Numerical and Experimental Crashworthiness Studies of
Foam-Filled Frusta
By
Chun (Phillip) Hou
A thesis submitted in conformity with the requirements
for the Degree of Master of Applied Science,
Department of Mechanical and Industrial Engineering
University of Toronto
© Copyright by Chun Hou 2013
ii
Numerical and Experimental Crashworthiness Studies of
Foam-Filled Frusta
Chun (Phillip) Hou
Master of Applied Science
Department of Mechanical and Industrial Engineering
University of Toronto
Abstract
Thin-walled metallic components have been widely used as energy absorbers. One key
drawback is the high initial crippling load, which typically results in passenger injuries. It is the
objective of this study to introduce taper angle to thin-walled prisms, and to examine the
crushing response of thin-walled frusta. Nonlinear finite element models of thin-walled frusta
of different cross-sectional geometries were developed. Experimental investigations were
conducted to validate these models. The effects of key design parameters on the energy
absorption characteristics of frusta were explored. Comparison between thin-walled prisms and
frusta show that taper angle helps to reduce the initial crippling load and increase the resistance
to global buckling. To take advantage of the interaction effects, a novel multi-frusta
configuration was developed and it was shown that the energy absorption efficiency is
significantly increased. The results of this work are valuable for enhancing the crashworthiness
performance of thin-walled metallic energy absorber.
iii
Acknowledgements
I offer my sincere appreciation and gratitude to Professor S.A. Meguid for his technical
guidance and financial assistance throughout the course of my research. His display of
confidence in my abilities and efforts is very much appreciated.
My fellow researchers in the Mechanics and Aerospace Design Laboratory (MADL) have been
a source of constant support and inspiration. It has been a great pleasure to work with all of
them, and I would like to specifically extend my heartfelt appreciation to Mr. Pieter Verberne
for his extended help at the latter stages of my thesis. In addition, I wish to thank Drs. Fan Yang,
Xuewen Yin, Jake Wernik, and Mr. Luke Maricic for all their help and friendship.
Finally, I would like to say thank you to my parents and my girlfriend, Miss Meng Lu. It is
difficult to express how much I appreciate their understanding and continued support during the
course of my studies.
iv
Table of Contents
Abstract ......................................................................................................................................... ii
Acknowledgements ...................................................................................................................... iii
Table of Contents ......................................................................................................................... iv
Notations ...................................................................................................................................... vii
List of Tables .............................................................................................................................. viii
List of Figures .............................................................................................................................. ix
Chapter 1 Introduction and Justification .............................................................................. 1
1.1 Introduction ......................................................................................................................... 1
1.2 Justification of the Study ..................................................................................................... 4
1.3 Research Objectives ............................................................................................................ 6
1.4 Method of Approach ........................................................................................................... 6
1.5 Thesis Layout ...................................................................................................................... 8
Chapter 2 Literature Review .................................................................................................. 9
2.1 Axial Collapse of Thin-Walled Sections ............................................................................. 9
2.1.1 Collapse Load and Energy Absorption ................................................................ 10
2.1.2 Stability ................................................................................................................ 12
2.1.3 Crashworthiness by Plastic Collapse ................................................................... 13
2.1.4 Quasi-static Finite Element Modelling of Thin-Walled Columns ....................... 19
2.1.5 Dynamic Finite Element Modelling of Thin-Walled Columns ............................ 20
2.2 Metallic Foams .................................................................................................................. 21
2.2.1 Mechanical Properties of Foam Materials ........................................................... 23
2.2.2 Finite Element Modelling of Foam Materials ...................................................... 26
2.3 Foam-Filled Columns ....................................................................................................... 28
Chapter 3 Finite Element Modelling .................................................................................... 34
3.1 Geometry of the Foam-Filled Frusta ................................................................................. 34
3.2 Constitutive Modelling of Materials ................................................................................. 36
3.2.1 Column Wall ........................................................................................................ 36
3.2.2 Aluminium Foam Core ........................................................................................ 37
3.2.3 Loading Platen ..................................................................................................... 40
v
3.3 Discretization Process ....................................................................................................... 41
3.4 Boundary and Contact Conditions .................................................................................... 42
3.5 Triggering Mechanism ...................................................................................................... 44
3.6 Explicit Finite Element Modelling Using LS-DYNA ....................................................... 45
3.7 Mass Scaling ..................................................................................................................... 46
3.8 Sensitivity Analysis ........................................................................................................... 46
3.8.1 Mesh Convergence ............................................................................................... 47
3.8.2 Contact Convergence ........................................................................................... 47
3.8.3 Trigger Section ..................................................................................................... 49
3.8.4 Mass Scaling ........................................................................................................ 52
Chapter 4 Experimental Investigations ............................................................................... 54
4.1 Physical Characteristics of Column and Foam ................................................................. 54
4.1.1 Materials Selected ................................................................................................ 54
4.1.2 Manufacturing Process ......................................................................................... 55
4.1.3 Annealing Procedure of Frustum ......................................................................... 60
4.2 Quasi-static Crushing of Thin-walled Frusta .................................................................... 61
Chapter 5 Results and Discussions ....................................................................................... 63
5.1 Validation of FE Models ................................................................................................... 63
5.1.1 Quasi-Static Collapse of Thin-Walled Circular Frusta ........................................ 63
5.1.2 Quasi-Static Collapse of Thin-Walled Multi-Frusta ............................................ 67
5.1.3 Quasi-Static Collapse of Thin-Walled Square Prisms ......................................... 70
5.2 Quasi-Static Collapse of Thin-Walled Frusta ................................................................... 73
5.2.1 Effect of Semi-Apical Angle ................................................................................ 73
5.2.2 Effect of Length, Width, and Thickness .............................................................. 78
5.2.3 Effect of Slenderness Ratio .................................................................................. 81
5.2.4 Effect of Cross Sectional Geometries .................................................................. 85
5.2.5 Effect of Foam-Filling ......................................................................................... 86
5.2.6 Effect of Friction between Column Wall and Foam Filler ................................... 89
5.3 Quasi-Static Collapse of Multi-Frusta .............................................................................. 90
5.3.1 Effect of Column Spacing .................................................................................... 91
5.3.2 Effect of Column-Die Spacing ............................................................................. 93
5.3.3 Effect of Semi-Apical Angle ................................................................................ 94
5.3.4 Effect of Foam-Filling ......................................................................................... 97
Chapter 6 Conclusions and Future Work ........................................................................... 99
6.1 Problem Statement ............................................................................................................ 99
6.2 Objectives.......................................................................................................................... 99
6.3 Conclusions ..................................................................................................................... 100
vi
