evolution of pressure as a function of the position along the prismatic
bar after the first time step and for different tolerance values in the
D R solver. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6-5
Stress gradient result for a tessellation under compression using dynamic relaxation and the Neo-Hookean material model. . . . . . . . .
6-6
. . . . . . . . . . . . . . . . .
64
Dynamic relaxation convergence study for sample A for both the NeoHookean and the cortical material model. . . . . . . . . . . . . . . . .
6-9
64
Force vs. displacement for a 1x1x1 single tessellation unit in compression using the cortical material model.
6-8
63
Force vs. displacement for a 1x1x1 single tessellation unit in compression using the Neo-Hookean material model. . . . . . . . . . . . . . .
6-7
62
65
Dynamic relaxation convergence study for sample A for both the NeoHookean and the cortical material model. . . . . . . . . . . . . . . . .
66
6-10 Evolution of the stress in the z direction o-2, as a function of the strain
in the z direction E2, for a quasi-static test conducted on sample A of
the biofidelic model using the cortical bone material.
. . . . . . . . .
67
6-11 Evolution of the stress o-2 for sample A under compression with a
maximum displacement of 5.025x10-6 using dynamic relaxation and
the cortical material model.
7-1
. . . . . . . . . . . . . . . . . . . . . . .
67
Representation of Message Passing Interface (MPI) application setup
for an 8 processor simulation with a tessellated unit cube. Processors
are numbered left to right in the x-direction, then in the y-direction,
and the z-direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7-2
71
Mesh of biomimetic bone tessellation comprised of 1,440 tessellation
units that corresponds to a total sample size of 11.7 mm x 11.2 mm x
11.1 mm. The current image is a partial view of the entire structure,
displaying 20 of the 72 processors using Paraview. . . . . . . . . . . .
13
73
7-3
Spatial distribution of the stress components azz in the biomimetic
bone microstructure. The current image is a partial view of the entire structure, displaying 20 of the 72 processors using Paraview and
demonstrating displacement. . . . . . . . . . . . . . . . . . . . . . . .
7-4
73
Evolution of the stress component ozz as a function of the strain component czz from the dynamic simulation conducted on the biomimetic
bone tessellation with an initial velocity of 10 m/s applied to both
extremities of the structure along the z-axis. . . . . . . . . . . . . . .
7-5
74
Evolution of the stress component ozz as a function of the strain component czz from the dynamic simulation conducted on sample A at 10
m/s in loading directions x, y, and z. . . . . . . . . . . . . . . . . . .
7-6
76
Spatial distribution of the stress component along specific loading directions x (a), y (b), and z (c) within sample A of the biofidelic microstructure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7-7
Simulation result showing uCT sample B under dynamic compression
with 10 m/s applied to the z extremities at time 2.89 x 10-5 seconds.
7-8
Evolution of stress azz as a function of strain ezz for sample B
simulations run at different strain rates.
7-9
76
77
CT
. . . . . . . . . . . . . . . .
78
Evolution of stress orzz as a function of strain ezz for pCT simulation on
sample B with a strain rate of 330/s demonstrating post-yield softening
due to buckling.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
7-10 Spatial distribution of stresses uz2 from dynamic simulation on biofidelic sample B with a velocity boundary condition of 1.10 m/s. .....
80
7-11 Spatial distribution of the stress component Uzz within the trusses of
one single unit-cell of the biomimetic model of trabecular bone after
10 and 30 time steps of a dynamic simulation with DG. The opened
interface elements characterizing cracks are highlighted in green in the
contour plots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
82
7-12 Evolution of stress ozz with respect to strain
Ezz
for a dynamic simu-
lation on sample A using both continuous Galerkin (CG) and discontinuous Galerkin (DG) in order to compare the effect of allowing for
discontinuities.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
83
7-13 Spatial distribution of stresses oz for a DG dynamic simulation of
sample A under compression at a time of 2.531x10- 6 s. . . . . . . . .
84
7-14 Visualization of the fracture surfaces within sample A of trabecular
bone for a compressive dynamic simulation at time 2.531x10-6 S. Interfaces with a minimal separation of 3.3x10-6 between elements are
highlighted in red.
8-1
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
84
Single time-step showing several iterations of dynamic relaxation up
to convergence, applied to sample A with 67,114 tetrahedral elements
prior to mesh scaling. . . . . . . . . . . . . . . . . . . . . . . . . . . .
8-2
90
Four time-steps showing several iterations of dynamic relaxation up
to convergence, applied to sample A with 67,114 tetrahedral elements
prior to mesh scaling. The legend indicated number of interface element openings and average opening distance in meters. . . . . . . . .
8-3
Evolution of stress o-z for the converged solution of a prismatic bar
run with dynamic relaxation and DG. . . . . . . . . . . . . . . . . . .
8-4
91
Dynamic simulation of a full tibia bone consisting of 145, 669 tetrahedral elements and 1, 531, 768 nodes. . . . . . . . . . . . . . . . . . . .
8-5
91
92
Dynamic simulation on full tibia mesh with fracture. Bulk elements
are shown in gray and fractured cohesive elements are shown in red. .
93
A-1 Full assembly of three layers of processors showing the spatial relation-
A-2
ship between one another. . . . . . . . . . . . . . . . . . . . . . . . .
96
Neighbor numbers for processors based on their relation to "X".
96
15
. . .
16
List of Tables
2.1
Example Material Properties for Bone, Kraft et. al [21] . . . . . . . .
2.2
Data from trabecular bone extracted from experiment curves in literature showing strain rate i, stiffness, and maximum stress o-.
27
Data
for bone volume to total volume percentage, sample length, sample diameter, species, and anatomical origin of specimen were given in each
source. For the stiffness, these results give an average stiffness of about
0.9 GPa with a standard deviation of 0.398 GPa.
4.1
. . . . . . . . . . .
Material properties specified by the user to describe trabecular bone,
calibrated with McElhaney experiments [15]. . . . . . . . . . . . . . .
7.1
31
52
Data from trabecular bone simulations conducted on sample B showing
strain rate i, stiffness, maximum stress o-, bone volume to total volume
percentage, sample length, and sample diameter. . . . . . . . . . . . .
17
78
18
Chapter 1
Introduction
1.1
Motivation
Improvised Explosive Devices, known as IEDs, present a dangerous threat to the
safety of military troops. In the case known as an underbelly blast, an improvised
explosive device is detonated under a military vehicle. This results in a high amplitude
short duration pressure wave [26] which propagates through the floor of the vehicle
and impacts the foot and lower leg region of the occupant. The impact can sometimes
incapacitate a soldier instantaneously [26].
These anti-vehicular mines hinder the ability of the military to react quickly. They
diminish the strength of troops and even lower morale. Explosive devices not only
injure soldiers, but also block access paths for help to arrive and increase costs associated with medical air transportation [38].
A study of the injuries sustained by soldiers in Operation Iraqi Freedom (OIF) and
Operation Enduring Freedom (OEF) reported a total of 3, 575 extremity combat
injuries between October 2001 and January 2005. 75% of extremity injuries to soldiers
in OIF and OEF were caused by improvised explosive devices [32].
26% of these
injuries involved bone fractures, with half of the fractures sustained in the lower
extremities, most commonly in the tibia and the fibula.
19
These circumstances are costly, difficult, and sometimes impossible to replicate experimentally. Therefore, it would be valuable to be able to predict the fracture of
bone using 3D finite element simulations.
1.2
Advanced Computational Mechanics Capabilities for Bone Fracture Applications
Trabecular bone is an active area of research. It was first investigated in the 1847, but
then more consistently in the 1960s. Since then, research in this area has been steadily
increasing to over 700 publications in the last year alone. A search (on pubmed.gov)
yielded 11,105 papers about trabecular bone. Only about 5% of those papers dealt
with numerical studies. An even smaller amount, 1.03%, incorporated bone fracture
or crack propagation into their studies and 0.8% analyzed fracture in 3D. There were
no publications found that utilized Message Passing Interface (MPI) or discontinuous
Galerkin (DG). Despite the small number of fracture publications, many sources [12]
[43] [26] stress the importance of bone analysis in fracture related cases, both for blast
scenarios and for falls experienced by the elderly.
Part of the reason there is a shortage of numerical simulations is because the beginnings of ongoing trabecular bone research coincide with the beginning of discontinuous
Galerkin finite elements and commercially available parallel computers.
Finite element development began with Alexander Hrennikoff and Richard Courant
[34] in the early 1940s. They both used mesh discretization and the idea of elements.
In 1947, Olgierd Zienkiewicz consolidated the various methods into finite elements
[44]. Discontinuous Galerkin methods were not developed until 20 years later. Parallel
computing was introduced in the mid 1950s. However, it wasn't until April 1958 that
it was considered for use in finite element calculations [52].
This purpose of this study is to understand the fracture of trabecular microstructures, but also to make use of computational advances by incorporating MPI and DG
20
finite elements. Therefore, there are interests in both the bone application and the
computational approach.
1.3
Objectives
The objective of this study is to model the mechanical response of trabecular bone at
high strain rates. Towards that goal, a material model that can accurately represent
bone is included in the model. Meshes generated from real bone specimens and from
idealized geometries serve as samples for analysis. The focus is to look at the stiffness,
yield, and fracture response of the samples.
Experiments can be expensive, limited in quantity, and sometimes difficult to reproduce for explosive cases. Therefore, fracture is more feasible to model than to study
experimentally. Proper modeling of trabecular bone behavior in this study will serve
as a tool for further investigation and for design of protective gear.
1.4
Outline
The consequences of lower extremity injuries serve to highlight the importance of this
study. This first section established goals, both mechanical and numerical, that need
to be attained in order to better understand the response of bone tissue. In the next
section, bone characteristics are described in detail, first from a biological point of
view and then from a mechanical point of view.
Then, morphological modeling of bone is explained. For one sample, a biofidelic mesh
was generated from images of real trabecular bone. For another sample, an idealized
microstructure was created based on the Cuitifno geometry [50]. Next, the details of
the constitutive model and its implementation are discussed.
Lastly, static and dynamic simulations using a finite element code developed within
the Radovitzky group [37] are conducted on the sample meshes described earlier.
21
The resulting behavior, including plasticity, buckling, and fracture are analyzed and
discussed.
