Mechanical response of aluminum honeycombs under ...€¦ · sister, Afrin Saber Linda and brother,...

Mechanical Response of Aluminum Honeycombs Under

Indentation and Combined Compression-Shear Loadings

A. S. M. Ayman Ashab

A thesis submitted in total fulfilment of the requirements for

the degree of Doctor of Philosophy

2016

Faculty of Science, Engineering and Technology

Swinburne University of Technology, Melbourne, Australia

i

Abstract

Hexagonal aluminum honeycombs are widely known for their excellent

properties such as their high flexural stiffness to weight ratio. They can undergo

large plastic deformation to absorb high energy. Owing to their distinctive

mechanical properties, prismatic aluminum honeycombs have been used as core

materials for several decades in different industrial applications. The mechanical

response of aluminum honeycombs subjected to in-plane and out-of-plane

compression load at different loading velocities has been widely studied. However,

the study on the mechanical response of aluminum honeycombs subjected to out-

of-plane indentation and combined compression-shear loads is limited. Therefore,

it becomes important to fill the research knowledge gaps of aluminum honeycombs

under these particular loading conditions. The current study is comprised of two

parts: the first part is the experimental work and numerical simulation of aluminum

honeycombs under indentation, the second part is the experimental work and

numerical simulation of aluminum honeycombs under combined compression-

shear.

In the first part of this study, mechanical responses of aluminum honeycombs

subjected to both quasi-static and dynamic out-of-plane indentation and

compression loads were investigated experimentally. Different strain rates (10‐3 to

102 s‐1) were achieved in quasi-static and dynamic tests. Plateau stress and energy

dissipated of different aluminum honeycombs were calculated at different strain

rates. The effect of impact velocity on the plateau stress and total dissipated energy

was analyzed. The total dissipated energy under indentation loads was different

from that that under compression loads; it is the sum of energy to compress and tear

ii

honeycombs. The effect of strain rate on the tearing energy per unit fracture area

was also analyzed and empirical formulae were proposed. The experimental results

indicate that both the plateau stress and energy dissipated increased with strain

rate. Numerical analysis using ANSYS LS-DYNA was carried out for out-of-plane

dynamic indentation and compression loads. The verified FE models were used in

comprehensive finite element analysis of different aluminum honeycombs at

various strain rates from 102 s-1 to 104 s-1. The effects of strain rate and 𝑡/𝑙 ratio on

the plateau stress, dissipated energy and tearing energy were discussed. An

empirical formula was proposed to describe the relationship between the tearing

energy per unit fracture area, and the relative density and strain rate for

honeycombs. Moreover, it was found that a generic formula can be used to describe

the relationship between tearing energy per unit fracture area and relative density

for both aluminum honeycombs and foams.

In the second part of this study, combined compression-shear loads were

applied on aluminum honeycombs experimentally at different strain rates (10‐3 to

102 s‐1) and at loading angles of 15, 30 and 45 in two different plane orientations

(TL and TW). The deformation of aluminum honeycombs, crushing force, plateau

stress and energy absorption were analyzed. The effects of loading plane, loading

angle and loading velocity were discussed. An empirical formula was proposed to

describe the relationship between plateau stress and loading angle. Furthermore,

numerical simulation of honeycombs subjected to combined compression-shear

was carried out using ANSYS LS-DYNA. The verified FE models were used to

calculate the compressive and shear stresses of honeycombs at various loading

angles and loading velocities. Crushing envelopes of honeycombs were proposed.

iii

The effects of honeycomb cell wall to edge length ratio (𝑡/𝑙) and loading velocity on

the crushing envelope were discussed as well.

iv

Acknowledgements

I would like to offer thanks to Allah (SWT) for giving me the opportunity to do

my PhD at Swinburne University of Technology and guiding me with knowledge,

strength and endurance throughout my study.

I would like to express my sincere gratitude to my supervisor A/Prof. Dong Ruan

for her guidance, support and encouragement during my research study. Her

research ideas, valuable suggestions and right guidance helped me greatly to

complete my study on time. I also would like to thank Prof. Guoxing Lu, who is the

co-author of my publications, for sharing his professional knowledge and providing

valuable advice on my work. I also owe my gratitude to my Co-Supervisor Dr Yat

Choy Wong and Associate Supervisor Prof. Cuie Wen for their ongoing suggestions

and assistance during my research work.

I am thankful to my colleagues Dr Shanqing Xu (Eric), Arafat Ahmed Bhuiyan,

Md. Rezwanul Karim, Dr Mohd Azman Yahaya, Dr Gayan Rathnaweera, Mr Rafea

Dakhil Hussein, Mr Martin Vcelka and Mr Stephen Guillow for their generous

support. My special thanks also goes to the technical staff; Meredith Jewson, Warren

Gooch, David Vass, Messieurs Fejas Xhaferi, Sanjeet Chandra, Rasekhi Kia and

Michael Culton for their help throughout my experimental work.

I take this opportunity to sincerely acknowledge the Swinburne University of

Technology for providing financial assistance in the form of a PhD scholarship which

buttressed me while I performed my research work.

I would like to thank all of my friends, brothers and sisters for their eternal

support and encouragement here in Melbourne; something I will never forget.

v

I am heartily thankful to my father, A.S.M. Alamgir, my mother, Hasina Banu, my

sister, Afrin Saber Linda and brother, Farhan Shahriar, for their love, care,

encouragement and support which was invaluable to me as I completed my study.

Thanks again for your prayers. I am grateful to my father-in-law, Md. Rezaul Karim,

mother-in-law, Ozifa Imroz and brother-in-law, Wasif Karim for their love and

encouragement. Last but not least, I am deeply thankful to my wife, Rajoanna Karim

Mowly for her unconditional love, care, encouragement and support in the entire

journey of this PhD. This thesis would not have been possible without her presence

beside me.


vi

Declaration

This research work has been done by the candidate and does not contain any

materials extracted from elsewhere or from a work published by anybody else. The

work for this thesis has not been presented elsewhere by the author for any other

degree or diploma.

To the best of the candidate’s knowledge and belief this thesis contains no material

previously published by any other person except where due acknowledgement has

been made.

All work presented in this thesis is primarily that of the author under the

supervision of A/Prof. Dong Ruan. Portions of some chapters have been published

in journals and conferences and others are expected to be published also.

Signature:


Melbourne, Australia

February 2016

vii

List of publications

Papers published:

1. Ashab, A., Wong, Y. C., Lu, G. X., and Ruan, D., 9-11 Sept. 2013, "Indentation tests of

aluminum honeycombs", International Symposium on Dynamic Deformation and

Fracture of Advanced Materials (D2FAM 2013), Loughborough University, UK. DOI:

10.1088/1742-6596/451/1/012003.

2. Ashab, A., Ruan, D., Lu, G. X., and Wong, Y. C., 2014, "Combined Compression-Shear

Behavior of Aluminum Honeycombs", Key Engineering Materials, Vol. 626, pp. 127-

132.

3. Ashab, A., Ruan, D., Lu, G. X., and Wong, Y. C., 14-20 Nov. 2014, "Analysis of

mechanical response of aluminum honeycomb subjected to indentation", The

Proceedings of the ASME 2014 International Mechanical Engineering Congress &

Exposition (IMECE2014), Montreal, Quebec, Canada, paper number IMECE2014-

36620.

4. Ashab, A., Ruan, D., Lu, G. X., Xu, S., and Wen, C., 2015, "Experimental investigation

into the mechanical behavior of aluminum honeycomb under quasi‐static and

dynamic indentation", Materials and Design, Vol. 74, pp. 138‐149.

5. Ashab A., Ruan, D., Lu, G. X., and Wong, Y. C., 2016, “Quasi‐static and dynamic

Experiments of aluminum honeycombs under combined compression‐shear

loading”, Materials and Design, Vol. 97, pp. 183‐194.

6. Ashab A., Ruan, D., Lu, G. X., Bhuiyan A. A., 2016, “Finite element analysis of

aluminum honeycombs subjected to dynamic indentation and compression loads”,

Materials, Vol 9(3), pp. 162‐177.

Paper accepted:

1. Ashab, A., Ruan, D., Lu, G. X., “Numerical simulation of aluminum honeycomb

subjected to combined compression-shear Loads,” Applied Mechanics and Materials,

accepted on 25 January 2016.

viii

Table of contents

Abstract ....................................................................................................................................................... i

Acknowledgements ................................................................................................................................... iv

Declaration ................................................................................................................................................ vi

List of publications .................................................................................................................................. vii

Table of contents ..................................................................................................................................... viii

List of Figures ........................................................................................................................................... xi

List of tables ........................................................................................................................................... xvii

CHAPTER 1. Introduction ............................................................................................................................1

1.1. Motivation ........................................................................................................................................ 1

1.2. Lightweight aluminum honeycombs............................................................................................ 12

1.3. Research questions and methodology ......................................................................................... 15

1.4. Structure of this thesis .................................................................................................................. 18

CHAPTER 2. Literature review ................................................................................................................. 22

2.1. Aluminum honeycombs ................................................................................................................ 22

2.2. The mechanical response of honeycombs subjected to compression ....................................... 24

2.2.1. In-plane compression of aluminum honeycombs ............................................................... 26

2.2.2. Out-of-plane compression of aluminum honeycombs ........................................................ 28

2.2.3. Factors that affect the crushing behavior of honeycombs .................................................. 30

2.2.3.1. Effect of cell wall material ............................................................................................................. 30

2.2.3.2. Effect of t/l ratio or relative density (ρ*/ ρs) ................................................................................ 32

2.2.3.3. Effect of strain rate, shock wave and inertia ................................................................................ 34

2.2.3.4. Effect of entrapped air ................................................................................................................... 40

2.3. The mechanical response of honeycombs subjected to other types of loadings ...................... 43

2.3.1. Indentation of honeycombs .................................................................................................. 43

2.3.2. Shear of honeycombs ............................................................................................................ 46

2.3.3. Combined compression-shear .............................................................................................. 49

CHAPTER 3. Experimental investigation of the mechanical behavior of aluminum honeycombs

under quasi-static and dynamic indentation .......................................................................................... 65

3.1. Experiment set-up ......................................................................................................................... 65

3.1.1. Aluminum Honeycomb Specimens ....................................................................................... 65

3.1.2. Fixtures ................................................................................................................................... 69

3.2. Experimental Results and Discussions ........................................................................................ 73

3.2.1. Deformation of Aluminum Honeycombs Subjected to Compression and Indentation .... 73

ix

3.2.2. Experimental Data Processing ............................................................................................. 77

3.2.3. Reproducibility of test results .............................................................................................. 78

3.2.4. Plateau Stress ........................................................................................................................ 80

3.2.5. Energy Absorption ................................................................................................................ 88

3.2.6. Tearing Energy in Indentation ............................................................................................. 91

3.3. Summary...................................................................................................................................... 100

CHAPTER 4. Finite element analysis of aluminum honeycombs subjected to dynamic indentation

and compression loads ........................................................................................................................... 103

4.1. Finite element (FE) modelling ................................................................................................... 103

4.2. Validation of FE models .............................................................................................................. 107

4.2.1. Deformation patterns ......................................................................................................... 107

4.2.1. Stress-strain curves ............................................................................................................ 111

4.3. Results and discussions .............................................................................................................. 116

4.3.1. The effect of t/l ratio ........................................................................................................... 116

4.3.2. The effect of strain rate, ε̇ ................................................................................................... 123

4.3.2.1. Plateau stress ................................................................................................................................123

4.3.2.2. Energy dissipation ........................................................................................................................125

4.3.3. Deformation pattern of aluminum honeycombs subjected to compression and

indentation .................................................................................................................................... 128

4.4. Summary...................................................................................................................................... 131

CHAPTER 5. Quasi-static and dynamic experiments of aluminum honeycombs under combined

compression-shear loading .................................................................................................................... 134

5.1. Experiment details ...................................................................................................................... 135

5.1.1. Specimens ............................................................................................................................ 135

5.1.2. MTS and high speed INSTRON machines .......................................................................... 138

5.1.3. Fixtures ................................................................................................................................ 139

5.1.4. Triaxial load cell set-up ....................................................................................................... 143

5.2. Results ......................................................................................................................................... 145

5.2.1. Deformation patterns ......................................................................................................... 145

5.2.2. Rotation of cell walls ........................................................................................................... 150

5.2.3. Force-displacement curves and the effects of loading angle and plane on crushing force

........................................................................................................................................................ 153

5.2.4. The effect of loading angle on plateau stress .................................................................... 168

5.2.5. The effect of loading velocity on the plateau stress .......................................................... 173

5.2.6. Energy dissipation under combined compression-shear load ........................................ 180

5.2.7. Measurement of normal compression and shear forces using a triaxial load cell ......... 184

5.3. Summary...................................................................................................................................... 189

x

CHAPTER 6. Numerical simulation of aluminum honeycomb subjected to combined compression-

shear loads ............................................................................................................................................... 193

6.1. Finite element modelling ............................................................................................................ 193

6.2. Validation of the FE models ........................................................................................................ 198

6.2.1. Deformation model .............................................................................................................. 198

6.2.2. Rotation of cell walls ........................................................................................................... 200

6.2.3. Force- Displacement curves ................................................................................................ 203

6.2.4. Plateau stress ....................................................................................................................... 207

6.3. Results and discussions .............................................................................................................. 209

6.3.1. Force distribution ................................................................................................................ 209

6.3.2. Vertical and horizontal force .............................................................................................. 213

6.3.3. Normal compressive and shear stresses............................................................................ 215

6.3.4. Crushing envelopes ............................................................................................................. 217

6.3.5. Effect of t/l ratio ................................................................................................................... 218

6.3.6. Effect of loading velocity ..................................................................................................... 225

6.4. Summary ...................................................................................................................................... 228

CHAPTER 7. Conclusions and recommendations for future work ....................................................... 231

7.1. Conclusions .................................................................................................................................. 231

7.2. Recommendations for future work ............................................................................................ 235

References................................................................................................................................................ 238

xi

List of Figures

Figure 1.1. The Inventory of U.S. Greenhouse Gas Emissions in 2013 = 6,673 Million

Metric Tons of CO2 equivalent [2]. ............................................................................ 2

Figure 1.2. Greenhouse gas emissions attributable to transportation from 1990 to 2013

[2]. ............................................................................................................................... 3

Figure 1.3. Greenhouse gas emissions in Australian states and territories in 2013 [3]. ...... 4

Figure 1.4. CO2 emissions from transport in the Australian states and territories in 2013

[3]: (a) New South Wales; (b) Victoria; (c) Queensland; (d) Western Australia;

(e) South Australia; (f) Tasmania; (g) Australian Capital Territory; (h) Northern

Territory. .................................................................................................................... 7

Figure 1.5. Fuel consumption per seat of different aircrafts [5-11]. ...................................... 8

Figure 1.6. Major advanced materials proposed for Airbus A380 [12]. ................................ 9

Figure 1.7. Different materials used in Boeing 787 airframe components [13]. ................ 10

Figure 1.8. Effect of mass-reduction technology on CO2 emission rate for constant

performance [14]. .................................................................................................... 11

Figure 1.9. Materials used by Lotus for mass reduction of its vehicles [14]. ...................... 12

Figure 1.10. Schematic diagram of aluminum honeycombs. ................................................ 13

Figure 1.11. (a) Crushable aluminum honeycombs in the Apollo 11 landing module [18]

(b) Aluminum honeycomb core in Boeing 787 Dreamliner [9], (c) aluminum

honeycombs in automotive industry [19]. ............................................................. 15

Figure 1.12. Research workflow. ............................................................................................ 17

Figure 2.1. Aluminum honeycombs manufacturing process: (a) expansion process of

honeycomb manufacture [20]; (b) corrugated process of honeycomb

manufacture [20]. .................................................................................................... 23

Figure 2.2. Typical stress-strain curves of honeycombs in compression: (a) in-plane; (b)

out-of-plane [21]; (c) a sketch of stress-strain curve which shows the three

deformation regimes [25]. ...................................................................................... 26

Figure 2.3. Stress-strain curves of Al 3003 honeycombs in compression: (a) in-plane (W);

(b) in-plane (L) direction [24]. ............................................................................... 28

Figure 2.4. Stress-strain curves of Al 3003 honeycombs in compression loaded in the out-

of-plane (T) direction [24]. ..................................................................................... 29

Figure 2.5. Stress-deformation curves of honeycombs: (a) aluminum honeycombs; (b)

stainless steel honeycombs [36]. ............................................................................ 31

Figure 2.6. Influence of relative density on the plateau stress of honeycombs in the out-of-

plane compression [39]. .......................................................................................... 34

xii

Figure 2.7. Influence of t/l ratio and strain rate on the stress enhancement of different

honeycombs loaded in out-of-plane direction [39]. .............................................. 37

Figure 2.8. Influence of specimen dimension on the force-displacement curves in

indentation [23]. ....................................................................................................... 45

Figure 2.9. (a) Schematic of test specimen; (b) Load-displacement curve of transverse

shear deformation [101]. ......................................................................................... 48

Figure 2.10. Universal Biaxial Testing Device (UBTD) developed by Mohr and Doyoyo

[106, 107] for combined compression-shear load. ................................................ 50

Figure 2.11. (a) The load–displacement curve of a type I honeycomb specimen under a

pure compressive load (b) The load histories of honeycomb specimen with

β=90° under a combined load with ∅=90° [108, 109]. .......................................... 51

Figure 2.12. Schematic diagram of the combined compression-shear loading device [111].

.................................................................................................................................... 53

Figure 2.13. Dynamic pressure-crush curves in TW plane at different loading angles

under combined compression-shear load [111]. ................................................... 54

Figure 3.1. Three types of aluminum hexagonal honeycomb specimens used in: (a)

indentation tests; (b) compression tests. ............................................................... 67

Figure 3.2. Schematic diagram of hexagonal honeycomb. .................................................... 69

Figure 3.3. The specially designed circular plate with holes for entrapped air to escape in

both indentation and compression tests. ............................................................... 71

Figure 3.4. Out-of-plane indentation tests on aluminum honeycomb specimens (4.2-3/8-

5052-.003N): (a) quasi-static test set-up on MTS machine; (b) dynamic test set-

up on INSTRON machine. ......................................................................................... 72

Figure 3.5. Photographs of deformed specimens after compression tests at a velocity of 5

ms-1. ........................................................................................................................... 74

Figure 3.6. Photographs of deformed specimens after indentation tests at a velocity of 5

ms-1: (a) honeycomb Type H31; (b) honeycomb Type H42; (c) honeycomb Type

H45. ............................................................................................................................ 76

Figure 3.7. Reproducibility of experiments on Type H31 honeycomb specimens under

indentation loads: (a) quasi-static loading at 5×10-3 ms-1; (b) dynamic loading at

5 ms-1. ........................................................................................................................ 79

Figure 3.8. Quasi-static out-of-plane stress-strain curves of three types of honeycombs

under different loading conditions: (a) indentation at 5×10-5 ms-1; (b)

compression at 5×10-5 ms-1; (c) indentation at 5×10-4 ms-1; (d) compression at

5×10-4 ms-1; (e) indentation at 5×10-3 ms-1; (f) compression at 5×10-3 ms-1. ...... 83

xiii

Figure 3.9. Dynamic out-of-plane stress-strain curves of three types of honeycombs with

different nominal density and t/l ratio under different loading conditions: (a)

indentation at 5×10-1 ms-1; (b) compression at 5×10-1 ms-1; (c) indentation at 5

ms-1; (d) compression at 5 ms-1............................................................................... 85

Figure 3.10. Effect of strain rate on the plateau stress of three types of honeycombs with

different nominal density under different loading conditions: (a) compression;

(b) indentation. ........................................................................................................ 87

Figure 3.11. Strain rate effect on the total dissipated energy of three types of honeycombs

under different loading conditions: (a) compression; (b) indentation. .............. 89

Figure 3.12. Specific energy-strain rate curves of three types of honeycombs under

different loading conditions: (a) compression; (b) indentation. ......................... 91

Figure 3.13. Tearing energy-strain rate curves of three types of honeycombs. ................. 93

Figure 3.14. Tearing energy per unit fracture area-strain rate curves of three types of

honeycombs. ............................................................................................................. 95

Figure 4.1. Typical FE models of honeycomb H31: (a) indentation; (b) compression. .... 107

Figure 4.2. Comparison between experimental and simulated deformation mode of

honeycomb H31 under compression: (a) experimental result; (b) FEA result; (c)

experimental post-test specimen; (d) FEA post-test specimen. ........................ 109

Figure 4.3. Comparison between experimental and FEA deformation pattern of

honeycomb H42 under indentation: (a) experimental post-test specimen; (b)

FEA post-test specimen. ........................................................................................ 111

Figure 4.4. Experimental and FEA stress-strain curves of two types of honeycombs at 5

ms-1: (a) indentation of H31; (b) compression of H31; (c) indentation of H42; (d)

compression of H42. .............................................................................................. 113

Figure 4.5. The effect of 𝑡/𝑙 ratio on the plateau stresses of honeycombs under

compression and indentation loads at a strain rate of 1×103 s-1. ....................... 120

Figure 4.6. The relationship between of the tearing energy per unit fracture area and

relative density of honeycomb at a strain rate of 1×103 s-1. ............................... 121

Figure 4.7. Normalized tearing energy per unit fracture area-relative density of different

cellular materials. .................................................................................................. 122

Figure 4.8. Effect of strain rate on the plateau stresses of two types of honeycombs

subjected to: (a) indentation; (b) compression. .................................................. 124

Figure 4.9. Normalized plateau stress of honeycomb-strain rate of honeycombs in

compression. .......................................................................................................... 125

Figure 4.10. Effect of high strain rate on the total dissipated energy of two types of

honeycombs: (a) indentation; (b) compression. ................................................. 126

xiv

Figure 4.11. Effect of strain rate on the tearing energy of different honeycombs. ........... 127

Figure 4.12. The dependency of tearing energy per unit fracture area of honeycombs and

strain rate. ............................................................................................................... 128

Figure 4.13. Deformation of honeycomb H31 at 5 ms-1: (a) compression; (b) indentation.