6.4 Thesis Contributions ....................................................................................................... 101
6.5 Future Work .................................................................................................................... 101
Reference .................................................................................................................................... 102
vii
Notations
trigger size mass matrix
area of contact segment fully plastic moment
effective width of column wall penalty factor
stress wave speed load ratio
damping matrix stroke efficiency
Displacement specific energy absorption
stroke length thickness of outer rigid die
diameter of column wall at mid-span total energy absorption
bottom diameter of column wall impact velocity
top diameter of column wall relative velocity of contact surface
exponential decay coefficient volume
elastic true strain relative volume
true strain densified relative volume plastic true strain width of outer rigid die
elastic modulus vector of externally applied loads
energy efficiency vector of nodal displacement
instantaneous collapse load ̇ vector of nodal velocity
mean collapse load ̈ vector of nodal acceleration
initial crippling force time step
mean collapse load of tube semi-apical angle of column wall
mean collapse load of foam frictional coefficient
mean collapse load of interaction tube-tube distance
mean collapse load of foam-filled tube tube-die distance
static frictional coefficient density
dynamic frictional coefficient foam density
shear modulus parent material density
thickness of column wall engineering strain
half fold length volumetric strain
contact stiffness engineering stress
stiffness matrix flow stress
bulk modulus of contact segment flow stress of foam
length of column wall true stress
Le effective length of the element ultimate tensile stress
Mass yield stress
viii
List of Tables
Table 3.1 Mechanical properties of A6061-T4 (after [85]) ......................................................... 36
Table 5.1 Failure modes for circular frusta of various slenderness ratios ................................... 83
Table 5.2 Failure modes for square frusta of various slenderness ratios ..................................... 83
Table 5.3 Failure modes for hexagonal frusta of various slenderness ratios ............................... 83
ix
List of Figures
Fig. 1.1 Crashworthiness testing revealed bad performance of lighter vehicle (after [7]) ............ 1
Fig. 1.2 Applications of thin-walled columns in the design of (a) automotive crash box,
(b) rail vehicle driver’s cab, and (c) aircraft seat (after [11-14]) .................................... 3
Fig. 1.3 Crash-box design in rail cab ............................................................................................ 5
Fig. 1.4 Schematics of multi-tube crash-box design: (a) current configuration,
(b) proposed configuration .............................................................................................. 5
Fig. 1.5 Method of approach ......................................................................................................... 7
Fig. 2.1 Common modes of collapse of thin-walled columns under axial compression:
(a) progressive buckling, (b) external inversion, and (c) axial splitting (after [27]) ....... 9
Fig. 2.2 Typical response of thin-walled columns: (a) instantaneous force vs. displacement,
(b) internal energy vs. displacement, and (c) mean force vs. displacement
(after [29]) ..................................................................................................................... 11
Fig. 2.3 Idealized axisymmetric crushing mode of cylindrical shell (after [33]) ....................... 14
Fig. 2.4 Deformation modes of thin-walled circular tubes: (a) concertina,
(b) diamond (after [36]) ................................................................................................. 15
Fig. 2.5 Deformation stages in an axial crushing of thin-walled square column (a) local
buckling, (b) rising of stress concentrations, (c) edge yielding, and
(d) plastic collapse (after [23]) ...................................................................................... 15
Fig. 2.6 Basic collapse elements: (a) type-I, (b) type II (after [39]) ........................................... 16
Fig. 2.7 Deformation modes of thin-walled square columns: (a) symmetrical,
(b) extensional, and (c) asymmetric mixed mode B (after [39]) ................................... 18
Fig. 2.8 Modes of deformation in static and dynamic loading conditions:
(a) dynamic plastic buckling, (b) progressive folding (after [8]) .................................. 20
Fig. 2.9 Metallic foams: (a) open cell, (b) open cell (after [53]) ................................................ 22
Fig. 2.10 Typical compressive stress-strain response of aluminium foam (after [59]) ............... 23
Fig. 2.11 Skeleton cube model: (a) initial geometry, (b) cell edge bending during
deformation (after [60]) ................................................................................................. 25
Fig. 2.12 Truncated cube model: (a) assembly of the cell, (b) basic folding element
(after [63]) ..................................................................................................................... 25
Fig. 2.13 FE model of the foam cell using truncated cube unit model (after [63]) ..................... 26
x
Fig. 2.14 Modified truncated cube model: (a) unit cell, (b) FE model of multiple cells
(after [64]) ..................................................................................................................... 27
Fig. 2.15 Unit cell model account for morphological imperfections: (a) curved walls,
(b) corrugated walls (after [15, 65]) .............................................................................. 27
Fig. 2.16 Effect of foam filling on the collapse load of thin-walled columns (after [43]) .......... 28
Fig. 2.17 FE model geometry and contact conditions (after [72]) ............................................... 30
Fig. 2.18 Comparison of deformation mode under quasi-static compression:
(a) empty column, (b) foam-filled column (after [74]) ................................................. 31
Fig. 2.19 Mode of collapse of empty and foam-filled circular aluminium columns
(after [61]) ..................................................................................................................... 32
Fig. 2.20 Results from static crushing of foam-filled components (after [81]) ........................... 33
Fig. 3.1 Detailed geometry representation of the foam-filled circular frustum .......................... 34
Fig. 3.2 Engineering stress-strain curve for A6061-T4 (after [85]) ........................................... 37
Fig. 3.3 Illustration of the yield curve in LS-DYNA material model #26 (after [82]) .............. 38
Fig. 3.4 Compressive stress-strain response of 10% aluminium foam and the yield
curve which was input into the LS-DYNA foam model (after [85]) ............................ 39
Fig. 3.5 FE model of the foam-filled square frustum ................................................................. 42
Fig. 3.6 Exaggerated view of the trigger in the FE model of square frustum with θ = 0°.......... 44
Fig. 3.7 Mean collapse load of various element sizes used to discretize tube wall .................... 47
Fig. 3.8 Initial crippling force levels for various penalty scale factors ...................................... 48
Fig. 3.9 Penetration depth as a percentage of element size for various penalty scale factors .... 49
Fig. 3.10 Mode of collapse of empty square column: (a) without geometrical trigger,
(b) with geometrical trigger (correct mode) .................................................................. 50
Fig. 3.11 Mean collapse load of empty square column with elements of reduced density ......... 50
Fig. 3.12 Mode of collapse of empty square column with elements of reduced density:
(a) 10 elements, (b) 100 elements ................................................................................. 51
Fig. 3.13 Mean collapse load of empty square column with elements of reduced thickness ...... 51
Fig. 3.14 Mode of collapse of empty column with elements of reduced thickness:
(a) 10 elements, (b) 100 elements ................................................................................. 52
xi
Fig. 3.15 Load-displacement response of empty square column using mass scaling of
1, 10, 100 and 1000 ....................................................................................................... 53
Fig. 3.16 Internal and kinetic energy of empty square column using mass scaling of
1, 10, 100 and 1000 ....................................................................................................... 53
Fig. 4.1 Photo of 10% relative density aluminium foam ............................................................ 55
Fig. 4.2 Aluminium cone: (a) engineering drawing, (b) solid model ......................................... 56
Fig. 4.3 Illustration of wire EDM to cut foam samples .............................................................. 57
Fig. 4.4 Pictures of manufactured circular frusta: (a) empty, (b) foam-filled ............................ 57
Fig. 4.5 Square-packed multi-frusta with outer die: (a) schematic drawing with detailed
dimension, (b) photo of the tested sample ..................................................................... 59
Fig. 4.6 Preliminary testing results: (a) original geometry, (b) deformation mode of as
received circular column, (c) deformation mode of annealed circular column ............. 60
Fig. 4.7 Stress-strain relationships of the received and annealed A6061-T6 specimens ............ 61
Fig. 4.8 Computer controlled Instron 8522 servohydraulic testing system ................................ 62
Fig. 5.1 Comparison of deformation mode of empty circular frustum: (a) experimental
result, (b) FE prediction ................................................................................................. 64
Fig. 5.2 Comparison of instantaneous and mean collapse load of empty circular frustum:
experimental result vs. FE prediction ............................................................................ 65
Fig. 5.3 Comparison of deformation mode of foam-filled circular frustum:
(a) experimental result, (b) FE prediction ..................................................................... 66
Fig. 5.4 Comparison of instantaneous and mean collapse load of foam-filled circular
frustum: experimental result vs. FE prediction ............................................................. 67