22
Chapter 2
Trabecular Bone Properties
2.1
Biological Hierarchical Structure of Trabecular Bone
2.1.1
Observation of Microstructure
Significant efforts have been made to study and identify the key geometrical and
mechanical features of bone. It is known that length and thickness of trabeculae
have a strong impact on the overall macroscopic behavior. These dimensions are
quantified through observations and measurements of medical images.
The most
popular techniques include various forms of computed tomography such as peripheral
high-resolution computed tomography (HR-pQCT) and micro-computed tomography
(puCT)[6] [41] [13] [29]. Magnetic resonance (MR) and Scanning Electron Microscopy
are among other methods that have been used[27] [51]. The methods mentioned have
been applied in either in vivo or post-mortem cases [24].
23
2.1.2
Characterization of Microstructure
In order to fully model underbelly blast problems, it is important to understand the
structural characteristics of bone. By understanding its properties and constitutive
behavior, we can create more accurate models to numerically simulate events that
happen in combat.
The structure of bone allows it to function as a high strength, low weight support
mechanism. Bone consists of a hard outer layer, known as the cortical, and a porous
inner core known as the trabecular.
Cortical bone tissue can be considered a composite material and is known for its
heirarchical structure.
It is composed of hydroxyapatite, collagen, proteoglycans,
noncollagenous proteins and water [5].
At the microscopic level, cortical bone is
composed of Haversian canals that are aligned longitudinally along the bone. These
canals are connected transversely by Volkmanns canals with capillaries and nerves.
Cylindrical lamellae surround the Haversian canals to create osteons. Cement lines
surround each osteon in order to separate Haversian canals from the interstitial bone
[5] [1]. These microstructures are densely packed. In fact, cortical bone has about
5-10% porosity [5].
Bone contains a porous inner core consisting of trabecular tissue. This is also known
as cancellous bone. At the microscale, trabecular bone is composed of individual
trabeculae struts in three dimensional space which form an open cellular structure
[9]. This cellular structure contains interconnected rods and plates at a variety of
angles. Unlike the densely packed cortical bone, trabecular bone has a porosity of
50-95% [5]. Figure 2-1 shows the dense cortical surrounding the open, sponge-like
trabecular architecture.
Trabecular bone properties can vary with age and with anatomical location. The
orientations and sizes of the cells develop according the directions and magnitudes of
the loads it needs to support [9] [19].
24
(b)
Figure 2-1: Structure of cortical bone (a) 3D Sketch of cortical bone, (b) Haversian system cut, (c) Haversian system photomicrograph, Fridez P. Modelisation de
l'adaptation osseuse externe. PhD thesis, In Physics Department, EPFL, Lausanne,
1996
At different anatomical locations, bone will have cortical and trabecular sections,
however the trabecular composition will vary as stated previously. Therefore, the
porous interconnected architecture of trabecular bone is optimized to support heavy
loads while maintaining a low weight. With age, individual trabeculae also become
thinner, thus altering its material properties [43]. Trabecular bone is heterogeneous,
in the sense that volume fraction, moduli, and strength can vary spatially across a
single bone specimen [19]. For this reason, it is desirable to understand the spatial
distribution of localized stresses at the bone microscale in addition to the global stress
response.
2.2
2.2.1
Mechanical Behavior of Trabecular Bone
Most Relevant Characteristics
Trabecular bone is an open cellular material composed of interconnected rods and
plates, as visible in Figure 2-2. Its most important characteristic is relative density,
or the fraction of solid to total volumetric space.
25
Cell size plays a lesser role in
determining mechanical properties. Nevertheless, cell shape is important because it
captures the anisotropy of bone. Topology is yet another important characteristic.
The 3D open cell structure of the trabecular provides information about edges, faces,
and connectivities that in turn affect mechanical properties [9].
properties can be seen in Table 2.1.
Sample material
Bone has been characterized as anisotropic,
visco-elastic, and visco-plastic. Trabecular bone only makes up 15-20% of the entire
bone material. However, its lightweight cellular structure provides important shock
absorbing capabilities and added strength [39].
The microstructure of trabecular
bone can vary based on anatomical location. It can develop to support the loads
encountered by the human body and by doing so, it can establish a direction along
which stiffness and strength are greatest [19].
(b)
(a)
Figure 2-2: Scanning electron micrograph of low density (a) rod-like trabecular bone
and high density (b) plate-like trabecular bone taken from the femoral head of a
human specimen [9].
2.2.2
Relative Density and Anisotropy
One of bone's most important characteristics is the ratio of the density of trabecular
bone to the desity of solid bone. Important values such as Young's modulus and yield
stress, are dependent on relative density. As seen in Figure 2-3 relative density can
26
Bone Type
Material Property
Cortical
E = 19.0 GPa
v = 0.3
Trabecular
p = 1810kg/m 3
132MPa
c=
E
300 MPa
v = 0.45
p = 600kg/m
ac
3
= 1.5MPa
Table 2.1: Example Material Properties for Bone, Kraft et. al [21]
have a strong effect on strength. Several attempts have been made to develop power
laws to characterize the relationship between density and modulus. One analysis, by
Lotz et al. [23], formulated some relationships describing the elastic modulus and
critical stress for varying levels of bone density p shown in Equations 2.1-2.4:
E(MPa) = 1904 p1 .64 axial direction
(2.1)
E(MPa) = 1157 p1 .78 transverse direction
(2.2)
0c(MPa) = 40.8 p1 .89 axial direction
(2.3)
ac(MPa) = 21.4 - p1 37 transverse direction
(2.4)
It is important to note that the relationships developed by Lotz are different for the
axial and transverse directions, thereby capturing the anisotropy in bone.
Another quantifiable property used to characterize bone is ash fraction. Ash fraction
allows you to account for bone mineral content, a property which influences mechanical behavior. [14]. They note that many models do not account for the difference
between bone volume fraction
B
and ash fraction a, and present a way to relate
them with Equation 2.5:
27
p
(2.5)
=B (1.41 + 1.29a)
TV
It is important to note that although these relationships can characterize mechanical
properties of bone, they do not take into account the effect of microstructural architecture. Therefore, a correlation between trabecular microstructure and macrostructure
would provide a more rigorous understanding of the mechanical behavior.
102
F,.,.. and4.6i.
Cr
ier aves(1077)
rbial
~
103
eh e.ins-We41 md 5"
Fesnu.a.4 t".
aCarern d Havesf1677
soin
1?.
(19574
Mist
ylabass
o Hgy.4 nd Cat. (416)
ta
00
C.i
anCarterONff61
Snia o
i dlS4)
baml~ 7165
W.*w .n1
0.1
102
W.I~.
Wff.
-
ia'
tat
S
ILnr0X
On
G':
411161
am Chtalmers
I.~
Ca
0
L OKN
.,
0
,
+
0.01
0.10*4
*c4*
*ai
JO
Or
c4k
**
1'
22
0.001
0.01
0.1
0.001
0.01
1
Relative density, jp 'p
0.11
Reltive defrsitV.
prfp
(a)
(b)
Figure 2-3: Plot of compressive strength vs density of trabecular bone as presented
by [10]. The plots show earlier (a) and later (b) studies conducted human bone
specimens, unless specified otherwise. Relative compressive strengths are normalized
by aus
182 MPa and relative densities are normalized by p,8
1800 kg/in 3 .
2.2.3
Strain Rate Effect on Mechanical Behavior
Bone has been known to exhibit time dependent behavior [19]. Thus, efforts have
been made to characterize the response of bone loaded under various strain rates.
28
While many experiments have been conducted at low strain rates, few have been
done at strain rates higher than 10 s-'. One experiment, conducted by McElhaney,
used an air gun type setup to apply compressive forces to a trabecular bone specimen.
They achieved strain rates of up to 4,000 s 1 and captured the stress-strain response
curves for bone at different strain rates [25]. Their stress-strain curves showed an
increase in bone modulus as strain rate increased. It was also noted that different
fracture response were exhibited for the strain rates tested. At strain rates below
1s-', the types of fractures were shear failures and cone failures, whereas at higher
strain rates the fractures were splinters in the longitudinal direction.
In an experiment conducted by Bonfield and Clark, it was also reported that Young's
modulus increased with strain rate almost linearly before plateauing at a bounded
value [2]. In yet another paper by J.D. Currey, the trend between strain rate and
modulus was plotted and it was deduced that by increasing the strain rate 1,000-fold,
the Young's modulus increased by 10%.
2.2.4
Fracture Mechanism
Bone fracture can occur due to impact associated with trauma scenarios or due to
muscle contraction of fatigued bones in elderly people [5] [19]. Bone has the ability
to dissipate energy in order to prevent fracture. This has been associated with its
geometrical features, but also with constitutive properties such as visco-plastic flow
[35].
The anisotropy found in bone affects the way in which fracture patterns propagate.
There is a preferential longitudinal direction along which cracks propagate
more smoothly than across transverse cross sections of the bone. It has been found
that for angles between 0-50' cracks are smooth, whereas for angles 50-90' cracks are
zig-zagged and require a higher fracture energy. Transversely, microstructures in the
cortical region, such as cement lines, can divert the direction of crack propagation and
even stop it [1]. The anisotropy of bone allows the fracture energy between directions
29
to change by two orders of magnitude [35].
2.2.5
Quasi-static Experimental Results
In the literature, several quasi-static experimental results for trabecular bone samples
have been documented. The most relevant information including sample origin, dimension, strain rate, apparent stiffness, and maximum stress have been summarized
in Table 2.2. Of the sources that provided stress plots, the data has been visually
extracted and plotted together in Figure 2-4 for comparison. The results in the literature give an average stiffness of about 0.9 GPa with a standard deviation of 0.398
GPa. The average maximum strength from the values reported is 15.74 MPa, with a
standard deviation of 4.96 MPa.
25
Stress vs. Strain for Quasi-Static Experiments
20 -
15-
N
10-
5
- -+
. -
0
00
0.02
0.04
0.06
Johnson A [2010]
Johnson B [2010]
Johnson C [2010]
Ramezani [2012]
Kelly [2011]
Harrison [2008]
0.08
0. LO
Strain
Figure 2-4: Stress results found in literature for quasi-static experimental tests on
trabecular bone samples.
