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UNIVERSITY OF CINCINNATI Date:___________________
I, _________________________________________________________, hereby submit this work as part of the requirements for the degree of:
in:
It is entitled:
This work and its defense approved by:
Chair: _______________________________ _______________________________ _______________________________ _______________________________ _______________________________
DESIGN AND DEVELOPMENT OF AN ENERGY ABSORBING SEAT AND
BALLISTIC FABRIC MATERIAL MODEL TO REDUCE CREW INJURY
CAUSED BY ACCELERATION FROM MINE/IED BLAST
A Thesis submitted to the
Division of Research and Advanced Studies
of the University of Cincinnati
in partial fulfillment of the
requirements for the degree of
MASTER OF SCIENCE
in the Department of Mechanical, Industrial and Nuclear Engineering
of the College of Engineering
2006
by
Gaurav Nilakantan
Bachelor of Engineering (B.E.)
Visveswaraiah Technological University, India, 2003
Committee Chair: Dr. Ala Tabiei
Abstract
Anti tank (AT) mines pose a serious threat to the occupants of armored vehicles.
High acceleration pulses and impact forces are transmitted to the occupant
through vehicle-occupant contact interfaces, such as the floor and seat, posing
the risk of moderate injury to fatality.
The use of an energy absorbing seat in conjunction with vehicle armor plating
greatly improves occupant survivability during such an explosion. The axial
crushing of aluminum tubes over a steel rail constitutes the principal energy
absorption mechanism. Concepts to further reduce the shock pulse transmitted
to the occupant are introduced during the study, such as the use of a foam
cushion and an inflatable airbag cushion.
The explicit non-linear finite element software LS-DYNA© is used to perform all
numerical simulations. Vertical drop testing of the seat structure with the
occupant are performed for comparison with experimental data after which
simulations are run, that utilize input acceleration pulses comparable to a mine
blast under an armored vehicle. The occupant is modeled using a 5th percentile
HYBRID III dummy. Data such as lumbar load, neck moments, hip and knee
moments, and head and torso accelerations are collected for comparison with
known injury threshold values to assess injury.
Numerical simulations are also conducted of the impact of a dummy’s feet by
a rigid wall whose upward motion is comparable to an armored vehicle’s
reaction to a mine blast directly underneath it. A 50th percentile HYBRID III
dummy is used in various seated positions. The input pulses that control the
motion of the rigid wall are varied in a step wise manner to determine the effect
on extent of injury. Data such as hip and knee moment, femoral force and foot
acceleration are collected from the dummy and compared to injury threshold
values from various references. By numerically simulating the mine blast under a
vehicle, the significant cost of conducting destructive full scale tests can be
avoided.
A simple numerical formulation is presented, to predict the deceleration
response during dynamic axial crushing of cylindrical tubes. The formulation
uses an energy balance approach and is coded in the high level language
MATLAB©. It can track the histories of plastic work, kinetic energy, and dynamic
crushing load during the crushing process, and finally yields the peak
acceleration magnitude, which can then be calibrated and used for injury
assessment and survivability studies by comparing with allowable values for
human occupants. Further, the geometric and material properties of the tube
can be varied to study its response during the dynamic axial crushing.
The impact resistance of high strength fabrics makes them desirable in
applications such as protective clothing for military and law enforcement
personnel, protective layering in turbine fragment containment, armor plating of
vehicles, and other similar applications involving protection resistance against a
high velocity projectile. Such fabrics, especially Kevlar©, Zylon©, and Spectra©,
can be used in the energy absorbing seat as a cushion cover for the high
density foam, to prevent tearing by unexpected shrapnel during an explosion
underneath the armored vehicle. The protective fabric can also be used as a
protective vest for the dummy occupant and as a liner inside the vehicle hull. A
material model has been developed to realistically simulate ballistic impact of
loose woven fabrics with elastic crimped fibers. It is based upon a
micromechanical approach that includes the architecture of the fabric and the
phenomenon of fiber reorientation, and excludes strain rate sensitivity as the
yarns are simplified as elastic members. The material model is implemented as
a FORTRAN© subroutine and integrated into the explicit, non-linear dynamic
finite element code LSDYNA© as a user-defined material model (UMAT). Results
of axial fabric tests run in LSDYNA© using this material model agree well with
other models. This justifies the use of a simplistic, computationally inexpensive
material model to realistically simulate ballistic impact.
Acknowledgements
I am indebted to my advisor Prof. Ala Tabiei for giving me a chance to work with
him on all his fascinating research, for believing in me and constantly guiding
and encouraging me.
I express my utmost gratitude to my parents S. Nilakantan and Nirmala
Nilakantan for all that they have done for me, for all their love, support, sacrifice,
and encouragement.
I sincerely thank committee members, Prof. Jay Kim, and Prof. David Thompson
for their presence on my committee and their suggestions.
I am also grateful to the University of Las Vegas-Nevada for funding part of this
research, as well as the Ohio Supercomputing Center for their high-speed
computing support.
I thank my colleague Srinivasa Vedagiri Aminijikarai for all his technical advice
and support.
Contents
a. List of Figures…………………………………………………………………………….... i
b. List of Tables……………………………………………………………………………….. vi
1. Introduction 1
1.1 Background………………………………………………………………………….. 1
1.2 Literature Review…………………………………………………………………….. 2
1.2.1 Energy Absorbing Seat………………………………………………….. 2
1.2.2 Foot Impact during IED/Mine blast…………………………………….. 4
1.2.3 Human Injury Criteria……………………………………………………. 5
1.2.4 Dynamic Axial Crushing of Circular Tubes……………………………. 13
1.2.5 Ballistic Impact of Dry Woven Fabrics…………………………………. 14
1.3 Scope of Work……………………………………………………………………….. 26
1.4 Outline of Thesis……………………………………………………………………… 27
2. Energy Absorbing Seat 29
2.1 Preliminary Design…………………………………………………………………… 29
2.2 Dynamic Axial Crushing of the Aluminum Tubes………………………………… 32
2.2.1 Techniques to reduce the initial crushing load of a tube…………... 36
2.3 Additional energy absorbing elements…………………………………………… 39
2.3.1 Low Density Foam Cushion…………………………………………….. 39
2.3.2 Airbag Cushion………………………………………………………….. 42
2.4 Shock Pulses Applied to the Structure…………………………………………….. 44
2.4.1 Impact After Free Fall…………………………………………………… 44
2.4.2 Mine Blast………………………………………………………………… 45
2.5 Filtering of Data……………………………………………………………………… 46
2.6 Validation of Initial EA Seat Simulations…………………………………………… 47
3. Results and Discussion: Energy Absorbing Seat 49
3.1 Test Matrix…………………………………………………………………………….. 49
3.2 Simulation Setup…………………………………………………………………….. 50
3.3 EA seat with GEBOD dummy subjected to vertical drop testing………………. 51
3.4 EA seat with HYBRID III dummy subjected to vertical drop testing…………….. 54
3.5 EA seat with GEBOD dummy subjected to mine blast testing…………………. 56
3.6 EA seat with HYBRID III dummy subjected to mine blast testing……………….. 59
3.7 Improved Modeling of the EA seat structure…………………………………….. 60
3.8 Effect of Aluminum Yield Strength on the Simulations…………………………... 61
3.9 Stages of Crushing of the Aluminum Crush Tube………………………………... 62
3.9.1 Stages of crushing for the original EA seat model…………………… 63
3.9.2 Stages of crushing for the improved EA seat model………………... 64
3.9.3 Shape of the Crushed Tube when Modeled with Solid Elements….. 64
3.10 Final EA Seat Design for use in full scale Vertical Drop Testing and mine
Blast Testing……………………………………………………………………………….
65
3.10.1 Vertical Drop Testing…………………………………………………… 66
3.10.2 Mine Blast Testing………………………………………………………. 68
3.11 New EA Mechanism……………………………………………………………….. 69
3.12 Conclusions………………………………………………………………………… 71
3.13 Scope for Further Work…………………………………………………………….. 73
4. Impact of Foot during IED/Mine Blast 74
4.1 Numerical Setup and Methodology………………………………………………. 74
4.2 Numerical Results and Discussion…………………………………………………. 78
4.2.1 Hybrid III dummy in a sitting straight position…………………………. 80
4.2.2 Hybrid III dummy in a driving position…………………………………. 84
4.3 Parametric Study…………………………………………………………………….. 89
4.4 Conclusions………………………………………………………………………….. 93
4.5 Scope for Further Work……………………………………………………………… 94
5. Dynamic Axial Crushing of Circular Tubes: Numerical Formulation 95
5.1 Need for a Simple Numerical Formulation……………………………………….. 95
5.2 Theory and Formulation…………………………………………………………….. 97
5.3 Results and Discussion……………………………………………………………… 107
5.4 Conclusions………………………………………………………………………….. 114
5.5 Scope for further work………………………………………………………………. 115
6. Ballistic Impact of Woven Fabrics 116
6.1 Description of the Material Model………………………………………………… 116
6.2 The Representative Volume Cell of the Model…………………………………... 116
6.3 Elastic Model………………………………………………………………………… 118
6.4 Numerical Results - Fabric Strip Testing…………………………………………… 125
6.4.1 Elastic model fabric strip test………………………………………….. 127
6.4.2 Viscoelastic model fabric strip test……………………………………. 129
6.4.3 Comparison between Elastic and Viscoelastic model results……… 130
6.5 Conclusions………………………………………………………………………….. 132
6.6 Scope for Further Work……………………………………………………………… 133
Appendix I
Source code for the numerical formulation of dynamic axial crushing of circular
tubes………………………………………………………………………………………. 134
Appendix II
Source code for the incremental constitutive equation used in the Elastic
material model to derive the stress-strain relationship………………………………. 146
References……………………………………………………………….. 148
i
List of Figures
Chapter 1
1.1 Crashworthy seat for commuter aircraft………………………………... 3
1.2 Evaluation of an OH-58 pilot’s seat……………………………………… 4
1.3 Dummy lower leg models used in the lower leg impact studies…….. 5
1.4 Numerical dummies developed by LSTC………………………………. 5
1.5 Axially crushed aluminum tube………………………………………….. 14
1.6 Numerical simulation of ballistic impact of fabric in LSDYNA………… 22
Chapter 2
2.1 Preliminary EA seat design………………………………………………... 29
2.2 Specifications of the rail substructure…………………………………… 31
2.3 Stress Vs. Strain curve for the aluminum crush tubes………………….. 32
2.4 Static and dynamic axial crushing load of cylindrical aluminum
tubes with a D/t ratio of 30.7……………………………………………...
35
2.5 Annular grooves on a circular crush tube………………………………. 36
2.6 Weakening the FE mesh of the crush tube…………………………….. 37
2.7 Heat treatment curve used during the annealing process…………… 38
2.8 Static performance of plain and wasted tubes……………………….. 38
2.9 Nominal stress Vs. strain curve for the low density foam material……. 39
2.10 Gravity settling of the dummy against the foam cushion…………….. 40
2.11 Contoured foam cushion headrest to minimize head injury…………. 41
2.12 Effect of foam cushion on HYRBID III head acceleration……………... 41
2.13 FE mesh of the EA seat with airbag cushion and GEBOD dummy…... 43
2.14 Input parameters of the airbag cushion………………………………... 44
2.15 Acceleration pulse used in vertical drop testing………………………. 45
2.16 Acceleration pulse representing a mine blast…………………………. 46
2.17 Filtering of data……………………………………………………………. 47
ii
List of Figures… continued
2.18 Validation of initial EA seat simulations………………………………….. 48
Chapter 3
3.1 EA seat with a GEBOD dummy………………………………………….. 50
3.2 EA seat with a HYBRID III dummy………………………………………… 51
3.3 Results of EA Seat with GEBOD dummy subject to vertical drop
testing……………………………………………………………………….. 53
3.4 Results of EA Seat with HYBRID III dummy subject to vertical drop
testing……………………………………………………………………… 56
3.5 Results of EA Seat with GEBOD dummy subject to mine blast testing.. 58
3.6 Results of EA Seat with HYBRID III dummy subject to mine blast
testing……………………………………………………………………….. 60
3.7 Improved modeling of the EA seat structure…………………………… 61
3.8 Effect of Aluminum yield strength on the simulations…………………. 62
3.9 Stages of tube crushing for original seat model………………………. 63
3.10 Stages of tube crushing for improved seat model……………………. 64
3.11 Shape of the crushed tube modeled with solid elements……………. 65
3.12 Final model of the EA seat structure…………………………………….. 66
3.13 Deceleration pulses………………………………………………………. 66
3.14 Dynamic axial crushing force of the tube, and Dummy-seat
contact force……………………………………………………………… 67
3.15 Contact force between the foot and floor…………………………….. 67
3.16 Acceleration pulses……………………………………………………….. 68
3.17 Dynamic axial crushing force of the tube, and Dummy-seat
contact force……………………………………………………………… 68
3.18 Contact force between the foot and floor…………………………….. 69
3.19 New honeycomb EA mechanism………………………………………. 70
iii
List of Figures… continued
3.20 Interior view of the EA mechanism……………………………………… 71
Chapter 4
4.1 Experimental setup of lower leg impact……………………………….. 74
4.2 Numerical setup in ‘Sitting Straight’ position……………………………. 75
4.3 Numerical setup in ‘Driving’ position…………………………………….. 76
4.4 Prescribed velocity of the wall…………………………………………… 77
4.5 Validation of femur axial compressive force with test db2a…………. 78
4.6 Validation of foot acceleration with test db2a………………………… 79
4.7 Validation of femur axial compressive force with test db3a…………. 79
4.8 Validation of foot acceleration with test db3a………………………… 80
4.9 Foot (z) acceleration……………………………………………………… 81
4.10 Hip flexion-extension moment for wall speeds 1 ft/s - 15 ft/s…………. 82
4.11 Hip flexion-extension moment for wall speeds 25 ft/s - 35 ft/s………... 82
4.12 Lower leg (z) acceleration………………………………………………... 83
4.13 Femur axial compressive force………………………………………….. 83
4.14 Knee flexion-extension moment………………………………………… 84
4.15 Foot (z) acceleration……………………………………………………… 85
4.16 Hip flexion-extension moment…………………………………………… 86
4.17 Lower leg (z) acceleration………………………………………………... 86
4.18 Femur axial compressive force………………………………………….. 87
4.19 Knee flexion-extension moment…………………………………………. 87
4.20 Ankle dorsi-plantar flexion moment for wall speeds 1 ft/s - 10 ft/s…… 88
4.21 Ankle dorsi-plantar flexion moment for wall speeds 15 ft/s - 35 ft/s….. 88
4.22 Variables used in the parametric study…………………………………. 89
iv
List of Figures… continued
4.23 Variation of peak foot acceleration with peak wall speed for a
dummy in a driving position……………………………………………… 91
4.24 Variation of peak femur force with peak wall speed for a dummy in
a driving position…………………………………………………………... 92
4.25 Variation of peak femur force with wall speed and knee angle for
various dummy positions…………………………………………………. 92
Chapter 5
5.1 Applied deceleration pulse simulating impact after freefall…………. 98
5.2 Formation of a basic folding element………………………………….. 100
5.3 Comparison of impactor velocity time history…………………………. 108
5.4 Comparison of energy transformation during the impact event……. 109
5.5 Comparison of dynamic crushing load………………………………... 110
5.6 Velocities from the numerical formulation……………………………… 111
5.7 Unfiltered EA seat acceleration data…………………………………… 111
5.8 FFT of the relative velocity of the EA seat………………………………. 112
5.9 Comparison of acceleration response…………………………………. 112
5.10 Comparison of peak acceleration magnitude………………………... 113
Chapter 6
6.1 Representative Volume Cell (RVC) of the model……………………… 117
6.2 Pin-joint bar mechanism………………………………………………….. 118
6.3 One Element Elasticity Model……………………………………………. 118
6.4 Equilibrium position of the central nodes……………………………….. 120
6.5 Yarn stress-strain response of viscoelastic model……………………… 124
6.6 Yarn stress-strain response of elastic model……………………………. 124
6.7 Numerical setup of fabric axial strip test………………………………... 125
v
List of Figures… continued
6.8 Von-Mises stress distribution for strip with 30 s-1 strain rate…………… 126
6.9 Axial strip tests of Elastic model………………………………………….. 128
6.10 Bias strip tests of Elastic model…………………………………………… 129
6.11 Axial strip tests of Viscoelastic model……………………………………. 129
6.12 Bias strip tests of Viscoelastic model…………………………………….. 130
6.13 Comparison of bias tests of elastic and viscoelastic models at
different strain rates……………………………………………………….. 131
6.14 Comparison of axial tests of elastic and viscoelastic models at
different strain rates……………………………………………………….. 132
vi
List of Tables
Chapter 1
1.1 Human tolerance limits to acceleration ……………………………….. 6 1.2 Abbreviated Injury Scale (AIS) and sample injury types for two body
regions……………………………………………………………………….
7
1.3 HIC for various dummy sizes……………………………………………… 9 1.4 Critical values for various dummies used in the calculation of NIC…. 11
1.5 Recommended injury criteria for landmine testing……………………. 13
Chapter 2
2.1 Dimensions and material properties of the cylindrical aluminum
tubes used…………………………………………………………………..
33
2.2 Axial crushing parameters of the cylindrical aluminum tubes used…. 35
Chapter 3
3.1 Test matrix for EA seat design…………………………………………….. 49
Chapter 5
5.1 Human tolerance limits to acceleration………………………………… 96 5.2 Characteristics of the shell and impactor………………………………. 107
Chapter 6 6.1 Material and geometric properties of the Kevlar© fabric strip……….. 127
1
Chapter 1
Introduction
1.1 Background
Efforts are continually underway to maximize occupant safety during
peacekeeping efforts. Anti Tank (AT) mines and Improvised Explosive Devices
(IED) pose a serious threat to the occupants of armored vehicles. High
acceleration pulses and impact forces are transmitted to the occupant through
vehicle-occupant contact interfaces, such as the floor and seat, posing the risk
of moderate to fatal injury. The use of an energy absorbing (EA) seat in
conjunction with vehicle armor plating greatly improves occupant survivability
during such an explosion. The U.S. Army does not currently have an effective EA
seat in use. The only additional protection offered to the occupant so far is the
seat cushion.
The design of such an EA seat will need to include a suitable energy absorbing
device that proves to be both effective and feasible to incorporate into current
armored vehicle designs. The EA seat will then need to be rigorously tested
against explosive ordnance. The dynamic axial crushing of aluminum tubes is
an extensively used energy absorbing element in crashworthiness studies
because of numerous advantages such as high energy absorption and a
reasonably constant operating force.
The occupant lower leg impact by the vehicle floor during an IED explosion is
also of interest in occupant survivability studies. There currently exists very little
experimental data of lower leg impact, and consequently the injury
mechanisms are still not fully understood and validation of numerical studies
becomes difficult.
2
Efforts are also on to accurately model the ballistic impact of high strength
fabrics and to understand their complex behavior by virtue of their fabric
architecture. Such fabrics have high applicability to occupant safety, especially
for their anti-penetration resistance to projectiles. Different models have been
presented over the years, but a single comprehensive model that captures all
the fabric phenomena during ballistic impact does not currently exist. Simplistic
models however have been presented that capture the most important
features with good accuracy and at the least computational expense.
With the advent of supercomputing and advanced commercial finite element
codes, the emphasis is on conducting numerical simulations of real world
phenomena, to reduce the high costs of destructive testing while still preserving
the accuracy of the problem. This is the rationale behind this research which
involves conducting numerical simulations of mine blast testing of the energy
absorbing seat, occupant lower leg impact by the vehicle floor during the
explosion of an IED, and the development of a material model to realistically
simulate ballistic impact.
1.2 Literature Review
1.2.1 Energy Absorbing Seat
Concepts that are used in the crashworthiness analysis of aircraft seats are quite
similar to those used in crew protection against mine blasts. In 1988, Fox [1]
performed a feasibility study for an OH-58 helicopter energy attenuating crew
seat. Energy attenuating concepts included a pivoting seat pan, a guided
bucket, and a tension seat. In 1989, Simula Inc. prepared an Aircraft Crash
Survival Design Guide [2] for the Aviation Applied Technology Directorate. The
guide outlined various injury criteria, and energy absorbing devices amongst
3
other such related topics. In 1990, Gowdy [3] designed a crashworthy seat for
commuter aircraft using a wire bending energy absorber design as seen in
Figure 1.3. This design was sub-optimal but provided satisfactory results for
vertical decelerations between 15-32 Gs.
Figure 1.1 Crashworthy seat for commuter aircraft [3]
In 1993, Laananen [4] performed a crashworthiness analysis of commuter
aircraft seats during full scale impact using SOM-LA (Seat Occupant Model –
Light Aircraft). He concluded that those current designs did not meet the then
standards for occupant safety and that vertical direction energy absorbing
devices needed to be implemented. In 1994, Haley Jr. [5] evaluated a retrofit
OH-58 pilot’s seat to study its effectiveness in preventing back injury, as seen in
Figure 1.2. In 1996, Alem et al. [6] evaluated an energy absorbing truck seat to
evaluate its effectiveness in protection against landmine blasts. In 1998, the
Night Vision and Electronic Sensors Directorate published a report on Tactical
Wheeled Vehicles and Crew Survivability in Landmine Explosions [7]. Keeman [8]
has briefly summarized the approach adopted during the design of vehicle
crashworthy structures that utilize joints and thin walled beams. In 2002, Kellas [9]
designed an energy absorbing seat for an agricultural aircraft using the axial
crushing of aluminum tubes as the primary energy absorber.
4
Figure 1.2 Evaluation of an OH-58 pilot’s seat [5]
1.2.2 Foot Impact during IED/Mine blast
Joss [10] described how anti-personnel landmines have become a global
epidemic. Khan et al. [11] studied the type of hind foot injuries caused by
landmine blasts and surgical techniques available to treat it. Horst et al. [12]
experimentally and numerically studied occupant lower leg injury due to
landmine detonations under a vehicle. Horst and Leerdam [13] presented
further research being conducted into occupant safety for blast mine
detonations under vehicles. Dummies are used to study the lower leg impact,
and data such as accelerations and forces are measured along the lower leg,
from which injury criteria are assessed. Figure 1.3 shows some of the dummy leg
models used during these studies.