Fig. 5.5 Deformation mode of crushed square-packed foam-filled frusta:
(a) top view, (b) bottom view ........................................................................................ 68
Fig. 5.6 Deformation mode of crushed square-packed foam-filled frusta: side view ................ 68
Fig. 5.7 Comparison of deformation mode of square-packed foam-filled frusta
(bottom view): (a) experimental result, (b) FE prediction ........................................... 69
Fig. 5.8 Comparison of instantaneous and mean collapse load of square-packed foam-filled
frusta: experimental result vs. FE prediction ................................................................. 69
Fig. 5.9 Comparison of instantaneous and mean collapse load: multi-frusta vs. 4 × single
foam-filled frusta ........................................................................................................... 70
xii
Fig. 5.10 Comparison of collapse mode of empty square column: (a) experimental result
(after [85]), (b) FE prediction ........................................................................................ 71
Fig. 5.11 Comparison of instantaneous and mean collapse load of empty square column:
experimental result (after [85]) vs. FE prediction ......................................................... 71
Fig. 5.12 Comparison of collapse mode of foam-filled square column: (a) experimental
result (after [85]), (b) FE prediction .............................................................................. 72
Fig. 5.13 Comparison of instantaneous and mean collapse load of foam-filled square prisms:
experimental result (after [85]) vs. FE prediction ......................................................... 73
Fig. 5.14 Instantaneous and mean collapse load for circular frusta of various semi-apical
angles ............................................................................................................................. 74
Fig. 5.15 Instantaneous and mean collapse load for square frusta of various semi-apical
angles ............................................................................................................................. 74
Fig. 5.16 Instantaneous and mean collapse load for hexagonal frusta of various semi-apical
angles ............................................................................................................................. 74
Fig. 5.17 Effect of semi-apical angle on the initial crippling force and SEA of frusta ............... 75
Fig. 5.18 Effect of semi-apical angle on load ratio of empty frusta ............................................ 76
Fig. 5.19 Modes of collapse of circular frusta with semi-apical angles:
(a) 0°, (b) 5°, and (c) 10° ............................................................................................... 77
Fig. 5.20 Modes of collapse of square frusta with semi-apical angles:
(a) 0°, (b) 5°, and (c) 10° ............................................................................................... 77
Fig. 5.21 Modes of collapse of hexagonal frusta with semi-apical angles:
(a) 0°, (b) 5°, and (c) 10° .............................................................................................. 77
Fig. 5.22 Tube inversion mode of collapse observed from circular frustum with θ = 35° ......... 78
Fig. 5.23 Effect of length on the initial crippling force and SEA of frusta ................................. 79
Fig. 5.24 Load-displacement responses for square prisms of length 250 mm and 350 mm ....... 79
Fig. 5.25 Effect of thickness on the initial crippling force and SEA of frusta ............................ 80
Fig. 5.26 Effect of effective width on the initial crippling force and SEA of frusta ................... 81
Fig. 5.27 Load-displacement response for circular prisms with various slenderness ratios........ 82
Fig. 5.28 Development of plastic hinges in Euler buckling mode for:
(a) circular prism, (b) square prism ............................................................................... 84
Fig. 5.29 Modes of collapse for circular frusta with l/2R = 6: (a) θ = 5°, (b) θ = 0° ................... 85
xiii
Fig. 5.30 Instantaneous and mean collapse load of circular, square and hexagonal prisms ........ 86
Fig. 5.31 Interaction effect between square column wall and foam filler ................................... 87
Fig. 5.32 Instantaneous and mean collapse load for foam-filled circular frusta of
various semi-apical angles ............................................................................................. 87
Fig. 5.33 Instantaneous and mean collapse load for foam-filled square frusta of
various semi-apical angles ............................................................................................. 88
Fig. 5.34 Instantaneous and mean collapse load for foam-filled hexagonal frusta of
various semi-apical angles ............................................................................................. 88
Fig. 5.35 Effect of semi-apical angle on load ratios of foam-filled frusta................................... 89
Fig. 5.36 Mean collapse load for varying frictional coefficient on the tube-foam interface of
foam-filled square prism ............................................................................................... 90
Fig. 5.37 Geometry representation of the multi-frusta configuration with θ = 0° ....................... 91
Fig. 5.38 Variation of SEA with column spacing ........................................................................ 92
Fig. 5.39 Variation of SEA with column-die spacing ................................................................. 93
Fig. 5.40 Mode of collapse of multi-frusta with θ = 5°, same orientation .................................. 94
Fig. 5.41 Mode of collapse of multi-frusta with θ = 5°, alternative orientation .......................... 94
Fig. 5.42 Variation of SEA with semi-apical angle for square multi-frusta ................................ 95
Fig. 5.43 Square multi-frusta configuration with larger column spacing
(to prevent overlap of frusta on the diagonal) ............................................................... 96
Fig. 5.44 Circular multi-frusta configuration with smaller column spacing ............................... 96
Fig. 5.45 Variation of SEA with semi-apical angle for circular multi-frusta .............................. 97
Fig. 5.46 Variation of SEA with semi-apical angle for foam-filled square multi-frusta ............. 98
1
Chapter 1
Introduction and Justification
1.1 Introduction
The desire to improve crashworthiness cannot be overestimated. The National Highway Traffic
Safety Administration revealed that 30,000 fatalities and over 1.5 million injuries were reported
in 2010 due to motor vehicle traffic crashes [1]. Steel has been extensively used in energy
absorbing systems, due to its relatively low price and high ductility [2]. Due to legislative
pressure, automobile manufacturers are increasingly designing subcompact vehicles to reduce
fuel consumption and gas emissions. In addition, there is increased interest in the use of
lightweight alloys, such as aluminium and magnesium, in automotive design [3].
However, a number of reports have revealed the poor crashworthy performance of lightweight
vehicles [4-6]. The insurance Institute for Highway Safety recently published the results of a
head-on collision between a midsized Toyota Camry and a subcompact Yaris [7]. There was far
more intrusion into the compartment of the Yaris than the Camry. During the impact, the test
dummies in both vehicles struck their heads on the steering wheels. However, the dummy in the
Yaris experienced severe head injury. There was also extensive force on the neck and right leg,
plus a deep gash at the right knee of the dummy in the minicar.
Fig. 1.1 Crashworthiness testing revealed bad performance of lighter vehicle (after [7])
2
There are various energy absorbers available to improve the crashworthiness of these vehicles,
without drastically increasing the weight. In particular, progressive folding collapse of thin-
walled structures is an efficient energy absorber because it can undergo large displacement at
near constant stress. Moreover, it is also economical and easy to manufacture [8]. Metallic
foams are great candidate for crashworthiness applications, as they can undergo large plastic
deformation at a relatively constant load level [9]. It has been proven that by filling thin-walled
sections with foam, the energy absorption capacity can be improved significantly. The increase
is mainly attributed to the interaction between the foam filler and the tube walls.
Thin-walled sections are widely used as energy absorbers in civil, mechanical, marine and
aeronautical applications [10]. In modern automotive design, thin-walled columns have been
used in the design of crash boxes and bumper beams [11, 12]. For rail vehicles, thin-walled
sections have been installed upfront to reduce the crippling impact load [13]. In the aircraft
industry, seats have been redesigned by placing two thin-walled aluminum tubes over the seat
rails and between two seat brackets, to reduce severe neck and spinal injuries [14, 15]. These
retrofitted seats were less than 3% heavier than the original design, but they helped to decrease
the acceleration experienced by the dummy occupant by 50% during a vertical drop test. Thin-
walled sections could also be installed at the foot of elevator shaft as an energy dissipating
shock absorber [16]. Fig. 1.2 illustrates the above applications of thin-walled columns.
Computational mechanics has become increasingly important in the assessment of the crash
behaviour of various structures [17]. Numerical simulation techniques, such as Finite Element
Method (FEM), have been implemented in the design process to optimize the design before
conducting prototype testing. Specifically, explicit finite element packages such as LS-DYNA
[18] and PAM-CRASH [19] have gained profound applications in analyzing complex non-linear
dynamic impact problems.
3
Fig. 1.2 Applications of thin-walled columns in the design of (a) automotive crash box,
(b) rail vehicle driver’s cab, and (c) aircraft seat (after [11-14])
4
1.2 Justification of the Study
In the axial collapse of thin-walled metallic sections, the kinetic energy is dissipated as plastic
strain energy during the formation of plastic hinges and extensional deformation of tube walls.
The design of thin-walled sections are challenging due to the large deformation process and the
material non-linearity. An ideal energy absorber must possess the following three characteristics:
(i) High energy absorption,
(ii) Acceptable crippling load, and
(iii) Repeatable mode of collapse.
