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