(a) (b) (c)
5
(d) (e)
Figure 1.3 a) Prosthetic leg model b) MADYMI detailed leg c) MADYMO Thor Lx
leg d) Interior view of the modeled leg e) HYBRID III Denton leg [13] 1.2.3 Human Injury Criteria
In order to determine the effectiveness of a design that protects occupants
against injury caused by crash and mine blasts, certain injury criteria need to be
defined. Occupant crash data such as forces, moments and accelerations are
collected from dummies used experimental tests and simulations and then
compared to these injury criteria to assess Occupant Survivability and Human
Injury. Figure 1.4 displays numerical dummies developed by LSTC for use in the
commercial finite element code LSDYNA©.
(a) (b)
Figure 1.4 a) GEBOD dummy b) HYBRID III dummy
6
a) Generalized Human Tolerance Limits to Acceleration
Table 1.1 displays the human tolerance limits for typical crash pulses along
three mutually orthogonal axes, for a well restrained young male. These values
provide a general outline of the safe acceleration limit for a human during a
typical crash. However, the time duration of the applied acceleration pulse has
not been specified. Higher acceleration pulses can be sustained for shorter
durations compare to lower acceleration pulses for longer durations, thus the
time duration in question is important [14].
Direction of Accelerative Force
Occupant’s Inertial Response Tolerance Level
Headward (+Gz) Eyeballs Down 25 G Tailward (-Gz) Eyeballs Up 15 G Lateral Right (+Gy) Eyeballs Left 20 G Lateral Left (-Gy) Eyeballs Right 20 G Back to Chest (+Gx) Eyeballs-in 45 G Chest to Back (-Gx) Eyeballs-out 45 G
Table 1.1 Human tolerance limits to acceleration [14]
b) Injury Scaling
Injury scaling is a technique for assigning a numerical assessment or severity
score to traumatic injuries in order to quantify the severity of a particular injury.
The most extensively used injury scale is the Abbreviated Injury Scale (AIS)
developed by the American Association for Automotive Medicine and originally
published in 1971. The AIS assigns an injury severity of “one” to “six” to each injury
according to the severity of each separate anatomical injury. Table 1.2 provides
the AIS designations and gives examples of injuries for two body regions. The
primary limitation of the AIS is that it looks at each injury in isolation and does not
provide an indication of outcome for the individual as a whole. Consequently,
the Injury Severity Score (ISS) was developed in 1974 to predict probability of
survival.
7
AIS Severity Head Spine 0 None - - 1 Minor Headache or Dizziness Acute Strain (no fracture or
dislocation)
2 Moderate Unconsciousness less than 1 hr., Linear fracture
Minor fracture without any cord involvement
3 Serious Unconscious, 1-6 hrs., Depressed fracture
Ruptured disc with nerve root damage
4 Severe Unconscious, 6-24 hrs., Open fracture
Incomplete cervical cord syndrome
5 Critical Unconscious more than 24 hr, Large hematoma , (100cc)
C4 or below cervical complete cord syndrome
6 Maximum Injury (virtually non-survivable)
Crush of Skull
C3 or above complete cord syndrome
Table 1.2 Abbreviated Injury Scale (AIS) and sample injury types
for two body regions [14]
The ISS is a numerical scale that is derived by summing the squares of the three
highest body region AIS values. This gives a score ranging from 1 to 75. The
maximal value of 75 results from three AIS 5 injuries, or one or more AIS 6 injuries.
Probabilities of death have been assigned to each possible score. Table 1.2
provides the AIS designations and gives examples of injuries for two body
regions. [14]
c) Dynamic Response Index (DRI)
The DRI is representative of the maximum dynamic compression of the vertebral
column and is calculated by describing the human body in terms of an
analogous, lumped-mass parameter, mechanical model consisting of a mass,
spring and damper. The DRI model assesses the response of the human body
to transient acceleration-time profiles. DRI has been effective in predicting
spinal injury potential for + Gz acceleration environments in ejection seats. DRI is
acceptable for evaluation of crash resistant seat performance relative to spinal
injury, if used in conjunction with other injury criteria including Eiband and
Lumbar Load thresholds. [14]
8
d) Lumbar Load Criterion
The maximum compressive load shall not exceed 1500 pounds (6672 N)
measured between the pelvis and lumbar spine of a 50th-percentile test
dummy for a crash pulse in which the predominant impact vector is parallel to
the vertical axis of the spinal column. This is one of the most widely used
criterions in vertical crash and impact testing. If the spinal cord is severely
compressed or severed, it can lead to either instant paralysis or fatality. [1, 9, 13-
16]
e) Head Injury Criterion (HIC)
HIC was proposed by the National Highway Traffic Safety Administration (NHTSA)
in 1972 and is an alternative interpretation to the Wayne State Tolerance Curve
(WSTC).[14, 15] It is used to assess forehead impact against unyielding surfaces.
Basically, the acceleration-time response is experimentally measured and the
data is related to skull fractures. Gadd [17] had suggested a weighted-impulse
criterion (GADD Severity Index, GSI) as an evaluator of injury potential defined as:
(1.1)
where
SI = GADD Severity Index
a = acceleration as a function of time
n = weighting factor greater than 1
t = time
Gadd plotted the WSTC data in log paper and an approximate straight line
function was developed for the weighted impulse criterion that eventually
became known as GSI. The Head Injury Criteria is given by
n
tSI a dt= ∫
9
(1.2)
where
a(t) = acceleration as a function of time of the head center
of gravity
t1,t2 = time limits of integration that maximize HIC
FMVSS 208 (Federal Motor Vehicle Safety and Standards) originally set a
maximum value of 1000 for the HIC and specified a time interval not exceeding
36 milliseconds. HIC equal to 1000 represents a 16% probability of a life
threatening brain injury. HIC suggests that a higher acceleration for a shorter
period is less injurious than a lower level of acceleration for a higher period of
time. As of 2000, the NHTSA final rule specified the maximum time limit for
calculating the HIC as 15 milliseconds. [4, 9, 17-23] Table 1.3 shows the HIC for
various dummy sizes.
Dummy Type
Large size Male
Mid size Male
Small size Female
6 year old child
3 year old child
1 year old infant
HIC15 Limit 700 700 700 700 570 390
Table 1.3 HIC for various dummy sizes [15]
f) Head Impact Power (HIP)
A recent report included the proposal of a new HIC entitled Head Impact Power
(HIP) It considers not only kinematics of the head (rigid body motion of the skull)
but also the change in kinetic energy of the skull which may result in
deformation of and injury to the non-rigid brain matter. The Head Impact Power
(HIP) is based on the general rate of change of the translational and rotational
kinetic energy. The HIP is an extension of previously suggested “Viscous Criterion”
1
2
2.5
2 1( ) ( )t
t
HIC t t a t dt⎡ ⎤
= − ⎢ ⎥⎢ ⎥⎣ ⎦∫
10
first proposed by Lau and Viano in 1986, which states that a certain level or
probability of injury will occur to a viscous organ if the product of its compression
‘C’ and the rate of compression ‘V’ exceeds some limiting value [14].
g) Injury Assessment Reference Values (IARS)
This rule adopts new requirements for specifications, instrumentation, test
procedures and calibration for the Hybrid III test dummy. [14]. The regulation’s
preamble has a detailed discussion of the injury mechanisms and the relevant
automotive mishap data for each of the injury criteria associated with the Hybrid
III ATD. Military test plans should implement these criteria.
h) Neck Injury Criterion (NIC)
The NIC considers relative acceleration between the C1 and T1 vertebra and is
given by [24]:
(1.3)
with
(1.4)
NIC must not exceed 15 m2/s2. [25]Another criteria NIC50 refers to NIC at 50mm
of C1-T1 (cervical-thoracic) retraction. Newly proposed Nij criteria by NHTSA
combines effects of forces and moments measured at occipital condyles and
is a better predictor of cranio-cervical injuries. Nij takes into account NTE (tension-
extension), NTF (tension-flexion), NCE (compression-extension), NCF (compression-
flexion). FMVSS specification No.208 requires that none of the four Nij values
exceed 1.4 at any point. The generalized NIC is given by [26]:
2( ) 0.2 ( ) [ ( )]rel relNIC t xa t V t= +
1
1
( ) ( ) ( )
( ) ( ) ( )
T Headrel x x
T Headrel x x
a t a t a t
v t a t a t
= −
= −∫ ∫
11
(1.5)
where
Fz = Upper Neck Axial Force (N),
My = Moment about Occipital Condyle
Fzn = Axial Force Critical Value (N), and
Myn = Moment Critical Value (N-m).
In FMVSS 208 (2000) final rule a neck injury criterion, designated Nij, is used. This
criterion is based on the belief that the occipital condoyle-head junction can
be approximated by a prismatic bar and that the failure for the neck is related
to the stress in the ligament tissue spanning the area between the neck and the
head. Nij must not exceed 1.0. [16, 22, 24, 26] Table 1.4 displays the critical
values for various dummies used in the calculation of Nij [15].
Dummy Type
Fzc (N) Flexion
Fzc (N) Extension
Myc (Nm) Flexion
Myc (Nm) Extension
Comments
3 year old dummy
2120 2120 68 27
Peak tension force < 1130 N Peak compression force < 1380 N
50th percentile
6806 6160 310 135
Peak tension (Fz) < 4170 N Peak extension (Fz) < 4000 N
Table 1.4 Critical values for various dummies used in the calculation of NIC [15]
i) Chest Criteria
Peak resultant acceleration will not exceed 60 G’s for more than 3 milliseconds
(Mertz, 1971) as measured by a Tri-axial accelerometer in upper thorax. Also, the
chest compression will be less than 3 inches for the Hybrid III dummy as
measured by a chest potentiometer behind the sternum [14, 15].
Z Yij
ZC YC
F MN F M⎛ ⎞ ⎛ ⎞= +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
12
j) Viscous Criterion
Viscous Criterion (V*C) – defined as the chest compression velocity (derived by
differentiating the measured chest compression) multiplied by the chest
compression and divided by the chest depth. This criterion has been mentioned
for the sake of completeness of information; however it is not widely used [14].
k) Femur Force Criterion
This criterion states that the compressive force transmitted axially through each
upper leg should not exceed 2,250 pounds or 10,000 N. Impulse loads that
exceed this limit can cause complete fracture of the femoral bone as well as
sever major arteries that can cause excessive bleeding. In numerical dummies,
discrete spring elements of known stiffness are included within the leg model,
from which the femur axial compressive force is easily extracted. In actual
dummies, load cells are placed on the dummy’s leg, which are calibrated to
provide the compressive force at the femur. [12, 13, 14, 27].
l) Thoracic Trauma Index (TTI)
The Thoracic Trauma Index is given by:
(1.6)
GR is the greater of the peak accelerations of either the upper or lower rib,
expressed in G’s. GLS is the lower spine peak acceleration, expressed in G’s. The
pelvic acceleration must not exceed 130 G’s [14].
13
m) Mine Blast Injury Criteria
U.S. Army’s Aberdeen Test Center has established injury criteria for mine blast
testing of high mobility wheeled vehicles. The injury criteria can also provide
guidance in standard crash impact testing orientations. These criteria are
comprehensive and provide a good assessment of injury that takes into
account the entire occupant’s body subject to any combination of external
stimuli associated with a mine blast. A few criteria are listed in Table 1.5 [14].
HYBRID III Simulant Response Parameter
Symbol (units) Assessment Reference Values
Head Injury Criteria HIC 750 ~5% risk of brain injury Lumbar spine axial compression force Fz (N) 3800 N (30ms) Femur or Tibia axial compression force Fz (N) 7562 N (10ms) Seat (Pelvis) vertical DRI DRI – Z(G) 15, 18, 23 G (low, med, high risk) Tibia axial compressive force combined with Tibia bending moment
F (N) M (N-m)
F/Fc – M/Mc < 1 where Fc=35,584N and Mc=225N-m
Table 1.5 Recommended injury criteria for landmine testing [14]
1.2.4 Dynamic Axial Crushing of Circular Tubes
The axial crushing of circular tubes by progressive plastic buckling has been the
subject of an extensive study over the years [28-47]. Perhaps one of the most
widely referred to technical paper in this field is that of Abramowicz and Jones
[29]. Gupta et al. [46, 47] studied the axisymmetric folding of tubes under axial
compression and incorporated both the change in tube thickness and yield
stress values of tension and compression into their model. Karagiozova et al.
[37, 38] studied the inertia effects, and dynamic effects on the buckling and
energy absorption of cylindrical shells under axial impact.
Galib and Limam [35] investigated the static and dynamic crushing of circular
aluminum tubes both experimentally and numerically using the commercial
software RADIOSS©. Bardi et al. [33] compared experimental results of tubes
under axial compression to nuemerical studies using the commercial software
14
ABAQUS©. Numerical analyses sometimes contain inaccuracies due to the high
mesh-sensitivity of the impact simulation. There is a difference in shell response
when simulating impact as a moving mass striking the stationery shell as
commonly observed in laboratory conditions, and as a the moving shell striking
a stationery rigid wall. Also, inappropriately filtering the data can lead to
significant under estimation of results such as crushing load [38]. Alghamdi [48]
reviewed common shapes of collapsible energy absorbers and different modes
of deformation of the most common ones. Nilakantan [49] presented a
numerical formulation to study the dynamic axial crushing of circular tubes
based on an energy balance approach. Figure 1.9 displays an axially crushed
aluminum tube, modeled in LSDYNA© [49, 50].
Figure 1.5 Axially crushed aluminum tube
1.2.5 Ballistic impact of dry woven fabrics
I) Modeling of the ballistic impact
Over the past few decades, many different techniques have been used to
derive the constitutive relations and model the overall fabric behaviour for use in
ballistic impact applications. Different models include various effects and
phenomena associated with the ballistic impact of fabrics, however there is no
15
single comprehensive model that reproduces and represents all phenomena at
the same time. However many simplistic models have been found to yield
results that are realistic.
a) Classification according to underlying theory
Researchers adopt different ways to approach the modeling of ballistic
response of dry woven fabrics. The methodology is discussed in later
paragraphs. This section simply enlists the approaches adopted by various
authors over time.
i) Analytical
Analytical methods make use of general continuum mechanics equations and
laws such as the conservation of energy and momentum. Governing equations
are set up using various parameters involved during the impact process.
Analytical methods are useful to handle simple physical phenomena, but
become increasingly complicated as the phenomena become more complex
and involve many variables.
This includes work by Vinson et al. [51], Taylor et al. [52], Parga-Landa et al.[53],
Chocron-Benloulo et al.[54], Navarro [55], Billon et al. [56], Gu [57], Hetherington
[58], Cox et al. [59], Naik et al. [60], Phoenix and Porwal [61, 62], Walker [63],
and Xue et al. [64].
ii) Semi-Empirical and empirical
Empirical studies rely on the analysis of data obtained through experimental
work in order to examine the fabric response and obtain constitutive relations
and failure criterion. This includes curve fitting, non-linear regression analysis of
experimental data, and the use of statistical distributions. Parametric equations
relate the various parameters studied during the experiment. The method is
16
useful when there are small numbers of variables to correlate [65]. Further, the
shortcoming is that the accuracy of the obtained model will depend on the
accuracy and completeness of the collected data. This includes work by
Cunniff [66], Shim et al. [67], and Gu [68].
iii) Numerical
This approach relies on techniques such as finite element and finite difference
methods, and the use of commercial packages such as ABAQUS©, DYNA3D©,
and LSDYNA© to conduct the analysis or simulation. Contact between and
amongst the yarns and projectile is better handled through the use of
commercial software. Further the fabric yarns may be modeled explicitly. This
includes work by Lomov et al. [69], Johnson et al. [70], Billon et al. [56], Lim et al.
[71], Shim et al. [72], Tan et al. [73, 74], Lim et al. [71], Roylance [75, 76], Hearle
[77], Boisse et al.[78], D’Amato et al. [79, 80], Duan et al. [81], Gu et al. [82],
Simons et al. [83], Teng et al. [84], Tarfaoui et al.[85, 86].
iv) Micromechanical
In a micromechanical approach, the fabric geometry is usually represented by
a representative volume cell or RVC, which by repeated translation will yield the
entire fabric structure. This RVC is then analyzed through equilibrium of forces,
variational potential energy methods, et cetera. to compute displacements,
stresses and strains. This includes work by Tabiei et al. [87, 88], Sheng et al. [89],
Dasgupta et al. [90], Tan et al. [91], Vandeurzen et al. [92] and Xue et al. [93].
v) Multi-scale constitutive
Multi-scale approaches make different assumptions of fabric behavior at
different scales. This arises due to the inherent multi-scale nature of fabrics which
17
are constructed from micro-scale fibrils. For example, the fabric behaves as a
continuum membrane at the macro scale; and at the micro scale, the
behavior is accounted for by constitutive modeling of the yarns as elastic or
viscoelastic members. This includes work by Nadler et al. [94] and Zohdi and
Powell [95].
vi) Variational
Variational principles include the Reissner variational principle, Galerkin method,
Rayleigh-Ritz method, and principal of minimum potential energy. These yield
governing differential equations which can then be solved using finite element
and finite difference methods. This includes work by Leech et al. [96], Roy et al.
[97], Sheng et al. [89], and Sihn et al. [98].
vii) Experimental
In order to validate the results from theoretical approaches, experimental data
is required. Further, by experimentally studying the ballistic impact of woven
fabrics, many new mechanisms of energy absorption and failure become
apparent, and effect of various parameters on the ballistic response can be
studied. This includes work by Starratt et al. [99], Susich et al. [100], Field et al.
[101], Wilde et al. [102], Prosser [103, 104], Cunniff [105], Shockey et al. , Wang
et al. [106, 107], Shim et al. [108], Lundin [109], Rupert [110], Orphal et al. [111],
and Manchor et al. [112].
b) Research based on number of fabric plies studied
Majority of the literature available today dealing with ballistic impact of fabrics
focuses on experimental and theoretical work of a single fabric layer. Few
literature deals with the ballistic impact of armor composed of multiple identical
18
layers of fabric such as Chocron-Benloulo et al.[54], Hearle et al. [77], Parga-
Landa et al. [53], Vinson et al. [51], Taylor et al. [52], Barauskas et al. [113],
Porwal and Phoenix [62], Sheng et al. [89], Vandeurzen et al. [92], Zohdi et al.
[114], Lomov et al. [69], Navarro et al. [55], Billon et al. [56], Tan et al. [74], Lim et
al. [74], Cunniff [115-117] and Schweizerhof et al. [118]. There is very limited work
on ballistic impact of fabrics composed of multiple layers of different fiber
material such as Cunniff et al. [119], Hearle [120] and Porwal and Phoenix [62].
c) Commercial finite element software packages used for analysis
With the advent of supercomputing, commercial finite element packages are
gaining popularity, because of the low cost alternative offered to costly
experimentation and destructive testing, as well as the potential testing of
materials not yet developed. Finite element packages also offer the option of
using of user-defined material models in place of the standard material models
and thereby provide a useful platform for the testing of new theories utilizing a
numerical form of solution. Finite element codes also can handle interaction
between the projectile and fabric, penetration, contact and friction between
yarns, and the deformation and failure of the fabric. Thus it is a very useful tool
for the simulation of ballistic impact of woven fabrics.
In the ballistic impact testing of fabrics, the most commonly used commercial
finite element packages are ABAQUS© by ABAQUS Inc. which involves the
ABAQUS/Standard and ABAQUS/Explicit solvers, DYNA3D© which is a part of a set
of public codes developed in the Methods Development Group at Lawrence
Livermore National Laboratory (LLNL) [121], and LS-DYNA© by Livermore Software
Technology Corporation [122, 123].
A few examples of research into ballistic impact of woven fabrics that use
these finite element packages are; ABAQUS© used by Xue et al. [64] and Diehl
et al. [124], DYNA3D© used by Shockey et al. [125, 126] and Lim et al. [71], and
LSDYNA© used by Tabiei et al. [87, 88, 127, 128], Gu et al. [57, 68], Shockey et
19
al. [129-131] and Duan et al. [81, 132-134], Shahkarami et al. [135], and
Schweizerhof et al [118].
d) Computer software and codes for solid modeling and computing
properties of textile composites
Brown et al. [136] describes a technique to automatically generate a solid
model of the representative volume element (RVE) of the fabric structure. The
solid model is generated using a program file written in I-deas® Open
Language. Cox et al. [59] lists various codes used in the computation of textile
composites properties, especially macroscopic stiffness, strength and
occasionally damage tolerance. These include μTEX-10 and μTEX-20 by Marrey,
R. V. et al, TEXCAD by Naik, Rajiv A., PW, SAT5, SAT8 by Raju, I. S., SAWC by
Whitcomb, J., CCM-TEX by Pochiraju, K., WEAVE by Cox, B., and BINMOD by
Cox, B. et al.
e) Approaches to modeling, based on author(s)
Vinson and Zukas [51] and Taylor and Vinson [52] modeled the fabric as conical
isotropic shells. The model treated the fabric as isotropic and did not
differentiate between warp and weft directions leading to a conical shaped
transverse deflection of the fabric, which is contrary to experimental findings.
Leech et al. [96] and Hearle et al. [77] modeled the fabric as a net. Prosser
[103] derived a mathematical model for the FSP-nylon system in his study of
ballistic impact of nylon panels by 0.22 caliber FSPs. He stated that for a set of
Vc determinations, plots of Vr (residual) and Vs (striking) can be adequately
represented by parabolas. There are periods in the plots of the squared V50
velocities and number of layers, where the plot linearity signifies that the
mechanism of penetration is constant. Cunniff [119] examined system effects
that occur during ballistic impact of woven fabrics by developing a conceptual
20
framework that relates ballistic impact mechanics of a single yarn to ballistic
impact mechanics of the fabric. Ting J. et al. [137] extended on the work of
Roylance et al. [138] and provided for contact between adjacent plies of a
multi-ply target and introduced slippage at yarn cross over points. Their model
predicted an increase in the ballistic limit when the friction of slippage
increases. Cunniff and Ting [139] developed a numerical model that treated
yarns as elastic rod elements, based on the work of [76]. Walker [63] developed
a constitutive model for an anisotropic fabric sheet based on elastic
deformations of the fibers. The centerline deflection of the fabric sheet was
solved with an approximate analytical solution that yields the final deformed
fabric shape and a simple equation for the force-displacement curve. Ting et
al. [137] and Shim et al. [72] modeled the fabric material as an orthogonal grid
of pin-jointed member elements. Shim et al. [67] used a three-element spring-
dashpot model to represent the viscoelastic behavior of the fibers and capture
its strain-rate sensitivity. The model accounts for yarn crimp. Roylance et al. [75]
modeled the fabric as an orthogonal mesh assembly of nodes interconnected
by flexible fiber members. A finite difference method was applied at the yarn
crossover points to simulate ballistic impact. Artificial buck up springs in the
transverse direction play a significant role in the ballistic limit determination. The
model lacks contact surfaces to interact with the projectile. Johnson et al. [70]
modeled the fabric with both pin-jointed members and thin membrane shells.