For thin-walled metallic energy absorbers, the main drawback is the high initial crippling load.
This is the first peak force in the load-displacement response, which is typically 2.5-3 times of
the mean crush load. This will cause passenger injuries and is unacceptable for crashworthiness
applications [20]. Previous studies have concentrated on introducing specially designed triggers,
such as indentations, to reduce this peak load [21-22]. However, these triggers are hard to
manufacture and do not significantly reduce the initial crippling load. In light of this, several
researchers have studied tapered columns in order to reduce the initial crippling force [23-25].
However, search of the literature indicated that there is no comprehensive study of thin-walled
frustum.
Moreover, no study of multi-frusta configuration has been reported. Research has shown that
multi-tube designs have been extensively used in engineering applications. Fig. 1.3 shows the
crash-box design in front of the rail cab [26]. However, using traditional straight columns in the
multi-tube structure could lead to significant initial crippling load. Multi-frusta configuration, on
the other hand, could be a solution to this problem. Fig. 1.4 shows the schematics of the current
crash-box design, as well as the proposed design for future applications. It was with this in mind
that the research was undertaken. The outcome of the research should provide guidelines to
increase the crashworthy performance of thin-walled metallic energy absorbers.
5
Fig. 1.3 Crash-box design in rail cab (after [26])
(a) (b)
Fig. 1.4 Schematics of multi-tube crash-box design: (a) current configuration,
(b) proposed configuration
6
1.3 Research Objectives
The objective of this thesis is to study the crashworthiness of thin-walled frusta of various cross-
sectional geometries and propose design criteria of frusta. Specifically, it is desired to:
(i) develop FE models of single and multi-packed thin-walled frusta of various cross-
sectional geometries using nonlinear finite element package ANSYS/LS-DYNA,
(ii) evaluate the effect of geometry and foam-filling on the energy absorption
characteristics of thin-walled frusta,
(iii) compare the response of thin-walled prisms with frusta,
(iv) evaluate the response of multi-frusta configurations and examine interaction effects,
and
(v) experimentally validate the FE models by testing single and multi-packed thin-walled
frusta.
1.4 Method of Approach
The research described in this thesis consists of four main sections, as illustrated in Fig. 1.5. The
first involves the development of non-linear FE model of thin-walled frusta. In the second,
efforts are devoted to the experimental validations of the developed FE model. In the third,
parametric studies are conducted to study the effect of geometry and foam-filling on the
crashworthiness of thin-walled frusta. The final part involves studying multi-frusta
configurations and their energy absorption characteristics.
7
Fig. 1.5 Method of approach
8
1.5 Thesis Layout
The thesis is divided into six chapters. Chapter One justifies the undertaking of the study and
outlines the adopted methodology. Chapter Two presents a review of the existing literature on
the thin-walled sections, metallic foams, and foam-filled structures. Chapter Three is devoted
to the FE modelling process of foam-filled columns. Details of the geometry, element selection,
material constitutive relationship, contact, and explicit finite element modelling techniques are
provided. A sensitivity analysis of key parameters in the foam-filled model is presented.
Chapter Four describes the experimental works conducted to validate the developed FE models.
Chapter Five summarizes the results of the finite element studies and outlines the major
findings of the work. Chapter Six concludes the work and presents areas for possible future
exploration.
9
Chapter 2
Literature Review
Summary: In this chapter, a review of relevant literature in the crashworthiness of foam-filled
structures is provided. The review is divided into three sections: plastic collapse of thin-walled
columns, metallic foams and their properties, and the crushing behaviour of foam-filled
structures.
2.1 Axial Collapse of Thin-Walled Sections
The three most common modes of collapse for axial compression of thin-walled sections are
progressive buckling, external inversion and axial splitting [27], as shown in Fig. 2.1. In this
review, the focus is on the progressive buckling of metallic components. Fibre reinforced
composites behave quite differently, often failing by fracture and tearing [28], and will not be
discussed.
(a) (b) (c)
Fig. 2.1 Common modes of collapse of thin-walled columns under axial compression: (a)
progressive buckling, (b) external inversion, and (c) axial splitting (after [27])
10
2.1.1 Collapse Load and Energy Absorption
In order to evaluate the crashworthiness of a component, several performance indicators have
been reported and utilized in previous studies. The collapse load is an important design
parameter in the field of crashworthiness, as it is directly related to the accelerations
experienced by the passengers. The instantaneous load-displacement response is obtained by
measuring the load on the component-striker interface and the displacement of the striker. The
area under the load-displacement curve gives the total energy absorption (TEA) of the
component, i.e.:
( ) ∫ ( )
( )
where F(d) is the instantaneous crushing force, and d is the displacement of the striker. In order
to evaluate the specific energy absorption (SEA), the total energy absorption is divided by the
mass of the component:
( )
∫ ( )
( )
where mc is the mass of the component. In the designing of energy absorbers, the oscillation of
instantaneous load is often ignored and the mean collapse load is computed as:
( )
∫ ( )
( )
The typical response of a thin-walled section is shown in Fig. 2.2.
11
Fig. 2.2 Typical response of thin-walled columns: (a) instantaneous force vs. displacement,
(b) internal energy vs. displacement, and (c) mean force vs. displacement (after [29])
As shown above in the instantaneous load-displacement curve, the load level will start to rise
steeply at a displacement , which renders the energy absorber unusable because of the
extremely high acceleration associated with the deformation process. Therefore, represents
the maximum useful displacement of the component, or the stroke length. The stroke efficiency
is defined as the maximum useful displacement divided by the original length of the energy
absorber:
( )
12
where L is the initial length of the energy absorber. An ideal energy absorber should attain
maximum force immediately after material yields, and maintain a constant force level
throughout the entire length of the component. As a result, the energy efficiency is defined as
follow:
( )
( )
2.1.2 Stability
Earlier work has established the fact that the slenderness ratio (L/D or L/c) dictates whether the
column will fail under progressive folding or Euler’s mode of buckling. Andrews et al. [30]
classified the mode of collapse of cylindrical tubes under quasi-static loading condition.
Particularly, the influence of L/D ratio on the crushing mode and energy absorption was
discussed. It was found that when Eulerian mode happened, the energy absorption level dropped
significantly due to a drastic drop of crushing force level. A design chart for the global buckling
was developed that gives the critical L/D ratio for specific thicknesses over diameter ratio (h/D).
Abramowicz and Jones [31] conducted an experimental study on the static and dynamic axial
crushing of mild steel columns of circular and square geometry. The critical slenderness ratios
were found for both cross-sectional geometries. It was concluded that the non-symmetric
deformation could cause inclination of the component, which could introduce global bending.
Jensen et al. [21] carried out numerical simulations using LS-DYNA to study the transition
between progressive folding and global buckling of axially loaded aluminium extrusions
AA6060, in both T4 and T6 temper conditions. It was found that when the impact velocity
increased, the critical slenderness ratio first increased, and then decreased due to loss of global
stability. Moreover, it was determined that smaller critical slenderness ratio was associated
aluminium columns with lower strength.
The transition between progressive folding and global buckling also depends on the coaxiality
of the load and the column. Han et al. [32] examined the influence of load angle on the response
13
of square steel box columns. Results indicated that there was a critical load angle beyond which
global buckling would occur. After the critical load angle, the value of the mean crushing load
dropped to about 40% of the mean crush load in pure axial collapse.
2.1.3 Crashworthiness by Plastic Collapse
A thin-walled circular tube, when subjected to axial compressive forces, may develop either
axisymmetric buckles (concertina) or non-axisymmetric buckles (diamond). It was indicated
that thicker tubes with R/h < 40-45, deformed axisymmetrically. On the other hand, thinner
tubes with larger R/h values deformed non-axisymmetrically [8]. Alexander [33] developed the
first analytical expression of the quasi-static axial collapse of thin-walled cylindrical shells. A
concertina mode was assumed in which the tube folded by forming axisymmetric rings. In the
analysis, the work done in deforming the metal was split into two parts, those required for
bending at the plastic hinges, and those required for stretching the tube wall between hinges. Fig.