30
Experimental Values for Quasi-Static Tests
CA,
Source
Ramezani [2012]
Johnson Specimen A [2010]
Johnson Specimen B [2010]
Johnson Specimen C [2010]
Kelly [2011]
Harrison [2008]
Verhulp Specimen 1 [2008]
Verhulp Specimen 2 [2008]
Verhulp Specimen 3 [2008]
Verhulp Specimen 4 [2008]
Verhulp Specimen 5 [2008]
Verhulp Specimen 6 [2008]
Verhulp Specimen 7 [2008]
0.027
0.01
0.01
0.01
0.0104
0.001
0.00017
0.00017
0.00017
0.00017
0.00017
0.00017
0.00017
Stiffness [GPa]
0.5
0.6
0.935
0.75
0.31
1.65
1.34
1.24
1.07
0.94
0.73
1.23
0.41
Max o [MPa]
13.5
7.7
13.2
12.5
14.2
24.5
24.55
21.27
18.27
14.41
19.08
21.83
10.36
BV/TV [%]
14.4 %
21.0%
26.6 %
34.1%
31.5%
28.5%
35.6%
34.3%
30.3%
18.4%
1 [mm]
10.0
3.773
3.843
3.779
8.0
10.0
5.0
5.0
5.0
5.0
5.0
5.0
5.0
d [mm]
7.0
7.175
7.263
7.275
8.0
7.0
5.35
5.35
5.35
5.35
5.35
5.35
5.35
Species
Human
Bovine
Bovine
Bovine
Bovine
Ovine
Bovine
Bovine
Bovine
Bovine
Bovine
Bovine
Bovine
Anatomy
Femoral Head
Femur
Femur
Femur
Proximal Tibia
L5 Vertebrae
Proximal Tibia
Proximal Tibia
Proximal Tibia
Proximal Tibia
Proximal Tibia
Proximal Tibia
Proximal Tibia
Table 2.2: Data from trabecular bone extracted from experiment curves in literature showing strain rate i, stiffness, and
maximum stress o. Data for bone volume to total volume percentage, sample length, sample diameter, species, and anatomical
origin of specimen were given in each source. For the stiffness, these results give an average stiffness of about 0.9 GPa with a
standard deviation of 0.398 GPa.
32
Chapter 3
Morphological Modeling of
Trabecular Bone Microstructure
3.1
3.1.1
Periodic Biomimetic Microstructures
State of Art
Periodic structures can be used to represent cellular solids [11] and study the behavior of trabecular at the microscale. By tessellating a design that represents the
architecture of trabecular bone, a full network of interconnected trabeculae can be
created.
Studying the bone structure at this level of detail will allow us to observe any localized behavior that is not seen at the macroscale. In fact, it has been found that
microstructural strains can be up to 2.2 times larger than the continuum strains
[40].
Periodic tessellations of unit cells have been used to represent trabecular bone in
several cases. In a study done by Guo and Kim [12], different unit cells were used to
represent rod-like and plate-like trabecular sections for the purpose of understanding
the effect of bone loss on mechanical response. Another study by Sander [40] used
33
rectangular open cell and offset rectangular open cell structures to model vertebral
trabecular bone.
Periodic unit cells were even applied to model cortical bone in
addition to trabecular bone [3]. This was done by tessellating square unit cells and
varying the porosity by choosing different geometrical configurations to fill the unit
square.
I've chosen to assemble and design the meshing for a particular structure studied
by Cuitiio et al. [50] because it has several good qualities. It can be generated by
tessellating a single unit cell, it describes an open cellular structure, and it has been
shown to capture anisotropy at different orientations.
3.1.2
Geometrical Structure of a Single Unit Cell
To capture the microstructure of trabecular bone, a unit cell was created inspired by
the Cuitifio model using Gmsh [8]. The attributes of the unit cell (angle 0, length 1,
and radius r) are specified by the user for added flexibility. By tessellating the unit
cell, we created a geometry that represents that of bone.
The cell consists of 4 cylinders which fuse at the origin. The top three cylinders are
referred to as struts, and the bottom cylinder is referred to as the handle. To create
the unit, I began by creating a point at the origin and considering all other points
that will lie on the z = 0 plane. Since there are 3 struts, there will be 3 points on
the z=0 plane at the location where the sides of the cylinders meet. These points (2,
6, 7) were created with 120 degrees of separation to fit the three cylinders. Next, I
created points (3, 4, 5) that "dip" a little below the z=0 plane. At these points the
struts from the top will join with the handle at the bottom.
To completely define the intersection, I created point to establish the height at which
the 3 struts fuse. The distance between that point and the origin is labeled by h.
To create the handle, the points that lie on the z = 0 plane were extruded in the
negative z-direction a distance 1. The base of the handle (10, 11, 12) was duplicated
and rotated about the y-axis so that the angle with the z-axis was 0, as requested by
34
Figure 3-1: Bottom part of Cuitiio unit
the user. This creates the first strut. By rotating the strut by 120 and 240 degrees
the last two struts were created. To finish the unit, circular arcs, ellipses, and lines
were added to create surfaces which describe the unit volume.
2r
Figure 3-2: Single Cuitinio unit
Once the uniit cell is created, it can be
joined
with the other unit cells through
translations or rotations. If joined through the handle, the lower unit must be rotated
about the z-axis by 60 degrees, as shown in Figure 3-3.
35
YX
Figure 3-3: Two Cuitiio unit cells joined together.
3.1.3
Geometrical Structure of a Tessellation Unit Cell
As mentioned previously, single unit cells can be combined. A tessellation unit cell,
shown in Figure 3-4, is composed of 12 single unit cells. The tessellation unit can
be replicated in 3D space to create a open large cellular structure simply by moving it by an established dx, dy, dz distance without needing rotations or complex
algorithms.
3.1.4
Geometrical Structure of the Tessellation
By simply moving the tessellation unit cell by dx, dy, or dz, an entire block of tessellated material can be created without the need for a complex algorithm. For the
tessellation unit chosen, the vector translations below allow us to fill the 3D space
36
w
z
IY
Figure 3-4: Tessellation unit cell containing 73,121 tetrahedral elements and 12,634
nodes, generated using gmsh.
while maintaining connectivity between the units.
dx
=
[4lsin(O)(sin(7r/6)+ 1), 0, 0]
(3.1)
dy = [0,4lsin(O)cos(7r/6),0]
(3.2)
dz = [0, 0, 3(21)(1 + cos(O))]
(3.3)
(3.4)
The tessellated structure shown in Figure 3-5 contains a tessellation in every processor
which is made up of multiple tessellation units. It measures 4.06 mm, 2.84 mm, and
1.85 mm in length in x, y, and z, respectively.
37
Figure 3-5: Tessellation based on the Cuitifio model consisting of 564,104 tetrahedral
elements. Each color represents a different processor.
3.1.5
Trabecular Dimensions
The program which generates the idealized microstructure tessellations takes the radius, length, and angle of unit as inputs. Therefore, it is important to prescribe values
which will realistically represent the bone anatomy of our specimen. Individual trabeculae length has been documented by several sources [43] [40] [281.
Figure 3-6: Slice of pCT viewed in ImageJ [42] consisting of 384 x 384 pixels and
representing an 8mm x 8mm cross-sectional area.
In the publication by Silva and Gibson [9], trabeculae thickness in the longitudinal
38
and transverse directions are calculated from 2D images of a vertebral bone specimen.
They tabulate results for trabeculae dimensions based on age. In this case, we have
information about the age of the patient that our sample was extracted from. The
images used in this study are from a 53 year old patient and the closest tabulated
values are those for a 55 year old patient. To verify that these values can be used to
describe our sample, the bone volume fractions were compared. The volume fraction
of our sample is 0.2160. The value reported in the paper was 0.127, which based on
the difference in age, weight, and anatomical location is reasonably close especially
It is also a closer match for our
when you consider the possible range 0.05-0.7[9].
sample than those reported by other sources.
Length: The value for trabeculae length was reported to be 0.196 mm for a human
trabecular sample taken from a patient whose age was close to that of the patient
whose trabcular was used for this study. This will be the input for 1 in the Cuitifio
geometry.
Radius: Given that we know the length from the literature and the bone volume
fraction from our sample, the value for the radius can be calculated based on Equation
3.5 for bone volume fraction which represents the Cuitino geometry [50]. Using this
equation, the calculated value for the radius is 0.0684 mm.
BV
2
solid
- 2
L2
VeVn
-2
(3.5)
C = 3v/57r/16
(3.6)
D = 3v"67r/32 - 23V2/96
(3.7)
Angle: The angle applied was chosen to be 54.74' to represent the isocline state.
Therefore, each of the struts are aligned with the same angle with respect to the z
axis.
Summary of Properties:
39
3.2
Length:
0.196 mm
Radius:
0.0684 mm
Angle:
54.740
Bone Volume Fraction:
0.2160
Image-based Biofidelic Microstructure
Various methods of bone observation exist in the biomedical community, as described
in Section 2.1.1. Of those, Micro-Computed Tomography (pCT) has been used multiple times [46 [6] [13] [31] for the purposes of generating biofidelic bone geometries
and is used here to reconstruct a bone mesh from a compilation of images.
The first step is to get a vertical stack of 2D images, or zstack, as shown in 3-9. Each
image is a pixel in thickness and several hundred pixels in width and length. For
this study, slices of pCT images were obtained from Prof. Niebur from Notre Dame
University.
The stack of images can be processed using various techniques in order to create a
three-dimensional finite element mesh. This allows us to get a solid representation of
bone that is anatomically correct.
To mesh the images, the Marching Cubes algorithm for 3D surface reconstruction
was used. It was developed by Lorensen and Cline [22] and included as the algorithm
in the Amira meshing software. This method was used because it results in smoother
meshes. It works by creating a cube between two slices of images such that the cube
vertices are at the midpoints of the pixels, as shown in Figure 3-7. It then assigns
a label, either 1 or 0 to each vertex, based on the color. The cube is composed of 8
vertices, 4 for the top slice and 4 for the bottom slice.
Depending on the labels of the marching cube, a triangulation is chosen to represent
that box of volume in the mesh. Figure 3-8 shows some triangulation choices for
40
Slice k + 1
Z
,1,k+1)
1,.i+ 1
i+1,',
0, , k+11)
0 (i,j+ ,k)
Slice k
1)
1
(l+1,j+
(,Jk(+,k
pixel
Figure 3-7: Marching cube used for mesh triangulation placed between two slices of
images, as shown by Lorensen and Cline [22].
certain combinations of labels. Therefore, this method takes into account the surrounding labels in order to create triangulations at angles that best represent the true
shape of the solid.