The computational model used a constitutive strength and fracture model that
depended on individual fiber characteristics. Bi-linear stress strain relationship is
assumed for the bar members to simulate yarn crimp. Shell elements provide
the contact surface and shear stiffness.
Shockey et al. [125, 126, 129-131] used finite solid elements to explicitly model
individual yarns and combined them in an orthogonal weave to form the fabric.
The model was found to be computationally very expensive; and became
unstable as the number of elements used to discretize the yarns crosses a
certain value. However the explicit yarn modeling allowed for observation of
21
phenomena such as yarn-yarn interaction and yarn pull-out. Chou et al. [140]
reviewed recent advances in the fabrication and design of three dimensional
textile preforms. Their review detailed advances made towards realizing an
integrated approach in the design and manufacture of three dimensional
textile preforms. Rao et al. [141] experimentally and theoretically studied the
influence of twist on the mechanical properties of high performance fiber yarns
including Kevlar© 29, Kevlar© 49, Kevlar© 149, Vectran© HS, Spectra© 900,
and Technora©. A model based on composite theory was developed to
highlight the decrease in modulus as a function of degree of twist and elastic
constants of the fibers. They concluded the existence of an optimal twist angle
of around 7° where all fibers exhibit their maximum tensile strength. At higher
angles of twist, the fibers get damaged reducing their tensile strength. The study
of Gasser et al. [142] aimed at recalling the specificity of the mechanical
behavior of dry fabrics and to understand the local phenomena that influence
the macroscopic behavior. A 3-d finite element analyses was compared to
biaxial tests on several fabrics. The developed model helped understand the
main aspects that lead to the specific behavior of woven fabrics and also help
design new fabrics by varying mechanical and geometric parameters. Billon et
al. [56] considered the fabric to be a collection of pin jointed members. Both an
analytical method and direct step finite element method were used and their
results were compared to experimental results. The input to the analytical model
includes fabric material properties, a constitutive relation and a failure criterion.
The model then predicts the ballistic limit and residual velocity. Lim et al. [71]
modeled fabric armor composed of Twaron© fibers in the finite element code
DYNA3D, using membrane elements under the continuum assumption of fabric.
A standard isotropic strain-rate dependant elastic-plastic model was used to
incorporate the strain-rate dependency of the Twaron fibers studied in [67].
Since the fabric architecture such as yarn crimp and cross section was not
considered, and the material was treated as isotropic, the deformation of the
fabric was conical when in fact it should have been pyramidal. Cheeseman
22
and Bogetti [143] reviewed the factors that influence ballistic performance,
specifically, the material properties of the yarn, fabric structure, projectile
geometry and velocity, far field boundary conditions, multiple plies and friction.
Ivanov and Tabiei [144] considered the fabric to be a grid of pin jointed bar
elements in their micromechanical approach. Tabiei et al.[87, 88, 144-146]
modeled the fabric as thin shells and developed their own material model for
use with the shell elements, that included effects of fiber reorientation and
locking angle, and fabric architecture such as crimp. The trellis mechanism
behavior of the flexible fabric in a free state before the packing of the yarns is
achieved by discounting the shear moduli of the yarn material. The fibers were
treated as viscoelastic members with a strain-rate based failure. The model was
implemented as a user defined subroutine in LSDYNA©. Contact forces at the
fiber cross over points were used to determine the rotational friction that
dissipated a part of the energy during reorientation.
Figure 1.10 Numerical simulation of ballistic impact of fabric in LSDYNA© by
Tabiei [144]
Gu [68] explicitly modeled individual yarns and combined them to form the
fabric mesh. A bimodal Weibull distribution was used to form the tensile
constitutive equations of the Twaron© yarn at high strain rates. Diehl et al. [124]
23
used ABAQUS/Standard and ABAQUS/Explicit to model structural performance of
systems containing woven fabrics. They investigated the limitations and
numerical problems of classical orthotropic lamina models, and introduced an
improved generalized cargo-net approach, models for membrane-only and
general shell behaviors, and experimental measurements utilized to obtain
effective modeling constants and parameters. Termonia [147] formulates the
mechanics of wave propagation in terms of impulse-momentum balance
equations, which are solved at each fiber cross over using a finite difference
technique. The model accounts for projectile characteristics such as shape,
mass and velocity, and also fiber properties such as denier, modulus and tensile
strength. The model also considers yarn slippage through the clamps, which is
often seen in experimental work. Termonia [148] also numerically investigates
the puncture resistance of fibrous structures by driving a needle shaped
projectile through a single fabric ply at a constant velocity of 100 m/s. Termonia
et al. [149] theoretically studied the influence of the molecular weight on the
maximum tensile strength of polymer fibers.
Barauskas and Kuprys [150] developed a model that could handle the collision
between fabric yarns in woven structures, where the longitudinal elastic
properties of each yarn are presented as a system of non-volumetric springs.
Their collision and response algorithm worked in a 3-d space and was based on
tight fitting of the yarns by using oriented bounding boxes, with a separation axis
theorem to handle collision detection between the oriented bounding boxes.
They assumed the yarn cross-sectional area to be constant and elliptical in
shape, with changing lengths of axes. Their system is characterized by a
significant reduction in degrees of freedom while still preserving the volumetric
behavior of the structure, when compared to traditional models that consider
yarns as fully deformable volumetric bodies. Phoenix and Porwal [61] developed
a membrane model based on an analytical approach to study the ballistic
response and V50 performance of multi-ply fibrous systems. They developed
solution forms for the tensile wave and curved cone wave considering constant
24
projectile velocity, and obtained an approximate solution for the membrane
response using matching boundary conditions at the cone wave front. Then
projectile deceleration due to membrane reactive forces was considered to
obtain other results such as cone velocity, displacement, and strain
concentration versus time. A later study by Porwal and Phoenix [62] based on
the above membrane model, studied the system effects in ballistic impact of a
cylindrical projectile into flexible, multi-layered targets with no bonding between
the layers. Each layer was assumed to have in-plane, isotropic, and elastic
mechanical properties.
II) Constitutive modeling of yarn
The fibers used in the ballistic impact resistant fabrics are viscoelastic. During
their constitutive modeling, it is important to account for their strain-rate
sensitivity. Properties such as the elastic modulus are dynamic and vary non-
linearly with strain. If static values are used during the analysis of the ballistic
impact of fabrics, it will lead to inconsistencies between numerical and
experimental results, as was observed in [76].
i) Based on the three element spring-dashpot model
Lim et al. [71] and Ivanov et al. [144] used a three-element spring dashpot
model to represent the viscoelastic behavior of the Twaron fibers. Twaron© fibers
are very similar to Kevlar© fibers as both belong to the Aramid family and have
identical static properties.
The viscoelasticity exists as a property of all materials but it is significant at room
temperature for polymeric materials mainly. The creep and the stress relaxation
are the results of the viscoelastic behavior of materials. For impact simulations,
we do not need the long-term effects of the viscoelasticity, so that the material
behavior can be simply described by a combination of one Maxwell element
25
without the dashpot and one Kelvin-Voigt element. The differential equation of
viscoelasticity can be derived from the model equilibrium in the form
(1.7)
where σ , ε , and ε are the stress, strain and strain rate respectively. Constants
Ka, Kb and μb can be derived experimentally and vary according to the material.
The principal behind the response of the fibers at different strain rates is as
follows. At low strain rate, below the transition strain rate, the dashpot offers little
resistance as damping is proportional to the velocity. The dashpot and parallel
connected spring are free to move according to spring stiffness Kb. Since Ka >
Kb, spring A remains rigid and spring B displaces preferentially. However at higher
strain rates, above the transition strain rate, the dashpot offers a resistance
higher than the stiffness of spring A. Now spring A moves preferentially
compared to the dashpot-spring B assembly, which remains rigid. In reality,
spring A represents the primary or intramolecular covalent bonds of the fiber
microstructure while spring B represents the secondary bonds which are the Van
der Waal forces and hydrogen bonds. The failure associated with these bonds is
discussed in later sections. The transition strain rate for Twaron© CT716 was
experimentally observed by [71] to be 410s-1. Based on their numerical
modeling, Ivanov et al. [144] observed the transition strain rate of 840 denier
Kevlar© 129 to be 100s-1.
ii) Based on Weibull distribution
Gu used a Weibull distribution of yarn strength to describe the stress-strain
response of Twaron fibers, based on [151, 152]. He used a two modal Weibull
distribution using the observation form [153] that aramids have a distinct skin-
core structure and that defects in the skin and core are the two main factors
( )a b b a b b aK K K K Kσ μ σ ε μ ε+ + = +
26
that influence the yarn strength composed of filaments without twist. From this
Gu obtained the following constitutive relation
(1.8)
The scale (m) and shape (σ) parameters were calculated from tensile
experimental data of yarn filaments [57] with the Levenberg-Marquardt
nonlinear least square estimation method [154]. Different constitutive relations
were obtained based on the strain rate. Wang et al. [106, 107] also used a
bimodal Weibull statistical distribution model to describe the strain-rate
dependence of Kevlar© 49 aramid fiber bundles for strain rates varying from
10-4 s-1 to 103 s-1.
1.3 Scope of Work
The stages involved in this research are as follows, but not necessarily in that
order
1) Extensive review of literature
2) Preliminary design of energy absorbing seat
a. Modeling and meshing of structure in HYPERMESH©
b. Setup of simulation inputfile in LS-PREPOST©
3) Validation of design by comparing simulation data with experimental
data of vertical drop testing of energy absorbing seat
4) Conducting numerical simulations of the mine blast on the energy seat
and studying occupant survivability
a. Use of prescribed acceleration pulses simulating a mine blast
b. Extraction of dummy data and comparison with injury criteria
1 2
01 02
expm m
E EE ε εσ εσ σ
⎡ ⎤⎛ ⎞ ⎛ ⎞⎢ ⎥= − −⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦
27
5) Final energy absorbing seat design including additional energy absorbing
concepts
a. High density foam / airbag cushion
b. Use of GEBOD and HYBRID III dummy
6) Conducting numerical simulations of the impact of occupant lower leg
by the vehicle floor during IED explosion under an armored vehicle
a. Use of GEBOD and HYBRID III dummy
b. Variation of wall speed – 1, 5, 10, 15, 25, 35 ft/s
c. Seated Straight and Driving position
7) Development of a numerical formulation to study the response of
dynamic axial crushing of circular tubes and to predict occupant
survivability during impact events
a. Program coded in MATLAB©
b. Comparison with experimental data for validation
c. Used in two different configurations
8) Development of a material model to realistically simulate ballistic impact
of loose woven fabric with elastic crimped fibers, and integration of the
material model into LS-DYNA©
a. Utilizes a micromechanical approach
b. Subroutine coded in FORTRAN© and integrated into LS-DYNA© as
a User Defined Material Model.
c. Comparison of axial testing of the elastic fabric model with the
viscoelastic fabric model.
1.4 Outline of Thesis
Chapter 1 introduces the subject of this research and extensively reviews the
previous literature. The steps followed during the course of this research are
briefly presented.
28
Chapter 2 looks at the design of the energy absorbing seat and the setup of the
numerical model. The various components of the design and the input pulses
used are studied. Initial simulation results of vertical drop testing are compared
with experimental results for validation.
Chapter 3 presents the detailed results of all the numerical simulations
conducted with the energy absorbing seat in accordance with the test matrix.
The crushing of aluminum tubes is studied. The final EA seat design is then
presented. A new mechanism using a honeycomb structure is briefly
introduced.
Chapter 4 studies the occupant lower leg impact during a mine blast. The
numerical setup is explained. A series of floor impact simulations are conducted
and numerical results are studied. A parametric study is also introduced.
Chapter 5 presents a numerical formulation using an energy balance approach
to study the dynamic axial crushing of circular tubes. The formulation is
implemented as a program and results are compared to experiments.
Chapter 6 presents a micromechanical model to study the ballistic impact of
loose woven fabrics with elastic crimped fibers. Fabric axial tests at various strain
rates are numerically simulated and results are compared to the viscoelastic
model.
Appendix I lists the source code for the dynamic axial crushing of circular tubes
and Appendix II lists the source code used to derive the incremental stress-strain
relationship for the elastic material model.
29
Chapter 2
Energy Absorbing Seat
2.1 Preliminary Design
The crashworthy commuter aircraft seat used in [9] forms the basis for this
design. Figure 2.1 displays the preliminary design of the energy absorbing seat
structure.
Figure 2.1 Preliminary EA seat design
The support structure rigidly holds two cylindrical steel rails inclined at a 20° angle
to the vertical. A set of upper and lower cylindrical brackets which slide along
the rails are attached to the seat. A steel collar is rigidly attached to each rail.
The aluminum crush tubes are coaxial with the steel rails and are positioned
30
between the upper bracket and collar. During vertical drop testing, the upper
brackets move downwards causing the crushing of the aluminum tubes against
the collars, which is the primary energy absorption principal used here. During a
mine blast, the entire support structure along with the attached collars move
upwards causing the crushing of the tubes against the upper brackets. For the
initial testing without a numerical dummy, the density of the seat material is
scaled to include the weight of an occupant. Later on, the occupant is
modeled using both a GEBOD dummy and a 5th percentile HYBRID III dummy.
An initial time delay of 50 ms in all simulations allows for gravity settling of the
dummy against the seat to ensure proper contact. In addition to the aluminum
crush tubes, further energy absorbing elements such as high density foam
cushions and airbag cushions are added to the design.
While modeling the structure in LSDYNA, certain simplifications are made to the
model. This facilitates the replacement of detailed structures and designs with
equivalent simplistic representations. The two inclined steel rails are attached to
the support structure by creating a ‘Node Set’ consisting of nodes belonging to
the rail and structure at the joint location and then using this node set in the
*CONSTRAINED_NODAL_RIGID_BODY keyword definition which ensures a rigid
joint between the rail and structure. The seat structure is modeled using shell
elements and a rigid material model. The reason for using rigid material defined
by the *MAT_RIGID keyword is that they are computationally efficient when
representing parts that do not deform or do not need to be monitored during
the study. Rigid elements are bypassed during the element processing in
31
LSDYNA. The set of four brackets are also attached to the seat by creating a
node set and then using this node set in a nodal rigid body definition. Figure 2.2
displays the linear dimensions of the rail, brackets, collar and crush tube.
Figure 2.2 Specifications of the rail substructure
Contact definitions are created in LSDYNA to specify contact between the rail,
crush tube, brackets, and collar. *CONTACT_AUTOMATIC_SURFACE_TO_SURFACE
and *CONTACT_AUTOMATIC_GENERAL are used to this effect. By using the
AUTOMATIC specification, the orientation of the shell segment normals is
automatic. The SOFT=2 option can be used to activate a different contact
formulation and causes the contact stiffness to be calculated considering the
global time step and nodal masses. This approach is generally more effective
when creating contact definitions between components of different mesh
densities and material stiffness.
The crush tube has the finest mesh as it deforms the most during the simulation
thus controlling the time step, and constitutes the energy absorbing member. If
32
Stress Vs. Strain for *Mat 24 - Aluminum
440004500046000470004800049000500005100052000
0 0.1 0.2 0.3 0.4 0.5 0.6
Strain (%)
Stre
ss (M
Pa)
*Mat 24 - Aluminum
the mesh is too fine, the time step falls to very small values causing the
simulation to run indefinitely. However LSDYNA Material Model 24 which is
*MAT_PIECEWISE_LINEAR_PLASTICITY allows the user to specify a minimum time
step for the material and when the simulation time step falls below this defined
value, the controlling element with this material model is deleted. Thus the
overall minimum time step of the problem can be controlled without using Mass
Scaling which adds mass to the component to prevent the time step from
falling below a certain value. Also, this material model allows an arbitrary stress
versus strain curve as well as arbitrary strain rate dependency to be defined,
which is illustrated by the curve shown in Figure 2.3 below. The yield stress needs
be specified for this model and is given a value of 145 MPa corresponding to
Aluminum 3003.
Figure 2.3 Stress Vs. Strain curve for the aluminum crush tubes
2.2 Dynamic Axial Crushing of the Aluminum Tubes
Axial crushing of cylindrical tubes became a very popular choice of impact
energy absorber because of its energy absorption capacity. It provides a
reasonably constant operating force, has high energy absorption capacity and
stroke length per unit mass. Further a tube subjected to axial crushing can
33
ensure that all of its material participates in the absorption of energy by plastic
work. [33, 35]. Classification of axial crushing of cylindrical tubes under quasi
static loading includes sequential concertina mode, sequential diamond
mode, Euler mode, concertina and diamond mode, simultaneous concertina
mode, simultaneous diamond mode, and tilting of tube axis mode. [155] The
D/t ratio of the cylindrical tubes used in the design determines the mode in
which the tubes will crush. Experimental observations of Alghamdi [48] showed
that thick cylinders (small D/t ratio, D/t<80-90) undergo a concertina
(axisymmetric) mode of deformation. Table 2.1 lists the dimensions and material
properties of the cylindrical aluminum tubes used in our design.
Inner diameter (Di) 26.437 mm Outer diameter (Do) 28.215 mm Mean diameter (D) 27.326 mm
Thickness (t) 0.889 mm Yield Strength (Y) 145 MPa
Length (L) 228.6 mm Density (ρ) 2.610E-09 ton/mm3
Young’s modulus of Elasticity 68948.000 N/mm2 Poisson’s ratio 0.33
Tangent modulus Refer Figure [1] LSDYNA Material model Piecewise Linear Plasticity Material model number 24
TABLE 2.1 Dimensions and material properties of the cylindrical aluminum tubes
used
Figure 2.3 displays the stress-strain plot of the aluminum material used for the
cylindrical tube, from which the Tangent modulus can be computed.
Substituting the mean diameter and thickness values, we obtain the D/t ratio as
30.7 which is much less than 80. Thus the tubes are classified as thick and will
deform in a concertina mode. It has been reported that the concertina mode
of deformation results in a higher specific energy absorption than the diamond
mode of deformation (high D/t ratios, non-axisymmetric) [48].
34
According to Alexander [31], the mean crushing load of a cylindrical tube is
given by the following expression:
(2.1)
where
Pav = Mean crushing load
Y = Yield strength
t = Tube thickness
D = mean tube diameter
It provides a good prediction for D/t<30. This equation is the most famous in
axial crushing of tubes. Abramowicz and Jones [29] improved Alexander’s
analysis and proposed the following expression:
1 2(6( ) 3.44 )avP Yt Dt t= + (2.2)
As can be seen from Equations 2.1 and 2.2, there is a dependence of the
mean crushing load on the yield strength and geometric properties of the crush
tubes. This is important in the design of the aluminum crush tubes as the
dynamic crushing load must not exceed the critical value that can cause
crushing of the lumbar column. Table 2.2 displays a few parameters associated
with the axial crushing of the cylindrical aluminum tubes, such as static and
dynamic crushing load, number of folds possible during crushing, total plastic
energy absorbed during the formation of each fold, and effective crushing
distance, for the tubes we used whose properties are listed in Table 2.1. Four
different yield strengths were selected, whose values ranged from 100 MPa to
300 MPa. The formulae for these parameters are obtained from [29]. These
properties are useful in determining the overall energy attenuation capability of
the aluminum crush tubes and can also be used to predict the response
1 26 ( )avP Yt Dt=
35
(acceleration pulse) of the EA seat during impact. The maximum compressive
lumbar load that can be sustained without injury is 6672 N. This must be kept in
careful consideration while selecting the tube material and geometric
properties.
100 MPa 145 MPa 220 MPa 300 MPa
Static crushing load (N) 3316.6 4809.1 7296.5 9949.8 Dynamic crushing load (N) 4151.1 6019.1 9132.5 12453 Number of folds possible 52 52 52 52
Total energy absorbed / fold (J) 24.54 35.58 53.98 73.61 Effective crushing distance (mm) 6.91 6.91 6.91 6.91
TABLE 2.2 Axial crushing parameters of the cylindrical aluminum tubes used
Figure 2.4 displays the static and dynamic axial crushing loads of cylindrical
aluminum tubes with a D/t ratio of 30.7 as a function of yield strength.
0
2000
4000
6000
8000
10000
12000
14000
0 50 100 150 200 250 300 350
Yield Strength (MPa)
Load
(N)
Static Crushing Load
Dynamic Crushing Load
Figure 2.4 Static and dynamic axial crushing load of cylindrical aluminum tubes
with a D/t ratio of 30.7
36
2.2.1 Techniques to reduce the initial crushing load of a tube
It has been observed that while the crushing of the tubes occurs under a
reasonably constant operating force, there always exists an initial peak that
corresponds to the formation of the first plastic hinge. This peak usually is about
1.5 to 2 times larger than the average crushing load of the tube and will be
dangerous to the occupant’s survivability if not attenuated.
a) Introduction of annular grooves in the crush tube
Research conducted by Daneshi et al. [34] shows that the introduction of
annular grooves in the crush tube will force plastic deformation to occur at
regular intervals along the tube, thereby causing uniform energy absorption and
a uniform deceleration pulse thus resulting in a controllable energy absorption
element, without any spike in the load-deformation plot that is usually
associated with the initial crushing force required for plastic buckling. Thus far,
only quasi static axial crushing tests have been performed, but have yielded
promising results.