.................................................................................................................................. 131

Figure 5.1. A photograph of aluminum honeycomb (4.2-3/8-5052-.003N). T is the out-of-

plane direction. L and W are the in-plane directions........................................... 135

Figure 5.2. A photograph of three types of honeycomb specimens used in combined

compression-shear tests. ....................................................................................... 138

Figure 5.3. Photographs of three sets of fixtures used on MTS machine for combined

compression-shear tests at three different loading angles of: (a) 15; (b) 30; (c)

45. ........................................................................................................................... 141

Figure 5.4. Photographs of two sets of fixtures used on high-speed INSTRON machine for

combined compression-shear tests at two different loading angles of: (a) 15;

(b) 30. ..................................................................................................................... 142

Figure 5.5. A photograph of testing fixture showing sliding guide rods. ........................... 142

Figure 5.6. Experimental set-up of combined compression-shear tests on the MTS

machine. .................................................................................................................. 143

Figure 5.7. A photograph of Kistler triaxial load cell (3-component force link-type 9377C).

.................................................................................................................................. 145

Figure 5.8. Crushing process of H31 honeycomb in TL plane at 45 loading angle under

combined compression-shear load at a velocity of 5×10-3 ms-1. Displacement

indicated is vertical cross-head movement. ......................................................... 147

Figure 5.9. Deformation of H31 honeycomb crushed in TL plane at 45loading angle at

5×10-3 ms-1. ............................................................................................................. 148

Figure 5.10. Photographs of three types of honeycombs tested under combined

compression-shear loads at different loading angles and in different planes. .. 149

Figure 5.11. Effect of loading velocity on rotational angle β of honeycomb H31 loaded at

30 in: (a) TL plane; (b) TW plane......................................................................... 151

Figure 5.12. Effect of loading angle on rotational angle β of honeycomb H31 at a velocity

of 5×10-3 ms-1 in the: (a) TL plane; (b) TW plane. ................................................ 153

Figure 5.13. Vertical force-displacement curves for honeycomb subjected to combined

compression-shear loads at a loading angle of 15: (a) H31 in TL plane; (b) H31

in TW plane; (c) H42 in TL plane; (d) H42 in TW plane; (e) H45 in TL plane; (f)

H45 in TW plane. .................................................................................................... 156

xv




H45 in TW plane. .................................................................................................... 159




H45 in TW plane. .................................................................................................... 162

Figure 5.16. Effect of loading angle on the vertical force-displacement curves of different

honeycombs at a loading velocity of 5×10-3 ms-1: (a) H31 loaded in the TL plane;

(b) H31 loaded in the TW plane; (c) H42 loaded in the TL plane; (d) H42 loaded

in the TW plane; (e) H45 loaded in the TL plane; (f) H45 loaded in the TW plane.

................................................................................................................................. 166

Figure 5.17. Sketch of the force components in a combined compression-shear test. .... 168

Figure 5.18. Effect of loading angle on plateau stress ratio for different honeycombs. ... 173

Figure 5.19. Effect of loading velocity on plateau stress at different loading angles and in

different planes: (a) H31 in TL plane; (b) H31 in TW plane; (c) H42 in TL plane;

(d) H42 in TW plane; (e) H45 in TL plane; (f) H45 in TW plane. ....................... 176

Figure 5.20. Effect of loading velocity on normalized plateau stress ratio at different

loading angles for honeycombs: (a) H31 in TL plane; (b) H31 in TW plane; (c)

H42 in TL plane; (d) H42 in TW plane; (e) H45 in TL plane; (f) H45 in TW plane.

................................................................................................................................. 180

Figure 5.21. Effect of loading velocity on specific energy at different loading angles and in

different planes (a) H31 in TL plane; (b) H31 in TW plane; (c) H42 in TL plane;

(d) H42 in TW plane; (e) H45 in TL plane; (f) H45 in TW plane. ....................... 184

Figure 5.22. Force-displacement curves of honeycombs subjected to combined

compression-shear load at 15° loading angle: (a) H31 in the TL plane; (b) H31 in

the TW plane; (c) H42 in the TL plane; (d) H42 in the TW plane; (e) H45 in the

TL plane; (f) H45 in the TW plane. ....................................................................... 187

Figure 6.1. A finite element model of honeycomb (H31). ................................................... 195

Figure 6.2. Finite element models of honeycombs subjected to combined compression-

shear loads at three different loading angles: (a) 15; (b) 30; (c) 45. ............ 198

Figure 6.3. Comparison between experimental and simulated results of deformation

model of honeycomb (H31): (a) experimental result at loading angle 15; (b)

xvi

simulated result at loading angle 15; (c) experimental result at loading angle

30; (d) simulated result at loading angle 30. .................................................... 200

Figure 6.4. Rotation of honeycomb cell walls subjected to combined compression-shear

load at 15 and 5 ms-1: (a) experiment; (b) FEA. ................................................. 201

Figure 6.5. Comparison between experimental and simulated rotational angle-

displacement of honeycomb H31 loaded at a velocity of 5 ms-1: (a) at 15 loading

angle; (b) at 30 loading angle. .............................................................................. 202

Figure 6.6. Comparison between experimental and simulated force-displacement curves

of different honeycombs loaded at 15 loading angle and a velocity of 5 ms-1: (a)

H31-TL plane; (b) H31-TW plane; (c) H42-TL plane; (d) H42-TW plane. ......... 205

Figure 6.7. Comparison between experimental and simulated force-displacement curves

of different honeycombs loaded at 30 loading angle and a velocity of 5 ms-1: (a)

H31-TL plane; (b) H31-TW plane; (c) H42-TL plane; (d) H42-TW plane. ......... 207

Figure 6.8. Vertical force-displacement curves of honeycomb H31 at different loading

angles and a velocity of 5 ms-1: (a) 15 loading angle in the TL plane; (b) 15

loading angle in the TW plane; (c) 30 loading angle in the TL plane; (d) 30

loading angle in the TW plane; (e) 45 loading angle in the TL plane; (f) 45

loading angle in the TW plane. .............................................................................. 212

Figure 6.9. Force-displacement curves of a single H31 honeycomb under combined

compression-shear loads in the TL plane at 5 ms-1: (a) vertical force; (b)

horizontal force. ...................................................................................................... 214

Figure 6.10. Crushing envelopes of honeycombs H31 in the normal stress-shear stress

coordinate system when they are subjected to combined compression-shear

loads at 5 ms-1: (a) TL plane; (b) TW plane. ......................................................... 218

Figure 6.11. Effect of t/l ratio in the normal compressive plateau stress-shear plateau

stress curves of honeycombs at a loading velocity of 5 ms-1 in the TL plane. ... 222

Figure 6.12. The effect of t/l ratio on the plateau stresses of honeycombs at a loading

velocity of 5 ms-1 in the TL plane: (a) pure compressive plateau stress; (b) shear

plateau stress. ......................................................................................................... 224

Figure 6.13. Effect of strain rate in the normal compressive plateau stress-shear plateau

stress curves of honeycombs at a loading velocity of 5 ms-1 in the TL plane. ... 226

Figure 6.14. The effect of strain rate on the plateau stresses of honeycombs at a loading

velocity of 5 ms-1 in the TL plane: (a) pure compressive plateau stress; (b) shear

plateau stress. ......................................................................................................... 227

xvii

List of tables

Table 2.1. Summary of previous work conducted on honeycombs (mainly aluminum

honeycombs) ............................................................................................................ 56

Table 3.1. Specification of aluminum honeycombs ............................................................... 68

Table 3.2. Summary of all quasi-static and dynamic experimental results ......................... 96

Table 4.1. Specification of aluminum honeycombs [20] ..................................................... 104

Table 4.2. Material properties used in the FE model of aluminum honeycombs [132] ... 105

Table 4.3. Material properties used in the FE model of rigid bodies [132] ....................... 105

Table 4.4. Comparison between FEA and experimental results at 5 ms-1 ......................... 115

Table 4.5. FEA results of honeycombs with constant cell wall thickness and different cell

size .......................................................................................................................... 117

Table 4.6. FEA results of honeycombs with constant cell size and different cell wall

thickness ................................................................................................................. 118

Table 5.1. Specification of three types of aluminum hexagonal honeycombs [20] .......... 137

Table 5.2. Quasi-static and dynamic experimental results for honeycomb H31 .............. 169



Table 5.5. Normal compressive and shear results of honeycomb H31 at different loading

velocities ................................................................................................................. 188


velocities ................................................................................................................. 189


velocities ................................................................................................................. 189

Table 6.1. Specification of aluminum honeycombs simulated [20] ................................... 194

Table 6.2. Material properties of aluminum honeycombs and blocks used in the finite

element analysis [38] ............................................................................................ 196

Table 6.3. Comparison between experimental and simulated results of honeycombs H31

and H42 at 5 ms-1 loading velocity ........................................................................ 208

Table 6.4. (a) Normal compressive and shear plateau forces and stresses of honeycomb

H31 at 5 ms-1 .......................................................................................................... 216

Table 6.5. Normal compressive and shear plateau stresses of honeycombs with different

t/l ratios at a loading velocity of 5 ms-1 ................................................................ 219

Table 6.6. Pure compressive and shear plateau stresses of honeycombs for different t/l

ratios at a loading velocity of 5 ms-1 ..................................................................... 223

xviii

Table 6.7. Pure compressive and shear plateau stresses of honeycombs for different strain

rates ......................................................................................................................... 226

1

Chapter 1. Introduction

1.1. Motivation

Nowadays, it is a trend to use lightweight materials to reduce weight in the

aircraft, automobile, naval architecture and other manufacturing industries. The

aim of this weight reduction is to enhance the fuel efficiency (reduce fuel

consumption) and also to minimize the materials cost. Capehart [1] mentioned in

his book that by reducing weight by approximately 1 %, the fuel consumption was

reduced approximately 0.75 %. With the enhancement of fuel efficiency, the MPG

(miles per gallon) value increases. As a result the combustion of fuel will be reduced

which has significant impact on decreasing carbon dioxide (CO2) emissions in the

environment. According to the statistics published by United States Environmental

Protection Agency [2] in 2013, the greenhouse gas emissions from the

transportation sector accounted for approximately 27% of the total U.S. greenhouse

gas emissions, which was the second largest contributor to greenhouse gas

emissions in the U.S. after ‘Electricity’ as shown in Fig. 1.1.

2

Figure 1.1. The Inventory of U.S. Greenhouse Gas Emissions in 2013 = 6,673 Million

Metric Tons of CO2 equivalent [2].

The majority of these gas emissions resulted from CO2 emissions caused by the

combustion of petroleum-based products, such as gasoline, in internal combustion

engines. More than half of the CO2 emissions were from passenger cars and light-

duty trucks, sport utility vehicles, pickup trucks, and minivans etc. The remaining

CO2 emission sources came from other types of transportation, including freight

trucks, commercial aircraft, ships, boats, trains, etc. The greenhouse gas emission

attributable to transportation has increased by approximately 16% since 1990 as

shown in Fig. 1.2. The increasing demand for travel and the limited achievements in

fuel efficiency across the U.S. transportation have caused this historic increase.

3

Figure 1.2. Greenhouse gas emissions attributable to transportation from 1990 to

2013 [2].

In 2013, the Department of the Environment, Australian Government [3]

published reports on greenhouse gas emissions in the different states and territories

of Australia. According to the statistical analysis (see Figure 1.3), Queensland, New

South Wales and Victoria are more responsible for greenhouse gas emissions than

the other states and territories.

4

Figure 1.3. Greenhouse gas emissions in Australian states and territories in 2013

[3].

According to the Australian Department of the Environment, the possible

sources for CO2 emissions are: stationary energy, transport, fugitive emissions,

industrial processes and industrial product use, agriculture, waste, land use, land-

use change and forestry, etc. Again transport is one of the major sources of

CO2 emissions in all the states and territories as shown in Fig. 1.4. There are many

possible options to reduce gas emissions and one of the major possible ways is to

improve the fuel efficiency of vehicles by considering materials, design, and

technologies. A significant reduction in mass of the vehicles could be achieved by

employing lightweight materials, advanced design of vehicle structures and new

technologies.

5

(a)

(b)

(c)

6

(d)

(e)

(f)

7

(g)

(h)

Figure 1.4. CO2 emissions from transport in the Australian states and territories in

2013 [3]: (a) New South Wales; (b) Victoria; (c) Queensland; (d) Western Australia;

(e) South Australia; (f) Tasmania; (g) Australian Capital Territory; (h) Northern

Territory.

Like other transport sectors, the aircraft industry is also responsible for

CO2 emissions in the environment and consequently it is aiming to reduce

CO2 emissions by increasing fuel efficiency. Both the renowned aircraft companies,

Airbus and Boeing, are taking efforts to manufacture larger commercial and cargo

8

aircrafts to increase capacity. The weight of the aircraft is a vital factor that affects

fuel consumption, so it is essential to develop lightweight aircraft structures. From

1960 to 2000 the overall fuel efficiency of jet aircrafts (based on Boeing 707)

improved by approximately 55% [4]. Figure 1.5 shows the improvement of fuel

consumption per seat (L/km) for different aircrafts.

Figure 1.5. Fuel consumption per seat of different aircrafts [5-11].

The use of lightweight materials in the aircraft industry is increasing day by day.

Among different lightweight materials, carbon and glass fibre-reinforced plastics,

aluminum, aluminum alloys, titanium, and composites are still the priority for

material selection in building modern commercial aircrafts for weight reduction

purposes. For example, in the Airbus A380, different lightweight materials

(aluminum, aluminum alloys, carbon fibre-reinforced materials, thermoplastic

materials, etc.) were used in building different components of the aircraft. Jerome

9

[12] described the innovation that was introduced in designing the A380 aircraft.

Lower airframe weight and better aerodynamics structure were translated into

lower fuel burn, reduced emissions into the atmosphere, and in reduction of

operating costs. The major advanced materials that were considered for Airbus

A380 are shown in Fig. 1.6.

Figure 1.6. Major advanced materials proposed for Airbus A380 [12].

The Boeing 787 Dreamliner is one of the latest commercial aircraft where

approximately 50 % of its structural components are made from composite

materials, 20 % from aluminum and aluminum alloys, 15 % from titanium, 10 %

from steel (primary landing gear) and 5 % from other materials [13]. The use of

these lightweight materials in the airframe components reduces the total mass of

the aircraft significantly and this led to the Boeing 787 being named as the most fuel

efficient passenger plane in aircraft manufacturing history. The materials used in

the Boeing 787 structure are shown in Fig. 1.7.

10

Figure 1.7. Different materials used in Boeing 787 airframe components [13].

The concept of mass reduction technology is being employed in the automobile

industry extensively for every part of the entire vehicle. All automobile companies

are striving to improve the vehicle’s performance and fuel efficiency or reduce

carbon dioxide (CO2) emissions, both of which are related to the mass of vehicles.

Figure 1.8 shows the influence of the mass reduction on the CO2 emission rate for

conventional vehicles reported by Lutsey [14]. It was found that a mass reduction of

approximately 10 % of a conventional vehicle resulted in a decrease in fuel

consumption of 6 % to 8 %.

11

Figure 1.8. Effect of mass-reduction technology on CO2 emission rate for constant

performance [14].

Ford intends to reduce vehicle weight by 250-750 lb from 2011 to 2020 [15],

Mazda aims to reduce vehicle weight by about 220 lb by 2016 [16], Nissan has stated

their intention to reduce weight by about 15 % by 2015 [17]. Similarly, Toyota has

indicated its intention to reduce the weight of the Corolla and other mid-size

vehicles by 30 % and 10 % respectively [16]. All other luxury automobile companies

are also focusing on mass reduction for their vehicles. For example, Lotus set up a

lightweight structure division for its automotive sector. They are developing

material compositions for their vehicle structure. They are implementing lower

density materials, such as aluminum, magnesium, plastics, etc., to optimize vehicle’s

mass as shown in Fig. 1.9 [14]. Similarly, BMW invested millions of dollar into a

carbon fibre manufacturing process while Jaguar made the decision to use

lightweight aluminum for its luxury vehicles.

12

Figure 1.9. Materials used by Lotus for mass reduction of its vehicles [14].

Structural components used in aircrafts and vehicles may be subjected to

dynamic loads in crash accidents. Materials and structures may deform differently

under dynamic and quasi-static loads. Therefore it is essential to understand the

mechanical properties of materials and structures under both quasi-static and

dynamic loads.

1.2. Lightweight aluminum honeycombs

Honeycomb is a type of cellular material (Fig. 1.10) with periodical isotropic or

anisotropic unit cells, which may have different shapes such as triangle, square,

equilateral triangle, isosceles, parallelogram, regular hexagon and, irregular

hexagon. Depending on the required characteristics and intended application,

honeycomb structures can be made of different materials, e.g., polymers, Nomex,

metals and ceramics. Among these, aluminum hexagonal honeycombs have

13

outstanding properties. They have a high strength ratio and they can undergo large

plastic deformation under almost constant force so that they are able to absorb a

large amount energy when they deform.

Figure 1.10. Schematic diagram of aluminum honeycombs.

They are widely used in various fields of engineering such as aerospace, aircraft,

automotive and naval engineering. Below are three example applications of

aluminum honeycombs.

Crushable aluminum honeycombs were used as shock absorbers located inside

the primary strut of the landing gear of the Apollo 11 lunar module [18] (Fig.

1.11a).

Aluminum honeycombs were used in landing gear doors and flaps of aircraft. In

the Boeing 787 Dreamliner, aluminum honeycombs were used as the core

material in the window frames, and interior components including stowage bins,

class dividers, partitions and crew rests (Fig. 1.11b).