3.2.1
Meshing Procedure
The meshing process is outlined in detail below. It begins with the p-CT images
which, as shown in Figure 3-9 capture the anatomical details in every slice. Here,
we can see 5 of the 371 layers of images that make up the vertical representation.
The first part of the preparation and meshing process was done using Amira and the
second part was done using Ansys.
Steps in the meshing process:
1. Import pCT images into ImageJ and convert to binary.
2. Import resulting piCT images into Amira.
3. Within Amira create a z-stack of images, which together represent the full 3D
volume. Here, the orthoslice and bounding box can be viewed to check that the
physical volume is represented by the images as desired.
41
O~1~2
1301
Figure 3-8: Triangulated cubes based on label combinations taken from each vertex
of the cube, as shown by Lorensen and Cline [22].
4. Label each voxel by selecting an RGB color threshold to distinguish empty space
from solid space.
5. Generate a surface and apply smoothing to create a 2D surface mesh.
6. Prepare and correct the surface mesh using original geometry information. The
prepare tetragen options allows you to correct issues related to aspect ratio,
dihedral angle, intersecting elements, and coplanar triangles.
7. Export the geometry as vrml.
8. Import the geometry into Ansys.
9. Fill the surface mesh to create a 3D volume mesh using tetrahedral elements.
10. Improve the 3D mesh by removing disconnected mesh sections and hanging
elements. Correct intersecting elements and aspect ratio.
42
Figure 3-9: Layered stacks of segmented binary images of trabecular bone taken from
pCT and imported in AMIRA. Each slice consists of 384x384x1 pixels. There are 371
slices in the z direction.
11. Convert Ansys mesh into Summit format for use with the Summit code.
Figure 3-10 shows the surface reconstruction of the geometry created from trabecular bone pCT images as well as the final mesh that results from processing those
images.
There are two mesh samples used throughout the study. The samples come from the
femoral neck of a 53 year old male. Sample A is the mesh sample shown in Figure
3-11(a) and measures 2 mm in width and 2.5 mm in length. Sample B is shown in
Figure 3-11(b) and measures 8 mm in diameter and 10 mm in length.
The process outlined in Section 3.2.1 allows us to achieve smooth, high quality meshes
such as the one shown in Figure 3-12(b). If instead the mesh is automatically filled,
43
the resulting mesh will contain imperfections and cavities such as the one shown in
Figure 3-12(a).
Working on every aspect of mesh generation, beginning with the
image and ending with the full three-dimensional mesh, eventually results in a high
quality mesh that can capture the details of the trabecular architecture. According to
an Ansys evaluation, 23.981% of the elements have near perfect aspect ratios falling
between 0.891 and 0.9405, with an aspect ratio of 1.0 representing an equilateral
geometry. Additionally, this higher quality mesh can progress further in applications
without reaching stability issues.
44
(a) Side view of trabecular bone surface reconstruction from pCT images in
Amira
(c) Top view of trabecular bone surface reconstruction from pCT images in Amira
b) Side view of trabecular bone mesh in Ansys
(d) Top view of trabecular bone mesh in Ansys
Figure 3-10: Trabecular bone mesh after processing pCT images in Amira and then
meshing in Ansys to create a mesh consisting of 2, 379, 141 linear tetrahedral elements.
45
(a) Sample A consisting of 67, 114 tetrahedral elements generated from a subset of pCT images which
measure 1/4 of the total image dimension in directions x and y.
#Z
(b) Sample B consisting of 2,379,141 tetrahedral
elements generated from the full pCT images.
Figure 3-11: Trabecular bone meshes for sample A and sample B.
46
(a) Three-dimensional trabecular bone mesh generated directly from images without AMIRA processing. There are visible gaps and cavities along the
surface.
(b) Three-dimensional trabecular mesh generated by
segmenting the binary images, creating a surface
mesh with AMIRA, and filling the volume in Ansys.
Figure 3-12: Zoomed in image of part of the trabecular bone mesh in the same vicinity
for meshes created using different processes. The mesh that results from AMIRA and
Ansys is a higher quality mesh.
47
48
Chapter 4
Material Behavior Modeling of
Trabecular Bone
4.1
Previous Models in Literature
Several material models have been reported in the literature that try to capture the
constitutive behavior of bone. Small deformation in bone has been studied using
the St.Venant-Kirchhoff model. Although this is a hyper-elastic model, it extends to
the nonlinear range and has been used to study the material behavior of bone for
deformations of up to 0.5% strain
[45].
Other models have been developed by using finite deformation kinematic equations.
These models consist of a purely elastic linear material model combined with a plasticity model. Yield surface information from trabecular bone has been used to implement
plasticity models such as Drucker-Prager and Mohr-Coulonb [20]. Another version
of a plasticity model [49] used a negative post-yield modulus to capture the softening
behavior. More advanced models have also incorporated rate sensitivity
49
[4] [7].
4.2
Constitutive Equations
Recent work by Johnson, et al. ?? characterizes cortical and trabecular response at
different strain rates by using a visco-elastic, visco-plastic constitutive model. They
also plot their simulation results alongside experimental values to show the correlation
[15].
350
300
* 1500 S
o s300s
250 -
*
200
0
1
o 0.1s
0.01 s~
0
*
o 0.001s
150
n
o
*
o
0
100
50
0
0.002
0.004
0.006
0.008
0.01
Strain
0.012
0.014
0.016
0.018
Figure 4-1: Stress-strain solid curve fits from the Johnson constitutive model compared with data points from the McElhaney experiment. [15]
The model has been calibrated and shown to capture the correct behavior of cortical
and trabecular bone [15]. The schematic shown in Figure 4-2 represents the model
used to capture the material behavior of bone and illustrates the elastic, visco-elastic,
and visco-plastic sections.
To capture the visco-elastic behavior of bone, two visco-elastic regimes are required.
The first, q1, represents viscosity in the low strain rate regime. The second, q2, has a
shorter time constant and is intended to capture the stiffening behavior that arises in
high strain rate cases. The model used by Johnson is Maxwell-Weichert and can be
described by Equation 4.1, where VE represents the visco-elastic component of the
50
total strain:
o-(t) =
Est
EoiVEt + TI
VE( 1 - e
)
-Egt
r/2VE2
-E
Subscript 0 represents the elastic or long-term equilibrium mechanism and subscripts
1 and 2 represent the first and second visco-elastic mechanisms, respectively. Due to
the observed dependence of yield stress on strain rate, a visco-plastic component modeled by the Ramberg-Osgood equation was also incorporated by Johnson in Equation
4.2:
j.P
T0(
o)
lo-
So
(4.2)
The VP exponent represents the visco-plastic mechanism. So is the Ramberg-Osgood
plasticity stress and m is the Ramberg-Osgood exponent. To completely describe the
material model, the strain contributions from each of the mechanisms are added in
Equation 4.3 to fully represent the model:
5=
Eo
(4.3)
+V VP
E,
Ez
111
112
E
So, m
Viscoelastic
Viscoplastic
Figure 4-2: Constitutive model schematic representing visco-elastic, visco-plastic bone
from [15]
51
Eo [GPa]
16.2
E1 [GPa]
4.4
E 2 [GPa]
23.5
Material Properties
71 [MPa s] 772 [kPa s]
132
227
So [MPa]
222
m
18.24
p [kg/m
1810
Table 4.1: Material properties specified by the user to describe trabecular bone,
calibrated with McElhaney experiments [15].
4.3
Implementation in Summit
The User-defined MATerial model subroutine (UMAT) was previously developed for
use in Abaqus. In order to make it functional for the Summit finite element framework, a few changes had to be made.
First, a new material was defined within the Materials Class and assigned an identification value at the top level of the material code. Then, a C interface was developed
and used to communicate between the Fortran UMAT and the C and C++ code.
Methods were defined to call the Fortan constitutive methods with data from the
Summit code used here. Arguments used here were consistent with the arguments of
the material subroutine. Lastly, variables were created to store properties read from
the user-defined materials file. Properties can be seen in Table 4.1.
52
3
Chapter 5
Modeling Framework
5.1
Computational Framework
In order to model dynamic impact problems, the Summit computational solid mechanics solver was used. This Lagrangian finite element solid solver was developed
by the Radovitzky group as described in
[37]
and has been implemented in parallel.
It was designed following the discontinuous Galerkin formulation and can therefore
allow for fracture. In addition to the source code, several constitutive models have
been added to describe tissues and biological structures.
5.2
Finite Deformation Numerical Formulation
In the continuum framework, the deformation gradient tensor F relates displacements
in the current configuration to displacements in the reference configuration via the
following relation:
Fi1 =
dx.
*(5.1)
j
dX1
In equation 5.1, i represents the current finite displacement configuration and I repre53
sents the reference configuration. The deformation mapping <p (X, t) is also a measure
of the current displacement as shown in equation 5.2.
Fi1
_
OXI
dxi
dXj
(5.2)
The Jacobian J of the deformation is defined as the determinant of the deformation
gradient tensor J = det(F). It represents the ratio of the change in volume in the
current configuration to the change in volume in the reference configuration. The First
Piola-Kirchhoff stress tensor relates stresses in the current configuration to areas in
the reference configuration, as shown in equation 5.3. o is the Cauchy stress.
Pa =Jo-isF7j
T
(5.3)
The problem is governed by the continuum equation for linear momentum balance.
The strong form is presented in equation 5.4. Here Vo is a material gradient, po is
initial density, and N is the unit surface normal in the reference configuration.
po'
=
V
o.pT +poB in Bo
(5.4)
Displacements are specified on Dirichlet boundary conditions: p = < on &DBo
Tractions are specified on Neumann boundary conditions: P - N =
on ONBO
Next, the weak form is presented in equation 5.5, integrated in every element and
summed over all elements. Here ch is an elementwise-continuous polynomial approximation of the deformation mapping.
6
Ph
54
represents the trial functions. Both are
discontinuous across element interfaces.
SP j - 4ho + Ph Vo6oh)dV
pjoB -dV
+E j
Ph -
6
(5.5)
NdS
"NB
eB
5.3
oL Ph h- NdS
-(
Discontinuous Galerkin
Discontinuous Galerkin (DG) is used to model separation between elements and allow
for fracture. The final formulation for DG is shown in equation 5.6 as presented in
[37].