Figure 2.5 Annular grooves on a circular crush tube
37
b) Weakening the finite element mesh of the crush tube
Simulations showed that by removing periodic shell elements along the
periphery of the crush tube, local buckling (and plastic deformation) can be
induced at a desired location, which will result in a lower initial deceleration
pulse. The energy absorption rate remains unaffected. By reducing the initial
deceleration pulse, we can make sure that at no point in the simulation, the
deceleration reaches the critical value causing injury. In Figure 2.6, red
elements correspond to the rail and blue elements correspond to the crush
tube.
Figure 2.6 Weakening the FE mesh of the crush tube
c) Heat treatment and wasting of the crush tube
Research conducted showed that by first subjecting the crush tube to an
Annealing cycle and then Wasting it by introducing a wrinkle around the tube’s
perimeter via a pipe cutting tool could reduce the crush initiation load and
deceleration pulse by as much as 50% as well as the initial peak in the load-
displacement curve. Figure 2.7 displays the heat treatment curve during the
annealing of the aluminum crush tube.
38
Figure 2.7 Heat treatment curve used during the annealing process [9]
As can be observed from the Figure 2.8, there is a great difference in both the
Crush Initiation Load as well as the Mean Sustainable Crushing Load when the
tube is subjected to different combinations of Annealing and Wasting. The tube
that was annealed and then wasted was found to be most suitable for the
simulations and had the closest desired crush initiation load [9]. When a wrinkle is
created on the tube’s periphery, strain hardening occurs due to plastic
deformation. Annealing helps remove this and restores the softness back to the
material
Figure 2.8 Static performances of plain and wasted tubes [9]
39
2.3 Additional energy absorbing elements
In addition to the aluminum crush tubes, additional energy absorbing elements
are added to the design to help further attenuate and delay the shock pulse as
well as to offer the occupant additional cushioning. These are discussed below.
2.3.1 Low density foam cushion
A low density foam cushion that covers the upper and lower sections of the seat
has been utilized to provide additional cushioning. The material model used in
LS-DYNA for the foam cushion is MAT_LOW_DENSITY_FOAM. It is used for
modeling highly compressible low density foams and its main applications are
for seat cushions and padding on the Side Impact Dummies (SID) [122]. This
foam model is not crushable, thus problems usually encountered in simulations
such as negative volume, invalid Jacobian and element inversion associated
with crushable foam has been avoided. Figure 2.9 displays the nominal stress
versus strain for the low density foam material used. The data was obtained from
the NHTSA Side Impact dummy model.
Figure 2.9 Nominal stress Vs. strain curve for the low density foam material
Low Density Foam
00.20.40.60.8
11.21.41.61.8
0 0.2 0.4 0.6 0.8 1
Strain
Nom
inal
Str
ess
NominalStress Vs.Strain
40
While specifying contact between the foam cushion and the seat, the card
*CONTACT_TIED_NODES_TO_SURFACE_OFFSET is also used to attach (tie) the
nodes of the bottom surface of the solid foam cushion elements to the seat so
that there is no chance of slippage during the impact, similar to how the
cushion is stitched to the seat frame in reality. The keyword
*CONSTRAINED_EXTRA_NODES can also be used to attach the foam cushion to
the seat.
As can be seen from Figure 2.10, the dummy first settles against the cushion by
the action of gravity before the acceleration pulse is applied in the simulation.
This provides proper contact between the dummy’s lower torso and the foam
cushion. The foam cushion provides additional cushioning during both the mine
blast and vehicle slam down.
Figure 2.10 Gravity settling of the dummy against the foam cushion
In order to keep the rearward head impact with the seat to a minimum, a
contoured headrest is used as displayed in Figure 2.11, which minimizes the
distance the head is thrown backwards and correspondingly any injury this may
41
cause. It has resulted in a reduction of the head G-forces of impact from 123
G’s without the foam cushion, to 45 G’s with the original foam cushion design, to
a final 25 G’s with the new contoured design, as can be seen from Figure 2.12.
Figure 2.11 Contoured foam cushion headrest to minimize head injury
-20
0
20
40
60
80
100
120
140
40 60 80 100 120
Time (ms)
Acc
eler
atio
n (G
)
No Foam Cushion
Orig Foam Cushion
Contoured FoamHeadrest
Figure 2.12 Effect of foam cushion on HYRBID III head acceleration
42
2.3.2 Airbag cushion
In addition to the aluminum crush tubes, an inflatable airbag cushion is added
to the upper surface of the rigid seat. The inflation is controlled by a sensor that
triggers inflation at a user defined acceleration level. During the mine blast and
subsequent vertical displacement of the armored vehicle and occupant, the
aluminum tubes are partly consumed. Thus, there is limited cushioning during
vehicle slam down. The purpose of the airbag cushion is provide cushioning to
the occupant during the mine blast and additional cushioning to the
occupant’s torso once the vehicle impacts the ground during slam down. The
inflation of the airbag is controlled in LS-DYNA using a user defined load curve.
LS-DYNA offers 11 airbag models as follows:
1) SIMPLE PRESSURE VOLUME
2) SIMPLE AIRBAG MODEL
3) ADIABATIC GAS MODEL
4) WANG NEFSKE
5) WANG NEFSKE JETTING
6) WANG NEFSKE MULTIPLE JETTING
7) LOAD CURVE
8) LINEAR FLUID
9) HYBRID
10) HYBRID JETTING
11) HYBRID CHEMKIN
The simple airbag [122] model in LS-DYNA has been utilized in these simulations.
Figure 2.13 displays the GEBOD dummy seated on a partially inflated airbag
cushion. The constant inflation of the airbag ensures proper contact at all times
with the dummy’s lower torso.
43
Figure 2.13 FE mesh of the EA seat with airbag cushion and GEBOD dummy
The user input for the simple airbag model is the Volume Vs. Time plot which
controls the airbag inflation. An initial filled volume can be specified at time t=0,
so that the airbag remains partially inflated and acts as a seat cushion during
normal operation. Whenever a mine blast occurs, the inbuilt LSDYA sensor is
triggered and the airbag rapidly inflates, thereby providing additional
cushioning. Figure 2.14 displays some of the properties of the airbag used such
as the rate of input airflow and rate of change of mass with time.
Airbag Volume
0
1
2
3
4
5
0 0.01 0.02 0.03 0.04 0.05 0.06
Time (s)
Vol
ume
(x 1
E+0
7)
Airbag Volume
(a)
44
dM/dT IN
0
5
10
15
20
0 0.01 0.02 0.03 0.04 0.05 0.06
Time (s)
dM/d
T IN
(x 1
E+03
)
dM/dT IN
(b)
Figure 2.14 a) Rate of airbag input airflow b) Rate of change of airbag mass
with time
2.4 Shock pulses Applied to the Structure
Two types of input pulses have been used in the simulations to represent freefall
under gravity, and a mine blast under an armored vehicle. These pulses
prescribe structural accelerations that correspond to the actual physical
phenomenon. This is a more robust method compare to displacement or
velocity control. A load curve that specifies acceleration versus time is used in
LS-DYNA to prescribe the motion of the structure. There is a time delay before
the application of the input pulse to allow for gravity settling of the occupant
against the structure to ensure proper contact. The individual pulses are
explained in the following sections.
2.4.1 Impact after Free Fall
Based on data from [9], the vertical impact after free fall is simulated by
applying a deceleration pulse to the base of the structure. This is accomplished
in LS-DYNA by applying the load curve that prescribes acceleration versus time
to all nodes at the lowermost surface of the structure. The structure is also given
45
an initial velocity in the downward direction corresponding to the vertical impact
velocity. The shock waves propagate from the base of the structure through the
brackets, onto the seat, and finally to the occupant. Figure 2.15 displays a
sample input deceleration pulse applied to the structure.
-20
0
20
40
60
80
100
0 20 40 60 80
Time (ms)
Acc
eler
atio
n (G
)
Vertical Drop TestPulse
Figure 2.15 Acceleration pulse used in vertical drop testing
2.4.2 Mine Blast
An acceleration pulse is used to simulate the effect of a mine blast under a
vehicle. Figure 2.16 displays the acceleration pulse that is applied to the
bottom surface of the structure in the upward direction. This throws the seat and
occupant upwards and after the reaching the peak vertical displacement, the
seat and occupant return to the ground via freefall. The center of mass of the
structure is constrained against any rotation to ensure the structure displaces
vertically along a straight line. The pulse gets transmitted from the structure to
the occupant through the energy absorbing mechanism. This pulse includes a
peak acceleration of 180 G for a 5 ms duration. This is followed by a 85 ms
duration of negative acceleration to put the final velocity at zero and final
46
displacement at its maximum vertical position. After that the acceleration
stabilizes at -1 G (freefall) until displacement is zero [156].
-50
0
50
100
150
200
0 50 100 150
Time (ms)
Acc
eler
atio
n (G
) Mine Blast Pulse
Figure 2.16 Acceleration pulse representing a mine blast
2.5 Filtering of Data
All numerical data contains noise and needs to be filtered to remove the high
excitations of nodal data which leads to the numerous peaks normally seen in
unfiltered data. The raw data is first subjected to a fast Fourier transform of FFT to
determine the cut-off or filtering frequency. The data is then filtered using a low
pass Butterworth filter with this cut-off frequency. The choice of filter and
frequency is important as the use of too low a filtering frequency can lead to
severe under prediction of the results and vice versa. Figure 2.17 displays
sample unfiltered data and the corresponding filtered data using a BW filter at
180 Hz.
47
(a)
(b)
Figure 2.17 a) Unfiltered data b) Filtered data
2.6 Validation of Initial EA Seat Simulations
Before simulations are created and run according to the test matrix, the initial
test results are first validated by comparison with experimental results. Numerical
results from the Energy Absorbing seat simulations have been compared with
experimental observations and data from [9]. The results from our simulations
are in very good agreement with the experimental data as can be seen from
48
Figure 2.18. There is high variance and correspondingly low repeatability of the
experimental curves and therefore the correspondence of results seen from the
simulation and experiment in Figure 2.18 is considered as very good validation
of the data. It is important to note the scarcity of further available experimental
data for such vertical drop testing of energy absorbing seats with a dummy
occupant.
-60
-40
-20
0
20
40
0 10 20 30 40 50 60
Time (s)
Acc
eler
atio
n (G
)
Experiment
Simulation
(a)
-20
0
20
40
0 10 20 30 40 50 60Time (s)
Acc
eler
atio
n (G
)
Experiment
Simulation
(b)
Figure 2.18 Comparison of experimental and simulation results (a) seat pulse
(b) occupant pulse
49
Chapter 3
Results and Discussion: Energy Absorbing
Seat
3.1 Test matrix
The test matrix is shown in Table 3.1. Due to the high volume of test results,
individual test data for all tests are not shown; rather comparison plots have
been presented in the subsequent sections.
Test GEBOD dummy
HYBRID III dummy
Plain Seat Foam Cushion
Airbag Cushion
Vertical Drop
Mine Blast
T1 x x x
T2 x x x
T3 x x x
T4 x x x
T5 x x x
T6 x x x
T7 x x x
T8 x x x
T9 x x x
T10 x x x
Table 3.1 Test matrix for EA seat design
50
The simulations were run on the Ohio Supercomputer Center using a Beowulf
cluster consisting of Intel P4 chips running on a Linux based OS. When using a
numerical dummy in the simulations, serial processing was used and in all other
simulations, parallel processing utilizing 8 nodes was used. The data was
processed using LS-PREPOST© to extract the results.
3.2 Simulation Setup
The setup of the simulation in LSDYNA is shown below. Figure 3.1 displays an EA
seat with the additional energy absorbing element which may be a foam
cushion or airbag cushion, and a GEBOD dummy occupant. Figure 3.2 displays
the same seat with a 5th percentile HYBRID III dummy.
Figure 3.1 EA seat with a GEBOD dummy
51
Figure 3.2 EA seat with a HYBRID III dummy
3.3 EA seat with GEBOD dummy subjected to vertical
drop testing
Comparison between EA Plain Seat / Foam / Airbag Cushion Seat w ith Gebod dummy subject to prescribed deceleration pulse during impact after freefall
-Head (x) Acceleration
-30
-20
-10
0
10
20
30
40
0 20 40 60 80 100 120
Time (ms)
Acce
lera
tion
(G)
Foam Cushion: Head (x) Acc
Plain Seat : Head (x) Acc
Airbag Cushion Seat: Head (x) Acc
period where gravity isused to settle the dummy
(a)
52
Comparison between EA Plain Seat and Foam Cushion Seat w ith Gebod dummy subject to prescribed deceleration pulse during impact after freefall
-Neck (x) Acceleration
-10
0
10
20
30
40
50
0 20 40 60 80 100 120
Time (ms)
Acc
eler
atio
n (G
)Plain Seat: Neck (x) Acc
Foam Cushion:Neck (x) Accperiod where gravity isused to settle the dummy
(b)
Comparison between EA Plain Seat / Foam / Airbag Cushion Seat w ith Gebod dummy subject
to prescribed deceleration pulse during impact after freefall -Head (y) Acceleration
-10-505
10152025303540
0 20 40 60 80 100 120
Time (ms)
Acc
eler
atio
n (G
)
Plain Seat: Head (y) Acc
Foam Cushion:Head (y) Acc
Airbag Cushion Seat: Head (y) Acc
period where gravity isused to settle the dummy
(c)
Comparison between EA Plain Seat / Foam / Airbag Cushion Seat w ith Gebod dummy subject to
prescribed deceleration pulse during impact after freefall -M id Torso (y) Acceleration
-60-40-20
020406080
100
0 20 40 60 80 100 120Time (ms)
Acc
eler
atio
n (G
)
Plain Seat: Mid Torso (y) Acc
Foam Cushion: Mid Torso (y) Acc
Input Pulse
Airbag Cushion: Mid Torso (y) Acc
(d)
53
Comparison between EA Plain Seat and Foam Cushion Seat with Gebod dummy subject to prescribed deceleration pulse during impact after freefall
-Force (x) between the Lower Torso and Seat
-10000
100020003000400050006000
0 20 40 60 80 100 120
Time (ms)
Forc
e (N
)
Plain Seat: Low er Torso - Seat : y force
Foam Cushion: Low er Torso - Seat : yforce
(e)
Comparison between EA Plain Seat / Foam / Airbag Cushion Seat w ith Gebod dummy subject to prescribed deceleration pulse during impact after freefall
-Lower Torso (y) Acceleration
-200
204060
80100
0 20 40 60 80 100 120Time (ms)
Acc
eler
atio
n (G
)
Plain Seat: Low er Torso (y) Acc
Input Pulse
Foam Cushion: Low er Torso (y) Acc
Airbag Cushion: Low er Torso (y) Acc
(f)
Comparison between EA Plain Seat and Foam Cushion Seat with Gebod dummy subject to prescribed deceleration pulse during impact after freefall
-Force (x) between the Upper Torso and Seat
-10000
0
10000
20000
30000
40000
0 20 40 60 80 100 120
Time (ms)
Forc
e (N
)
Plain Seat: Upper Torso - Seat : x force
Foam Cushion: Upper Torso - Seat : xforce
(g)
Figure 3.3 Results of EA Seat with GEBOD dummy subject to vertical drop testing
54
3.4 EA seat with HYBRID III dummy subjected to vertical
drop testing
Comparison between EA Plain Seat and Foam Cushion Seat with Hybrid III dummy subject to prescribed deceleration pulse during impact after freefall
-Head (x) Acceleration
-200
20406080
100120140
0 20 40 60 80 100 120
Time (ms)
Acc
eler
atio
n (G
) Plain Seat: Head X Acc
Foam Cushion: Head (x) Acc
period where gravity isused to settle the dummy
(a)
Comparison between EA Plain Seat and Foam Cushion Seat with Hybrid III dummy subject to prescribed deceleration pulse during impact after freefall
-Neck (x) Acceleration
-200
20406080
100120140
0 20 40 60 80 100 120
Time (ms)
Acc
eler
atio
n (G
)
Plain Seat: Neck (x) Acc
Foam Cushion: Neck (x) Acc
period where gravity isused to settle the dummy
(b)
55
Comparison between EA Plain Seat and Foam Cushion Seat with Hybrid III dummy subject to prescribed deceleration pulse during impact after freefall
-Neck Moment
-51000
-41000
-31000
-21000
-11000
-1000
9000
0 20 40 60 80 100 120
Time (ms)
Nec
k M
omen
t (N
-mm
)
Plain Seat: Neck Moment
Foam Cushion: Upper NeckMoment
Foam Cushion: Lower NeckMomentperiod where gravity is
used to settle the dummy
(c)
Comparison between EA Plain Seat and Foam Cushion Seat with Hybrid III dummy subject to prescribed deceleration pulse during impact after freefall
-Rigid Seat Acceleration (y) Pulse
-60
-40
-20
0
20
40
60
80
100
0 20 40 60 80 100 120
Time (ms)
Acc
eler
atio
n (G
)
Foam Cushion: Rigid Seat (y) AccInput Deceleration PulsePlain Seat: Rigid Seat (y) Acc
(d)
56
Comparison between EA Plain Seat and Foam Cushion Seat with Hybrid III dummy subject to prescribed deceleration pulse during impact after freefall
-Lower Torso (y) Acceleration
-20-10
0102030405060708090
0 20 40 60 80 100 120
Time (ms)
Acc
eler
atio
n (G
)
Foam Cushion: Lower Torso (y) AccInput Deceleration PulsePlain Seat: Lower Torso (y) Acc
(e)
Figure 3.4 Results of EA Seat with HYBRID III dummy subject to vertical drop
testing
3.5 EA seat with GEBOD dummy subjected to mine blast
testing
Comparison between EA Airbag Cushion Seat and Foam Cushion Seatwith GEBOD dummy subject to prescribed Mine Blast
-Head Acceleration
-8-4048
12162024
0 20 40 60 80 100 120 140
Time (ms)
Acc
eler
atio
n (G
)
Head (x) Acc - Foam
Head (y) Acc - Foam
Head (x) Acc - Airbag
Head (y) Acc - Airbag
(a)
57
Comparison between EA Airbag Cushion Seat and Foam Cushion Seatwith GEBOD dummy subject to prescribed Mine Blast
-Input and Lower Torso Pulses
-50
0
50
100
150
200
0 20 40 60 80 100Time (ms)
Acc
eler
atio
n (G
)
Low er Torso (y) Acc - Foam
Rigid Seat (y) Acc
Input Pulse
Low er Torso (y) Acc - Airbag
(b)
Comparison between EA Airbag Cushion Seat and Foam Cushion Seatw ith GEBOD dummy subject to prescribed Mine Blast
-Input and Middle Torso Pulses
-50
0
50
100
150
200
0 20 40 60 80 100Time (ms)
Acc
eler
atio
n (G
)
Mid Torso (y) Acc - Foam
Rigid Seat (y) Acc
Input Pulse
Mid Torso (y) Acc - Airbag
(c)
58
Comparison between EA Airbag Cushion Seat and Foam Cushion Seatwith GEBOD dummy subject to prescribed Mine Blast
-Upper Torso and Seat (x) Force
-20000
2000400060008000
1000012000
0 20 40 60 80 100 120 140
Time (ms)
Forc
e (N
)
Foam Cushion SeatAirbag Cushion Seat
(d)
Comparison between EA Airbag Cushion Seat and Foam Cushion Seatwith GEBOD dummy subject to prescribed Mine Blast
-Lower Torso and Seat (y) Force
-4000-2000
02000400060008000
10000
0 20 40 60 80 100 120 140
Time (ms)
Forc
e (N
)
Foam Cushion Seat
Airbag Cushion Seat
(e)
Figure 3.5 Results of EA Seat with GEBOD dummy subject to mine blast testing
59
3.6 EA seat with HYBRID III dummy subjected to mine blast
testing
Comparison between EA Plain Seat and Foam Cushion Seatwith Hybrid III dummy subject to prescribed Mine Blast
-Head Acceleration
-20
0
20
40
60
80
100
0 20 40 60 80 100 120
Time (ms)
Acc
eler
atio
n (G
) Plain Seat: Head (x) Acc
Foam Cushion: Head (x) Acc
period where gravity isused to settle the dummy
(a)
Comparison between EA Plain Seat and Foam Cushion Seatwith Hybrid III dummy subject to prescribed Mine Blast
-Input and Lower Torso Pulses
-50
0
50
100
150
200
0 20 40 60 80 100 120
Time (ms)
Acc
eler
atio
n (G
) Plain Seat: Low er Torso (y)AccInput Deceleration Pulse
Foam Cushion: Low er Torso(y) Acc
(b)
60
Comparison between EA Plain Seat and Foam Cushion Seatwith Hybrid III dummy subject to prescribed Mine Blast
-Input and Middle Torso Pulses
-50
0
50
100
150
200
0 20 40 60 80 100 120
Time (ms)
Acc
eler
atio
n (G
) Plain Seat: Mid Torso (y) Acc
Input Deceleration Pulse
Foam Cushion: Mid Torso (y)Acc
(c)
Figure 3.6 Results of EA Seat with HYBRID III dummy subject to mine blast testing
3.7 Improved Modeling of the EA seat structure
So far the energy absorbing seat structure used was modeled based on
dimensions from [9]. However, after making a few simple modifications to the
structure, better results were obtained. The run time of the simulation can be
drastically reduced by making all elements excepting the crush tube elements
as rigid, so that they are bypassed during element processing. Then they can
be switched back to deformable to study the bending of the rails and overall
structural deformability and strength. 4-noded shell elements (quad) are
computationally less expensive than 8-noded solid elements (brick) and are the
preferred choice wherever possible. When using shell elements, it is important to
take the thickness of the shell into consideration while modeling the mid-
surface. Figure 3.7 displays an improved modeling of the EA seat structure.
61
Figure 3.7 Improved modeling of the EA seat structure
3.8 Effect of Aluminum Yield Strength on the Simulations
According to Alexander, the mean axial crushing load of aluminum tubes is
given by
Pav = 6Yt(Dt)1/2 (3.1)
This implies that Pav is directly proportional to the yield strength Y. However this
applies to the static crushing of tubes. In these simulations, there is an 80 G
impulse load applied to the crush tube in a 10-20 ms period. The variation of
Rigid Seat deceleration pulse is around only 1-2 G for yield stresses ranging from
124 MPa to 230 MPa. This clearly demonstrates the dynamic effect of high
impulse axial loading of aluminum tubes, which results in approximately similar
energy absorption and deceleration pulse for the studied range of yield stresses.
