modeling and characterization of motor vehicle collisions
TRANSCRIPT
Modeling and Characterization of Motor Vehicle Collisions: Analytical, Numerical and Experimental Investigations
by
Mohamed Taher Zaki Hassan, MSc
A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy
Mechanical and Industrial Engineering University of Toronto
© Copyright by Mohamed Taher Zaki Hassan 2019
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Modeling and Characterization of Motor Vehicle Collisions:
Analytical, Numerical and Experimental Investigations
Mohamed Taher Zaki Hassan, MSc
Doctor of Philosophy
Mechanical and Industrial Engineering
University of Toronto
2019
Abstract
The increasing number of fatalities and injuries in motor vehicle accidents highlights the
importance of studying occupants’ response during collisions; specifically, the head and neck
due to their vulnerability. In this research program, we study the occupant kinematic response
and kinetic behavior during frontal, rear and lateral motor vehicle collisions. In view of its
severity and commonality, rear-end collisions leading to whiplash, have been given additional
attention. The work, which is conducted analytically, numerically and experimentally, is divided
into four integrated sections. First, an analytical multibody dynamics model (MBD) of the head
and the neck was developed to determine the head response during simulated frontal, lateral and
rear impacts. Second, extensive nonlinear dynamic finite element (FE) simulations were carried
out to study the occupant response in the aforementioned impact scenarios using detailed vehicle
and occupant numerical models. Third, a novel head-neck prototype was developed to
experimentally validate the newly developed MBD and FE models. Fourth, a novel shock
absorber was developed using foam-filled frusta. The FE simulations were extended to examine
a number of safety strategies, involving seat belt, head restraint, airbag and shock absorber on the
occupant response during rear-end collisions.
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The outcomes of this work provide greater understanding of occupants’ neck injury mechanism
in vehicle collisions. Our results further reveal that in frontal impacts, the capsular ligament (CL)
and the interspinous ligament (ISL) are vulnerable to injury, while in lateral collisions, the
highest ligament elongation was reported for the CL, exceeding a stipulated injury threshold. In
rear-end collisions, the anterior longitudinal ligament was at risk of injury, as well as the ISL and
the CL during neck flexion. Our work reveals that the frontal airbag plays an important role in
preventing excessive neck flexion in rear impact, and that the newly proposed shock absorber
can lead to improved occupant safety. Additionally, it further shows that the experimentally
developed head-neck 3D printed prototype response is in good agreement with the MBD and FE
predictions. The prototype can be used as the core for developing head-neck models in future
anthropomorphic test dummies.
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Dedication
To my family…
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Acknowledgments
First, I would like to offer my sincerest appreciation and gratitude to Professor Shaker Meguid
for his continuous support as a mentor and a supervisor. This research would not have been
conducted without his guidance and assistance. Specifically, I wish to acknowledge the efforts of
Professor Meguid in directing my research, proof reading my thesis, and for his expert and
instructive input to the research. I also wish to thank Professors Craig Simmons and Lidan You
for their input.
I am also very thankful to all my colleagues in the Mechanics and Aerospace Design Lab who
were helpful and encouraging, specially: Mr. Gabriel Shi, Mr. Pieter Verberne, Mr. Prayers Roy,
Dr. Ahmed Alian and Mrs. Valerie Meguid.
My family has been my source of encouragement, support and love all the way. I will be thankful
to you all my life.
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Table of Contents
Abstract ......................................................................................................................................................................... ii
Dedication .................................................................................................................................................................... iv
Acknowledgments ......................................................................................................................................................... v
Table of Contents ......................................................................................................................................................... vi
Co-Authorship and List of Publications ....................................................................................................................... xi
List of Tables .............................................................................................................................................................. xiii
List of Figures ............................................................................................................................................................. xv
List of Appendices ...................................................................................................................................................... xxi
List of Abbreviations ................................................................................................................................................. xxii
Chapter 1. Introduction and Justification ............................................................................................................ 1
1.1. Introduction ..................................................................................................................................................... 1
1.2. Justification of the Study ................................................................................................................................. 3
1.3. Research Objectives ........................................................................................................................................ 4
1.4. Method of Approach ....................................................................................................................................... 5
1.5. Thesis Layout .................................................................................................................................................. 6
References................................................................................................................................................................ 7
Chapter 2. Literature Review.............................................................................................................................. 9
2.1. Definition of Spine Anatomy .......................................................................................................................... 9
Vertebral Column ............................................................................................................................................ 9
Vertebrae ......................................................................................................................................................... 9
Intervertebral Disc ......................................................................................................................................... 10
Ligaments ...................................................................................................................................................... 11
Muscles 12
2.2. Overview of Motor Vehicle Collision Injuries .............................................................................................. 12
2.2.1. Frontal Collisions ............................................................................................................................ 12
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2.2.2. Rear Collision Injuries .................................................................................................................... 14
2.2.3. Lateral Collisions ............................................................................................................................ 15
2.3. Neck Injury Criteria ...................................................................................................................................... 17
2.4. Multibody Dynamics Models ........................................................................................................................ 19
2.5. Finite Element Models .................................................................................................................................. 20
2.6. Experimental studies ..................................................................................................................................... 22
2.6.1. In Vivo ............................................................................................................................................ 23
2.6.2. In Vitro ............................................................................................................................................ 24
2.6.3. Anthropometric Test Dummies ....................................................................................................... 25
2.7. Crashworthiness and Energy Absorption ...................................................................................................... 26
2.7.1. Collapse of circular thin-walled structures ..................................................................................... 26
2.7.2. Collapse Load and Energy Absorption ........................................................................................... 27
2.7.3. Aluminum foam .............................................................................................................................. 29
2.7.4. Foam-filled thin-walled structures .................................................................................................. 30
References.............................................................................................................................................................. 33
Chapter 3. Paper #1: Nonlinear Multibody Dynamics and Finite Element Modeling of Occupant
Response: Part I - Rear Vehicle Collision ............................................................................................................. 43
Abstract .................................................................................................................................................................. 43
3.1. Introduction ................................................................................................................................................... 44
3.2. Multibody Dynamics Model ......................................................................................................................... 45
3.2.1. Single DOF Model .......................................................................................................................... 48
3.2.2. Two DOF Model ............................................................................................................................. 53
3.2.3. Rotational Limits of Intervertebral Joints ....................................................................................... 55
3.2.4. Solver .............................................................................................................................................. 55
3.3. Finite Element Modeling ............................................................................................................................... 56
3.3.1. FE Modeling of Vehicular Collision ............................................................................................... 56
3.3.2. FE Modeling of Occupant Response ............................................................................................... 58
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3.4. Results and Discussion .................................................................................................................................. 61
3.4.1. Validation of MBD and FE models ................................................................................................ 61
3.4.2. Occupant Response in vehicle-to-vehicle impact ........................................................................... 64
3.5. Conclusions ................................................................................................................................................... 69
Acknowledgment ................................................................................................................................................... 69
References.............................................................................................................................................................. 69
Addendum to Chapter 3 ............................................................................................................................................... 74
A3.1 Occupant Protection ...................................................................................................................................... 74
A3.2 Results and Discussion.................................................................................................................................. 75
Appendices ............................................................................................................................................................ 77
Appendix 3.1: Population of Matrices of 1 DOF Model ............................................................................... 77
Appendix 3.2: Matrices population of 2 DOF model .................................................................................... 78
Chapter 4. Paper #2: Nonlinear Multibody Dynamics and Finite Element Modeling of Occupant
Response: Part II – Frontal and Lateral Vehicle Collisions ................................................................................... 82
Abstract .................................................................................................................................................................. 82
4.1. Introduction ................................................................................................................................................... 83
4.2. Multibody Dynamics Modeling .................................................................................................................... 84
4.3. Finite Element Modeling ............................................................................................................................... 88
4.3.1. Vehicle Crash Simulation ............................................................................................................... 88
4.3.2. Occupant Response ......................................................................................................................... 90
4.4. Results and Discussion .................................................................................................................................. 92
4.4.1. Validation of MBD model .............................................................................................................. 92
4.4.2. Occupant Response ......................................................................................................................... 95
4.4.3. Risks of Injury .............................................................................................................................. 100
4.5. Conclusions ................................................................................................................................................. 108
Acknowledgment ................................................................................................................................................. 109
References............................................................................................................................................................ 109
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Chapter 5. Experimental Characterization of Cervical Spine Kinematics in Whiplash Trauma using a
Sled System 114
Abstract ................................................................................................................................................................ 114
5.1. Introduction ................................................................................................................................................. 114
5.2. Details of the Head-Neck Prototype ............................................................................................................ 116
5.2.1. Skull and Vertebrae ....................................................................................................................... 116
5.2.2. Intervertebral Disc ........................................................................................................................ 118
5.2.3. Ligaments...................................................................................................................................... 119
5.2.4. Facet Joint ..................................................................................................................................... 121
5.2.5. Neck Stabilization System ............................................................................................................ 121
5.3. Impact Sled.................................................................................................................................................. 122
5.4. Imaging and Sensory ................................................................................................................................... 124
3.1.1 High-speed Imaging ...................................................................................................................... 124
3.1.2 Accelerometers ............................................................................................................................. 125
5.5. Results and Discussion ................................................................................................................................ 126
5.5.1. Comparison to Multibody Dynamics and Finite Element Simulations ......................................... 127
5.6. Conclusions ................................................................................................................................................. 131
References............................................................................................................................................................ 132
Chapter 6. Paper #3: Effect of Seat Belt and Head Restraint on Occupant’s Response during Rear-End
Collision 135
Abstract ................................................................................................................................................................ 135
6.1. Introduction ................................................................................................................................................. 135
6.2. Model and Materials.................................................................................................................................... 137
6.3. Results and Discussion ................................................................................................................................ 141
6.4. Application of Neck Injury Criteria ............................................................................................................ 150
6.5. Conclusions ................................................................................................................................................. 153
Acknowledgment ................................................................................................................................................. 154
References............................................................................................................................................................ 154
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Chapter 7. Paper#4: Effect of Interfacial Friction and Fold Penetration on the Progressive Collapse of
Foam-Filled Frustum using Kinematically Admissible Model ............................................................................ 156
Abstract ................................................................................................................................................................ 156
7.1. Introduction ................................................................................................................................................. 156
7.2. The kinematically admissible model for foam-filled frustum ..................................................................... 159
7.3. Improved foam-shell interaction ................................................................................................................. 165
7.3.1. Foam-Shell Interaction Mechanism .............................................................................................. 165
7.3.2. Interaction I: penetration of the shell into the foam ...................................................................... 166
7.3.3. Interaction II: interfacial friction between the shell and the foam ................................................ 168
7.4. Results and Discussions .............................................................................................................................. 170
7.4.1. Validation of the proposed kinematically admissible model ........................................................ 170
7.4.2. Effect of the revised fold proportion for interaction on the crush behaviour ................................ 174
7.4.3. Effect of the foam/shell friction on energy absorption.................................................................. 175
7.5. Conclusions ................................................................................................................................................. 178
Acknowledgements .............................................................................................................................................. 178
References............................................................................................................................................................ 179
Chapter 8. Conclusions, Contributions and Future Work ............................................................................... 181
8.6. Summary of Research Findings .................................................................................................................. 181
8.7. Thesis Contributions ................................................................................................................................... 182
8.8. Future Work ................................................................................................................................................ 183
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Co-Authorship and List of Publications
This is to certify that the work presented in this thesis was conducted by me, Mohamed T.Z.
Hassan. Dr. Shaker Meguid is my thesis supervisor and a co-author of all papers included in this
dissertation. He was instrumental in setting up the entire project and the ideas behind it,
examining progress, proof reading the manuscripts and contributing to response to reviewers.
Mr. M. G. Shi is a co-author of papers # 1 and 2 that make up part of this thesis. He was a Master
student working in MADL with a specific focus on the overall behaviour of car collisions. In
addition to the guidance offered by Dr. Meguid, I spent considerable hours teaching him the
fundamentals of multibody dynamics, and the theory and practice of conducting high fidelity 3D
finite element simulations involving numerical human model using LS-DYNA. He provided
some limited input to the modeling and analysis phases of the research, and contributed to the
technical content and validity of the hypotheses considered in Chapters 3 and 4. I can testify that
I conducted the analyses and prepared the initial drafts of the manuscripts of papers 1, 2 and 3,
and contributed significantly to all aspects of the research in paper #4, which involved Dr. Yang
from Tongji University. For paper #4, I carried out extensive experimental work to validate the
analytical model, provided necessary technical and theoretical input during the preparation and
revision of the manuscript collaboratively with the co-authors. In particular, I conducted
mechanical tests using empty and foam filled frusta under displacement control as well as
compiled the test data and compared my crashworthiness findings with the numerical
predictions. I have also contributed to the response to reviewers. In almost all the revisions, the
reviewers acknowledged both the originality and importance of the work to the future of motor
vehicle safety. This thesis is a compilation of my articles that have been published during the
tenure of my doctorate.
LIST OF PUBLICATIONS:
ARCHIVED JOURNAL PUBLICATIONS:
1. Mohamed T.Z. Hassan, Mo Gabriel Shi, and S.A. Meguid, Nonlinear multibody dynamics
and finite element modeling of occupant response: part I—rear vehicle collision, International
Journal of Mechanics and Materials in Design, 15, pp. 3–21, 2019.
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2. Mo Gabriel Shi, Mohamed T.Z. Hassan, and S.A. Meguid, Nonlinear multibody dynamics
and finite element modeling of occupant response: part II—frontal and lateral vehicle
collisions, International Journal of Mechanics and Materials in Design, 15, pp. 23–41, 2019.
3. Mohamed T.Z. Hassan and S. A. Meguid, Effect of Seat Belt and Head Restraint on
Occupant’s Response During Rear-End Collision”, International Journal of Mechanics and
Materials in Design, 14, pp. 231–242, 2018.
4. F. Yang, M. Wang, M. T. Z. Hassan, S. A. Meguid, A. M. S. Hamouda. Effect of Interfacial
Friction and Fold Penetration on the Progressive Collapse of Foam-Filled Frustum using
Kinematically Admissible Model, International Journal of Crashworthiness, 23, pp. 581–592,
2018.
CONFERENCES PUBLICATIONS AND PRESENTATIONS
1. S. A. Meguid, Mohamed T. Z. Hassan, Strategies for Improved Vehicle Safety: Survivability
of Occupants. 6th International Conference Integrity-Reliability-Failure (IRF2018), July
2018, Lisbon, Portugal (Keynote Lecture).
2. Mohamed T.Z. Hassan, S. A. Meguid, Effect of Impact Severity on Occupant’s Response
during Rear-End Collisions, 6th International Conference Integrity-Reliability-Failure
(IRF2018), July 2018, Lisbon, Portugal.
3. Mohamed T.Z. Hassan, Mo Gabriel Shi, S. A. Meguid, Multibody Dynamic Analysis of
Whiplash, 6th International Conference Integrity-Reliability-Failure (IRF2018), July 2018,
Lisbon, Portugal.
4. Mohamed T.Z. Hassan, S. A. Meguid, Viscoelastic Multibody Dynamics of Whiplash.
Proceedings of the 25th Canadian Congress of Applied Mechanics (CANCAM2015), June
2014, London, Ontario, Canada.
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List of Tables
Table 2.1 Whiplash Associated Disorders [35] ........................................................................................................... 15
Table 3.1 Geometry of the MBD models [37] ............................................................................................................. 48
Table 3.2 Masses and moment of inertia of cervical vertebrae [33,38–41] ................................................................. 50
Table 3.3 Intervertebral rotational stiffness curve coefficients in sagittal plane [42] .................................................. 51
Table 3.4 Axial stiffness of intervertebral joints [46] .................................................................................................. 53
Table 3.5 Maximum angles of rotation at each intervertebral joint [48,49] ................................................................ 55
Table 3.6 Material properties of the steels used in the vehicle structure ..................................................................... 57
Table 3.7 Displacement and rotational response of the head center of mass ............................................................... 67
Table 4.1 Geometry of the MBD model [42] .............................................................................................................. 86
Table 4.2 Masses and moment of inertia of cervical vertebrae [43–46] ...................................................................... 86
Table 4.3 Intervertebral rotational stiffness curve coefficients in sagittal and frontal planes [47–51] ........................ 87
Table 4.4 Maximum angles of rotation at each intervertebral joint [39,50,52,55] ...................................................... 88
Table 4.5 Peak displacements and rotational response of the head center of mass and their time of occurrence in
frontal and lateral collisions ........................................................................................................................................ 97
Table 4.6 Sub-failure injury percent elongations of cervical ligaments [71–73] ....................................................... 103
Table 5.1 Mechanical properties of the bone in human vertebrae and polyethylene terephthalate glycol modified
[38–41] ...................................................................................................................................................................... 117
Table 5.2 Peak head displacements and their time of occurrence for the MBD, FE and experimental models ........ 130
Table 6.1 Seat and head restraint dimensions ............................................................................................................ 140
Table 6.2 Seat arrangements used in the simulations ................................................................................................ 141
Table 7.1 Geometric dimensions of the foam-filled frustum sample ........................................................................ 170
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Table 7.2 Material constants for the FE model .......................................................................................................... 172
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List of Figures
Figure 1.1 Vehicle collision: (a) frontal [8], (b) lateral [9] and rear [10] ...................................................................... 2
Figure 1.2 Experimental study using volunteer, anthropomorphic test dummy and human subject [13] ...................... 3
Figure 1.3 Likelihood of injury or death for different impact types [34] ...................................................................... 4
Figure 1.4 Block diagram showing adopted method of approach ................................................................................. 6
Figure 2.1 Atlas and axis [4] ....................................................................................................................................... 10
Figure 2.2 Cervical vertebrae C3-C7 [5] ..................................................................................................................... 10
Figure 2.3 Structure of the intervertebral disc [6] ....................................................................................................... 11
Figure 2.4 Ligaments in the spine [1] .......................................................................................................................... 11
Figure 2.5 Muscles in the neck [9] .............................................................................................................................. 12
Figure 2.6 Frontal crash test [11] ................................................................................................................................. 13
Figure 2.7 Rear crash test [36] ..................................................................................................................................... 14
Figure 2.8 Side crash test [49] ..................................................................................................................................... 16
Figure 2.9 Working guidelines for NDC in the BioRID P3 for low-speed rear impacts ............................................. 18
Figure 2.10 Multibody human model of Himmetoglu et al. [75] ................................................................................ 19
Figure 2.11 Model of Esat et al. [87] ........................................................................................................................... 20
Figure 2.12 Finite element model of Zhang et al. [96] ................................................................................................ 21
Figure 2.13 The 50th percentile GHBMC FE model [108] .......................................................................................... 22
Figure 2.14 Motion of head, neck and torso during whiplash [40] .............................................................................. 23
Figure 2.15 Cervical spine model with muscle force replication system [139] ........................................................... 25
Figure 2.16 Collapse modes of thin-walled circular tubes: (a) concertina, (b) diamond and (c) global buckling
[168,169] ..................................................................................................................................................................... 27
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Figure 2.17 Typical response of a thin-walled tube collapsing by progressive folding (after [167]) .......................... 28
Figure 2.18 Compressive stress strain curve of closed cell foam [173]....................................................................... 29
Figure 2.19 Aluminum foam: (a) open cell and (b) closed cell [175].......................................................................... 30
Figure 2.20 Effect of foam filling on the collapse load of thin-walled columns (after [176]) ..................................... 31
Figure 2.21 Axisymmetric collapse mode of thin-walled cylinder proposed by Alexander [177] .............................. 32
Figure 3.1 A schematic of (a) human cervical spine and (b) MBD model in the sagittal plane .................................. 46
Figure 3.2 MBD system investigated: (a) a generalized joint containing two links and (b) coordinate system and sign
convention used ........................................................................................................................................................... 47
Figure 3.3 A schematic of generalized (a) single DOF and (b) two DOF MBD models showing two adjacent links
meeting at a viscoelastic joint ...................................................................................................................................... 48
Figure 3.4 Setup of vehicle collision FE simulation showing bullet and target vehicles ............................................. 58
Figure 3.5 GHBMC 50th percentile numerical male occupant FE model seated on the vehicle seat .......................... 60
Figure 3.6 Stress-strain curve of polyurethane foam material of the seat .................................................................... 60
Figure 3.7 Experimental T1 acceleration profile after [68] ......................................................................................... 62
Figure 3.8 Validation of MBD model head response against PMHS sled test [68]: (a) horizontal head displacement
and (b) head rotation .................................................................................................................................................... 63
Figure 3.9 Experimental horizontal sled acceleration profile by [9] ........................................................................... 63
Figure 3.10 Head center of mass horizontal displacement with respect to the seat of the FE model compared to
experimental volunteer test [9] .................................................................................................................................... 64
Figure 3.11 Driver seat velocities resulting from 32 km/h rear-end collision ............................................................. 65
Figure 3.12 Response of 50th percentile male occupant without headrest and restrained using 3-point seatbelt, with
32 km/h rear collision velocity profile applied to the seat ........................................................................................... 66
Figure 3.13 Head center of mass displacements during 32 km/h rear-end collision: (a) Horizontal and (b) vertical .. 66
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Figure 3.14 Rear collision response of the MBD models and the FE model representing an occupant with no
headrest and restrained using a 3-point seatbelt and 32 km/h rear-end collision velocity profile applied to the seat .. 68
Figure A3.1 Setup of rear impact simulations showing (a) the GHBMC 50th percentile male model seated on a seat
rig equipped with a seat belt, a head restraint and an airbag, and (b) the folded airbag embedded in the steering
wheel ........................................................................................................................................................................... 74
Figure A3.2 Head (a) horizontal and (b) vertical displacement with respect to T1 vertebra for the different seat
configurations: no head restraint (NoHR), with head restraint (HR), and with head restraint and airbag (HR&AB) . 75
Figure A3.3 Occupant head during neck flexion: (a) unsupported and (b) supported by the airbag ........................... 75
Figure A3.4 Percent elongation of ALL, PLL, ISL, LF and CL normalized against percent elongation to sub-failure
for the different seat configurations: without head restraint (NoHR), with head restraint (HR), and with head restraint
and airbag (HR&AB) .................................................................................................................................................. 76
Figure 4.1 Schematic of the MBD model showing the applied DOF (indicated by the grey arrows) for the (a) single
DOF and (b) two DOF models .................................................................................................................................... 85
Figure 4.2 Setup of vehicle collision FE simulation: (a) frontal and (b) lateral .......................................................... 90
Figure 4.3 FE model of a seated 50th percentile male occupant restrained using a 3-point seatbelt, collision velocity
profiles applied to the floor and the door ..................................................................................................................... 92
Figure 4.4 Frontal collision head center of mass (a) horizontal and (b) vertical displacements, and lateral collision
head center of mass (c) horizontal and (d) vertical displacements .............................................................................. 94
Figure 4.5 Velocity profiles recorded for the driver seat in the target vehicle for (a) frontal and (b) lateral collisions
..................................................................................................................................................................................... 95
Figure 4.6 Response of a 50th percentile male occupant restrained using 3-point seatbelt during 32 km/h frontal
collision showing maximum neck flexion ................................................................................................................... 96
Figure 4.7 MBD and FE models head center of mass response: frontal collision (a) horizontal and (b) vertical
displacements, and lateral collision (c) horizontal and (d) vertical displacements ...................................................... 97
Figure 4.8 Frontal and lateral collisions response of MBD and FE models resulting from a 32 km/h collision ......... 99
Figure 4.9 Response of a 50th percentile male occupant restrained using 3-point seatbelt during 32 km/h lateral
collision velocity profile applied to the floor and door.............................................................................................. 100
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Figure 4.10 IV-NIC values calculated in (a) frontal, (b) rear, and (c) lateral collisions ............................................ 101
Figure 4.11 ALL, PLL, LF, ISL, and CL maximum elongations normalized against injurious elongation thresholds
in (a) frontal, (b) rear, and (c) lateral collisions ......................................................................................................... 104
Figure 4.12 Cortical bone peak von Mises stress of cervical vertebrae in frontal, rear, and lateral 32 km/h collisions
................................................................................................................................................................................... 106
Figure 4.13 Peak von Mises stress in the cortical bone of (a) C3 vertebra in frontal collision, (b) C7 vertebra in rear
collision and (c) C6 vertebra in lateral collision ........................................................................................................ 107
Figure 5.1 Head-neck prototype: (a) geometry obtained from the GHBMC FE model and (b) 3D printed skull and
vertebrae .................................................................................................................................................................... 117
Figure 5.2 Detailed geometry of 3D printed vertebrae: (a) C1, (b) C2 and (c) T1 .................................................... 117
Figure 5.3 One of the molds used to develop the IVDs ............................................................................................. 118
Figure 5.4 The IVD attached to the vertebrae. The articular process of each vertebra is covered with neoprene rubber
(black) ........................................................................................................................................................................ 119
Figure 5.5 Force - Elongation curve of the rubber ligament developed compared to literature data for the ALL at the
C2-C5 levels by Yoganandan et. al [42] .................................................................................................................... 120
Figure 5.6 The anterior longitudinal ligament (ALL) attached to the vertebra and the neck stabilization system of the
head-neck prototype .................................................................................................................................................. 121
Figure 5.7 Neck stabilization system wire tension control: (a) illustration and (b) photo of the system assembled on
the sled ....................................................................................................................................................................... 122
Figure 5.8 Exploded view of the sled used for impact simulation [44] ..................................................................... 123
Figure 5.9 The head-neck prototype mounted on the sled ......................................................................................... 124
Figure 5.10 A typical frame from the video captured using the high-speed camera showing the tracking points (green
crosses) of the head’s reference target ....................................................................................................................... 125
Figure 5.11 ADXL345 accelerometers connection circuit ........................................................................................ 126
Figure 5.13 Experimentally measured horizontal acceleration recorded at the neck base (T1 vertebra)................... 127
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Figure 5.12 The head-neck FE model extracted from the GHBMC FE (a) without muscles, skin and flesh and skin-
flesh and (b) with muscles, skin and flesh. Acceleration was applied at T1 vertebra and the yellow highlighted nodes
................................................................................................................................................................................... 128
Figure 5.14 Comparison between the responses of the experimental results, MBD and FE predictions................... 129
Figure 5.15 Head displacements with respect to T1 vertebra for the experimental results, and MBD and FE
predictions: (a) horizontal and (b) vertical ................................................................................................................ 130
Figure 5.16 Head rotation measured experimentally and predicted using MBD and FE .......................................... 131
Figure 6.1 The GHBMC FE model seated (a) without a headrest and without seat belt, (b) with a poorly adjusted
headrest and a seat belt, and (c) with a properly adjusted headrest and a seat belt .................................................... 138
Figure 6.2 Seat material compressive stress-strain curve by Grujicic et al. [25] ....................................................... 140
Figure 6.3 Relative head CG displacement with respect to T1 vertebra for the GHBMC FE model using seat
arrangement A compared to cadaver test by Prasad et al. [26] (a) horizontal and (b) vertical .................................. 142
Figure 6.4 For the five seat arrangements: (a) relative head horizontal displacement with respect to T1, (b) relative
head vertical displacement with respect to T1, (c) head horizontal acceleration and (d) head vertical acceleration . 143
Figure 6.5 (a) Change in head rotation in the sagittal plane with respect to time for the five seat arrangements and (b)
the deformation of the neck and head position for seat arrangements A, B and E .................................................... 145
Figure 6.6 Contact between the head and the headrest for (a) poorly adjusted headrest (seat arrangements B and C)
and (b) for properly adjusted headrest (seat arrangements D and E) ......................................................................... 146
Figure 6.7 (a) schematic diagram of the ramping effect and (b) vertical displacement of the CG of the hip without
seat belt ...................................................................................................................................................................... 147
Figure 6.8 Occupant’s response during the entire simulation for seat arrangement A. Figures (a)-(e) show occupant’s
position with respect to the car seat at 0 ms, 75 ms, 150 ms, 270 ms and 350 ms .................................................... 148
Figure 6.9 Capsular ligament elongation at each intervertebral level for the different seat arrangements ................ 150
Figure 6.10 The NIC, Nij and Nkm injury criteria evaluation for the five seat arrangements ..................................... 152
Figure 7.1 A schematic diagram of the foam-filled frustum under axial loading ...................................................... 160
xx
Figure 7.2 Experimental setup of the progressive crushing of the foam-filled frustum, with the enlarged views
showing the sample before and after the test ............................................................................................................. 171
Figure 7.3 Stress versus strain curves for the material of (a) the frustum shell, and (b) the foam core, the dashed line
indicating the characteristic yield stress .................................................................................................................... 171
Figure 7.4 FE model for the progressive crushing of the foam-filled frustum, (a) half of the model, and (b) the
collapsed configuration ............................................................................................................................................. 173
Figure 7.5 Comparison of our analytical model with the experiments and FE simulations for (a) instantaneous
crushing force, and (b) mean crushing force ............................................................................................................. 174
Figure 7.6 Effect of the fold portion update on the mean crushing force contributed by (a) shell penetration, and (b)
interfacial friction for different fold length h and different folding parameter m ...................................................... 175
Figure 7.7 Instantaneous force contributed by different sources versus the crushing distance ................................. 175
Figure 7.8 Variation of (a) mean crushing force, and (b) fold length with the folding parameter m for different
foam/shell interfacial conditions................................................................................................................................ 176
Figure 7.9 Comparison of the predicted fold length with the experiment result ....................................................... 177
Figure 7.10 Effect of taper angle on the instantaneous crushing force contributed by (a) the penetration of foam by
shell, and (b) the friction between the foam and shell ............................................................................................... 178
xxi
List of Appendices
Appendix 3.1: Population of Matrices of 1 DOF Model ............................................................................................. 77
Appendix 3.2: Matrices population of 2 DOF model .................................................................................................. 78
xxii
List of Abbreviations
Abbreviations
ALL Anterior Longitudinal Ligament
ATD Anthropomorphic Test Dummy
CL Capsular Ligament
DOF Degree of Freedom
EuroNCAP European New Car Assessment Programme
FE Finite Element
GHBMC Global Human Body Model Consurtium
HUMOS Human Model for Safety
IAR Instantaneous Axis of Rotation
IIHS Insurance Institute for Highway Safety
ISL Interspinous Ligament
IVD Intervertebral Disc
MBD Multibody Dynamics
NHTSA National Highway Traffic Safety Administration
NSS Neck Stabilization System
PCD Polycrystalline Diamond
PETG Polyethylene Terephthalate Glycol Modified
PLL Posterior Longitudinal Ligament
SEA Specific Energy Absorption
TEA Total Energy Absorbed
THUMS Total Human Model for Safety
ViVA Virtual Vehicle-safety Assessment
1
Chapter 1.
Introduction and Justification
Summary: In this chapter, we describe the details of the study, justify the reasons for
undertaking it, outline the research objectives, and present the method of approach adopted to
treat these objectives. Additionally, a summary of the layout of the thesis is provided.
1.1. Introduction
According to the World Health Organization, road traffic crashes is the top cause of death
worldwide for people aged 15-29 [1]. Besides death, motor vehicle crashes may result in
disabilities and/or chronic injuries. Of the injuries suffered in vehicular impacts, head injuries,
mostly commonly due to impacts with the vehicle interior, are some of the most frequently
observed injuries suffered by vehicle occupants. Such injuries are most commonly found in the
frontal and lateral impacts [2–4]. In rear impacts, the neck is most frequent site of injury, with
more than 80% of injuries suffered in rear impacts being cervical whiplash [5]. Whiplash results
in neck pain, limited neck movement, visual disturbance and dizziness. According to the
National Highway Traffic Safety Administration (NHTSA) the number of injuries resulting from
rear-end collisions increased during 2007-2015 from 485,000 incidents to 556,000 incidents
becoming the most common reason for injuries in motor vehicle collisions [6,7].