Cohesive elements are used, along with a traction separation law (TSL), to
model fracture between elements. Fracture is initiated when the stress exceed the
critical stress for the material.
Cohesive tractions depend on the contribution of
the displacement jump to the local state of deformation. Complete decohesion (zero
traction) occurs when the displacement jump 6 is greater than a user-specified critical
displacement jump. The binary operator a is 0 when there is no fracture and 1 when
there is fracture.
Z j (Po' h -ho
(1- a
=
- N-dS
[Ph] - N-
+
1B
f61B
poB -OphdV +(
e~6
5.4
J61 B
(Ph
e)E
T([ h]) [6h]dS+
+ Ph Vo6ph)dV + a
B
e
B
Aoh -T -NdS
j
C > [h]- N-dS)
<
hs
(5.6)
NB
Solver: Newmark Scheme
To solve the structural dynamics problem, a Newmark time stepping algorithm is
used as expressed in equations 5.7, 5.8 and 5.9.
55
xn+1
n
5X++I" =
,+
+ Ati"
+ At2
+ At[(1
-
-
3e +
#in+(
Y):R" + yKn+ 1 ]
#
(59)
= M-[fext - fi"tin+1
Here, M is the lumped mass matrix, fext is the external force, and
ternal force.
(5.8)
fint
is the in-
and -y are Newmark parameters which have been set to 0 and 0.5,
respectively.
56
Chapter 6
Numerical Simulation of
Quasi-Static Test
6.1
6.1.1
Numerical Method: Dynamic Relaxation Solver
Implementation in Summit
Dynamic relaxation (DR) is an explicit iterative scheme used to obtain static solutions
in structural mechanics. This method uses the assumption that the static solution is
the steady state part of a transient response [48]. In other words, a damped dynamic
system with a constant applied force is assumed to reach static equilibrium. This is
achieved with both viscous and kinetic damping. Previous sources have investigated
optimal damping coefficients for convergence [16] [33].
Dynamic relaxation is commonly used for problems with geometrical and material
non-linearities. If the problem was numerically treated as a static problem, it would
be described by using equation 6.1. However, since the problem is treated as the equilibrium solution of a dynamic problem, the solution is obtained by solving equation
6.2. Here, M is the mass matrix, C is the damping matrix, K is the stiffness matrix,
57
and F is the applied force. P is equivalent to Kx.
Kx=F
(6.1)
Kx = F
(6.2)
Ms + Ci + P(x) = F
(6.3)
M + C+
Central difference is used to evaluate derivatives.
(--z"-
in-1/2
" =
+ X)/At
(6.4)
(-i n-1/2 + in+1/2)/At
(6.5)
The value for x can be obtained by the average value:
S=
in-1/2
2
(in-1/2 +
=n 2,-
(6.6)
in+1/2)
-n+1/2
(6.7)
The next displacement in the iterative procedure can be found using the following
expression presented in [48]:
z1/2
zn+1/2
zn+1
=AtM(F0
-
2 + cAt
n
- P4)/2
in-1/2
+ 2AtM 1(Fn
+ Atn+1/2
-
P"2)/(2 + cAt)
,if n
-
0
(6.8)
,if n
#
0
(6.9)
, for all n
(6.10)
The value for in-1/2 in equation 6.7 can be plugged in to equation 6.5 to obtain
the expression in equation 6.11. Then equation 6.11 can be plugged in to the next
displacement equation shown in 6.10. Finally this gives us the form shown in equation
58
6.12 that was used when implementing this iterative method.
in+1/2 = (zAt + 2i")/2
xn+1
xn
(6.11)
+ -z"At2 + z"At
2.
(6.12)
For rapid converge, the damping coefficient presented in [16] was used, where R is
the residual:
(xn )T ( F c= 2 (X
n§T
iR
)
1/2
.j))/(6.13)
The steps followed in the dynamic relaxation algorithm are as follows:
1. Apply boundary conditions
2. Calculate the initial residual force Ro and set current residual R to the highest
value a double can hold
3. Calculate a residual ratio R/Ro
4. While R/Ro >user set tolerance, continue, otherwise end
5. Update velocities and displacements
6. Calculate the damping coefficient c
7. Update accelerations and velocities
8. Evaluate R and R/Ro
9. n = n + 1, Return to 4
In addition to the ratio of the current residual to the initial residual R/Ro, several
other criteria are used for convergence. The rate of decrease of the R/Ro ratio is
also considered. This helps determine early convergence in continuous Galerkin cases
where the residual ratio is monotonically decreasing but is not yet close enough to
an asymptotic value. If the slope is sufficiently low, it can be considered converged.
59
Another criteria is the residual itself. If R is small to begin with, the system is already
equilibrated and the iteration loop is bypassed. Lastly the kinetic energy is used to
improve convergence. For discontinuous Galerkin, fractured elements may sometimes
detach from the mesh body with a velocity. If high kinetic energies are detected,
velocities are set to zero as suggested by [36] [33]. This essentially removes the effect of
viscous damping in the equation of motion and also contributes to convergence.
6.1.2
Example: Convergence of Prismatic Bar
Dynamic relaxation was tested with a simulation conducted on a prismatic bar shown
in Figure 6-1. The prismatic bar measures 5 m in length and 1 m in width. It consists
of 78 linear tetrahedral elements. The progression of convergence within the dynamic
relaxation scheme was observed for one time step. Within every simulation time step,
there are multiple dynamic relaxation iterations that take place until the problem has
converged to the static solution. Initially, the displacement of the nodes for which
Dirichlet boundary conditions are prescribed will create an imbalance among the
forces. Over many iterations, one can see how the pressure in Figure 6-2 and the
displacement in Figure 6-3 settle to the converged solution.
Displacements (m) Z
-0.0002
.00025
-0.0001
0
-0.0001
1-0.0002
-0.00025
Figure 6-1: Spatial distribution of the displacement field along the z direction after
convergence of DR solver, for a displacement prescribed at both extremities of the
prismatic bar along the z direction.
As mentioned before, the tolerance for convergence can be set by the user. In Figure
6-4, a few tolerance values were investigated to determine the sufficient one needed
to reach equilibrium. It was determined that values less than or equal to 1x10-'
60
1.0
1.0
Prismatic Bar with Dynamic Relaxation
1e7
1-
-
-
10 Iterations
-
15 Iterations
120 Iterations
349 Iterations
0.8
A,
0.6
1A
0.4
0.2
T113w-U1
-
0
1
3
2
Distance Along Bar (m)
4
Figure 6-2: Evolution of the pressure as a function of the position along the prismatic
bar at different iterations of the DR solver: the response reaches static equilibrium
as the number of DR iterations increases.
were reasonable choices for the tolerance and allowed the system to reach similar
equilibrium solutions for this problem.
61
0.0004
0.0002
E
E 0.0000
20
-0.0002
'~
-0.0004
0
1
2
.--
15 iterations
120 Iterations
-
349 Iterations
3
45
Distance Along Bar (m)
Figure 6-3: Evolution of the displacement as a function of the position along the
prismatic bar at different iterations of the DR solver: the response reaches static
equilibrium as the number of DR iterations increases.
a-
Prismatic Bar with Dynamic Relaxation
1
1
Tolerance le-8
0
Tolerance le-6
1500000
1450000
.-.
e-.
1400000
1
Tolerance le-5
Tolerance 5e-5
9-o Tolerance le-4
*--eTolerance
le-2
o! 1350000
1300000
1250000
1200000
0.00002
0.00004
0.00006 0.00008 0.00010
Distance Along Bar (m)
0.00012
Figure 6-4: Evolution of pressure as a function of the position along the prismatic
bar after the first time step and for different tolerance values in the DR solver.
62
Quasi-Static Simulation of Biomimetic Trabec-
6.2
ular Bone
In this section, a single tessellation unit was placed under compression. The problem
was run with both Neo-Hookean and cortical constitutive models, for comparison.
The qualitative behavior observed using Neo-Hookean is shown in Figure 6-5, although both cortical and Neo-Hookean material models exhibited a similar behavior.
In Figure 6-5(b) and Figure 6-5(c), the compressive and tensile stresses on either side
of curved trabeculae are apparent. The buckling response can be observed. Global
stress responses are shown for the Neo-Hookean case in Figure 6-6 and for the cortical
case in Figure 6-7. Unlike the Neo-Hookean case, in the cortical case the effect of geometrical softening is possible due to the buckling as well as due to the visco-plasticity
in the constitutive model. The constitutive model is key in being able to describe the
trabecular mechanical response, both qualitatively and quantitatively.
Stresses
Z (Pa)
Stresses Z (Pa)
r 000000
0
-2e+6
I-2e+6
-4000000
-4000000
(b) 1.43% applied strain
(a) 0.48% applied strain
Stresses Z (Pa)
Stresses Z(Pa)
00000
000000
r
-0
I-2e+6
2e+6
4000000
.7
-4000000
.71
(d) 6.65% applied strain
(c) 3.79% applied strain
Figure 6-5: Stress gradient result for a tessellation under compression using dynamic
relaxation and the Neo-Hookean material model.
63
Displacement vs. Force for 1xix1 Tessellation in Compression
-500
-1000
N -1500
-2000
-25001
Residual Force
Linear Reference
U
16
-0.14
-0.12
-0.08
-0.06
-0.10
Displacement (m)
-0.04
-0.02
0.00
Figure 6-6: Force vs. displacement for a ix1xi single tessellation unit in compression
using the Neo-Hookean material model.
Displacement vs. Force for ixix1 Tessellation in Compression
-200-400-600-
z
-800-
N
0
U.
-1000-1200-1400
-- Residual Force
Linear Reference
-1600
-180-.07
-0.06
-0.05
-0.03
-0.04
Displacement (m)
-0.02
-0.01
0.00
Figure 6-7: Force vs. displacement for a 1xix1 single tessellation unit in compression
using the cortical material model.