2.3 presents the idealized model in this study. The equation to calculate the mean collapse load
was proposed as:
√ ( )
where is the yield strength of the material, K is a constant best determined by fitting
experimental data. For the case of mild steel with yield strength of 70,000 psi, excellent fit was
obtained when K = 6.
14
Fig. 2.3 Idealized axisymmetric crushing mode of cylindrical shell (after [33])
Pugsley et al. [34] later studied the collapse mechanism of thinner cylindrical shells. In their
analysis, energy was assumed to be absorbed by plastic bending and shear of the diamond
pattern. They proposed a theoretical estimate of the mean axial crushing load for diamond mode
of collapse:
( ) ( )
Pugsley [35] later proposed another model for the diamond mode based on the folding of a row
of n diamonds. Using the same plastic hinge analysis as shown in Alexander’s work, the average
crushing force was evaluated as:
( )
where n is the number of diamonds formed during crushing, which depends on the D/h ratio.
Generally speaking, n increases for large D/h ratio [36]. Fig. 2.4 shows the concertina and
diamond modes of collapse for thin-walled circular columns.
15
(a) (b)
Fig. 2.4 Deformation modes of thin-walled circular tubes: (a) concertina, (b) diamond (after [36])
For square prisms, Kitagawa et al. [23] describes the folding mechanisms as follows. The
graphical illustration is shown in Fig. 2.5.
(i) local buckling begins at the weakest point of the column, and a slight wave appears on
the column wall,
(ii) stress concentrations rise at the edges of the column wall as the collapse continues,
(iii) the edges of the column wall yield, and the collapse load reaches its maximum value,
(iv) the first part of the plate folds in an accordion like mode, and
(v) the same mechanism is followed by subsequent local buckling and folding.
(a) (b) (c) (d)
Fig. 2.5 Deformation stages in an axial crushing of thin-walled square column (a) local buckling,
(b) rising of stress concentrations, (c) edge yielding, and (d) plastic collapse (after [23])
The theoretical analysis for the static buckling of thin-walled square columns follows the same
general procedure compared to circular columns. One of the first models was proposed by
Wierzbicki and Abramowicz [37, 38] using superfolding element. In their method, a
16
kinematically admissible model was developed as a basic folding mechanism. This model was
later modified by Abramowicz and Jones [39] to study the progressive buckling of square tubes
with mean width c and wall thickness h. Two basic collapse elements were identified, as
illustrated in Fig. 2.6. Based on these two elements, four deformation modes were predicted: one
symmetric mode, one extensional mode, and two asymmetric modes.
(a) (b)
Fig. 2.6 Basic collapse elements: (a) type-I, (b) type II (after [39])
To find the mean collapse load of each deformation mode, the external work of axial crushing
force was equated to the internal work required to form either one layer of lobes with four basic
collapse elements, or two adjacent layers of lobes with eight basic collapse elements. The four
types of deformation modes are described below:
Symmetric Mode
In the symmetric mode of deformation, two lobes on opposite sides fold inward and the other
two folds outward. This mode consists of four type-I basic collapse element in one layer. This
particular crushing mode is predicted to form in thin square tubes with c/h > 40.8 approximately
[40]. Assuming an effective crushing distance of 0.73 [39], the mean crushing force and the half
fold length (H) of this mode are predicted as [40]:
( ⁄ ) ( )
( )
17
where is the fully plastic moment of the wall per unit length, given by:
( )
where is the flow stress of the material. For an ideal elastic-perfectly plastic material, the
flow stress is equivalent to the yield stress. The flow stress can be modified for strain hardening
materials in the present formula, though at the expense of considerably more complicated
calculations [37]. Abramowicz et al. [41, 42] theoretically developed the following equivalent
flow stress, denoted by , to be substituted for the plastic flow stress in the theoretical
expressions developed by Abramowicz and Jones [40].
(
) ⁄
(
) ⁄
( )
where n is the strain hardening constant, and is the ultimate tensile strength.
Hanseen et al. [43] used this formula to evaluate the mean flow stress and substituted into
Equation 2.9 to calculate mean crushing force of square extrusions made of aluminium alloy
AA6060. Excellent agreement was found between theoretical and experimental results.
Extensional Mode
Extensional mode is predicted to form in thick square tubes with c/h < 7.5 approximately [40]. It
consists of four type-II folding elements in one layer. The mean crushing force and the half fold
length are predicted as:
( ⁄ ) ⁄ ( )
( )
Asymmetric Mixed Mode A
Asymmetric mixed mode A-type consists of two layers with six type-I and two type-II basic
folding elements. This mode of deformation is kinematically possible and has been observed by
18
Abramowicz and Jones [39]. However, it is less often reported in literature than the previous
two modes of collapse. The mean crushing load and the half fold length are evaluated as:
( ⁄ ) ( ⁄ ) ( )
( )
Asymmetric Mixed Mode B
The asymmetric mixed mode B-type progressive buckling is idealized as two adjacent layers of
lobes having seven type-I and one type-II basic folding elements. It is virtually indistinguishable
with the symmetric mode of collapse [40]. This type of crushing is predicted to occur within the
range of 7.5 ≤ C/H ≤ 40.8. The mean crushing force and the half fold length are given as:
( ⁄ ) ( ⁄ ) ( )
( )
The predictions of the mean crushing force associated with the four types of deformation modes
agree reasonably well with the experimental results reported by Abramowicz and Jones [39]. It
is important to note that the exact mode of collapse observed from experiment also depends on
the imperfections of the sample and is therefore difficult to predict.
(a) (b) (c)
Fig. 2.7 Deformation modes of thin-walled square columns: (a) symmetrical, (b) extensional,
and (c) asymmetric mixed mode B (after [39])
19
2.1.4 Quasi-static Finite Element Modelling of Thin-Walled Columns
FEM has been widely applied to study the collapse of thin-walled columns. Fyllingen et al. [44]
compared the numerical solutions of tube crushing modelled by both shell and solid elements. It
was pointed out that shell element could be a good compromise for modelling thin walled tubes,
especially considering the significant reduction in computational time. Meguid et al. [45]
studied the dynamic collapse of square aluminium columns using LS-DYNA explicit finite
element solver. The influence of symmetric boundary and initial conditions on the resulting
mode of collapse and the effect of using different platforms on the results were investigated. It
was shown that numerical errors in untriggered models could lead to erroneous collapse modes.
Moreover, half column model was suggested since it could capture global buckling mode of
collapse and was sufficiently constrained along one plane of symmetry. In this way, spurious
deformation modes encountered in the untriggered full model configuration could be eliminated.
Marzbanrad et al. [46] studied the effect of triggering on the crashworthiness of circular
aluminium tubes using explicit FEM. A variety of trigger types was generated including notches,
holes and plastic folds. It was concluded that the initial crippling force could be reduced by
using these triggers. However, no optimal solution was given among all the alternatives.
FEM has also been used in the designing of novel energy absorbers. For instance, Stangl et al.
[20] developed a symmetrically stepped circular thin-walled tube to provide the battery tray
inside conventional GM G-Van with necessary compliance. The collapse load and energy
absorption were predicted using DYNA3D and verified by experimental testing. Following the
previous study, Stangl et al. [47] further analyzed the effect of fillet radii on the response of
stepped circular energy absorber. Zhang et al. [48] developed a multi-cell shock absorber by
introducing internal webs to thin-walled square columns. Numerical results using LS-DYNA
showed that the SEA can be greatly improved by using the multi-cell configuration.