The limit of the study was 230 MPa as once the yield stress reaches that of the
steel rail which is 440 MPa, the rail begins to bend excessively before the
crushing of the aluminum tubes. Similarly 124 MPa was chosen as the starting
62
yield stress as this corresponds to a soft version of Al 3003 grade. As the strain
rate decreases, the difference in response of aluminum of varying yield stresses
becomes more prominent. This means for a 20-30 G impulse load, the
difference in the deceleration pulse will be around 5-10 G.
Effect of Aluminum yield strength on the Rigid SeatDeceleration Pulse
-2
0
2
4
6
8
10
0 0.01 0.02 0.03 0.04 0.05
Time (seconds)
Acc
eler
atio
n (G
)
Al 124 MPaAl 200 MPaAl 230 MPaAl 145 MPa
Figure 3.8 Effect of Aluminum yield strength on the simulations
3.9 Stages of Crushing of the Aluminum Crush Tube
Snapshots of the crushing of the aluminum tube during the simulation are shown
below. It has been reported that based on the strain rate, the onset of crushing
will occur at different locations. For low strain rates (0.01 to 1 s-1), the crushing
usually initiates at the point of impact of the impactor and the crush tube. For
very high strain rates (10 to 40 s-1), the crushing usually initiates at the end of the
tube which is rigidly supported. The presence of imperfections in the tube will
also greatly influence the location of initiation of the crushing and the mode in
which the crushing will occur, i.e. sequential concertina or diamond mode. Shell
elements prove to be much more computationally efficient than solid elements
when modeling the crush tube. However to get a proper shape of the fold
formation, a very fine mesh of the order of 1 mm needs to be used, which is not
feasible. Thus a coarser mesh is used, which will result in a non-axisymmetric
63
mode of crushing, or a crumpling effect, as seen in Figures 3.9 and 3.10.
However the amount of energy absorbed and the shell response was observed
to be identical to the case when solid elements were used to model the crush
tube. The shape of the crushed tube when using solid elements with the same
material model can be seen in Figure 3.11.
3.9.1 Stages of crushing for the original EA seat model
(a) (b)
(c)
Figure 3.9 a) Plastic buckling first occurs at the bottom bracket b) Onset of
plastic buckling at the top bracket c) Completely crushed aluminum crush tube
64
3.9.2 Stages of crushing for the improved EA seat model
(a) (b)
(c)
Figure 3.10 a) Plastic buckling first occurs at the top bracket b) Onset of plastic
buckling at the bottom bracket c) Completely crushed aluminum crush tube
3.9.3 Shape of the Crushed Tube when Modeled with Solid
Elements
Figure 3.11 displays a snapshot of the shape of the crushed tube when
modeled with solid elements. The time step fell below 1.00E-08 leading to
automatic deletion of the solid elements by the material model, to prevent the
65
simulation from running indefinitely. This is a disadvantage of using
computationally expensive elements and a very fine mesh in order to get the
most accurate representation of the physical phenomenon.
Figure 3.11 Shape of the crushed tube modeled with solid elements
3.10 Final EA Seat Design for use in full scale Vertical Drop
Testing and Mine Blast Testing
Figure 3.12 displays the final model of EA seat structure. The headrest has been
given a contoured shape to better support the head. Closer tolerances have
been used in all dimensions along with a finer mesh for the crush tubes. In
previous models, results from only the head, lumbar and torso regions were
concentrated on. Now the base of the structure has been extended till
underneath the dummy’s feet in order to extract the foot acceleration and
impact force along with the rest of the other simulation data. Finally, all the
important points of impact between the dummy and the vehicle have been
66
covered and can now be analyzed for occupant survivability. Since the data
that can be extracted from the HYBRID III dummy is more extensive and reliable
than the GEBOD dummy, it is the preferred dummy of choice. A 5th percentile
dummy has been used.
Figure 3.12 Final model of the EA seat structure
3.10.1 Vertical Drop Testing
-40
-20
0
20
40
60
80
100
0 20 40 60 80 100 120
Time (ms)
Acc
eler
atio
n (G
)
Low er Torso
Seat
Structure
Figure 3.13 Deceleration pulses
67
-2000
0
2000
4000
6000
8000
10000
12000
14000
16000
0 20 40 60 80 100 120
Time (ms)
Forc
e (N
)
Tube Crushing
Dummy-Seat Contact
Figure 3.14 Dynamic axial crushing force of the tube, and Dummy-seat contact
force
-5000
0
5000
10000
15000
20000
25000
0 20 40 60 80 100 120
Time (ms)
Forc
e (N
)
Foot-Floor Contact
Figure 3.15 Contact force between the foot and floor
68
3.10.2 Mine Blast Testing
-50
0
50
100
150
200
0 20 40 60 80 100 120
Time (ms)
Acc
eler
atio
n (G
)
Low er Torso
Seat
Structure
Figure 3.16 Acceleration pulses
-2000
0
2000
4000
6000
8000
10000
12000
14000
16000
0 20 40 60 80 100 120
Time (ms)
Acc
eler
atio
n (G
)
Tube Crushing
Dummy-SeatContact
Figure 3.17 Dynamic axial crushing force of the tube, and Dummy-seat contact
force
69
-2000
0
2000
4000
6000
8000
10000
12000
14000
0 20 40 60 80 100 120
Time (ms)
Acc
eler
atio
n (G
)
Foot-Floor Contact
Figure 3.18 Contact force between the foot and floor
3.11 New EA Mechanism
A conventional energy absorbing axially crushable aluminum tube and steel
rail is used to provide occupant protection against vertical acceleration
pulses. However during a mine blast, there are significant lateral forces and
accelerations as the occupant and vehicle are thrown sideways and
backwards. It is not feasible to design a horizontal steel rail-aluminum crush
tube to protect against this. Further all 360° in the horizontal plane needs to
be protected as depending on the location of the mine blast under the
vehicle, the occupant may be accelerated in any direction. This means that
cylindrical crush tubes will have to be placed in a circular pattern in as many
directions as possible, which is not feasible. Instead, a circular annular disk
having a honeycomb structure can be utilized to protect against sideward
and lateral accelerations. LSDYNA offers a Honeycomb Material Model. A
rigid thin steel cylinder placed in the annulus will move laterally according to
the vehicle’s motion as it is thrown up and sideways in a mine blast. As the
steel cylinder moves laterally, it begins to crush the aluminum honeycomb
70
which contacts it throughout the circumference. Another rigid steel cylinder
on the outer periphery provides the surface against which the honeycomb is
compressed against. Figure 3.19 displays the new honeycomb EA
mechanism.
Figure 3.19 New honeycomb EA mechanism
By using many layers of this honeycomb structure, a high degree of
acceleration pulse attenuation can be accomplished in the lateral direction,
reducing injury criteria such as HIC, NIC and forces between the torso and
seat. Rigid circular steel covers are placed above and below the
honeycomb structure and thin steel cylinder in order to contain the entire
mechanism and ensure there is only movement in the lateral direction.
Figure 3.20 displays the interior view of the honeycomb EA design.
71
Figure 3.20 Interior view of the EA mechanism
3.12 Conclusions
1. The crushing of aluminum tubes has proven to be a very effective energy
absorption technique and can attenuate input deceleration pulses to
survivable levels.
2. The initial static crushing strength of the aluminum material used
influences the peak deceleration pulse felt at the rigid seat and brackets for
low strain rates. However for higher strain rates, the dynamic effect comes
into play and there is little difference in the response of the aluminum crush
tube.
3. The ideal energy absorber for the energy absorbing seat mechanism
would be a very long aluminum crush tube with a low yield stress, however
size constraints are imposed upon the design because of the limited length
available for crushing below the seat pan.
72
4. A foam cushion modeled with either low density foam or crushable foam
helps in increasing occupant survivability by:
a) Delaying the onset of the applied deceleration pulse to the
occupant
b) Attenuating the peak magnitude of the transmitted deceleration
pulse from the seat
c) Extending the time duration of application of the pulse to the
occupant, thereby spreading the shock over a longer duration
5. The responses of the GEBOD and HYBRID III dummy to the same boundary
and loading conditions is different.
6. The H3OUT database corresponding to the HYBRID III dummy contains
data of the various joint moments and forces, and serves as an essential
source of data for comparison with known injury data references to assess
injury.
7. The GEBOD dummy is mainly used for visualization purposes. Only the
accelerations and contact forces of various dummy segments can be
extracted, thus providing for a very limited source of data for comparison
with known injury data references to assess injury.
8. The design that incorporates the aluminum crush tubes and foam cushion
is a robust, efficient and cost-effective design for an energy absorbing seat
mechanism.
9. The usage of prescribed acceleration pulses to simulate vertical drop tests
and mine blasts is an accurate representation of reality.
73
10. It is not possible to assess the extent of occupant injury, merely the
probability of occurrence of an injury. Further work is required in the field of
Injury Criteria.
11. By improving the modeling of the seat structure, viz. better tolerances
and longer crush tubes, the energy absorption can be maximized while the
peak deceleration pulse at the rigid seat can be minimized.
12. Current versions of LSTC Hybrid III dummy do not allow for extraction of
lumbar load data and so we have to wait for future versions from LSTC to
obtain such data.
13. The use of the HYBRID III dummy to model a human occupant during
vertical drop testing and mine blast testing has been properly validated.
3.13 Scope for Further Work
Airbags and occupant restraint systems can be included into the design.
More advanced numerical dummies such as the BioRID and Thor-Lx with
better detailed structures can be used in place of the HYBRID III dummy. The
EA seat mechanism can be incorporated into the mesh of the actual
armored vehicle provided it can be made available from the U.S. Army.
LSDYNA supports ALE formulations, and instead of using prescribed structural
acceleration pulses to simulate mine blasts, actual mine explosions can be
simulated using keywords such as *LOAD_BRODE and *LOAD_BLAST which
creates pressure waves around the structure similar to an actual IED blast, by
specifying parameters such as amount of TNT used and distance of
explosive device from the structure. Other EA mechanisms such as the
Honeycomb structure mentioned in preceding sections can be further
explored.
74
Chapter 4
Impact of Foot during IED/Mine Blast
4.1 Numerical Setup and Methodology
The experimental set up used in [12] formed the basis for the numerical
simulation and is displayed in Figure 4.1. A rigid seat comprised of thin shell
elements is modeled and rigidly fixed in 3-d space. The occupant is simulated
by a 50th percentile HYBRID III dummy which is internally created by LS-DYNA
[122] during the initialization of the simulation run. The dummy is seated on this
rigid seat.
Figure 4.1 Experimental setup of lower leg impact
75
The position and orientation of the arms is not important and is therefore left at
its default values as seen in Figures 4.2 and 4.3, as it plays no role during the
simulation. Similarly an occupant restraint mechanism such as a seatbelt has
not been modeled as there is no significant middle or upper torso movement
during the simulation. The main region of activity lies between the foot and hip
of the dummy.
Figure 4.2 Numerical setup in ‘Sitting Straight’ position
76
Figure 4.3 Numerical setup in ‘Driving’ position
A rigid horizontal wall modeled with thin shell elements simulates the impacting
vehicle floor. The motion of the wall is controlled by prescribing its velocity.
Contact is specified between the dummy-seat and dummy-wall interfaces.
Before the wall starts moving upwards and imparts a high acceleration pulse to
the dummy’s foot; gravity is applied to the dummy so that it properly settles into
the seat and the foot properly contacts the wall. While modeling the wall, care is
taken to position the wall as close as possible to the feet, as even the smallest
gap between the wall and feet can significantly alter extracted data such as
femur axial compressive force. Since it is not possible during modeling to ensure
exact contact, gravity is applied which ensures the foot initially settles against
the wall. As the feet are thrown upwards during the application of the input
77
prescribed velocity pulse to the wall, data such as foot acceleration, lower leg
acceleration, hip moment, knee moment and femur axial compressive force
are measured. This data is later extracted and filtered using LSPREPOST© and
graphical plots for visualization are created. Filtering of data is done with a low
range Butterworth filter, with a cut-off frequency at 300 Hz. The data is then
compared to reference values to assess injury. A series of six simulations are run
for each case of occupant seated position viz. sitting straight and driving
position. For the driving position, the knee flexion-extension angle is changed
from its default value of 90 degrees to 55 degrees as seen in Figure 4.3. The
peak wall impact speed is increased in a step wise manner as follows: 1 ft/s, 5
ft/s, 10 ft/s, 15 ft/s, 25 ft/s, 35 ft/s. The velocity control curve that prescribes the
wall motion is shown in Figure 4.4 below.
0
2
4
6
8
10
12
0 20 40 60 80
Time (ms)
Wal
l Spe
ed (m
/s)
1 ft/s
5 ft/s
10 ft/s
15 ft/s
25 ft/s
35 ft/s
gravitysettling
Figure 4.4 Prescribed velocity of the wall
78
4.2 Numerical Results and Discussion
First the data from the simulations is validated against experimental data from
[12]. It is important to note the scarcity of available data due to limited research
conducted, especially on human cadavers, and the classified nature of such
work. Data such as foot acceleration and femur axial compressive force have
injury criteria [7, 13, 14, 15, 27] associated with them and are therefore used for
validation. However data such as knee, hip and ankle moments do not have
associated injury criteria yet and further research needs to be conducted into
this. This data has still been presented in our study as it is important and can
serve as a reference in the future. As can be seen from Figures 4.5 to 4.8, our
data from numerical simulations is in very good agreement with the
experimental data.
-8000
-7000
-6000
-5000
-4000
-3000
-2000
-1000
0
1000
0 5 10 15 20Time (ms)
Forc
e (N
)
experiment
simulation
Figure 4.5: Validation of femur axial compressive force with test db2a
79
-40
-20
0
20
40
60
80
100
120
140
160
0 5 10 15 20 25Time (ms)
Acc
eler
atio
n (G
) experiment
simulation
Figure 4.6: Validation of foot acceleration with test db2a
-12000
-10000
-8000
-6000
-4000
-2000
0
2000
0 5 10 15 20Time (ms)
Forc
e (N
)
experiment
simulation
Figure 4.7: Validation of femur axial compressive force with test db3a
80
-100
-50
0
50
100
150
200
250
0 5 10 15 20 25Time (ms)
Acc
eler
atio
n (G
) experiment
simulation
Figure 4.8: Validation of foot acceleration with test db3a
4.2.1 Hybrid III dummy in a sitting straight position
Figures 4.9 to 4.14 display the data extracted from the HYBRID III dummy in the
sitting straight position, subject to foot impact at six varying peak impact
speeds. All data are seen to display a trend that can be accurately interpolated
or extrapolated to predict results, eliminating the need of running further
simulations. There is no noticeable activity at the upper torso and the main
region of interest is between the foot and hip. The ankle moment has not been
extracted as the foot maintains its inclination of 90 degrees with the lower leg
throughout the simulation. The femur axial compressive load exceeds the safety
limit of 10,000 N between the 25 and 35 ft/s peak wall impact speeds. The
81
exact critical speed can be easily interpolated and is found to be
approximately 27.5 ft/s for the given nature of prescribed input acceleration
pulse. The hip flexion-extension moments for the cases of 25 ft/s and 35 ft/s wall
impact speeds are displayed separately in Figure 4.11, as their magnitudes are
greater than the hip moments for the cases of 1 ft/s – 15 ft/s by an order of
about 100. Correspondingly, the knee flexion-extension moment values for
these higher range impact speeds, of 25 ft/s and 35 ft/s have been truncated
such that the data lines fit within the limits of the plot. The exact significance of
these high moments at the knee and hip is not fully understood at this time as
there is no associated injury for this data. They have been presented as a future
reference.
Hybrid III - Sitting PositionFoot (z) Acceleration
-500
50100150200250300
0 10 20 30 40 50 60 70
Time (ms)
Foot
Acc
eler
atio
n (G
)
1 ft/s Wall Speed5 ft/s Wall Speed10 ft/s Wall Speed15 ft/s Wall Speed25 ft/s Wall Speed35 ft/s Wall Speed
Figure 4.9 Foot (z) acceleration
82
Hip (y) Flexion-Extension MomentHybrid III - Sitting Position
-500000-400000-300000-200000-100000
0100000
0 10 20 30 40 50 60 70
Time (ms)
Hip
Mom
ent N
-mm
1 ft/s Wall Speed5 ft/s Wall Speed10 ft/s Wall Speed15 ft/s Wall Speed
Figure 4.10: Hip flexion-extension moment for wall speeds 1 ft/s - 15 ft/s
Hip (y) Flexion-Extension MomentHybrid III - Sitting Position
-50000000
-40000000
-30000000
-20000000
-10000000
0
10000000
0 10 20 30 40 50 60 70
Time (ms)
Hip
Mom
ent N
-mm
25 ft/s Wall Speed35 ft/s Wall Speed
Figure 4.11: Hip flexion-extension moment for wall speeds 25 ft/s - 35 ft/s
83
Hybrid III - Sitting PositionLower Leg (z) Acceleration
-500
50100150200250
0 10 20 30 40 50 60 70
Time (ms)
Low
er L
eg
Acc
eler
atio
n (G
)
1 ft/s Wall Speed5 ft/s Wall Speed10 ft/s Wall Speed15 ft/s Wall Speed25 ft/s Wall Speed35 ft/s Wall Speed
Figure 4.12: Lower leg (z) acceleration
Hybrid III - Sitting PositionFemur Axial Compressive Force
-15000-12000-9000-6000-3000
03000
0 10 20 30 40 50 60 70
Time (ms)
Fem
ur A
xial
C
ompr
essi
ve F
orce
(N)
1ft/s Wall Speed5ft/s Wall Speed10ft/s Wall Speed15ft/s Wall Speed25ft/s Wall Speed35ft/s Wall Speed
Figure 4.13: Femur axial compressive force
84
Knee (y) Flexion-Extension MomentHybrid III - Sitting Position
-500-400-300-200-100
0100
0 10 20 30 40 50 60 70
Time (ms)
Knee
Mom
ent N
-mm
(x
1E-
06) 1 ft/s Wall Speed
5 ft/s Wall Speed10 ft/s Wall Speed15 ft/s Wall Speed
Figure 4.14: Knee flexion-extension moment
4.2.2 Hybrid III dummy in a driving position
Figures 4.15 to 4.21 display the data extracted from the HYBRID III dummy in the
driving position, subject to foot impact at six varying peak impact speeds. All
data are seen to display a trend that can be accurately interpolated or
extrapolated to predict results, eliminating the need of running further
simulations. There is no noticeable activity at the upper torso and the main
region of interest is between the foot and hip. Even at a 35 ft/s peak impact wall
speed, the femur axial compressive force stays below the safety limit of 10,000
N, implying no injury to the foot. Since the axis of the femoral bone is inclined to
the direction of motion of the impacting wall, the force transmitted to the lower
leg is limited to the vertical component of the total force. This explains the
significantly lower compressive axial femur force as compared to the sitting
85
straight position. The critical impact speed can be easily obtained through
extrapolation. Initially, the heel of the foot is the only contact point between the
dummy and the wall. As the wall moves upwards, the foot rapidly changes its
initial inclination angle of 90 degrees with the lower leg, as the ankle joint
rotates. Significant ankle moments can be observed for the higher wall impact
speeds compare to the negligible values for speeds below 15 ft/s. Thus the
ankle moments for the cases of 15 ft/s – 35 ft/s are displayed separately in
Figure 4.21. The ankle moments for the peak impact speeds of 1 ft/s – 10 ft/s do
not display any trend, however their magnitudes are of the order of 1.00E-11
and can therefore be treated as negligible. It is not known at what peak dorsi-
plantar flexion moment the ankle joint will fail, and whether this failure can be
numerically simulated by the HYBRID III dummy.
Hybrid III - Driving PositionFoot (z) Acceleration
-50
0
50
100
150
200
0 10 20 30 40 50 60 70
Time (ms)
Foot
Acc
eler
atio
n (G
)
Wall Speed 1ft/sWall Speed 5ft/sWal Speed 10ft/sWall Speed 15ft/sWall Speed 25ft/sWall Speed 35ft/s
Figure 4.15 Foot (z) acceleration
86
Hybrid III - Driving PositionHip (y) Flexion-Extension Moment
-80000
-60000
-40000
-20000
0
20000
0 10 20 30 40 50 60 70
Time (ms)
Hip
Mom
ent N
-mm
Wall Speed 1ft/sWall Speed 5ft/sWall Speed 10ft/sWall Speed 15ft/s
Figure 4.16: Hip flexion-extension moment
Hybrid III - Driving PositionLower Leg (z) Acceleration
-50
0
50
100
150
200
0 10 20 30 40 50 60 70
Time (ms)
Low
er L
eg
Acce
lera
tion
(G)
Wall Speed 1ft/sWall Speed 5ft/sWall Speed 10ft/sWall Speed 15ft/sWall Speed 25ft/sWall Speed 35ft/s
Figure 4.17: Lower leg (z) acceleration
87
Hybrid III - Driving PositionFemur Axial Compressive Force
-10000-8000-6000-4000-2000
02000
0 10 20 30 40 50 60 70
Time (ms)
Fem
ur F
orce
(N)
Wall Speed 1ft/sWall Speed 5ft/sWall Speed 10ft/sWall Speed 15ft/sWall Speed 25ft/sWall Speed 35ft/s
Figure 4.18: Femur axial compressive force
Hybrid III - Driving PositionKnee (y) Flexion-Extension Moment
-505
10152025
0 10 20 30 40 50 60 70
Time (ms)
Knee
Mom
ent N
-mm
(x 1
E-0
6)
Wall Speed 1ft/sWall Speed 5ft/sWall Speed 10ft/sWall Speed 15ft/s
Figure 4.19: Knee flexion-extension moment
88
Hybrid III - Driving PositionAnkle (y) Dorsi-Plantar Flexion
-4E-11-2E-11
02E-114E-116E-118E-11
0 10 20 30 40 50 60 70
Time (ms)
Ank
le M
omen
t N-m
m
Wall Speed 1ft/sWall Speed 5ft/sWall Speed 10ft/s
Figure 4.20: Ankle dorsi-plantar flexion moment for wall speeds 1 ft/s - 10 ft/s
Hybrid III - Driving PositionAnkle (y) Dorsi-Plantar Flexion
-10000000
-8000000
-6000000
-4000000
-2000000
0
2000000
0 10 20 30 40 50 60 70
Time (ms)
Ank
le M
omen
t N-m
m
25 ft/s Wall Speed35 ft/s Wall Speed
Figure 4.21 Ankle dorsi-plantar flexion moment for wall speeds 15 ft/s - 35 ft/s
89
4.3 Parametric Study
The data analyzed from the simulations showed a trend and therefore a simple
parametric study was undertaken to determine if it is possible to predict the
lower leg and foot response without having to run a numerical simulation or
conduct a destructive test. In a simple case study, there is one independent
variable viz. wall speed (as dictated by the input pulse) and three dependant
variables in the simulations involving dummy foot impact by a rigid moving wall,
as seen in Figure 4.22. They are:
1) Knee angle (θ1)
2) Ankle angle(θ2)
3) Dummy Size (5th, 50th or 95th percentile)
There are other variables which can also be considered such as hip angle,
dummy gender and type, et cetera; however this will lead to a very complex
study, beyond the scope of the current undertaking.