64
6.3
Quasi-Static Simulation of Biofidelic Trabecular Bone
Quasi-static simulations were run on sample A. In addition to using a tolerance of
1x10-5 for convergence, the rate of the residual ratio was also considered. Figure 6-8
shows how the residual ratio decreases as number of dynamic relaxation iterations
increases. This is shown for different constitutive models and for two simulation time
steps. On the other hand, Figure 6-9 shows the difference between the current residual
ratio and the previous residual ratio for every 1, 000 steps taken. The tolerance criteria
was set such that when the residual ratio difference reaches 6x10--6 per 1, 000 iteration
steps, then convergence is reached. Therefore, between one step and the next, the
difference should be less than 6x10- 9 .
-
102
Dynamic Relaxation Residual Convergence
'
i
e-.
e-
I
CC
uCT/4NeohookeanTimeStep0
uCT/4 Neohookean, Time Step 1
uCT/4 Cortical, Time Step 0
uCT/4 Cortical, Time Step 1
10-4[
10-5[
10-6
2000
4000
6000
8000
10000 12000
Number of Iterations
14000
16000
Figure 6-8: Dynamic relaxation convergence study for sample A for both the NeoHookean and the cortical material model.
Using the convergence criteria described above, sample A was run under compression.
Boundary conditions were applied to all nodes within 0.00033 m of the z extremities.
The evolution of stress is shown in Figure 6-10. The initial apparent stiffness is 6.92
65
1
,Dynamic
Relaxation Residual Change
I-T uCT/4 Neohookean, Time Step 0
.-.
uCT/4 Neohookean, Time Step 1
01
*-.
uCT/4 Cortical, Time Step 0
uCT/4 Cortical, Time Step 1
10
10
10
1000
20000
30000
40000
5000
60600
7000
Number of Iterations
Figure 6-9: Dynamic relaxation convergence study for sample A for both the NeoHookean and the cortical material model.
GPa. After a deformation of about 0.01 mm, which corresponds to 0.004 strain, the
response is no longer linear. This stress curve is in the vicinity of results reported in
Section 2.2.5 from the literature, but slightly stiffer.
Sample A is smaller than a representative volume element (RVE) in terms of morphological characteristics, such as volume fraction, and in terms of mechanical response.
In fact, the response of samples that are not representative tends to be stiffer than
the response of an RVE [17] [30].
Qualitative
results for the simulation are shown in
Figure 6-11. There are not as many opportunities for buckling as there are in the
larger trabecular sample. Additionally, the few trabeculae in this section are more
plate-like than rod-like and have a strong vertical orientation. These all factors which
contribute to the high stress response pertaining to this problem.
66
Dynarnic Relaxation Simulation of Trabecular Section
35
30
25
0~
-...... ..
1-
- ........
-.
20
-......
15
-
-.-.
......
-....
..
-..
..
...
......I
........
-...
.....
I,'
aJ
..
..
... -..
...
-..
....
-..
-..
-..
.....
...
10
-......-
-.
.....
..... ..
5
0'000
0.001
0.002
0.003
Strain
0.004
0.005
0.006
Figure 6-10: Evolution of the stress in the z direction o- as a function of the strain
in the z direction Ezz for a quasi-static test conducted on sample A of the biofidelic
model using the cortical bone material.
Stresses Z (Pa)
.313e+8
1e+8
-0
-1e+8
-2e+8
-3e+8
-3.8e+8
Figure 6-11: Evolution of the stress czz for sample A under compression with a maximum displacement of 5.025x10-6 using dynamic relaxation and the cortical material
model.
67
6.4
Discussion
The dynamic relaxation scheme which was implemented here allows us to solve large
parallel problems explicitly. By solving for the equilibrium response of a damped
dynamic system, a static solution was found. Dynamic relaxation was used to quantify
the stress response of a trabecular tessellation as well as a trabecular mesh based on
real bone geometry.
As explained in Section 6.3, the stress response for sample A has a stiffer response
than it does for sample B. The quasi-static result for sample A is 22.45% larger
than the stress shown in the Harrison curve [13].
However, this is an artifact of
the geometry. If we consider sample B which is larger, there is more porosity and
interconnectedness that contributes to compliance. Even in dynamics, it was shown
that the stresses for sample B are lower than those for the quasi-static case for sample
A. Dynamic relaxation can be a useful tool to investigate large quasi-static problems
but fuller geometries should be considered in order to approach the real behavior
more closely.
68
Chapter 7
Numerical Simulation of Dynamic
Test
7.1
Numerical Method: Parallel Design for Periodic Structures in Summit
Periodic microstructures possess many useful characteristics such as energy absorption
capabilities and high structural efficiency. This makes them suitable for representing
biological materials, foams, trusses, and even composites [9] [40]. Due to their wide
range of scientific applications, a computational design was created in Radovitzky's
computational framework, Summit, that allows users to model such structures. The
following section presents a design that allows the user to model a dynamics problem
for any possible periodic unit that can be tessellated in 3D space via x, y, and z
translations.
In our computational framework, each periodic unit mesh is loaded by a processor.
The result is exactly the same as if an entire tessellation of units was loaded and
then partitioned as it is done by Metis [18] for each processor. Hence, each processor
keeps a different part from the global division of the entire mesh and each processor
69
supports the resolution of the sub-problem corresponding to this sub-mesh while
communicating information with the processor neighbors.
The difference between Metis and the process outline here is that in this case, periodic
units are directly loaded to different processors as opposed to loading an entire mesh
through the use of a single processor and then distributing the partitions to the other
processors. In the latter version, the efficiency of the application is limited by the
memory available to the processor which initially loads the large mesh. This creates
a bottleneck in the system which either slows it down or prohibits the program from
continuing.
Periodicity is used here in order to individually load each processor with its portion
of the mesh, which would be a unit cell identical to that held in every processor.
Together all the unit cells make up the full tessellation.
To use this application,
communication maps must be written for the unit cell. These maps must then be
read and implemented by every partition in the application.
An application was written to handle these requirements.
Its inputs include: the
number of processors to be used in x, y, and z directions, the mesh containing the
unit cell, a file containing translation values, and a maps file. The translation file
and the maps file are automatically generated when generating the Cuitifno geometry
described previously. Another program was also created to generate a maps file for
any periodic geometry, for future extensions of this application.
For simplicity, the tessellation of a cube is shown in Figure 7-1. Processors are numbered in increasing order in the x direction, y direction, and lastly z direction.
70
(b) Adding the 6th processor
(a) Adding the 5th processor
'7e
(c) Adding the 7th processor
(d) Adding the 8th processor
Figure 7-1: Representation of Message Passing Interface (MPI) application setup for
an 8 processor simulation with a tessellated unit cube. Processors are numbered left
to right in the x-direction, then in the y-direction, and the z-direction.
7.2
Dynamic Simulation of Biomimetic Trabecular
Bone
To mimic the geometrical characteristics of trabecular bone, a periodic tessellation
was used based on the Cuitifio model. It was implemented using the parallel design
described in Section 7.1, which was developed specifically for this purpose. It was
created using 3, 4, and 6 processors in the x, y, and z directions, respectively, for a
total of 72 processors. Each processor loaded 20 tessellation units, loading a total of
1,440 tessellation units for the entire simulation. The processor numbers were chosen
such that the final size of the assembled structure was at least as large as the bone
sample used for pCT observation, i.e. 11.7 mm x 11.2 mm x 11.1 mm. Individual
71
trabeculae used in this test measure 0.196 mm in length, 0.0474 mm in radius, and
54.740 of inclination, for a bone volume fraction of 0.1086.
Automotive impact velocities can be in the range of 2 - 6 m/s. Military vehicle
peak impact velocity has been documented at 12 m/s, with a threshold for injury
of 9.7 m/s [26]. In order to be above the injury threshold, a velocity of 10 m/s was
applied at both extremities of the tessellation along the z axis such that the sample
was placed in compression. Lateral movement was not restricted. The displacement
and force components along the z axis were saved at each time step. Strain was
computed by dividing the displacement by the original tessellation length and stress
was computed by averaging the reaction force over the nominal area of the specimen.
Figure 7-2 shows the biomimetic tessellation prior to any applied velocities. Figure
7-3 shows the spatial distribution of the stress field of the component a22 within
the biomimetic bone microstructure. The presence of negative and positive stresses
indicate the existence of buckling which can be seen along individual trabeculae.
Figure 7-4 presents the dynamic response of the tessellation under compression by
plotting the stress component o-2 as a function of the strain component 622.
The oscillations observed at smaller deformations, at the beginning of the simulation,
are characteristic of a dynamic simulation and hence did not appear in the results
from the quasi-static simulations previously introduced in Chapter 6.
The initial apparent stiffness is calculated by diving rzz by c2 in the pre-yield stress
response. We obtain a stiffness between 0.2 and 0.6 GPa. Yield occurs at approximately 0.33 mm of deformation which corresponds to a peak stress of 4 MPa, after
which the structure buckles and strain softening is apparent.
72
ii:
(
Figure 7-2: Mesh of biomimetic bone tessellation comprised of 1,440 tessellation units
that corresponds to a total sample size of 11.7 mm x 11.2 mm x 11.1 mm. The current
image is a partial view of the entire structure, displaying 20 of the 72 processors using
Paraview.
Stresses Z(Pa)
1
+8
0
j-2e+8
-4e+8
Figure 7-3: Spatial distribution of the stress components a,, in the biomimetic bone
microstructure. The current image is a partial view of the entire structure, displaying
20 of the 72 processors using Paraview and demonstrating displacement.
73
Eu
N
fLn
Strain
Figure 7-4: Evolution of the stress component rz, as a function of the strain component ezz from the dynamic simulation conducted on the biomimetic bone tessellation
with an initial velocity of 10 m/s applied to both extremities of the structure along
the z-axis.
74
7.3
Dynamic Simulation of Biofidelic Trabecular
Bone
7.3.1
Trabecular Bone Sample A
Dynamic simulations were first conducted on sample A, a real trabecular bone structure which is 64 times smaller than sample B in volume, via an image-based biofidelic
microstructural model. The anisotropy of the mechanical response was investigated
by conducting three distinct simulations where the same velocity of 10 m/s was applied under compression but along each of the three axes x, y, and z. Figure 7-5 shows
the dynamic response of sample A with the evolution of the stress component along
the loading direction as function of the strain component along the same direction
for each of the simulations with loading in the x, y, and z directions. Here the biofidelic trabecular bone microstructure is anisotropic with a largest and lowest stiffness
along the z and x-axis, respectively. Indeed, the stiffnesses along the x and z axes
are 5.4 and 2.0 GPa and are measured using the same method presented previously
in Section 7.2. Maximum stresses are 16.14, 26.28, and 33.18 MPa in the x, y, and z
directions. Figure 7-6 shows the spatial distribution of stress components along the
loading direction within sample A of the biofidelic bone microstructure.