In the automotive industry, for optimal weight distribution and safety, the

chassis is made out of carbon fibre with aluminum honeycombs [19] (Fig. 1.11c).

14

(a)

(b)

15

(c)

Figure 1.11. (a) Crushable aluminum honeycombs in the Apollo 11 landing module

[18] (b) Aluminum honeycomb core in Boeing 787 Dreamliner [9], (c) aluminum

honeycombs in automotive industry [19].

1.3. Research questions and methodology

A large body of research has been conducted on aluminum honeycombs to study

their mechanical responses when subjected to different types of loadings. A

comprehensive literature review can be found in Chapter 2, the Literature Review.

Some research gaps were identified and this present research work aims to fill in

some of these gaps. In summary, this research will study experimentally and

numerically the mechanical responses of aluminum honeycombs subjected to out-

of-plane indentation and combined compression-shear loads respectively. The

research work is divided into four parts. The first part is the experimental

indentation and compression tests on aluminum honeycombs by using MTS and

high speed INSTRON machines at different loading velocities. The energy absorption

of different aluminum honeycombs is investigated. The second part is the finite

16

element analysis (FEA) of aluminum honeycombs subjected to indentation and

compression loads. Parametric studies are conducted using FEA to investigate the

effects of different parameters such as dimensions of cells and loading velocity on

the energy absorption of honeycombs. The third part presents the experiments on

aluminum honeycombs subjected to combined compression-shear loads by using

MTS and high speed INSTRON machines at various loading velocities and angles. The

fourth part is the finite element analysis of the combined compression-shear

crushing of aluminum honeycombs, from which the compressive and shear forces

can be calculated. Crushing envelopes are developed and the effects of dimensions

of honeycomb cells and loading velocity are discussed. A workflow of this research

work is shown below in Fig. 1.12.

17

Figure 1.12. Research workflow.

18

1.4. Structure of this thesis

Chapter 1: A brief introduction of the applications of lightweight materials

including aluminum honeycombs in industries, research questions of this PhD work

and the structure of this thesis.

Chapter 2: It is a comprehensive literature review of the mechanical behavior and

deformation mechanism of aluminum honeycombs subjected to various loadings.

The review includes experimental, numerical and theoretical analyses on

honeycombs loaded in both in-plane and out-of-plane directions. The factors that

affect the material properties of honeycomb are also discussed. The research gaps

are identified and the scope of this research work is determined based on the

literature review.

Chapter 3: Experimental investigation of the mechanical behavior of three types of

aluminum hexagonal honeycombs under quasi-static and dynamic indentation in

the out-of-plane directions are conducted using MTS and high speed INSTRON

machines. The effect of strain rate on both plateau stress and energy absorption is

studied. The tearing energy is calculated as the difference in energy dissipated in

indentation and compression of the same type of honeycomb. Empirical formulae

are proposed for tearing energy with relation to the strain rate.

19

Chapter 4: Finite element analysis of aluminum hexagonal honeycombs with

different 𝑡/𝑙 ratios is carried out by using ANSYS/LS-DYNA. The FEA models are

verified by experimental results obtained in Chapter 3. The verified FEA models are

then used to study the effect of strain rate and 𝑡/𝑙 ratio on the plateau stress,

dissipated energy and tearing energy. An empirical formula is proposed to describe

the relationship between tearing energy per unit fracture area and the strain rate

and 𝑡/𝑙 ratio or relative density.

Chapter 5: The mechanical behaviour of three types of aluminum hexagonal

subjected to combined compression-shear loads are experimentally studied. Both

quasi-static and dynamic tests are conducted at three different loading angles of 15,

30 and 45 and five different velocities respectively. The deformation, crushing

force, plateau stress and energy absorption of honeycombs are presented. The

effects of loading plane, loading angle and loading velocity are discussed. An

empirical formula is proposed to describe the relationship between plateau stress

and loading angle. Furthermore, a triaxial load cell is employed in the modified

experimental technique to measure the normal compressive and shear forces at 15

loading angle and three different low velocities.

20

Chapter 6: Finite element analysis of aluminum honeycombs subjected to combined

compression-shear loads is carried out by ANSYS LS-DYNA. The FEA facilitates the

measurement of the vertical and horizontal forces from which the normal

compressive and shear forces can be calculated. Crushing envelopes are established

accordingly. The effects of loading velocity, dimensions of honeycombs (𝑡/𝑙) and

loading plane (TL and TW) on the crushing envelope are discussed. Empirical

formulae are derived to describe crushing envelopes for honeycombs subjected to

combined compression-shear loads.

Chapter 7: In this chapter, the findings of this research work are summarised and

possible future work is also recommended.

22

Chapter 2. Literature review

Honeycombs made of aluminum, copper, stainless steel, Nomex paper, etc. may

be subjected to different types of loadings (such as in-plane and out-of-plane

compression, indentation, shear and combined compression-shear loadings) in

their applications. Their mechanical behaviour has been investigated

experimentally, numerically and theoretically under a wide range of quasi-static and

dynamic loading velocities. In this chapter, the literature review of the research

conducted on the mechanical response of honeycombs (mainly aluminum

honeycombs) is presented.

2.1. Aluminum honeycombs

Man–made aluminum honeycombs can be manufactured by two different

approaches: expansion process and corrugated process [20]. In the expansion

process, very thin aluminum sheets are tacked up with adhesive in-between to form

HOBE (HOneycomb Before Expansion) blocks. Then HOBE blocks are expanded to

become honeycomb structures. The expansion process is shown in Fig. 2.1(a). In the

corrugated process, aluminum sheets are rolled to form into a corrugated shape and

then adhesive is applied to the corrugated nodes, after which the blocks are formed

by stacking the corrugated sheets together as shown in Fig. 2.1(b).

23

(a)

(b)

Figure 2.1. Aluminum honeycombs manufacturing process: (a) expansion process

of honeycomb manufacture [20]; (b) corrugated process of honeycomb manufacture

[20].

24

2.2. The mechanical response of honeycombs subjected to compression

The deformation and crushing mechanism of aluminum honeycombs subjected

to compressive loads has been carried out theoretically, experimentally and

numerically in the two in-plane (L-ribbon or W-transverse) directions and one out-

of-plane (T) direction. Gibson and Ashby [21, 22] and Zhou and Mayer [23] classified

three different deformation regimes of a typical crushing of honeycombs as: (1)

linear elasticity (2) plateau region and (3) densification in stress-strain curves of

honeycombs loaded in both the in-plane and out-of-plane directions as shown in Fig.

2.2. The average stress over this region is known as the plateau stress, 𝜎𝑝𝑙 (Fig. 2.2c).

In a typical crushing of honeycombs, compression initiates by linear elastic

deformation of the cells with a rise in stress, and the cell walls bend due to the stress.

After this, a plastic buckling of the cell walls occurs with a roughly constant plateau

stress and finally the stress rises which is considered as densification of cell walls. It

has been demonstrated that the deformation, plateau stress and energy absorption

of aluminum honeycombs depends on not only the geometrical configuration and

mechanical properties of cell wall materials [21, 22, 24], but also the loading

velocity. Cell wall thickness to cell edge length ratio, 𝑡 𝑙⁄ or relative density, 𝜌∗ 𝜌𝑠⁄ ,

strain rate, and ε̇, are the important factors that affect plateau stress as well as the

energy absorption of aluminum honeycombs.

25

(a)

(b)

26

(c)

Figure 2.2. Typical stress-strain curves of honeycombs in compression: (a) in-

plane; (b) out-of-plane [21]; (c) a sketch of stress-strain curve which shows the

three deformation regimes [25].

2.2.1. In-plane compression of aluminum honeycombs

Zhou and Mayer [23] tested two types of aluminum honeycombs subjected to in-

plane compression load. They observed that the stress-strain curves were similar

for both the two in-plane directions but crushing strength in the ribbon direction

(L) was larger of that in the transverse direction (W). Khan et al. [24, 25] also

observed similar phenomena in their experimental in-plane compression tests on

aluminum 3003 alloy honeycombs. They discussed that the stresses remained the

same up to a certain strain in both the two in-plane directions, but in the plateau

region, honeycombs behaved like elastic-perfectly-plastic material with no work

hardening. Folding and shear band of the cells were observed due to the in-plane

crushing process. Furthermore, they indicated that the stress fluctuated due to the

27

shear band with subsequent folding of cells when loaded in the W- direction. The

shear band was not entirely developed when honeycombs were loaded in the L-

direction and as a result stress was not fluctuated. The stress-strain curves of in-

plane crushing of honeycombs reported by Khan et al. [24, 25] are shown in Fig. 2.3.

Papka and Kyriakides [26, 27] studied the in-plane crushing of honeycombs

experimentally and numerically. They characterized the crushing response of

honeycombs by the collapse mechanism of cells under displacement control loading.

The honeycomb cells uniformly collapsed in the form of row by row destabilization

until densification of the whole honeycomb specimen occurred. During crushing in

the in-plane directions, Klintworth and Stronge [28] observed an interaction

between the plastic collapse and elastic buckling, and they developed constitutive

equations for the deformation of transversely crushed honeycombs. Hönig and

Stronge [29] reported numerical in-plane crushing of aluminum honeycombs and

identified the elastic wave propagation effects on the crushing initiation which

causes reinforce or delay crushing.

(a)

28

(b)

Figure 2.3. Stress-strain curves of Al 3003 honeycombs in compression: (a) in-plane

(W); (b) in-plane (L) direction [24].

2.2.2. Out-of-plane compression of aluminum honeycombs

Gibson and Ashby [21, 22] reported that the stiffness and strengths were larger

in the out-of-plane (T) direction than in the two in-plane (L and W) directions. Khan

et al. [24, 25] also experimentally observed that in the out-of-plane direction,

honeycombs behaved the strongest. They observed a sharp peak in the elastic region

in the stress-strain curve of honeycomb crushed in the out-of-plane direction. The

stress-strain curves of out-of-plane crushing of honeycombs reported by Khan et al.

[24, 25] are shown in Fig. 2.4. The yield stress was also found to be significantly

higher than that in the two in-plane directional crushing. They identified that both

global and local deformation occurred simultaneously in the honeycomb cell walls

loaded in the out-of-plane direction. These deformations were caused by the cell

walls buckling. They also observed difference in the shear and axial deformation

29

around the specimen’s boundaries and inner cells respectively. This difference was

caused by constrains between the inner cell walls and the neighbouring cell walls.

Figure 2.4. Stress-strain curves of Al 3003 honeycombs in compression loaded in

the out-of-plane (T) direction [24].

Wierzbicki [30] reported that the angle of cell walls was mainly responsible for

the buckling of hexagonal honeycomb cells. He mentioned that the deformation of

honeycomb cells was developed by stationary and moving plastic hinges. Wang [31]

theoretically determined the out-of-plane elastic collapse stress of hexagonal

honeycombs and found good agreement with their experimental data and the semi-

empirical formulae derived by Gibson and Ashby [32]. Mohr and Doyoyo [33] and

Aktay et al. [34] also reported the plastic deformation process of the cell walls while

compressed in the out-of-plane direction. Deqiang et al. [35] conducted numerical

analyses of honeycombs in the out-of-plane direction and found that the progressive

plastic buckling deformation initiated from the top surface which moved

30

downwards and the magnitude of force peak to peak value in the force-displacement

curve fluctuated with the folding process.

2.2.3. Factors that affect the crushing behavior of honeycombs

In the literature it was found that there are number of factors that have an effect

on the crushing strength, energy absorption and deformation mechanism of

aluminum honeycombs [26, 27, 30, 35-46] under compressive loads. The major

factors indicated in the literature are: cell wall material, 𝑡 𝑙⁄ ratio or relative density

(𝜌∗ 𝜌𝑠⁄ ), loading velocity or strain rate (𝜀̇) , shock wave, inertia and entrapped air.

All these factors are reviewed in the following subsections.

2.2.3.1. Effect of cell wall material

Different types of honeycombs have been studied by the researchers [36, 47-52]

both experimentally and numerically. The mechanical properties of different types

of honeycombs vary according to the cell wall material. Quasi-static and dynamic

compression loads have been applied to high density metal honeycombs such as

stainless steel honeycombs and aluminum honeycombs by Baker et al. [36]. They

observed different deformation patterns of the honeycombs constructed from two

different cell wall materials. They also found differences in the stress-deformation

curves due to the effects of material strength and density as shown in Fig. 2.5.

Aminanda et al. [47] reported an experimental and numerical compression study of

honeycombs made from drawing paper, Nomex paper and aluminum. They

observed similar load-displacement curves but differences in magnitude and

deformation due to the behavior of different materials. Other types of honeycombs

made from various materials were studied by many researchers: Papka and

Kyriakides [26], Porter [48], Foo et al. [49], Dharmasena et al. [50], Gpoichand et al.

31

[51], Engilner et al. [52] and so on. These honeycombs were RIP steel TRIP-matrix

composite honeycombs, Nomex honeycombs, ceramic honeycombs, copper

honeycombs, poly-carbonated honeycombs, etc. Due to the different cell wall

material properties, the crushing strength, energy absorption and deformation

mechanisms varied. In general, the stronger the cell wall material, the stronger the

honeycombs.

(a)

(b)

Figure 2.5. Stress-deformation curves of honeycombs: (a) aluminum honeycombs;

(b) stainless steel honeycombs [36].

32

2.2.3.2. Effect of t/l ratio or relative density (𝜌∗ 𝜌𝑠⁄ )

McFarland [42] conducted the pioneer work and derived a semi-empirical

formula to calculate the mean crushing strength of honeycombs under axial

compression. It was found that the plateau stress increased with the increase of 𝑡 𝑙⁄

ratio. The effect of cell wall thickness to edge length ratio (𝑡 𝑙⁄ ) on the mean plateau

stress was more precisely identified by Wierzbicki [30], where the plateau stress for

a regular hexagonal honeycomb was found to increase with 𝑡 𝑙⁄ ratio by a power law

with the exponent of 5/3. He derived an equation relating the stress and 𝑡 𝑙⁄ ratio of

the hexagonal honeycombs for collapse due to plastic buckling as Eq. 2.1.

𝜎𝑝𝑙∗ = 𝐶𝑜𝜎𝑦𝑠(𝑡/𝑙)5/3 (2.1)

where, 𝜎𝑝𝑙∗ = Plateau Stress, 𝑡 𝑙⁄ = Cell wall thickness to cell wall length ratio, 𝐶𝑜 =

6.6, 𝜎𝑦𝑠=yield stress.

Later, Yamashita and Gotoh [40] conducted quasi-static and dynamic compression

tests on aluminum 5052 honeycombs to study the compressive strength and energy

dissipation in the out-of-plane direction. They found that the crushing strength

increases with the 𝑡 𝑙⁄ ratio, which also followed the power of 5/3 of the 𝑡 𝑙⁄ ratio as

derived by Wierzbicki [30]. Xu et al. [38, 39] also found the same relationship

between the plateau stress and 𝑡 𝑙⁄ ratio in their experimental and finite element

analyse. This 𝑡 𝑙⁄ ratio also significantly influenced the densification strain where

with the higher value of 𝑡 𝑙⁄ ratio, the densification strain is smaller; as reported by

Gibson and Ashby [21], Bulson [53], Masters and Evans [54], Balawi and Albot [55].

Papka and Kyriakides [26, 27] conducted experimental and numerical analyses

to investigate the effect of cell wall thickness to the plateau stress of honeycombs

33

loaded in the in-plane direction. They found that for the same cell size of aluminum

honeycombs the plateau stress increased with the cell wall thickness (t). Later, Ruan

et al. [43] employed finite element software ABAQUS for in-plane compression of

honeycombs and found a good correlation between the plateau stress and the cell

wall thickness edge length ratio 𝑡 𝑙⁄ by a power law. Deqiang and Weihong [37]

reported finite element analysis of Double-walled hexagonal honeycomb cores

(DHHCs) subjected to in-plane impact load. They also found a similar power law

relationship between the in-plane plateau stress and 𝑡 𝑙⁄ ratio. Hu and Yu [56] also

discussed the effect of cell wall thickness on the crushing strength of honeycombs.

From their analytical and numerical study they found that the crushing strength is

1.3 times larger for the double thickness honeycombs than the single thickness

honeycombs.

The relative density (𝜌∗ 𝜌𝑠⁄ ) of honeycombs is a function of 𝑡 𝑙⁄ ratio. Gibson and

Ashby [21] presented the relationship between the 𝑡 𝑙⁄ ratio, relative density

(𝜌∗ 𝜌𝑠⁄ ) for hexagonal honeycombs as Eq. 2.2.

𝜌∗

𝜌𝑠=

𝑡 𝑙⁄ (ℎ 𝑙+2)⁄

2 𝑐𝑜𝑠 𝜃 (ℎ 𝑙+𝑠𝑖𝑛 𝜃)⁄ (2.2a)

where, ℎ and 𝑙 are the cell edge lengths respectively, 𝜃 is the expanding angle. For

perfect hexagonal honeycombs (ℎ = 𝑙, 𝜃 = 30), the above Eq. 2.2a reduces to

𝜌∗

𝜌𝑠=

2

√3 𝑡

𝑙 (2.2b)

Foo et al. [57] conducted numerical analyses by ABAQUS. They increased the

density of honeycombs by adjusting the cell wall thickness, t, and the node width, b

and found that the crushing peak load increased with the honeycombs density. Xu.

34

et al. [39] identified the influence of relative density (𝜌∗ 𝜌𝑠⁄ ) in their experimental

analysis, and observed that for the smaller cell size honeycombs the plateau stress

is higher than the larger cell size honeycombs, as shown in Fig. 2.6. This might be

caused by the actual relative density as the relative density of smaller cell size

honeycombs is higher than the larger cell size honeycombs.

Figure 2.6. Influence of relative density on the plateau stress of honeycombs in the

out-of-plane compression [39].

2.2.3.3. Effect of strain rate, shock wave and inertia

In both in-plane and out-of-plane loading conditions, it had been determined

that the strain rate or loading velocity, shock wave and inertia effect influences the

plateau stress of the honeycombs.

35

The plateau stress increases with the increase of compression velocity as well as

the strain rate. To investigate the effect of impact velocity on the out-of-plane

crushing strength of aluminum honeycombs Goldsmith and Sackman [45]

conducted dynamic crushing tests at different velocities up to approximately 35 ms-

1 and they observed that the plateau stress increased by 20 % - 50 % at higher

velocities. Wu and Jiang [46] reported that the out-of-plane crushing strength of

aluminum honeycombs was proportional to the initial striking velocity of the

projectile. They also found that the crushing strength was significantly enhanced

under impact loading conditions in comparison with the quasi-static conditions,

which might result from the inertia effect and material strain rate sensitivity. Baker

et al. [36] utilized a high pressure gas gun apparatus to study uniaxial dynamic

compression deformation of aluminum honeycombs. They found the plateau stress

in the dynamic test was approximately 50 % higher than that in the quasi-static test.

Zhao and Gary [44] used a modified Split Hopkinson Pressure Bar (SHPB) technique

to observe macroscopic rate sensitivity. They observed significant enhancement in

the plateau stress by approximately 40 % when the loading velocity increased from

quasi-static to dynamic (2-28 ms-1). Zhou and Mayer [23] reported an out-of-plane

quasi-static and dynamic compression analysis of aluminum hexagonal honeycomb

specimens. They tested two different types of honeycombs with different cell sizes

(19.1 mm and 6.4 mm) and found that the plateau strength was sensitive to strain

rate. Zhao and Abdennadher [58] reported strength enhancement of metal tubes by

using the Split Hopkinson Pressure Bar (SHPB) with the intention to analyse the

plastic folding behavior of aluminum honeycombs. They observed the crushing

strength enhancement in the specimens due to strain rate sensitivity and lateral

inertia effect of the edges. Xu et al. [38, 39] investigated quasi-static and dynamic

36

out-of-plane compressive behavior of different aluminum honeycombs at a wide

range of strain rates experimentally and numerically. They found the plateau

strength increased with both the ratio of cell wall thickness to edge length (t/l) of

honeycombs and strain rate. They derived the following relationship between

plateau stress and strain rate in order to explicitly consider the strain rate

contribution:

𝜎𝑝𝑙∗ = 𝐶1𝜎𝑦𝑠(𝑡/𝑙)𝑘3(1 + 𝐶2𝜀̇)𝑝 (2.3)

where, 𝜎𝑦𝑠 is the yield stress of aluminum, 𝐶1, and 𝑘3 can be obtained for different

strain rates, 𝑘3 and p are non-dimensional coefficients.