20
2.1.5 Dynamic Finite Element Modelling of Thin-Walled Columns
It has been commented by Norman Jones [8] that dynamic plastic buckling happens in dynamic
axial loading conditions, which is due to the structural inertia effects. In this circumstance, the
deformed shape of the structure might be very different from the progressive buckling profile, as
illustrated in Fig. 2.8. In dynamic plastic buckling, the shell is wrinkled over the entire length
and the lateral displacement field is associated with high mode numbers [8].
(a) (b)
Fig. 2.8 Modes of deformation in static and dynamic loading conditions:
(a) dynamic plastic buckling, (b) progressive folding (after [8])
There are two major effects associated with dynamic loading: strain rate effect and inertia effect.
The strain rate effect is usually taken into account of by using material constitutive models with
strain-rate sensitivity, such as the Johnson-Cook model [49]. In this model, the Von-Mises flow
stress depends on the plastic strain, strain rate, and temperature.
Abramowicz and Jones [40] conducted experiments on the impact of mild steel square tubes in
the velocity range of 0 – 12 m/s. It was concluded that the mode of collapse under dynamic
loading was the same compared to quasi-static, but an increase in the mean load was observed.
This was attributed to the strain-rate sensitivity of steel. The Cowper-Symonds uniaxial
constitutive equation that accounts for strain-rate effect was used. The following equations were
21
proposed to evaluate the mean collapse forces of symmetric and extensional deformation modes
under dynamic loading conditions:
[ ( ) ]( ) ( )
[ ( ) ][ ( ) ] ( )
where V is the impact velocity, p and D are the Cowper-Symonds constants. The predicted mean
collapse force agreed well with their experimental findings.
Because aluminium is strain rate insensitive material, only inertia effects are observed in the
dynamic loading of aluminium thin-walled sections. Langseth et al. [50] experimentally studied
the energy absorption and the collapse behaviour of square thin-walled columns made of
aluminium alloy AA6060 subjected to dynamic and quasi-static loading conditions. The
experimental results showed that the dynamic mean collapse load was significantly higher than
the corresponding quasi-static load for the same axial displacement, which indicated a strong
inertia effect. Moreover, uncertainties in the mode of collapse were found in thinner specimens
with more temper treatment. In another case, Langseth et al. [51] analyzed the sensitivity of
axial collapse load of square aluminium columns to impact velocity and mass ratio. It was
concluded that the mean collapse load kept increasing with an increase in impact velocity, and
that the mass ratio between the projectile and the specimen had no influence on the mean
collapse load.
2.2 Metallic Foams
Cellular materials are typically used in cushioning, damping, insulation, construction, and many
other applications [52]. Three-dimensional cellular structures are called foams. Foams may be
either open-cell, where gas pockets connect with each other, or closed-cell, where gas forms
discrete pockets, each completely surrounded by solid material. Fig. 2.9 shows both types of
foams.
22
(a) (b)
Fig. 2.9 Metallic foams: (a) open cell, (b) open cell (after [53])
Metallic foams possess a good combination of mechanical, electrical, thermal and acoustic
properties [54]. In particular, the mechanical strength, stiffness and energy absorption capacity
of metallic foams are much higher than those of polymer foams [55]. This presents a unique
opportunity to cost-effectively increase the crashworthiness of vehicles without adding much
weight penalty. Metallic foams are attractive materials for automotive, aerospace, military and
many other applications [56]. They have been used in the design of helmet and car bumper
system [57, 58] .
Metallic foam can undergo large plastic deformation at a relatively constant load level. Fig. 2.10
illustrates a typical compressive stress-strain response. The response closely resembles an ideal
energy absorber in that the stress attains the maximum value quickly, and maintains the level
over a large range of strain.
23
Fig. 2.10 Typical compressive stress-strain response of aluminium foam (after [59])
The compressive stress-strain curve for typical closed-cell metallic foam exhibits three distinct
regions: linear elastic, plateau, and densification. Linear elasticity is controlled by cell wall
bending and cell face stretching. The plateau region is associated with the collapse of cells and
formation of plastic hinges. It is worth noting that since the foam cells collapse as the foam is
squeezed, the axial compression produces very little lateral spreading, resulting in a close-to-
zero Poisson’s ratio during the plastic collapse. Also, the compression of gas within each cell
during the plateau stage gives a slowly rising stress-strain curve. In the densification range, cells
have been completely collapsed with opposing cell walls touching each other. Further strain
compresses the solid itself, giving the final region of rapidly increasing stress.
2.2.1 Mechanical Properties of Foam Materials
A number of researchers have investigated the mechanical properties of foam materials. Ashby
and Gibson showed that the mechanical properties of foams depend on their relative density [60].
A power law relationship relating foam properties to the relative density was proposed:
(
)
( )
0 1
Str
ess
Strain
Elastic
Stress Plateau
Densification
24
where Pf is the property of the foam, Ps is the property of the solid, is the foam density, is
the density of the solid material, C and n are the proportionality and exponential constant,
respectively.
The power law equation given above has been fit to experimental data by a number of
researchers using the method of least squares in order to evaluate the values of the coefficients.
For example, Hanssen et al. [61] obtained the following empirical relationship to evaluate the
flow stress by matching with the experimental result of Hydro closed-cell aluminium foam:
( )
( )
where is the flow stress of foam expressed in MPa. In another case, Reyes et al. [62]
developed the following equation to provide a best fit to their uniaxial compressive testing data
of Hydro closed-cell aluminium foam:
( )
( )
Ashby and Gibson [60] pioneered the analytical modeling of metallic foams. For open-cell
foams, they evaluated the flow stress of the foam based on the initial bending collapse of the
skeleton cube model. Fig. 2.11 illustrates the skeleton cube model as well as the mechanism for
initial bending collapse. The formula to calculate the flow stress was proposed as:
( )
( )
25
(a) (b)
Fig. 2.11 Skeleton cube model: (a) initial geometry, (b) cell edge bending during deformation
(after [60])
where is the yield stress of the solid material. For closed-cell foam, Santosa and Wierzbicki
[63] developed a truncated cube model as the basic folding element, which was composed of
pyramidal and cruciform sections, see Fig. 2. 12. The crushing resistance of each section was
calculated analytically based on energy considerations in conjunction with the minimum
principle in plasticity. The following formula was proposed to determine the foam flow stress in
uniaxial loading condition:
( )
( )
(a) (b)
Fig. 2.12 Truncated cube model: (a) assembly of the cell, (b) basic folding element (after [63])
26
The equation above showed good agreement with experimental data. However, the prediction of
flow stress value influenced the result significantly, which was the main difficulty in using this
equation.
2.2.2 Finite Element Modelling of Foam Materials
Santosa and Wierzbicki [63] used PAM-CRASH to simulate the deformation of three truncated
cube models as shown in Fig. 2.13, one on top of another. Very good correlation was found
between their analytical and numerical results.
Fig. 2.13 FE model of the foam cell using truncated cube unit model (after [63])
Meguid et al. [64] replaced the pyramidal section of the truncated cube model with hemispheres,
stating that metallic foam cells are mostly spherical due to high surface tension present during
the solidification process. A multi-cell FE model with 125 unit cells was simulated with in-plane
density variations of cells following a statistical distribution of the Gaussian type. The modified
27
unit cell model and the assembly of cells are shown in Fig. 2.14. The nominal stress-strain curve
obtained from the simulation was found to agree well with experimental findings.
(a) (b)
Fig. 2.14 Modified truncated cube model: (a) unit cell, (b) FE model of multiple cells (after [64])
Attia et al. [65] modified Meguid’s model to account for morphological imperfections at the
cellular scale. It was found that cell wave corrugations severely reduced both peak collapse load
and energy absorption level of the structure. On the other hand, employing curved cell walls
only had a minor effect on the crashworthiness performance of the foam.