2
Figure 1.1 Vehicle collisions: (a) frontal [8], (b) lateral [9] and rear [10]
Three main approaches are utilized to evaluate how the occupant responds in various impact
scenarios. The first is the experimental approach. A number of experimental studies have been
conducted on volunteers to evaluate how the occupants respond to different impact accelerations
[11–15]. Although volunteer studies provide the most accurate response, the impact severity
must be limited to avoid injuring the volunteers. In order to overcome the limited impact severity
barrier, full or partial post mortem human surrogates (PMHS) may be used instead [16–18].
However, the use of PMHS in testing is also subject to stringent ethical considerations [13],
limiting its practical value. In the past few decades, experimental studies have been conducted
primarily through the use of anthropomorphic test dummies (ATDs) to evaluate the safety of
motor vehicles crashes such as Hybrid III [19], Test device for Human Occupant Restraint
(THOR) [20] and BioRid II [21].
3
Figure 1.2 Experimental study using volunteer, anthropomorphic test dummy and human
subject [13]
The second approach to study the human response is the use of multibody dynamics (MBD). In
these models, the bones were modeled as rigid bodies connected through different types of joints
and the soft tissues were modeled as viscoelastic elements [22–26]. These multibody dynamics
model studied the occupant response to different types of loading. Many models focused on the
head/neck region while some other efforts modeled the entire human body such as MADYMO
(MAthematical DYnamic Model) [27].
As a result of an increase in availability of computational power over the past few years, the
finite element (FE) method has been extensively utilized to provide biofidelic models of the
human body to be used under different types of loading. To analyze and understand the human
response in motor vehicle crashes, a number of FE models were developed, whether focusing on
the head/neck region [28,29] or full human body models for both male and female occupants
such as the Global Human Body Model Consortium (GHBMC) [30], Total Human Model for
Safety (THUMS) [31] HUMOS (HUman MOdel for Safety) [32] and ViVA (Virtual Vehicle
Safety Assessment) [33]. These models can be used for in depth studies of injury mechanisms
during crashes. However, they come at a high computational cost.
1.2. Justification of the Study
Despite the significant enhancement in safety of motor vehicles in the last few decades, the
number of fatalities/injuries resulting from motor vehicle collisions remains a serious injury and
a source of trauma which indicates that there is still room for further improvements in the field of
vehicle occupant protection.
4
According to the National Highway Traffic Safety Administration (NHTSA) in the US [34], in
2016, the highest fatality rate was observed for oblique motor vehicle collisions, followed by
frontal, then rear and finally lateral impacts, as shown in Figure 1.3. However, for the same year,
~ 694,000 injury cases were reported resulting from rear impacts, which is the highest number of
fatalities/injuries of all impact types. Therefore, in this work, we consider frontal, rear and lateral
collisions, with emphasize on rear-end collision due to its commonality and being the main
source of occupant injury.
Figure 1.3 Likelihood of injury or death for different impact types [34]
In order to provide better protection for motor vehicle occupants, it is crucial to study the
occupant’s dynamic response and injury mechanisms during vehicle collisions as this will enable
us to develop the appropriate strategies to mitigate injury. From the conducted literature survey,
it is concluded that additional research is needed to further study the dynamic response of
occupants during collisions using multibody dynamics. Furthermore, the use of numerical full
body human models is essential to conduct realistic FE simulations, which will help in better
understanding of occupants’ injury mechanisms during motor vehicle collisions.
1.3. Research Objectives
The objectives of this research are to examine occupant’s kinematic response and kinetic
behaviour during frontal, rear and lateral motor vehicle collisions. Specifically, the focus of the
work can be summarized as follows:
0%
10%
20%
30%
40%
Frontal Rear Lateral
Fatal Injury
5
i. Develop multibody dynamics model to capture the occupant’s head response in frontal,
rear and lateral end collisions accounting for variable intervertebral rotational stiffness,
ii. Conduct detailed and realistic nonlinear dynamic finite element analysis to simulate
motor vehicle collisions, obtain realistic collision accelerations and determine the
occupant response resulting from these collisions,
iii. Develop 3D printed head-neck prototype and conduct experimental validation of the
multibody dynamics and the finite element simulations of occupant’s kinematic response
to rear collisions, and
iv. Design and develop a novel shock absorber in the form of foam-filled frusta to enhance
the crashworthiness of motor vehicles and limit occupants’ injury.
1.4. Method of Approach
This research is divided into two main sections as shown in Figure 1.4. The first is concerned
with studying the vehicle response during impact by conducting nonlinear finite element
simulations and enhancing the vehicle crashworthiness through the development of a novel
shock absorber. The second aspect of this work is to study the occupant response during
simulated motor vehicle collisions using multibody dynamics, dynamic nonlinear finite element
simulations and experimentally. The outcome will enable us to understand occupant’s response
during collisions and the strategies needed to ensure his/her protection.
6
Figure 1.4 Block diagram showing adopted method of approach
1.5. Thesis Layout
The thesis is divided into eight chapters. In Chapter 1, we introduce the work; justify its
undertaking, outline the research objectives and the method of approach adopted in achieving
these objectives. In Chapter 2, we provide a review of the current state of literature about the
techniques used to study occupant response in motor vehicle collisions along with modelling of
collapsible thin walled tubes as energy absorbers. Chapters 3-7 are provided in the form of
published articles. The development of multibody dynamics model to capture occupant’s
response during rear impact as well as finite element simulation are provided in Chapter 3. In
Chapter 4, we expand our multibody dynamics model to study frontal and lateral collisions,
conduct finite element simulations of these collisions, and evaluate the probability of injury for
frontal, rear and lateral impacts. Chapter 5 discusses the experimental validation of occupant
Strategies to Mitigate Occupant Injury
Motor Vehicle
Finite Element simulation of frontal, rear and lateral collisions
Realistic Seat Acceleration
Design of Novel Shock
Absorber: Analytical, FE, Exp.
Reduce energy
transferred to occupant
Occupant – Vehicle Interaction
Multibody
Dynamics
Model
Capture
occupant
response
Finite Element
Simulations
Parameters affecting
occupant response:
Head restraint, seat
cushion, seat stiffness,
seat belt, airbag
Experimental:
Head-Neck
Prototype
Validation
7
response using a developed human head-neck prototype. In Chapter 6, we conduct FE
simulations to study the effect of seat belt, headrest and seat cushion stiffness on the occupant
response in rear-end collision. The design and development of a novel shock absorber is
discussed analytically, numerically and experimentally in Chapter 7. In Chapter 8, the research
conclusions, contributions and future work are stated and discussed.
References
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[2] S. Kuppa, J. Wang, M. Haffner, and R. Eppinger, Lower extremity injuries and associated injury criteria,
17th ESV Conf., 4, 1–15, 2001.
[3] D. C. Viano and C. S. Parenteau, Injury risks in frontal crashes by delta V and body region with focus on
head injuries in low-speed collisions., Traffic Inj. Prev., 11, 382–390, 2010.
[4] J. Augenstein et al., Injury Patterns in Near-Side Collisions, SAE 2000 World Congr., 2000.
[5] K. Ono and M. Kanno, Influences of the physical parameters on the risk to neck injuries in low impact speed
rear-end collisions, Accid. Anal. Prev., 28, 493–499, 1996.
[6] National Highway Traffic Safety Adminstration - Department of Transportation, Traffic Safety Facts 2007:
A Compilation of Motor Vehicle Crash Data from the Fatality Analysis Reporting System and the General
Estimates System, Washington, DC, 2007.
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A Compilation of Motor Vehicle Crash Data from theFatality Analysis Reporting System and the General
Estimates System, Washington, DC, 2017.
[8] Upper Austria: Frontal impact collision for boys – VIENNA.AT – shilfa. [Online]. Available:
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2019].
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[11] J. A. Pramudita, K. Ono, S. Ejima, K. Kaneoka, I. Shiina, and S. Ujihashi, Head / Neck / Torso Behavior
and Cervical Vertebral Motion of Human Volunteers During Low Speed Rear Impact : Mini-sled Tests with
Mass Production Car Seat, in 2007 International IRCOBI Conference on the Biomechanics of Injury, 2007,
201–217.
[12] S. Kumar, Y. Narayan, and T. Amell, Analysis of low velocity frontal impacts, Clin. Biomech., 18, 694–703,
Oct. 2003.
[13] S. M. Beeman, A. R. Kemper, M. L. Madigan, C. T. Franck, and S. C. Loftus, Occupant kinematics in low-
speed frontal sled tests: Human volunteers, Hybrid III ATD, and PMHS, Accid. Anal. Prev., 47, 128–139,
2012.
[14] H. Choi et al., Experimental and numerical studies of muscular activations of bracing occupant, Int. Tech.
Conf. Enhanc. Saf. Veh., 2005.
[15] C.-Y. Chang, J. D. Rupp, M. P. Reed, R. E. Hughes, and L. W. Schneider, Predicting the effects of muscle
activation on knee, thigh, and hip injuries in frontal crashes using a finite-element model with muscle forces
from subject testing and musculoskeletal modeling., Stapp Car Crash J., 53, 291–328, Nov. 2009.
8
[16] J. Ash, C. G. Shaw, D. Lessley, and J. Crandall, PMHS restraint and support surface forces in simulated
frontal crashes, JSAE Annu. Congr., 4, 41–46, 2013.
[17] F. J. Lopez-Valdes et al., The Six Degrees of Freedom Motion of the Human Head, Spine, and Pelvis in a
Frontal Impact, Traffic Inj. Prev., 15, 294–301, Apr. 2014.
[18] F. A. Pintar, N. Yoganandan, and D. J. Maiman, Lower cervical spine loading in frontal sled tests using
inverse dynamics: potential applications for lower neck injury criteria, Stapp Car Crash J., 54, 133, 2010.
[19] J. K. Foster, J. O. Kortge, and M. J. Wolanin, Hybrid III-A Biomechanically-Based Crash Test Dummy, in
21st Stapp Car Crash Conference, 1977.
[20] T. Keon, Alternative Approaches to Occupant Response Evaluation in Frontal Impact Crash Testing, SAE
Int. J. Transp. Saf., 4, 2016-01–1540, Apr. 2016.
[21] J. Davidsson, BioRID II Final Report, 1999.
[22] A. T. Dibb et al., Pediatric Head and Neck Dynamics in Frontal Impact: Analysis of Important Mechanical
Factors and Proposed Neck Performance Corridors for 6- and 10-Year-Old ATDs, Traffic Inj. Prev., 15,
386–394, May 2014.
[23] D. Bose, J. R. Crandall, C. D. Untaroiu, and E. H. Maslen, Influence of pre-collision occupant parameters on
injury outcome in a frontal collision, Accid. Anal. Prev., 42, 1398–1407, Jul. 2010.
[24] R. Meijer, H. Elrofai, and J. Broos, Evaluation of an Active Multi-Body Human Model for Braking and
Frontal Crash Events, 23rd Int. Tech. Conf. th Enhanc. Saf. Veh., 1–12, 2013.
[25] T.-L. Teng, F.-A. Chang, Y.-S. Liu, and C.-P. Peng, Analysis of dynamic response of vehicle occupant in
frontal crash using multibody dynamics method, Math. Comput. Model., 48, 1724–1736, Dec. 2008.
[26] M. Turkovich and L. Van Roosmalen, A Preliminary Study on the Effects of Obesity on Occupant Response
in Frontal Impact, in RESNA Annual Conference, 2010.
[27] Y. Huang, A. King, and J. Cavanaugh, A MADYMO Model of Near-Side Human Occupants in Side
Impacts, J. Biomech. Eng., 116, 228–235, 1994.
[28] M. B. Panzer, J. B. Fice, and D. S. Cronin, Cervical spine response in frontal crash, Med. Eng. Phys., 33,
1147–1159, Nov. 2011.
[29] A. Wittek, J. Kajzer, E. Haug, and K. Ono, Finite Element Modeling of the Muscle Effects on Kinematic
Responses of Head-Neck Complex in Frontal Impact at High Speed., JSME Int. J. Ser. C, 44, 379–388,
2001.
[30] J. J. Combset, Current statues and future plans of the ghbmc (global human body models consortium), in 6th
International Symposium: Human Modeling and Simulation in Automotive Engineering, 2016.
[31] J. L. Forman, R. W. Kent, K. Mroz, B. Pipkorn, O. Bostrom, and M. Segui-Gomez, 41. Predicting rib
fracture risk with whole-body finite element models: development and preliminary evaluation of a
probabilistic analytical framework., Ann. Adv. Automot. Med., 56, 109–24, 2012.
[32] J. Mordaka, R. Meijer, L. van Rooij, and A. E. Żmijewska, Validation of a Finite Element Human Model for
Prediction of Rib Fractures, SAE Tech. Pap. 2007-01-1161, Apr. 2007.
[33] J. Östh, M. Mendoza‐Vazquez, A. Linder, M. Y. Svensson, and K. Brolin, The VIVA OpenHBM Finite
Element 50th Percentile Female Occupant Model: Whole Body Model Development and Kinematic
Validation, in IRCOBI Conference 2017, 2017, IRC-17-60, 443–466.
[34] National Highway Traffic Safety Adminstration - Department of Transportation, Traffic Safety Facts: A
Compilation of Motor Vehicle Crash Data from the Fatality Analysis Reporting System and the General
Estimates System. [Online]. Available: https://cdan.nhtsa.gov/tsftables/tsfar.htm#. [Accessed: 18-Sep-2018].
9
Chapter 2.
Literature Review
Summary: For the sake of completeness and continuity, we provide herein a literature review
covering the background for the entire thesis. The reader must recognize that as a result of using
predominantly paper-based format in this dissertation, there may be some overlap in the
literature review of the relevant chapters. This literature review on modeling and characterization
of motor vehicle collisions is divided into seven sections. The first provides definition of cervical
spine anatomy. The second summarizes injuries resulting from frontal, rear and lateral collisions.
The third addresses current neck injury criteria. The fourth discusses modeling of occupant
response using multibody dynamics. In the fifth section we attend to finite element efforts to
develop head-neck or full human body models. The sixth addresses experimental testing of
human response in the different impact scenarios. The last section provides a brief review of
relevant literature of crashworthiness of foam-filled thin-walled frusta.
2.1. Definition of Spine Anatomy
Vertebral Column
The human spine consists of the cervical spine (seven vertebrae), thoracic spine (twelve
vertebrae), lumbar spine (five vertebrae), sacrum (five fused vertebrae) and the coccyx (three to
four fused segments). The spine looks straight in the frontal plane but in the sagittal plane there
are four curvatures in the spine. These curvatures provide flexibility and shock absorbing
capacity [1]. The spine is responsible for transferring moments from head and trunk to the pelvis,
allowing sufficient motion between head, trunk, and pelvis and protecting the spinal cord.
Vertebrae
Since the scope of our study is the head and neck, we will focus only on the cervical part of the
spine. The cervical spine consists of seven vertebrae (C1-C7). The first two vertebrae C1 and C2
10
are different from other vertebrae anatomically, as shown in Figure 2.1. C1 is called the atlas and
C2 is called the axis [2]. The atlas supports the head and provides the principal sagittal rotation
of the head, while axis articulates with the atlas to provide the horizontal rotation of the head [3].
The remaining vertebrae (C3-C7) are similar in shape, as shown in Figure 2.2.
Figure 2.1 Atlas and axis [4]
Figure 2.2 Cervical vertebrae C3-C7 [5]
Intervertebral Disc
The intervertebral disc is located between the bodies of two adjacent vertebrae. It is responsible
for load transfer from one vertebra to the inferior one [5]. It consists of three parts, as shown in
Figure 2.3. The first is the nucleus pulposus which consists of a network of fine fibrous strands in
mucoprotein gel. The water content of the nucleus pulposus is 70-90%. The second is the
annulus fibrosis which is the outer shell of the disc and consists of concentric laminated bands of
collagen fibers oriented at 30 degrees from disc plane. The fibers of each laminate are oriented at
an opposite direction from the two adjacent laminates. At the inner zone of the disc the annulus
fibers are attached to the end-plates while at the outer zone they are attached to the osseous
tissues of the vertebral body. The outer attachment is much stronger than inner attachment. The
third is the cartilaginous end-plate which is a hyaline cartilage that separates the annulus fibrosis
and nucleus pulposus from the vertebral body.
11
Figure 2.3 Structure of the intervertebral disc [6]
Ligaments
Ligaments connect bones together by transferring tensile loads between them and are responsible
for the stability of the spine. The spinal ligaments, shown in Figure 2.4, are:
• The anterior (ALL) and posterior longitudinal ligaments (PLL) are attached to the anterior
and posterior surfaces of the vertebra body, respectively, and are also attached to the
intervertebral disc.
• The capsular ligament (CL) provides flexion stability in the cervical spine.
• The ligamentum flavum (LF) has the highest elasticity of all soft tissues in the human body
allowing high recoverable deformation.
• The interspinous ligament (ISL) is a thin weak tissue with high collagen content. It joins
adjacent spinous processes and is not present in all adults. It blends with the supraspinous
ligament posteriorly.
Figure 2.4 Ligaments in the spine [1]
12
Muscles
Muscles provide the movement to the head and neck. They are attached to the skull, vertebrae,
rib cage and clavicles [7].Tension of the muscles can be classified as active and passive. In active
tension of the muscle, the force is generated from actin and myosin fibers in the sarcomeres,
while in passive tension, the force is created from the elongation of the connective tissues of the
muscletendon unit (stretching). This passive tension can be very large and responsible for
muscular weakness. Passive tension affects the range of motion of the joints and the increase in
that tension which limits the range of motion is called passive insufficiency [8]. The muscles in
the neck region are shown in Figure 2.5.
Figure 2.5 Muscles in the neck [9]
2.2. Overview of Motor Vehicle Collision Injuries
2.2.1. Frontal Collisions
Frontal impacts are considered the most common and fatal forms of vehicular collisions.
According to the National Highway Traffic and Safety Administration (NHTSA), in 2017 more
than 61% of fatal passenger vehicle collisions were caused by frontal impacts [10].
13
Figure 2.6 Frontal crash test [11]
For unrestrained occupants, most injuries are caused by impacts with the steering wheel.
Although the seat belt can reduce the possibility of serious injuries by up to 65% [12], seat belt
loading can be the primary cause of thorax and abdomen injuries. The compressive loading
exerted by the seat belt could result in ribcage and sternum fractures as well as kidney and liver
injuries [13,14]. The liver is particularly vulnerable in restrained occupants but only when the
frontal airbag is not deployed [15,16]. It must be noted that unrestrained drivers impacting the
steering wheel account for 68% of all abdominal injuries, while the seat belt accounts for only
17% [14], indicating that the seatbelt is still effective at protecting against abdominal injuries,
despite its associated risks. Older individuals, individuals with higher mass, as well as female
occupants are more likely to suffer more from severe thoracic injuries [17–20]. During frontal
collisions, knee injuries are also common, but they are short term injuries, mainly knee sprain
[21]. The pelvis is also prone to injury due to the knee contact with the front dashboard [22,23].
Although frontal impact injuries primarily occur at the thorax, the head and the lower extremities
are also at significant risk [24]. About 22% of all neck injuries resulting from motor vehicle
collisions is caused by frontal collisions [25]. Fracture of cervical vertebrae and ligament tearing,
specifically the ISL and LF at the mid cervical spine [26], are common injuries in frontal impacts
[27]. The occupant head is also prone to injury in frontal collisions [21]. Contact with vehicle
14
interior and windshield is the main reason for head and brain injuries [28] even at low impact
velocities [29].
2.2.2. Rear Collision Injuries
A significant number of rear-end impacts results in whiplash which is the most common injury in
motor vehicle collisions. About 95% of rear-end accidents resulted into minor injuries, of which
80% were concentrated on the neck [30]. For drivers suffering from Whiplash-Associated
Disorders (WAD), rear impacts cause ~52% of the injuries. Whiplash patients suffer from
headache, neck pain, limited neck motion, visual disturbance, weakness and dizziness [31–33].
Most whiplash patients recover during the first few weeks after the injury; however, about 40%
of the patients suffer from post-whiplash syndrome which is characterized by unexplained
cognitive and physical symptoms [34]. The Quebec Task Force [35] classified WAD in order to
help deciding the treatment of whiplash injury and symptomatology as shown in Table 2.1.
Figure 2.7 Rear crash test [36]
15
Table 2.1 Whiplash Associated Disorders [35]
Grade Clinical Presentation
0 No complaint about the neck
No physical signs
I Neck complaint of pain, stiffness or tenderness only
No physical signs
II Neck complaint
Musculoskeletal signs
III Neck complaint
Neurological sings
IV Neck complaint
Fracture dislocation
Spasm and/or narrowing in the arteries cause blood flow alteration which was found in whiplash
patients suffering from chronic headache, dizziness, blurred vision and tinnitus [37]. Kalawy et
al. [38] found that patients suffering from chronic neck pain after whiplash have high blood flow
in regions associated with localized pain. During rear impacts, the lower cervical spine is under
extension leading to a decrease in the spinal canal diameter and volume resulting in high risk of
cord injury. A study by Ito et al. [39] shows that people with normal canal diameter will not
suffer from cord compression; however, patients with narrow spinal canal are vulnerable for cord
injury during whiplash. The change in the volume of the spinal canal may generate pressure
gradient between the inside and the outside of the canal due to resistance to blood flow which
directly loads the dorsal root ganglia and nerve roots [37,40].
Female occupants are at 50% higher risk to suffer from whiplash compared to their male
counterparts [41] due to the reduced neck cross-sectional area, muscle strength and ligament
stiffness in female occupants. [42,43].
2.2.3. Lateral Collisions
Lateral collisions can be classified either near-side or far side based on the position of impact
with respect to the occupant location in the vehicle. The occupant is at high risk of injury in near-
side collisions [44]. [45]. The angle of impact greatly affects the risk of occupant injury.
Perpendicular impacts result in more significant injuries than oblique impacts [46,47], as well as
higher fatality risks [48]. Therefore, side impact research primarily focuses on the case of
perpendicular lateral impacts.
16
Figure 2.8 Side crash test [49]
In near-side impacts, the head, chest, and pelvis are most prone to injuries where pelvic fracture,
and kidney and liver injuries are most common. Near-side collisions can also be fatal due to
injuries to the brain, aorta, heart and ribs [50].
The main cause of injury in near-side collisions is the occupant contact with the vehicle interior,
primarily the side door [47,51]. The injury severity does not only depend on the collision
velocity but also on the occupant-door interaction [52]. Over 80% of injuries to the thorax, neck
and pelvis resulting from near-side lateral impacts are due to contact with the intruding vehicle
structure [53]. In near-side impacts, seat belt use is not associated with decreased risks of pelvic
fractures or thoracic injuries and may itself serve as a source of injury. During a near-side
collision, the seat belt restraints the occupant’s lateral movement holding the occupant in place
while the intruding door impacts the occupant [48,54–57].On the other hand, the seat belt
reduces the possibility of injury for far-side occupants in lateral impacts [45,47,54], in particular
brain injuries and skull fractures [58–60].
Although the risk of occupant injury due to a far-side collision is relatively low [61], it can still
result in injuries to the head and chest due to impacts with the adjacent seat and/or occupant [62].
However, neck injury in a far-side collisions is unlikely [53,63].
17
2.3. Neck Injury Criteria
Many injury criteria were developed to determine the possibility of injury occurrence under
certain loading conditions. Here, we review some of the most commonly used neck injury
criteria.
The Neck injury criterion (NIC) developed by Boström et al [64] is one of the commonly used
injury criteria and is mainly used to assess whiplash injury. It simulates the transient pressure
gradient which is generated in the spinal canal due to the change in the volume of the spinal
canal caused by the relative horizontal motion between the head and the torso which occurs
during the first 100 ms of impact. The NIC value is calculated using the following equation:
𝑁𝐼𝐶 = 0.2 𝑎𝑟𝑒𝑙 + 𝑣𝑟𝑒𝑙2 < 15
𝑚2
𝑠2 (2.1)
where arel and vrel are the relative acceleration and velocity between C1 and T1, respectively. The
human threshold for NIC value is 15 m2/s2. A lower conservative threshold of 14.4 m2/s2 was
suggested by Ivancic and Sha [65]. NIC is sensitive to factors like crash pulse, seat deflection
characteristics and head to head restraint distance [66].
The Nij Neck injury criterion was proposed to assess neck injuries during frontal impacts [67]. It
combines the effects of force and moment measured at the occipital condyles. The criterion is
given by:
𝑁𝑖𝑗 =𝐹𝑧
𝐹𝑖𝑛𝑡+
𝑀𝑦
𝑀𝑖𝑛𝑡 (2.2)
where Fz is the axial force, My is the flexion/extension bending moment, Fint is the load critical
intercept value, and Mint is the moment critical intercept value. The values of Fint and Mint depend
on the dummy used in testing. The threshold for this criterion is 1.
The Neck Protection Criterion (Nkm) [68] is used to assess injuries during rear impacts and is
evaluated by:
𝑁𝑘𝑚 =𝐹𝑥𝐹𝑖𝑛𝑡
+𝑀𝑦
𝑀𝑖𝑛𝑡 (2.3)
18
where Fx is the shear force, My is the flexion/extension bending moment, Fint is the load critical
intercept value, and Mint is the moment critical intercept value. This injury criterion focuses on
the loads at the occipital condyles while most of whiplash injuries occur at the lower cervical
spine (C5-C7) which makes its accuracy in determining the possibility of injury occurrence
questionable.
Viano et al. [69] proposed the Neck Displacement Criterion (NDC) which studies the kinematics
of the head relative to T1 vertebra. The criterion classifies the behavior of the curves of the
vertical displacement, horizontal displacement and head rotation relative to T1 as excellent,
good, acceptable or poor as shown in Figure 2.9.
Figure 2.9 Working guidelines for NDC in the BioRID P3 for low-speed rear impacts
The Intervertebral Neck Injury Criterion (IV-NIC) [70] proposes that injury will occur when the
extension/flexion angle at an intervertebral level exceeds its physiological limits. Exceeding the
physiological limit may cause injury to the soft tissues like ligaments, muscles and discs. The IV-
NIC is evaluated by:
𝐼𝑉 − 𝑁𝐼𝐶𝑖 =𝜃𝑡𝑟𝑎𝑢𝑚𝑎,𝑖
𝜃𝑝ℎ𝑦𝑠𝑖𝑜𝑙𝑜𝑔𝑖𝑐𝑎𝑙,𝑖 (2.4)
where θtrauma is the intervertebral motion under traumatic loading and θphysilogical is the
physiological range of motion. This injury criterion does not address the axial loads nor the
effect of acceleration on the head and neck.
19
2.4. Multibody Dynamics Models
Multibody dynamics (MBD) is considered one of the simplest and computationally efficient
techniques to estimate the human response. One of the first MBD models was developed in 1968
by Martinez et al. [71] to simulate the head-neck response during whiplash using rigid body
dynamics. In another simple model, the human head and torso were modeled as two-link system
connected through a revolute joint [72].
Other MBD models segmented the human body and neck into rigid segments which are
connected with through springs, dampers and/or viscoelastic elements [73–76], as shown in
Figure 2.10. In other efforts, the modeling approach focused only on the human head-neck [77–
80]. McKenzie et al. [81] modeled the neck as rigid vertebrae connected by viscoelastic solid
beams to represent the intervertebral discs.
A mathematical model of the head and neck was developed by De Jager [82] in which the rigid
head and vertebrae were connected by linear viscoelastic discs, non-linear viscoelastic elements,
frictionless facets and Hill-type muscles. However, that model did not account for variable
rotation stiffnesses at each intervertebral joint. This model was later integrated in other full
human body MBD models [83,84] or was used to develop other head-neck models with an
increased complexity [85,86].
Figure 2.10 Multibody human model of Himmetoglu et al. [75]
20
Esat et al. [87] modeled the vertebrae as rigid bodies which are connected by linear viscoelastic
intervertebral discs and nonlinear viscoelastic ligaments, as shown in Figure 2.11. The muscle
elements behave actively and passively.
Figure 2.11 Model of Esat et al. [87]
2.5. Finite Element Models
Finite element models are superior to multibody dynamics models because they do not only
study the kinematics but also the stresses and strains in the soft tissues. A good finite element
model requires a geometry that can accurately represent the problem under investigation. The
cervical spine has an intricate geometry compared to other engineering problems which makes
the generation of the geometry an exhaustive process. A common process to develop the
geometry of human body parts is using computerized topography (CT) scan which provides
accurate 3D models. Several FE models were developed to study the cervical spine under
different types of loading. Brolin et al. [88] used a FE model of the upper cervical spine to study
the effect of ligaments material properties on the kinematics of the spine. Teo et al. [89]
developed a FE model for the lower cervical spine (C4-C6) to predict its response under different
loading configuration. The model consists of the bony vertebrae, facets, intervertebral discs and
ligaments. DeWit et al. [90] used a detailed FE model of the C5-C7 spine segment to investigate
tissue failure under different loading conditions. Many full models of the cervical spine were
21
developed by others [91–95]. Zhang et al. [96] modeled the endplates and annulus of the disc and
the ligaments using elastic properties while applied viscoelastic properties for the nucleus of the
disc and for muscles (see Figure 2.12).
Figure 2.12 Finite element model of Zhang et al. [96]
In most mathematical and FE models the passenger’s head is assumed to be looking straight
ahead producing forces in the sagittal plane only while this is unlikely in most accidents. For
instances, in a rear collision, the impact force will cause a slightly rotated head to rotate more
before extension [6]. Shateri [97] and Fice et al. [98] developed detailed FE models to study the
response of the neck during out of position rear impacts. Another detailed model developed by
van der Horst [99] simulates the head and neck in frontal, lateral and rear impacts. Hasegawa et
al. [100] developed a FE model to study different injury mechanism during whiplash. This model
incorporated the spinal cord, nerve roots and cerebrospinal fluid. Cronin [101] used a FE model
to investigate possible source of pain in rear impacts. A number of studies used FE to investigate
injuries in the spinal cord [102–105]. Stemper et al. [106] used a numerical model to evaluate the
effect of reflex muscle contraction on the kinematics of the spine during whiplash. Contraction of
muscles decreases the angular rotation of spine segments especially at shorter reflex delays and
at low impact severity. The effect of muscle contraction decreases as the impact severity
increases.
22
Since the occupant’s response is greatly affected by the interaction with the car seat, it is crucial
to study the response of the entire occupant’s body instead of response of head and neck only. In
further efforts to develop full human body FE models, the Total Human Model for Safety
(THUMS) [107] and the Global Human Body Model Consortium (GHBMC) [108] developed a
number of detailed FE models of the human body for both male and female occupants and
pedestrians, the HUman MOdel for Safety (HUMOS) developed a model for the male occupant
[109], the Virtual Vehicle-safety Assessment (ViVA) project for Open-source Human Body
Models (OpenHBM) addressing gender diversity developed a FE model of the 50th percentile
female occupant [110] and the PIPER project which developed a scalable FE model of a child
occupant [111]. These newly developed FE models reflect the importance of using full body
models in impact simulations.
Figure 2.13 The 50th percentile GHBMC FE model [108]
2.6. Experimental studies
The experimental studies can be categorized as: in vivo testing of human volunteers, in vitro
testing of cadavers, and tests using anthropometric test dummies.
23
2.6.1. In Vivo
In order to characterize the human kinematic response in frontal collisions, Ewing et al. [112]
subjected seated volunteers using sled test to impact accelerations up to 10 g. The kinematics
response was captured using high-speed imagery as well as accelerometers attached to the head
and torso. The outcome of that study was used later to validate multiple analytical, numerical and
experimental studies.
A study conducted on volunteers to study the kinematics of the neck during whiplash [113]
shows that initially, the seat presses the volunteer back straightening the spine followed by an
upward and forward push of the occupant back compressing the cervical spine due to the head
inertia. The cervical spine then forms an S-shape with small flexion occurring in the upper part
and extension in the lower part. This is followed by the extension of the cervical spine. The torso
then moves downward to the front but is prevented by the seat belt which is responsible for
increasing whiplash injury.