They have experimented on three different honeycomb specimens with different

𝑡 𝑙⁄ ratios and derived equations to estimate the strength enhancement of

honeycombs by relating 𝑡 𝑙⁄ ratio and strain rate under out-of-plane compression

load, as shown in Fig. 2.7. The effect of strain rate on the plateau stress of

honeycombs in the out-of-plane direction was experimentally and numerically

investigated by other researchers such as: Zhao and Gary [44], Yamashita and Gotoh

[40], Zhao et al. [59], Foo et al. [57], Deqiang et al. [35], Hou et al. [60], Wang et al.

[61], Li et al. [62], Akatay et al. [63] and so on. They all observed a power law

relationship between the plateau stress and strain rate.

37

Figure 2.7. Influence of 𝑡 𝑙⁄ ratio and strain rate on the stress enhancement of

different honeycombs loaded in out-of-plane direction [39].

Alavi Nia et al. [64] conducted an experimental crushing analysis on foam-filled

aluminum hexagonal honeycombs under the out-of-plane compression load. They

discussed the influence of strain rate on the crushing strength and energy

absorption. They found that bare honeycombs were more sensitive to strain rate

than foam-filled honeycombs and that in both cases crushing strength increased.

Hozhabr Mozafari et al. [65] employed ABAQUS software to discuss the structural

behavior of foam-filled honeycomb models. They observed that mean crushing

strength and energy absorption capacity of foam filled honeycombs was greater

than the sum of strengths of bare honeycombs and foam separately.

Honeycombs are also strain-rate sensitive when they are crushed in the in-plane

directions. Ruan et al. [43] reported numerical in-plane crushing of aluminum

38

honeycombs and they found that both the in-plane plateau stresses increased with

the impact velocity by a square law. Zheng et al. [66] also observed similar

enhancement in the in-plane plateau stress of honeycombs due to the influence of

increasing loading velocity. Deqiang and Weihong [37] employed different impact

velocities from 3 ms-1 to 250 ms-1 to crush the Double-walled hexagonal honeycomb

cores (DHHCs) in the in-plane direction and reported an enhancement of plateau

stress with loading velocity. Hu et al. [67-69] also discussed the strain rate

sensitivity of honeycombs subjected to in-plane compression loads.

Stress enhancement is not a material characteristic. Formation of shock wave

occurs due to high impact speed which induces strength enhancement on the

cellular material. So, for the quasi-static loading case, the effect of shock wave is

negligible. Reid and Peng [70, 71] reported the formation of shock waves at high

impact velocity (about 50 ms-1). They discussed the concept of possible shock front

formation in the densification part of cellular materials. At high impact velocity the

strength behind the shock front was found to be larger than that of the shock front

before. Zhao et al. [59, 72] studied the impact behavior of aluminum foam made

with different manufacturing techniques by employing the Split Hopkinson

Pressure Bar (SHPB). They applied high impact velocities up to 50 ms-1 and

observed significant rate sensitivity on the strength enhancement of aluminum

foams. They also found a good agreement with Reid and Peng’s [70] theory of shock

wave formation that has an effect on the stress enhancement. Tan et al. [73-75]

studied the strength properties of aluminum foams at high impact velocities (10 to

210 ms-1). They estimated a ‘shock-type’ deformation response occurs at high

39

impact velocities. The influence of the impact velocity on the strength of aluminum

foams was also discussed by other researchers [76-80].

Xu et al. [38] carried out finite element analyses of aluminum honeycombs to study

the effect of shock wave at high impact velocity. They identified the critical impact

velocity (approximately 100 ms-1) for the shock wave enhancement. They proposed

a relationship between stress and high impact velocity based on Reid and Peng’s

[70] theory as follows:

𝜎𝑝𝑙∗ = 𝜎𝑦𝑠 −

𝑝0𝑣2

𝜀𝑑 (2.4)

where, 𝜎𝑦𝑠, 𝑝0, 𝜀𝑑, 𝑣 are the quasi-static stress, honeycombs density, densification

strain and impact velocity, respectively.

Klintworth [81] discussed the micro-inertia effect on the enhancement of the

crushing stress of aluminum honeycombs. At a strain rate of 100 s-1, he found that

crushing force increased about 50 % in the case of the uniaxial strain condition

which is smaller than the uniaxial stress condition due to the inertia effect. Hönig

and Stronge [29, 82] also found that the inertia effect was stronger in the uniaxial

strain condition than in the uniaxial stress. From the numerical study they discussed

that the stress enhancement with the increase of loading velocity was mainly caused

by the translational micro-inertia and not the micro-rotational inertia. Wang et al.

[83] experimentally and numerically studied the inertia effect on the plateau stress

and energy absorption of aluminum honeycombs. They found that plateau stress

increased significantly below the impact load 30 ms-1 and slowly between 30 – 80

ms-1. They also stated that the strength and crashworthiness became stronger when

the impact mass was heavier.

40

Calladine and English [84] observed that under impact compression the buckling

of the column structure occurred at a delayed time, where the critical buckling force

was higher than the quasi-static force. They mentioned that the delayed time was

due to the inertia effect under impact load. Su et al. [85, 86] reported that for the

structures made of rate-sensitive materials, along with the strain rate sensitivity

effect, the inertia effect acted as an dominating factor on the impact behavior of the

structures. Langseth et al. [87, 88] experimented with steel and aluminum square

tubes under static and dynamic axial crushing, and observed an increase in strength

due to the lateral inertia effect. Tan et al. [75], Zhao et al. [59] and Lee et al. [89] also

reported a similar inertia effect in their crushing analysis of aluminum foams.

2.2.3.4. Effect of entrapped air

Along with the 𝑡 𝑙⁄ ratio, strain rates, 𝜀̇, and inertia effect, the air entrapped in

honeycomb cells also contributes to the strength enhancement. Researchers [40,

41, 59, 90, 91] observed that internal pressure of the cellular structures raised by

the effect of entrapped air. Zhao et al. [59] tested both aluminum honeycombs and

foams and confirmed a contribution of entrapped air on the strength enhancement.

Zhang and Yu [91] employed axial load on thin-walled circular tubes and analysed

the pressurized condition. They observed that the strength enhancement was due

to the direct effect of the increased air pressure as well as an indirect effect which

resulted because of interaction between air pressure and tube wall buckling. They

estimated the effect of entrapped air by proposing a semi-empirical relation which

was fitted to the experimental results.

Zhao et al. [92] reported a Multi-scale analysis of aluminum honeycombs that

had been conducted to investigate the effects of pressure of the entrapped air in the

41

honeycomb cells. They demonstrated that the amplitude of the air pressure was

approximately twice the initial pressure at 50 % strain (only 1.1 times at 10 %

strain). They found the stress enhancement in the honeycombs was due to the effect

of entrapped air. Yamashita and Gotoh [40] reported an air pressure effect on the

crushing of honeycombs loaded in the out-of-plane direction. They assumed that the

effect of air pressure is relatively small in the beginning stage of crushing. They

estimated the increase of pressure ∆P from the following Eq. 2.5:

∆𝑃 = 𝑃0 (𝑉0

𝑉− 1) (2.5)

where,

𝑃0= Initial Pressure (Atmospheric pressure),

𝑉0 = Initial Volume,

𝑉 = Volume during compression.

Then they measured the compression stress during the impact test from air

pressure and crush stress. This resulted in Eq. 2.6:

𝜎𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑 = 𝜎𝑐 + ∆𝑃 = 𝜎𝑐 + 𝑃0 (𝐻0

𝐻0−𝑆− 1) (2.6)

where,

𝜎𝑐= Crush stress during compression,

𝐻0= Initial height of honeycomb core,

𝑆 = Compressive stroke.

Xu et al. [41] determined the effect of entrapped air on the mean plateau stress

of different honeycombs. They introduced the entrapped air pressure inside the

honeycomb specimens by varying the hole percentage from 0 to 100% through a

42

layer of GMS Composites EP-280 films. The films were used to seal both ends of the

specimen. They found that along with the other factors such as 𝑡 𝑙⁄ ratio or relative

density (𝜌∗ 𝜌𝑠⁄ ) and strain rates, 𝜀̇ entrapped air effect on the crushing strength of

the honeycombs. They introduced a leaking rate, �̇�, through an analytical solution

to demonstrate the entrapped air effect which shows that the leaking is dependent

on the strain rate and hole percentage, and independent of cell size and 𝑡 𝑙⁄ ratio.

The leakage of air during crushing was been calculated by the following Eq. 2.7.

�̇� = 1 − 𝑃𝑉

𝑃0𝑉0 (2.7)

Where, 𝑃0 and 𝑃 are the initial pressure (Atmospheric pressure) and the pressure

after displacement reached a certain value respectively [41]. Finally they derived a

constitutive equation (Eq. 2.8) relating stress, 𝑡 𝑙⁄ ratio, strain, strain rates, 𝜀̇, and

leaking rate, �̇�.

𝜎 = 𝐶1𝜎𝑦𝑠(𝑡 𝑙⁄ )𝑘3(1 + 𝐶2𝜀̇)𝑝 + 𝑃0(1

1−𝜀− 1)(1 −

�̇�

�̇�) (2.8)

Where, the value of C1, C2, k3, p and δ̇ were calculated from the experimental results.

43

2.3. The mechanical response of honeycombs subjected to other types of

loadings

2.3.1. Indentation of honeycombs

Klintworth and Stronge [93] studied the stress and deformation characteristics

of transversely crushed ductile honeycombs. By applying indentation load in the in-

plane direction they estimated a lower indentation force from the comparison

between the peak strength underneath the indenter and the yield strength of the

aluminum honeycombs, and an upper indentation force by linking the indentation

dissipated rate with the work rate. They also stated that the ductile honeycombs

subjected to compression strain softened after yielding occurred, causing localized

deformation in the crushing cells.

Lee and Tsotsis [94, 95] reported that the localized deformation and damage

behavior of the honeycomb core material was difficult to analyse because the stress

field varied from highly localized stress on the top surface to a stress-free bottom

surface of the honeycombs subjected to out-of-plane indentation load. They

discussed that the indentation strength strongly depended on the core density and

this phenomena was valid for both core and panel failure. The indentation failure

mechanism had been observed using two basic core failure modes, which were

compression and shear. When core compression was the controlling failure mode,

the onset indentation failure pressure Pc was expressed as,

𝑃𝑐 =𝜎𝑐𝑜𝑟𝑒

𝑀𝑎𝑥 𝜎𝑧 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 𝑎𝑝𝑝𝑙𝑖𝑒𝑑 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 (2.9)

where, σcore is the core compression strength. Similarly when the core shear

performance dominated, Pc was

44

𝑃𝑐 =𝜏𝑐𝑜𝑟𝑒

𝑀𝑎𝑥 𝜏𝑥𝑧 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 𝑎𝑝𝑝𝑙𝑖𝑒𝑑 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 (2.10)

where, σcore is the core transverse shear strength. In either case Pc was proportional

to the core strength (compression or shear) and inversely proportional to the

severity of stress concentration. From the experimental results it was observed that

the maximum σz per unit applied pressure for different cores depended on indenter

diameter and maximum core transverse shear stress τxz per unit applied pressure

increased with the decreasing skin thickness or increasing core density reported by

Lee and Tsotsis [95]. Zhou and Mayer [23] reported indentation punch tests of

aluminum honeycombs varying in cell size (6.4 mm and 19.1 mm) at relatively low

loading velocity. A typical indentation test involved both compression of the

material underneath the indenter and tearing (fracturing) of the honeycombs

around the edge of the indenter. The deformation was classified as four primary

deformation mechanisms: shear, tearing initiation, tearing, and compression.

During crushing these four mechanisms interacted with each other. Likewise, these

four phases contributed to the indentation resistance. Initially they used four types

of indenter with different shapes (rectangular, square, triangular and circular) and

found similar failure appearance in the honeycomb specimens. Later, they employed

different specimen sizes to study the influence of specimen dimension on the force-

displacement curve as shown in Fig. 2.8. It was observed that larger specimens had

lower total punch force and a flatter force. These were caused by the boundary

constraints of the larger surroundings of the specimen around the intender. To

calculate the tearing strength in indentation, they used the following force balance

equation.

𝐹𝑇 = 𝐹𝑐 + 𝐹𝑡 (2.11)

45

where, 𝐹𝑇 , 𝐹𝑐 and 𝐹𝑡 are the total punching force, compression force and tearing force

respectively. The compression force was calculated from the average strength, σc

and cross-sectional area, 𝐴𝑐 of the indenter, 𝐹𝑐 = 𝜎𝑐𝐴𝑐. By using Eq. 2.11, the tearing

force, 𝐹𝑡, was calculated and then the tearing strength was obtained from the tearing

force and circumferential length of the indenter.

Figure 2.8. Influence of specimen dimension on the force-displacement curves in

indentation [23].

To the best of the author’s knowledge, no study has been reported in the

literature so far on the indentation of aluminum hexagonal honeycombs subjected

to dynamic loading. The influence of 𝑡 𝑙⁄ ratio or relative density (𝜌∗ 𝜌𝑠⁄ ) and loading

velocities or strain rates, 𝜀̇, on the plateau stress, energy dissipation, especially on

tearing energy, has not been studied yet.

46

2.3.2. Shear of honeycombs

Many researchers [21, 96-101] reported the shear properties of aluminum

honeycombs. Kelsey et al. [96] and Gibson and Ashby [21] discussed the linear

elastic shear behavior of aluminum honeycombs. Kelsey et al. [96] mentioned the

variation of the shear moduli of honeycombs with cell angle and angle of applied

shear stress.

Zhang and Ashby [98] comprehensively discussed the shear behavior of

honeycombs. They found that under out-of-plane shear, the cell walls buckle and

bulge, and this buckling load could be determined using the second moment of

inertia and width of the wall. The nature of buckling during shear was not similar to

that in the case of uniaxial load. Moreover, this behavior was not a feasible result for

cases as it varied with the bonding properties of core and face. The de-bonding

failure mechanism of adhesives was another problem which was difficult to analyse.

Generally, flexible honeycombs were weaker than epoxy adhesives. The Crack

Propagation effect on the energy consumption of honeycombs was initiated by

defects. Grèdiac [97] employed finite element analysis to study the out-of-plane

shear moduli of honeycomb cells. He found that the shear modulus decreased with

the increase of the cell wall thickness of honeycombs. Pan et al. [101] studied

aluminum alloy 5056 honeycombs to investigate Out-of-plane shear behavior and

deformation mechanism by conducting a single block shear test as shown in Fig. 2.9.

They classified the deformation into four regions from the load-displacement curves

of honeycombs under transverse shear load: Stage І, the load increased to the first

peak due to the elastic deformation that includes bending and shear deformation;

Stage II, the plastic deformation that involves wrinkling and fracture of cell walls;

47

Stage III, load increased to the second peak where the fracture and de-bonding of

cell walls occurred; Stage IV, the load decreased rapidly which inclined to zero. From

the analytical model they demonstrated the existence of a relationship between the

bending deformation of cell walls to the transverse shear modulus and strength

which depended on the core height. For a lower core height, the contribution of

bending deformation to the transverse shear modulus and strength of the cell walls

was considerable but for the higher core height, it was negligible. They considered

the cell walls as isotropic thin plate and derived an expression for the elastic shear

stress of the inclined walls as:

𝜏𝑠𝑐𝑟 = 𝐾

𝐸𝑠

1−𝑣𝑠2 (

𝑡2

𝑙2)2 (2.12)

where, 𝐸𝑠 is the Young’s elastic modulus of honeycombs, 𝐾 is the constraint factor,

t and l are the cell wall thickness and cell wall length respectively.

(a)

48

(b)

Figure 2.9. (a) Schematic of test specimen; (b) Load-displacement curve of

transverse shear deformation [101].

Lee et al. [100] reported out-of-plane shear properties of Nomex honeycombs.

They found that the shear strength of the honeycombs depended on the loading

direction: the shear strength of honeycombs loaded along the longitudinal

directions was higher than that of the honeycombs loaded along the transverse

directions due to the orientation of the different thicknesses of cell walls. They also

found similar deformation phases as described by Pan et al. [101] for honeycombs

subjected to shear load. Cote et al. [102] tested square honeycombs made of

stainless steel. They found a linear relationship between the shear strength and

relative density of the honeycombs and that the shear strength was nearly isotropic.

Zhou and Mayer [23] reported both out-of-plane and in-plane shear properties

of aluminum honeycombs. They described different deformation mechanisms of

honeycombs subjected to different shearing loads.

49

Qiao et al. [103] employed finite element analysis to study the effectiveness of

in-plane shear stiffness of thin-walled composite honeycomb cores. They defined

optimized geometries for different honeycomb cores to improve the in-plane shear

stiffness properties. Shi and Tong [104] discussed the influence of geometry on the

in-plane shear stiffness of a honeycomb sandwich panel and proposed an improved

lower limit for the equivalent shearing stiffness honeycombs.

2.3.3. Combined compression-shear

In the applications of aluminum honeycombs, they are not only subjected to pure

compressive or indentation load but sometimes also subjected to combined

compression-shear load. The mechanical response and crushing behavior under the

combined compression-shear loading condition have been studied by several

groups of researchers. Mohr and Doyoyo [105] introduced a standard ARCAN

apparatus in their out-of-plane compression-shear test of honeycombs. The function

of the apparatus was to control all the displacement at the boundaries of the

specimen. They used an additional load cell to measure additional horizontal force

that was developed by the previous clamped configured investigation. Mohr and

Doyoyo [106, 107] further looked into biaxial loading to study the post-yield

behavior under combined out-of-plane loading by developing a modified testing

device called the Universal Biaxial Testing Device (UBTD); shown is Fig. 2.10. They

employed different loading angles, ranging from 0°-90°. As a function of the applied

loading angle, they classified the deformation of honeycombs into five regions:

Elastic I, Elastic II, Nucleation, Softening and Crushing. They discussed the

relationship between normal and shear stresses for each of the different

deformation regions and also proposed an expression for the elliptic envelope in the

50

nucleation region to describe the crushing mechanism in the normal and shear

stress planes under combined compression-shear load.

Figure 2.10. Universal Biaxial Testing Device (UBTD) developed by Mohr and

Doyoyo [106, 107] for combined compression-shear load.

Hong et al. [108, 109] established a bi-axial quasi-static loading function in their

compression-dominant combined load system. They introduced two actuators to

apply horizontal and vertical forces for shear and compressive loads respectively.

Their bi-axial loading methods were able to analyse the combined out-of-plane

compression-shear and/or in-plane bi-axial behavior of honeycombs with face

plates that were used to avoid slippage. Furthermore, based on their quasi-static

method they introduced a dynamic in-plane compression-dominant inclined load in

the same multi-axial test machine. The high impact velocities were measured by

laser beams and receiver which might have an effect on measurement accuracy.

51

They discussed a comparison between pure compressive load-displacement and

combined loads-time curves shown in Fig. 2.11.

(a)

(b)

Figure 2.11. (a) The load–displacement curve of a type I honeycomb specimen

under a pure compressive load (b) The load histories of honeycomb specimen with

β=90° under a combined load with ∅=90° [108, 109].