7.3.2
Trabecular Bone Sample B
In order to underline the size effects of the trabecular bone microstructure and to
obtain a mechanical response which is closer to the one of a representative volume element, dynamic simulations were conducted on the entire trabecular bone microstrueture observed via pCT. The same boundary condition velocity of 10 m/s, which was
described previously in Section 7.2 for sample A, was applied to sample B. In addition, velocities of 3.3 m/s and 1.1 m/s were also considered in order to observe the
strain rate dependency in the dynamic response of sample B.
75
N
oloo
0.0os
0.015
0.0'io
Strain
0.020
Figure 7-5: Evolution of the stress component o-z as a function of the strain component ez from the dynamic simulation conducted on sample A at 10 in/s in loading
directions x, y, and z.
Stresses
X Pa)
688+8
Stresses
Z (Pa)
Stresses
V (Pa)
I
2e+8
0
2e+8
-2e+8
-2e+B
-4e+8
-4e+8
4.86e+8
d4e+8
-5.(1e+8
(a) x Compression
2e+8
0
-4.89e+8
(b) y Compression
(c) z Compression
Figure 7-6: Spatial distribution of the stress component along specific loading directions x (a), y (b), and z (c) within sample A of the biofidelic microstructure.
Each simulation was run on 50 processors and took approximately 24, 52, and 64
hours of real computation time for applied velocities of 10 m/s, 3.3 m/s and 1.1 m/s,
respectively. Velocities 10 m/s, 3.3 m/s and 1.1 m/s correspond to strain rates of
1, 000/s, 330/s, and 110/s, respectively.
Figure 7-7 shows the spatial distribution of the stress component
-zz within the
microstructure for a velocity of 10 m/s. Buckling of the truss is observed within the
microstructure, with tensile and compressive stresses on either side of the trabeculae.
The simulations conducted at different strain rates have similar qualitative features
76
with respect to post-yield behavior. Figure 7-8 presents the evolution of the stress
along the z-axis as a function of the strain along the same direction for the three
different velocities.
The peak of stress is equal to 19.62, 17.20, and 15.86 MPa for velocities of 10, 3.3, and
1.1 m/s, respectively which demonstrates the influence of the strain rate dependency
on the dynamic response of the trabecular bone, manifested by an increase in the
peak stress with respect to the velocity. Initial stiffnesses for velocities of 10, 3.3,
and 1.1 m/s fall in the range of 2.1-3.3, 1.5-2.4, and 2.0-2.2 GPa, respectively. We
can see that as the strain rate of the simulation increases, the upper bound of the
apparent stiffness also increases, thus reflecting the relationship between increased impact velocity and decreased compliance. Furthermore, for larger deformations, some
geometric softening can be observed in Figure 7-9 which results from the buckling
observed in the microstructure.
Stresses Z (Pa)
e+8
2e+8
0
-2e+8
-4e+8
Figure 7-7: Simulation result showing uCT sample B under dynamic compression
with 10 m/s applied to the z extremities at time 2.89 x 10- seconds.
Figure 7-10 shows the deformation of the microstructure pertaining to sample B for
different time steps in a simulation with an applied velocity of 10 m/s. The thin layer
of unstressed region at the extremities along the z axis represents the 0.0005 m section
that boundary conditions were applied to. Trabeculae thicknesses vary throughout
77
Stress Result for Full uCT Dynamic Simlation
20
15
N
...............
..
....... ..............
10 F-
.. .. ..
L,
5
-
I
0.005
0.010
uCT v=10m/s dE/dt=1000/s
uCT v=3.3m/s dE/dt=330/s
uCT v=1.1m/s dE/dt=110/s
0. 015
0.020
Strain
0.025
0.030
0.0 35
Figure 7-8: Evolution of stress o as a function of strain e,2 for sample B pCT
simulations run at different strain rates.
Numerical uCT Results
uCT
uCT
uCT
uCT
L] BVTV [%]
10.86%
10.86%
10.86%
Mesh
I[ ]
Simulation 1 1,000
Simulation 2 330
Simulation 3 110
1 [mm]
10.0
10.0
10.0
d [mm]
8.0
8.0
8.0
Table 7.1: Data from trabecular bone simulations conducted on sample B showing
strain rate i, stiffness, maximum stress -, bone volume to total volume percentage,
sample length, and sample diameter.
sample B and there is a tendency to buckle towards the positive x direction since the
trabeculae on the opposite side are thinner. The results obtained for each simulation
are summarized in Table 7.1.
78
Stress vs. Strain for uCT with v=3.3m/s dE/dt=3:
10 -
2
N
A
-
6
2 -
oo
0.01
0.02
Strain
Figure 7-9: Evolution of stress -,, as a function of strain E,2 for pCT simulation
on sample B with a strain rate of 330/s demonstrating post-yield softening due to
buckling.
79
L
Stresses Z (Pa)
Stresses Z(Pa)
10
+8
0
I-2e+8
+8
-2e+8
-4e+8
-4e+8
-59+8
-5e+8
(b) Time = 2.88721x10-
(a) Time = 0 s
5
s
I
Stresses Z (Pa)
Stresses Z (Pa)
0
-2e+8
e+8
0
-2e+8
-4e+8
-5e+8
-4e+8
-5e+8
.X
Y
(d) Time = 7.21798x10-
(c) Time = 5.77439x10-5 s
5
s
Stresses Z (Pa)
Stresses Z (Pa)
e+8
0+8
0
-2e+8
-2e+8
-4e+8
-5e+8
-4e+8
-59+8
(f) Time = 0.000129923 s
(e) Time = 0.00010827 s
Figure 7-10: Spatial distribution of stresses o-22 from dynamic simulation on biofidelic
sample B with a velocity boundary condition of 1.10 m/s.
80
7.4
Dynamic Fracture Simulations using Discontinuous Galerkin
7.4.1
Fracture Simulation of Biomimetic Unit Cell
In order to test the constitutive material bone behavior with the DG method for
initiating cracks during the simulation, dynamic simulations were conducted using
DG on a single unit-cell of our biomimetic morphological model of trabecular bone.
In this simulation, an impact velocity of 10 m/s was applied to the z-axis extremities
of a unit cell mesh consisting of 4,734 tetrahedral elements.
Figure 7-11 shows the spatial distribution of the stress component o22 within the
single unit-cell after 10 and 30 time steps corresponding to 9.2x10- 7 and 2.6x10-6 S.
These contour plots display cracked interface elements highlighted in the figures and
allow us to spatially localize the cracks inside the trusses of the unit-cell. It is apparent
that cracks initially develop in crevices such as the sides where the struts meet the
handle of the unit cell. There is also some crack development at the top where all
three struts meet. The DG framework previously developed and implemented in
Summit is running correctly with the constitutive material behavior and hence will
be used on both, the larger biomimetic model and the biofidelic model of the bone
morphology.
7.4.2
Fracture Simulation of Biofidelic Trabecular Bone
Here, a dynamic simulation with DG was conducted on sample A of the biofidelic
model of trabecular bone morphology by applying an impact velocity of 10 m/s along
the z axis extremities and running on 4 processors. Appropriate fracture properties
based on age were chosen to be c, = 43.5 MPa for critical stress and 6c = 0.033 mm
for the critical displacement opening [47].
Figure 7-12 shows the evolution of the resulting stress o-22 as a function of the strain
81
Stresses Z (Pa)
0000000
Stresses Z (Pa)
10000000
7500000
17500000
5000000
5000000
2500000
2500000
00
-200000
(a) 10 Timesteps
(b) 30 Timesteps
Figure 7-11: Spatial distribution of the stress component a-z within the trusses of
one single unit-cell of the biomimetic model of trabecular bone after 10 and 30 time
steps of a dynamic simulation with DG. The opened interface elements characterizing
cracks are highlighted in green in the contour plots.
6zz from the dynamic simulation run with DG. In addition, the dynamic response
previously obtained using the classical continuous Galerkin (CG) method is also plotted. As expected, the dynamic CG response coincides with the dynamic DG response
at the first steps of deformation while cracks have not yet appeared. At a deformation less than 1%, the dynamic DG response becomes less stiff and manifests more
softening which is characteristic of cracks that appear in the microstructure.
The maximum stress is determined from the curves and values obtained are 29.51
MPa for the CG response and 25.34 MPa for the DG response, showing that the
apparition of cracks inside the microstructure has an effect not only on the evolution of the stiffness but also on the maximum stress before yielding. In fact, yield
and post-yielding softening occur sooner in DG than in CG and softening is more
significant in DG due to the material softening caused by the cracks. The material
softening that appears in DG with the apparition of cracks is added to the structural
softening that also appears in CG and is due to some buckling of the trusses in the
microstructure.
Figure 7-13 shows the spatial distribution of the stress component -zz within sample
A of the biofidelic bone model from the dynamic simulation with DG, at a time step
of 16,000 corresponding to 2.531x10- 6 s. More precisely, it captures the compressive
82
Stress vs. Strain
30
0
N 15
n
-
-
-
-
. . ... .
....
Ln
0
. . . ..
.. . . . ...
... ....
-
5
CG v=10m/s dE/dt=4000/s
DG v=10m/s dE/dt=4000/s
ooo
0.005
0.010
Strain
0.015
0.020
Figure 7-12: Evolution of stress o22 with respect to strain Ezz for a dynamic simulation
on sample A using both continuous Galerkin (CG) and discontinuous Galerkin (DG)
in order to compare the effect of allowing for discontinuities.
and tensile stresses on either side of the plate-like trabeculae which are responsible for
the buckling response. Fractured sections have been visualized in red in Figure 7-14
in order to highlight locations in which element separation has exceeded 3.3x10- 6 m
and make them more apparent. Although the critical separation distance was set to
3.3x10-5 m, highlighting a separation of 3.3x10- 6 m includes both fractured interfaces
and interfaces soon to be fractured. Some of the fractured areas appear to be either
on the tensile side of bent trabeculae or propagating radially outward from closed
cellular regions.