Figure 2.14 Motion of head, neck and torso during whiplash [40]
Another study by Kaneoka et al. [114] on volunteers using a sled shows similar response of the
neck during whiplash. The study shows that during whiplash, the instantaneous axis of rotation
(IAR) shifts upward compared to normal motion. Amevo et al. [115] investigated the normal
positions of IARs of the cervical motion segments. Other studies investigated the response of the
head, neck and muscle response during rear impacts [116–118]. The effect of occupant
awareness and the role of the cervical muscle activation on the occupant’s response during
different impact scenarios was addressed in other efforts [119–126]. These studies show that
occupant awareness or bracing for impact reduces the resultant head velocity and acceleration.
Although in vivo studies provide the most accurate human response, the impact severity in these
studies are quite limited to prevent the injury of volunteers participating in the study. Therefore,
24
these studies do not provide much information about injury mechanisms or occupant’s behavior
for higher impact accelerations which arises the importance of using other numerical and
experimental techniques to better understand occupant injuries in collisions.
2.6.2. In Vitro
In vitro tests are conducted using post-mortem human surrogates (PMHS), whether in full or
using isolated body parts. PMHS sled tests are similar to volunteer sled tests, although injurious
acceleration pulse may be utilized. The kinematics of the body are the recorded using
accelerometers and high-speed imagery [127–133]. This method is useful for characterizing
whole-body responses of the PMHS subjects, as well as the interactions between occupants,
restraint devices, and vehicle interior parts.
Yoganandan et al. [134] conducted a study on an isolated osteoligamentous cervical spine
extracted from human cadavers to capture the dynamic response during whiplash by applying
velocity at the lower part of the neck and capture the motion from retroreflective targets attached
to the vertebrae. Similar studies on cadavers was conducted by Grauer et al. [135], Panjabi et al.
[136,137] and Cholewicki et al. [138].
Ivancic et al. [139] developed a model of the cervical spine to simulate whiplash, as shown in
Figure 2.15. They used whole cervical spine specimens and attached muscle force replication
system to enhance the response of the model. The motion of the model was captured using high-
speed imaging. This model was used in a number of whiplash studies to analyze the kinematics
of facet joints [140], to study the possibility of injury of the anterior longitudinal ligament [141],
to study the injury mechanisms of the intervertebral discs [142], to determine the coefficients of
the dynamic sagittal flexibility of the neck [143] and to study the possibility of spinal canal
narrowing during whiplash and its effect on pinching the spinal cord [39]. Nibu et al. [144]
studied the effect of whiplash trauma on the vertebral artery using cadavers. Siegmund et al.
[145,146] investigated the injury of the facet capsule during whiplash.
25
Figure 2.15 Cervical spine model with muscle force replication system [139]
A number of studies used PMHS specimens in lateral impact test. In these tests, the seated
PMHS specimen was impacted laterally with a vertical wall and the kinematic response was
captured using high-speed photography [147–152] .In some studies, the wall was equipped with
load cells to determine the contact loads between the wall and the different body parts.
[149,153]. It must be noted that in these studies, the seat belt effect was not accounted for, which
greatly affects the occupant response.
2.6.3. Anthropometric Test Dummies
Testing dummies are widely used to study and investigate the effect of different variables such as
the vehicle’s crashworthiness, seat properties and impact severity on the occupant during
impacts. Multiple organizations such as the National Highway Traffic and Safety Administration
(NHTSA), the Insurance Institute for Highway Safety (IIHS) and the European New Car
Assessment Programme (Euro NCAP) require the use ATDs as a standard way to evaluate the
crashworthiness and safety of new vehicles.
A number of dummies have been developed for different occupant size, sex and collision type.
One of the most common used ATDs is the Hybrid III which is used for frontal and rear collision
test [133,154]. It must be noted that in sled tests, the Hybrid III ATD shows excessive spinal
stiffnesses in the thoracic and cervical regions [78,155,156] and its cervical response in the
sagittal plane is questionable [127,157].
26
The EuroSID IIre [158,159] and the SID-IIs [160,161] are two ATDs developed specifically for
lateral collision tests. The BioRID I (Biofidelic Rear Impact Dummy) was first developed in
1998 to study occupant response in low-speed rear-end collisions [162]. The dummies are
equipped with accelerometers and load cells to capture the kinematics as well as kinetics of the
head and neck. Using these data, injury is evaluated using injury criteria discussed in Section 2.3.
Siegmund et al. [163] used BioRID II test dummy seated on a testing sled to study the effect of
impact acceleration on different neck injury criteria. ATDs are usually used to evaluate the
performance of car seats [164–166].
The response of the ATD during a test is evaluated using the kinematics of the dummy as well as
the forces at different body locations, such as the neck loads. Although ATDs are commonly
used for standardized testing and evaluation of vehicles, they do not provide us with a clear
understanding of injury mechanisms in neck soft tissues during motor vehicle collisions.
2.7. Crashworthiness and Energy Absorption
In motor vehicle collisions, the kinetic energy from the impacting (bullet) vehicle is transformed
into three parts: kinetic energy of the impacted (target) vehicle, plastic deformation of both
vehicles and energy lost in friction during the collision. In order to protect the occupant in the
target vehicle, it is of great importance to reduce the kinetic energy transferred during the
collision. This can be achieved by increasing the amount of energy transformed into plastic work
of the vehicle body. One of the efficient methods to enhance the vehicle’s crashworthiness is
through the progressive folding of thin-walled structures. In this work, our focus is thin walled
tubes/frusta.
2.7.1. Collapse of circular thin-walled structures
Thin-walled circular tubes under axial compressive loading generally deform in one of the
following modes: (i) axisymmetric folding (concertina), (ii) asymmetric folding (diamond
shape), (iii) mixed mode of the two aforementioned modes, or (iv) global buckling [167]. The
different collapse modes are shown in Figure 2.16
27
(a) (b) (c)
Figure 2.16 Collapse modes of thin-walled circular tubes: (a) concertina, (b) diamond and
(c) global buckling [168,169]
The collapse mode, whether global buckling or progressive folding, is mainly affected by the
slenderness ratio (L/D) of the tube. Andrews et al. [170] showed that the Eulerian collapse mode
leads to a significant drop in the energy absorption level. A design chart was developed to
identify the critical L/D ratio for specific thickness over diameter ratio that leads to global
buckling. Abramowicz and Jones [171] obtained the critical slenderness ratio for circular and
square steel columns under quasi-static and dynamic axial loading. Their study shows that the
asymmetric folding mode could cause inclination of the column leading to global buckling. For
thin-walled cylinders collapsing under axial compressive load, the ratio between the cylinder’s
radius to the thickness R/t dictates the mode of collapse. Thick cylinders with R/t <40-45
collapse in axisymmetric mode while thinner cylinders collapse in asymmetric mode [172].
2.7.2. Collapse Load and Energy Absorption
Here, we summarize the indicators used to evaluate the performance of a shock absorber. An
important design parameter is the collapse load of the absorber since it directly affects the
acceleration which the vehicle occupant experiences during collision. The first indicator is the
total energy absorbed (TEA) by the absorber which is evaluated by the area under the load versus
crushing distance:
𝑇𝐸𝐴(𝑑) = ∫ 𝐹(𝑑)𝑑𝑥𝑑
0
(2.5)
28
where F(d) is the instantaneous crushing force, and d is the crushing displacement. Another
parameter is the specific energy absorption (SEA), which is the total energy absorbed divided by
the mass of the absorbed, given by:
𝑆𝐸𝐴(𝑑) =1
𝑚𝑎∫ 𝐹(𝑑)𝑑𝑥
𝑑
0
(2.6)
where ma is the mass of the absorber. Since the instantaneous crushing load oscillates, the mean
collapse load Fm is used instead which is calculated by:
𝐹𝑚(𝑑) =1
𝑑∫ 𝐹(𝑑)𝑑𝑥
𝑑
0
(2.7)
The typical response of a thin-walled tube collapsing by progressive folding is shown in Figure
2.17. The load increases until the first fold is formed. This load is called the crippling load or the
maximum load. The fluctuation in the load after the crippling load represents the progressive
folding of the tube.
Figure 2.17 Typical response of a thin-walled tube collapsing by progressive folding (after
[167])
29
The instantaneous crushing load will increase rapidly after a crushing distance dmax when the
absorber is fully collapsed after which the absorber is unusable. Using the maximum crushing
distance, the stroke efficiency of the absorber Se can be evaluated as:
where L is the initial length of the energy absorber.
2.7.3. Aluminum foam
Aluminum foam is a lightweight cellular material which is used in crashworthiness applications
due to its high energy absorption capability. It can undergo large deformation without significant
increase in the applied load, as shown in Figure 2.18, which is an important characteristic of a
good shock absorber.
Figure 2.18 Compressive stress strain curve of closed cell foam [173]
Aluminum foam is produced by introducing bubbles in the molten metal by increasing the
molten metal viscosity to entrap the bubbles inside then stir it with a foaming agent to produce
the gas bubbles [174]. It can be either open-cell or closed-cell foam, depending on the gas
pockets in the foam whether they are connected or discrete, as shown in Figure 2.19.
𝑆𝑒 =𝑑𝑚𝑎𝑥
𝐿 (2.8)
30
(a) (b)
Figure 2.19 Aluminum foam: (a) open cell and (b) closed cell [175]
2.7.4. Foam-filled thin-walled structures
It was found that the energy absorbed by foam-filled shells exceeds the sum of the energies
absorbed by the shell and the foam independently. This is attributed to the energy dissipated due
to the interaction between the foam and the shell such that the mean collapse force of the foam-
filled absorber 𝐹𝑚𝑡𝑜𝑡 is given by:
where 𝐹𝑚𝑐 is the mean collapse load of the empty column, 𝐹𝑚
𝑓 is the mean crushing load of the
foam and 𝐹𝑚𝑖𝑛𝑡 is the force increase resulting from the interaction between the column and the
foam. The role of the interaction effect on increasing the mean total collapse load is shown in
Figure 2.20.
𝐹𝑚𝑡𝑜𝑡 = 𝐹𝑚
𝑐 + 𝐹𝑚 𝑓
+ 𝐹𝑚𝑖𝑛𝑡 (2.9)
31
Figure 2.20 Effect of foam filling on the collapse load of thin-walled columns (after [176])
Numerous analytical models were developed to study the collapse of thin-walled structures.
Alexander [177] developed an analytical model for the deformation of thin-walled hollow
cylinder in concertina mode. This model, shown in Figure 2.21, assumed that the energy
dissipated in deforming the cylinder is composed of two parts: work required for bending the
plastic hinges and work required to stretch the cylinder wall between hinges. The resulting mean
collapse load was deduced to be:
where σy is the material’s yield strength and K is an experimentally determined constant.
𝐹𝑚 = 𝐾𝜎𝑦 𝑡1.5√𝐷 (2.10)
32
Figure 2.21 Axisymmetric collapse mode of thin-walled cylinder proposed by Alexander
[177]
Later, this model was improved to include the effect of the partial inside, partial outside folding
mechanism [178], the effect of the curved fold geometry [179], and the effect of the taper angle
[180]. Reid et al. [181] and Abramowicz and Wierzbicki [182] studied the effect of the
interaction between the foam and the shell for foam filled frusta analytically. Hanssen et al. [14]
deduced the crushing force of foam filled tubes using an empirical model based on experimental
results.
Finite element is one of the tools commonly used to analyze thin-walled tubes under axial
compression. According to Fyllingen et al. [184] shell elements are efficient in modeling thin-
walled tubes at a low computational cost compared to solid elements. A finite element study by
Marzbanrad et al. [185] showed that the initial crippling load of circular aluminum tubes can be
reduced by triggering the progressive folding of the tube using notches, holes or plastic folds.
During axial compressive dynamic loading of thin-walled tubes, two major effects can be
observed: the inertia effect and the strain rate effect. Aluminum is insensitive to strain rate,
therefore, only the effect of inertia can be observed. The study of the quasi-static and dynamic
loading by Ahmad and Thambiratnam [186] revealed that the crushing force is not much affected
33
by the impact velocity. Numerous other studies investigated the performance and the mode of
collapse of foam filled thin walled structures, whether experimentally [187–189] or numerically
using FE [173,190,191].
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aluminium foam filler, Int. J. Mech. Sci., 41, 967–993, Aug. 1999.
[184] Ø. Fyllingen, O. S. Hopperstad, A. G. Hanssen, and M. Langseth, Modelling of tubes subjected to axial
crushing, Thin-Walled Struct., 48, 134–142, Feb. 2010.
[185] J. Marzbanrad, A. Abdollahpoor, and B. Mashadi, Effects of the triggering of circular aluminum tubes on
crashworthiness, Int. J. Crashworthiness, 14, 591–599, Nov. 2009.
[186] Z. Ahmad and D. P. Thambiratnam, Dynamic computer simulation and energy absorption of foam-filled
conical tubes under axial impact loading, Comput. Struct., 87, 186–197, Feb. 2009.
[187] S. R. Guillow, G. Lu, and R. H. Grzebieta, Quasi-static axial compression of thin-walled circular aluminium
tubes, Int. J. Mech. Sci., 43, 2103–2123, Sep. 2001.
[188] A. G. Hanssen, M. Langseth, and O. S. Hopperstad, Static and dynamic crushing of circular aluminium
extrusions with aluminium foam filler, Int. J. Impact Eng., 24, 475–507, May 2000.
[189] S. A. Meguid, M. S. Attia, and A. Monfort, On the crush behaviour of ultralight foam-filled structures,
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aluminium columns, Int. J. Impact Eng., 34, 348–359, Feb. 2007.
42
[191] S. A. Meguid, J. C. Stranart, and J. Heyerman, On the layered micromechanical three-dimensional finite
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43
Chapter 3.
Paper #1: Nonlinear Multibody Dynamics and Finite Element Modeling of Occupant Response: Part I - Rear
Vehicle Collision
This chapter has been published in International Journal of Mechanics and Materials in Design,
15, 3-21, 2019. Available at: https://doi.org/10.1007/s10999-019-09449-x
Abstract
With the rise in vehicle ownership, the need to reduce the risk of injury among vehicle occupants
that arises from vehicle collisions is important to occupants, insurers, manufacturers and policy
makers alike. The human head and neck are of special interest, due to their vulnerable nature and
the severity of potential injury in these collisions. This work is divided into two parts: In Part I,
we focus our attention to modeling rear collision that could lead to whiplash. Specifically, two
multibody dynamics (MBD) models of the cervical spine of the 50th percentile male are
developed using realistic geometries, accelerations and biofidelic variable intervertebral
rotational stiffness. Furthermore, nonlinear finite element (FE) simulations of two generic
compact sedan vehicles in rear collision scenario were performed. Using the acceleration profiles
measured at the driver’s seat of the colliding vehicles, FE simulation of a seated and restrained
occupant in rear collision was performed to determine the occupant response. The resultant
accelerations, measured at the T1 vertebra of the occupant model, were used as an input to the
MBD models to obtain their kinematic response. Validation of the MBD models shows great
agreement with experimentally published data. Comparison between the MBD and FE
simulations for a 32 km/h vehicle-to-vehicle impact shows similar trends in head trajectory.
However, the MBD models reported less peak head displacements compared to the FE model.
This is attributed to the failure of the anterior longitudinal ligament at the mid cervical spine
leading to increased intervertebral rotation in the FE model.
44
Keywords whiplash; rear impact; nonlinear; finite element; multibody dynamics; occupant
kinematics
3.1. Introduction
Despite the significant enhancement in safety of motor vehicles in the last few decades,
whiplash, resulting from rear collisions, remains a serious injury and a source of trauma. It is
estimated that more than 800,000 whiplash injuries occur annually in the United States alone [1],
resulting in neck pain, limited neck movement, visual disturbance and dizziness. According to
the National Highway Traffic Safety Administration (NHTSA) the number of injuries resulting
from rear-end collisions increased during 2007-2015 from 485,000 incidents to 556,000 incidents
becoming the most common reason for injuries during motor vehicle collisions [2,3]. In order to
provide better protection for occupants, it is important to identify and compute the occupant
response during collisions. Many techniques have been devised to study occupant response
during rear-end collisions. A number of experimental studies were conducted on volunteers [4,5].
However, to limit injury to those volunteers, the level of impact severity was reduced to an
acceptable level. As a result, such studies are very limited. To overcome this limitation, cadavers
have been used in other experimental studies [6,7]. Although the use of cadavers allows testing at
higher impact forces, the cadavers lack responsiveness due to the absence of muscle activation.
Anthropomorphic test dummies (ATDs), such as HybridIII-TRID [8] and BioRID2 [9], have
been widely used not only in motor vehicles crash tests but also for railway crashes [10] and
military applications [11]. However, these ATDs too have many limitations.
This does not in any way reduce the great strides and advance made as a result of these efforts in
developing car safety strategies for the passenger car. Advance in computer modeling and
simulation has added significantly to these strategies. Numerous numerical techniques exist in
the literature. Examples include numerous multibody dynamics (MBD) models, in which the
body parts are modeled as rigid links connected through a number of springs, dampers and/or
viscoelastic elements; see, e.g., Refs. [12–15].
The finite element (FE) method has been used extensively in biomechanical and biomedical
applications such as design and analysis of implants [16,17], study of tumors [18,19] and sports
applications [20] to name a few. Due to the complexity of the neck geometry and the need not
45
only to obtain the dynamic response, but also the stresses in the soft tissues, the FE method has
been successfully used to develop more realistic models of whiplash.
A number of FE models of the head and neck have been developed to study whiplash by
applying the loads to the torso (T1 vertebra), as explained in Refs. [21–24]. However, the
accuracy of these models depends greatly on the representation of the loads transferred to the
torso during a collision. Furthermore, these models do not provide any details about the overall
behavior of the occupant; such as ramping, interaction with the seat and the seatbelt. Therefore, it
is essential to study the interaction of the occupant with the car seat for better understanding of
the dynamic response.
In an effort to increase the accuracy of numerical modeling, research groups developed accurate
numerical models of full humans. Examples include the Total Human Model for Safety
(THUMS) [25] and the Global Human Body Model Consortium (GHBMC) [26] who developed
a number of detailed FE models of the human body for both male and female occupants, and
pedestrians. The HUman MOdel for Safety (HUMOS) developed a model of the male occupant
[27], the Virtual Vehicle-safety Assessment (ViVA) project for Open-source Human Body
Models (OpenHBM) addressing gender diversity developed a FE model of the 50th percentile
female occupant [28] and the PIPER project which developed a scalable FE model of a child
occupant [29]. These newly developed numerical models reflect the importance of using full
body models in collision simulations.
In this study, two MBD models of the head and neck are developed to capture the head response
during rear collision by making use of a vehicle-to-vehicle rear impact FE simulation to acquire
realistic seat acceleration for the MBD model and the FE simulation of a seated full human body
model of a male occupant.
3.2. Multibody Dynamics Model
The MBD models developed focus only on the head and the cervical spine of a seated 50th
percentile male occupant. The models assumed that all motion occur in the sagittal plane. Figure
3.1 demonstrates the idealization of the cervical spine as a series of rotating rigid links. The
human cervical spine consists of 7 vertebrae, joined by intervertebral discs, ligaments and
46
muscles. The cervical spine begins with the C7 vertebra at the base of the neck and ends with the
C1 vertebra at the base of the skull. Each pair of adjacent vertebrae constitutes a functional
spinal unit (FSU) [30]. The upper vertebra of each FSU rotates about an instantaneous axis of
rotation (IAR) located in the lower vertebra [30–32]. In the current MBD models, the cervical
spine was idealized as a series of rigid links joined by viscoelastic joints. Two models were
considered: single and two degrees of freedom (DOF) models. In the single DOF model, the
intervertebral joints were assumed to allow rotation only. In the two DOF model, axial extension
of the links was added to represent the axial flexibility of the FSUs.
Figure 3.1 A schematic of (a) human cervical spine and (b) MBD model in the sagittal plane
The Kelvin-Voigt material model is commonly adopted to model the passive response of soft
tissues to external loads [33–35]. The material model contains a viscous damper and a spring in
parallel. This material model was used in the current MBD models to represent the axial and
rotational viscoelastic properties of the intervertebral joints.
Figure 3.2 represents a generalized system of 2 links and depicts the coordinate system used in
the development of the MBD models for the sagittal plane. The coordinate system of the model
is attached to the T1 vertebra. The positive horizontal (x) and vertical (z) directions are in the
upward and forward directions, respectively, and the direction of positive rotation is in the
47
counter-clockwise direction. Using this coordinate system, the locations of the joints with respect
to T1 vertebra are described by the following sums in the x and z directions:
𝑥𝑖 = −∑𝑙𝑗 sin 𝜃𝑗
𝑖
𝑗=1
(3.1)
𝑧𝑖 = ∑𝑙𝑗 cos 𝜃𝑗
𝑖
𝑗=1
(3.2)
where lj is the length of the ith segment and θj is the rotational angle of the jth joint, measured
from the vertical axis.
In the sagittal plane rotation, the IARs of the middle and lower cervical vertebrae are located
within the lower vertebra of each FSU [31,32]. The IAR of C1 vertebra is found within the dens
[30] and the IAR of the C0-C1 joint at the base of the skull is located at the occipital condyles
[36]. The IAR locations of the MBD models were calculated from the geometry of the 50th
percentile male cervical spine [37]. The dimensions and initial angle of the rigid links are given
inTable 3.1.
Figure 3.2 MBD system investigated: (a) a generalized joint containing two links and (b)
coordinate system and sign convention used
48
Table 3.1 Geometry of the MBD models [37]
Link Lower IAR Upper IAR Lower joint
level
Length
(mm)
Initial
angle (°)
1 C7 C6 C7-T1 21.5 -20
2 C6 C5 C6-C7 16.9 -5
3 C5 C4 C5-C6 16.5 0
4 C4 C3 C4-C5 18.9 -5.5
5 C3 C2 C3-C4 16.9 5
6 C2 C1 C2-C3 46.9 8
7 C1 C0 C1-C2 7.8 8.5
8 C0 Head center C0-C1 57.9 -20
3.2.1. Single DOF Model
In the single DOF MBD model, the intervertebral joints were modeled as viscoelastic rotational
joints with only one rotational DOF. Figure 3.3(a) demonstrates a generalized single DOF model
with 2 links showing the positive direction of motion.
Figure 3.3 A schematic of generalized (a) single DOF and (b) two DOF MBD models
showing two adjacent links meeting at a viscoelastic joint
49
The equations of motion of the rigid links were derived using Euler-Lagrange equation of
motion; as follows:
𝑑
𝑑𝑡(𝜕𝐿
𝜕𝑞��) −
𝜕𝐿
𝜕𝑞𝑖+
𝜕𝑅
𝜕𝑞𝑖= 𝑄𝑖 (3.3)
where t is time, qi is the ith generalized coordinate, 𝑞�� is the first time derivative of the ith
generalized coordinate, Qi is the ith generalized force, R is the Rayleigh dissipation function, and
L is the Lagrangian given by:
𝐿 = 𝑇 − 𝑉 (3.4)
where T is the total kinetic energy of the system and V is the total potential energy of the system.
For the current system, 8 generalized coordinates were used to describe its state.
The linear velocities of the joints were found by differentiating equations (3.1) and (3.2) with
respect to time:
𝑥�� = −∑(𝑙𝑗��𝑗 cos 𝜃𝑗)
𝑖
𝑗=1
(3.5)
𝑧�� = −∑(𝑙𝑗��𝑗 sin 𝜃𝑗)
𝑖
𝑗=1
(3.6)
where ��𝑖 is the first time derivative of the ith angle of rotation θi. Using equations (3.5) and (3.6),
T is given as the sum of a rotational component Trot and a translational component Ttrans; viz.:
𝑇rot = ∑1
2𝐼𝑖𝜃��
28
𝑖=1
(3.7)
𝑇trans = ∑1
2𝑚𝑖(𝑥��
2 + 𝑧��2)
8
𝑖=1
(3.8)
50
where Ii is the rotational moment of inertia of the ith segment and mi is the mass of the ith
segment. The masses of the vertebrae were based on Hoover’s [33] estimate of the neck segment
mass for a 50th percentile male and the measurements of Lowrance and Latimer [38], and the
moments of inertia of the vertebrae were obtained from de Jager [39]. Plaga et al. [40] found the
mass of the 50th percentile male head to be 4.7 kg and Schneider et al. [41] gave the moment of
inertia of the 50th percentile male head as 0.0222 kg·m2 in the sagittal plane. The mass and
moment of inertia of each segment of the MBD model are summarized in Table 3.2.
Table 3.2 Masses and moment of inertia of cervical vertebrae [33,38–41]
Vertebra m (kg) I×10-3 (kg·m2)
C0 (head) 4.7 22.2
C1 0.12 0.22
C2 0.14 0.25
C3 0.25 0.24
C4 0.32 0.23
C5 0.37 0.23
C6 0.3 0.24
C7 0.29 0.22
The potential energy of the system is stored in the rotational joints. In the current system of
variable stiffness rotational springs, the total potential energy stored is given by:
𝑉 = ∑∫ 𝜙 𝑘𝑟𝑖(𝜙) 𝑑𝜙
𝜙𝑖
0
8
𝑖=1
(3.9)
where kri is the rotational stiffness of the ith joint and ϕi is the relative angles of rotation for the ith
joint, given by:
𝜙1 = 𝜃1 − 𝜃01 (3.10)
𝜙𝑖 = (𝜃𝑖 − 𝜃𝑖−1) − (𝜃0𝑖 − 𝜃0(𝑖−1)), 𝑖 ≥ 2 (3.11)
where θ0i is the initial angle of rotation of the ith link. Camacho et al. [42] found that the moment-
angle relationship of cervical FSUs follows equation of the form:
51
𝜙𝑖 =1
𝐵𝑖ln (
𝑀𝑖
𝐴𝑖+ 1) (3.12)
where Mi is the bending moment applied on the ith intervertebral joint, and Ai and Bi are
experimentally obtained coefficients for each intervertebral joint. Rearranging and differentiating
equation (3.12) with respect to ϕi yields the expression for the rotational stiffness of each joint:
𝑘𝑟𝑖(𝜙𝑖) =𝑑𝑀𝑖
𝑑𝜙𝑖= 𝐴𝑖𝐵𝑖𝑒
𝐵𝑖𝜙𝑖 (3.13)
Camacho et al. [42] calculated the stiffness of the C0-C1-C2 complex as a single entity. Testing
of isolated C0-C1 and C1-C2 intervertebral joints has shown that the C0-C1 level incorporates
for approximately 30% of the stiffness of the entire C0-C1-C2 complex, while the C1-C2 joint
incorporates for the remainder 70% [43,44]. This permits the calculation of separate C0-C1 and
C1-C2 rotational stiffnesses. The Ai and Bi values obtained in the sagittal plane for each
intervertebral level are shown in Table 3.3.
Table 3.3 Intervertebral rotational stiffness curve coefficients in sagittal plane [42]
Intervertebral
level
Flexion Extension
Ai
(N·m/°) Bi
Ai
(N·m/°) Bi
C0-C1 0.0193 0.3052 -0.0136 -0.3937
C1-C2 0.045 0.3052 -0.0317 -0.3937
C2-C3 0.1029 0.4714 -0.0037 -1.0137
C3-C4 0.0218 0.7503 -0.0068 -1.1416
C4-C5 0.113 0.3929 -0.0027 -1.641
C5-C6 0.0618 0.5587 -0.0126 -0.9581
C6-C7 0.1406 0.5607 -0.0125 -1.2366
C7-T1 0.6084 0.3949 -0.3105 -0.6489
The energy dissipated in the viscoelastic joints was accounted for using the Rayleigh dissipation
function, given by:
𝑅 =1
2𝑐𝑟1𝜃1
2+
1
2∑𝑐𝑟𝑖 (��𝑖
2− ��𝑖−1
2)
8
𝑖=2
(3.14)
52
where cri is the rotational damping coefficient of the ith joint. The rotational damping coefficient
of cervical FSUs was found to be 1.5 N·m·s/rad at each intervertebral level [39,45].
During rear impact, the torso accelerates forward due to the interaction between the occupant and
the vehicle seat. Within the T1 vertebra coordinate system, the acceleration of the T1 vertebra
was represented by a series of inertial forces applied to the head and the cervical vertebrae in the
opposite direction to the T1-acceleration, such that:
𝐹𝑖 = 𝑎𝑥𝑚𝑖𝑖 + 𝑎𝑧𝑚𝑖�� (3.15)
where 𝐹𝑖 is the inertial force applied at the ith segment, ax and az are the respective applied
accelerations in the x and z directions, and 𝑖 and �� are unit vectors in the x and z directions,
respectively. The virtual work done by the inertial forces in the entire system, δW, is given by:
𝛿𝑊 = ∑𝐹𝑖 ∙ 𝛿𝑟𝑖
𝑛
𝑖=1
(3.16)
where n is the number of generalized coordinates of the system, and 𝛿𝑟𝑖 is the virtual
displacement of the center of mass (CM) of the ith segment, defined as:
𝛿𝑟𝑖 = ∑𝜕𝑟𝑖
𝜕𝑞𝑗𝛿𝑞𝑗
𝑛
𝑗=1
(3.17)
and
𝑟𝑖 = 𝑥𝑖𝑖 + 𝑧𝑖�� (3.18)
is the location of the ith joint within the T1 coordinate system.
Substituting and rearranging the above expressions yields δW in terms of Qi:
𝛿𝑊 = ∑𝑄𝑖𝛿𝑞𝑖
𝑛
𝑖=1
(3.19)
where Qi is given by:
53
𝑄𝑖 = −𝑙𝑖(𝑎𝑥 cos 𝜃𝑖 + 𝑎𝑧 sin 𝜃𝑖)∑𝑚𝑗
8
𝑗=𝑖
(3.20)
The equations of motion of the rigid links system were obtained by substituting equations (3.7),
(3.8), (3.9), (3.14) and (3.20) into equation (3.3), producing a system of 8 second-order
differential equations. The resulting system of equations was re-arranged into the following
matrix form:
[𝐴]{��} + [𝐵]{��2} + [𝐶]{��} + [𝐷]{𝑞} = {𝑄} (3.21)
[A] matrix represents the masses and moments of inertia of the MBD system, [B] matrix
represents the Coriolis inertial force, [C] and [D] matrices contain the constant damping and
variable stiffness coefficients, respectively. {Q} is the generalized force vector defined in
equation (3.20). The matrices and vectors represented by equation (3.21) can be found in
Appendix 3.1.
3.2.2. Two DOF Model
In the two DOF model, we impart an axial displacement to the system based on the observation
that cervical FSUs demonstrate approximately linear force-displacement relationships in the
axial direction [46]. The two DOF model was given the added DOF of axial displacement as
shown in Figure 3.3(b). We used constant axial stiffness for the intervertebral joints. The
stiffness coefficient of each intervertebral joint is given in Table 3.4.
Table 3.4 Axial stiffness of intervertebral joints [46]
FSU Stiffness (×105 N/m)
C0-C1 4.18
C1-C2 1.72
C2-C3 1.86
C3-C4 11.44
C4-C5 10.01
C5-C6 6.67
54
C6-C7 4.54
C7-T1 12.32
Since li is variable in the two DOF model, the velocity equations (3.5) and (3.6) become:
𝑥�� = −∑(𝑙�� sin 𝜃𝑗 + 𝑙𝑗��𝑗 cos 𝜃𝑗)
𝑖
𝑗=1
(3.22)
𝑧�� = ∑(𝑙�� cos 𝜃𝑗 − 𝑙𝑗��𝑗 sin 𝜃𝑗)
𝑖
𝑗=1
(3.23)
where 𝑙�� is the first time derivative of the lj. By incorporating the energy stored in axial
elongations, the potential energy equation (3.9) becomes:
𝑉 = ∑(∫ 𝜙 𝑘𝑟𝑖(𝜙) 𝑑𝜙
𝜙𝑖
0
+1
2𝑘𝑒𝑖(𝑙𝑖 − 𝑙0𝑖)
2)
8
𝑖=1
(3.24)
where kei is the axial stiffness of the ith link and l0i is the initial length of the ith link.