52

From Fig. 2.11 it was observed that the normal plateau load under combined

load is smaller than that under pure compressive load. The absorbed energy during

the plateau stress was defined from the sum of work done by normal load and shear

load. The energy absorption rate was defined by the following equation,

�̇� = 𝜎�̇� + 𝜏�̇� (2.13)

where, σ and τ are the normal crush and shear strength. They found that energy

absorption rate depends on the ratio of shear stress to compressive stress and

orientation angle. The normalized energy absorption rate increases with the shear

stress ratio and its value is less than 1 for a given set of shear stress ratios for 𝛽 =

90° and 𝛽 = 30°; the value is higher than 1 for 𝛽 = 0°.

It has been investigated that under pure compression, stacking of folds was

observed on the top of the cell walls, and inclined stacking patterns were observed

under combined loads which were described as a consequence of the asymmetric

location of horizontal plastic hinge lines due to shear load. It was found that with the

increase in impact velocity, the normal crushing strength increased but shear

strength remained approximately the same. This normalized crushing strength

under pure compressive loads was higher than the normalized crushing strength

under inclined loads. With the increase in impact velocity, the shape of macroscopic

yield surfaces changed but the progressive folding mechanisms in honeycomb

specimens obtained were similar for both inclined and pure compressive loads. The

authors also proposed a relationship between the macroscopic yield criterion and

the impact velocity.

Hou et al. [110, 111] conducted quasi-static and dynamic combined

compression-shear tests of aluminum honeycombs by using the Split Hopkinson

53

Pressure Bar (SHPB) technique. In order to apply combined dynamic shear-

compression loads, a special experimental set-up had been introduced in the

experiment as shown in Fig. 2.12. In their experimental analysis, different loading

angles from 0° to 60° and an impact velocity of 15 ms-1 were employed. A universal

INSTRON-3369 tension or compression had been used to conduct quasi-static and

dynamic combined shear-compression tests at five loading angles in the out-of-

plane (TW) direction where 0° angle for pure compression and 30°, 40°, 50° and 60°

angles for combined shear-compression were employed.

Figure 2.12. Schematic diagram of the combined compression-shear loading device

[111].

From the experiments, they found that under uniaxial out-of-plane

compression, the first peak in the pressure-crush curve represented the plastic

collapse and long plateau stress represented the successive folding process. On the

other hand, for combined shear-compression in the out-of-plane direction, the

initial peak of the curve varied with loading angle, 𝜃, where peak value decreased

with the increase of loading angle, as shown in Fig. 2.13. The stress in the plateau

54

region also decreased with the increase of angle, 𝜃. The deformation pattern they

observed by the high-speed camera was described as:

Difference in initial collapse: For uniaxial compression, collapse initiated either on

top or on bottom of the face but for combined shear-compression case, collapse

initiated simultaneously on both faces.

Variation in cell wall axes: Uniaxial compression kept the cell axes inclined and

combined shear-compression kept the cell axes perpendicular to the loading

surfaces.

Figure 2.13. Dynamic pressure-crush curves in TW plane at different loading angles

under combined compression-shear load [111].

Hou et al. [112] also employed numerical analyses of honeycombs under

combined compression-shear to investigate the normal and shear crushing of

honeycomb models under combined compression-shear loading. They found good

agreement between numerical results and experimental results. Also, they found

55

enhancement in the normal and shear behaviors and using a Levenberg–Marquardt

algorithm, they derived an elliptical criterion relating the normal and shear

strength.

Zhou et al. [113] conducted quasi-static combined compression-shear tests on

Nomex honeycombs and observed two different deformation modes: plastic

buckling and extension fracture of the cell walls. Tounsi et al. [114] developed a

numerical model of honeycombs to investigate the effects of loading angle and in-

plane orientation on the crushing response of aluminium honeycombs at 15 ms-1

loading velocity. Most recently, Tounsi et al. [115] conducted experiments to study

the effects of loading angle and in-plane orientation angle on the deformation mode

of honeycombs subjected to mixed shear-compression loading. They observed three

different deformation modes: Mode 1 (Fold formation on a single side), Mode 2 (Fold

formation on both sides) and Mode 3 (combination between Mode 1 and Mode 2).

However, under combined compression-shear load at different loading angles,

the effects of 𝑡 𝑙⁄ ratio or relative density (𝜌∗ 𝜌𝑠⁄ ) and strain rates, ε̇ on the plateau

stress and specific energy have not been reported in the literature. The normal

compressive and shear force components of the applied combined compression-

shear load are not directly measured by experimental analysis. The deformation

mechanism and cell wall rotation at different loading angles in two-plane

orientation (TL plane and TW plane) of the honeycombs have not been studied so

far.

A summary of research conducted on honeycombs (mainly aluminum

honeycombs) is presented in Table 2.1.

56

Table 2.1. Summary of previous work conducted on honeycombs (mainly aluminum honeycombs)

Pure compression

No. Researchers Velocity

(ms-1)

Loading

direction

Equipment and method

used

Type of

honeycombs

Honeycombs’ specification References

Cell size,

D

(mm)

Cell wall

thickness, t

(mm)

1 Khan et al. (2012) 8.33×10-6 In-plane Instron machine Aluminum

3003

honeycombs

8.2 0.1 [24]

2 Khan et al. (2012) 8.33×10-6 In-plane Instron and FEA Aluminum

3003

honeycombs

4.1, 6.3

and 8.2

0.05, 0.1, 0.15 [25]

3 Papka and Kyriakides

(1994)

4.13×10-5 In-plane Electromechanical testing

machine

Aluminum 5052-

H39 honeycombs

9.53 0.094, 0.119 and

0.145

[26]

4 Papka and Kyriakides

(1998)

4.13×10-5 In-plane Displacement control testing

machine and FEA

Aluminum 5052-

H39 honeycombs

9.53 0.145 [27]

5 Zhang and Ashby

(1992)

1×10-5 In-plane Experimental and FEA Nomex

honeycombs

3, 5, 6, 13 - [99]

6 Zhou and Mayer

(2002)

Quasi-static In-plane MTS machine Aluminum

3003

honeycombs

19.1 and

6.4

0.076 [23]

57

7 Foo et al. (2007) 4.17×10-4 In-plane Instron and FEA Nomex

honeycombs

13 0.3 [49]

8 Zhao and Gary (1998) 2, 10 and 28 In-plane SHPB Aluminum

honeycombs

4.7 and

6.2

0.08 [44]

9 Hönig and Stronge

(2002)

1, 2.5, 5, 10,

20 and 30

In-plane FEA and drop weight testing

machine

Aluminum 5052

honeycombs

9.53 0.145 [29, 82]

10 Hu et al. (2013) 7.8, 10, 60,

100 and 150

In-plane Instron and FEA Aluminum 5052

honeycombs

6.53, 6.9 0.083 [69]

11 Hu et al. (2014) 10,60 and

100

In-plane FEA and Theoretical Aluminum 5052

honeycombs

6.9 0.136, 0.249,

0.267, 0.322,

0.324 and 0.346,

[68]

12 Ruan et al. (2003) 3.5-280 In-plane FEA Aluminum

honeycombs

4.7 0.08, 0.2, 0.3, 0.4

and 0.5,

[43]

13 Balawi and Albot

(2008)

- In-plane electromechanical

and servo-hydraulic, and

FEA

Aluminum 5052

honeycombs

3.175,

4.76 and

1.59

0.0178, 0.0762

and 0.0381

[55]

14 Deqiang and Weihong

(2009)

3-250 In-plane FEA Double-walled

hexagonal

honeycombs

5.2 0.03, 0.05, 0.07,

0.08, 0.1, 0.12

and 0.15

[37]

15 Aminanda et al.

(2005)

8.33×10-6 Out-of-plane Instron and FEA Nomex

honeycombs

6 0.12 [116]

16 Khan et al. (2012) 8.33×10-6 Out-of-plane Instron machine Aluminum 8.2 0.1 [24]

58

3003

honeycombs

17 Khan et al. (2012) 8.33×10-6 Out-of-plane Instron and FEA Aluminum

3003

honeycombs

4.1, 6.3

and 8.2

0.05, 0.1, 0.15 [25]

18 Wu and Jiang (1997) 8.33×10-6

and 25.5

Out-of-plane Shimadzu

material-testing machine

and gas gun

Aluminum 5052-

H38, 5056-H38

honeycombs

3.175,

4.763,

0.0254 [46]

19 Aktay et al. (2008) 8.33×10-5 Out-of-plane Experimental and FEA Aluminum and

Nomex

honeycombs

13.5 0.07 [34]

20 Mohr and Doyoyo

(2003)

8.33×10-5 Out-of-plane Experimental and FEA Aluminum5056-

H39 honeycombs

5.4 0.033 [33]

21 Xu et al. (2014) 5×10-5-10 Out-of-plane FEA Aluminum 5052-

H39 honeycombs

3.175,

3.175

4.763 and

9.525

0.0254, 0.0508,

0.0254 and

0.0762,

[39, 41]

22 Alavi and Sadeghi

(2010)

4.17×10-5 Out-of-plane Instron Aluminum 5052-

H39 honeycombs

3.175

and 4.76

0.0178 and

0.0508

[117]

23 Alavi and Sadeghi

(2013)

4.17×10-5,

1×10-1 and

2×10-1

Out-of-plane Instron Aluminum 5052-

H39 honeycombs

3.175

and 4.76

0.0508 [64]

59

24 Hou et al. (2012) 3×10-5 ,

1×10-4 and

10-28

Out-of-plane universal

tension/compression testing

machine, SHPB and FEA

Aluminum 3003

and 5052

honeycombs

3.46,

4.33, 5.2,

and 4.76,

6.35, 9.52

0.04, 0.05, 0.06,

and 0.0762

[118]

25 Lee et al. (2002) 2×10-5 Out-of-plane Instron Nomex

honeycombs

9.5 0.22 [100]

26 Zhang and Ashby

(1992)

1×10-5 Out-of-plane Experimental and FEA Nomex

honeycombs

3, 5, 6, 13 - [98]

27 Zhao et al. (2005) 1×10-5 and

10

Out-of-plane SHPB

Aluminum 5052,

5056 honeycombs

3, 4.76, 6,

6.35, 7,

9.5 22

and 9.52

0.055, 0.076,

0.058, 0.076,

0.05, 0.08 and

0.076

[59]

28 Yang and Qiao (2008) 8.33×10-4,

1.67×10-4,

3.33×10-4

and 5×10-4

Out-of-plane MTS, FEA and Theoretical Aluminum

honeycombs

5.08 and

6.5

0.0762 [119]

29 Foo et al. (2007) 4.17×10-4 Out-of-plane Instron and FEA Nomex

honeycombs

13 0.3 [49]

30 Zhou and Mayer

(2002)

4.6×10-3,

4.8×10-3,

1.34, 1.48,

3.14, 4.9 and

5.08,

Out-of-plane MTS machine Aluminum 3003

honeycombs

19.1 and

6.4

0.076 [23]

60

31 Wang et al. (2013) 3×10-2 Out-of-plane MTS machine Aluminum 3003-

H18 honeycombs

9.525 0.05 [61]

32 Mahmoudabadi and

Sadighi (2011)

Quasi-static

and 3-5

Out-of-plane universal testing machine

(Zwick), drop

hammer and Theoretical

Aluminum 3003-

H18 foam filled

honeycombs

5 and 7 0.0508 and

0.0635

[120]

33 Zhao et al. (2006) Quasi-static

and 14

Out-of-plane SHPB Aluminum 5052

and 5056

honeycombs

- - [92]

34 Zhao and Gary (1998) 2, 10 and 28 Out-of-plane SHPB Aluminum

honeycombs

4.7 and

6.2

0.08 [44]

35 Yamashita and Gotoh

(2005)

10 Out-of-plane drop-hammer apparatus and

FEA

Aluminum

5052

honeycombs

9.525 0.02, 0.033 and

0.066, 0.02-0.12

[40]

36 Goldsmith and

Sackman (1992)

10-40 Out-of-plane Pneumatic gun Aluminum

5052 and Nomex

honeycombs

3.175

and 6.35

0.0254 and

0.0508

[45]

37 Wang et al. (2014) 20-80 Out-of-plane High-Speed

Crash System and FEA

Aluminum 5052-

H18 honeycombs

3.46 0.06 [83]

38 Alavi et al. (2008) 17-144 Out-of-plane Gas gun and analytical Aluminum 5052-

H39 honeycombs

3.175,

and 4.76

0.05, 0.076 and

0.038

[121]

39 Deqiang et al. (2010) 3-350 Out-of-plane FEA Double-walled

hexagonal

honeycombs

5.4 0.03, 0.05, 0.07,

0.08, 0.1, 0.12

and 0.15

[35]

61

40 Xu et al. (2014) 5-500 Out-of-plane Instron, MTS and FEA Aluminum 5052-

H39 honeycombs

9.525 0.00462-

0.03695

[38]

Indentation


(ms-1)

Loading

direction


used

Type of

honeycombs

Honeycombs specification References

Cell size,

D

(mm)

Cell wall

thickness, t

(mm)

41 Klintworth and

Stronge (1989)

Quasi-static In-plane Theoretical Aluminum

honeycombs

6.35 0.079 [93]

42 Zhou and Mayer

(2002)

Quasi-static Out-of-plane MTS machine Aluminum 3003

honeycombs

19.1 and

6.4

0.076 [23]

43 Foo et al. (2008) Low-velocity Out-of-plane Instron Dynatup, FEA and

Analytical

aluminum

3003-H19

honeycombs

6.35 0.0635 [57]

Shear


(ms-1)

Loading

direction


used

Type of

honeycombs


Cell size,

D

(mm)

Cell wall

thickness, t

(mm)

44 Cote et al. (2006) 8×10-7 Out-of-plane Screw driven test machine

and FEA

Stainless steel

square

honeycombs

6-17.5 0.3 [102]

62

45 Pan et al. (2008) 1.67×10-6 Out-of-plane Zwick universal test

Machine and Theoretical

Aluminum 5056

honeycombs

4.76 0.018 [101]

46 Grèdiac (1993)

- Out-of-plane FEA Metal

honeycombs

- - [97]

47 Lee et al. (2002) 2×10-5 Out-of-plane Instron Nomex

honeycombs

9.5 0.22 [100]

48 Hazizan and Cantwell

(2003)

1.67×10-5,

1.67×10-4

and 1.67×10-

3

Out-of-plane Kistler 5011 piezo-electric

load cell

Aeroweb 3003

honeycombs

6 - [122]

49 Zhang and Ashby

(1992)

1×10-5 Out-of-plane Experimental and FEA Nomex

honeycombs

3, 5, 6, 13 - [98]

50 Shi and Tong (1995) - Out-of-plane Theoretical Honeycomb core - - [104]

51 Zhou and Mayer

(2006)

Quasi-static Out-of-plane

and in-plane

MTS machine Aluminum 3003

honeycombs

19.1 and

6.4

0.076 [23]

52 Qiao et al. (2008) - In-plane FEA Honeycomb core - - [103]

Combined compression-shear


(ms-1)

Loading

angle


used

Type of

honeycombs


Cell size,

D

(mm)

Cell wall

thickness, t

(mm)

53 Zhou and Mayer

(2002)

Quasi-static 90° MTS machine Aluminum 19.1 and

6.4

0.076 [23]

63

3003

honeycombs

54 Mohr and Doyoyo

(2003-2004)

1.67×10-5 0° - 90° Arcan apparatus, Universal

biaxial testing device

Aluminum

5056-H39

honeycombs

8.3 and

5.36

0.033 and 0.033 [33, 106, 107]

55 Zhou et al. (2012) 1.67×10-5 0° - 90° Universal testing machine Nomex

honeycombs

4.76 0.065 [123]

56 Hong et al. (2006) 1×10-4 15° Instron Aluminum

5052-H38

honeycombs

9.5 0.025 [108]

57 Hong et al. (2008) 6.7 - 6.8 15° Gas Gun Aluminum

5052-H8

honeycombs

9.5 0.025 [109]

58 Hou et al. (2010–

2011)

1×10-4 and

15

0° - 60° SHPB and FEA Aluminum

5052

honeycombs

6.35 0.076 [111, 112]

59 Tounsi et al. (2013) 15 0° - 60° FEA Aluminum

5056

honeycombs

6.35 0.076 [114]

60 Tounsi et al. (2016) 15 0° - 60° SHPB Aluminum

5056

honeycombs

6.35 0.076 [115]

65

Chapter 3. Experimental investigation of the mechanical

behavior of aluminum honeycombs under quasi-static

and dynamic indentation

In this chapter, the dynamic behavior of aluminum honeycombs under out-of-

plane indentation at different loading velocities is investigated. Indentation and

compression tests of three types of HEXCELL® aluminum hexagonal honeycombs

were conducted using MTS and high-speed INSTRON machines at strain rates from

10-3 to 102 s-1 respectively. The tearing energy was calculated as the difference in

energy dissipated in indentation and compression of the same type of honeycomb.

It was found that tearing energy was affected by strain rate and nominal density of

honeycomb. Empirical formulae were proposed for tearing energy in terms of strain

rate.

3.1. Experiment set-up

3.1.1. Aluminum Honeycomb Specimens

Three different types of HEXCEL® hexagonal honeycombs with varying cell size,

cell wall thickness and nominal density were used in both the indentation and

compression experimental tests [20]. The specification of the honeycombs,

provided by the manufacturer, is listed in Table 3.1. Three different types of

honeycombs were named as H31, H42 and H45 for honeycombs 3.1-3/16-5052-

.001N, 4.2-3/8-5052-.003N and 4.5-1/8-5052-.001N, respectively. The cell wall

thicknesses were the same for honeycombs 3.1-3/16-5052-.001N and 4.5-1/8-

5052-.001N, but less than that for honeycomb 4.2-3/8-5052-.003N. The nominal 𝑡 𝑙⁄

ratios of honeycombs H42 and H45 are the same. However, the actual density

66

(measured and provided by the manufacturer) of honeycomb H42 is slightly lower

than that of honeycomb H45.

All specimens were carefully prepared to prevent deformation during the cutting

process. The photographs of each type of honeycomb specimen used in both the

compression and indentation tests are shown in Fig. 3.1. The height of all the

honeycomb specimens, h, is 50 mm, the same as the height of honeycomb panels

from which the specimens were cut. The in-plane dimensions of all the honeycomb

specimens used in indentation tests are 180 mm × 180 mm (Fig. 3.1a) and 90 mm ×

90 mm in compression tests (Fig. 3.1b). Such dimensions ensure that each specimen

has sufficient honeycomb cells so that the specimen-size effect is minimized and the

measured properties can represent the bulk properties. Onck et al. [124], Andrews

et al. [125], Deqiang et al. [35] and Xu et al. [39] indicated that a minimum number

of cells (9×9 or 7×7) should be included in honeycomb specimens in order to obtain

the bulk properties of honeycombs. Xu et al. [39] found a linear relationship between

the plateau stress and number of ‘‘Y’’ units which was similar to that Wu and Jiang

[46] and Alkkhader and Vural [126] reported. The dimensions of the indenter used

in indentation tests and the platens used in compression tests are the same, i.e., 90

mm × 90 mm. Therefore, in the present study, the number of cells in each specimen

under the indenter or compressive platen is 9 × 9 for the honeycomb with the largest

cell size (9.525 mm) and 28 × 28 for the honeycomb with the smallest cell size (3.175

mm).

67

(a)

(b)

Figure 3.1. Three types of aluminum hexagonal honeycomb specimens used in: (a) indentation tests; (b) compression tests.