(a) (b)
Fig. 2.15 Unit cell model account for morphological imperfections:
(a) curved walls, (b) corrugated walls (after [15, 65])
28
2.3 Foam-Filled Columns
The development of inexpensive closed-cell aluminium foams presents a unique opportunity to
increase the crashworthiness of vehicles in a cost-effective manner [66, 67]. By filling thin-
walled section with foam, the force level is significantly higher than the combined effect of
empty column and foam alone. Fig. 2.16 illustrates the interaction effect. The general expression
of the axial collapse force of a foam-filled structure is:
( )
where is the mean collapse load of the foam-filled column,
and
are the mean
collapse loads of empty column and foam, respectively, and is the increase in mean
collapse force due to the interaction effect. The mean collapse loads of column and foam are
solely dictated by the material and geometrical properties. The foam-column interaction
component relies on the relative stiffness of the interface and the penetration resistance of the
foam layer adjacent to the outer column.
Fig. 2.16 Effect of foam filling on the collapse load of thin-walled columns (after [43])
29
One of the earliest investigations on the crushing behaviour of thin-walled section filled with
foam was reported by Thornton [68]. He conducted quasi-static and dynamic axial compressive
testing of polyurethane foam-filled sections. It was found that foam filling was not weight
effective as compared to thickening of empty tube wall. This finding was later verified by
Lampinen and Jeryan [69] by a similar study. Reddy et al. [70] studied the effect of
polyurethane foam filler on the axial crushing of thin-walled aluminium alloy cans. It was found
that the stability of crushing was improved by the presence of filler. Mantena et al. [71]
analyzed the effect of foam density for three different polymeric structural foams as fillers
inside hollow steel tube. It was concluded that the lower density foams were not effective
because the crushing load did not show significant improvement.
Hanseen et al. [43] experimentally studied aluminium foam-filled square aluminium columns
made of 6060-T4 and 6060-T6. It was found that the transition from symmetric mode to
extensional mode is dependent on the foam density as well as the column wall thickness. An
empirical expression was developed to evaluate the mean collapse load of foam-filled column:
√ ( )
where is the width of the foam-filler and is a parameter to be determined by
experimental data in order to provide best fit. Note that on the right hand side of the equation,
the first term represents the column strength (same as Equation 2.9), the second term represents
the foam strength, and the third term denotes the interaction effect.
FEM has been used extensively to study the crushing behaviour of foam-filled sections. Meguid
et al. [72] investigated the crashworthiness of ultralight polymeric foam-filled structures
numerically and experimentally. Fig. 2.17 illustrates the FE model which was used for their
numerical simulations. The tube material was aluminium with circular cross section, with a PVC
foam filler of annulus cross section and density of 60 kg/m3. Comprehensive finite element
simulations were conducted to investigate the effects of key geometrical parameters on the
energy absorption characteristics of foam-filled columns. In their FE simulations, the foam
filler’s stiffness was varied by changing the cross-sectional dimensions through introducing
30
concentric through-thickness holes. The tube stiffness was controlled by changing the wall
thickness. It was found that the relative axial stiffness of the component has a major role in the
collapse behaviour of the foam-filled column. In addition, it was suggested that there exists an
optimum geometrical configuration in which the maximum value of SEA can be obtained.
Fig. 2.17 FE model geometry and contact conditions (after [72])
Santosa and Wierzbicki [73] used explicit finite element code PAN-CRASH to examine the
quasi-static axial crushing of AA6063-T7 aluminium square columns filled with aluminium
foam. In particular, the strengthening mechanism of the foam filler was examined. Empirical
relationships were derived to calculate the mean collapse force of honeycomb-filled column
(strong axis aligned with the compression axis) and that of aluminium foam filled column
(strong mechanical properties in all directions) based on their numerical results:
( )
( )
31
It was concluded that the strengthening contribution due to the lateral strength of the foam by
restraining the formation of inward folds of outer column was approximately equal to that
provided by the uniaxial strength of the foam. Moreover, it was suggested that a good way to
leverage the foam properties to provide better energy absorption was to increase the density and
strength of the foam, thus taking advantage of both mechanisms.
By foam-filling, the number of lobes on the tube wall will increase due to the elastic-plastic
foundation provided by the foams to the sidewalls [74], as shown in Fig. 2.18. Hanssen et al. [43]
conducted extensive experiments to study the behaviour of square AA6060 aluminium
extrusions filled with aluminium foams under quasi-static loading conditions. It was reported
that the number of symmetric lobes of aluminium square columns increased from 5-6 to 8-10
due to aluminium foam filling. A number of other researchers have drawn similar conclusions
[29, 75, 76].
(a) (b)
Fig. 2.18 Comparison of deformation mode under quasi-static compression:
(a) empty column, (b) foam-filled column (after [74])
When the foam core is filled within circular columns, it has been widely reported that the non-
axisymmetric mode of collapse transform to axisymmetric mode. This is due to the restraining
of the inner fold by the lateral strength of the foam. Hanssen et al. [61] experimentally tested
AA6060 aluminium extrusions filled with aluminium foam under quasi-static loading condition.
Fig. 2.19 presents one set of results corresponding to T4 temper condition and h = 1.43 mm. It
was concluded that low density foam filler (0.13 g/cm3) had no effect on the deformation mode
compared to the non-filled extrusions. On the other hand, noticeable shift from diamond to
concertina mode was observed for higher density foam fillers (0.25 g/cm3 and 0.35 g/cm
3).
32
Fig. 2.19 Mode of collapse of empty and foam-filled circular aluminium columns (after [61])
It has been shown that functionally graded foam material (FGFM) is a suitable candidate for
improving SEA over traditional uniform density foam (UDF) [77, 78]. Hooman [79] used LS-
DYNA to analyze the crushing behaviour of aluminium columns filled with discrete
functionally graded aluminium foam with different densities. The effects of various design
parameters on the energy absorption characteristics were examined, such as, density grading,
number of grading layers, and thickness of interactive layer. It was concluded that the SEA of
FGFM was improved over UDF.
Numerous researchers have investigated the effect of bonding on the response of the foam-filled
structure. Santosa et al. [80] conducted both numerical and experimental study of aluminium
foam-filled sections. It was noted that the mean crushing force in the case of bonded filling can
reach up to 60% higher than that of the unbounded case. However, the initial peak load also
increased significantly due to the additional stiffness provided by the adhesive. Hanssen et al.
[81] also conducted research on the crushing response of square aluminium extrusions filled
with aluminium foam filler. From their static crushing tests, it was reported that bonded foam
filling could result in global ruptures that would suppress progressive folds and result in a
drastic decrease in SEA. It was cautioned that strict requirements need to be placed on the
mechanical properties of foam filler, extrusion and adhesive in order to avoid fracture. A
summary of their experimental results is shown in Fig. 2.20.
33
Fig. 2.20 Results from static crushing of foam-filled components (after [81])
34
Chapter 3
Finite Element Modelling
Summary: In this chapter, we discuss the developed FE model used in simulating the crushing
behaviour of thin-walled frusta.
3.1 Geometry of the Foam-Filled Frusta
Three cross-sectional geometries are considered in this study: circular, square and hexagonal.
For illustration purpose, the geometry of foam-filled circular frustum is shown in Fig. 3.1.
Fig. 3.1 Detailed geometry representation of the foam-filled circular frustum
35
The geometrical parameters include length L, diameter at mid-span D, thickness h, and semi-
apical angle θ. For illustration purpose, top diameter DT and bottom diameter DB are also shown
in the figure. The relations between DT, DB, D and θ are expressed as:
( ) ( )
( ) ( )
( ) ⁄ ( )
It is important to mention that the developed FE models are able to simulate the crushing
behaviours of both prisms and frusta. Prisms are considered as a special case of frusta with θ =
0°. This was achieved by incorporating the semi-apical angle as a new design parameter in the
FE models. The frusta geometries were built by first specifying the diameter (width) at the mid-
span and the semi-apical angle. Next, the top and bottom diameters (width) were calculated
based on Equation 3.1 and 3.2. Finally, the top and bottom cross-sections were constructed and
the surface was created by connecting the top and bottom cross-sections.