By incorporating the energy dissipated in axial displacements, equation (3.14) is modified to:
𝑅 =1
2𝑐𝑟1𝜃1
2+
1
2∑𝑐𝑟𝑖 (��𝑖
2− ��𝑖−1
2)
8
𝑖=2
+1
2∑𝑐𝑒𝑖 (𝑙��
2)
8
𝑖=1
(3.25)
where cei is the axial damping coefficient of the ith joint with a value of 1000 N·s/m for all
intervertebral levels [47]. By substituting the above expressions into equation (3.3), 16 equations
of motion were obtained to describe the two DOF model. The equations of motion were arranged
into a matrix form shown in equation (3.21) and each matrix or vector can be found in Appendix
3.2.
55
3.2.3. Rotational Limits of Intervertebral Joints
The stiffness relationships described by equation (3.13) were only calculated for intervertebral
joints within the normal range of motion, limited by ligaments and bone-to-bone contact.
Through the use of isolated ligamentous spine models, Ivancic et al. [48] and Panjabi et al. [49]
recorded the maximum flexion and extension angles reached by each intervertebral level in rear
and frontal collisions of varying severities. These angles were greater than the angles of rotation
observed in voluntary motion [50,51], as rotation angles beyond the physiological limit may be
observed in severe collisions [48,49]. The maximum angles of rotation in both extension and
flexion for each intervertebral level are provided in Table 3.5.
Table 3.5 Maximum angles of rotation at each intervertebral joint [48,49]
Intervertebral level Flexion (°) Extension (°)
C0-C1 14.6 27.7
C1-C2 9.4 6.4
C2-C3 11.4 6.6
C3-C4 16.4 9.6
C4-C5 9.9 11.9
C5-C6 11.9 10.9
C6-C7 11.4 12.9
C7-T1 14.5 10.6
3.2.4. Solver
Equation (3.21) for both the single and two DOF models may be rearranged into the following
form:
{��} = [𝐴]−1({𝑄} − [𝐵]{��2} − [𝐶]{��} − [𝐷]{𝑞}) (3.26)
This system of n second-order differential equations was then rewritten as a system of 2n first-
order differential equations, where n indicates the number of generalized coordinates in each
system:
56
{{𝑑𝑞𝑖
𝑑𝑡}
{𝑑��𝑖
𝑑𝑡}
} = {{��𝑖}
[𝐴]−1({𝑄} − [𝐵]{��2} − [𝐶]{𝑞} − [𝐷]{𝑞})} (3.27)
This system was solved numerically in MATLAB. The single DOF system was solved using the
ODE45 function based on the Runge-Kutta numerical method of solving systems of first-order
differential equations. It was chosen for its capacity for adaptive step sizing, which allows for the
minimization of computational cost, while maximizing solution accuracy. For the two DOF
system, the ODE15s function, designed for stiff systems of differential equations, produced
results in significantly reduced time.
3.3. Finite Element Modeling
During a vehicle collision, the kinetic energy of the collision is partially absorbed by the plastic
deformation of the contacting regions of the vehicle body and the remainder to the struck vehicle
and occupant. The magnitude and duration of the collision loads have been directly linked with
cervical injury probability and the severity of this injury [52,53]. In this section, the finite
element method (FEM) was used to: (i) Simulate two-car collision in a rear impact scenario to
obtain realistic acceleration and velocity profiles at the driver’s seat of the impacted vehicle, (ii)
apply the collision velocity profiles as an input to a FE model of a seated occupant and (iii) to
use the resulting T1 vertebra velocity and acceleration profiles from the FE simulation in the
MBD analysis to obtain the kinematic cervical response of the occupant. The nonlinear dynamic
FE analysis was conducted using the explicit solver of LS-DYNA.
3.3.1. FE Modeling of Vehicular Collision
A detailed FE model of a generic compact sedan vehicle provided by NHTSA [54] and
extensively validated [55] was used to simulate central collisions between two identical vehicles.
The model consisted of 1.5 million shell, beam, and solid elements, with a mass of 1100 kg. The
model contains the metallic structure of the vehicle, the tires, as well as the interior components
such as the seats and steering wheel. The vehicle structure was mainly composed of steel with
elasto-plastic material models, using the piecewise linear plasticity material model in LS-DYNA.
57
The material properties are summarized in Table 3.6. Upon exceeding the yield stresses, the
stress-strain relationships of the materials were defined using plastic stress-strain curves. The
material model accounts for the strain rate-dependence of yield stress through the Cowper and
Symonds model, which scales the yield stress by:
𝜎𝑑
𝜎𝑠= 1 + (
휀
𝐶)
1𝑝
(3.28)
where σd is the dynamic yield stress, σs is the static yield stress, 휀 is the strain rate, and C and p
are material constants and were assigned the values of 8000 s-1 and 8, respectively.
Table 3.6 Material properties of the steels used in the vehicle structure
Young’s modulus (GPa) 200
Poisson’s ratio 0.3
Density (kg/m3) 7890
Yield stress (MPa) 180 - 350
The collision simulations consisted of the two vehicles colliding against one another in a rear
impact scenario, as shown in Figure 3.4. Both vehicles rested on a horizontal rigid plane. An
initial velocity was assigned to the colliding (bullet) vehicle, while the other (target) vehicle was
initially stationary. The vehicles were positioned such that their centers of gravity were aligned
in the direction of the impact velocity. In order to represent the rotation of the wheels of the
vehicle, an initial rotational velocity was also assigned to the wheels of the bullet vehicle.
58
Figure 3.4 Setup of vehicle collision FE simulation showing bullet and target vehicles
Due to the complex nature of vehicle collisions, it is difficult to identify the contact regions
during collisions. To overcome this difficulty, we initiated our FE analysis by assuming contact
of all parts of the vehicle against all other parts using an AUTOMATIC_SINGLE_SURFACE
contact definition in LS-DYNA. The model used segment-based two-way contact rather than the
standard penalty contact formulation, which checks for contact between segments rather than
nodes against surfaces to prevent nodes penetrating the surface.
An initial constant velocity was assigned to the bullet vehicles. In FMVSS 208 [56], the NHTSA
requires new vehicles to demonstrate sufficient occupant protection capability at 48 km/h in rear
collisions. However, the literature shows the high probability of cervical injury in even low-
velocity collisions [52,56,57]. Therefore, a lower impact velocity of 32 km/h was chosen in the
current study. Gravity was considered in the simulations.
3.3.2. FE Modeling of Occupant Response
The current study used the GHBMC 50th percentile male model to represent the occupant. The
model weighs 78 kg, is 174.9 cm tall, and contains 2,197,853 elements. The geometry of the
model was based on MRI and CT scan data of a 50th percentile 26-year-old male in good health
[58]. The GHBMC 50th percentile male model was previously validated by the authors in rear
collision [59] using data obtained through PMHS tests. The cervical spine, in particular, was
subjected to several studies that validated it against experimental data [60,61].
59
The GHBMC 50th percentile male model was seated on a FE model of a vehicle seat extracted
from the generic vehicle model. The seat model contained the driver’s seat and the vehicle floor.
The seatbelt plays a major role in shaping the response of the occupants [62–64]. In rear
collisions, the seatbelt prevents the ramping of the occupant along the seat back by restraining
the occupant’s hip. In order to model the effects of the seatbelt on the occupant response, a
validated 3-point seatbelt model developed by Östh et al. [28] was used to restrain the occupant
model. The seatbelt model was fitted around the numerical human body model to ensure no gap
existed between the seatbelt and the GHBMC model. Figure 3.5 shows the setup of the seated
and restrained GHBMC male FE model.
In the vehicle interior, the occupant primarily interacts with the polyurethane foam material of
the seat. The seat foam was modeled using a low-density foam material model with a density of
101 kg/m3 and the nonlinear stress-strain relationship shown in Figure 3.6. The foam of the
vehicle seat was modeled using constant stress tetrahedral solid elements because it is suitable
for large deformation of foam.
60
Figure 3.5 GHBMC 50th percentile numerical male occupant FE model seated on the
vehicle seat
Figure 3.6 Stress-strain curve of polyurethane foam material of the seat
61
Contact was modeled between the body of the occupant model and the vehicle seat. Because the
current study investigated the cervical response of the vehicle occupant due to movements of the
T1 vertebra, no contact was modeled between the head and the headrest. The kinematics of the
head and cervical spine depended entirely on the movement of the torso. Segment-based contact
was used between the human body and the vehicle seat. It is worth noting that segment-based
contact is the recommended contact treatment method for soft materials that may produce large
deformations [65], such as human soft tissue and polyurethane foam of the seat. Furthermore,
segment-based contact allows LS-DYNA to check edge-to-edge penetrations without
penetrations of nodes against surfaces and automatically calculates contact stiffness based on the
time step. The same contact algorithm was applied between the occupant torso and the seatbelt,
and between the occupant feet and the vehicle floor.
The friction between the occupant and the seat back had been observed to play a major role in
collision response of the occupant, with lower coefficients of friction producing higher ramping-
up of the occupant, resulting in more severe injuries [66]. In order to simulate a worst-case
scenario, the respective static and dynamic coefficients of friction were taken to be 0.577 and
0.360, to represent a low-friction vehicle seat [67].
Only gravity was initially applied to the seated occupant model. The seatbelt was moved into
position around the GHBMC model to represent a snug fit, while avoiding influencing its
posture. The collision velocity profile of the driver’s seat obtained from the vehicle-to-vehicle
FE simulation was applied to the seated occupant model. The velocity profile was applied to the
vehicle floor, as well as to the seatbelt end-points.
3.4. Results and Discussion
3.4.1. Validation of MBD and FE models
An important aspect of the work is to validate the two models developed in this study. The MBD
model is validated first followed by the FE validation. To validate the MBD models, the T1-
acceleration measured in experimental sled test was applied to the single and two DOF MBD
models. The rear acceleration profile, shown in Figure 3.7, was obtained from PMHS sled tests
62
representing 2.6 m/s collisions using isolated musculoskeletal head-neck systems by [68]. The
predicted head kinematic response was compared with the experimental results.
Figure 3.7 Experimental T1 acceleration profile after [68]
The head horizontal displacement and rotation of the single DOF and the two DOF MBD models
compared to the experimental results of [68] are shown in Figure 3.8. The results show that both
the horizontal head displacement and head rotation demonstrate good agreement with
experimental data. Because the experimental data ended before peak head displacement, peak
displacement values were not compared against the MBD model response. However, both single
and two DOF models response remained within the experimental response corridors up to 127
ms. Both MBD models presented nearly an identical response. This was attributed to the
relatively low collision velocities simulated by the acceleration profile, which was not sufficient
to distinguish between the models.
63
Figure 3.8 Validation of MBD model head response against PMHS sled test [68]: (a)
horizontal head displacement and (b) head rotation
The GHBMC male occupant FE model and the generic sedan vehicle FE model have been
individually extensively validated against experimental data [55,69–72]. However, the
interaction between the two models (simulated human and vehicle) require validation against
experimental results. To validate the FE model, the rear collision acceleration profile measured
experimentally in sled tests of restrained volunteers by [9], shown in Figure 3.9, was applied to
the vehicle floor of the seated GHBMC model. The experimental rear collision sled test included
a headrest which was not modeled in the current FE model. Contact between the head and the
headrest was observed at 94 ms. Therefore, only results between 0 and 94 ms were considered.
Figure 3.9 Experimental horizontal sled acceleration profile by [9]
Figure 3.10 shows the horizontal head response of the FE model in rear collision. At 94 ms, the
mean horizontal head displacement was 70 mm, while the FE model produced a displacement of
64
86 mm. In general, the FE results demonstrated similar trends as the volunteer experimental
results. The major sources of discrepancies between the FE model response and experimental
results are: (i) lack of muscle activation in the FE model, resulting in different cervical
kinematics and (ii) the difference in seat stiffness between the flexible seat with foam padding
used in the FE model and the rigid seats used in the experimental tests [9].
Figure 3.10 Head center of mass horizontal displacement with respect to the seat of the FE
model compared to experimental volunteer test [9]
3.4.2. Occupant Response in vehicle-to-vehicle impact
The velocity profile recorded for the driver’s seat from the vehicle-to-vehicle collision FE
simulation is shown in Figure 3.11. The seat accelerates until it reaches a maximum velocity of ~
18 km/h (5 m/s) in the horizontal forward direction (+x direction) during the first 100 ms after
impact. The velocity remains constant until 180 ms after which the seat velocity decreases. It was
observed that the separation between the two vehicles occurs at 180 ms. The difference between
the seat velocity of the target vehicle (18 km/h) and the impact velocity of the bullet vehicle (32
km/h) is attributed to (i) the energy dissipated in the plastic deformation of both vehicles during
collision, (ii) the energy dissipated in friction during the contact between the deforming parts,
and (iii) the friction between the tires and the ground. A much smaller velocity profile is
observed in the vertical direction (z direction) with a peak velocity of ±1.1 km/h (±0.3 m/s) while
insignificant change in the velocity is recorded in the lateral direction.
65
Figure 3.11 Driver seat velocities resulting from 32 km/h rear-end collision
In a rear collision, the vehicle seat is accelerated in the forward direction, applying a compressive
load to the back of the seated occupant. The head moves backward with respect to the seat due to
the head’s inertia, inducing extension in the cervical spine. The occupant response during
maximum neck extension is shown in Figure 3.12. The horizontal and vertical accelerations of
T1 vertebra recorded from the occupant-seat FE simulation were applied to the MBD models and
their responses are compared.
Figure 3.13 shows that the head displacement magnitudes of the MBD models were significantly
lower than that of the FE model. The MBD models also reached maximum displacement
significantly earlier than the FE model. The peak displacement magnitudes and times are given
in Table 3.7.
Figure 3.13(a) shows the horizontal displacement response of the MBD and FE models. The two
DOF MBD model produced a peak horizontal displacement that was approximately 4% higher
than the single DOF model. The FE model produced a peak horizontal displacement that was
approximately 39% above both MBD models. Figure 3.13(b) shows that the FE model vertical
displacement was approximately 187% higher than the MBD models. In addition, the single and
two DOF MBD models reached peak displacements 47 and 45 ms before the FE model,
respectively.
66
Figure 3.12 Response of 50th percentile male occupant without headrest and restrained
using 3-point seatbelt, with 32 km/h rear collision velocity profile applied to the seat
Figure 3.13 Head center of mass displacements during 32 km/h rear-end collision: (a)
Horizontal and (b) vertical
67
Table 3.7 Displacement and rotational response of the head center of mass
FE 1 DOF MBD 2 DOF MBD
Peak horizontal displacement (mm) 171.7 122.5 127.2
Peak vertical displacement (mm) 100.7 34.9 27.5
Peak rotation (°) 82.1 62.3 60.0
Time of peak displacement (ms) 190 143 145
The increased displacements in the FE model were the result of ligament failure in the FE model.
The Anterior Longitudinal Ligament (ALL) beam elements were observed to undergo near-
complete failure at the C3-C4 intervertebral level. Ligament failure reduces the rotational
stiffness of the intervertebral level, resulting in increased intervertebral rotation. Furthermore, the
intervertebral rotations of the MBD models exceeded the ranges of the nonlinear rotational
stiffness curves [42,73], leading to reduced model displacements.
Figure 3.13 shows that the head of the FE model initially sagged in flexion due to the counter-
clockwise rotation of the T1 vertebra as the occupant contacts the seat back, while the head
continued to move at a constant linear velocity with respect to the seat. The response of the MBD
models does not reflect this initial flexion, due to the exclusion of T1 rotational accelerations.
In Figure 3.14, the non-physiologic “S”-shaped curvature characteristic of whiplash [74] was
demonstrated by both the FE and MBD models between 76 and 114 ms. The “S”-shape is a
straightening of the physiological lordosis of the cervical spine and occurs when the head
translates horizontally in the posterior direction without rotation. Furthermore, the “S”-shape is
associated with an increased risk of cervical injury at the lower intervertebral levels [74].
68
Figure 3.14 Rear collision response of the MBD models and the FE model representing an
occupant with no headrest and restrained using a 3-point seatbelt and 32 km/h rear-end
collision velocity profile applied to the seat
The MBD model is an important tool which provides the kinematics of the head during rear
impacts. One of its main advantages it requires few seconds to evaluate the head/neck response
at a low computational cost. All MBD simulations were conducted using a single core with
running time < 1 min while the occupant-seat FE simulation was conducted using 64 cores with a
running time of ~2 days. The high computational cost of the FE simulations is necessary due to
the large number of elements and the non-linearity of the simulations. This high computational
cost can be reduced by simplifying the areas of the human body which are not of interest. Finite
element models of the human occupant provide crucial information about the kinematics,
kinetics and strains in necks’ soft tissues which can be very beneficial to whiplash evaluation
studies concerned with occupant’s safety.
69
3.5. Conclusions
In Part I of our work, two MBD models of the cervical spine were developed to simulate the
response during rear-end motor vehicle collision. The single DOF model contained only
rotational viscoelastic joints, while the two DOF model allowed axial extension. The models,
which were subject to realistic velocity and acceleration profiles, were used to determine the
kinematic response of the occupant head and cervical spine. In addition, a FE model of a seated
and restrained 50th percentile simulated male occupant was developed and validated against
experimental results. The MBD models showed agreement in their responses with experimental
published data.
For a 32 km/h rear-end collision, the driver seat in the target vehicle reported a peak velocity of
18 km/h due to energy dissipated mainly in vehicle plastic deformation. Using this seat velocity
profile, the comparison between the MBD and the occupant-seat FE models response shows that
peak horizontal head displacements in the MBD models were less than that in the FE model by
~39% and occurred sooner. Furthermore, the FE reported failure in the ALL at the C3-C4
leading to increased intervertebral rotations.
The MBD model is a beneficial tool that provides a quick estimation of the head’s kinematics
during rear collisions. However, finite element models are essential for more detailed kinetics
and stress/strain state of the soft tissues of the neck. The outcomes of our models are highly
beneficial in injury characterization resulting from vehicle collisions and can be used to study
and evaluate whiplash injury mechanisms for occupants subjected to rear-end impacts.
Acknowledgment
This publication was made possible by NPRP grant# (7-236-3-053) from the Qatar National
Research Fund (a member of Qatar Foundation). The statements made herein are solely the
responsibility of the author(s).
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Conference Proceedings (No. IRC-15-69), 2015.
[73] A. Trajkovski, S. Omerovic, S. Krasna, and I. Prebil, Loading rate effect on mechanical properties of
cervical spine ligaments, Acta Bioeng. Biomech., 16, 13–20, 2014.
[74] J. N. Grauer, M. M. Panjabi, J. Cholewicki, K. Nibu, and J. Dvorak, Whiplash produces and S-shaped
curvature of the neck with hyperextension at lower levels, Spine (Phila. Pa. 1976)., 22, 2489–2494, 1997.
74
Addendum to Chapter 3
In order to maintain the integrity of the article and to account for additional work conducted, we
have included this addendum. In it, we provide a detailed account of the influence of the frontal
airbag on the occupant response in rear-end vehicle collision.
A3.1 Occupant Protection
In order to protect the occupant during rear-end collisions, the neck extension and flexion should
be limited to prevent neck injury. This can be achieved by the proper adjustment of the head
restraint and deploying the frontal airbag.
The 50th percentile male GHBMC FE model was seated on the seat rig shown in Figure A3.1 and
was subjected to the acceleration profile resulting from a 32 km/h rear impact. The rig consists of
the seat with a head restraint, the seat floor and the steering wheel compound. Furthermore, the
rig is equipped with a seat belt and an airbag. The airbag deployment was initiated 100 ms after
the impact started to ensure that the occupant neck was already going into extension and the head
is far enough from the deployed airbag.
Figure A3.1 Setup of rear impact simulations showing (a) the GHBMC 50th percentile male
model seated on a seat rig equipped with a seat belt, a head restraint and an airbag, and (b)
the folded airbag embedded in the steering wheel
75
A3.2 Results and Discussion
The head horizontal and vertical head displacements with respect to T1 vertebra are shown in
Figure A3.2. Although the presence of the head restraint reduces the head displacement,
specifically the vertical one, during the neck extension, it increases the severity of the occupant
rebound. The head restraint increases the horizontal and vertical head displacements during the
neck flexion by 57% and 147%, respectively. Deploying the airbag during a rear-end collision
eliminates the excessive neck flexion, as shown in Figure A3.3. The airbag reduces the head
horizontal and vertical displacements during the neck flexion by 56.5% and 90%, respectively.
Figure A3.2 Head (a) horizontal and (b) vertical displacement with respect to T1 vertebra
for the different seat configurations: no head restraint (NoHR), with head restraint (HR),
and with head restraint and airbag (HR&AB)
Figure A3.3 Occupant head during neck flexion: (a) unsupported and (b) supported by the
airbag
76
The presence of the head restraint reduces the head rotation during neck extension by 70.5%.
However, it increases the head rotation during the neck flexion by ~176%. That excessive head
rotation during neck flexion is reduced by ~91% when the frontal airbag is deployed.
The possibility of injury of two of the cervical ligaments was assessed. The percent elongation of
the anterior longitudinal ligament and the interspinous ligament normalized against the sub-
failure percent elongation is shown in Figure A3.4.
Figure A3.4 Percent elongation of ALL, PLL, ISL, LF and CL normalized against percent
elongation to sub-failure for the different seat configurations: without head restraint
(NoHR), with head restraint (HR), and with head restraint and airbag (HR&AB)
The highest ALL elongation is observed when no head restraint is used due to excessive neck
extension. The ALL elongation exceeds the sub-failure threshold at the C2-C3 and C4-C5 levels.
The use of the head restraint reduces the ALL elongation at all intervertebral levels below the
injury threshold. However, the head restraint increases the ISL elongation which is subjected to
tensile loading during neck flexion. The ISL is prone to injury when no airbag is used at the mid
and lower cervical spine. The frontal airbag reduces the ISL elongation significantly below the
injury threshold.
Our work shows that the head restraint is not sufficient to provide adequate protection for the
neck in rear-end collisions. Even when using a properly adjusted head restraint, neck ligaments
are vulnerable to injury. Although most of the current research focuses on injury during neck
extension, our work shows the importance of protecting the occupant during neck flexion, too.
The lowest ligament elongation is achieved when the proposed safety strategy is applied by
deploying the frontal airbag, which limits excessive neck flexion.
77
Appendices
Appendix 3.1: Population of Matrices of 1 DOF Model
[𝐴] =
[ 𝑚1−n𝑙1
2 + 𝐼1 𝑚2−𝑛𝑙1𝑙2 cos(𝜃1 − 𝜃2) ⋯ 𝑚𝑛𝑙1𝑙𝑛 cos(𝜃1 − 𝜃𝑛)
𝑚2−𝑛𝑙1𝑙2 cos(𝜃1 − 𝜃2) 𝑚2−𝑛𝑙22 + 𝐼2 ⋯ 𝑚𝑛𝑙2𝑙𝑛 cos(𝜃2 − 𝜃𝑛)
⋮ ⋮ ⋱ ⋮𝑚𝑛𝑙1𝑙𝑛 cos(𝜃1 − 𝜃𝑛) 𝑚𝑛𝑙2𝑙𝑛 cos(𝜃2 − 𝜃𝑛) ⋯ 𝑚𝑛𝑙𝑛
2 + 𝐼𝑛 ]
[𝐵] = [
0 𝑚2−𝑛𝑙1𝑙2 sin(𝜃1 − 𝜃2) ⋯ 𝑚𝑛𝑙1𝑙𝑛 sin(𝜃1 − 𝜃𝑛)
−𝑚2−𝑛𝑙1𝑙2 sin(𝜃1 − 𝜃2) 0 ⋯ 𝑚𝑛𝑙2𝑙𝑛 sin(𝜃2 − 𝜃𝑛)⋮ ⋮ ⋱ ⋮
−𝑚𝑛𝑙1𝑙𝑛 sin(𝜃1 − 𝜃𝑛) −𝑚𝑛𝑙2𝑙𝑛 sin(𝜃2 − 𝜃𝑛) ⋯ 0
]
[𝐶] = [
𝑐1 + 𝑐2 −𝑐2 0 0−𝑐2 ⋱ ⋱ 00 ⋱ 𝑐𝑛−1 + 𝑐𝑛 −𝑐𝑛
0 0 −𝑐𝑛 𝑐𝑛
]
[𝐷] =
[ 𝑘1(𝜙1) + 𝑘2(𝜙2) −𝑘2(𝜙2) 0 0
−𝑘2(𝜙2) ⋱ ⋱ 0
0 ⋱ 𝑘𝑛−1(𝜙𝑛−1) + 𝑘𝑛(𝜙𝑛) −𝑘𝑛(𝜙𝑛)
0 0 −𝑘𝑛(𝜙𝑛) 𝑘𝑛(𝜙𝑛) ]
{𝑄} = {
−𝑚1−𝑛𝑙1(𝑎𝑥 cos 𝜃1 + 𝑎𝑧 sin 𝜃1)
−𝑚2−𝑛𝑙2(𝑎𝑥 cos 𝜃2 + 𝑎𝑧 sin 𝜃2)⋮
−𝑚𝑛𝑙𝑛(𝑎𝑥 cos 𝜃𝑛 + 𝑎𝑧 sin 𝜃𝑛)
}
{��} = {��1
⋮��𝑛
} {��2} = {𝜃1
2
⋮
𝜃��2} {𝑞} = {
𝜃1 − 𝜃01
⋮𝜃𝑛 − 𝜃0𝑛
} {𝜙1
⋮𝜙𝑛
} = {
𝜃1 − 𝜃01
(𝜃2 − 𝜃02) − (𝜃1 − 𝜃01)⋮
(𝜃𝑛 − 𝜃0𝑛) − (𝜃𝑛 − 𝜃0𝑛)
}
𝑚1−𝑛 = ∑ 𝑚𝑖𝑛𝑖=1 𝑙1−𝑛 = ∑ 𝑙𝑖
𝑛𝑖=1 𝑛 = 8 for 1 DOF
78
Appendix 3.2: Matrices population of 2 DOF model
[ 𝐴]=
[ 𝐼 1
+𝑚
1−𝑛𝑙 1
2𝑚
2−𝑛𝑙 1
𝑙 2co
s(𝜃 1
−𝜃2)
⋯𝑚
𝑛𝑙 1
𝑙 𝑛co
s(𝜃 1
−𝜃 𝑛
)
𝑚2−𝑛𝑙 1
𝑙 2co
s(𝜃 1
−𝜃2)
𝐼 2+
𝑚2−𝑛𝑙 2
2⋯
𝑚𝑛𝑙 2
𝑙 𝑛co
s(𝜃2−
𝜃 𝑛)
⋮⋮
⋱⋮
𝑚𝑛𝑙 1
𝑙 𝑛co
s(𝜃 1
−𝜃 𝑛
)𝑚
𝑛𝑙 2
𝑙 𝑛co
s(𝜃2−
𝜃 𝑛)
⋯𝐼 𝑛
+𝑚
𝑛𝑙 𝑛
2
0𝑚
2−𝑛sin( 𝜃
1−
𝜃2)𝑙 2
⋯𝑚
𝑛sin( 𝜃
1−
𝜃 𝑛)𝑙 𝑛
−𝑚
2−𝑛sin( 𝜃
1−
𝜃2)𝑙 1
0⋯
𝑚𝑛sin( 𝜃
2−
𝜃 𝑛)𝑙 𝑛
⋮⋮
⋱⋮
−𝑚
𝑛sin( 𝜃
1−
𝜃 𝑛)𝑙 1
−𝑚
𝑛sin( 𝜃
2−
𝜃 𝑛)𝑙 2
⋯0
0−
𝑚2−𝑛sin( 𝜃
1−
𝜃2)𝑙 1
⋯−
𝑚𝑛sin( 𝜃
1−
𝜃 𝑛)𝑙 1
𝑚2−𝑛sin( 𝜃
1−
𝜃2)𝑙 2
0⋯
−𝑚
𝑛sin( 𝜃
2−
𝜃 𝑛)𝑙 2
⋮⋮
⋱⋮
𝑚𝑛sin( 𝜃
1−
𝜃 𝑛)𝑙 𝑛
𝑚𝑛sin( 𝜃
2−
𝜃 𝑛)𝑙 𝑛
⋯0
𝑚1−𝑛
𝑚2−𝑛co
s(𝜃 1
−𝜃2)
⋯𝑚
𝑛co
s(𝜃 1
−𝜃 𝑛
)
𝑚2−𝑛co
s(𝜃 1
−𝜃2)
𝑚2−𝑛
⋯𝑚
𝑛co
s(𝜃2−
𝜃 𝑛)
⋮⋮
⋱⋮
𝑚𝑛co
s(𝜃 1
−𝜃 𝑛
)𝑚
𝑛co
s(𝜃2−
𝜃 𝑛)
⋯𝑚
𝑛]
79
[ 𝐵]=
[ 0
𝑚2−𝑛sin( 𝜃
1−
𝜃2) 𝑙
1𝑙 2
⋯𝑚
𝑛sin( 𝜃
1−
𝜃 𝑛) 𝑙
1𝑙 𝑛
−𝑚
2−𝑛sin( 𝜃
1−
𝜃2) 𝑙
1𝑙 2
0⋯
𝑚𝑛sin( 𝜃
2−
𝜃 𝑛) 𝑙
2𝑙 𝑛
⋮⋮
⋱⋮
−𝑚
𝑛sin( 𝜃
1−
𝜃 𝑛) 𝑙
1𝑙 𝑛
−𝑚
𝑛sin( 𝜃
2−
𝜃 𝑛) 𝑙
2𝑙 𝑛
⋯0
−𝑚
1−𝑛𝑙 1
−𝑚
2−𝑛𝑙 2
cos(
𝜃 1−
𝜃2)
⋯−
𝑚𝑛𝑙 𝑛
cos(
𝜃 1−
𝜃𝑛)
−𝑚
2−𝑛𝑙 1
cos(
𝜃 1−
𝜃2)
−𝑚
2−𝑛𝑙 2
⋯−
𝑚𝑛𝑙 𝑛
cos(
𝜃2−
𝜃 𝑛)
⋮⋮
⋱⋮
−𝑚
𝑛𝑙 1
cos(
𝜃 1−
𝜃 𝑛)
−𝑚
𝑛𝑙 2
cos(
𝜃2−
𝜃 𝑛)
⋯−
𝑚𝑛𝑙 𝑛
𝑚1−𝑛𝑙 1
𝑚2−𝑛𝑙 1
cos(
𝜃 1−
𝜃2)
⋯𝑚
𝑛𝑙 1
cos(
𝜃 1−
𝜃 𝑛)
𝑚2−𝑛𝑙 2
cos(
𝜃1−
𝜃2)
𝑚2−𝑛𝑙 2
⋯𝑚
𝑛𝑙 2
cos(
𝜃2−
𝜃 𝑛)
⋮⋮
⋱⋮
𝑚𝑛𝑙 𝑛
cos(
𝜃 1−
𝜃 𝑛)
𝑚𝑛𝑙 𝑛
cos(
𝜃2−
𝜃 𝑛)
⋯𝑚
𝑛𝑙 𝑛
0𝑚
2−𝑛sin( 𝜃
1−
𝜃2)
⋯𝑚
𝑛sin( 𝜃
1−
𝜃 𝑛)
−𝑚
2−𝑛sin( 𝜃
1−
𝜃2)
0⋯
𝑚𝑛sin( 𝜃
2−
𝜃 𝑛)
⋮⋮
⋱⋮
−𝑚
𝑛sin( 𝜃
1−
𝜃 𝑛)
−𝑚
𝑛sin( 𝜃
2−
𝜃 𝑛)
−⋯
0]
80
[ 𝐶]=
[ 𝑐 𝑟1+
𝑐 𝑟2
−𝑐 𝑟
20
00
00
0−
𝑐 𝑟2
⋱⋱
00
00
00
⋱𝑐 𝑟
𝑛−1+
𝑐 𝑟𝑛
−𝑐 𝑟
𝑛0
00\
00
0−
𝑐 𝑟𝑛
𝑐 𝑟𝑛
00
00
00
00
𝑐 𝑒1
00
00
00
00
𝑐 𝑒2
00
00
00
00
⋱0
00
00
00
0𝑐 𝑒
𝑛]
[ 𝐷]=
[ 𝑘𝑟1( 𝜙
1)+
𝑘𝑟2( 𝑠
2)
−𝑘
𝑟2( 𝜙
2)
00
00
00
−𝑘
𝑟2( 𝜙
2)
⋱⋱
00
00
0
0⋱
𝑘𝑟𝑛−1( 𝜙
𝑛−1)+
𝑘𝑟𝑛( 𝜙
𝑛)
−𝑘
𝑟𝑛( 𝜙
𝑛)
00
00
00
−𝑘
𝑟𝑛( 𝜙
𝑛)
𝑘𝑟𝑛( 𝜙
𝑛)
00
00
00
00
𝑘𝑒1
00
00
00
00
𝑘𝑒2
00
00
00
00
⋱0
00
00
00
0𝑘
𝑒𝑛]
81
{ 𝑄}=
ۓ۔ە−
𝑚1−𝑛𝑙 1
( 𝑎𝑥co
s𝜃 1
+𝑎
𝑧sin𝜃 1
)
−𝑚
2−𝑛𝑙 1
( 𝑎𝑥co
s𝜃2+
𝑎𝑧sin𝜃2)
⋮−
𝑚𝑛𝑙 1
( 𝑎𝑥co
s𝜃 𝑛
+𝑎
𝑧sin𝜃 𝑛
)
−𝑚
1−𝑛( 𝑎
𝑥sin𝜃 1
−𝑎
𝑧co
s𝜃 1
)
−𝑚
2−𝑛( 𝑎
𝑥sin𝜃2−
𝑎𝑧co
s𝜃2)
⋮−
𝑚𝑛( 𝑎
𝑥sin𝜃 𝑛
−𝑎
𝑧co
s𝜃 𝑛
)
{ 𝑞
}=
ቊ൛𝜃𝑖ൟ
൛𝑙𝑖ൟ
ቋ=
ۓ۔ە𝜃 1 𝜃2 ⋮ 𝜃𝑛 𝑙 1 𝑙 2 ⋮ 𝑙 𝑛
{ 𝑞
2}=
{ቄ𝜃𝑖2
ቅ
൛2𝜃𝑙ൟ
}=
ۓ۔ە𝜃 1
2
𝜃22 ⋮
𝜃 𝑛2
2𝜃 1
𝑙 12𝜃2𝑙 2
⋮2𝜃 𝑛
𝑙 𝑛
{ 𝑞
}=
ቊ൛𝜃ൟ 𝑙ቋ
=
ۓ۔ە𝜃 1 𝜃2 ⋮ 𝜃 𝑛 𝑙 1 𝑙 2 ⋮ 𝑙 𝑛
{𝑞}=
{{ 𝜃}
{ 𝑙}}
=
ۓ۔ە𝜃 1
−𝜃01
𝜃2−
𝜃02
⋮𝜃 𝑛
−𝜃0𝑛
𝑙 1−
𝑙 01
𝑙 2−
𝑙 02
⋮𝑙 𝑛
−𝑙 0
𝑛
{𝜙
1 ⋮ 𝜙𝑛
}=
{
𝜃 1−
𝜃01
( 𝜃2−
𝜃02)−
( 𝜃1−
𝜃01)
⋮( 𝜃
𝑛−
𝜃0𝑛)−
( 𝜃𝑛−
𝜃0𝑛)}
𝑚𝑖−
𝑛=
∑𝑚
𝑗𝑛 𝑖
𝑙 𝑖
−𝑛
=∑
𝑙 𝑗𝑛 𝑖
𝑛=
16
for
2 D
OF
82
Chapter 4.