68

Table 3.1. Specification of aluminum honeycombs

Type Material description* Cell size,

D

Single cell

wall

thickness, t

t/l

ratio

Nominal

Density,

ρ

Young’s

Modulus

No. of cells under the

Indenter or platen

mm mm kg/m3 GPa

H31 3.1-3/16-5052-.001N 4.763 0.0254 0.00924 49.66 0.52 19×19

H42 4.2-3/8-5052-.003N 9.525 0.0762 0.0139 67.28 0.93 9×9

H45 4.5-1/8-5052-.001N 3.175 0.0254 0.0139 72.09 1.03 28×28

*In the material description, 3.1, 4.2 and 4.5 are the nominal densities in pounds per cubic foot, 3/16, 3/8 and 1/8 are the cell size in

inches, 5052 is the aluminum alloy grade, 0.001 or 0.003 is the nominal foil thickness in inches and N denotes non-perforated cell walls.

Data were provided by the manufacturer.

69

The out-of-plane (T) direction and two in-plane (L-ribbon, W-transverse)

directions of a honeycomb are shown in Fig. 3.2. Each unit cell of the hexagonal

honeycomb consists of two double walls (bonded by adhesive) and four single walls.

The honeycomb strength in the out-of-plane direction is much greater than that in

the other two in-plane directions [21].

Figure 3.2. Schematic diagram of hexagonal honeycomb.

3.1.2. Fixtures

An MTS machine was used in quasi-static tests and a high-speed INSTRON

machine (VHS 8800) was used in dynamic tests; the latter was equipped with VHS

software, which enabled a constant velocity in each test. The indenter for the

indentation tests and the platens for the compression tests were both made of mild

steel. The indenter for the indentation tests was also used as the upper platen in

compression tests, which had a square cross-section with dimensions of 90 mm ×

90 mm. A specially designed circular plate (diameter = 200 mm) with holes

70

(diameter = 3.5 mm, which is the smallest our tool can machine) in the middle part

was used to allow air to escape from honeycomb specimens during the tests (Fig.

3.3). More than 270 holes were drilled in the circular plate to cover the specimen’s

loading area during indentation and compression. Therefore, in all the tests, the

effect of entrapped air was minimized and could be ignored. The circular plate was

fixed to the machines and all the specimens were placed on this fixed plate by using

a thin layer of double-sided sticky tape. The indenter was fixed on the moving piston

of both the MTS and INSTRON machines. On the MTS machine, the circular plate was

fixed on the bottom part of the machine and the indenter was fixed to the upper

cross head (Fig. 3.4a). On the INSTRON machine, the indenter was fixed on the

moving bottom piston of the machine and the circular plate was fixed on the upper

cross head with load cell (Fig. 3.4b). In dynamic tests, when the indenter crushed

the specimen at a certain velocity, the specimen’s outer edges tended to curl and lost

contact with the top circular platen. To maintain consistency with quasi-static

results (where the test specimen did not curl), two rubber bands were used (on

opposite sides) to prevent specimen edges from curling.

71

Figure 3.3. The specially designed circular plate with holes for entrapped air to

escape in both indentation and compression tests.

(a)

72

(b)

Figure 3.4. Out-of-plane indentation tests on aluminum honeycomb specimens

(4.2-3/8-5052-.003N): (a) quasi-static test set-up on MTS machine; (b) dynamic test

set-up on INSTRON machine.

For the quasi-static tests using the MTS machine, three different velocities, 5×10-

5 ms-1, 5×10-4 ms-1 and 5×10-3 ms-1, were applied in the indentation and compression

tests (Fig. 3.4a). The corresponding nominal strain rates were 10-3, 10-2 and 10-1 s-1,

respectively. Dynamic indentation and compression tests were conducted at two

different constant velocities, 5×10-1 ms-1 and 5 ms-1, by using the high-speed

INSTRON machine (VHS8800) (Fig. 3.4b). The corresponding nominal strain rates

were 10 and 102 s-1, respectively.

73

3.2. Experimental Results and Discussions

3.2.1. Deformation of Aluminum Honeycombs Subjected to Compression and Indentation

A digital camera and a high-speed camera were used in the quasi-static and

dynamic compressive and indentation tests, respectively, to observe the

deformation patterns of the three types of honeycomb specimens.

In both the quasi-static and dynamic compression tests, all three types of

honeycomb specimens deformed in a similar pattern. No significant difference was

observed in the deformation mechanism at different strain rates. During

compression, crushing was initiated by elastic buckling and then progressive plastic

buckling of the cell walls was observed from the lower and upper interfaces of the

specimen between two loading fixtures (i.e. indenter and circular plate).

Photographs of post-test specimens under compressive load of three types of

honeycombs are shown in Fig. 3.5. It can be seen that cell walls along the four edges

deformed in an irregular pattern but in the central portion all the cells deformed in

a uniform pattern for all the three types of honeycomb specimens.

Xu et al. [39] did not observe any significant difference in the plastic buckling

between dynamic and quasi-static compression. During deformation of the

honeycombs, single walls deformed in such a way to accommodate the deformation

of the adjacent double walls. The deformation pattern of the cell walls observed in

this study agrees well with the global collapse mode in [39].

However, in indentation, it was difficult to observe the crushing pattern of the

specimens because the indenter penetrated into the middle portion of the

honeycomb specimens and the surrounding un-deformed cells blocked the view.

74

Therefore, the deformation of honeycombs in indentation could only be investigated

by studying the deformed honeycomb specimens after tests. Photographs of

specimens that were taken after indentation tests are shown in Fig. 3.6 and no

evident difference was observed in the deformation pattern at different strain rates.

For all three types of honeycombs, due to a higher level of lateral constraints,

honeycomb cells in the central region buckled in a regular pattern. Irregular tearing

was found in honeycomb specimens under the edges of the indenter.

Figure 3.5. Photographs of deformed specimens after compression tests at a

velocity of 5 ms-1.

75

(a)

(b)

76

(c)

Figure 3.6. Photographs of deformed specimens after indentation tests at a velocity

of 5 ms-1: (a) honeycomb Type H31; (b) honeycomb Type H42; (c) honeycomb Type

H45.

The double walls of honeycombs were formed by gluing two single walls

together. In indentation tests, de-bonding of the double walls was observed along

the two edges of the indenter (see Fig. 3.6b) while tearing took place in single walls.

During indentation, crushing initiated with elastic buckling and was followed by

plastic buckling of the cell walls, which was associated with de-bonding of the

double walls. In the final stage, densification was governed by the plastic collapse.

In both the quasi-static and dynamic indentation, a similar buckling pattern was

observed. During the compression tests progressive buckling of the cell walls

initiated from the top or bottom surfaces between the indenter and circular fixed

77

platen, but in the indentation case it could not be seen due to the surrounding cells

of specimens.

It was also found that the deformed area of the specimens after indentation tests

is slightly larger than the cross-sectional area of the indenter. This was caused by

the irregular tearing and de-bonding of the cell walls along the four edges of the

indenter. Since the cell wall thickness of honeycomb Type H42 is three times of that

for honeycombs Types H31 and Type H45, honeycomb Type H42 deformed more

uniformly.

3.2.2. Experimental Data Processing

Force and displacement data were recorded by the data acquisition system

connected to the MTS and INSTRON machines. The nominal stress was calculated by

dividing the measured force by the area of the original cross-section of specimens

under the indenter or platen, which was 90 mm × 90 mm for all specimens. The

nominal strain was calculated by dividing the measured displacement by the

original height (50 mm) of specimens.

A typical indentation stress-strain curve (Fig. 3.7a) demonstrates three regions:

(1) linear region, where the stress increases linearly with strain; (2) plateau region,

where the stress is almost constant with the increase of strain; and (3) densification

region, where the stress increases significantly with strain and the honeycomb

becomes densified. The average stress in the plateau region is defined as the plateau

stress and the onset strain when the densification starts is defined as the

densification strain. Plateau stress and densification strain are the two critical

parameters of honeycombs that can be determined by various methods. Li et al.

[127], Tan et al. [22] , Avalle et al. [128] and Xu et al. [8] employed energy efficiency

78

methods to calculate densification strain, εd, and plateau stress, σpl. In the present

study, the plateau stress is calculated as the average stress between a displacement

of 5 mm and 38 mm, and total energy dissipated is calculated from a displacement

of 0 mm to 38 mm prior to the densification. The results obtained by this method

are found to concur with those by Xu et al. [20] using the energy efficiency method.

3.2.3. Reproducibility of test results

To exemplify the reproducibility of experimental results, three repeated

indentation tests on nominally identical specimens were performed under quasi-

static and dynamic loading conditions at velocities of 5×10-3 ms-1 and 5 ms-1,

respectively. The stress-strain curves of Type H31 (3.1-3/16-5052-0.001N)

honeycomb specimens are shown in Fig. 3.7. At 5×10-3 ms-1, the plateau stresses are

1.06 MPa, 1.05 MPa and 1.05 MPa for specimens H31-5-1, H31-5-2 and H31-5-3,

respectively. The maximum difference is approximately 0.94 %. Similarly, at 5 ms-1

the plateau stresses are 1.23 MPa, 1.22 MPa and 1.21 MPa for specimens H31-9-1,

H31-9-2 and H31-9-3, respectively. The maximum difference is approximately 1.6

%. It can be seen that the three curves at the same loading velocity are very close to

each other and the difference is negligible, which indicates that the experimental

results are consistent. Therefore, only one test was conducted at each loading

velocity in this study. Please note that in the dynamic tests, the INSTRON machine

was stopped at a displacement of 45 mm before the densification, for safety reasons.

Therefore, in Fig. 3.7(b) the densification region is not fully recorded.

79

(a)

(b)

Figure 3.7. Reproducibility of experiments on Type H31 honeycomb specimens

under indentation loads: (a) quasi-static loading at 5×10-3 ms-1; (b) dynamic loading

at 5 ms-1.

80

3.2.4. Plateau Stress

In both the quasi-static and dynamic tests, five different strain rates ranging

from 10-3 to 102 s-1, were applied in tests, respectively. The stress-strain curves

(quasi-static and dynamic) of three types of honeycomb specimens at different

strain rates are shown in Figs. 3.8-3.9. The honeycombs vary in cell size (D), cell wall

thickness (t) and t/l ratio. Honeycomb Type H45 has the greatest nominal density

(72.09 kg/m3) and t/l ratio, while honeycomb Type H31 has the smallest nominal

density (49.66 kg/m3) and t/l ratio. It is known that plateau stress increases with

nominal density and t/l ratio. Therefore, the plateau stress is the largest for

honeycomb Type H45 and smallest for honeycomb Type H31. From the stress-strain

curves (Figs. 3.8-3.9) it is observed that the plateau stresses did not increase before

densification, which confirmed that the entrapped air escaped successfully during

crushing [41]. At the strain rate of 102 s-1, fluctuation in stress due to the high impact

velocity has been observed and shown in Figs. 3.9(c) and 3.9(d). The source of that

fluctuation was due to the tiny gap between the load cell and circular fixed platen. A

high-speed camera was employed to identify this vibration. A thin metal sheet was

used between the load cell and circular fixed platen to reduce such vibration.

81

(a)

(b)

82

(c)

(d)

83

(e)

(f)

Figure 3.8. Quasi-static out-of-plane stress-strain curves of three types of

honeycombs under different loading conditions: (a) indentation at 5×10-5 ms-1; (b)

compression at 5×10-5 ms-1; (c) indentation at 5×10-4 ms-1; (d) compression at 5×10-

4 ms-1; (e) indentation at 5×10-3 ms-1; (f) compression at 5×10-3 ms-1.

84

(a)

(b)

85

(c)

(d)

Figure 3.9. Dynamic out-of-plane stress-strain curves of three types of honeycombs

with different nominal density and t/l ratio under different loading conditions: (a)

indentation at 5×10-1 ms-1; (b) compression at 5×10-1 ms-1; (c) indentation at 5 ms-

1; (d) compression at 5 ms-1.

86

At the same strain rate, it was found that plateau stress in the indentation test

was higher than that in the corresponding compression test. During indentation,

honeycomb cells were both compressed and torn simultaneously. A greater force

and more energy were therefore required. For example, at 102 s-1, the plateau

stresses in compression tests of honeycombs Types H31, H42 and H45 were 1.06

MPa, 1.64 MPa and 2.08 MPa, respectively. At the same strain rate, the

correspondingly plateau stresses in indentation tests of honeycombs Types H31,

H42 and H45 were 1.23 MPa, 2.13 MPa and 2.35 MPa, respectively.

The relationship between the plateau stress and strain rate is shown in Fig. 3.10

for both compressive and indentation loads. From Fig. 3.10 it can be seen that under

both compression and indentation the plateau stresses increases with the impact

velocity for all three types of honeycombs. The plateau stresses in compression

increase by 15.1 %, 14.6 % and 12.5 % for honeycombs Types H31, H42 and H45

respectively from strain rate 10-3 s-1 to 102 s-1. In the case of indentation, the

corresponding plateau stresses increase by 14.63 %, 13.15 % and 11.92 % for the

same range of strain rates for honeycombs Types H31, H42 and H45, respectively.

87

(a)

(b)

Figure 3.10. Effect of strain rate on the plateau stress of three types of honeycombs

with different nominal density under different loading conditions: (a) compression;

(b) indentation.

88

3.2.5. Energy Absorption

Energy absorbed by honeycomb specimens is the area under the force-

displacement curves, in compression tests, 𝐸𝑐, and in indentation, 𝐸𝐼 . The total

energy dissipated by the three types of honeycomb specimens is calculated to a

displacement of 38 mm prior to the densification at different strain rates from 10-3

s-1 to 102 s-1 under both compressive and indentation loads, and is shown in Fig.

3.11. Due to the lowest nominal density for honeycomb Type H31, its energy

absorption is found to be the smallest at all the strain rates. Similarly, honeycomb

Type H45 has the highest density and thus its energy absorption is found to be the

largest at all the strain rates. Moreover, it has been observed that the total energy

increases with strain rate for all three types of honeycombs under both compressive

and indentation loads.

In compression, when strain rate increases from 10-3 s-1 to 102 s-1, the total

energy absorbed increases by 14.97 %, 14.03 % and 12.48 % for honeycombs Types

H31, H42 and H45, respectively. Similarly, in indentation, the total energy increases

by 14.63 %, 13.15 % and 11.92 % respectively for honeycombs Types H31, H42 and

H45 when the strain rate increases from 10-3 s-1 to 102 s-1. Among the three types of

honeycombs the energy enhancement of honeycomb Type H31 is the highest and it

is the lowest for honeycomb Type H45, in both indentation and compression. The

enhancement percentage of the total energy dissipation in compression is higher

compared to that in indentation.

89

(a)

(b)

Figure 3.11. Strain rate effect on the total dissipated energy of three types of

honeycombs under different loading conditions: (a) compression; (b) indentation.

90

The specific energy is defined as the energy per unit mass. The average mass of

honeycomb specimens under the indenter or platen was 19.1 g, 28.5 g and 29.6 g for

honeycombs Types H31, H42 and H45, respectively. The values of specific energy

calculated at different strain rates are listed in Table 3.2. Specific energy is plotted

against strain rate in Fig. 3.12.

(a)

91

(b)

Figure 3.12. Specific energy-strain rate curves of three types of honeycombs under

different loading conditions: (a) compression; (b) indentation.

3.2.6. Tearing Energy in Indentation

Lu et al. [129] reported their experimental investigation into the tearing energy

of four corners of square tubes made of aluminum and mild steel, respectively. They

employed four rollers which were driven simultaneously to tear the four corners of

the tube. The motion of the rollers caused tearing and bending of the side walls. They

found that the tearing energy depended on not only the material properties but also

the thickness of the material. Lu et al. [130] conducted tensile tests on thin ductile

plates to measure the tearing energy. They noticed that tearing energy was not

constant. During the in-plane tearing of a thin plate, they assumed that total energy

dissipation consisted of two portions: one was the plastic deformation energy and

92

the other was tearing energy. They used an energy balance equation to determine

the tearing energy.

When studying the tearing energy of aluminum honeycombs, Zhou and Mayer

[23] employed force an equilibrium equation to calculate the tearing force (Ft)

around the four edges of the indenter. Since both tearing of cell walls along the edges

of the indenter and compression of honeycomb cells of the honeycomb specimens

occurred simultaneously during the indentation of honeycombs, the total energy

dissipated in tearing, 𝐸𝑡, was calculated by the difference of energy dissipated in

indentation, 𝐸𝐼 , and energy dissipated in compression, 𝐸𝑐. By using a similar energy

conservation equation to that used by Lu et al. [130], the tearing energy in the

indentation can be calculated as:

E𝑡 = EI- 𝐸𝑐 (3.1)

The magnitude of compressive energy, 𝐸𝑐, and tearing energy, 𝐸𝑡, at different

strain rates is listed in Table 3.2. Tearing energy- strain rate curves are shown in Fig.

3.13. It can be noticed from Fig. 3.13 that the tearing energy for all types of

honeycombs increases with strain rate from 10-3 s-1 to 102 s-1. The enhancement of

the compressive strength of honeycomb with the strain rate was summarised by Xu

et al. [41], while the mechanism of the tearing strength enhancement requires

further study. From Fig. 3.10 and Fig. 3.11 it has been seen that both magnitudes of

the plateau stress and total dissipated energy for honeycomb Type H45 are the

greatest due to its highest density. However in the case of tearing energy, the largest

magnitude of tearing energy is found for honeycomb Type H42, which has the

largest cell wall thickness. Similar to the findings by Lu et al. [129], honeycomb cell

93

wall thickness is also effecting on the tearing energy dissipated in indentation. The

average percentage of tearing energy in the total dissipated at different strain rates

of honeycombs Types H31, H42 and H45 are 17 %, 22 % and 11 %, respectively. The

contribution of tearing energy towards the total dissipated energy is found to be the

largest for Type H42 honeycomb, due to its largest cell wall thickness.

Initially, in order to determine whether the friction between the indenter and

cell walls had any effect on dissipated energy, we used grease to reduce the possible

friction of indenter edges when conducting indentation tests. However, no

significant difference was found in the energy dissipated. Therefore, friction has

little effect on the total energy as well as tearing energy. This is in good agreement

with the results observed by Zhou and Mayer [23].

Figure 3.13. Tearing energy-strain rate curves of three types of honeycombs.

94

Zhou and Mayer [23] calculated tearing strength from the surface area of the four

edges of the indenter. The same approach was used in the current study. The

fracture area, 𝐴𝑡 , was calculated as the product of the circumferential length of the

square shape indenter (90 mm × 4) and the displacement (38 mm) of the indenter.

The tearing energy per unit fracture area-strain rate curves is plotted in Fig. 3.14 for

three types of honeycombs. The best fitted lines are also plotted in Fig. 3.14 for

different types of honeycombs. It has been found that the tearing energy per unit

fracture area increases with the strain rate, for all types of honeycombs. The

relationship between the tearing energy per unit fracture area and strain rates is

described in Eq. (3.2).

𝐸𝑡

𝐴𝑡= 4.237 + 0.213𝜀̇0.266 for honeycomb H31 (3.2a)

𝐸𝑡

𝐴𝑡= 9.040 + 0.677𝜀̇0.175 for honeycomb H42 (3.2b)

𝐸𝑡

𝐴𝑡= 4.803 + 0.515𝜀̇0.136 for honeycomb H45 (3.2c)

95

Figure 3.14. Tearing energy per unit fracture area-strain rate curves of three types

of honeycombs.

Since the t/l ratio for H42 and H45 are the same, three types of honeycombs

which are experimentally studied here have only two different relative density or

t/l ratios. Therefore it is impossible to propose an empirical formula to describe

tearing energy per fracture area in terms of t/l ratio in this chapter. Finite element

analysis will be employed in the next chapter to derive the relationship between

tearing energy per fracture area and honeycomb density.