Figure 4.22 Variables used in the parametric study
90
The data extracted for assessment of injury comprises:
1) Axial Compressive Femur Force
2) Foot Acceleration
3) Lower Leg Acceleration
By running simulations while systematically varying the dependant variables, it is
possible to parametrically study the effect each variable has on the extracted
data. By using non-linear regression techniques and curve fitting, equations of
best fit can be derived that handle from one to all of the dependant variables
at a time. Further, it is possible to code the entire system of equations into a
software package, such that a user can input the required variables and extract
peak values of injury criteria, after which a comparison can be made to
allowable values to assess injury. This will further reduce the reliance on running
numerous numerical simulations, which can also prove costly. It will also provide
a useful ready reference during the design of occupant survivability
enhancement mechanisms to handle the explosion of AT mines.
Sample plots from the one variable parametric study of the data extracted from
the HYBRID III dummy are presented below. Figure 4.23 displays the variation of
peak foot acceleration with peak wall speed for a 50th percentile HYRBID III
dummy in the driving position. The equation of best fit used in a fifth order
polynomial (seen in the figure in red font) and corresponds to a perfect
coefficient of correlation (R2=1).
91
Variation of Foot Acceleration with Wall Speed for HYBRID III in the Driving Position
y = 3E-05x5 - 0.002x4 + 0.0492x3 - 0.445x2 + 5.654x - 0.2515R2 = 1
0
40
80
120
160
0 5 10 15 20 25 30 35 40
Peak Wall Speed (ft/s)
Peak
Foo
t Acc
eler
atio
n (G
)
curve of best fit
Figure 4.23 Variation of peak foot acceleration with peak wall speed for a
dummy in a driving position
Figure 4.24 displays the variation of peak femur force with wall speed for a
50th percentile HYRBID III dummy in the driving position. A fifth order
polynomial was used to curve fit the data. We observe the peak femur axial
compressive force does not exceed the safety limit of 10,000 N at the
maximum peak wall speed of 35 ft/s used in our study. However, we can
extrapolate the data using the best fit curve equation to determine at what
peak wall speed, the peak compressive load will reach 10,000 N. This is an
advantage of the parametric study. Further numerical simulations do not
need to be run to determine the above unknown value.
92
Variation of Femur Force with Wall Speed for HYBRID III in the Driving Position
y = -0.0002x5 + 0.015x4 - 0.4452x3 + 6.7427x2 - 307.46x - 13.749R2 = 1
-10000
-8000
-6000
-4000
-2000
0
0 5 10 15 20 25 30 35 40
Peak Wall Speed (ft/s)
Pea
k Fe
mur
Axi
al C
ompr
essi
ve
Forc
e (N
) curve of best fit
Figure 4.24 Variation of peak femur force with peak wall speed for a dummy in
a driving position
Figure 4.25 displays the variation of peak femur force with wall speed and knee
angle for the 50th percentile HYBRID III dummy in various seated positions.
Variation of Peak Femur Force with Wall Speed and Knee Anglefor the HYBRID III
-14000-12000-10000
-8000-6000-4000-2000
0
0 10 20 30 40Peak Wall Speed (ft/s)
Peak
Fem
ur F
orce
(N)
Knee Angle = 0 (Sitting)
Knee Angle = 55 (Driving)
Knee Angle = 27.5 (Extrapolated)
Extrapolated data
Figure 4.25 Variation of peak femur force with wall speed and knee angle for
various dummy positions
93
Here, both wall speed and knee angle are varied systematically, and the
resultant effect on the femur force is studied. While 4th and 5th order
polynomials are used to curve fit the Femur Force Vs. Wall Speed data, a
simple average is used to interpolate the effect of varying the Knee Angle
(seated dummy position). Though we did not run further simulations at this point,
by running more simulations with varying knee angles, a better fit parametric
equation for the knee angle can be derived. Similarly, the effect of the Ankle
Angle can also be included in the parametric study (three variable study) to
provide a comprehensive analysis of the data.
4.4 Conclusions
By comparing the dummy data extracted from the simulations with the known
allowable values, injury can be accurately assessed. The femur axial
compressive force is far higher in the case of sitting straight position than the
driving position as the entire compressive load is directly transmitted to the
femoral bone, since the direction of the compressive load coincides with axis of
the femoral bone. The ankle moments are significantly higher for the upper
range of wall impact speeds in the driving position, compare to the sitting
straight position where the complete lower surface of the foot maintains its flat
contact with the wall throughout the simulation leading to negligible ankle
moments. Thus occupant position plays an important role in the magnitude of
loads transmitted and injury severity. The use of the HYBRID III dummy for
occupant simulation during mine blast testing has been satisfactorily validated,
after comparison of foot acceleration and femur axial compressive load, with
experimental data. Further data extracted such as hip, knee and ankle
moments can therefore be used now for accurate injury assessment. It has
been reported in automobile crash testing that the HYBRID III legs are too stiff
94
which may lead to an underestimation of injury. The accuracy of results can be
optimized by using more advanced dummies that better model the human
body such as the Thor-Lx and Hybrid Denton leg [12]. However our simulations
have demonstrated the use of the HYBRID III dummy for occupant safety
assessment during a mine blast application with satisfactory results. There is a
scarcity of available data pertaining to lower leg impact during a mine blast
under armored vehicles and extensive research needs to be conducted,
especially testing on human cadavers in order to better understand the injury
assessment and establish a reliable and extensive source of experimental data.
Further, new injury criteria for the foot need to be developed. A simple
parametric study was presented to predict the occupant response during lower
leg impact. The parametric study used curve fitting of numerical data using 4th
and 5th order polynomials. A very good coefficient of correlation was seen,
justifying the use of a parametric approach to study trends in the occupant
response without having to resort to further numerical simulations and destructive
tests.
4.5 Scope for Future Work
More advanced dummies with detailed leg models can be used in place of
the HYBRID III dummy. The lower leg and foot response can be further studied for
different types of input pulses and impact speeds, as well as other occupant
seated positions. More injury criteria can be formulated that utilizes other data
apart from foot accelerations and compressive femur loads. The simple
parametric study presented to predict the occupant’s response to lower leg
impact can be further worked on by increasing the data set used for curve
fitting thereby ensuring better accuracy and more generality. Also, more input
parameters can be included in the regression analysis, such as occupant size,
weight, and gender which vary as per the numerical dummy used, thereby
varying the occupant’s lower leg response to foot impact.
95
Chapter 5
Dynamic Axial Crushing of Circular Tubes:
Numerical Formulation
5.1 Need for a Simple Numerical Formulation
Studies of the axial crushing of cylindrical shells involve analytical and numerical
approaches, which are then compared to experimental studies to validate the
proposed approach. Analytical studies are limited in their approach and
sometimes rely on the curve fitting of experimental data in order to obtain
expressions for certain parameters. Further, analytical approaches are usually
able to only predict either approximate or mean values of the parameters
studied, such as mean static and dynamic crushing load, approximate half
wavelength of crushing, and approximate energy absorbed per fold formation.
It is not possible to obtain the time histories of variables during the impact event,
such as instantaneous impactor velocity and deceleration, instantaneous
plastic work of crushing, and instantaneous dynamic crushing load. These
parameters are important in applications involving occupant safety, where they
can be compared with reference values to assess occupant survivability. For
example, the acceleration response of the impactor will yield the acceleration
response of a crew seat in a vertical drop test to simulate aircraft crash-landing.
This can then be compared to the peak acceleration that can be sustained by
a human to assess survivability. Table 5.1 displays the human tolerance limits for
typical crash pulses along three mutually orthogonal axes, for a well restrained
young male. These values provide a general outline of the safe acceleration
limit for a human during a typical crash. Higher acceleration pulses can be
96
sustained for shorter durations compare to lower acceleration pulses for longer
durations [14].
Direction of Accelerative
Force
Occupant’s Inertial Response Tolerance Level
Headward (+Gz) Eyeballs Down 25 G
Tailward (-Gz) Eyeballs Up 15 G
Lateral Right (+Gy) Eyeballs Left 20 G
Lateral Left (-Gy) Eyeballs Right 20 G
Back to Chest (+Gx) Eyeballs-in 45 G
Chest to Back (-Gx) Eyeballs-out 45 G
Table 5.1 Human tolerance limits to acceleration [14]
Numerical studies involve the use of commercial finite element packages, such
as the dynamic analysis finite element code ABAQUS© in [33, 37, 38], and
RADIOSS© in [35]. These analyses sometimes contain inaccuracies due to the
high mesh-sensitivity of the impact simulation. There is a difference in shell
response when simulating impact as a moving mass striking the stationery shell
as commonly observed in laboratory conditions, and as a the moving shell
striking a stationery rigid wall. Also, inappropriately filtering the data can lead to
significant under estimation of results such as crushing load [38]. However, the
results from numerical studies are still reliable and can be used to accurately
study the crushing of tubes. As the mesh density increases, so does the
computational intensity, and subsequently running numerous simulations to
study the response of different combinations of mesh densities, impactor
parameters, tube material and geometrical properties becomes very expensive
computationally, and time consuming.
Therefore, a simple numerical formulation was developed that extended on
analytical approaches and was able to accurately predict the instantaneous
response of the cylindrical shell under dynamic axial impact, but at a fraction of
97
the computational expense. The primary purpose of the numerical formulation
remained the response prediction for any combination of input parameters
such as impactor mass and velocity, and tube geometric and material
properties. The formulation is based on an energy balance approach that
primarily utilizes analytical work of [28, 29, 31], to predict the response history
such as the impactor velocity and deceleration, plastic work and kinetic energy
dissipated, and the dynamic crushing load. The formulation is implemented in
the high-level language MATLAB© and is run on a Pentium 1.3 GHz personal
computer. Run times average around 40 seconds for a tube approximately 228
mm in length impacted at 8 m/s by a 78 kg mass.
5.2 Theory and Formulation
Terminology:
h thickness of shell element / cylinder wall
2H initial distance between hinges on top and bottom of a basic
folding
element
α folding angle
v velocity of impacting mass
R mean radius of cylinder
D mean diameter of cylinder
l folding length
L length of cylindrical tube
ε circumferential strain
σo yield stress
NRB refers to a nodal rigid body that may either represent an impactor
or
a combination of a seat and a human occupant
98
m mass of NRB or impactor
Δt time step
M full plastic moment of the tube wall per unit length
The formulation can be used in two configurations, each of which will require
slight modifications be made to the program code. The first configuration
involves an impactor striking a stationery aluminum crush tube, whose farthest
end is supported by a stationery rigid wall. The second configuration involves the
application of the formulation in crashworthiness applications. Here, both the
aluminum tube and the impactor initially move with the same velocity. A large
deceleration pulse is applied to the supporting structure to simulate the impact
event, say impact after freefall as displayed in Figure 5.1.
-20
0
20
40
60
80
100
0 20 40 60 80
Time (ms)
Acc
eler
atio
n (G
)
Vertical Drop TestPulse
Figure 5.1 Applied deceleration pulse simulating impact after freefall
This deceleration pulse is transmitted from the support structure to the impactor
which begins to crush the tube. The final response of the impactor will indicate
the crushing response of the aluminum tube. Usually, the crew seat of an aircraft
represents the impactor, and thus its acceleration response will indicate the
99
deceleration pulse that will be transmitted directly to the human occupant, and
this forms the basis of the survivability study.
This formulation uses a simple energy balance equation in order to predict the
response history of the energy absorbing seat or EA seat, as the aluminum tubes
get axially crushed during impact. The initial kinetic energy of the NRB (here, EA
seat) is absorbed by various sources during the impact test and is given by
Equation 5.1
_ _ _. . .......init finalNRB NRB crushing friction rail bending seat frame defK E K E E E E E E= + + + + +
(5.1)
For the sake of simplicity, we only consider energy dissipated by the dynamic
axial crushing of the aluminum tubes, as this is the predominant source of
energy dissipation. If the NRB were only to represent an impactor, then the other
sources such as seat frame deformation and seat cloth tearing would
automatically be excluded as they would be non-applicable to the study.
Rewriting Equation 5.1 in iterative form, we get
( ) ( 1) ( ). .n n nNRB NRB crushingK E K E E−= −Δ
(5.2)
For the first step of the iteration, the crushing energy is zero and therefore the
kinetic energy of the NRB is given by
(1) 21.2NRB initK E mv=
(5.3)
Equation 5.4 to Equation 5.17 discusses the approach used to compute the
energy absorbed during the dynamic axial crushing of the tubes. Figure 5.2
100
depicts the formation of a basic folding element, which occurs progressively
until the entire tube has been crushed or the impactor has been brought to rest.
Figure 5.2 Formation of a basic folding element [29]
Johnson [157] computes the energy absorbed due to circumferential forces as
10
2H
oE h dAσ ε= ∫ (5.4)
with
[ ]{2 ( sin ) 2 } 2d R s R Rdtα
εΠ + − Π Π
= (5.5)
and
2 ( sin )dA R s dsα= Π + (5.6)
101
Alexander [31] however does not consider the variation of ‘s’ in the computation
of mean circumferential strain. It is also worthwhile to note that the simple
approaches adopted by Alexander [31], Abramowicz and Jones [29] do not
account for strain hardening of the material during the formation of lobes.
Substituting Equations 5.5 and 5.6 in Equation 5.4, we get, as per [29]
2 31
1sin4 cos
2 3odE H HE hdt R
ασ αα⎛ ⎞
= = Π +⎜ ⎟⎝ ⎠ (5.7)
The aim is to obtain the energy dissipated by the formation of a lobe at each
time increment, so that the dynamic response can be studied and time histories
of plastic work and NRB velocity can be extracted.
Writing Equation 5.7 in iterative form
2 3 ( )( ) ( ) ( ) ( )1
sin4 cos2 3
nn n n n
oH HE h t
Rασ α α
⎛ ⎞Δ = Π + Δ⎜ ⎟
⎝ ⎠ (5.8)
Disregarding the variation of mean circumferential strain with ‘s’, Equation 5.5
becomes
cosHRααε =
(5.9)
Equation 5.9 is solved using the non-linear Levenberg-Marquardt formulation, to
obtain the fold angle at each increment, since the strain rate at each time step
is known. Substituting rate of change of fold angle from Equation 5.9 into
Equation 5.8, we obtain
102
2 3 ( ) ( )( ) ( ) ( )1 ( )
sin4 cos2 3 cos
n nn n n
o n
H H RE h tR Hα εσ α
α⎛ ⎞
Δ = Π + Δ⎜ ⎟⎝ ⎠ (5.10)
The increment in time or time step is usually kept constant throughout the
analysis. The choice of time step is very important, as if the time step is too
large; it does not capture the folding process properly and can lead to drastic
underestimation of energy dissipated. Within the formation of each lobe, since
0°< α <90°, the sub-iterative process is stopped when the fold angle exceeds
90°. The next main iteration is started, for the formation of the subsequent fold.
Abramowicz and Jones [29] suggested an equation for the strain rate during the
axisymmetric or concertina mode of crushing. The velocity of the NRB and
consequently the strain rate of the aluminum tubes change (decrease) with
each time increment. Thus we need to apply an iterative method to their
equation to update the strain rate at each instant.
( 1)( ) 0.25
0.86 0.618 2
nn v
hR R
ε−
=⎡ ⎤−⎢ ⎥⎣ ⎦ (5.11)
In their derivation of Equation 5.11, Abramowicz and Jones assumed the mean
velocity of the striking mass as half of the initial striking velocity, based on an
approximation of observed experimental results. However, for this application,
based on experimental observations of [9] and simulations run using the explicit
dynamic analysis finite element software LSDYNA© by Nilakantan [50], the mean
velocity is approximately 0.70 - 0.85 times the initial impact velocity. Thus the
factor of 0.25 in Equation 5.11 needs to be suitably calibrated to yield the best
approximation. Values ranging between 0.35 and 0.48 were found to yield the
best application-specific fit. Substituting Equation 5.11 into Equation 5.10, we
103
obtain a final equation for the computation of increment in energy dissipated
by circumferential forces at each time step.
( )2 ( )
( ) ( 1) ( )1
sin 12 3 0.86 0.618 2
nn n n
oH HE hv t
R hR
ασ − ⎛ ⎞Δ = Π + Δ⎜ ⎟
⎝ ⎠ − (5.12)
Alexander [31] calculated the energy dissipated in three stationery
circumferential plastic hinges during the crushing of one lobe
2 4 ( sin )dE Md D hα α= Π + (5.13)
where
2
0243HM σ
= (5.14)
Differentiating Equation 5.13 with respect to time and writing it in iterative form,
as before,
( ) ( ) ( ) ( ) ( )2 4 ( sin )n n n n nE M D h tα α α⎡ ⎤Δ = Π + Δ⎣ ⎦ (5.15)
Thus, the total incremental energy absorbed due to axial crushing, at each time
increment is
( ) ( ) ( )
1 2n n nE E EΔ = Δ + Δ (5.16)
Equation 5.16 is used to update the total crushing energy as
104
( ) ( 1) ( )n n ncrushing crushing crushingE E E−= + Δ (5.17)
We can now compute the net instantaneous kinetic energy of the NRB as
( ) ( 1) ( ). .n n nNRB NRB crushingK E K E E−= −Δ
(5.18)
or
( ) ( ) ( ). .n initial nNRB NRB crushingK E K E E= −
(5.19)
From this, the velocity response of the NRB is computed as
( )( ) 2 . nn NRBK Ev
m=
(5.20)
The above sub-iterative procedure represented by Equation 5.2 to Equation
5.20 is repeated until the fold angle reaches 90°. Then, the entire procedure is
repeated ‘N’ times or until the net kinetic energy of the NRB falls below zero,
whichever occurs first. The former case indicates that the maximum number of
folds has been formed and the tube is fully crushed, while the latter case
indicates that the velocity of the NRB seat has been brought to zero before the
tube has been fully crushed. Here ‘N’ stands for the number of folds that can be
formed in a tube of length ‘L’ and is given as
/ 2N L l= (5.21)
where
105
‘l’ is the half fold length presented by Alexander [31] as
( )1 2/ 3l Rtπ=
(5.22)
which is based on the static yield stress of the material, and needs to be
corrected by a factor given by [32]
* 0.86 0.568 tl D= − (5.23)
The choice of the fold length is very important as it determines the simulation
run-time as well as the total energy dissipated (number of folds x energy
absorbed per fold) which controls the final velocity. Various authors have
presented different formulae to calculate the fold length, each considering
certain factors. These include Abramowicz and Jones [30]
( )1 21.76 / 2l Rt= (5.24)
which is based on the static yield stress, and Wierzbicki et al. [41]
( )1 22.62 / 2l Rt= (5.25)
which uses a flow stress equal to 92% of the ultimate tensile stress of the material
[38]. The formula of Wierzbicki et al. in Equation 5.25 provided the most
accurate results and was therefore chosen for our formulation. Galib and Limam
[35] have provided a brief tabular description of various analytical models.
Karagiozova et al. [37] have reported that for high energy impact and strain rate
sensitive shells, there is a significant axial crushing of the shell, represented by Δ,
which leads to a shortening of the shell. Thus, if ‘L’ is the initial length of the shell,
106
only ‘L- Δ’ will be available for the formation of folds. This may lead to an
overestimation of the number of folds, if the axial shortening of the shell is not
considered. In such a situation, the energy dissipated during the axial crushing
also needs to be considered into the energy balance, if its magnitude is
significant.
The following portion describes the changes that need to be made when using
the formulation in the second configuration. In the experimental vertical drop
tests outlined in [9], the support structure and platform are not instantaneously
brought to rest, rather, a deceleration pulse is applied as described in Figure
5.1, resulting in a rapid initial decrease in the initial velocity and then the
structure is gradually brought to rest. To account for this deceleration pulse into
the formulation, we need to use the relative velocity between the EA seat and
support structure instead of the instantaneous EA seat velocity, in all equations
where it is used and at each time step, given as
( ) ( ) ( )n n nrel inst strucv v v= −
(5.26)
The input deceleration pulse, similar to the one displayed in Figure 5.1, is time
integrated to obtain the instantaneous structure velocity. This is fed into the
formulation to determine the relative EA seat velocity at each time increment.
Finally, we obtain the entire relative velocity history of the EA seat. A Fast Fourier
Transform (FFT) is performed on the velocity history data to establish the cut-off
frequency that will be used for filtering the data. Then, a low pass Butterworth
filter is applied to the velocity data and it is differentiated with respect to time to
finally obtain the acceleration response of the EA seat. From this, the magnitude
of the maximum acceleration is obtained, which is used for injury assessment
and survivability studies by comparing it to allowable values. The choice of
filtering can have an adverse effect on the interpretation of some
crashworthiness parameters, such as the underestimation of energy dissipation
107
when integrating force-displacement history that has been filtered with low
frequency filters [38]. Similarly, the choice of filter can affect the acceleration
response of the EA seat and therefore care must be taken in choosing the right
filter and filtering frequency.
5.3 Results and Discussion
To validate the formulation, results have been compared with numerical results
from axial impact tests of cylindrical shells by Karagiozova et al. [37], using the
FE code Abaqus/Standard©. This corresponds to the use of the formulation in
the first configuration. Table 5.2 lists the parameters used in the numerical
simulation of [37].