Paper #2: Nonlinear Multibody Dynamics and Finite Element Modeling of Occupant Response: Part II –
Frontal and Lateral Vehicle Collisions
This chapter has been published in International Journal of Mechanics and Materials in Design,
15, 23-41, 2019. Available at: https://doi.org/10.1007/s10999-019-09450-4
Abstract
Due to the increased number of fatalities and injuries in motor vehicle accidents, it is crucial to
study the kinematic and kinetic occupant response during collisions, specifically the head and
neck response due to their vulnerable nature. In Part I, we have addressed rear end collisions. In
Part II, we examine occupant response in frontal and lateral collisions. Two multibody dynamics
models of the cervical spine of the 50th percentile male were developed and validated. The
cervical spine was modeled as a series of rigid links connected through single and two degrees of
freedom viscoelastic joints. In addition, finite element simulations of two compact sedan vehicle
were conducted to capture realistic crash acceleration of the driver seat in frontal and lateral
collision scenarios. Furthermore, finite element simulations were performed to capture the
kinematic and kinetic response of a seated restrained male occupant subjected to the realistic seat
accelerations in frontal and lateral collisions. Finally, the possibility of injury in frontal, lateral
and rear collisions was evaluated. The evaluation of ligament injury risk shows high risk of
injury at the interspinous ligament in frontal collision, at the anterior longitudinal ligament in
rear collision and at the near-side capsular ligament in lateral collision. The highest vertebral
fracture risks were found at the mid- and lower cervical spine in rear and lateral collisions. The
outcomes of this work provide a better understanding of occupant injury mechanism during
frontal, lateral and rear collisions which is essential to enhancing motor vehicle safety.
Keywords nonlinear; finite element; multibody dynamics; occupant kinematics, vehicular
collision
83
4.1. Introduction
According to the World Health Organization, road traffic crashes is the top cause of death
worldwide for people aged 15-29 [1]. Besides death, motor vehicle crashes may result in
disabilities and/or chronic injuries. Of the injuries suffered in vehicular impacts, head injuries,
mostly commonly due to impacts with the vehicle interior, are some of the most frequently
observed injuries suffered by vehicle occupants. Such injuries are most commonly found in the
frontal and lateral impacts [2–5]. In rear impacts, the neck is most frequent site of injury, with
more than 80% of injuries suffered in rear impacts being cervical whiplash [6]. Although there
have been great efforts in enhancing motor vehicle safety in the previous decades, the high
number of injuries/fatalities indicates that there is still room for further improvements in the field
of vehicle occupant protection. In order to provide better protection for motor vehicle occupants,
it is crucial to study the occupant’s responses injury mechanisms during crashes.
Three main approaches are utilized to evaluate how the occupant responds in various impact
scenarios. The first is the experimental approach. A number of experimental studies have been
conducted on volunteers to evaluate how the occupants respond to different impact accelerations
[7–12]. Although volunteer studies provide the most accurate response, the impact severity must
be limited to avoid injuring the volunteers. In order to overcome the limited impact severity
barrier, full or partial post mortem human surrogates (PMHS) may be used instead [9,13–19].
However, the use of PMHS in testing is also subject to stringent ethical considerations [9],
limiting its practical value. In the past few decades, experimental studies have been conducted
primarily through the use of anthropomorphic test dummies (ATDs) to evaluate the safety of
motor vehicles crashes such as Hybrid III [20], Test device for Human Occupant Restraint
(THOR) [21] and BioRid II [22].
The second approach to study the human response is the use of multibody dynamics (MBD). In
these models, the bones were modeled as rigid bodies connected through different types of joints
and the soft tissues were modeled as viscoelastic elements [23–30]. These multibody dynamics
model studied the occupant response to different types of loading. Many models focused on the
head/neck region while some other efforts modeled the entire human body such as MADYMO
(MAthematical DYnamic Model) [31].
84
Thanks to an increase in availability of computational power over the past few years, the finite
element (FE) method has been extensively utilized to provide biofidelic models of the human
body to be used under different types of loading. To analyze and understand the human response
in motor vehicle crashes, a number of FE models were developed, whether focusing on the
head/neck region [32,33] or full human body models for both male and female occupants such as
the Global Human Body Model Consortium (GHBMC) [34], Total Human Model for Safety
(THUMS) [35] HUMOS (HUman MOdel for Safety) [36] and ViVA (Virtual Vehicle Safety
Assessment) [37]. These models can be used for in depth studies of injury mechanisms during
crashes. However, they come at a high computational cost, both resources and time wise.
Many injury criteria were developed to assess the possibility of injury of occupants in different
impact scenarios. Many of these criteria use the occupant’s kinematics to determine the
possibility and the severity of injury such as the Neck Injury Criterion (NIC) [38] and the
Intervertebral Neck Injury Criterion (IV-NIC) [39].
In the current study, we are concerned with developing and validating an analytical multibody
dynamics model that is capable of capturing the head and neck kinematics during vehicular
collisions. In Part I of our work we examined occupant’s kinematics during rear collisions. Part
II is concerned with frontal and lateral collisions. Furthermore, FE simulations of impacting
vehicles were carried out in frontal and lateral collision scenarios in order to obtain realistic crash
accelerations and velocities to which the occupant is subjected during impact. A FE model of the
male occupant is subjected to the impact velocities resulting from vehicle-to-vehicle impact
simulations to determine the kinematic and kinetic response of the occupant. Finally, we
determine the possibility of occupant injury in frontal, lateral and rear impact scenarios.
4.2. Multibody Dynamics Modeling
The MBD models developed in this study represent the cervical spine of a seated 50th percentile
male vehicle occupant. The models assumed that in frontal collision the motion occurs in the
sagittal plane, while in lateral collision the motion occurs in the frontal plane. Since the
head/neck response during frontal collision occurs in the sagittal plane, the same model used for
rear collision in Part I was used for frontal collision. The parameters for the frontal and lateral
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models will be summarized here while the detailed derivation of the model can be found in Part
I.
The cervical spine and the head were modeled as a series of rigid links connected by viscoelastic
joints. For two adjacent vertebrae, the upper vertebra was assumed to rotate about an
instantaneous axis of rotation (IAR) located in the lower vertebra [40,41]. Two models were
developed in this work. In the single DOF model, the viscoelastic joints allowed only rotation
while in the two DOF model, axial extension of the links was also added to represent the axial
flexibility of the intervertebral joint. Figure 4.1 demonstrates a generalized model with 2 links
showing the concept of the single DOF and two DOF MBD models. The grey arrows indicate the
direction of motion. The IAR locations of the MBD models were determined from the geometry
of the 50th percentile male cervical spine [42]. The length and the initial angle of each of the 8
links of the MBD model in the sagittal and frontal planes are shown in Table 4.1. The mass and
moment of inertia of each vertebra are shown in
Table 4.2
Figure 4.1 Schematic of the MBD model showing the applied DOF (indicated by the grey
arrows) for the (a) single DOF and (b) two DOF models
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Table 4.1 Geometry of the MBD model [42]
Sagittal Plane Frontal Plane
Link Lower
IAR
Upper
IAR
Lower joint
level
Length
(mm)
Initial
angle (°)
Length
(mm)
Initial
angle (°)
1 C7 C6 C7-T1 21.5 -20 16.8 0
2 C6 C5 C6-C7 16.9 -5 16.5 0
3 C5 C4 C5-C6 16.5 0 18.8 0
4 C4 C3 C4-C5 18.9 -5.5 16.8 0
5 C3 C2 C3-C4 16.9 5 46.4 0
6 C2 C1 C2-C3 46.9 8 7.71 0
7 C1 C0 C1-C2 7.8 8.5 54.4 0
8 C0 Head
center C0-C1 57.9 -20
16.8 0
Table 4.2 Masses and moment of inertia of cervical vertebrae [43–46]
Vertebra m (kg) Sagittal plane
I×10-3 (kg·m2)
Frontal plane
I×10-3 (kg·m2)
C0 (head) 4.7 22.2 14.5
C1 0.12 0.22 0.22
C2 0.14 0.25 0.25
C3 0.25 0.24 0.24
C4 0.32 0.23 0.23
C5 0.37 0.23 0.23
C6 0.3 0.24 0.24
C7 0.29 0.22 0.22
The following expression was developed for the variable rotational stiffness of each
intervertebral joint based on the moment-angle relationships of cervical intervertebral joint:
𝑘𝑟𝑖(𝜙𝑖) = 𝐴𝑖𝐵𝑖𝑒𝐵𝑖𝜙𝑖 (4.1)
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where, and Ai and Bi are experimentally obtained coefficients for each intervertebral joint and ϕi
is the relative angles of rotation for the ith joint. Based on [47–49], the values for Ai and Bi values
were obtained in the sagittal plane. The same mathematical expression was used to describe the
moment-angle relationships of intervertebral joints in lateral bending [50,51]. The values for Ai
and Bi in the sagittal and frontal planes are summarized in Table 4.3.
The stiffness relationships described by equation (4.1) were only calculated for intervertebral
joints within the normal range of motion [47], limited by ligaments and bone-to-bone contact.
Through the use of isolated ligamentous spine models, Ivancic, et al. [52] and Panjabi, et al. [39]
recorded the maximum flexion and extension angles reached by each intervertebral level in rear
and frontal collisions of varying severities. These angles were greater than the angles of rotation
observed in voluntary motion [53,54]. Due to a lack of lateral bending data, physiological limits
found by [50] and [55] were used in the frontal plane. The rotational limits applied to the MBD
models at each intervertebral level are summarized in Table 4.4.
Table 4.3 Intervertebral rotational stiffness curve coefficients in sagittal and frontal planes
[47–51]
Intervertebral
level
Flexion Extension Lateral
Ai (N·m/°) Bi Ai (N·m/°) Bi Ai (×103
N·m/°) Bi
C0-C1 0.0193 0.3052 -0.0136 -0.3937 0.6152 1.655
C1-C2 0.045 0.3052 -0.0317 -0.3937 0.5652 2.323
C2-C3 0.1029 0.4714 -0.0037 -1.0137 0.2928 1.717
C3-C4 0.0218 0.7503 -0.0068 -1.1416 0.1089 2.062
C4-C5 0.113 0.3929 -0.0027 -1.641 0.1310 1.958
C5-C6 0.0618 0.5587 -0.0126 -0.9581 0.1754 2.725
C6-C7 0.1406 0.5607 -0.0125 -1.2366 0.4383 2.935
C7-T1 0.6084 0.3949 -0.3105 -0.6489 9.676 1.193
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Table 4.4 Maximum angles of rotation at each intervertebral joint [39,50,52,55]
Intervertebral
level Flexion (°) Extension (°)
Lateral
bending (°)
C0-C1 14.6 27.7 8
C1-C2 9.4 6.4 6.5
C2-C3 11.4 6.6 10
C3-C4 16.4 9.6 11
C4-C5 9.9 11.9 11
C5-C6 11.9 10.9 8
C6-C7 11.4 12.9 7
C7-T1 14.5 10.6 4
4.3. Finite Element Modeling
In the first section of the FE analysis, the frontal and lateral collisions of two passenger vehicles
were simulated using dynamic FE method and the passenger seat velocity profiles in the
impacted vehicle were recorded. In the second section of the analysis, the passenger seat velocity
profiles recorded in the first section were applied to the FE model of a restrained seated
occupant. Using the outputs of the second simulation, the kinematics as well as the injury risk of
the occupant were determined. Using the T1 vertebra accelerations, resulting from the occupant
FE simulation, as an input to the MBD models, the kinematics of the FE model will be compared
to the MBD models’ response. The nonlinear dynamic FE analysis was conducted using the
explicit solver of LS-DYNA.
4.3.1. Vehicle Crash Simulation
The same FE model of a generic compact vehicle used in Part I (rear collision) was used to
simulate frontal and lateral impacts. The model used is one of the Crash Simulation Vehicle
Models made available by NHTSA and it was validated against experimental data for various
crash scenarios [56]. The vehicle has a mass of 1100 kg and the model consists of ~1.5 million
elements including, beam, shell and solid elements.
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The collision simulations consisted of two identical vehicles colliding against one another in the
frontal and lateral directions, as shown in Figure 4.2. In each impact scenario, the vehicles were
positioned such that their centers of mass were aligned in the direction of impact. Both vehicles
rested on a horizontal rigid plane. An initial velocity was assigned to the colliding (bullet)
vehicle, while the other (target) vehicle was initially at rest. An initial rotational velocity was
applied to the wheels of the bullet vehicle to simulate wheel rotation. In FMVSS 208, the
NHTSA requires new vehicles to demonstrate sufficient occupant protection capability at 32 and
48 km/h in lateral and rear collisions, respectively. Due to the high probability of cervical injury
in even low-velocity collisions [57–59], the 32 km/h velocity was selected for the frontal and
lateral collisions, similar to the 32 km/h impact velocity used for rear collision in Part I. Gravity
was applied to both vehicles as 9.8 m/s2 downward acceleration.
Due to the complexity of determining the contacting parts and regions in vehicle collisions, all
parts of the vehicle were considered for contact against all other parts including self contact
using the AUTOMATIC_SINGLE_SURFACE contact definition in LS-DYNA. The model used
segment-based two-way contact formulation which checks for contact between segments rather
than nodes against surfaces reducing the risk of nodes penetrating surfaces.
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Figure 4.2 Setup of vehicle collision FE simulation: (a) frontal and (b) lateral
4.3.2. Occupant Response
In this section of the study, the kinematic and kinetic response of the occupant is determined
using the GHBMC FE model of the 50th percentile male occupant. The current study used the
GHBMC 50th percentile seated male model. The model weighs 78 kg, is 174.9 cm tall and
contains ~2.2 million elements. The GHBMC was extensively validated in frontal [60,61], rear
[62], and lateral [63,64] collisions using data obtained through PMHS tests.
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The male occupant model was placed on a FE model of a vehicle seat as shown in Figure 4.3.
The seat model contained the driver’s seat and the vehicle floor. In the case of lateral collisions,
the vehicle door was also included in the model to study the interactions between the human
body and the vehicle interior. A 3-point seatbelt was used to restrain the occupant model. To
ensure no gap existed between the occupant model and the seatbelt, the seatbelt was fitted around
the human body model.
Only gravity at 9.8 m/s2 was initially applied to the seated occupant model. Collision velocity
profiles obtained from vehicle-to-vehicle impact simulations were applied to the seated occupant
model. The profiles were applied to the vehicle floor, as well as to the seatbelt endpoints. In the
near-side lateral collision scenario, the near-side door was assigned the velocity measured at the
protruding armrest during the two-vehicle lateral collision simulation.
Contact was defined between the body of the GHBMC occupant model, the vehicle seat, and the
side door. Because the current study investigated the cervical response of the vehicle occupant
due to movements of the T1 vertebra, no contact was modeled between the head and the seat, or
between the head and the side door. The kinematics of the head and cervical spine entirely
depended on the movement of the torso. Segment-based contact was used between the human
body, and both the vehicle seat and the seatbelt because it is the recommended contact treatment
method for soft materials that may produce large deformations, such as human soft tissue and
polyurethane foam. The static and dynamic coefficients of friction were defined as 0.577 and
0.360, respectively to represent a low-friction vehicle seat [65].
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Figure 4.3 FE model of a seated 50th percentile male occupant restrained using a 3-point
seatbelt, collision velocity profiles applied to the floor and the door
4.4. Results and Discussion
4.4.1. Validation of MBD model
The MBD models were validated by applying T1-accelerations measured in sled tests conducted
with volunteers and PMHS specimens to the single and two DOF MBD models. The predicted
head kinematic response was compared with the experimental results to determine the accuracy
of predicted results. The frontal acceleration profile was obtained from volunteer sled tests
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representing 17 m/s collisions [66]. The lateral acceleration profile was obtained from volunteer
shoulder impact tests at 1.5 m/s [67].
Figure 4.4 shows the head horizontal and vertical displacements of the single and two DOF
MBD models obtained using the T1-acceleration profiles from the experimental studies. Figure
4.4(a) shows the frontal collision horizontal displacement results of the single and two DOF
MBD models in comparison with the experimental corridor. The peak mean experimental
horizontal displacement was 142.4 mm. The respective single and two DOF models produced
peak displacements of 138.7 mm and 155.6 mm, corresponding to 97.4% and 109.3% of the peak
mean experimental results, respectively. However, the single and two DOF models reached peak
displacement at 90 ms and 80 ms, respectively, while the experimental results reached peak
displacement at 130 ms. Figure 4.4(b) shows the vertical displacement response of the single and
two DOF models in frontal collision. The peak mean experimental vertical displacement was
147.0 mm. In comparison, the single and two DOF models produced peak vertical displacements
of 120.5 mm and 121.5 mm, respectively, corresponding to 82.0% and 82.7% of the peak mean
experimental results, respectively.
Figure 4.4(c) shows the lateral collision horizontal head displacement response of the single and
two DOF models in comparison with the experimental corridor. The peak mean experimental
horizontal displacement was 58.4 mm. The single and two DOF models produced peak
displacements of 66.2 mm and 65.2 mm, respectively, corresponding to 113.4% and 111.6% of
the peak mean experimental result. The respective peak displacement times of the single and two
DOF models were found at 132.8 ms and 133.4 ms, while the experimental peak displacement
was observed at 134.3 ms. Similarly, Figure 4.4(d) shows that in lateral collision, the single and
two DOF peak vertical head displacements were 102.8% and 118.3% of the peak mean
experimental displacement, respectively.
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Figure 4.4 Frontal collision head center of mass (a) horizontal and (b) vertical
displacements, and lateral collision head center of mass (c) horizontal and (d) vertical
displacements
The single and two DOF models presented nearly identical response in lateral collisions. This
was attributed to the relatively low collision velocities simulated by the acceleration profiles,
which were not sufficient to distinguish between the models. In frontal collision, more significant
differences between the two models could be found due to the higher collision velocity used. In
general, the MBD models show good agreement with the experimental data. The models were
able to capture the peak horizontal head displacement in the sagittal and frontal planes. The
discrepancy between the MBD model results and the experimental data may be attributed to: (i)
intervertebral rotations of the MBD models exceeding the ranges of the nonlinear rotational
stiffness curves leading to reduced model displacements and (ii) the MBD models not accounting
for cervical musculature.
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4.4.2. Occupant Response
The developed MBD and FE models were used to determine the cervical response of the
occupant in vehicular collisions. First, frontal and lateral collisions were simulated between two
compact sedan vehicles. The velocity of driver’s seat in the target vehicle in the horizontal,
vertical and lateral directions resulting from the two collision scenarios was recorded. The
velocity profiles were then applied to the FE model of the seated occupant. Figure 4.5 shows the
velocity profiles of the driver’s seat in the target vehicle for frontal and lateral collisions. The
sign convention and coordinate system used is similar to the one used in the MBD analysis: the
positive x direction is forward, positive y is in the lateral direction and the positive z direction is
in the vertical upward direction. As shown in Figure 4.5, the seat deceleration was higher for the
case of lateral collision due to the orientation of the wheels in this impact scenario. In this case,
the target vehicle wheels slide instead of rolling leading to higher deceleration.
In a frontal collision, the vehicle seat experiences a backward acceleration and the occupant
experiences a compressive load on the chest and abdomen due to the seatbelt. The head
continues to travel forward due to inertia, inducing flexion in the cervical spine. The occupant
response during maximum neck flexion is shown in Figure 4.6.
Figure 4.5 Velocity profiles recorded for the driver seat in the target vehicle for (a) frontal
and (b) lateral collisions
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Figure 4.6 Response of a 50th percentile male occupant restrained using 3-point seatbelt
during 32 km/h frontal collision showing maximum neck flexion
The T1-accelerations of the occupant FE model were applied to both the single and two DOF
MBD models to determine the kinematics of the occupant cervical spine. Figure 4.7(a) and (b)
shows the head center of mass horizontal and vertical displacements from the MBD and FE
models in frontal collision, respectively, calculated with respect to the T1 vertebra. The results
show that the FE and MBD model response followed the same general trends in frontal collision.
The cervical spine went into flexion until reaching peak displacement and rotation after which
the head rebounded backwards. However, the FE and MBD models presented different
magnitudes and times of peak displacements, as shown in Table 4.5.
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Figure 4.7 MBD and FE models head center of mass response: frontal collision (a)
horizontal and (b) vertical displacements, and lateral collision (c) horizontal and (d)
vertical displacements
Table 4.5 Peak displacements and rotational response of the head center of mass and their
time of occurrence in frontal and lateral collisions
Frontal Lateral
FE
1 DOF
MBD
2 DOF
MBD
FE
1 DOF
MBD
2 DOF
MBD
Horizontal (mm) 126.2 140.3 155.9 219.5 86.7 91.7
Vertical (mm) 82.0 123.6 128.9 55.1 26.4 23.2
Rotation (°) 54.0 90.2 90.9 67.9 43.2 42.4
Time (ms) 149 120 127 154 87 87
Figure 4.7(a) and Table 4.5 show that the single and two DOF models present comparable peak
horizontal displacements in frontal collision, with the two DOF model peak displacement
approximately 10% higher than the single DOF model. In comparison, the FE model produced a
peak displacement of 126.2 mm, 10% and 20% lower than the single and two DOF model
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response, respectively. Figure 4.7(b) shows the vertical displacement response of the MBD and
FE models in frontal collision. Similar to the horizontal displacement response, the two DOF
model peak displacement was 6% higher than that of the single DOF model due to the added
axial flexibility. The FE peak displacement was 34% and 36% lower than the single and two
DOF models, respectively.
The MBD models produced higher displacements than the FE model due to the simplicity and
the lack of cervical musculature in the MBD models. During cervical flexion movement, the
posterior cervical muscles, including the cervical trapezius, splenius capitis, and splenius
cervicis, limit the cervical range of motion [68]. Furthermore, the flexion range of motion of the
FE model was also limited by chin-chest contact as shown in Figure 4.6. Figure 4.8 shows good
agreement between the MBD and FE results. The two DOF MBD model produced greater
displacements than the single DOF model, as well as a delayed rebound.
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Figure 4.8 Frontal and lateral collisions response of MBD and FE models resulting from a
32 km/h collision
In near-side lateral collision, the seat is accelerated in the lateral direction and the seatbelt applies
a lateral load to the occupant neck and hips. Furthermore, the intruding vehicle door may impact
the shoulders and hips of the occupant, as shown in Figure 4.9. Figure 4.7(c) and (d) present the
horizontal and vertical displacements of the occupant head center of mass with respect to the T1
vertebra through the course of the lateral collision, respectively. The results show that while the
models initially followed the same trends in lateral bending, the MBD models produced
significantly reduced displacements compared with the FE model. The MBD models reached
maximum displacement 67 ms earlier than FE model. The peak head displacement and rotation
magnitudes, and times of occurrence are summarized in Table 4.5. The peak horizontal
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displacement of the two DOF model was approximately 6% higher than the single DOF model.
The significant discrepancy between the MBD and FE models in lateral collision is attributed to
the lateral rotational limits of the MBD models, which constrained the intervertebral rotations of
the models to the physiological range of motion in the frontal plane. Therefore, the MBD models
were unable to produce the involuntary range of motion under non-physiologic collision loading
[50,55].
Figure 4.9 Response of a 50th percentile male occupant restrained using 3-point seatbelt
during 32 km/h lateral collision velocity profile applied to the floor and door
Figure 4.8 shows that the MBD models demonstrated good agreement with the FE model during
the initial phase of the response. At 60 ms, all 3 models predicted that the head would translate
horizontally without rotation, while the cervical spine bends laterally. However, after 80 ms, the
FE model presented a significantly larger peak displacement magnitude than the MBD models.
4.4.3. Risks of Injury
The risks of injury at each cervical intervertebral level may be evaluated using the kinematic and
kinetic response of the occupant cervical spine. Here, a preliminary estimate of the injury risks
was obtained using the IV-NIC, the ligament elongations and the vertebral stresses resulting
from 32 km/h frontal, rear and lateral collisions.
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IV-NIC
The IV-NIC is based on the assumption that cervical injury occurs when intervertebral rotations
exceed physiological limits, with the risk of injury increasing with the increased rotation angle
[69]. The IV-NIC value at any intervertebral level is calculated as the ratio between the peak
intervertebral rotation measured at that level to the physiological level of that level. An IV-NIC
value greater than 1 indicates that this intervertebral joint exceeded the physiological limit, hence
there is a risk of injury at that level. Using the physiological angles of rotation found by [69], the
IV-NIC values in frontal, rear, and lateral collisions were calculated for each intervertebral level.
The results are shown in Figure 4.10.
Figure 4.10 IV-NIC values calculated in (a) frontal, (b) rear, and (c) lateral collisions
Figure 4.10(a) shows that in frontal collision, the highest IV-NIC value was found at the C4-C5
intervertebral level with a value of 5.55, calculated using the FE model. The second highest IV-
NIC value was at the C7-T1 intervertebral level, with a value of 4.59. Figure 4.10(b) shows that
in rear collision, the two highest IV-NIC values were found at the C3-C4 and C7-T1 joints, with
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values of 3.31 and 3.03, respectively. In both frontal and rear collisions, the probability of injury
was highest at the mid- and lower cervical spines.
Figure 4.10(a) and (b) show that in the frontal and rear collisions, the MBD and FE models
demonstrate a similarly increasing trend of injury risk toward the inferior end of the cervical
spine. However, the risk of injury at C4-C5 and C3-C4 in frontal and rear collisions,
respectively, were underestimated by the MBD models. In both frontal and rear collisions, the
discrepancies were caused by the intervertebral rotational limits of the MBD models. The C4-C5
joint flexion was limited to 9.9° and the C3-C4 extension was limited to 9.6°, both of which were
reached during the course of the collisions. As mentioned earlier in Part I of this work, the
Anterior Longitudinal Ligament (ALL) did undergo near-complete failure at the C3-C4 level
during the 32 km/h rear collision, which further contributed to the increased rotation in the rear
collision FE simulation. Furthermore, Figure 4.10(a) and (b) show that the MBD models
produced significantly greater rotations and injury risks than the FE model at the C0-C1 and C1-
C2 joints. The IV-NIC values predicted by the FE models were between 28% and 72% of those
calculated using the MBD models. The MBD models assumed that the C0-C1 and C1-C2 joints
are capable of significant rotation in sagittal plane motion, based on the experimental results
obtained by Grauer, et al. [70]. This assumption may require additional experimental
investigations in light of the current FE results.
Figure 4.10(c) shows that in lateral collision, the highest IV-NIC value was found at the C7-T1
joint, where the FE model produced a value of 2.60. The MBD models produced a peak IV-NIC
value of 1 at the C7-T1 joint. Because the intervertebral joint rotations of the MBD models were
constrained to the physiological ranges of motion, an IV-NIC of 1 indicates that the
intervertebral joint is likely to experience injury, without providing an estimate of the severity of
injury.
The IV-NIC values calculated using the FE and MBD models followed the same general trends,
with a few exceptions. In the sagittal plane, both FE and MBD models predicted high injury risks
at the lower and mid-cervical spine. In lateral collision, both the FE and MBD models predicted
injury at the C7-T1 joint, but only the FE model predicted high rotations at the C2-C3 joint.
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Ligaments
The primary site of osteoligamentous spinal injury in the frontal and rear collisions is the cervical
ligaments, which are composed of collagen fibers oriented along the longitudinal direction of
each ligament. The primary mode of loading of ligaments in cervical kinematics is tensile
displacement, and ligament failure was found to directly relate to the percent elongation [71].
Moreover, ligaments sustain sub-failure injuries at 62% of the failure elongation [72,73]. In the
current study, the percentage elongations of the ALL, Posterior Longitudinal Ligament (PLL),
Ligamentum Flavum (LF), Interspinous Ligament (ISL), and Capsular Ligament (CL) were
examined through the course of a collision. Sub-failure injurious ligament percent elongations,
shown in Table 4.6, were found by multiplying the ligament failure percent elongations by the
injury-to-failure elongation ratio of 62%.