83
Stresses Z (Pa)
(e+8
|e+8
i
10
-2e+8
-3e+8
V
%
Figure 7-13: Spatial distribution of stresses o-22 for a DG dynamic simulation of
sample A under compression at a time of 2.531x10-6 S.
Figure 7-14: Visualization of the fracture surfaces within sample A of trabecular bone
for a compressive dynamic simulation at time 2.531x10- 6 s. Interfaces with a minimal
separation of 3.3x10- 6 between elements are highlighted in red.
84
7.5
Discussion
When it comes to the mechanical behavior of human trabecular bone, a number
of challenges have been identified in the literature. While the results presented in
this study do not answer every open question, they are steps in the right direction.
The high strain rate response of trabecular bone, which is of the utmost interest
to soldiers on the field, is characterized and studied. With the model developed
in this study, parallel large scale finite element simulations are used to study the
behavior of bone with an appropriate visco-elastic visco-plastic constitutive model.
Large simulations conducted here use meshes consisting of up to 2.4 million elements.
Spatial stress distributions are observed to determine areas of buckling in individual
trabeculae. Global stresses are also studied to examine yield and post-yield behavior
such as geometrical softening. Effects of trabecular anisotropy and multi-axial failure
are among several issues identified as challenging areas in the literature [19]. While
multi-axial cases are not investigated in this study, the model developed here has
the ability to handle that case. As mentioned in Chapter 2 and as observed in our
simulations, the bone is anisotropic and the preferential direction generally matches
the main loading direction of the bone structure. Given that bone grows and develops
according to the forces it withstands, it is reasonable for the direction which supports
normal forces in the body to be the most stiff.
As expected, the dynamic results reach higher peak stresses than the quasi-static
experimental results presented in Chapter 2. What is of interest to note is that although the dynamic response reaches higher stresses, it is still within the vicinity of
the results from the literature. The average stress in the literature was 15.74 MPa.
For the highest strain rate tests, 1, 000/s, the maximum stress was 19.62 MPa. This
is 24.7% higher than what was reported for a quasi-static case. Medium and low
strain rates resulted in maximum stresses which were only 9.28% and 0.76% higher
than the average quasi-static response in the literature. Therefore, the slower dynamic experiments yield results that approach the static response more closely. This
indicates that the results mentioned here are within the scope of accepted stress re85
sponses for trabecular bone, especially when you consider variations in sample origins
and mnicrostructure.
The results presented in study demonstrate how high impact loads can crush the
trabecular structure and result in fractures, similar to the lower extremity injuries
occurring on the battlefield. This is accomplished because the model incorporates
fracture through the use of discontinuous Galerkin elements and a traction separation
law. Modeling fracture in the 3D bone structure opens new opportunities for analysis
in that area. More information about the number of cracks, direction of the crack, and
evolution of the crack path can be gathered from the simulations presented here.
86
Chapter 8
Conclusions
8.1
Discussion
Improvised explosive devices can be extremely harmful to soldiers on the field. Statistical reports from Operation Iraqi Freedom and Operation Enduring Freedom indicate
that a significant number of lower extremity injuries result in fracture. The Institute
for Soldier Nanotechnologies (ISN), in collaboration with the Army Research Laboratory (ARL), has focused on investigating bone under high strain rates to better
understand the mechanics and to be able to devise improved protective gear in the
future.
Experimental replications of conditions experienced on the battlefield are practically
unfeasible and very costly. In such situations, computational models can be extremely
useful. Even if the model is simplified in comparison to reality, it can still be used
to study the effect of different impact rates, loading directions, and geometrical samples.
Towards the aim of understanding lower extremity injuries, a computational model
was developed which uses a biofidelic trabecular bone geometry constructed from
pCT medical images. A biomimetic trabecular structure is also investigated based on
the the model proposed by Wang and Cuitifio
87
[50].
The model is run using Summit,
a suite of computational solid mechanics solvers developed by the Radovitzky group
[37].
The trabecular bone samples were placed in quasi-static loading conditions
and dynamic loading conditions at 10m/s, 3.3m/s, and 1.1 m/s. The samples were
modeled past the point of fracture initiation, which was regarded as a challenge in
the literature. Additionally, samples were loaded in different directions in order to
observe the effect of anisotropy.
Limited quantities of quasi-static experimental data were available for comparison.
Although samples range in volume fraction, anatomical site, and loading rate, our
results fall in the same stress range.
The average stress in the literature was 15.74 MPa. However, those results are quasistatic and it is expected the dynamic simulations should reach higher stresses. Not
only is that the case, but it is interesting to note the difference by which the dynamic
results exceed the quasi-static results. For a test with a strain rate of 1, 000/s, the
maximum stress exceeds the quasi-static average by 24.7%. For strain rates of 330/s
and 110/s, the stress exceeds the quasi-static average by only 9.28% and 0.76%,
respectively. The slowest loading rate used in the dynamic experiments exceeds the
average stress reported in literature by less than one percent. Considering the wide
variation in sample origins and microstructural architecture, this result suggests that
the model can be used to represent bone mechanical behavior.
In addition to reaching peak stresses, yield and post-yield behavior is captured by
the model. Buckling is apparent within the structure. It can both be observed on
the trabeculae through visualizations and through its impact on the stress response.
The initiation of fracture and, more importantly, the propagation of fracture can be
analyzed in the 3D structure. The difference in the mechanical response between
bone structures that are and are not allowed to fracture is highlighted.
The results presented in this study suggest that the model captures the response
of bone, with the stress results falling within a reasonable range in comparison to
published data. In addition, details which had not been captured before about postyield fracture behavior were presented. This model serves as the base for adding more
88
levels of hierarchy and ultimately devising protective solutions to mitigate the effects
of blast waves.
8.2
8.2.1
Recommendations for Future Work
Quasi-static Fracture Simulation
Dynamic relaxation can be used to model quasi-static problems that involve fracture.
Although progress has been made in terms of optimizing convergence rates through
damping coefficients, issues remain when the problem involves fracture. If the residual
ratio is the only convergence criteria used, problems arise when fracture occurs. As
cracks grow, there is an imbalance between the internal and external forces.
The
residual will decrease until it cannot reconcile the forces in the cracked problem. It
will then oscillate as shown in Figure 8-1, contrary to the continuous Galerkin case
in which the residual monotonically decreases. If the residual tolerance is decreased,
this oscillation problem will still emerge as displacement openings between elements
increase, as shown in Figure 8-2.
As suggested by [33], the kinetic energy of the system can be monitored such that
when it increases the velocities can be reduced. Following this approach, convergence
can be achieved even in the discontinuous Galerkin case. A converged prismatic bar
with fracture is shown in Figure 8-3. Although the method is functional, improvements to the convergence rate must be made in order for larger problems to become
feasible.
8.2.2
Large Scale Bone Modeling
This study focused on the small scale characterization of bone in the trabecular
region. The work done here can be extended to include macroscale studies of the
tibia. The tibia can include both trabecular and cortical regions. The regions would
89
0.0045
Dynamic Relaxation Convergence, Quarter uCT
1
Residual Force
0.0040 -0.0035
-
--
-
.
-
-
-
-
0.0030
z
0.0025 0.0020 -
-
0.0015
.
-
.-
0.0010
0.0000 0
10000
20000
30000
40000
Number of Iterations
50000
60000
Figure 8-1: Single time-step showing several iterations of dynamic relaxation up to
convergence, applied to sample A with 67,114 tetrahedral elements prior to mesh
scaling.
require different material properties, but the same constitutive model developed by
the Socrate group may be used [15].
Preliminary work in this area was done as
part of this study. The stress response for a spall case is shown in Figure 8-4. Of
particular interest is the midsection of the bone. Velocities were chosen such that
the stress waves propagating through the bone would amplify at the center and reach
a value slightly above the critical stress. This would cause fracture in the center of
the bone. Figure 8-5 demonstrates this mechanism and highlights the elements which
have been fractured in red. Fractured elements have exceeded a critical separation
distance. This work can be extended to develop a connection between bone at the
macroscale and trabecular at the microscale.
90
0.0045
Dynamic Relaxation Convergence, Quarter uCT
Step
.Step
Step
Step
0.00400.0035
0,
1,
2,
3,
0 openings
25718 openings (2.5e-8)
25718 openings (5.4e-8)
25718 openings (7.87e-8)
0.0030
-
150.0025
--
--
-
.........
.....
-.........--.
0.0020
. . . . . .-.
0.0015
-
.
0.0010
0.0005
0.0000
10000
20000
50000
40000
30000
Number of Iterations
60000
70000
80000
Figure 8-2: Four time-steps showing several iterations of dynamic relaxation up to
convergence, applied to sample A with 67,114 tetrahedral elements prior to mesh
scaling. The legend indicated number of interface element openings and average
opening distance in meters.
Stresses Z (Pa)
1367.886
4000
I2000
13000
1000
0
-888.939
Figure 8-3: Evolution of stress o2 for the converged solution of a prismatic bar run
with dynamic relaxation and DG.
91
stresses Magnitude
910530
6e+7
4e+e7
2e+7
0
Time: 0.00789 ms
Figure 8-4: Dynamic simulation of a full tibia bone consisting of 145, 669 tetrahedral
elements and 1, 531, 768 nodes.
92
(a) Time = 49ps
(b) Time = 69ps
(c) Time = 80p~s
(d) Time = 99ps
Figure 8-5: Dynamic simulation on full tibia mesh with fracture. Bulk elements are
shown in gray and fractured cohesive elements are shown in red.
93
94
Appendix A
Neighbor Numbers for
Communication Map
For the biomimetic tessellation application, neighboring processor communication
maps were written after the geometry was created. This section explains how the
neighbor numbers were assigned. This assumes your point of reference is the "X" in
the middle of the blue layer shown in Figure A-2(c). It is possible for you to have
neighbors above, below, and on the same z-plane. Figure A-1 demonstrates how all
three layers fit together. Here, each small piece represents a different processor. In
relation to the blue "X", Figure A-2 shows which neighbor numbers correspond to
which physical relation, i.e. directly above (neighbor #12),
left (neighbor #20).
95
diagonally below to the
Figure A-1: Full assembly of three layers of processors showing the spatial relationship
between one another.
yI
3
(a) Axis
(b) Top Layer
(c) Middle Layer
(d) Bottom Layer
Figure A-2: Neighbor numbers for processors based on their relation to "X".
96
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