96

Table 3.2. Summary of all quasi-static and dynamic experimental results

Test no. Specimen Test type Velocity Strain

Rate

Plateau

Stress

Total

Dissipated

energy

Tearing

energy

Tearing

energy per

unit

fracture

area

Specific

energy

ms-1 s-1 MPa J J kJ/m2 J/g

H31-1 3.1-3/16-

5052-.001N

Indentation 5×10-5 0.00

1

1.05 325 58 4.24 17.02

H31-3 3.1-3/16-

5052-.001N

Indentation 5×10-4 0.01 1.08 329 59 4.31 17.23

H31-5 3.1-3/16-

5052-.001N

Indentation 5×10-3 0.1 1.13 336 60 4.39 17.59

H31-7 3.1-3/16-

5052-.001N

Indentation 5×10-1 10 1.19 353 63 4.61 18.48

H31-9 3.1-3/16-

5052-.001N

Indentation 5 100 1.23 382 68 4.97 20.00

97

H31-2 3.1-3/16-

5052-.001N

Compression 5×10-5 0.00

1

0.9 267 - - 13.98

H31-4 3.1-3/16-

5052-.001N

Compression 5×10-4 0.01 0.92 270 - - 14.14

H31-6 3.1-3/16-

5052-.001N

Compression 5×10-3 0.1 0.95 276 - - 14.45

H31-8 3.1-3/16-

5052-.001N

Compression 5×10-1 10 0.98 290 - - 15.18

H31-

10

3.1-3/16-

5052-.001N

Compression 5 100 1.06 314 - - 16.44

H42-1 4.2-3/8-

5052-.003N


1

1.85 554 125 9.14 19.44

H42-3 4.2-3/8-

5052-.003N

Indentation 5×10-4 0.01 1.92 578 129 9.43 20.28

H42-5 4.2-3/8-

5052-.003N

Indentation 5×10-3 0.1 2.01 601 131 9.58 21.09

H42-7 4.2-3/8-

5052-.003N

Indentation 5×10-1 10 2.05 613 136 9.94 21.51

98

H42-9 4.2-3/8-

5052-.003N

Indentation 5 100 2.13 644 145 10.60 22.60

H42-2 4.2-3/8-

5052-.003N


1

1.4 429 - - 15.05

H42-4 4.2-3/8-

5052-.003N

Compression 5×10-4 0.01 1.49 449 - - 15.75

H42-6 4.2-3/8-

5052-.003N

Compression 5×10-3 0.1 1.53 470 - - 16.49

H42-8 4.2-3/8-

5052-.003N

Compression 5×10-1 10 1.56 477 - - 16.74

H42-

10

4.2-3/8-

5052-.003N

Compression 5 100 1.64 499 - - 17.51

H45-1 4.5-1/8-

5052-.001N


1

2.07 623 69 5.04 21.05

H45-3 4.5-1/8-

5052-.001N

Indentation 5×10-4 0.01 2.19 657 68 4.97 22.20

H45-5 4.5-1/8-

5052-.001N

Indentation 5×10-3 0.1 2.23 667 72 5.26 22.53

99

H45-7 4.5-1/8-

5052-.001N

Indentation 5×10-1 10 2.26 686 75 5.48 23.18

H45-9 4.5-1/8-

5052-.001N

Indentation 5 100 2.35 712 79 5.77 24.05

H45-2 4.5-1/8-

5052-.001N


1

1.82 623 - - 18.72

H45-4 4.5-1/8-

5052-.001N

Compression 5×10-4 0.01 1.92 657 - - 19.90

H45-6 4.5-1/8-

5052-.001N

Compression 5×10-3 0.1 1.98 667 - - 20.10

H45-8 4.5-1/8-

5052-.001N

Compression 5×10-1 10 2.03 686 - - 20.64

H45-

10

4.5-1/8-

5052-.001N

Compression 5 100 2.08 712 - - 21.39

100

3.3. Summary

In this experimental study, both out-of-plane indentation and compression tests

at constant velocities from 5×10-5 ms-1 to 5 ms-1 have been conducted using MTS and

high-speed INSTRON machines. A specially designed fixture (i.e., a circular plate

with a number of small holes) has been employed in order to allow the entrapped

air in the honeycombs to escape, which minimizes the effect of entrapped air in

honeycombs. Three repeated tests have been conducted at 5×10-5 ms-1 and 5 ms-1,

respectively, and the results matched very well for the same loading rate. Therefore

only one test has been conducted under each loading condition.

In order to minimize the effect of specimen dimensions, all the specimens tested

have been carefully prepared with a minimum number of 9 × 9 cells. Force-

displacement data have been recorded by the computer connected to both the MTS

and INSTRON machines. For all the compression and indentation tests, stress-strain

curves, total energy-strain rate curves, specific energy-strain rate curves and tearing

energy-strain rate curves, have been calculated accordingly and presented. The

calculated plateau stress and energy absorption varied with loading condition. Due

to the effect of tearing, the magnitudes of the plateau stress and total dissipated

energy have been found to be higher in indentation than those in the compression

tests. For example, at 5 ms-1 velocity, the plateau stress and total dissipated energy

of honeycomb Type H42 in indentation are 2.13 MPa and 644 J, while in

compression they are 1.64 MPa and 499 J, respectively. The plateau stress and total

energy dissipated of all the types of honeycombs increased with strain rate.

Moreover, both the plateau stress and dissipated energy increased with the nominal

density of honeycombs. Tearing energy has been calculated by working out the

101

difference between the total energy in the indentation tests and compression tests.

The tearing energy also increased with the strain rate. The percentages of tearing

energy in the total dissipated energy in indentation have been calculated and found

to be 17 %, 22 % and 11 % for Type H31, H42 and H45, respectively. A high value

of tearing energy has been observed for honeycomb Type H42 compared to those of

the other two types of honeycombs. This was due to the thicker cell wall thickness

of honeycomb Type H42. During indentation, tearing of single cell walls was found

along the two edges of the indenter and de-bonding in double cell walls has been

observed along the other two edges of the indenter for all types of honeycombs.

Furthermore, the tearing energy per unit fracture area increased with strain rate for

all honeycombs studied. Due to the honeycombs available and the limitation of

testing machines, it was not possible to conduct tests on honeycombs with other t/l

ratios and at velocities higher than 5 ms-1. The finite element analysis will be

employed for further investigation in the subsequent chapter (Chapter 4).

103

Chapter 4. Finite element analysis of aluminum

honeycombs subjected to dynamic indentation and

compression loads

The previous chapter (Chapter 3) discussed the experimental investigation of

aluminum honeycombs under out-of-plane indentation at different low and

intermediate loading velocities. In this chapter, the dynamic behavior of aluminum

hexagonal honeycombs subjected to out-of-plane indentation and compression

loads will be investigated numerically using ANSYS LS-DYNA. The finite element

(FE) models will be verified by the experimental results in terms of deformation

pattern, stress-strain curve and energy dissipation. Then, the verified FE models will

be used in the comprehensive finite element analysis of different aluminum

honeycombs. Plateau stress, 𝜎𝑝𝑙, and dissipated energy (EI for indentation and Ec for

compression) will be calculated at different strain rates ranging from 102 s-1 to 104

s-1. The effects of strain rate and 𝑡/𝑙 ratio on the plateau stress, dissipated energy

and tearing energy will be discussed. Thereafter, the relationship between the

tearing energy per unit fracture area, relative density and strain rate for

honeycombs will be proposed. Moreover, a generic formula will be stated that can

be used to describe the relationship between tearing energy per unit fracture area

and relative density for both aluminum honeycombs and foams.

4.1. Finite element (FE) modelling

In the present study, numerical analysis of aluminum honeycombs was carried

out using ANSYS LS-DYNA [131]. Two types of honeycombs, differing in cell size and

cell wall thickness, were simulated. The specification of the honeycombs, provided

by the manufacturer, is listed in Table 4.1. The dimension of each honeycomb model

104

is the same as that of the actual specimen used in the previous experiments (chapter

3). The height of all the honeycombs, h, was 50 mm. The in-plane dimensions of all

honeycomb specimens were 180 mm × 180 mm in indentation simulation (Fig. 1a)

and 90 mm × 90 mm in compression simulation (Fig. 1b).

Table 4.1. Specification of aluminum honeycombs [20]

Type Material

description*

Cell size,

D

Single cell

wall thickness,

t

Cell wall

thickness to edge

length ratio, t/l

mm mm

H31 3.1-3/16-5052-.001N 4.763 0.0254 0.00924

H42 4.2-3/8-5052-.003N 9.525 0.0762 0.0139

*In the material description, 3.1 and 4.2 are the nominal densities in pounds per cubic

foot, 3/16 and 3/8 are the cell size in inches, 5052 is the aluminum alloy grade, 0.001 or

0.003 is the nominal foil thickness in inches, and N denotes non-perforated cell walls. Data

were provided by the manufacturer, HEXCEL®. The relation between cell size, D end cell

edge length, l is: 𝐷 = √3𝑙.

Aluminum honeycomb walls were simulated using a bilinear kinematic

hardening material model. The corresponding material properties are listed in

Table 4.2. Belytschko-Tsay Shell 163 elements with five integration points were

employed to simulate the honeycomb cell walls for high computational efficiency

[132]. In each honeycomb cell, single wall thickness was employed for the four

oblique walls and double wall thickness was employed for the two vertical walls. To

identify the optimum element size, a convergence test was carried out. Five different

element sizes, 2.1 mm, 1.4 mm, 0.7 mm, 0.3 mm and 0.15 mm were used to simulate

105

compression of honeycombs at 5 ms-1. No significant difference (less than 7 %) was

observed between the results for element sizes 0.7 mm and 0.15 mm. Therefore in

this FE analysis of aluminum honeycombs, an element size of 0.7 mm was employed.

Since tearing of cell walls happened in honeycombs under indentation,

MAT_ADD_EROSION failure criteria with a maximum effective strain of 0.3 [133]

was used in the indentation models. All degrees of translational freedom of one node

at a corner of the honeycomb were fixed to keep the honeycomb in place (i.e., no

rigid body movement).

Table 4.2. Material properties used in the FE model of aluminum honeycombs [132]

Material

Properties

Mass Density

(ρ)

Young’s

Modulus

(E)

Poisson’s

ratio (υ)

Yield

Stress

(σys)

Tangent

modulus

(Etan)

Magnitude 2680 kg/m3 69 GPa 0.33 292 690 MPa

In physical experiments, honeycomb specimens were placed on a fixed lower

plate and crushed by the upper plate (in compression) or indenter (in indentation).

In FE models, the plates and indenter were simulated by rigid bodies. The material

properties used for the plates and indenter are listed in Table 4.3.

Table 4.3. Material properties used in the FE model of rigid bodies [132]

Material

Properties

Mass Density (ρ) Young’s Modulus

(E)

Poisson’s ratio

(υ)

Magnitude 7830 kg/m3 207 GPa 0.34

106

For the fixed lower plate, all degrees of freedom were fixed. For the upper plate

(in compression) and indenter (in indentation) all three rotational movements and

two transitional movements in the X and Z directions were fixed. The upper plate or

indenter could move in the negative Y direction at a constant velocity to compress

or indent honeycombs.

A tiny gap (0.1 mm) between the fixed lower plate and the honeycomb was

employed to avoid the initial penetration at the beginning of the simulation. For the

same reason, an initial gap of 5 mm was also introduced between the upper plate or

the indenter and the honeycomb. SURFACE_TO_SURFACE contacts were employed

between the plates or indenter and honeycomb. Typical finite element models of

indentation and compression of honeycombs in the out-of-plane direction are

shown in Fig. 4.1.

(a)

107

(b)

Figure 4.1. Typical FE models of honeycomb H31: (a) indentation; (b) compression.

4.2. Validation of FE models

4.2.1. Deformation patterns

Figure 4.2 shows the comparison between the experimental and simulated

deformation of honeycomb H31 in compression at 5 ms-1. An identical deformation

mode in both the experiments [39] and Finite Element Analysis (FEA) was observed:

when the honeycomb was compressed in the out-of-plane (T) direction, buckling of

cell walls initiated from both the top and bottom ends and propagated to the middle

of the honeycomb (Figs. 4.2a and b). Figs. 4.2 (c) and (d) show the deformed

honeycomb H31 after crushing in the experiment and FEA respectively. Almost

identical deformation patterns were found in the experimental and FEA results. Due

to the stronger lateral constraints in the central part of the honeycomb, the

108

honeycomb deformed in a much more regular pattern in the central part. However

along the four edges of the indenter, honeycomb cell walls deformed in an irregular

pattern. Similar deformation patterns and mechanisms were observed for

honeycomb H42 in compression.

(a)

(b)

109

(c)

(d)

Figure 4.2. Comparison between experimental and simulated deformation mode of

honeycomb H31 under compression: (a) experimental result; (b) FEA result; (c)

experimental post-test specimen; (d) FEA post-test specimen.

110

Figure 4.3 shows comparison between experimental and the FEA deformation

pattern of honeycomb H42 subjected to out-of-plane indentation at a velocity of 5

ms-1. Similar irregular tearing patterns were observed in both the experiment and

FEA. The FEA results of another type of honeycomb H31, also showed a similar

deformation pattern to that observed in the previous experiments [8].

(a)

111

(b)

Figure 4.3. Comparison between experimental and FEA deformation pattern of

honeycomb H42 under indentation: (a) experimental post-test specimen; (b) FEA

post-test specimen.

4.2.1. Stress-strain curves

FEA and experimental stress-strain curves of two types of honeycombs are

shown in Fig. 4.4. Similar general trends in the stress-strain curves were found for

both honeycombs in indentation and compression.

112

0.0 0.2 0.4 0.6 0.80

1

2

3

4

5St

ress

(M

Pa)

Strain

H31-FEA

H31-ExperimentIndentation at 5 ms-1

(a)

0.0 0.2 0.4 0.6 0.80

1

2

3

4

5

Stre

ss (

MP

a)

Strain

H31-FEA

H31-ExperimentCompression at 5 ms-1

(b)

113

0.0 0.2 0.4 0.6 0.80

1

2

3

4

5

6Indentation at 5 ms-1

Stre

ss (

MP

a)

Strain

H42-FEA

H42-Experiment

(c)

0.0 0.2 0.4 0.6 0.80

1

2

3

4

5Compression at 5 ms-1

Stre

ss (

MP

a)

Strain

H42-FEA

H42-Experiment

(d)

Figure 4.4. Experimental and FEA stress-strain curves of two types of honeycombs

at 5 ms-1: (a) indentation of H31; (b) compression of H31; (c) indentation of H42;

(d) compression of H42.

114

The plateau stress is defined as the average stress between displacements from

5 mm to 38 mm. The total dissipated energy is the area under the force-

displacement curves up to 38 mm, which is described by 𝐸𝑐 in compression and 𝐸𝐼

in indentation. Tearing of the cell walls along the four edges of the square indenter

occurred simultaneously during the indentation. Tearing energy, 𝐸𝑡, was calculated

using the following energy conservation equation:

E𝑡 = EI- 𝐸𝑐 (4.1)

where, 𝐸𝑡 is the dissipated energy in tearing, 𝐸𝐼 is the energy dissipated in

indentation and, 𝐸𝑐 is the energy dissipated in compression.

Comparisons between the FEA and experimental results in terms of plateau

stress and dissipated energy are listed in Table 4.4. For two different types of

honeycombs (H31 and H42), the simulated plateau stresses and total dissipated

energies were found to be slightly lower than the corresponding experimental

values in both indentation and compression. The differences were between 4.71 %

and 11.62 %, which was acceptable.

115

Table 4.4. Comparison between FEA and experimental results at 5 ms-1

Test type Honeycombs

material

Plateau stress Dissipated energy

Exp FEA Difference Exp FEA Difference

MPa J % MPa J %

Indentation H31 1.23 1.17 4.88 382 364 4.71

Indentation H42 2.13 1.89 11.26 644 571 11.33

Compression H31 1.06 0.99 6.66 314 294 6.36

Compression H42 1.64 1.45 11.58 499 441 11.62

116

4.3. Results and discussions

4.3.1. The effect of 𝑡 𝑙⁄ ratio

The effect of 𝑡 𝑙⁄ ratio on the mechanical properties of honeycomb is studied in

this section. Firstly, the thickness of honeycomb cell walls was fixed as 0.0254 mm.

Five different cell sizes, 3.175 mm, 3.969 mm, 4.763 mm, 6.35 mm and 9.525 mm,

were employed. A constant strain rate of 1 ×103 s-1 was used in the simulation. The

FEA results are listed in Table 4.5. Both in indentation and compression, it was found

that the plateau stress decreased with the increase in cell size for a constant cell wall

thickness. Similar to the plateau stress, dissipated energy and tearing energy also

decreased with the increase in cell size.

Secondly, honeycomb cell size, D, was kept constant as 4.763 mm (the

corresponding cell edge length was 2.75 mm), the thickness of the cell wall varied

from 0.0178 mm to 0.1524 mm, where the corresponding 𝑡 𝑙⁄ ratios ranged from

0.00647 to 0.05542. The simulation results at a constant strain rate of 1 ×103 s-1 are

listed in Table 4.6.

The plateau stresses of honeycombs subjected to out-of-plane compression and

indentation were found to increase with 𝑡 𝑙⁄ ratio (Fig. 4.5) by power laws. The

exponents are 1.47 for compression (Eq. 4.2a) and 1.36 for indentation (Eq. 4.2b).

Xu et al. [39, 132] also found a similar power law relationship between plateau

stress and 𝑡 𝑙⁄ ratio, with an exponent of 1.49.

117

Table 4.5. FEA results of honeycombs with constant cell wall thickness and different cell size

Test Type FEA

no.

Cell size,

D

Cell Wall

thickness, t

𝒕 𝒍⁄ ratio Plateau

stress

Dissipated

energy

Tearing

energy

mm mm MPa J J

Indentation

CS-I-1 3.175 0.0254 0.01388 3.74 1262 309

CS-I-2 3.969 0.0254 0.01109 2.81 955 224

CS-I-3 4.763 0.0254 0.00924 2.18 731 183

CS-I-4 6.35 0.0254 0.00692 1.39 472 153

CS-I-5 9.525 0.0254 0.00462 0.73 256 78

Compression

CS-C-1 3.175 0.0254 0.01388 2.95 953 -

CS-C-2 3.969 0.0254 0.01109 2.26 731 -

CS-C-3 4.763 0.0254 0.00924 1.69 548 -

CS-C-4 6.35 0.0254 0.00692 0.99 319 -

CS-C-5 9.525 0.0254 0.00462 0.55 178 -

118

Table 4.6. FEA results of honeycombs with constant cell size and different cell wall thickness

Test Type FEA no. Cell Wall

Thickness, t

Cell

Edge

Length, l

𝒕 𝒍⁄ ratio Plateau

Stress

Dissipated

Energy

Tearing

Energy

mm mm MPa J J

Indentation

TL-I-1 0.0178 2.75 0.00647 1.05 378 91

TL-I-2 0.0254 2.75 0.00924 2.16 727 187

TL-I-3 0.0381 2.75 0.01386 3.72 1260 315

TL-I-4 0.0508 2.75 0.01847 4.91 1642 387

TL-I-5 0.0635 2.75 0.02309 5.98 2005 449

TL-I-6 0.0762 2.75 0.02771 7.24 2415 526

TL-I-7 0.0889 2.75 0.03233 8.09 2695 678

TL-I-8 0.1016 2.75 0.03695 9.21 3082 829

TL-I-9 0.127 2.75 0.04618 11.39 3823 1064

TL-I-10 0.1524 2.75 0.05542 13.24 4451 1370

TL-C-1 0.0178 2.75 0.00647 0.89 287 -

TL-C-2 0.0254 2.75 0.00924 1.66 540 -

TL-C-3 0.0381 2.75 0.01386 2.93 945 -

119

Compression

TL-C-4 0.0508 2.75 0.01847 3.91 1255 -

TL-C-5 0.0635 2.75 0.02309 4.82 1556 -

TL-C-6 0.0762 2.75 0.02771 6.24 1889 -

TL-C-7 0.0889 2.75 0.03233 6.65 2017 -

TL-C-8 0.1016 2.75 0.03695 7.34 2253 -

TL-C-9 0.127 2.75 0.04618 8.86 2759 -

TL-C-10 0.1524 2.75 0.05542 9.86 3081 -

120

0.01 0.10.1

1

10

FEA- indentation

FEA- compression

Best fitted line (indentation)

Best fitted line (compression)

Pla

teau

str

ess

(MP

a)

t/l ratio

Figure 4.5. The effect of 𝑡 𝑙⁄ ratio on the plateau stresses of honeycombs under

compression and indentation loads at a strain rate of 1×103 s-1.