Table 4.6 Sub-failure injury percent elongations of cervical ligaments [71–73]
Tissue C1-C5 C5-T1
ALL 18.4% 32.3%
PLL 22.3% 21.3%
CL 64.6% 64.6%
LF 62.6% 54.9%
ISL 40.3% 42.3%
The ligament elongations produced by the FE model were normalized against the sub-failure
injurious elongations in Table 4.6, in order to evaluate the risks of ligament injury. The
normalized percent elongations of the cervical ligaments are shown in Figure 4.11. In the
cervical spine FE model, the ALL, PLL, and LF extend from T1 to C2 and the ISL extends from
T1 to C1. The CL is found at all intervertebral levels between T1 and C0. The horizontal line in
each plot indicates the normalized injurious elongation threshold.
Figure 4.11 (a) shows that the lower cervical spine was most at risk of injury in frontal collisions.
The highest risks of injury were found at the C6-C7 and C7-T1 vertebral joints, where the ISL
experienced 2.4 and 3.2 times the injurious percent elongation, respectively. The CL at the C7-
T1 joint was also at a high risk of injury, where it was subjected to 1.8 times the injurious percent
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elongation. The C4-C5 joint was also at risk of ISL and CL tears, which experienced 2.14 and
1.26 times the injurious percent elongation, respectively.
Figure 4.11 ALL, PLL, LF, ISL, and CL maximum elongations normalized against
injurious elongation thresholds in (a) frontal, (b) rear, and (c) lateral collisions
Comparison between Figure 4.10(a) and Figure 4.11(a) shows that the risk of ligament injury in
frontal collision is related to the IV-NIC value of each intervertebral level. Both Figure 4.10(a)
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and Figure 4.11(a) predict the highest risks of injury at the lower cervical spine, and both predict
elevated injury risks at the C4-C5 joint.
Figure 4.11(b) shows that in rear collision, the mid-cervical spine is most at risk of ligament
injuries. The maximum risk of injury was found at the C3-C4 joint, where the ALL experienced
1.8 times the injurious percent elongation. At the C2-C3 and C4-C5 joints, the ALL was
subjected to 1.29 and 1.31 times the injurious percent elongation, respectively. No other
ligaments were at risk of injury. Figure 4.10(b) and Figure 4.11(b) show that correlations can
also be found between the IV-NIC and the relative risks of ligament injury in rear collision. Both
figures predicted the highest injury probability at the C3-C4 joint, as well as the high injury
probabilities at the C2-C3 and C4-C5 joints. However, the IV-NIC predicted high risks of injury
at the C7-T1 joint, which was not predicted to suffer from ligament injuries in Figure 4.11(b).
Figure 4.11(c) shows that in lateral collision, the injury risk is evenly distributed among the mid-
and lower cervical spine. Between the C3-C4 and the C7-T1 joints, the far-side CL was subjected
to between 1.8 (C3-C4) and 2 (C7-T1) times the injurious ligament percent elongations. At the
C4-C5 level, the ISL experienced 1.07 times the injurious percent elongation. No other ligaments
were at risk of injury. Comparison between Figure 4.10(c) and Figure 4.11(c) shows that the
correlation between IV-NIC and the risk of ligament injury also exists in lateral collision. Both
figures predicted the highest risk of injury at the C7-T1 joint, as well as a small rise in injury risk
at the C4-C5 joint compared with adjacent joints. However, the high ligament injury risks at C3-
C4 and C5-C6 were not reflected in the IV-NIC.
Bones
Cortical bone is the primary load-bearing component of bone [74] and the von Mises failure
criterion is commonly used to evaluate the risk of bone fracture in cortical bone [75–77]. In the
current work, the von Mises stresses of the vertebral cortical bone were calculated through FE
modeling to evaluate the risks of bone fractures in the vertebrae. The yield strength of cortical
bone lies between 105 and 129 MPa, with an average experimental value of 116 MPa [78–80].
Figure 4.12 shows the maximum von Mises stresses in vertebral cortical bone during frontal,
rear, and lateral collisions, evaluated through FE simulations. Results indicate that in frontal
collision, only the C3 vertebra was at risk of bone fracture, reaching a maximum von Mises
106
stress of 208.6 MPa. The highest von Mises stress was found at the articular process and was
caused by the tensile loading of the CL during flexion of the upper cervical spine. Figure 4.13(a)
shows the stress distribution of the C3 vertebra when the peak von Mises stress was reached. In
vertebrae of the mid- and lower cervical spine, maximum cortical bone stress was primarily
found at the anterior surfaces of the vertebral bodies, caused by compressive loading between
adjacent vertebrae. In more severe collisions, this compressive loading was found to cause
flexion teardrop fractures in the vertebral bodies [81].
Figure 4.12 Cortical bone peak von Mises stress of cervical vertebrae in frontal, rear, and
lateral 32 km/h collisions
107
Figure 4.13 Peak von Mises stress in the cortical bone of (a) C3 vertebra in frontal collision,
(b) C7 vertebra in rear collision and (c) C6 vertebra in lateral collision
Figure 4.12 shows that rear collision had the highest risk of bone fracture found throughout the
cervical spine. At the lower cervical spine, the primary cause of cortical bone stress was the
bone-to-bone contact observed at the C5-C6 and C6-C7 joints. Contact occurred between
adjacent spinous processes at peak extension and induced local stresses in the vertebrae, leading
to high risks of clay-shoveler fractures [82]. In addition, the ALL experienced high tensile
loading during neck extension. Because the ALL is attached to the anterior surfaces of the
vertebral bodies, additional tensile stresses were introduced to the cortical bone, leading to high
risks of extension teardrop fractures [83]. The highest von Mises stress in rear collision was
found at the C7 vertebra, where stresses were caused by both ALL tension and bone-to-bone
contact. The von Mises stress distribution of the C7 vertebra is shown in Figure 4.13(b).
108
Figure 4.12 shows that the risk of bone fracture in lateral collision was similar to those of rear
collision, with the highest fracture risks found in the mid- to lower cervical spine. The primary
cause of stress was bone-to-bone contact between adjacent near-side articular processes, leading
to high risks of lateral mass fractures [84]. Bone-to-bone contact was found at the mid- to lower
cervical spine, at the C4-C5, C6-C7, and C7-T1 joints, corresponding with the locations of
highest von Mises stresses in lateral collision shown in Figure 4.12. The highest von Mises stress
was found at the C6 vertebra, with a peak value of 208.9 MPa which is shown in Figure 4.13(c).
In the rear and lateral collision scenarios, Figure 4.12 shows some correlation with the IV-NIC
trends shown in Figure 4.10. Both IV-NIC and cortical bone stress increased toward the lower
cervical spine. Furthermore, in frontal collision, the increased rotation at the C2-C3 intervertebral
joint corresponds with the high risk of C3 fracture. However, because the risk of bone fracture is
primarily dependent on bone-to-bone contact, the IV-NIC, designed to predict soft tissue injuries,
is not a reliable predictor of cervical bone fracture injuries.
4.5. Conclusions
In the current study, two MBD models of the cervical spine were developed and validated. The
single DOF model contained only rotational viscoelastic joints, while the two DOF model
allowed axial extension. In Part I, the occupant response in rear collision was analyzed. Here in
Part II, the kinematic response of the occupant head and cervical spine was determined in frontal
and lateral collisions. In addition, a FE model of a seated and restrained 50th percentile simulated
male occupant was developed to determine the kinematic and kinetic response of the occupant
head and cervical spine for the aforementioned collisions.
Finite element simulations from Part I and Part II reveal that in frontal collision, the highest risk
of ligament injury was found at the lower cervical spine, where the ISL was predicted to
experience 3.2 times the injurious elongation, and the CL was predicted to experience 1.75 times
the injurious elongation. The current work also revealed that frontal collision presented a risk of
fracture at the C3 vertebra, due to tensile loading in the CL. In rear collision, the highest risks of
ligament injury were found at the upper to mid cervical spine, where the ALL was predicted to
experience 1.8 times the injurious elongation. No other ligaments were at risk of injury. Rear
collision presented high risks of vertebral fracture, both due to bone-to-bone contact at the mid-
109
to lower cervical spine and ALL tension at the upper to mid cervical spine. In lateral collision,
the highest risks of ligament injury were found throughout the mid- to lower cervical spine
between C3 and T1, where the near-side CL was predicted to experience between 1.8 and 2 times
the injurious elongation. The ISL at C4-C5 was predicted to experience 1.1 times the injurious
elongation, but no other ligaments were at risk of injury. Lateral collision presented high risks of
vertebral fracture, primarily at the lower cervical spine, due to bone-to-bone contact between
adjacent articular processes.
If we were to consider collisions of the same velocity, frontal collision posed the highest
ligament injury risk while rear collision posed the highest bone fracture risk. Correlation was
found between the IV-NIC value of each intervertebral level and the risks of ligament injuries
and bone fractures. The outcomes of this work can be very beneficial in enhancing motor vehicle
safety to prevent and lessen occupant injury during collisions. The MBD model is a useful tool to
provide a quick estimation of the head/neck kinematics during various collision scenarios.
Acknowledgment
This publication was made possible by NPRP grant# (7-236-3-053) from the Qatar National
Research Fund (a member of Qatar Foundation). The statements made herein are solely the
responsibility of the author(s).
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Chapter 5.
Experimental Characterization of Cervical Spine Kinematics in Whiplash Trauma using a Sled System
Abstract
A prototype of the head, the cervical spine and T1 vertebra of the 50th percentile male was
developed to experimentally characterize occupant’s kinematics in whiplash trauma. The head
and the vertebrae were 3D printed of materials of comparable properties to those of humans,
while the anterior and posterior ligaments were modeled using viscoelastic rubber sheets. The
intervertebral discs were developed from urethane rubber. Furthermore, a Neck Stabilization
System was introduced to hold the head in the driving posture prior to collision. The head-neck
model was mounted on a testing sled to simulate a rear impact case. The response of the model
was captured using high-speed imagery and compared to a previously developed multibody
dynamics (MBD) and finite element (FE) models. The results reveal that the head-neck
displacements and rotations as well as their time of occurrence were in good agreement with the
MBD and FE predictions.
5.1. Introduction
Despite the enormous amount of research conducted to understand whiplash injury mechanisms
and enhancing motor vehicle and occupants’ safety, occupants still suffer from whiplash injury.
According to the National Highway Traffic Safety Administration (NHTSA) [1], injuries
resulting from rear-end collisions in 2016 was the highest among all impact types with more than
690,000 injuries. This indicates the importance of understanding the mechanisms associated with
whiplash injury and the development of new techniques to characterize occupant’s kinematics in
rear-end collisions.
Numerous techniques have been used to model the occupant response in motor vehicle
collisions. The first being multibody dynamics (MBD) of the occupant head and neck [2–4] or
115
the entire occupant body [5,6]. The second approach being the finite element (FE) method.
Multiple models have been developed of either the head-neck [7–10] or of the entire occupant’s
body such as the Total Human Model for Safety (THUMS) [11], the Global Human Body Model
Consortium (GHBMC) [12], the HUman MOdel for Safety (HUMOS) [13], and the Virtual
Vehicle-safety Assessment (ViVA) project for Open-source Human Body Models (OpenHBM)
[14].
The third approach is to simulate the occupant response experimentally. Experimental
investigations can be in-vivo, where volunteers are seated on impact sleds and subjected to a
simulated rear impact. A number of studies were carried to capture (i) occupant’s kinematics in
rear collisions [15,16], (ii) the response of the head, neck and muscles in rear collisions [17–19]
and (iii) the effect of occupant awareness on the kinematic response [20–22]. However, the
impact pulse to which the volunteers are subjected has to be limited to prevent their injury, which
is only helpful in understanding low impact injury mechanisms.
Human cadavers, whether in full or partial, were used to capture the occupant’s dynamic
response in whiplash [23–27]. Ivancic et al. [28] developed a model of the cervical spine by
attaching a muscle force replication system to an entire cervical spine specimen in order to
enhance the accuracy of the model. This model was used later to analyze facet joints kinematics
[29], injury of the anterior longitudinal ligament [30], injury mechanisms of the intervertebral
disc [31], the dynamic sagittal flexibility of the neck [32] and to study the possibility of spinal
canal narrowing during whiplash [33]. Although this approach can provide a response similar to
occupant response during collision, it lacks muscle activation and furthermore there are ethical
concerns that hinders this approach, especially for children.
Anthropomorphic Test Dummies (ATDs) such as such as HybridIII-TRID [34] and BioRID2
[35] are used to simulate the occupant head response in rear collisions by capturing the
occupant’s kinematics as well as the neck forces using load cells. Although ATDs response is
evaluated using neck injury criteria to determine the possibility of injury, they lack the ability to
identify the injury mechanism during the collisions.
In this work we intend to develop a novel human head-neck model that is capable of providing
the occupant’s kinematics and can be used in the future to analyze and understand whiplash
injury mechanisms.
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In order to carry out appropriate comparisons between the predicted kinematic behavior of the
head-neck and the prototype, it is necessary to ensure that: (i) the geometrical features are the
same, (ii) the physical properties are similar, (iii) the material properties are comparable and (iv)
muscles and soft tissues are comparable. Whilst the first three items can be achieved, attempts
have also been made to account for tissue and muscle behavior in our prototype. The details are
explained in the following sections.
5.2. Details of the Head-Neck Prototype
5.2.1. Skull and Vertebrae
The skull, the cervical vertebrae (C1–C7) and the first thoracic vertebra (T1) were developed
using 3D printing. The geometry used in the process was obtained from the GHBMC 50th
percentile male FE model. The GHBMC model is provided in the form of LS-DYNA keyword-
file. The geometry was converted to STL format, which is necessary for 3D printing. The
geometry of the head-neck prototype obtained from the GHBMC FE model and the 3D printed
model are shown in Figure 5.1. Figure 5.2 shows the 3D printed C1, C2 and the T1 vertebrae.
The material used for 3D printing the parts is polyethylene terephthalate glycol modified (PETG)
which is a thermoplastic polymer resin known for its impact resistance [36]. A comparison
between the mechanical properties of bone in human vertebrae and PETG shows that they have
comparable properties, as shown in Table 5.1. The mass of the head in the experimental
prototype was ~3 kg, while the human head mass ranges from 2.8-6.5 kg with an average of 4.7
kg [37].
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(a) (b)
Figure 5.1 Head-neck prototype: (a) geometry obtained from the GHBMC FE model and
(b) 3D printed skull and vertebrae
(a) (b) (c)
Figure 5.2 Detailed geometry of 3D printed vertebrae: (a) C1, (b) C2 and (c) T1
Table 5.1 Mechanical properties of the human vertebral bone and polyethylene
terephthalate glycol modified [38–41]
Bone PETG
Cortical Cancellous
Density (kg/m3) 1800-1900 180-630 1380
Young’s modulus (GPa) 20 0.151-0.487 2.2
Tensile strength (MPa) 80-150 2.54 53
Compressive strength (MPa) 90-280 2.22 55
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5.2.2. Intervertebral Disc
In order to develop the intervertebral disc (IVD) for the neck model, artificial discs which are
used in replacement surgeries were considered to develop a similar disc for the neck model.
BRYAN® cervical disc is one of such discs, which has a polyurethane rubber core enclosed by
titanium alloy shells. The disc is then fixed to the vertebrae by screwing the titanium alloy shells
to the vertebrae.
In this study, the IVDs were developed using ReoFlex 40 urethane rubber, which has a shore A
hardness of 40A, a specific gravity of 1.02 and a tensile strength of 3.4 MPa. Discs of a height of
8 mm were produced with variable diameters ranging from 14-18 mm in 1 mm increment
depending on each intervertebral level. Molds were developed to produce the desired discs with
the different diameters. Figure 5.3 shows the mold that was used to produce the 17 mm and 18
mm discs. The urethane rubber was poured into the molds and left to cure overnight (more than
the curing time of the rubber of 16 hours). Before pouring the liquid rubber into the molds, the
molds were sprayed by mold release.
Figure 5.3 One of the molds used to develop the IVDs
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At each intervertebral level, the rubber IVDs were attached to the superior and inferior vertebrae
using a cyanoacrylate-based glue. The attached IVDs were tested to ensure that the disc will fail
before glue debonding. Figure 5.4 shows the IVD attached to vertebrae.
Figure 5.4 The IVD attached to the vertebrae. The articular process of each vertebra is
covered with neoprene rubber (black)
5.2.3. Ligaments
Two ligaments were considered in this study: The Anterior Longitudinal Ligament (ALL) and
the Posterior Longitudinal Ligament (PLL). These two ligaments were considered because the
ALL is the ligament under tension during the neck extension and they are two of the main
ligaments responsible for the stability of the spine.
The ligaments were cut from 1/8” thick shore hardness 80A rubber sheets in the desired
dimensions. The ALL has a width of 24 mm while the PLL has a width of 20 mm. Both
ligaments run from T1 to the base of the skull covering the entire length of the neck. The
dimensions of the ligaments developed were selected to ensure that the ligaments provide the
same response as human ligaments. Therefore, the mechanical response of the rubber ligaments
was compared to human ligaments tested by Yoganandan et. al [42]. In their study, human
cervical ligaments were loaded in tension at a rate of 10 mm/s and the change in the tensile force
with displacement was reported. The rubber ligaments were tested under tensile loading for the
same loading rate using a 5 kN cell Instron 5965 universal testing machine. The response of the
rubber ligament compared to the human ligament is shown in Figure 5.5.
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Figure 5.5 Force - Elongation curve of the rubber ligament developed compared to
literature data for the ALL at the C2-C5 levels by Yoganandan et. al [42]
The ligaments were attached to the vertebrae using a cyanoacrylate-based glue and were tested to
ensure that the ligaments will fail before glue debonding. The ALL attached to the head-neck
model is shown in Figure 5.6. The PLL is not visible since it is attached to the posterior of the
vertebral body.
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Figure 5.6 The anterior longitudinal ligament (ALL) attached to the vertebra and the neck
stabilization system of the head-neck prototype
5.2.4. Facet Joint
In order to reduce the friction at the facet joint between two adjacent vertebrae during neck
extension/flexion, neoprene rubber was attached to the articular process of each vertebra and the
surface of the rubber was lubricated. The rubber was cut from 0.8 mm thick sheets in the desired
shape and was attached to the articular process using cyanoacrylate-based glue. The attached
rubber facet surface is shown in Figure 5.4.
5.2.5. Neck Stabilization System
The occupant’s neck muscles are responsible for maintain the driving posture. Here, we assume
that the driver is looking forward without embracing for impact. In order to maintain a similar
posture for the head-neck model, a Neck Stabilization System (NSS) was developed based on a
muscle replication system developed by Panjabi et al. [43] and Ivancic et al. [28].
The NSS consists of four wires attached to the bottom of the skull and run along the neck. The
wires were attached to the skull and run along the neck anteriorly, posteriorly and along both
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lateral sides. The posterior wire runs through wire guides, which were attached to the spinous
process of each vertebra from C2 to C7. The anterior wire runs through one wire guide, which
was attached to the vertebral body of C4 vertebra. Both lateral wires run through the transverse
foramen of C2-C7 vertebrae. The NSS is shown in Figure 5.6.
The other end of the wires run through tension springs and were attached to M6 bolts which were
used to control the tension in the wires by tightening or untightening the bolts, as shown in
Figure 5.7. Tapped through holes were drilled into the sled to which the bolts were fastened.
After tightening a bolt to achieve the desired tension in the wire, the bolt was secured by means
of a nut as shown in Figure 5.7(a).
Figure 5.7 Neck stabilization system wire tension control: (a) illustration and (b) photo of
the system assembled on the sled
5.3. Impact Sled
To simulate a rear impact, the inclined impact sled available in the Mechanics & Aerospace
Design Lab was used. An exploded view of the sled is shown in Figure 5.8. The details of the
sled can be found in [44]. When released, the sled slides along the guide rails and is stopped by
means of springs at the end of the guide rails. Dampers were added at both ends of the springs to
reduce high frequency vibrations. The impact severity is controlled by the height at which the
sled is released.
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Figure 5.8 Exploded view of the sled used for impact simulation [44]
To mount the head-neck model on the sled, the T1 vertebra was enclosed in a block of epoxy,
which was then fixed to the sled by means of two M6 screws. During mounting the T1 vertebra
in the epoxy, the inclination of T1 vertebra was measured to maintain an inclination of 11° from
the horizontal plane which is the same inclination of the T1 vertebra in the GHBMC FE model
[45]. The head-neck model mounted on the sled is shown in Figure 5.9.
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Figure 5.9 The head-neck prototype mounted on the sled
5.4. Imaging and Sensory
3.1.1 High-speed Imaging
During the impact test, the motion of the head-neck model was captured using a Fastec Imaging
TSHRMS high-speed camera at a rate of 1000 fps which allows capturing the motion of the head
and neck every 1 ms. A MATLAB code was developed to analyze the video capturing the
motion of the head and the neck during the impact. The code assigns points to the targets at the
head and T1 vertebra (see Figure 5.10) and tracks these points providing their coordinates for
each frame. Using these coordinates the position of the head with respect to T1 vertebra as well
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as the head’s angle of rotation can be calculated. Reference targets were attached to the CG of
the head and the T1 vertebra to be tracked.
Figure 5.10 A typical frame from the video captured using the high-speed camera showing
the tracking points (green crosses) of the head’s reference target
3.1.2 Accelerometers
Two 3-axis ADXL345 accelerometers were used to capture the sled and the head accelerations
during the test. The accelerometers have a range of ± 16 g, a resolution up to 13-bit and a
measurement rate up to 3200 Hz. The accelerometers were controlled through an Arduino UNO
board. The circuit connecting the accelerometers to the Arduino board is shown in Figure 5.11.
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Figure 5.11 ADXL345 accelerometers connection circuit
Both accelerometers were calibrated before testing. To calibrate the accelerometers, the gain and
the offset of each accelerometer in the three axes was determined. The earth gravity was used for
the calibration process. For each direction (X, Y or Z), the accelerometers were aligned such that
the desired direction is subjected to 1 g and -1 g and the corresponding reading is recorded each
time. The gain and the offset error for each direction were then determined.
5.5. Results and Discussion
The experimentally measured horizontal sled acceleration is shown in Figure 5.12. The
acceleration reaches a peak value of 21 m/s2 at 96 ms. The negative acceleration recorded
between 200-400 ms is due to the rebound of the sled caused by the springs at the end of the
guide rails. This acceleration profile was applied to the T1 vertebra in the MBD and FE models.
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Figure 5.12 Experimentally measured horizontal acceleration recorded at the neck base
(T1 vertebra)
5.5.1. Comparison to Multibody Dynamics and Finite Element
Simulations
The response of the experimental prototype was compared to the developed MBD model of the
head and neck [2], and the GHBMC FE model [12]. To facilitate the comparison, the
acceleration recorded at the T1 vertebra during the experimental test was applied to the MBD
and FE simulations. For the FE, two models of the head and the neck were extracted from the
GHBMC human model, as shown in Figure 5.13. The first contained the head, the vertebrae, the
IVDs and the spinal ligaments, while all muscles, skin and flesh were removed. The second FE
model included the neck muscles and skin/flesh. The geometrical features of the head-neck
prototype were very similar to the one used in the MBD and FE simulations. Additionally, the
mass of the head in both MBD and FE simulations was changed to be similar to that of the
experimental prototype. In the FE model, the mass of the head was adjusted by changing the
density of the head elements. For the FE simulations, the acceleration was applied at the T1
vertebra; the nodes highlighted in Figure 5.13.
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Figure 5.13 The head-neck FE model extracted from the GHBMC FE (a) without muscles,
skin and flesh and skin-flesh and (b) with muscles, skin and flesh. Acceleration was applied
at T1 vertebra and the yellow highlighted nodes
A sequence of the head-neck prototype response during the experimental test using the high-
speed camera is shown in Figure 5.14 and compared to response of the MBD and FE models.
The response of the experimental model agrees with response of the MBD model and the FE
models, specifically the FE model without muscles.
The head horizontal and vertical displacements with respect to T1 vertebra for the MBD and FE
predictions, and the experimental findings are shown in Figure 5.15. The theoretical predictions
and the experimental results show that the neck goes into extension until the head reaches its
peak displacement followed by the rebound phase. The peak horizontal and vertical
displacements and their time of occurrence are provided in Table 5.2.
The FE model with muscles shows the least head displacements, while the FE model without
muscles shows the most severe response. The FE predicted response forms a corridor bound by
the stiffest neck repose (with muscles) and the most flexible response (without muscle). Both the
MBD predictions and the experimental results agree more with the response of the FE model
129
without muscle. The differences between the FE predictions without muscles and the
experimental results are attributed to the NSS used in the experimental work, which increases the
stiffness of the neck. However, this increase in the stiffness is not comparable to the presence of
neck muscles. It must be noted that the tension in the wires of the NSS was maintained at a
minimum level to hold the head in its position without introducing further tension in these wires.
Figure 5.14 Comparison between the responses of the experimental results, MBD and FE
predictions
130
Figure 5.15 Head displacements with respect to T1 vertebra for the experimental results,
and MBD and FE predictions: (a) horizontal and (b) vertical
Table 5.2 Peak head displacements and their time of occurrence for the MBD, FE and
experimental models
Peak horizontal
displacement (mm)
Peak vertical
displacement (mm) Time (ms)
MBD 121 33 217
FE 178 83 273
FE with muscles 85 13 214
Experimental 121 78 256
The head rotation of the MBD and FE predictions, and experimental results is shown in Figure
5.16. The peak head rotation in extension for the experimental, MBD, FE without muscles and
FE with muscles models are 42.5°, 59.4°, 75.8° and 29.7°, respectively. Analogous to the head
displacements, the maximum head rotation was reported from the FE model without the muscles,
while the least head rotation was reported from the FE model with muscles. The head rotation
response of the MBD predictions and experimental results is within the corridor formed by the
FE models. The NSS in the experimental effort limited the maximum head rotation when
compared with the predicted response of MBD and FE without muscles models.
131
Figure 5.16 Head rotation measured experimentally and predicted using MBD and FE
Only the FE predictions without muscles show a small initial flexion rotation of 5.5° between 50-
100 ms under the weight of the head before the acceleration of the T1 vertebra. This initial
flexion was prevented in the experimental effort by the NSS and it was eliminated by the
presence of muscles in the FE analysis.
Overall, the MBD and the FE predictions without muscles show a comparable response to
experimental results. The differences in the response is attributed to the NSS, which is not
accounted for in the MBD and FE analysis as well as the differences in the properties of the
materials used in the experimental prototype, when compared to human soft tissues. In this work,
the NSS was used to stabilize the model before simulation but it was not intended to provide the
passive muscle response expected in human model. However, this can be achieved by controlling
the tension in the wires through the spring mechanism used in the model. Furthermore, the
experimental prototype only accounted for the ALL and PLL Moreover, the NSS can be further
developed to simulate the active muscle response by adding actuators to control the tension in the
wires. The head-neck prototype presented here can be used as a core development of head-neck
models in future ATDs.
5.6. Conclusions
A novel head-neck prototype was developed to characterize occupant kinematics in whiplash
testing. The model was subjected to 2.14 g rear impact acceleration. The response was compared
to previously developed MBD and GHBMC neck FE predictions. The experimental results show
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good agreement with the MBD and FE predictions. Two FE neck models were considered in the
study: one with neck muscles and the other without. The response of these two bounding cases
form a corridor representing the stiffest and most flexible neck response. This study reveals that
the experimental findings of the head-neck kinematics are within the corridor formed by the FE
predictions. Although the NSS was used to stabilize the neck without affecting the response, it
can be further developed to simulate muscles active and passive responses. Moreover, the newly
proposed experimental set can incorporate other neck ligaments and be used to simulate frontal
and lateral collisions.
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Chapter 6.
Paper #3: Effect of Seat Belt and Head Restraint on Occupant’s Response during Rear-End Collision
This chapter has been published in International Journal of Mechanics and Materials in Design,
14, 231-242, 2018. Available at: https://doi.org/10.1007/s10999-017-9373-6
Abstract
Current neck injury criteria used to evaluate whiplash injuries are based on the kinematics or
kinetics of the occupant’s head and neck during rear impacts. The occupant’s response is affected
by many factors including impact severity, seat design and occupant related factors such as
gender and posture. Most of the current finite element models are concerned with modeling the
head and neck, ignoring the interaction of the seat with the occupant during rear collision. In this
work the Global Human Body Model Consortium (GHBMC) finite element model was used to
study these interaction effects with emphases on the effect of seat belt, headrest and seat stiffness
on the occupant’s response during rear-end collisions and evaluate the response using three neck
injury criteria. The study shows the dramatic importance of the occupant’s seat restraint and head
rest upon occupant safety. Specifically, the occupant ramping during rear impacts can be
prevented by using the seat belt. Furthermore, the headrest reduces the head displacement and
rotation. Our work further reveals that the head displacement reduction can lead to higher
moments, axial and shear forces at the neck, especially for cases involving poorly adjusted or
stiffer headrest.
Keywords: whiplash, injury, rear-impacts, finite element, seat belt, headrest
6.1. Introduction
Rear-end impacts represent approximately 24% of all multiple vehicles crashes in the United
States [1]. A significant number of these crashes result in whiplash which is the most common
136
injury in car accidents. According to the National Highway Traffic Safety Administration
(NHTSA), about 806,000 whiplash injuries occur annually in the US costing over $9 billion [2].
Despite the advancement in car safety, whiplash injuries remain a serious problem. The number
of whiplash injuries or Whiplash Associated Disorders (WAD) is increasing [3]. Whiplash
patients suffer from headache, neck pain, limited neck motion, visual disturbance, weakness and
dizziness [4–6]. The WAD severity and duration depends on many factors like occupant’s
gender, posture, awareness, seat and headrest geometry, stiffness and collision severity [7–9].
In order to enhance car seat design for better protection against whiplash injuries, it is essential to
study the occupant’s response during collisions. A number of studies have been conducted to
evaluate the human response during rear impacts. The first relies on volunteers [5,10] to provide
a realistic indication of the human response to rear-end collisions. However, the impact severity
in these studies is rather limited to prevent injury of the test subjects. The second approach
utilises cadavers, but they lack the muscle response. The third employs anthropomorphic test
dummies (ATDs) such as HybridIII-TRID [11] and BioRID2 [12] which have been specifically
designed for rear-end collisions testing. Although ATDs have been shown to approximate the
head motion and loading during whiplash, the biofidelity of the neck is quite limited. The fourth
relies on the development of numerical models to overcome the complexity and the cost of
experimental studies. Multibody dynamics is one of the numerical techniques used to simulate
whiplash in which the body is modeled using a number of rigid elements connected using
revolute joints, elastic springs, dampers, and/or viscoelastic elements [13–16]. The cervical spine
has an intricate geometry and studying the kinematics only does not provide a detailed
description of injury mechanisms of whiplash. The stresses and strains in soft tissues are a key
factor in understanding whiplash injury, which multibody dynamics does not address. Therefore,
we need to consider another modelling technique such as finite element method (FEM) to
construct a more detailed model of the human body. Several FE models were developed to study
the cervical spine under different types of loading [17–21]. In these FE efforts, advanced
material models were used to describe the mechanical properties of the intervertebral discs,
ligaments, muscles and facet joints which in return provide a better understanding of injury
mechanisms in soft tissues.
Currently, most FE models only focus on the head and neck, ignoring the interaction between the
occupant and the car seat during rear-end collisions. One of the main reasons is the complexity in
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modeling the entire human body and the burden of the high computational cost associated with
large numerical models. Studies focusing on the head and neck may not provide an accurate
description of whiplash. For instance, neglecting the interaction of the torso with the car seat can
alter the kinematics of the head and neck [22]. Torso ramping can be included in the head-neck
models by adding additional loads at the T1 vertebra [23] to prevent using full body models and
reduce the computational cost. However, this raises a concern about the accuracy of the response
of these models compared to full body models and the accuracy of the loads applied at T1 for
each impact scenario. Therefore, it is crucial to capture the response of the entire body for better
understanding of the interactions that affect whiplash injuries which is indeed the motivation
behind the current study. In this work, we intend to study the interaction between the occupant
and the car seat showing the effect of seat belt and headrest on the occupant’s response during
rear-impacts using a full human body FE model.