Shell Radius (mm) 11.875
Shell Thickness (mm) 1.65
Shell Length (mm) 106.68
Young’s Modulus (GPa) 72.4
Yield Stress (MPa) 295
Impact Speed (m/s) 4
Impactor Mass (kg) 262.5
Table 5.2 Characteristics of the shell and impactor [37]
Figure 5.3 compares the instantaneous velocity of the impactor from the
numerical simulation and the formulation. The total simulation time, which is
either the time taken to completely crush the tube or bring the impactor to rest,
is accurately predicted by the formulation. The formulation accurately predicts
the instantaneous impactor velocity in the initial period of the simulation.
108
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04Time (s)
Velo
city
(m/s
)
Karagiozova et al.(Numerical)
Formulation
Figure 5.3 Comparison of impactor velocity time history
A slight deviation is observed towards the end of the simulation. This can be
attributed to the strain hardening effect of the material which is captured in the
simulation by using an appropriate material model. However strain hardening is
not accounted for in the formulation, as it is based on the simple uniaxial yield
stress of the aluminum. Since the velocity is calculated from the residual kinetic
energy of the NRB, any deviation in the computed energy will reflect in the
computed velocity.
Figure 5.4 compares the kinetic energy of the impactor (Tk) and the energy
dissipated in plastic deformations (Tp) from the numerical simulation and
formulation. The results from the formulation are in very good agreement with
the numerical simulation.
109
0
0.5
1
1.5
2
2.5
0 0.01 0.02 0.03 0.04Time (s)
Ener
gy/W
ork
(KJ)
Tk - Numerical
Tw - Numerical
Tk - Formulation
Tw - Formulation
Figure 5.4 Comparison of energy transformation during the impact event
Figure 5.5 compares the dynamic crushing load obtained from the numerical
simulation and the formulation. It is not possible to get an exact match in the
shapes of both plots, which was also experienced in [35]; during a comparison
of dynamic crushing load from numerical simulation and from experiments.
However we can observe that the peaks in both cases are in reasonable
agreement in terms of the occurrence and magnitudes. Moreover, the mean
dynamic crushing load as computed from the formulation agrees well with the
numerical simulation.
110
0
10
20
30
40
50
60
0 0.01 0.02 0.03 0.04Time (s)
Cru
shin
g Fo
rce
(KN
)
Numerical
Formulation
Figure 5.5 Comparison of dynamic crushing load
The formulation is now used in the second conFigureuration, to compare results
with experimental data from [9]. To model the system, the EA seat and crush
tube are initially moving with the same velocity. The deceleration pulse is then
applied to one end of the crush tube, and the response of the impactor, in this
case an EA seat, at the other end is observed. The formulation tracks all the
instantaneous velocities. Figure 5.6 displays the velocities of the structure, and
instantaneous and relative EA seat velocity.
111
-10000
100020003000
40005000600070008000
900010000
0 0.02 0.04 0.06 0.08TIme (s)
Velo
city
(mm
/s) Instantaneous EA
Seat Velocity
Structure Velocity
Relative EA SeatVelocity
Figure 5.6 Velocities from the numerical formulation
The relative EA seat velocity is integrated with respect to time to obtain the
acceleration data in its raw or unfiltered form, as displayed in Figure 5.7.
-2.00E+06-1.00E+060.00E+001.00E+062.00E+063.00E+064.00E+065.00E+066.00E+067.00E+068.00E+06
0 0.02 0.04 0.06 0.08
Time (s)
Acc
eler
atio
n (m
m/s
2)
Unfiltered data
Figure 5.7 Unfiltered EA seat acceleration data
112
On performing an FFT of the relative velocity data, the cut-off frequency was
chosen as 20 Hz, as illustrated in Figure 5.8.
0
2000
4000
6000
8000
10000
12000
0 50 100 150 200 250 300 350
Frequency (Hz)
FFT
(rel
ativ
e ve
loci
ty) FFT
Figure 5.8 FFT of the relative velocity of the EA seat
The unfiltered acceleration data is then filtered using a Butterworth filter with a
cut-off frequency of 20 Hz. The maximum magnitude is determined to be
approximately 30.5 G. Figure 5.9 compares the EA seat acceleration response
from experiment and numerical formulation.
-20
-10
0
10
20
30
40
0 10 20 30 40 50 60 70 80 90
Time (ms)
Acc
eler
atio
n (G
)
ExperimentFormulation
Figure 5.9 Comparison of acceleration response
113
Figure 5.10 compares the peak acceleration magnitudes of EA seat from
experiment, simulation and formulation. All factors involved in the actual
experiment could obviously not be replicated in this simple formulation, and
thus there is a small difference in the peak accelerations.
Experimental Simulation Formulation0
5
10
15
20
25
30
35
Acc
eler
atio
n (G
)
Figure 5.10 Comparison of peak acceleration magnitude
The simple formulation presented herein utilizes uniaxial yield stress and therefore
does not include the plastic strain and strain hardening effect of a material. In
order to do so, the tangent modulus needs to be introduced into the
calculation of increment stress from increment strain at each step, and this must
be used to update the strain energy. This is automatically taken care of by
Material Model 24 in LSDYNA which allows the user to input an arbitrary stress
versus strain curve, as well as for an arbitrary strain rate dependency to be
defined. The energy dissipation due to bending and cracking of the seat
structure and rail, tearing of the seat cloth, and friction during sliding of the seat
brackets against the rail, has not been considered in the energy balance
formulation. The above contributions do not drastically later the energy balance,
but should be considered for accurate comparison between experiments and
114
the formulation. The support structure velocity was loaded from a data file
obtained through experiments. Since the time step used in the formulation is
1.00E-05 seconds, the same frequency must be used during the collection of
experimental/simulation data, since this data is used in the calculation of the EA
seat relative velocity. However the frequency of experimental and simulation
data was only 10,000 cycles/sec, and that obtained from the data file was
about 1000 data points for each second, leading to a slight step formation
which can be observed in the plot of relative EA seat velocity. This problem
however was slightly offset during the filtering of the data. Taking into account
the points mentioned above, the results of the simple numerical formulation
were in reasonable agreement with the results from experiments when used in
the second configuration, as can be seen from Figure 5.11. With appropriate
modifications, the formulation proves to be a reliable, low-cost method to
ultimately predicting crew response caused by acceleration from crash.
5.4 Conclusions
The simple numerical formulation presented herein can accurately predict
results when used in the first configuration. When used in the second
configuration, there is a slight overestimation of the predicted peak
acceleration response, due to the current non-reproducibility of all experimental
conditions into the formulation. However, the more the number of terms
considered in the energy balance equation represented by Equation 5.1, the
more will be the accuracy of the formulation. In summary, the formulation is
accurately able to predict key impact event variables that are essential in the
study of injury assessment in crashworthiness applications, and design of energy
absorbing devices.
115
5.5 Scope for Further Work
The simple numerical formulation presented herein utilizes the simple uniaxial
yield stress in the energy computations. However, the aluminum crush tube
displays a strain hardening effect, and this needs to be included into the
calculations, by also incorporating the tangent modulus of the material. The
formulation can also be extended from 1-d to multi-dimensions as well as cases
where the impacting mass is offset from the axis of the circular tubes. The
calculation of the dynamic crushing force using the impulse-momentum
equation needs to be refined for improved accuracy.
116
Chapter 6
Ballistic Impact of Woven Fabrics
6.1 Description of the Material Model
A computational micro-mechanical material model of loosely woven fabric for
non-linear finite element impact simulations is presented in this chapter. The
model is a mechanism incorporating the crimping of the fibers as well as the
trellising. The equilibrium of the mechanism allows the straightening of the fibers
depending on the fiber tension. The contact force at the fiber crossover points
determines the rotational friction which dissipates a part of the impact energy.
The stress-strain relationship is elastic based on a one-element spring model. The
failure of the fibers is strain rate independent. The model is implemented as user
defined subroutine in the transient finite element code LS-DYNA. The reader is
encouraged to refer [144] for the basic terminology and formulation of the
material model, since most parts remain similar to the work presented here.
6.2 The Representative Volume Cell of the Model [144]
The representative volume technique, vastly used in the micro-mechanical
models, is utilized hereafter. A current deformed state of the fabric is
considered. The Representative Volume Cell (RVC) of the loosely woven fabric
material model is extracted from the deformed pattern of the material, as seen
in Figure 6.1. The RVC consists of an undulated fill yarn crossed over an
undulated warp yarn as seen in Figure 6.1. The parameters of the RVC are: the
117
yarn span, s, the fabric thickness, t, the yarn width, w, and the yarn cross-
sectional area, S.
warp yarn
fill yarn
t
s
w
S
Figure 6.1 Representative Volume Cell (RVC) of the model
The complex geometry of the yarns is simplified and they are represented as a
pin-joint mechanism of straight elastic bars connected at the middle crossover
point by a rigid link as seen in Figure 6.2. The end nodes of the yarns are always
in the plane of the shell element xy. The distance between the central nodes of
the yarns in z-direction is provided by the rigid link of length t/2, which is always
normal to the xy-plane, and their z-coordinates depend on the equilibrium of
the stretching forces in the yarns. The pin-joints have axes always parallel to the
xy-plane and perpendicular to the bars. The mechanism allows the in-plane
rotation of the yarns about the rigid link as a trellis mechanism and the
straightening of the zig-zag undulated yarns depending on their tension. The
deformation of the yarns as a result of the contact between the yarns is
neglected. The in-plane orientation of the yarns is determined by the unit
vectors, q’s, or the braid angles, θ’s, measured with respect to the axis x of the
RVC coordinate system. The subscript f denotes the fill yarn and the subscript w
denotes the warp yarn.
118
In explicit finite element codes as LS-DYNA, the material model has to determine
the stress response of the material to the strain increment obtained at each
time step of the explicit time integration. Assuming that the RVC coordinate
system is the shell element local coordinate system, the stress response of the
woven fabric RVC to the strain increment passed to the model in the RVC
coordinate system has to be developed by the micro-mechanical approach.
fill yarn
warp yarn
θfqf
θwqw
x
y
z
t/2
rigid link
Figure 6.2 Pin-joint bar mechanism
6.3 Elastic Model
Figure 6.3 One Element Elasticity Model
σ, ε σ, ε
Ka
119
The governing equation of elasticity for the one element elastic model can be
derived from equilibrium as
aKσ ε= (6.1)
Then Equation 6.1 can be written in incremental form for the time step n+1 as
follows:
( ) ( ) ( ) ( )( )n n n n
aKσ σ ε ε+ Δ = + Δ (6.2)
We can determine the stress increment from the last equation
( ) ( ) ( ) ( )n n n na aK Kσ ε ε σΔ = + Δ − (6.3)
Where Ka is the spring stiffness and is equal to E1, the static Young’s modulus of
elasticity. The only failure mode available is when the fiber strain reaches the
failure strain or
maxaK
σε ε= > (6.4)
The input parameters for the Elasticity model are Hookean spring coefficient Ka
and static ultimate strain εmax. We now consider the equilibrium position of the
central nodes as seen in Figure 6.4..
120
Figure 6.4 Equilibrium position of the central nodes
We consider the equilibrium of the central nodes (the crossover point) of the
yarns at time step n+1 because the incremental elasticity equations of the
yarns are written for this instant. Again we assume that this state is linear
interpolation of the states at time step n and time step n+1. The equilibrium
state is given in Figure 6.4 for the fill yarn (upper scheme) and for the warp yarn
(lower scheme). The span between the yarns and the length of the bars can be
calculated for each time step of interest as follows:
ssss nw
nw
nf
nf
)1()1()1()1( , ++++ Λ=Λ= (6.5)
( ) ( )2)()(2)1(
)1(2)()(
2)1()1(
2,
2nn
w
nwn
wnn
f
nfn
f hs
Lhs
L δδ −+⎟⎟⎠
⎞⎜⎜⎝
⎛=++
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛=
++
++ (6.6)
The vertical components of the yarn forces can be determined as follows:
121
( )( ) ( ) ( ) ( )
( ) ( )( 1) ( 1)
n n n nf f n n
f f f fn nf f
h hF N S
L Lδ δ
σ σ+ +
+ += = + Δ (6.7)
( )( ) ( ) ( ) ( )
( ) ( )( 1) ( 1)
n n n nn nw w
w w w wn nw w
h hF N SL L
δ δ σ σ+ +
− −= = + Δ (6.8)
where S is the cross-sectional area of the yarns. The equilibrium of the
mechanism is reached when
wf FF 22 = (6.9)
Developing Equation (24) by plugging in Equations (14) and (16) we get:
( )
( ) ( )( ) ( ) ( )
1 2n n
f n n na f a f fn
f
hK K S
Lε ε σ
+
+ ∂⎡ ⎤+ Δ −⎣ ⎦
= ( )
( ) ( )( ) ( ) ( )
1 2n n
n n nwa w a w wn
w
h K K SL
ε ε σ+
+ ∂ ⎡ ⎤+ Δ −⎣ ⎦ (6.10)
The strain increments of the yarns are determined by the expressions:
( ) ( )1( )
n nf fn
f
L LL
ε+ −
Δ = and ( ) ( )1
( )n n
n w ww
L LL
ε+ −
Δ = (6.11)
Where L is the initial length of the bars calculated by the formula:
2 2
2 4s tL ⎛ ⎞ ⎛ ⎞= +⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠ (6.12)
122
Substituting yarn strain increments of Equation 6.11 in Equation 6.10 and
plugging Equations 6.6 in, we can get the final equation after some small
simplifications:
( )( )
( )21
2( ) ( ) ( ) ( )
2
nn n n nw
f wsh h
+⎛ ⎞+ ∂ + −∂⎜ ⎟⎜ ⎟
⎝ ⎠ x
( )
( ) ( )21
2( ) ( ) ( ) ( )22
nf nn n n na
a f f f f
sKK h LL
ε σ+⎡ ⎤⎛ ⎞⎛ ⎞⎢ ⎥⎜ ⎟− + + + ∂ −⎜ ⎟⎢ ⎥⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎢ ⎥⎝ ⎠⎣ ⎦
+ ( )( )
( )21
2( ) ( ) ( ) ( )
2
nfn n n n
w f
sh h
+⎛ ⎞∂ − + + ∂⎜ ⎟
⎜ ⎟⎝ ⎠
x
( )
( ) ( )21
2( ) ( ) ( ) ( )22
nnn n n na w
a w w w wK sK h LL
ε σ+⎡ ⎤⎛ ⎞⎛ ⎞⎢ ⎥⎜ ⎟− + + −∂ −⎜ ⎟⎜ ⎟⎢ ⎥⎜ ⎟⎜ ⎟⎝ ⎠⎢ ⎥⎝ ⎠⎣ ⎦
= 0
(6.13)
Equation (6.13) can be solved numerically for ( )n∂ by means of Newton-
Raphson method. The vertical position change of the central nodes is
constrained in order to avoid the snap-through behavior of the mechanism, ( )4 4nt tδ− ≤ ≤ . In this way, the buckling of the yarns in compression is
represented by the structural buckling of the membrane shell element model.
The vertical positions of the central nodes, initially set to 4/)0()0( thh wf == , are
finally updated:
( 1) ( ) ( )n n nf fh h+ = + ∂ , ( 1) ( ) ( )n n n
w wh h+ = + ∂ (6.14)
From here on, the calculation of strain and corresponding stress response
follows the formulation from [144] and is therefore not presented.
123
Because the behavior of the yarn has been represented by a one-element
linear spring system, there is no strain rate or viscoelastic effect present. This
implies that regardless of the strain rate in the fabric material during loading or
deformation, the stress-strain response will remain the same. However in reality,
all high strength and high modulus fabrics exhibit varying degrees of
viscoelasticity. But representing a yarn as a simple elastic member greatly
simplifies the formulation as well makes the numerical model less
computationally expensive with good accuracy. Moreover, it acts as a simple
platform on which various modifications and additions can be made to the
model such as adding yarn pullout effect, remote yarn failure, and other
complex phenomena, after which the yarn behavior can then be easily
switched back to being viscoelastic. Figure 6.5 displays the yarn stress-strain
curves for varying strain rates during the axial testing of a fabric strip that used
the viscoelastic model from [144]. It is evident that the stress-strain response
varies as the strain rate varies. In addition, a transition strain rate can be
observed at 100 s-1, where the mode of failure changes from primary yarn
failure to secondary yarn failure. The curves exhibit non-linearity because of the
viscoelastic nature of the yarn. Figure 6.6 displays the stress-strain response of a
yarn for the same fabric strip test using the elastic model. Perfect linearity is
observed in the plot as a one-element linear spring has been used. No matter
what the strain rate is, the response remains the same. Moreover, this elastic
model has only one mode of yarn failure, unlike the two-mode yarn failure
observed in the viscoelastic model.
124
Figure 6.5 Yarn stress-strain response of viscoelastic model
125
Figure 6.6 Yarn stress-strain response of elastic model
6.4 Numerical Results – Fabric Strip Testing
To test the elastic material model, axial and bias tensile tests are conducted on
a Kevlar© strip measuring 203.2 mm x 50.4 mm. Axial tests refer to the case
when the strip is pulled along the direction of the primary yarn. Bias tests refer to
the case when the strip is pulled along a 45 degree angle to the primary yarn.
The numerical setup is displayed in Figure 6.7. The strip is modeled using shell
elements which utilize the elastic material model. During the running of the
simulation, LSDYNA makes external calls to the user defined material model and
passes the strain increments to the model, which then returns the stress
increments to the main program. The set of nodes along the width of one end
of the strip are constrained completely. A displacement control is prescribed for
the corresponding set of nodes on the other end of the strip.
Figure 6.7 Numerical setup of fabric axial strip test
126
Because fabrics have very low out of plane stiffness, using extremely small
values of shear moduli can cause numerical instabilities and are therefore
scaled higher in the material model. This does not later the behavior of the
fabrics in any way. Table 6.1 lists the material properties of the Kevlar© fabric
strip used. Axial and bias tests are run using strain rates ranging from 1 s-1 to 300
s-1. The numerical results are then compared to data from the axial strip tests
using the viscoelastic model outlined in [144]. Element stresses and strains,
fabric deformation shape and stress contours are observed. Figure 6.8 displays
the Von-Mises stress distribution for a Kevlar© strip test with a prescribed strain
rate of 30 s-1.
Figure 6.8 Von-Mises stress distribution for strip with 30 s-1 strain rate
127
Longitudinal Young’s modulus 25 GPa Locking angle 10 deg
Transverse Young’s modulus 1.5 GPa Initial braid angle 45 deg
Shear modulus (G12) 1 GPa Coefficient of friction (μ) 1.7
Shear modulus (G23) 1 GPa Bulk modulus (K) 500 GPa
Shear modulus (G) 300 GPa Primary spring stiffness (Ka) 50 MPa
Yarn failure strain (εfail) 0.30 Yarn width (w) 0.32750 mm
Fabric real thickness 0.23 mm Yarn cross sectional area (S) 4.63E-02 mm2
Fabric effective thickness 0.125 mm Yarn span (s) 0.74710 mm
Table 6.1 Material and geometric properties of the Kevlar© fabric strip
6.4.1 Elastic model fabric strip test
Figure 6.9 displays the stress-strain curves for an element at the middle of the
strip during the axial tests. For very low strain rates ranging from 0.01 s-1 to 1 s-1,
an initial almost flat portion is seen in the plot, as the element stresses are very
low and take time to start building up. The stress wave propagates extremely
slowly through the fabric strip from the end being pulled to the constrained end.
There is very little geometric non-linearity for such low strain rates and as a
consequence, in accordance with the material model which uses one spring
element to represent the yarn behavior, the stress-strain curve is linear. As the
strain rate increases, there is increased geometric non-linearity in the strip due to
excessive deformation of the fabric and the stress-strain curve assumes a
curved shape. Such non-linearity is common in problems where the direction of
forces change as the structure shape deforms, such as pressure on the inner
surface of an elastic hemispherical membrane. When the failure strain is
reached for a particular element, or it warps beyond the maximum shell
warpage angle, it is deleted from the mesh.
128
-500
0
500
1000
1500
2000
2500
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
Strain
Stre
ss (M
Pa)
Elastic-Axial-1
Elastic-Axial-30
Elastic-Axial-100
Elastic-Axial-300
Figure 6.9 Axial strip tests of Elastic model
Figure 6.10 displays the stress-strain curves for an element at the middle of the
strip during the bias tests. Because the fabric is being pulled in the bias
direction, the yarns first realign along the bias direction until the locking angle is
reached and then start getting stretched. This is similar to a trellis mechanism
and is referred to as a scissoring effect. Once the fabric locking angle as
specified in the material model is reached, element stresses start building up.
The locking angle used in these tests was 10 degrees and this roughly
corresponds to a strain of 0.145 in the global reference coordinate system.
Remember, this frame of reference is different from both the RVE coordinate
system and the yarn axes, and thus while a strain in the fabric seems to exist
even with zero stress up to a value of 0.145 in the global system, the yarn strain
is actually zero.
129
-500
0
500
1000
1500
2000
2500
0 0.1 0.2 0.3 0.4
Strain
Stre
ss (M
Pa)
Elastic-Bias-1
Elastic-Bias-30
Elastic-Bias-100
Elastic-Bias-300
Figure 6.10 Bias strip tests of Elastic model
6.4.2 Viscoelastic model fabric strip test
Figures 6.11 and 6.12 display the stress-strain curves for an element at the
middle of the strip during the axial and bias tests respectively, using the
viscoelastic model. Both strain rate sensitivity and non-linearity can be observed
in the plots. For the bias tests, there is yarn realignment until the locking angle is
reached after which the yarns start getting stressed.