The geometries of the column were selected based on the practical dimensions of manufactured
aluminum columns with the intention of preventing global buckling of the column. This was
achieved by selecting ratios of L/c and c/h such that stable progressive buckling mode could be
ensured. The following values of geometrical parameters were considered in this study:
(i) Semi-apical angle (θ): 0, 5, 10, 15 degrees
(ii) Length (L): 250, 300, 350 mm
(iii) Effective width at mid-span (c): 60, 70, 90 mm
(iv) Thickness (h): 1.5, 2.5, 3.5 mm
Equal mass was ensured when comparing the crashworthy performance of foam-filled structures
of different cross-sectional geometries. In order to study the effect of foam-filling, 10%
aluminum closed-cell foam was selected as the filler. This was because preliminary studies
revealed desirable interaction effect between aluminium column and 10% aluminium foam.
36
3.2 Constitutive Modelling of Materials
3.2.1 Column Wall
The constitutive relationship of column wall was modelled using piecewise linear plasticity
model with isotropic hardening and Von-Mises yield criterion, corresponding to material model
#24 in LS-DYNA theoretical manual [82]. This material model has been widely applied to study
the crushing behaviors of thin-walled metallic components [22, 29, 83]. The elastic relationship
is defined by the input of Young’s modulus, Poisson’s ratio and yield stress. The plastic
relationship is defined using effective plastic strain, effective stress pairs, which are used to
form piecewise linear yield envelope.
The material selected for this study was aluminium alloy A6061-T4. A6061 is a very popular
alloy used in automotive structures [84]. T4 temper condition was chosen because of its greater
ductility compared to T6 temper condition, in spite of its lower strength. The mechanical
properties of A6061-T4 are summarized in Table 3.1, which was obtained from the
experimental studies published in the literature [85]. Fig. 3.2 presents the engineering stress-
strain data of A6061-T4, which was incorporated into the developed FE model to accurately
simulate the crushing behaviour of this material.
Table 3.1 Mechanical properties of A6061-T4 (after [85])
Material Properties Value
Elastic modulus, E (GPa) 70
Poisson's ratio, ν 0.334
Yield stress, y (MPa) 145
Ultimate strength, u (MPa) 245
37
Fig. 3.2 Engineering stress-strain curve for A6061-T4 (after [85])
To determine the true plastic strain-true stress curve, the following relationships were used:
( ) ( )
( ) ( )
( )
where ε, are the engineering strain and stress, is the true stress, is the total true strain,
and is the plastic true strain.
3.2.2 Aluminium Foam Core
The aluminium foam used in this analysis is 10% aluminium closed-cell foam received from
Cymat Technologies. In LS-DYNA, material model #26 is appropriate for the modelling of such
metallic foams [82]. In this model, the constitutive relationship of the metallic foam is
simplified by replacing the actual behaviour with an approximately equivalent honeycomb
model, having the same properties in all three directions. The material is based on fully
uncoupled behaviour on the six components of stress and strain. In the model, the yield curves
38
for the six stress components are defined independently. Fig. 3.3 illustrates a typical yield curve
for this material model [82].
Each stress component is defined as a function of volumetric strain. Due to the fact that there is
no interaction of the stress components in this material, the plastic Poisson’s ratio is zero. This
corresponds well with the fact that the lateral expansion of the foam during compressive testing
is negligible. Therefore, assuming isotropic foam, the experimental uniaxial compressive stress-
strain curve can be input for the yield curves governing the normal stress components. The
mechanical properties of the foam was obtained from mechanical testing performed by
Heyerman [85]. Fig. 3.4 illustrates the compressive stress-strain response of 10% aluminium
foam and the corresponding yield curve, which was used for the foam model.
Fig. 3.3 Illustration of the yield curve in LS-DYNA material model #26 (after [82])
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Fig. 3.4 Compressive stress-strain response of 10% aluminium foam and the yield curve which
was input into the LS-DYNA foam model (after [85])
During compression, the elastic moduli of the foam vary from their initial values to the fully
compacted values linearly with the relative volume Vrelative:
( )
( )
( ) ( )
( )
( )
( )
40
where Eij and Gij are the elastic and shear moduli in the ij direction, Ejiu and Giju are the elastic
and shear moduli of the original configuration (uncompressed), Edensified and Gdensified are the
elastic and shear moduli of the densified foam. The factor β is defined as:
[ (
) ] ( )
where Vrelative is the current relative volume, and Vdensified is the densified relative volume. The
definition of relative volume and volumetric strain ( ) are given as follows:
( )
( )
At each time step, the stress components are updated based on the elastic interpolated moduli.
The updated stress is checked against the yield curve. If any stress component exceeds
permissible value, it will be scaled back to the yield curve. When the densification strain is
reached, the material will behave as a solid and switches to an isotropic elastic-perfectly plastic
material with von-Mises yield criterion. In this analysis, a densification strain of 0.9 was input
into the material model.
3.2.3 Loading Platen
Rigid body material was defined for the steel loading platen in order to reduce the
computational cost required to perform an explicit analysis. In a rigid body, all the degrees of
freedom of the nodes are coupled to the body’s centre of mass, which means that a rigid body
only has six degrees of freedom. At each time step, the motion of the body is calculated at the
centre of mass, and the solution is transferred to all the nodes. Preliminary analysis showed that
the response of thin-walled sections was not sensitive to the impact mass, which matched well
with previous studies [51, 85]. Square steel block with cross-sectional geometry of 150 × 150
mm and thickness of 25 mm was used as loading platen.
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3.3 Discretization Process
Shell 163 element with Belytschko-Lin-Tsay formulation was used to model the column wall.
The Belytschko-Tsay element formulation is the fastest type of all the explicit dynamics shell
elements provided in LS-DYNA. Previous analysis has shown its robustness in the simulation of
thin-walled columns under axial loading condition [44, 72]. Reduced integration was used to
prevent volumetric locking. Generally for plastic behaviour, more than 3 integration points
through thickness are required [86]. In this study, six integration points were used to fully
account for the variation of stress field through thickness. To discretize the foam filler, eight-
node solid element (Solid 164 element in LS-DYNA) with one-point integration was used.
Furthermore, stiffness-based hourglass control was used to prevent spurious zero-energy mode
due to reduced integration.
Preliminary analysis revealed acceptable computational cost of the developed FE model for the
crushing of thin-walled frusta. Therefore, symmetry was not used to save computational cost,
since the full model could capture all deformation modes, including non-symmetric Euler
buckling. For illustration purposes, Fig. 3.5 shows a section view of the developed FE model of
foam-filled square frustum.
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Fig. 3.5 FE model of the foam-filled square frustum
3.4 Boundary and Contact Conditions
Several boundary conditions were used to properly model the crushing behaviour. The bottom
surfaces of the column and foam were clamped by restraining all degrees of freedom. The top
surfaces of the column and foam were fully constrained except for the crushing direction to
prevent unrealistic deformation. A velocity of 1 m/s was applied to the top striker. The velocity
was ramped to the final value over 100 ms in order to avoid a large step force during the initial
phase of the contact between striker and tube wall.
In addition, several contact algorithms were used in the developed model. Automatic single
contact was applied to the column wall to prevent self-penetration of developed folds during the
crushing process. Automatic node-to-surface contact was applied to the interface between the
column and the loading platen. Finally, automatic surface-to-surface contact was used between
43
the column and the foam filler, as well as between the foam-filler and the loading platen.
Automatic contacts were used in the analysis since these contact types have no orientation
requirement, which means that the contact search algorithm checks for penetration from both
sides of the shell mid-plane [82]. This is important since the column wall can fold over and
change orientation.
All contact algorithms mentioned above use a penalty formulation in order to calculate the
contact force on the interface. In this formulation, the penetration between two bodies depends
on the contact stiffness; the resulting contact force is the product of penetration and contact
stiffness. The contact stiffness in ANSYS/LS-DYNA is determined by the following