6.2. Model and Materials
The Global Human Body Model Consortium (GHBMC) 50th percentile male FE model was
seated on a car seat, as shown in Figure 6.1. The 26 years old male model weight and height are
78 kg and 174.9 cm, respectively. The GHBMC model consists of 988 parts discretized using
2.18 million elements including solid, shell, beam and discrete elements. The neck subregion of
the model consists mainly of the vertebrae which were discretized using shell and brick elements,
the intervertebral discs which were modeled using shell and brick elements, cartilages which
were discretized using brick elements, muscles which were discretized using 1-D and brick
elements, and ligaments for which 1-D elements were used.
138
Figure 6.1 The GHBMC FE model seated (a) without a headrest and without seat belt, (b)
with a poorly adjusted headrest and a seat belt, and (c) with a properly adjusted headrest
and a seat belt
Three different seat arrangements were used in this study: one without a headrest and without a
seat belt, the second with a poorly adjusted headrest and a seat belt, and the third with a properly
adjusted headrest and a seat belt. For the poorly adjusted headrest, the top of the headrest was
110 mm below the top of the head while for the properly adjusted headrest the top of the headrest
139
was 24 mm above the top of the head. The distance between the headrest and the back of the
head (backset) was 68 mm for both headrests. The headrest position is defined as poor or proper
based on the rating by the Insurance Institute for Highway Safety [24]. The seat back of the three
arrangements was inclined by 25° from the vertical axis. The seat dimensions are shown in Table
6.1. In the current study the seat back was considered rigid. The three seat arrangements were
discretized using tetrahedron elements and their material model was selected to be low density
foam. The seat material is a polyurethane foam and its mechanical properties are the same as
those used by Grujicic et al. [25]. The density was assumed to be 23 kg/m3 and the nominal
compressive stress-strain curve used for this material model is shown in Figure 6.2. Two degrees
of stiffness were used for this material model. The stiffness was changed by varying the ordinate
values (stress) of the constitutive law shown in Figure 6.2. The soft material has the stress-strain
curve shown in Figure 6.2. For the stiff material considered, the ordinate values were increased
by a factor of 100. The seat stiffness for both degrees of stiffness was determined through quasi-
static testing (QST) of the seat. For the QST, the lower nodes of the seat frame were fixed and a
rigid dummy of the torso was pressed against the seat backrest. A constant velocity of 1m/s was
applied to the dummy horizontally while no constraints were applied in the vertical and lateral
directions. The seat stiffness was calculated from the force - displacement curve of the rigid
dummy. For the soft and stiff materials the corresponding seat stiffness is 275 kN/m and 8360
kN/m, respectively. The effect of gravity was applied as a body load to the FE model. In all
simulations the occupant and the seat were initially at rest. During a rear impact the acceleration
of the target (front) car is transferred from the car frame to the lower part of the seat frame.
Therefore, in order to simulate a rear impact, the velocity was applied horizontally to the lower
part of the seat frame while constraining its motion in the vertical and lateral directions. No
constraints were applied to the backrest or to the headrest. The nonlinear dynamic analysis was
conducted using the explicit solver of LS-DYNA commercial software package (Livermore
Software Technology Corporation LSTC).
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Figure 6.2 Seat material compressive stress-strain curve by Grujicic et al. [25]
Table 6.1 Seat and head restraint dimensions
Backrest length – from headrest to cushion (cm) 64
Cushion length – up to the backrest (cm) 49
Backrest – total width including wings (cm) 44
Backrest – inside wings (cm) 26
Cushion width – total width including wings (cm) 52
Cushion width – inside wings (cm) 27
Headrest length – top to bottom (cm) 20
Headrest width (cm) 29
The GHBMC FE model was first validated by comparing its response with an experimental
study conducted on cadavers by Prasad et al. [26]. In this validation test, the seat arrangement
without a headrest (seat arrangement A) was used (see Figure 6.1(a)) and a velocity profile was
applied horizontally to the seat frame to simulate a rear-end collision. This velocity profile is the
same one used in the experimental study [26]. The seat was accelerated from zero to 7 m/s in 150
ms (an average acceleration of 4.8 g). This simulation was repeated using a similar velocity
profile, but the maximum velocity was limited to 3.5 m/s (an average acceleration of 2.4 g).
In order to study the effect of the headrest and the seat belt on the occupant’s response, the 4.8 g
velocity profile was applied to four different seat arrangements: B, C, D and E. Arrangements B
and C had a poorly adjusted headrest (see Figure 6.1(b)) and were assigned the stiff and soft
material models, respectively, while arrangements D and E had a properly adjusted headrest (see
141
Figure 6.1 (c)) and were assigned the stiff and soft material models, respectively. The different
seat arrangements used in this study are shown in Table 6.2.
Table 6.2 Seat arrangements used in the simulations
Seat
Arrangement
Belt and Head
Restraint
Seat Stiffness
(kN/m) Head restraint topset
A No 275 -
B Yes 8360 110 mm below head top (Poor)
C Yes 275 110 mm below head top (Poor)
D Yes 8360 24 mm above head top (Proper)
E Yes 275 24 mm above head top (Proper)
6.3. Results and Discussion
To validate the GHBMC FE model, the head response of the FE model seated as per arrangement
A was compared with the experimental results from tests on cadavers by Prasad et al. [26].
Figure 6.3 shows the relative horizontal and vertical displacements of the head center of gravity
(CG) with respect to the T1 vertebra for the FE model compared to the cadaver test. The results
indicate that the FE model’s response conforms greatly to that of the cadavers’ results. During
the first 60 ms the lower cervical spine of the FE model was under extension, while the upper
cervical spine was under flexion; forming the well-recognized S-shape curvature for whiplash as
reported by Grauer et al. [27]. The entire cervical spine then underwent extension until the
maximum head displacement was reached around 150 ms after the impact which was followed
by the rebound phase.
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Figure 6.3 Relative head CG displacement with respect to T1 vertebra for the GHBMC FE
model using seat arrangement A compared to cadaver test by Prasad et al. [26] (a)
horizontal and (b) vertical
The head displacements for the five seats arrangements are shown in Figure 6.4(a and b). The
presence of headrest, even if poorly adjusted, reduced the head displacements significantly.
During the extension of the neck, the horizontal relative head displacements with respect to the
T1 vertebra were reduced by ~50% for arrangement B, ~54% for arrangement C and ~68% for
both arrangements D and E. The horizontal relative head displacement curve for arrangement D
shows that the head bounces from headrest after 70 ms from impact and comes into contact with
the headrest once again after 140 ms. The reason for this behavior is the lack of cushioning and
the inability of the headrest to absorb some of the impact energy.
The headrest reduced the vertical relative head displacement for arrangements B, C, D and E by
~82%, ~96%, ~94% and ~95%, respectively. The vertical relative head displacement trajectory
for all cases show an initial compression of the cervical spine during the first 75 ms. The inclined
seat back leads to a force component that drives the torso (T1 vertebra) upwards. In view of the
inertia of the head, its dynamic response will be delayed leading to the compression of the
cervical spine.
143
Figure 6.4 For the five seat arrangements: (a) relative head horizontal displacement with
respect to T1, (b) relative head vertical displacement with respect to T1, (c) head horizontal
acceleration and (d) head vertical acceleration
The horizontal and vertical head accelerations for the five seat arrangements are shown in Figure
6.4(c and d). The peak horizontal head acceleration when no headrest was used (seat
arrangement A) is more than twice the seat acceleration (4.8 g). The results show that the
presence of the headrest subjected the head to higher accelerations compared to the case where
no headrest was used. The significant increase in the acceleration is attributed to the sudden stop
of the head when it comes into contact with the headrest. The peak horizontal head acceleration
for seat arrangements A, B, C, D and E are ~10 g, ~ 43 g, ~26 g, ~54 g and ~34g, respectively.
The peak vertical head acceleration for the aforementioned arrangements are ~9 g, ~29 g, ~12 g,
~11 g and ~9 g. Results show that for arrangements with the same headrest position,
144
arrangements having lower seat stiffness reported lower head acceleration. Seats with lower
stiffness have higher energy absorption capabilities; hence they reduce the acceleration to which
the head is subjected. Comparing arrangements having same seat stiffness (B and D, or C and E)
shows that the properly adjusted headrest subjected the head to higher horizontal acceleration
compared to a poorly adjusted headrest. However, the properly adjusted headrest reduced the
vertical head acceleration.
The rotation of the head in the sagittal plane is shown in Figure 6.5(a). For seat arrangement A,
the maximum head rotation was determined to be 83° at ~150 ms after impact. Seat
arrangements B and C limited the maximum head rotation to 26° occurring at ~93 ms after
impact, reducing the head rotation by some ~69%. The properly adjusted headrest in seat
arrangements D and E led to a maximum head rotation of only 13° at ~100 ms after impact,
reducing head rotation by ~84%. The deformation of the neck is shown in Figure 6.5(b) for seat
arrangements A, B and E. The headrest of arrangements B and E reduced the overall neck
deformation with higher reduction reported for the properly adjusted headrest (arrangement E).
Figure 6.6 shows the contact between the head and the headrest for seat arrangements C and E.
The head came into contact with the top edge of the headrest of seat arrangements B and C
which did not provide good support of the head and allowed its further rotation after contact with
the headrest. On the other hand, seat arrangement C provided the proper support of the head
which limited the rotation of the head. It is evident that the reduction in head displacements and
rotation is higher in the case of a properly adjusted headrest compared to a poorly adjusted one.
145
Figure 6.5 (a) Change in head rotation in the sagittal plane with respect to time for the five
seat arrangements and (b) the deformation of the neck and head position for seat
arrangements A, B and E
146
Figure 6.6 Contact between the head and the headrest for (a) poorly adjusted headrest (seat
arrangements B and C) and (b) for properly adjusted headrest (seat arrangements D and
E)
After the initial compression of the neck, the head moves superiorly. When no seat belt was used,
the torso and the hip moved superiorly due to the ramping effect of the seat back. A similar
response was reported by Davidsson et al. [12]. The ramping of the occupant during rear impacts
depends on the inclination of the seat back and its acceleration. A schematic of the ramping
effect is shown in Figure 6.7(a). The body of the occupant is assumed to have a mass m and the
seat back is inclined by an angle θ from the vertical axis. In order to study the relative motion of
the occupant with respect to the car seat during ramping, we will consider the non-inertial frame
of reference in which the seat is not moving and the occupant is moving with an acceleration a to
the left (in a direction opposite to the actual direction of motion of the seat). The respective
forces acting on the body are mg and ma due to the gravity g and the seat acceleration a.
Resolving these two forces in a plane parallel to the seat back gives 𝑚 𝑎 𝑠𝑖𝑛 𝜃 acting upwards
and 𝑚 𝑔 cos 𝜃 acting downwards. The friction force between the body and the seat is given by:
𝐹𝜇 = 𝑚 𝜇( 𝑎 cos 𝜃 + 𝑔 sin 𝜃) (6.1)
147
where μ is the coefficient of friction between the body and the seat. Hence, the resultant ramping
force Framp along the inclined seat back can be given by:
𝐹𝑟𝑎𝑚𝑝 = 𝑚(𝑎 sin 𝜃 − 𝑔 cos 𝜃 − 𝜇( 𝑎 cos 𝜃 + 𝑔 sin 𝜃)) (6.2)
For the body to ramp upwards, the resultant force Framp must be positive. Otherwise, the body of
the occupant will not move vertically. The acceleration threshold for no ramping can be
determined from Eq. (6.2) at the onset of ramping, where the ramping force Framp = 0. Ignoring
the effect of friction between the seat and the occupant, the acceleration threshold athreshold = g
cot(θ). For the seat used in the current study θ = 25° for which athreshold = 2.14 g. The
acceleration threshold indicates that the body will ramp for both accelerations applied in the
current study (4.8 g and 2.4 g).
Figure 6.7 (a) schematic diagram of the ramping effect and (b) vertical displacement of the
CG of the hip without seat belt
Figure 6.7(b) shows the vertical displacement of the CG of the hip of the human FE model for
the two accelerations (4.8 g and 2.4 g) without using a seat belt (seat arrangement A). For both
cases the hip ramped upwards during the acceleration of the seat (the first 150 ms after impact)
until the seat reaches and maintains its maximum velocity (the acceleration is zero) after which
148
the hip moves downwards under the effect of gravity. The maximum hip displacement is directly
proportional to the seat acceleration. For instance, the peak hip displacements for the 4.8 g and
2.4 g seat accelerations are 275 mm and 112 mm, respectively.
The response of the occupant during the entire simulation using seat arrangement A with an
average seat acceleration of 4.8 g is shown in Figure 6.8. Initially the occupant was at rest (Fig.
8(a)) when the seat began accelerating. During the seat acceleration (Figure 6.8(b)) the body of
the occupant is ramped along the back of the seat. The maximum ramping position was reached
150 ms after impact when the seat reached and maintained its maximum velocity (Figure 6.8(c))
after which the occupant moved downward under his own weight. After 270 ms, the seat came to
rest (Figure 6.8(d)). Since the occupant was not restrained to the car seat, the occupant continued
moving forward leaving the car seat as depicted clearly in Figure 6.8(e).
Figure 6.8 Occupant’s response during the entire simulation for seat arrangement A.
Figures (a)-(e) show occupant’s position with respect to the car seat at 0 ms, 75 ms, 150 ms,
270 ms and 350 ms
Ramping of the body changes the position of the occupant’s head vertically. Therefore, even in
the presence of a perfectly adjusted headrest, the head may not be supported effectively, if the
149
seat belt is not used during rear-end collisions, which indicates the important role of the seat belt
in protecting against whiplash trauma. Furthermore, the seat belt plays another important role in
protecting the occupant during rear-end collisions. During a typical rear-end collision, the car
seat accelerates forward with the occupant’s body followed by deceleration of the seat, until it
reaches a standstill position. During the deceleration phase, the occupant’s body will continue its
forward motion unless the occupant is restrained to the car seat. If the seat belt is not used, the
occupant may hit the steering wheel or the windshield, depending on the impact severity. In the
simulation conducted using seat arrangement A, at 4.8 g acceleration, the seat came to rest some
270 ms after the impact. At 350 ms after impact the body of the occupant left the car seat in a
way similar to the occupant’s response to a frontal impact. Using the seat belt restraints the
occupant to the car seat and provides protection against impacting the steering wheel or the
windshield.
The elongation of the capsular ligament (CL) was used to evaluate the occupant’s response for
the different seat arrangements since it is the most common reason for neck pain [6]. Figure 6.9
shows the CL elongation at each intervertebral level of the cervical spine. The highest CL
elongation (~31%) was reported at the C2C3 level for the case of arrangement A. For the same
arrangement, the elongation at other levels was considerably low except for the C6C7 level.
Arrangements B, C and D increased the elongation of the CL compared to arrangement A at all
levels except the C2C3 level. The following pattern can be noticed that for these three
arrangements (B, C and D): at all intervertebral levels arrangement B reported the highest CL
elongation followed by arrangement C and then arrangement D. Arrangement E reported the
lowest CL elongation at all levels with almost eliminating the elongation at C3C4 and C4C5
levels.
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Figure 6.9 Capsular ligament elongation at each intervertebral level for the different seat
arrangements
6.4. Application of Neck Injury Criteria
The kinematics of the occupant during collisions are not sufficient to determine whether
whiplash injury will occur or not. Therefore, neck injury criteria are necessary to evaluate the
possibility of whiplash injury. Furthermore, since there is no specific injury mechanism
responsible for whiplash injury, different injury criteria covering different injury mechanisms
should be considered for occupant’s safety. Each injury criterion is based on a specific injury
mechanism. Three neck injury criteria were used to evaluate the occupant’s response for the five
aforementioned cases. The first criterion is the Neck Injury Criterion (NIC) [28] which
determines the NIC value using the relative velocity and acceleration of the head as given by Eq.
(6.3) below:
𝑁𝐼𝐶 = 0.2 𝑎𝑟𝑒𝑙 + 𝑣𝑟𝑒𝑙2 < 15
𝑚2
𝑠2
(6.3)
The threshold for NIC is 15.
151
The second criterion is the normalized Neck Injury Criterion Nij [29] which is evaluated using
the axial force Fz and the bending moment My in the sagittal plane at the occipital condyle; as
follows:
𝑁𝑖𝑗 =𝐹𝑧
𝐹𝑖𝑛𝑡+
𝑀𝑦
𝑀𝑖𝑛𝑡
(6.4)
The intercept values for the Nij criterion are Fint = ±4500 N for tension and compression, and Mint
= 310 N.m for flexion and 125 N.m for extension. The third criterion is the Nkm injury criterion
[30], which is evaluated using the horizontal shear force Fx and the bending moment in the
sagittal plane at the occipital condyle; as follows:
𝑁𝑘𝑚 =𝐹𝑥𝐹𝑖𝑛𝑡
+𝑀𝑦
𝑀𝑖𝑛𝑡
(6.5)
The intercept values for Nkm are Fint = ±845 N for the positive and negative shear force, and Mint
= 88.1 N.m for flexion and 47.5 N.m for extension. The injury threshold for both Nij and Nkm is
1.0.
The evaluation of the injury level using the three neck injury criteria is shown in Figure 6.10.
Although Nij and Nkm distinguish clearly between the three different cases, NIC does not show
significant difference between them. When no headrest was used (seat arrangement A), the peak
NIC value exceeded the threshold indicating possibility of injury. Using seat arrangements B and
D increased the NIC value, while using seat arrangements having a lower stiffness of 275 kN/m
(arrangements C and E) decreased the NIC peak value. It must be noted that unlike other seat
arrangements, arrangement E reduced the NIC value below the threshold indicating that this seat
arrangement provided the occupant with protection against whiplash. Evaluation using Nij and
Nkm shows that none of the cases exceeded the injury threshold. When compared to arrangement
A, all other seat arrangements increased the Nij and Nkm values except for arrangement E which
maintained the same Nij value and decreased the Nkm value. Therefore, according to the three
neck injury criteria used in the study, seat arrangement E is the optimum of the five introduced
arrangements to protect the occupant against whiplash. Comparing seat arrangements having the
same stiffness (B and D) shows that the headrest position did not affect their NIC values.
However, the properly adjusted headrest (arrangement D) reported lower Nij and Nkm values
compared to the poorly adjusted headrest (arrangement B). Considering arrangements with the
152
same headrest position (B and C, or D and E) shows clearly that headrests with a lower stiffness
reported a reduction in their NIC, Nij and Nkm values.
Figure 6.10 The NIC, Nij and Nkm injury criteria evaluation for the five seat arrangements
153
The significant difference between the evaluation of possibility of injury of NIC on one hand and
Nij and Nkm on the other hand indicates that despite the great amount of research conducted on
whiplash, no current neck injury criterion can provide an accurate evaluation of whiplash injury.
The difference between the injury criteria may be related to the injury mechanism upon which
each criterion was based and developed. NIC is based on the transient pressure inside the spinal
canal during whiplash while Nij and Nkm are based on the individual’s neck tolerance to tension,
compression, shear, flexion and extension. The results of Nij and Nkm criteria conform to the
results from the CL elongation where seat arrangement B reported the severest response followed
by arrangement C then D then A and finally arrangement E providing best protection for the
occupant.
Although the headrest is expected to protect the head and neck during rear impacts, the values of
the three neck injury criteria and the CL elongation show that a stiff or poorly adjusted headrest
may increase the possibility of neck injury. In general, the headrest reduces the head
displacements and rotation but may subject the head to high accelerations and subject the neck to
high moments, axial and shear forces. Headrest materials with good energy storage capabilities
are necessary to provide better protection for the head. The position of the headrest with respect
to the head at the time of impact is a key factor in occupant protection against whiplash. If the
headrest is close enough to the head at the moment of impact, it will provide immediate support
to the head and will prevent its initial backward acceleration. Decreasing the head acceleration
will decrease the forces to which the neck is subjected and hence reduce the possibility of injury.
6.5. Conclusions
The present study shows the importance of the use of head restraint and seat belt for the effective
safety of the occupant during rear-end collision. The head restraint reduced the head
displacement and rotation during rear impact significantly. Although the head displacements and
rotation were reduced, the head acceleration increased due to the sudden stop by the headrest.
Our simulations show clearly that the headrest position and material play a crucial role in
occupant safety; softer headrest materials reduce the head acceleration and capsular ligament
elongation compared with stiffer materials, which in return reduces the forces and moments
acting on the neck. Properly adjusted headrest is a key factor in reducing the possibility of injury.
The headrest may not be sufficient to protect the occupant from whiplash, if the seatbelt is not
154
engaged. The seat belt prevents the vertical ramping motion of the body, which maintains the
head in a relatively safe position with respect to the headrest. Furthermore, the seat belt prevents
the occupant from the forward and uncontrolled motion that could lead to impacting the
windshield or the steering wheel during rear-end collisions.
Acknowledgment
This paper was made possible by NPRP grant #6 - 292 - 2 - 127 from the Qatar National
Research Fund (a member of Qatar Foundation). The statements made herein are solely the
responsibility of the authors. The authors also wish to acknowledge the Global Human Body
Model Consortium (exclusively distributed by Elemance LLC Winston Salem, NC, USA) for
using the 50th percentile seated male FE model. Finally, the authors wish to thank Dr. Stewart
McLachlin for his help obtaining the GHBMC FE model.
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Chapter 7.
Paper#4: Effect of Interfacial Friction and Fold Penetration on the Progressive Collapse of Foam-
Filled Frustum using Kinematically Admissible Model
This chapter has been published in International Journal of Crashworthiness, 23, 581-592, 2018.
Available at: https://doi.org/10.1080/13588265.2018.1489337
Abstract
In this paper, we extend our earlier analytical efforts of the progressive collapse of foam filled
conical frustum with the aid of a kinematically admissible folding mechanism. The instantaneous
crushing force as well as the mean crushing force was derived from the principle of energy
conservation accounting for the typically overlooked term of foam/shell interaction. Specifically,
in this study, we accounted for the effect of two critical parameters upon the accuracy of our
upper bound solution. The first is concerned with interfacial shell-foam friction. The second is
with a more realistic proportion of the fold involved in the foam/shell interaction. The results of
the analytical model are compared with nonlinear elasto-plastic finite element collapse
predictions as well as validation with our crush test results. The results reveal the effects of the
interfacial friction and the modified fold proportion upon the accuracy of the analytical model
and its ability in predicting the crushing force curve and the fold length.
Keywords: Progressive collapse; Foam filled; Frustum; Interfacial friction
7.1. Introduction
Cellular materials such as aluminum foams possess superior properties for energy absorption as
they are lightweight and can undergo large deformation at nearly constant load. However, their
relative low strength limit their potential to be used alone as energy absorbers. Usually,
aluminum foams are used as fillers in thin-walled light weight structures. As reported in the
literature, foam filled thin-walled structures can exhibit enhanced specific energy absorption and
157
crushing stability compared to their empty counterparts. The enhancement effect depends on the
structural parameters such as the foam density and the shell thickness. In addition, introducing
the foam into the shell does not require additional space. For these reasons substantial efforts
have been devoted to investigating the crush behaviour of foam-filled thin-walled structures for
energy absorption applications. It has been proven that the energy absorption of foam-filled
column exceeds the numerical sum of energy absorption of the filler and the column due to the
complex interaction effects that exist at the column -foam interface. Chen and Wierzbicki [9]
investigated multi-cell columns with foam filling and found that the interaction effects between
the foam and the column wall made the total crushing resistance increase by 140% and 180% of
the direct foam resistance for double and triple cells, respectively. The fluctuation amplitude of
the crushing force curve can also be decreased after foam filling, indicating better crushing force
efficiency for foam-filled structures.
Besides the specific energy absorption, there are two more criteria for evaluating energy absorber
performance: the peak crushing force and the repeatability of the collapse mode. The peak
crushing force should not be too high so as to avoid possible damage caused by extreme
acceleration transferred to the occupants. The collapse mode, on the other hand, should be
repeatable and global buckling should be avoided. In this regard, frustum columns have certain
advantages over the most commonly used straight cylindrical columns. First, the frustum can
lower the initial crippling force while maintaining the same level of total energy absorption.
Second, global buckling is less likely to happen for the frustum than for the straight column,
especially for large slenderness ratio and oblique loading. The collapse of frusta during crushing
is more likely to take place in a progressive manner. For this reason, we focus our attention on
the crush behaviour of foam-filled frusta. Specifically, our attention is mainly focused on the
effect of the interaction between the foam and the shell on the energy absorption.
A number of efforts have been devoted to investigate the energy absorption performance and the
collapse mode of foam-filled thin-walled structures. This includes the extensive experimental
work contributed by Guillow et al. [11], Hanssen et al. [15][16], Meguid et al. [21], extensive
numerical efforts using finite element method [4][10][22][23], mesh free methods [6][32][33],
and various optimization schemes [7][29]. For the loading condition, both the quasi-static and the
dynamic loading through impacting with an incident mass were investigated. Ahmad and
Thambiratnam [4] investigated both the dynamic response and the quasi-static response of the
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empty and foam filled tubes, and found that the impact velocity has minor influence on the
crushing force.
Besides the experimental and numerical work, several analytical models were also developed to
investigate the energy absorption of thin walled columns. The first attempt was made by
Alexander [5] for the analytical expression of thin-walled cylinder deformed in concertina mode.
This model was later improved and extended to incorporate the partly inside and partly outside
folding scenario [13], the curved fold geometry [1], and the introduction of taper angle [12][19].
Besides the circular tube, multicorner cross-sectional columns were also analyzed. Wierzbicki
and Abramowicz [30] proposed a basic folding mechanism based on the kinematic continuity for
the multicorner cross-sectional columns. Later, Abramowicz and Jones [2] applied this
mechanism for square tubes, and predicted four deformation modes: one symmetric, one
extensional, and two asymmetric modes. Mahmoodi et al. [20] theoretically investigated the
effects of the taper angle, the wall thickness and the number of cells on the crashworthiness of a
tapered multi-cell tubes. Hong et al. [17] derived the crushing force and the absorbed energy for
the tapered triangular tubes. A few theoretical studies were also conducted for the foam filled
cases. Reid et al. [28] took the effect of the foam/shell interaction into account by amending the
expression of mean force of the tube shell considering the shortened folding distance limited by
the locking strain of foam. Abramowicz and Wierzbicki [3] also considered the shortened fold
compression caused by the foam filling, and additionally amended the energy dissipation
contributed by the foam through the volume reduction considering the penetration of the shell.
Niknejad et al. [26][27] deduced the crushing force by assuming the foam/shell interaction
contributes to the same [26] or 80% [27] of the axial compressive resistance of the polyurethane
foam itself. Hanssen et al. [14] gave an empirical model of power law expression for the
crushing force of foam filled tubes based on the extensive experimental results.
In spite of these efforts made by the academic and industrial communities, challenges still exist
for accurately predicting the crushing force response and the folding geometry of the foam-filled
energy absorbers. Existing efforts in the literature are mainly on the numerical and experimental
approaches. Analytical approaches are relatively few, especially for the foam-filled frustum
configuration. Most of the existing analytical models gave the mean crushing force instead of the
instantaneous crushing force. Meanwhile, energy dissipation due to the interaction between the
159
foam and the shell is calculated via empirical relations determined from experimental or
numerical results.
Based on the above literature review, a thorough theoretical investigation into the foam/shell
interaction and its effect on the progressive deformation behaviour is necessary for the
development of foam-filled energy absorbers. Recently, we [24][25][31] developed a theoretical
model in which the foam-shell interaction was treated as a uniform pressure equaling to the foam
yield stress applied on the inside part of the fold. The model was able to give both the
instantaneous and the mean crushing force, and could correctly predict the effect of the
foam/shell interaction on the folding configuration. In this paper, we expand our work by
incorporating major improvements to carefully account for the comprehensive description of the
foam/shell interaction. Specifically, the proportion of the fold involved in the interaction is
updated, and the interfacial friction between the foam and the shell wall is taken into account.
This paper is organized as follows. Following this introduction, we give a brief introduction of
our analytical model based on the kinematically admissible mechanism in Section 7.2. In Section
7.3, we present the two modifications to our model and investigate their effects on the obtained
results. Some discussions and comparisons with the numerical and experimental work are
provided in Section 7.4, and the paper is concluded in Section 7.5.
7.2. The kinematically admissible model for foam-filled
frustum
The problem envisaged is shown in Figure 7.1. It contains a thin-walled frustum with taper angle
α filled with aluminum foam axially crushed by a rigid platen moving downwards. To simplify
the problem, a kinematically admissible progressive folding mechanism is followed based on the
following assumptions. The deformation is assumed to be axisymmetric. Each fold is composed
of several straight limbs, joined by the plastic hinges. The thickness of the frustum shell remains
constant during the deformation. The foam material is assumed to be perfectly plastic with zero
Poisson’s ratio. And the effect of foam densification is neglected.
160
Figure 7.1 A schematic diagram of the foam-filled frustum under axial loading
According to our previous analysis, the three-limb outward-inward folding is the most energy-
favorable folding scenario [31], and is taken for investigation in this work. As shown in Figure
7.1, each fold contains three limbs and four plastic hinges, except for the first fold which only
contains three plastic hinges. The letters a to f denote the control points, in which b, d and f are
neutral points that do not their radii during the folding process. Taking the first fold as an
example, the radii at the neutral points can be determined from the undeformed profile as:
𝑅𝑏 = 𝑅𝑎 + 𝑚ℎ1 𝑠𝑖𝑛 𝛼
(7.1)
𝑅𝑑 = 𝑅𝑏 + (1 − 𝑚)(ℎ1 + ℎ2) 𝑠𝑖𝑛 𝛼 (7.2)
𝑅𝑓 = 𝑅𝑏 + [(1 − 𝑚)ℎ1 + ℎ2 + 𝑚ℎ3] 𝑠𝑖𝑛 𝛼 (7.3)
where, m is the folding parameter defined as the ratio of the inward part of each fold limb, hi,
i=1,2,3 is the length of the ith fold limb.
The following relations are obtained comparing the deformed profile with the undeformed
profile. The detailed derivations are omitted and can be found in [18].
ℎ1 = 𝐾ℎ2, 𝐾 =1+𝑠𝑖𝑛𝛼
1−𝑠𝑖𝑛𝛼 (7.4)
ℎ3 = ℎ1 (7.5)
sin(𝜃2 − 𝛼) = 𝐾 sin(𝜃1 + 𝛼) − (𝐾 + 1) sin 𝛼 (7.6)
161
𝜃3 = 𝜃1 (7.7)
In the above equations, θi, i=1,2,3 is the rotation angle of the ith fold limb. The crushing distance,
uz, for the collapse of the ith fold can be determined viz.,
𝑢𝑧 = 𝑢𝑧(𝑖−1)
+ ℎ2[(1 + 𝐾(1 + 𝑎1𝑚)) 𝑐𝑜𝑠 𝛼 − 𝐾(1 + 𝑎1𝑚) 𝑐𝑜𝑠(𝜃1 + 𝛼)
− 𝑐𝑜𝑠(𝜃2 − 𝛼)]
(7.8)
where, ( )1−i
zu is the crushing distance at the moment that the previous (i -1)th fold is fully
collapsed. a1 is a switch parameter to differentiate the first fold from other folds.