-500
0
500
1000
1500
2000
2500
3000
3500
0 0.1 0.2 0.3 0.4 0.5
Strain
Stre
ss (M
Pa)
Viscoelastic-Axial-1Viscoelastic-Axial-30Viscoelastic-Axial-100Viscoelastic-Axial-300
Figure 6.11 Axial strip tests of Viscoelastic model
130
-500
0
500
1000
1500
2000
2500
3000
3500
0 0.1 0.2 0.3 0.4 0.5
Strain
Stre
ss (M
Pa)
Viscoelastic-Bias-1Viscoelastic-Bias-30
Viscoelastic-Bias-100Viscoelastic-Bias-300
Figure 6.12 Bias strip tests of Viscoelastic model
6.4.3 Comparison between Elastic and Viscoelastic model results
Figure 6.13 and 6.14 compare the results from the two models. A very good
correlation can be seen between the results of the elastic and viscoelastic
model for the tensile fabric strip tests at varying strain rates. The viscoelastic
model is seen to fail at a higher strain than the elastic model, since the
viscoelastic model has a two mode yarn failure, which occurs at different strain
rates. This good correlation enforces the fact that sometimes simplistic models
can prove to be as reliable as complex models that try to represent all fabric
phenomena to the highest degree possible. The advantage of simple models
lay in their reduced computational requirements.
131
-50
0
50
100
150
200
250
300
350
400
0 0.05 0.1 0.15 0.2 0.25
Strain
Stre
ss (M
Pa)
Elastic-Bias-1Viscoelastic-Bias-1
-500
0
500
1000
1500
2000
2500
0 0.1 0.2 0.3 0.4
Strain
Stre
ss (M
Pa)
Elastic-Bias-30
Viscoelastic-Bias-30
(a) (b)
-500
0
500
1000
1500
2000
2500
3000
0 0.1 0.2 0.3 0.4
Strain
Stre
ss (M
Pa)
Elastic-Bias-100
Viscoelastic-Bias-100
-500
0
500
1000
1500
2000
2500
3000
3500
0 0.1 0.2 0.3 0.4 0.5
Strain
Stre
ss (M
Pa)
Elastic-Bias-300
Viscoelastic-Bias-300
(c) (d)
Figure 6.13 Comparison of bias tests of elastic and viscoelastic models at
different strain rates a) 1s-1 b) 30-1 c) 100-1 d) 300-1
132
-200
0
200
400
600
800
1000
1200
1400
1600
1800
2000
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
Strain
Stre
ss (M
Pa)
Elastic-Axial-1Viscoelastic-Axial-1
-500
0
500
1000
1500
2000
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
Strain
Stre
ss (M
Pa)
Elastic-Axial-30Viscoelastic-Axial-30
(a) (b)
0
500
1000
1500
2000
2500
3000
0 0.1 0.2 0.3 0.4
Strain
Stre
ss (M
Pa)
Elastic-Axial-100Viscoelastic-Axial-100
0
500
1000
1500
2000
2500
3000
3500
0 0.1 0.2 0.3 0.4 0.5
Strain
Stre
ss (M
Pa)
Elastic-Axial-300Viscoelastic-Axial-300
(c) (d)
Figure 6.14 Comparison of axial tests of elastic and viscoelastic models at
different strain rates a) 1s-1 b) 30-1 c) 100-1 d) 300-1
6.5 Conclusions
The elastic model is able to capture phenomena such as fabric reorientation,
fabric locking, and rotational frictional effects. A simple failure mode has been
used for the yarns. The results compare very well to the viscoelastic model. Non-
linearity is observed in the Elastic model in spite of using a linear spring to model
the yarn behavior. There is no strain rate sensitivity associated with the yarn
133
behavior. However, the fabric response shows slight strain rate sensitivity
because of the inherent complex architecture of the fabric and inter-yarn
interactions. The model is very computationally inexpensive while still
maintaining reasonable accuracy. Further testing of the elastic model is
required in the form of transverse impact tests and draping tests before full
validity can be claimed.
6.7 Scope for further work
The phenomena of yarn pullout and remote yarn failure can be included in the
model, as well as the energy dissipated by inter-yarn friction during pullout. A
constant cross section of the yarn has been assumed and therefore the
transverse compression of the yarns has not been accounted for. This can be
overcome by including the compression of the yarns into the equilibrium of the
central nodes of the RVE. Further testing of the Elastic model is required and has
been outlined in the preceding section. A more comprehensive strain based
failure can be incorporated instead of a simple two mode failure. Dynamic
mechanical properties of the fabrics can be fed into the formulation using data
tables, rather than using static properties, as it is well established that the
mechanical properties of the yarns is highly strain rate dependent.
134
Appendix I
Source code for the numerical formulation of dynamic
axial crushing of circular tubes
Configuration I
%********************************************************************** % Analysis of Dynamic Axial Crushing of Thin Cylindrical Tubes (and) % Energy Balance formulation to predict EA Seat response % % Author(s): Dr. Ala Tabiei and Gaurav Nilakantan % Date: March, 2006 (Final version) % email: [email protected] / [email protected] % % Notice: The authors assume no responsibility for the veracity, % validity, accuracy or applicability of the formulations or results % obtained herein. % Suggestions and comments are welcome. % % Copyright (C) 2006 % by Tabiei, Nilakantan % All rights reserved % %********************************************************************** % % List of variables: % sigmay - uniaxial yield stress % r - mean radius of the cylindrical tube % d - mean diameter of the cylindrical tube % h - wall thickness of the cylindrical tube % L - axial length of the cylindrical tube % v - final velocity of structure at the time of impact % vinst - instantaneous velocity of EA seat % vrel - relative velocity of EA seat % m - total mass of EA seat and occupant % Mo - full plastic moment of the tube wall per unit % length % H - initial half distance between the plastic hinges % at the top and bottom of a basic folding element % xm - intermediate folding variable % delta - effective crushing distance % l - folding length % n - number of folds that can be formed in distance L % a - angle of fold % da - increment in fold angle % E - total plastic energy absorbed by the tube % dE - increment in E % E1 - energy dissipated due to circumferential forces% % dE1 - increment in E1
135
% E2 - energy dissipated during crushing of three % stationery % circumferential plastic hinges % dE2 - increment is E2 % KEi - initial kinetic energy % KEnet - net kinetic energy % Ps - mean static crushing load % Pd - mean dynamic crushing load % i - counter variable % j - counter variable % t - time % dt - increment in time % epsilon_dot - strain rate % alpha_dot - rate of change of fold angle %********************************************************************** % System of consistent units (N,mm,ton,s,degree) %********************************************************************** clear clc %*************************** % User defined data %*************************** %sigmay=input('Enter the uniaxial yield stress of the material (MPa): %'); %r=input('Enter the mean radius of the cylindrical tube (mm): '); %h=input('Enter the thickness of the cylindrical tube (mm): '); %v=input('Enter the final velocity of the structure before impact %(mm/s): '); %m=input('Enter the total mass of the EA Seat and Occupant (ton): '); %L=input('Enter the length of the cylindrical tube (mm): '); %dt=input('Enter the time step for the problem (s): '); global xo global epsilon_dot global r global dt global H %********************************************************************** % Pre defined data - Based on Karagiozova Pg 1095 "Inertia Effects...." %********************************************************************** sigmay=295.000; r=11.875; h=1.65; v=4000.0; m=0.2625; L=106.68; dt=0.00001; %********************************************************************** d=r*2; Mo=(2/sqrt(3))*(sigmay*h^2)/4; xm=0.28*H/2; delta=2*H*(0.86-0.543*sqrt(h/d)); %******************************************************************* % Based on Wierzbicki % Comment: Number of folds greatly effects total run time. The % formulae of Wierzbicki are chosen as it yields the longest % fold length and thus yields the least number of folds l=2.62*sqrt(r*h/2);
136
H=1.853*r*sqrt(h/r); %******************************************************************* n=L/(2*l) % Kinetic Energy is in Joules KEi=(0.5*m*(v^2))/1000; %********************************************************************** % From Abramowicz and Jones, as referenced in D. Al Galib et al. Ps=Mo*(25.230*(sqrt(d/h))+15.09); %********************************************************************** vinst=v; vinstarray(1,1)=v; E1=0; E2=0; KEnet=KEi; KE(1,1)=KEi; counter=2; time(floor(n),1)=zeros; cntr(floor(n),1)=zeros; energy_per_fold(floor(n),1)=zeros; totaltime(1,1)=0; totaltime(2,1)=dt; force(1,1)=0; Earray(1,1)=0; % failure flag is activated when net K.E falls below zero failflag=0; for i=1:1:floor(n) a=0; alphainst(1,i)=0; while(a<(pi/2) && KEnet>0) %************************************************************* % Calculation of strain rate from Abramowicz and Jones epsilon_dot=0.5*vinst/r/(0.86-0.618*sqrt(h/d)); xo=alphainst(counter-1,i); % Call function to solve for instantaneous fold angle options=optimset('NonlEqnAlgorithm','lm'); alpha=fsolve(@myfun,0,options); alphainst(counter,i)=alpha; a=alphainst(counter,i); % Computation of fold energies dE1=(4*Mo*pi*(a-xo)*(d+h*sin(a)))/1000; dE2=(4*pi*sigmay*h*(H/2+H^2*sin(a)/(3*r))*epsilon_dot*r*dt)... /1000; E1=E1+dE1; E2=E2+dE2; E=E1+E2; Earray(counter,1)=E; dE=dE1+dE2; energy_per_fold(i,1)=energy_per_fold(i,1)+dE; KEnet=KEnet-dE; if(KEnet<0) failflag=1; end vinst=sqrt(2*KEnet*1000/m); vinstarray(counter,1)=vinst; KE(counter,1)=KEi-E; % Impulse-momentum equation force(counter,1)=m*(vinstarray(counter-1,1)-vinst)/dt;
137
time(i,1)=time(i,1)+dt; cntr(i,1)=cntr(i,1)+1; totaltime(counter,1)=totaltime(counter-1,1)+dt; counter=counter+1; if(failflag==1) break end end if(failflag==1) break end end xaxis=[0:dt:(counter-2)*dt]'; %********************************************************************** % Plot of Instantaneous Velocity Vs Time %********************************************************************** subplot(2,2,1); plot(xaxis,vinstarray) title('Instantaneous Velocity Vs Time') %********************************************************************** % Plot of Energy Vs Time %********************************************************************** subplot(2,2,2); plot(xaxis,KE,xaxis,Earray) legend('K.E.','Plastic Work') title('Transformation of Energy w.r.t Time') %********************************************************************** % Plot of Crushing Force Vs Time %********************************************************************** subplot(2,2,3); xaxis2=[0:dt:(counter-2)*dt]'; plot(xaxis2,force) title('Crushing Force w.r.t Time') %**********************************************************************
138
Configuration II
%********************************************************************** % Analysis of Dynamic Axial Crushing of Thin Cylindrical Tubes (and) % Energy Balance formulation to predict EA Seat response % % Author(s): Dr. Ala Tabiei and Gaurav Nilakantan % Date: March, 2006 (Final version) % email: [email protected] / [email protected] % % Notice: The authors assume no responsibility for the veracity, % validity, accuracy or applicability of the formulations or results % obtained herein. % Suggestions and comments are welcome. % % Copyright (C) 2006 % by Tabiei, Nilakantan % All rights reserved % %********************************************************************** % % List of variables: % sigmay - uniaxial yield stress % r - mean radius of the cylindrical tube % d - mean diameter of the cylindrical tube % h - wall thickness of the cylindrical tube % L - axial length of the cylindrical tube % v - final velocity of structure at the time of impact % vinst - instantaneous velocity of EA seat % vrel - relative velocity of EA seat % m - total mass of EA seat and occupant % Mo - full plastic moment of the tube wall per unit % length % H - initial half distance between the plastic hinges % at the top and bottom of a basic folding element % xm - intermediate folding variable % delta - effective crushing distance % l - folding length % n - number of folds that can be formed in distance L % a - angle of fold % da - increment in fold angle % E - total plastic energy absorbed by the tube % dE - increment in E % E1 - energy dissipated due to circumferential forces% % dE1 - increment in E1 % E2 - energy dissipated during crushing of three % stationery % circumferential plastic hinges % dE2 - increment is E2 % KEi - initial kinetic energy % KEnet - net kinetic energy % Ps - mean static crushing load % Pd - mean dynamic crushing load % i - counter variable
139
% j - counter variable % t - time % dt - increment in time % epsilon_dot - strain rate % alpha_dot - rate of change of fold angle %********************************************************************** % System of consistent units (N,mm,ton,s,degree) %********************************************************************** clear clc %********************************************************************** % User defined data % If needed, remove the comments symbol to allow the user to enter % their own data %********************************************************************** %sigmay=input('Enter the uniaxial yield stress of the material (MPa): %'); %r=input('Enter the mean radius of the cylindrical tube (mm): '); %h=input('Enter the thickness of the cylindrical tube (mm): '); %v=input('Enter the final velocity of the structure before impact %(mm/s): '); %m=input('Enter the total mass of the EA Seat and Occupant (ton): '); %L=input('Enter the length of the cylindrical tube (mm): '); %dt=input('Enter the time step for the problem (s): '); %******************************** % Declaration of global variables %******************************** global xo global epsilon_dot global r global dt global H %************************** % Pre defined data %************************** sigmay=145.000; r=13.667; h=0.890; v=8919.0; m=0.07214; L=228.6; dt=0.00005; %************************** d=r*2; Mo=(2/sqrt(3))*(sigmay*h^2)/4; xm=0.28*H/2; delta=2*H*(0.86-0.543*sqrt(h/d)); %******************************************************************* % Based on Wierzbicki l=2.62*sqrt(r*h/2); H=1.853*r*sqrt(h/r); %******************************************************************* n=L/(2*l); %********************************** % NOTE: Kinetic Energy is in Joules %********************************** KEi=(0.5*m*(v^2))/1000;
140
%********************************************************************** % From Abramowicz and Jones, as referenced in D. Al Galib et al. Ps=Mo*(25.230*(sqrt(d/h))+15.09); %********************************************************************** vinst=v; E1=0; E2=0; KEnet=KEi; KE(1,1)=KEi; counter=2; time(floor(n),1)=zeros; cntr(floor(n),1)=zeros; vinstarray(1,1)=vinst; %*********************************************************** % Load file containing velocity profile of support structure %*********************************************************** load strucbot.dat vrel=v-strucbot(1,1); totaltime(1,1)=0; totaltime(2,1)=dt; energy_per_fold(floor(n),1)=zeros; Earray(1,1)=0; % failure flag is activated when net K.E falls below zero failflag=0; for i=1:1:floor(n) a=0; alphainst(1,i)=0; while(a<(pi/2) && KEnet>0) % Calculation of strain rate epsilon_dot=0.5*vrel/r/(0.86-0.618*sqrt(h/d)); xo=alphainst(counter-1,i); % Call function to solve for instantaneous fold angle options=optimset('NonlEqnAlgorithm','lm'); alpha=fsolve(@myfun,0,options); alphainst(counter,i)=alpha; a=alphainst(counter,i); % Computation of fold energies dE1=(4*Mo*pi*(a-xo)*(d+h*sin(a)))/1000; dE2=(4*pi*sigmay*h*(H/2+H^2*sin(a)/(3*r))*epsilon_dot*r*dt)... /1000; E1=E1+dE1; E2=E2+dE2; E=E1+E2; dE=dE1+dE2; KEnet=KEnet-dE; if(KEnet<0) break end % Calculation of instantaneous EA seat velocity vinst=sqrt(2*KEnet*1000/m); % Locate the instantaneous support structure velocity for location=1:1:80 if ((totaltime(counter,1)*1000)>strucbot(location,1) || ... (totaltime(counter,1)*1000)==strucbot(location,1))&&... ((totaltime(counter,1)*1000)<strucbot(location+1,1) ||...
141
(totaltime(counter,1)*1000)==strucbot(location+1,1)) vstruc=strucbot(location,2); vstrucarray(counter,1)=vstruc; break end end if(KEnet<0) failflag=1; end vrel=vinst-vstruc; vinstarray(counter,1)=vinst; vrelarray(counter,1)=vrel; KE(counter,1)=KEi-E; velocity(counter,1)=vrel; Earray(counter,1)=E; energy_per_fold(i,1)=energy_per_fold(i,1)+dE; counter=counter+1; time(i,1)=time(i,1)+dt; cntr(i,1)=cntr(i,1)+1; totaltime(counter,1)=totaltime(counter-1,1)+dt; if(failflag==1) break end end if(failflag==1) break end end %********************************************************************** % Plot of Relative Velocity Vs Time %********************************************************************** subplot(2,2,1); xaxis=[0:dt:(counter-2)*dt]'; plot(xaxis,velocity) xlabel('Time (s)') ylabel('Relative Velocity (mm/s)') title('Relative Velocity Vs Time') %********************************************************************** % Plot of Instantaneous Velocity Vs Time %********************************************************************** subplot(2,2,2); plot(xaxis,vinstarray) xlabel('Time (s)') ylabel('Instantaneous Velocity (mm/s)') title('Instantaneous Velocity Vs Time') %********************************************************************** % Plot of Kinetic Energy Vs Time %********************************************************************** subplot(2,2,3); plot(xaxis,KE) xlabel('Time (s)') ylabel('Kinetic Energy (J)') title('Kinetic Energy Vs. Time') %********************************************************************** % Plot of Plastic Work Vs Time %**********************************************************************
142
subplot(2,2,4); plot(xaxis, Earray) xlabel('Time (s)') ylabel('Plastic Work (J)') title('Plastic Work Vs. Time') %**********************************************************************
143
Function ‘myfun’ used in Configuration I and II
%********************************************************************** % Analysis of Dynamic Axial Crushing of Thin Cylindrical Tubes (and) % Energy Balance formulation to predict EA Seat response % % Author(s): Dr. Ala Tabiei and Gaurav Nilakantan % Date: March, 2006 (Final version) % email: [email protected] / [email protected] % % Notice: The authors assume no responsibility for the veracity, % validity, accuracy or applicability of the formulations or results % obtained herein. % Suggestions and comments are welcome. % % Copyright (C) 2006 % by Tabiei, Nilakantan % All rights reserved % %********************************************************************** function F=myfun(alpha) global xo global epsilon_dot global r global dt global H F=(alpha-xo)*cos(alpha)-(epsilon_dot*r*dt/H); end
144
Contents of the ‘strucbot’ data file
0.000000000 8918.999023 0.999868847 8918.999023 1.999737695 8918.999023 2.999963937 8918.999023 3.999832552 8918.999023 4.999701399 8918.999023 5.999927875 8918.999023 6.999796722 8918.999023 7.999665104 8918.999023 8.999891579 8918.999023 9.999760427 8918.999023 10.9999869 8903.797852 11.99985575 8909.385742 12.9997246 8911.800781 13.99995107 8912.78125 14.99981992 8913.760742 15.99968784 8914.741211 16.99991524 8915.720703 17.99978316 8916.701172 18.99965294 8917.680664 19.99987848 8917.517578 20.99974826 8911.72168 21.9999738 8894.895508 22.99983986 8865.357422 23.9997115 8788.709961 24.999924 8650.976563 25.99980123 8456.224609 26.99979954 8130.765625 27.99988911 7654.873535 28.99986878 7093.491211 29.99978885 6399.95166 30.99999577 5641.805664 31.99985623 4954.264648 32.99997374 4430.261719 33.99997577 4053.821533 34.99996662 3777.724365 35.99990532 3489.584473 36.99990735 3158.296387 37.99990937 2764.373291 38.99997473 2302.322266 39.99994323 1877.890137 40.999908 1509.176025 41.99997336 1270.943848 42.99993068 1161.104248 43.99995133 1135.381592 44.99999061 1126.782959 45.99997401 1047.84314 46.99999094 903.3579102 47.99992219 753.4771118 48.99991304 621.5859985 49.99994114 556.0709839 50.99992454 576.8318481 51.99996755 650.6864624 52.99992487 771.6782837
145
53.99993807 881.1166382 54.99996245 925.1243897 55.99997565 942.7033691 56.9999665 883.3707886 57.999935 820.6410523 58.99997428 790.6043091 59.99993533 796.6851807 60.99992245 849.2549439 61.99995056 934.7791138 62.99994886 1016.16272 63.99992108 1045.249146 64.99996781 1035.973145 65.99995494 1018.441101 66.99997187 991.5888062 67.99998879 964.7364502 68.99996102 954.8067017 69.9999854 963.5032959 70.99998742 984.4906006 71.99994475 1018.152466 72.99996913 1048.165405 73.9999935 1062.786255 74.99996573 1059.519531 75.99996775 1040.250244 76.99998468 1014.58783 77.99999416 987.2261963 78.99996638 978.5117188 79.99999076 977.9264526 80.99997789 998.3884888 81.99997991 1016.932434 82.99996704 1033.3125 83.99996907 1036.335205 84.99999344 1036.62085 85.99995822 1031.807617 86.9999826 1023.998779 87.99996227 1016.104065 88.9999941 1010.756287 89.99998868 1018.790222 90.00004828 0.000
146
Appendix II
Source code for the incremental constitutive equation
used in the Elastic material model to derive the stress-
strain relationship
%********************************************************************** % Program to calculate Stress-Strain in a Loose Woven Fabric with % linearly elastic crimped fibers using a 1-element Spring Model. % Micro Mechanical approach has been utilized. % % Author(s): Dr. Ala Tabiei and Gaurav Nilakantan % Date: February, 2005 (Final version) % email: [email protected] / [email protected] % % Notice: The authors assume no responsibility for the veracity, % validity, accuracy or applicability of the formulations or results % obtained herein. % Suggestions and comments are welcome. % % Copyright (C) 2005 % by Tabiei, Nilakantan % All rights reserved % %********************************************************************** clear all %Defining Strain Increment Step Size de=0.000005; %Defining Parameters of the 1-element Spring Model %K1 - Stiffness of Spring 1 that rep a combination of Primary- %Secondary Bond Strength %E1max - Static Ultimate Strain for Spring 1 K1=input('Enter the Spring Stiffness : '); E1max=input('Enter the Static Ultimate Strain of the Spring : '); %Initialize Stress and Strain at t=0 s=0; e=0; ds=0; %Failure Mode: 0-Safe; 1-Fail Fail=0; i=1; while Fail==0 ds=(K1*e)+(K1*de)-(s); s=s+ds;
147
e=e+de; if (e>E1max) Fail=1; end stress(i)=s; strain(i)=e; i=i+1; end %Plotting Results plot(strain,stress); grid; xlabel('Strain'); ylabel('Stress, MPa'); title('Stress-Strain curves for Linearly Elastic fibers at ANY Strain rate');
148
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