𝑎1 = {1, first fold
0, subsequent fold (7.9)
The analytical expressions of the crushing force are derived from the basic principle of energy
conservation, i.e., the input work equals to the energy dissipation W during the progressive
folding of the foam-filled frustum. Since the input work is the integral of the crushing force F
over the crushing distance uz, the following equation holds:
∫𝐹𝑑𝑢𝑧 = 𝑊 (7.10)
The energy dissipation W comes from three sources: (i) the plastic deformation of the frustum
shell Wshell, (ii) the resistance of the foam Wfoam, and (iii) the interaction between the foam and
frustum shell Winter, i.e.,
𝑊 = 𝑊shell + 𝑊foam + 𝑊inter (7.11)
Wshell can be further decomposed into the part from the bending of the plastic hinges Wb, and the
part from the circumferential strain of the shell wall Wc. The first part is calculated as:
𝑊𝑏 = 𝑎1𝑑𝑊b_𝑏 + 𝑑𝑊b_𝑐 + 𝑑𝑊b_𝑒 + 𝑑𝑊b_𝑓 (7.12)
162
= 2𝜋𝑀𝑝 [∫ ((1 − 𝑎1)𝑅𝑏 + 𝑅𝑐)𝑑𝜃1
𝜃1
0
+ ∫ (𝑅𝑐 + 𝑅𝑒)𝑑𝜃2
𝜃2
0
+ ∫ (𝑅𝑒 + 𝑅𝑓)𝑑𝜃3
𝜃3
0
]
In the above equation, Mp is the plastic bending moment per unit circumferential length given as,
𝑀𝑝 =𝜋𝜎𝑦𝑡𝑡0
2
2√3 (7.13)
where, yt is the tensile yield strength of the shell and 0t is the thickness of the shell.
The energy dissipated by the circumferential strain is given as:
𝑊𝑐 = 𝑑𝑊c_Limb1 + 𝑑𝑊c_Limb2 + 𝑑𝑊c_Limb3
= ∫ (𝑎1𝑟∫ 𝜎yt𝑡0 |𝑑휀1
𝑑𝜃1| 𝑑𝐴1
𝑚ℎ1
0
+ ∫ 𝜎yt𝑡0 |𝑑휀1
𝑑𝜃1| 𝑑𝐴1
0
−(1−𝑚)ℎ1
)𝑑𝜃1
𝜃1
0
+ ∫ (∫ 𝜎yt𝑡0 |𝑑휀2
𝑑𝜃2| 𝑑𝐴2
(1−𝑚)ℎ2
0
+ 𝑟∫ 𝜎yt𝑡0 |𝑑휀2
𝑑𝜃2| 𝑑𝐴2
0
−𝑚ℎ2
)𝑑𝜃2
𝜃2
0
+ ∫ (𝑟 ∫ 𝜎yt𝑡0 |𝑑휀3
𝑑𝜃3| 𝑑𝐴3
𝑚ℎ3
0
)𝑑𝜃3
𝜃3
0
(7.14)
In the above equation, Ai, i=1,2,3 is the wall area of the ith fold limb. r is the ratio of the
compressive yield strength to the tensile yield strength.
The crushing force due to the resistance of the foam is given as Eq. (7.15).
𝐹foam = 𝐴foam𝜎foam (7.15)
where, foam is the plateau yield stress of the foam, Afoam is the cross-section of the frustum that
is under crushing. During development of each fold, Afoam is approximated to be constant foamA .
��foam =
1
4𝜋(𝑅𝑏 + 𝑅𝑓)
2
(7.16)
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where, bR is the radius of the smaller end. The dissipated energy due to the resistance of the
foam can be expressed as,
𝑊foam = ∫ 𝐹foam𝑑𝑢𝑧
𝑢𝑧
0
(7.17)
The interaction between the shell and the foam is represented by an internal pressure equal to the
yield strength of the foam foam acting on the inward portion of the fold. The energy
corresponding to the shell-foam interaction is calculated as the positive work by the pressure.
During the crushing of each fold, the energy increment is given by,
𝑑𝑊inter = 𝑑𝑊inter_Limb1 + 𝑑𝑊inter_Limb2 + 𝑑𝑊inter_Limb3
= ∫ 𝑝𝑑𝐴1(𝒏1 ⋅ 𝑑𝒖1)𝑚ℎ1
0
+ ∫ 𝑝𝑑𝐴2(𝒏2 ⋅ 𝑑𝒖2)0
−𝑚ℎ2
+ ∫ 𝑝𝑑𝐴3(𝒏3 ⋅ 𝑑𝒖3)𝑚ℎ3
0
(7.18)
where, ni, i=1,2,3 is the unit vector normal to the shell surface of the ith limb. ui, i=1,2,3 is the
displacement of an arbitrary point at the ith limb.
The energy contributed by the shell-foam interaction is then determined by,
𝑊inter = ∫𝑑𝑊inter (7.19)
The instantaneous crushing force can be derived from Eq. (7.10) as,
𝐹 =
𝑑𝑊
𝑑𝑢𝑧
(7.20)
Similar to the decomposition of the energy dissipation, the instantaneous crushing force can also
be decomposed into three parts corresponding to the contributions from the shell, the foam, and
the foam-shell interaction; viz.,
𝐹 = 𝐹shell + 𝐹foam + 𝐹inter (7.21)
where
164
𝐹shell =𝑑𝑊shell
𝑑𝑢𝑧 , 𝐹foam =
𝑑𝑊foam
𝑑𝑢𝑧 and 𝐹inter =
𝑑𝑊inter
𝑑𝑢𝑧 (7.22)
During the collapse of each fold, foamF is assumed to be constant foam foamA . The other two parts
can be further deduced as Eq.(7.23) noting that θ2 relates to θ1 and is not an independent variable.
𝐹shell =
𝑑𝑊shell
𝑑𝜃1
𝑑𝑢𝑧
𝑑𝜃1
, 𝐹inter =
𝑑𝑊inter
𝑑𝜃1
𝑑𝑢𝑧
𝑑𝜃1
(7.23)
The mean crushing force is defined by Eq. (24) and can also be decomposed in the same way as
Eqs. (7.22) and (7.23).
�� =∫𝐹𝑑𝑢𝑧
𝛥𝑢𝑧
(7.24)
The above deductions involve two undetermined parameters: the fold length h (here h refers to
either h1 or h2 since they depend on each other) and the folding parameter m. According to the
upper bound theorem of plasticity, the selected m and h should minimize the mean crushing
force, i.e.
��selected = 𝑚𝑖𝑛(��(𝑚, ℎ)) (7.25)
Therefore, m and h can be determined from the following equations.
( ) ( )shell inter, ,0
F h m F h mF
h h h
= + =
(7.26)
𝜕��
𝜕𝑚=
𝜕��shell(ℎ,𝑚)
𝜕𝑚+
𝜕��inter(ℎ,𝑚)
𝜕𝑚= 0
(7.27)
The detailed expressions for the instantaneous and the mean crushing force can be found in our
previous paper [24].
165
7.3. Improved foam-shell interaction
7.3.1. Foam-Shell Interaction Mechanism
In this section, we extend our previous kinematically admissible analytical model by
incorporating two critical improvements in terms of the foam/shell interaction. First, the
proportion of the fold involved in the foam/shell interaction is improved. Second, the interfacial
friction between the shell and the foam is taken into account.
In our previous model, the interaction between the foam and shell is assumed to be applied on
the inside portion of the fold ligaments, i.e. sections ab, de, ef. From Figure 7.1, it can be seen
that two small sections indicated with red color are not taken into account in our previous model.
The actual fold portion involved in the foam/shell interaction is somewhat larger. Our current
improved model takes the two small sections into account in the interaction region. The
respective lengths of the two sections are determined using the following expressions:
𝐿1∗ = 𝑚𝑖𝑛 (
𝑠𝑖𝑛 𝛼
𝑠𝑖𝑛𝜃1
[ℎ1(𝑐𝑜𝑠 𝛼 − 𝑐𝑜𝑠(𝜃1 + 𝛼)) + ℎ2(𝑐𝑜𝑠 𝛼 − 𝑐𝑜𝑠(𝜃2 − 𝛼))], (1
− 𝑚)ℎ1) (7.28)
𝐿2∗ = 𝑚𝑖𝑛 (
𝑠𝑖𝑛 𝛼
𝑠𝑖𝑛𝜃2𝑚[ℎ1(𝑐𝑜𝑠 𝛼 − 𝑐𝑜𝑠(𝜃1 + 𝛼)) + ℎ2(𝑐𝑜𝑠 𝛼 − 𝑐𝑜𝑠(𝜃2 − 𝛼))], (1
− 𝑚)ℎ2) (7.29)
According to the literature [8][14], the interaction between the foam and the shell is responsible
for the enhanced energy absorption capacity of the foam-filled structures. However, the
mechanisms of the interaction have not been fully understood. Some authors attributed the
interaction effect to the increase in the number of folds, the local densification of the foam, or the
multi-axial instead of uniaxial deformation of the foam. These explanations are either
phenomenological or difficult to be quantified. In our previous work, the interaction was
represented by the work done in compressing the foam by the collapsing shell, in which case the
foam/shell interaction could be well quantified and its influence on the fold mode could be
166
correctly predicted. Another issue to be considered is the property of the interface which also
plays an important role on the energy absorption performance. Experiments showed that foam-
filled tubes with a fully bonded interface could absorb higher energy than the non-bonded
counterparts. The folding mode can also be influenced by the bonding condition of the interface.
Bearing in mind that the fully bonded interface is the extreme case of the interface friction
condition, we take into account the friction between the foam and shell in this regard. The energy
dissipation due to the foam/shell interaction is now composed of two parts: (i) part from the
penetration of the shell into the foam and (ii) part from the friction between the foam and the
shell, viz.:
𝑊inter = 𝑊indent + 𝑊frict (7.30)
The crushing force due to the foam/shell interaction can also be decomposed into two parts, viz.:
𝐹inter = 𝐹indent + 𝐹frict (7.31)
Next, we give detailed deductions for the crushing force from the two parts of energy dissipation,
respectively.
7.3.2. Interaction I: penetration of the shell into the foam
The energy dissipation due to the first mechanism of foam/shell interaction, i.e. the penetration
of the shell into the foam, can be calculated utilizing Eq. (7.18), viz.:
𝑑𝑊indent = ∫ 𝑝𝑑𝐴1(𝒏1 ⋅ 𝑑𝒖1)𝑚ℎ1
−𝐿1∗
+ ∫ 𝑝𝑑𝐴2(𝒏2 ⋅ 𝑑𝒖2)𝐿2∗
−𝑚ℎ2
+ ∫ 𝑝𝑑𝐴3(𝒏3 ⋅ 𝑑𝒖3)𝑚ℎ3
0
(7.32)
It is noted that the integration limits are updated from Eq. (7.18) to correctly represent the
portion of the fold subject to the foam/shell interaction. For the first limb,
𝒏1 = [−𝑐𝑜𝑠(𝜃1 + 𝛼)
−𝑠𝑖𝑛(𝜃1 + 𝛼)], (7.33)
167
𝒖1 = [𝑢𝑅
𝑢𝑧] = [
−𝐿(𝑠𝑖𝑛(𝜃1 + 𝛼) − 𝑠𝑖𝑛 𝛼)
(ℎ1 + 𝐿)(𝑐𝑜𝑠(𝜃1 + 𝛼) − 𝑐𝑜𝑠 𝛼) + ℎ2(𝑐𝑜𝑠(𝜃2 − 𝛼) − 𝑐𝑜𝑠 𝛼)],
𝑑𝐴1 = 2𝜋[𝑅𝑏 − 𝐿 𝑠𝑖𝑛(𝜃1 + 𝛼)]𝑑𝐿
For the second ligament,
𝒏2 = [−𝑐𝑜𝑠(𝜃2 − 𝛼)
𝑠𝑖𝑛(𝜃2 − 𝛼)],
𝒖2 = [𝑢𝑅
𝑢𝑧] = [
𝐿(𝑠𝑖𝑛(𝜃2 − 𝛼) + 𝑠𝑖𝑛 𝛼)
𝑚ℎ1(𝑐𝑜𝑠(𝜃1 + 𝛼) − 𝑐𝑜𝑠 𝛼) + (𝑚ℎ2 + 𝐿)(𝑐𝑜𝑠(𝜃2 − 𝛼) − 𝑐𝑜𝑠 𝛼)],
𝑑𝐴2 = 2𝜋[𝑅𝑑 + 𝐿 𝑠𝑖𝑛(𝜃2 − 𝛼)]𝑑𝐿
(7.34)
For the third ligament,
𝒏3 = [−𝑐𝑜𝑠(𝜃1 + 𝛼)
− 𝑠𝑖𝑛(𝜃1 + 𝛼)],
𝒖3 = [𝑢𝑅
𝑢𝑧] = [
−𝐿(𝑠𝑖𝑛(𝜃1 + 𝛼) − 𝑠𝑖𝑛 𝛼)
𝐿(𝑐𝑜𝑠(𝜃1 + 𝛼) − 𝑐𝑜𝑠 𝛼)],
𝑑𝐴3 = 2𝜋[𝑅𝑓 − 𝐿 𝑠𝑖𝑛(𝜃1 + 𝛼)]𝑑𝐿
(7.35)
Substituting the above expressions into Eq. (7.32), we get:
( ) ( ) ( )( ) ( )( )
( ) ( ) ( )( ) ( )( )
( )( )
1
*1
*2
2
2
2
1 1 1 1 2 1 2 2 1
2
indent 2 1 1 2 1 2 2 2 2
1 10
sin d sin sin d sin
d 2 sin sin d sin d sin
sin
mh
bL
L
dmh
mh
f
Ld h h R L dL
W p Ld mh mh R L dL
Ld R L dL
−
−
+ + + + − − +
= − + + − + − + − + − +
(7.36)
After a tedious deduction, we get:
168
( ) ( ) ( )( )
( ) ( )( )
( ) ( )
( ) ( )( )
( )
indent 1indent
1
2 2 3*2 *3
1 1 1 1 1
1
2 3* 2 * 3
2 2 2 222
1
21 1 2 2
11
2 *2
1 1*
1 1 1 1
d d
d d
2sin
2 2 3
dsin
d 2 3
d2sin sin *
dd d
sin sin2
z
b f
d
z
b
WF
u
mh L mh mh LR R
mh L mh LR
ph h
u
mh Lmh L R
=
− + + − +
− + + − −
+ + + −=
−+ + − +
( ) ( )( )
( )2 * 2
2 2*
2 2 2 2sin sin2
d
mh Lm mh L R
− − − + − −
(7.37)
7.3.3. Interaction II: interfacial friction between the shell and the foam
During the crushing of each fold, the energy increment contributed by the foam/shell interaction
is given by:
𝑑𝑊frict = 𝑑𝑊frict_Limb1 + 𝑑𝑊frict_Limb2 + 𝑑𝑊frict_Limb3
= ∫ 𝑐𝑓𝑝𝑑𝐴1|𝒕1 ⋅ 𝑑𝒖1|𝑚ℎ1
−𝐿1∗
+ ∫ 𝑐𝑓𝑝𝑑𝐴2|𝒕2 ⋅ 𝑑𝒖2|𝐿2∗
−𝑚ℎ2
+ ∫ 𝑐𝑓𝑝𝑑𝐴3|𝒕3 ⋅ 𝑑𝒖3|𝑚ℎ3
0
(7.381)
where, it , i=1,2,3 is the unit vector tangential to the shell surface of the ith limb. ui and Ai,
i=1,2,3 are given in Eqs. (7.33)-(7.35). For the first limb,
𝒕1 = [−𝑠𝑖𝑛(𝜃1 + 𝛼)
𝑐𝑜𝑠(𝜃1 + 𝛼)],
(7.39)
For the second limb,
𝒕2 = [−𝑠𝑖𝑛(𝜃2 − 𝛼)
− 𝑐𝑜𝑠(𝜃2 − 𝛼)],
(7.40)
For the third limb,
169
𝒕3 = [− 𝑠𝑖𝑛(𝜃1 + 𝛼)
𝑐𝑜𝑠(𝜃1 + 𝛼)],
(7.41)
Substituting the above expressions into Eq. (7.38), we get:
𝑑𝑊frict = 2𝜋𝑐𝑓𝑝{∫ (ℎ2 𝑠𝑖𝑛(𝜃1 + 𝜃2) 𝑑𝜃2)(𝑅𝑏 − 𝐿 𝑠𝑖𝑛(𝜃1 + 𝛼))𝑑𝐿
𝑚ℎ1
−𝐿1∗
+∫ (𝑚ℎ1 𝑠𝑖𝑛(𝜃1 + 𝜃2) 𝑑𝜃1)(𝑅𝑑 + 𝐿 𝑠𝑖𝑛(𝜃2 − 𝛼))𝑑𝐿𝐿2∗
−𝑚ℎ2
},
(7.42)
After a tedious deduction, we get:
𝐹frict =
𝑑𝑊frict
𝑑𝜃1
𝑑𝑢𝑧
𝑑𝜃1
=2𝜋𝑐𝑓𝑝 𝑠𝑖𝑛(𝜃1 + 𝜃2)
𝑑𝑢𝑧
𝑑𝜃1
ە
۔
𝑑𝜃2ۓ
𝑑𝜃1ℎ2 ((𝑚ℎ1 + 𝐿1
∗ )𝑅𝑏 −(𝑚ℎ1)
2 − 𝐿1∗ 2
2𝑠𝑖𝑛(𝜃1 + 𝛼))
+𝑚ℎ1 ((𝑚ℎ2 + 𝐿2∗ )𝑅𝑑 −
(𝑚ℎ2)2 − 𝐿2
∗ 2
2𝑠𝑖𝑛(𝜃2 − 𝛼))
(7.43)
If we neglect *
1L and *
2L , i.e., assuming the foam/shell interaction is applied to the three fold
segments ab, de, ef, the contribution due to penetration of the shell becomes exactly the same as
given in our previous work. The contribution due to friction between the shell and the foam can
be reduced as follows,
𝐹frict =2𝜋𝑐𝑓𝑝 𝑠𝑖𝑛(𝜃1 + 𝜃2)
𝑑𝑢𝑧
𝑑𝜃1 ە۔
ۓ𝑑𝜃2
𝑑𝜃1ℎ2 (𝑚ℎ1𝑅𝑏 −
(𝑚ℎ1)2
2𝑠𝑖𝑛(𝜃1 + 𝛼))
+𝑚ℎ1 (𝑚ℎ2𝑅𝑑 −(𝑚ℎ2)
2
2𝑠𝑖𝑛(𝜃2 − 𝛼))
(7.44)
170
7.4. Results and Discussions
7.4.1. Validation of the proposed kinematically admissible model
Prior to analyzing the effect of the revised foam/shell interaction on the progressive collapse
behavior, we preceded to validate our analytical models. In this regard, we compared our
theoretical results with our experiments and FE simulations. The experimental tests were carried
out using the universal testing machine model CSS-44000 from Changchun Testing Machine
Institute. The foam-filled frustum samples were provided by Shanghai Shili Machineries Co.,
Ltd. The tapered outer shell was made of aluminum AL6061T6, manufactured by a CNC lathe
machine with a polycrystalline diamond (PCD) tool. The tapered core was made of 10% closed-
cell AL6061 foam, also processed using the PCD tool. The foam core was inserted into the
frustum shell with an exact fit. No glue was used on the interface. The geometry parameters are
given in Table 7.1. The sample was annealed at a temperature of 500 Celsius degrees for 1 hour
followed by a gradual cooling at a rate of approximately 100 degrees per hour to room
temperature before testing. The test setup is shown in Figure 7.2. The sample was supported on a
smooth table and compressed by a rigid platen at a speed of 4 mm/min. In addition to the
crushing test, tensile test was conducted for a dog-bone sample of the shell material, and
compression test was conducted for a cube sample of the foam material to determine the material
properties. The obtained stress-strain curves are shown in Figure 7.3(a) and (b) for the shell and
the foam, respectively.
Table 7.1 Geometric dimensions of the foam-filled frustum sample
α (degree) Rtop (mm) L (mm) t0 (mm)
3 45 130 1.8
171
Figure 7.2 Experimental setup of the progressive crushing of the foam-filled frustum, with
the enlarged views showing the sample before and after the test
(a) (b)
Figure 7.3 Stress versus strain curves for the material of (a) the frustum shell, and (b) the
foam core, the dashed line indicating the characteristic yield stress
172
The FE simulations were carried out for the same geometry of the test sample using the explicit
FE code ANSYS/LS-DYNA. The frustum shell was discretized using Belytschko–Lin–Tsay 4-
noded shell elements (type Shell 163). The foam core was discretized using 8-noded solid
elements with reduced integration (type Solid 164). Different element sizes were tested for the
purpose of balancing the computational cost and accuracy. The adopted element size was
approximately t0 and 1.5t0 for the foam and the shell, respectively. The material properties of the
shell were modelled using the conventional von-Mises plasticity model with isotropic hardening.
The foam material was modeled using the honeycomb model that assumes uncoupled relations
between the corresponding components of the stress and strain tensors. All the constitutive
parameters were obtained from the experimental curves and are summarized in Table 2. The
configuration of the FE model is shown in Figure 7.4(a). The foam-filled frustum was clamped at
the bottom and loaded by a rigid platen that was moving downwards at speed of 1 m/s at the top.
The node-to-surface contact algorithm was used for the contact between the frustum shell and the
rigid platen. The surface-to-surface contact algorithm was used for the contact between the shell
and the foam filler, as well as between the foam filler and the rigid platen. In addition, all regions
of the frustum shell were examined via self-contact algorithm to prevent self-penetration. The
half configuration of the FE model is shown in Figure 7.4(a), with the collapsed sample shown in
Figure 7.4(b).
Table 7.2 Material constants for the FE model
Density
(103 kg/m3)
Elastic modulus
(GPa)
Poisson’s
ratio
Characteristic
yield stress
(MPa)
Shell 2.7 70 0.334 125
Foam 0.27 70 0 4
173
Figure 7.4 FE model for the progressive crushing of the foam-filled frustum, (a) half of the
model, and (b) the collapsed configuration
The theoretical results are predicted by our analytical model with the same geometry and
material constants as in the experiments and FE simulations. The complete instantaneous
crushing curve is obtained by repeatedly using Eqs. (7.21) - (7.23) to describe the deformation of
each fold and updating the geometrical parameters from one fold to another. Considering the
perfect plasticity assumption in the analytical model, the yield stress of the shell material is taken
as the characteristic yield stress σc, which is the average of the 0.2% yield stress σ0.2 and the
ultimate stress at breakage σu as proposed by Hanssen et al. [14]. The obtained theoretical results
of our analytical model are compared with the experimental data and the FE numerical results in
Figure 7.5(a). It shows that the crushing forces predicted by our analytical model are in good
agreement with those from the experimental tests and FE simulations. All curves are composed
of multiple fluctuations indicating the progressive collapse of the multiple folds. The crushing
force increases gradually from one fold to another because of the increase of the cross-sectional
area under crushing. The tendency is more clearly seen if we plot the history curves of the mean
crushing force in Figure 7.5(b). Some deviations exist between the analytical crushing force and
the experimental or numerical results due to the idealization of the folding geometry and
neglecting the densification of foam in the analytical model. Overall, the agreements between our
174
analytical predictions and the experimental or FE numerical results provide validation for our
analytical model.
(a) (b)
Figure 7.5 Comparison of our analytical model with the experiments and FE simulations
for (a) instantaneous crushing force, and (b) mean crushing force
7.4.2. Effect of the revised fold proportion for interaction on the crush
behaviour
The effect of updating the fold proportion for interaction can be seen in Figure 7.6. In this and
subsequent figures, the crushing force is normalized by 2𝜋𝑅𝑏𝑡0𝑟𝜎𝑦𝑡 + 𝜋𝑅𝑏2𝜎foam, and the fold
length is normalized by ℎ0 = √2𝑅𝑏𝑡0. It shows that both the crushing forces contributed by the
shell penetration and by the foam/shell interface friction increase after updating the fold
proportion involved in the foam-shell interaction. The difference is more evident for a larger fold
length and a smaller folding parameter m. Nevertheless, within the common ranges of the fold
length h (around h0) and folding parameter m (around 0.5), the difference is not significant.
Figure 7.7 plots the instantaneous force contributed by different sources versus the crushing
distance for h=0.8h0 and m=0.4 during collapse of one fold. The solid lines correspond to results
of the updated model, while the dash lines correspond to those of the previous model. It shows
that the effect of updating the interaction portion is negligible.
175
(a) (b)
Figure 7.6 Effect of the fold portion update on the mean crushing force contributed by (a)
shell penetration, and (b) interfacial friction for different fold length h and different folding
parameter m
Figure 7.7 Instantaneous force contributed by different sources versus the crushing
distance
7.4.3. Effect of the foam/shell friction on energy absorption
Next, we investigate the effect of the foam/shell interfacial friction. In Figure 7.5, we compare
the obtained instantaneous crushing curves for the modified model with friction and our earlier
model without friction. It shows that the crushing force is larger when considering the interfacial
friction. This result is consistent with the FE predictions.
176
In Figure 7.8(a), we plot the normalized mean crushing force versus folding parameter m for
different interfacial friction conditions. The empty frustum case is also added for comparison,
and only the contributions from the shell and the foam/shell interaction are taken into account for
comparison purposes. In this graph, we can see that a larger friction coefficient results in a higher
mean crushing force. Meanwhile, the mean crushing force takes its minimum at m around 0.5,
which is the energy favourable value of m according to the upper bound theorem of plasticity.
The obtained energy favourable m is smaller as the friction coefficient becomes larger. The
energy favourable m corresponding to the empty frustum is obviously larger than all the foam-
filled cases investigated.
(a) (b)
Figure 7.8 Variation of (a) mean crushing force, and (b) fold length with the folding
parameter m for different foam/shell interfacial conditions
Figure 7.8(b) plots the obtained fold length versus folding parameter m for different interfacial
friction conditions. It shows that a larger friction coefficient results in a smaller fold length and
the empty frustum leads to the longest fold length. As the folding parameter increases, a
minimum fold length exists at m close to its energy favourable value in Figure 7.8(a). It indicates
that a stronger interface friction can lead to a smaller folding parameter and a smaller fold length
from the energy viewpoint.
Figure 7.9 compares the predicted energy favourable fold length for different interfacial
conditions with the experiment result. It indicates that the analytical model without the inclusion
177
of the foam/shell interfacial friction over-estimates the fold length when compared with the
experimental results.
We also investigated the effect of the taper angle on the crushing force contributed by the
foam/shell interaction. Figure 7.10 plots the instantaneous crushing force versus the crushing
distance for different taper angles during the collapse of one fold. Figure 7.10(a) and (b)
correspond to the contributions of the penetration of the foam by the shell and the friction
between the foam and the shell, respectively. It shows that the penetration effect results in a
larger crushing force as the taper angle increases, while the friction effect is the opposite. In
addition, the instantaneous crushing force contributed by the penetration decreases
monotonically during the collapse of the fold. While the instantaneous crushing force contributed
by the interfacial friction first increase and then decrease, except for the zero taper angle case
which shows monotonic decrease.
Figure 7.9 Comparison of the predicted fold length with the experiment result
178
(a) (b)
Figure 7.10 Effect of taper angle on the instantaneous crushing force contributed by (a) the
penetration of foam by shell, and (b) the friction between the foam and shell
7.5. Conclusions
In this paper, we extend our previous analytical work on the progressive collapse of foam-filled
frustum by considering two critical parameters (i) the foam/shell interfacial friction and (ii) a
revised fold interaction region. The results indicate the insignificant influence of revised
interaction region on the energy absorption. However, consideration of the foam/shell interfacial
friction leads to an increase of the energy absorption and the crushing force, as well as a decrease
of the predicted folding parameter and fold length. The consideration of the foam/shell friction
improves the accuracy of the analytical predictions of the model. This paper presents a first
effort in the analytical modeling of the effect of foam/shell interface friction on the progressive
collapse of foam-filled frustum.
Acknowledgements
This research effort was made possible by NPRP Grant # (7-236-3-053) from the Qatar National
Research Fund (a member of Qatar Foundation), National Natural Science Foundation of China
(11402173; 11772231), and Fundamental Research Funds for the Central Universities
(1500219128). Additional support from the Natural Sciences and Engineering Research Council
of Canada and Shanghai Supercomputer Center is also gratefully acknowledged.
179
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Chapter 8.
Conclusions, Contributions and Future Work
8.6. Summary of Research Findings
The findings of this thesis can be summarized as follows:
1. Simulations of occupant response using 32 km/h frontal, rear and lateral collisions reveal
that in frontal impact the interspinous ligament and capsular ligament are vulnerable to
injury. In rear collision, the anterior longitudinal ligament was prone to injury at the upper
and mid cervical spine. Bone fracture was also observed in rear impact due to bone-to-bone
contact and anterior longitudinal ligament excessive elongation. In lateral collision, the
capsular ligament was at risk of injury in the mid and lower cervical spine. Vertebral
fracture was also observed at the lower cervical spine.
2. A novel experimental head-neck prototype was developed and used to validate the
multibody dynamics (MBD) and finite element (FE) models. The response of the
experimental prototype agrees well with the response of the MBD and FE predictions.
3. The proper positioning of the head restraint is of great importance to support the head during
neck extension. Poorly adjusted head restraint can increase the forces to which the neck is
subjected; thus, increasing the possibility of occupant injury. Furthermore, softer head
restraint materials reduce the head acceleration and neck ligament elongation compared to
stiffer materials. The seat belt is crucial in rear collisions to prevent occupant ramping along
the seat back and maintain a good position of the head with respect to the head restraint.
4. The addition of aluminum foam to thin-walled aluminum columns subjected to axial
compressive loading increases the amount of energy absorbed due to the energy absorbed by
the foam and the energy dissipated in the interfacial friction between the foam and the
column.
182
5. The aluminum foam increases the stability thin-walled cylinders during axial compression
and ensures an axisymmetric folding mode of collapse. It also maintains the stability of the
thin-walled cylinder under oblique loading and prevents global buckling.
6. In rear-end collisions, the head restraint protects the neck during extension. However, it
increases the severity of the neck flexion leading to possible injury of the interspinous
ligament and the intervertebral disc.
7. Deploying the front airbag during a rear collision can reduce the head horizontal and vertical
displacements in rebound by 56% and 90%, respectively.
8.7. Thesis Contributions
1. Developed novel one and two DOF multibody dynamics analytical models accounting for
non-linear joint stiffness and linear joint displacement, which led to relatively more accurate
occupant kinematics during frontal, rear and lateral collisions.
2. Developed high resolution and detailed non-linear FE simulations of vehicular rear
collisions accounting for: seat belt, airbag, seat cushion stiffness, and head restraint position.
3. Designed, developed and analyzed a novel shock absorber that would lead to a dramatic
reduction in energy transferred to the occupant.
4. My entire research work provides a detailed account of vehicular rear collisions that has
been not accurately modeled in the literature, for example the literature employs artificial
seat accelerations. In my work, realistic seat velocities and accelerations were obtained from
an experimentally validated FE model.
5. A realistic head-neck interface using 3D printing was developed to validate both the MBD
and FE predictions. This will pave the way for future research not merely concerned with
rear collision but also frontal and lateral collisions.
6. I have devised new instrumentation and measurement techniques to facilitate the
measurement of the trajectory of the center of mass of the head during rear collisions using
modified sled design.
183
7. The results of the entire work should lead to enhanced safety of motor vehicles.
8.8. Future Work
1. In this research effort, passive shock absorbers have been suggested. However, it is
suggested to consider active shock absorbers using magnetic rheological fluid to facilitate
varied stiffness requirements.
2. We have not considered the design of the seat as an integral part of the crashworthiness
analysis. It is suggested to carry out an integrated design approach in which the seat
behavior is considered amongst crashworthy devices such as airbag and magnetorheological
shock absorbers.
3. It is also suggested that future work could consider morphing of car body to accommodate
for greater energy absorption. For example, shock absorption could be avoided, if we are
able to create a crumbled zone during rear collisions in both the target and the bullet
vehicles.
4. Improve and further develop the head-neck experimental prototype. Specifically, improving
the Neck Stability System to replicate muscles active and passive responses and accounting
for neck ligaments other than the anterior and posterior longitudinal ligaments.
5. This study identified the kinetics and kinematics of occupant response during frontal, rear
and lateral collisions. It did not, however, relate the models output to current injury criteria.
This was beyond the scope of this study. Future work should consider the relationship of the
kinematic and kinetic predictions and measurements to injury criteria.