For compression, 𝜎𝑝𝑙 = 2.93𝜎𝑦𝑠(𝑡 𝑙⁄ )1.47 (4.2a)

For indentation, 𝜎𝑝𝑙 = 4.49𝜎𝑦𝑠(𝑡 𝑙⁄ )1.36 (4.2b)

The tearing energies were calculated using Eq. (4.1) and are shown in Tables 4.5

and 4.6. Tearing energy was also found to increase with the 𝑡 𝑙⁄ ratio. The fracture

area, 𝐴𝑡 , was calculated as the product of the circumferential length of the square

shape indenter (90 mm × 4) and the displacement (38 mm) of the indenter [23]. The

relationship between the tearing energy per fracture area, 𝐴𝑡 , and the relative

density, 𝜌 𝜌0⁄ , is shown in Fig. 4.6. It was found that with the increase of t/l ratio or

relative density, the tearing energy per unit fracture area increased.

121

0.01 0.1

10

100 Tearing energy per unit fracture area

Best fitted line

Et /

At ,

(kJ/

m2)

Relative density

Figure 4.6. The relationship between of the tearing energy per unit fracture area

and relative density of honeycomb at a strain rate of 1×103 s-1.

Previously, Zhou and Mayer [23] conducted quasi-static indentation tests on

different honeycombs. They discussed the influence of specimen and indenter

dimensions on the plateau strength of aluminum honeycombs. Moreover, other

researchers [134-136] conducted quasi-static indentation tests on aluminum foams.

Shi et al. [134] proposed a theoretical formula and an empirical formula between

tearing energy per unit fracture area and relative density. In order to compare these

two types of cellular materials (honeycomb and foam, which were made from

different aluminum alloys), tearing energy per unit fracture area was normalized by

the yield stress of the parent aluminum alloy for both honeycombs and foams. The

relation between the normalized tearing energy per unit fracture area and relative

density is shown in Fig. 4.7. Using yield stress 𝜎𝑦𝑠= 150 MPa for the foams in Shi et

al. [134], Olurin et al. [135] and Olurin et al. [136], the equation proposed by Shi et

122

al. [134], 𝛾 = 119.4�̅�, can be rewritten as 𝛾 = 0.79𝜎𝑦𝑠�̅�, where 𝛾, 𝜎𝑦𝑠 and �̅� are

tearing energy per unit area, yield of aluminum and relative density of foam

respectively. The normalized tearing energy per unit fracture area for both

aluminum foams and honeycombs are plotted together in terms of relative density

in Fig. 4.7. The equation of the best fitted line is as follows, which is very similar to

that for foams is as follows:

𝐸𝑡

𝐴𝑡= 0.80𝜎𝑦𝑠(𝜌 𝜌0⁄ ) (4.3)

10-3 10-2 10-1 10010-3

10-2

10-1

100

Indentation tests [Chapter 3] (honeycomb)

Indentation tests by Xiaopeng Shi et al. [134] (ALPORAS foam)

Indentation tests by Zhou and Mayer [23] (honeycomb)

Indentation tests by Olurin et al. [135] (ALPORAS foam)

Bearing tests by Olurin et al. [135] (ALPORAS foam)

No

rmal

ized

tea

rin

g en

ergy

per

un

it f

ract

ure

are

a [(

)/

ys]

Relative density

Figure 4.7. Normalized tearing energy per unit fracture area-relative density of

different cellular materials.

123

4.3.2. The effect of strain rate, �̇�

4.3.2.1. Plateau stress

In the previous experimental study (chapter 3), honeycombs were crushed at

low and intermediate strain rates (1 × 10−3 to 1 × 102 s-1). FEA was conducted on

honeycombs at high strain rates (1 × 102 to 1 × 104 s-1). Both experimental and FEA

results are shown in Fig. 4.8, which demonstrates the influence of strain rate on the

plateau stress of two different honeycombs subjected to out-of-plane indentation

and compression loadings respectively. For both types of honeycombs the plateau

stress increased with strain rate in both indentation and compression. Due to the

higher t/l ratio, the plateau stress is larger for honeycomb H42 than honeycomb

H31.

10-3 10-2 10-1 100 101 102 103 104

1

10

100 H31- FEA

H31-Experiment [chapter 3]

H42- FEA


Pla

teau

str

ess

(MP

a)

Strain rate (s-1)

Indentation

(a)

124

10-3 10-2 10-1 100 101 102 103 104

1

10

100 H31- FEA


H42- FEA


CompressionP

late

au s

tres

s (M

Pa)

Strain rate (s-1)

(b)

Figure 4.8. Effect of strain rate on the plateau stresses of two types of honeycombs

subjected to: (a) indentation; (b) compression.

Experiments and FEA of compression of aluminum honeycombs were conducted

by various researchers [35, 44, 45, 58-60, 83]. In previous experimental studies

(chapter 3), enhancement in the plateau stress was observed at low and

intermediate loading velocities. Wang et al. [83] reported remarkable enhancement

of plateau stress at high impact velocity (20-80 ms-1). Goldsmith and Sackman [45]

found a 50 % enhancement in plateau stress at dynamic velocities up to 35 ms-1.

Zhao and Gary [44] observed significant enhancement in the plateau stress by

approximately 40 % when the loading velocity increased from quasi-static to

dynamic (2-28 ms-1). Similar enhancement of plateau stress with the loading

velocity was also discussed by Hou et al. [60] and Zhao et al. [59]. In order to

compare these results with the current FEA, plateau stresses of honeycombs were

125

normalized as (σpl/σys)/(t/l)1.5 and plotted in Fig. 4.9 in terms of strain rate. These

current FEA results show significant enhancement of plateau stress at high impact

velocities, which agree very well with the FEA results of Deqiang et al. [35].

10-3 10-2 10-1 100 101 102 103 104

1

10

100 H31-FEA (present analyses)

Wang et al. [83], (experiment)

Deqiang et al. [35], (FEA)

Goldsmith & sackman [45], (experiment)

Chapter 3, (experiment)

Zhao & Gary [44], (experiment)

Hou et al. [60], (experiment)

Zhao et al. [59], (experiment)

No

rmal

ized

pla

teau

str

ess

in c

om

pre

ssio

n

[(

pl/

ys)/

(t/l

)1.5

] (M

Pa)

Strain rate (s-1)

Figure 4.9. Normalized plateau stress of honeycomb-strain rate of honeycombs in

compression.

4.3.2.2. Energy dissipation

Figure 4.10 shows the effect of strain rate on the dissipated energy of two types

of honeycombs under indentation and compression loadings respectively. Similar to

the plateau stress, for two types of honeycombs the dissipated energy increased

with strain rate in both indentation and compression, while for honeycomb H42, the

dissipated energies in both indentation and compression were found to be larger

than those of honeycomb H31 due to the higher t/l ratio.

126

10-3 10-2 10-1 100 101 102 103 104102

103

104

105

Indentation H31- FEA


H42- FEA


Dis

sip

ated

en

ergy

(J)

Strain rate (s-1)

(a)

10-3 10-2 10-1 100 101 102 103 104102

103

104

105

Compression H31- FEA


H42- FEA


Dis

sip

ated

en

ergy

(J)

Strain rate (s-1)

(b)

Figure 4.10. Effect of high strain rate on the total dissipated energy of two types of

honeycombs: (a) indentation; (b) compression.

127

Tearing energy, which is the difference between the total energies dissipated in

indentation and compression, was plotted in Fig. 4.11. Due to the higher t/l ratio the

magnitude of tearing energy is larger for honeycomb H42 than for honeycomb H31

at the same strain rate. For both honeycombs, tearing energy increases with strain

rate. The fitted curve for tearing energy per unit fracture area for honeycomb at

different strain rates is shown in Fig. 4.12.

10-3 10-2 10-1 100 101 102 103 104101

102

103

104

H31- FEA


H42- FEA


Tea

rin

g en

ergy

(J)

Strain rate (s-1)

Figure 4.11. Effect of strain rate on the tearing energy of different honeycombs.

128

102 103 104

1

10 Tearing energy per unit fracture area

Best fitted lineE

t /A

t -

1

Strain rate (s-1)

Figure 4.12. The dependency of tearing energy per unit fracture area of

honeycombs and strain rate.

The relationship between the tearing energy per unit fracture area and the

relative density at high strain rates (beyond 1×102 s-1) is described by the following

equation.

𝐸𝑡

𝐴𝑡 = 1.37 × 103(𝜌 𝜌0⁄ )1.32(1 + 8.77 × 10−4𝜀̇1.03) (4.4)

4.3.3. Deformation pattern of aluminum honeycombs subjected to compression and indentation

Fig. 4.13 shows the enlarged isometric and front (sectional plane) views of

honeycomb H31 under out-of-plane indentation and compression loads. Three

images of deformation were taken at displacements of 0 mm, 20 mm and 40 mm

respectively from the animation of FEA by using LS-PrePost software. In Fig. 4.13

(a) it is seen that the progressive buckling of the cell wall occurs from both ends of

the honeycomb simultaneously and propagates to the middle region of the

129

honeycomb, which is similar to that observed in the previous experimental study

(chapter 3). Deformation mode is found to be independent of strain rate. Xu et al.

[39] also observed a negligible effect of strain rate on the buckling of honeycomb

cells in the out-of-plane compression.

In the previous experimental study, it was impossible to observe the

deformation of honeycomb under the indenter. In the current FEA, the deformation

of honeycomb in indentation is observed from the front sectional plane view, as

shown in Fig. 4.13 (b). It is found that the progressive buckling of cell walls initiates

from the top end of the honeycomb, which is immediately beneath the indenter, and

propagates in the same manner till densification. Progressive buckling takes place

in the middle portion of the honeycomb model underneath the indenter, which is

associated with the tearing of cell walls along the four edges of the indenter. No

significant difference is observed in the buckling pattern at different strain rates.

130

(a)

131

(b)

Figure 4.13. Deformation of honeycomb H31 at 5 ms-1: (a) compression; (b)

indentation.

4.4. Summary

In this chapter, finite element analysis (FEA) of different honeycomb models has

been developed using ANSYS LS-DYNA to study the mechanical behavior of

honeycombs under out-of-plane indentation and compression loads over a wide

range of high strain rates from 1 × 102 to 1 × 104 s-1. The FE models have been

132

validated by the previous experimental results (compression and indentation) in

terms of deformation, stress-strain curves, plateau stress and dissipated energy. A

reasonable agreement between the FEA and experimental results has been found

for both honeycombs H31 and H42.

It is found that the plateau stress, dissipated energy and tearing energy increases

with 𝑡 𝑙⁄ ratio. For a constant strain rate of 1×103 s-1, the plateau stresses increase

with 𝑡 𝑙⁄ ratio by power laws with exponents of 1.47 and 1.36 for compression and

indentation respectively.

Moreover, the plateau stress, dissipated energy and tearing energy increase

gradually for low and intermediate strain rates. Significant enhancement in the

plateau stress, dissipated energy and tearing energy is observed at high strain rates

for honeycombs subjected to either compression or indentation loads. An empirical

formula is proposed for the tearing energy per unit fracture area in terms of strain

rate and relative density of honeycombs.

The current FEA reveals that at velocities beyond 5 ms-1, under indentation,

plastic buckling of the honeycomb cell walls occurs from the end which is adjacent

to the indenter, while under compression the buckling of honeycomb cell walls

occurs from both ends of the honeycomb.

It is found that under quasi-static indentation, the empirical formula proposed

by Shi et al. [8] can be used for honeycombs as well.

134

Chapter 5. Quasi-static and dynamic experiments of

aluminum honeycombs under combined compression-

shear loading

This chapter investigates experimentally the mechanical response of aluminum

hexagonal honeycombs subjected to combined compression-shear loads. Three

types of HEXCELL® 5052-H39 aluminum honeycombs with different cell sizes and

wall thicknesses are studied. Both quasi-static and dynamic tests are conducted at

five different loading velocities ranging from 5×10-5 ms-1 to 5 ms-1 by using an MTS

and a high-speed INSTRON machine. Honeycombs are loaded in both TL and TW

planes at three different loading angles of 15, 30 and 45. The deformation of

honeycombs, crushing force, plateau stress and energy absorption are presented.

The effects of loading plane, loading angle and loading velocity are also discussed.

An empirical formula is proposed to describe the relationship between plateau

stress and loading angle. Moreover, experimental investigation is also conducted by

employing a triaxial load cell, which is able to measure the normal compressive and

shear forces of honeycombs subjected to combined compression-shear loads. Such

experiments are conducted at three different low velocities (5×10-5 ms-1 to 5×10-3

ms-1) at a loading angle of 15. Tests using the triaxial load cell at loading angles of

30 and 45 are not conducted because the corresponding triaxial load cell fixtures

are too large to be housed into the machines.

135

5.1. Experiment details

5.1.1. Specimens

A honeycomb structure has two in-plane (L and W) directions and one out-of-

plane (T) direction, as shown in Fig. 5.1. HEXCELL® hexagonal honeycombs made

by the corrugated method from aluminum alloy 5052-H39 sheets are used in this

study [20]. Each unit cell of a hexagonal honeycomb consists of four single walls and

two double walls. The double walls are formed by adhesively bonding two single

walls.

Figure 5.1. A photograph of aluminum honeycomb (4.2-3/8-5052-.003N). T is the

out-of-plane direction. L and W are the in-plane directions.

136

In this experimental study, three different types of aluminum hexagonal

honeycombs with different cell sizes and cell wall thicknesses are employed. The

different types of honeycombs are labelled as H31, H42 and H45, where their

respective cell sizes (distance between the two vertical double walls) are 4.763 mm,

9.525 mm and 3.175 mm. The specification of each type of honeycomb is listed in

Table 5.1. Cubic specimens with lengths of 50 mm in the T, L and W directions are

used. These dimensions enabled a minimum number of five cells in both the L and

W directions (to minimize the size effect) and obtain the bulk properties of the

honeycombs. Both Onck et al. [124] and Xu et al. [39] indicated that a minimum

number of cells should be included in honeycomb specimens in order to obtain

realistic bulk properties of the honeycombs. All specimens were carefully prepared

from honeycomb panels supplied by the manufacturer to ensure the specific

dimension was achieved without any deformation. Photographs of the three

different types of honeycomb specimens used in these experiments are shown in

Fig. 5.2.

137

Table 5.1. Specification of three types of aluminum hexagonal honeycombs [20]

Type Designation* Cell size, D

(mm)

Cell wall

length, l

(mm)

Single cell wall

thickness, t

(mm)

Cell wall

thickness to

edge length

ratio, t/l

Nominal Density,

ρ (kg/m3)

H31 3.1-3/16-5052-.001N 4.763 2.75 0.0254 0.00924 49.66

H42 4.2-3/8-5052-.003N 9.525 5.5 0.0762 0.0139 67.28

H45 4.5-1/8-5052-.001N 3.175 1.83 0.0254 0.0139 72.09

*In the above designation, 3.1, 4.2 and 4.5 are the nominal densities in pounds per cubic foot; 3/16, 3/8 and 1/8 are the cell sizes

(distance between two vertical walls) in inches; 5052 is the aluminum alloy grade; 0.001 or 0.003 is the nominal single wall thickness in

inches and N denotes non-perforated cell walls. Data provided by the manufacturer [20].

138

Figure 5.2. A photograph of three types of honeycomb specimens used in combined

compression-shear tests.

5.1.2. MTS and high speed INSTRON machines

Quasi-static and dynamic combined compression-shear tests at different

velocities were conducted using an MTS machine and a high-speed INSTRON (VHS

8800) testing machine. Constant loading velocities were achieved on both machines

for all the tests conducted. Tests at velocities of 5×10-5 ms-1, 5×10-4 ms-1 and 5×10-3

ms-1 were conducted on the MTS machine. Tests at velocities of 5×10-1 ms-1 and 5

ms-1 were conducted on the high-speed INSTRON machine. The maximum load

capacity of the MTS and INSTRON machines were 250 kN and 100 kN respectively.

For the MTS machine, the loading block was connected to the top cross-head of the

machine and moved downwards to load specimens (Fig. 5.3). For the high-speed

INSTRON machine, the loading block was connected to the lower moving piston,

which moved upward to load the specimen against the upper support block as

shown in Fig. 5.4. For both the machines, two specimens were placed on the two

surfaces of the support block at the same distance away from the centre of the

fixture to avoid transverse loading. A thin layer of double-sided sticky tape was used

to fix one end of the specimen to the surface of the support block to ensure that the

139

two specimens were placed symmetrically on the support block and to prevent

slippage of the specimens during crushing. Both the MTS and INSTRON machines

were connected to computers which recorded the force and displacement histories.

5.1.3. Fixtures

Three different loading angles of 15, 30 and 45 were employed in the

combined compression-shear tests by using three different sets of fixtures made of

mild steel, AISI1045 (see Figs. 5.3 and 5.4). Two identical specimens were crushed

simultaneously to ensure no transverse load was applied to the testing machine and

to avoid potential damage to the machine. Each set of fixtures consisted of two

blocks—a support block and a loading block. Two sliding guide rods made from

stainless steel were located on the two sides of the blocks as shown in Fig. 5.5. These

two sliding guide rods were securely fitted to one block (support block), and the

other block (loading block) was able to slide in the vertical direction. The sliding

guide rods were to prevent any transverse movement of one block with respect to

the other. To reduce friction between the sliding guide rods and the loading block,

brass rings were press-fitted inside two cylindrical channels in the sliding loading

block. Lubricant oil was sprayed around the sliding rods and channels to minimize

friction.

The 15 and 30 angle fixtures were used on two different universal testing

machines (an MTS and a high-speed INSTRON) to conduct both quasi-static and

dynamic tests at different loading velocities. The 45 angle fixture was used only for

quasi-static tests (MTS machine), because the fixture dimensions were too large to

be housed in the high-speed INSTRON machine. Figs. 5.3 and 5.4 show the three

different sets of fixtures used for the MTS machine and the two sets of fixtures used

140

for the high-speed INSTRON machine. During testing, combined compression-shear

loads were applied in two different plane orientations—designated as TL plane and

TW plane in Fig. 5.1. Honeycomb specimens were compressed in the T direction and

sheared along either the L direction (known as TL plane) or in the W direction

(known as TW plane).

(a)

141

(b)

(c)

Figure 5.3. Photographs of three sets of fixtures used on MTS machine for combined

compression-shear tests at three different loading angles of: (a) 15; (b) 30; (c) 45.

(a)

142

(b)

Figure 5.4. Photographs of two sets of fixtures used on high-speed INSTRON

machine for combined compression-shear tests at two different loading angles of:

(a) 15; (b) 30.

Figure 5.5. A photograph of testing fixture showing sliding guide rods.

143

5.1.4. Triaxial load cell set-up

In this section a modified experimental set-up is proposed which enables the

measurement of normal compression and shear forces of honeycombs subjected to

a combined compression-shear load at 15 loading angle. In this experimental set-

up a triaxial load cell was employed which facilitates the measurement of forces

acting in different directions. Tests at constant loading velocities of 5×10-5 ms-1,

5×10-4 ms-1 and 5×10-3 ms-1 were conducted on the MTS machine. The experimental

set-up on the MTS machine is shown in Fig. 5.6. The support block was connected to

the top cross-head of the machine and fixed in that position (Fig. 5.6). The loading

block could move upward to load the specimens at constant velocities. The MTS

machine was connected to computers which recorded the force and displacement

histories.

Figure 5.6. Experimental set-up of combined compression-shear tests on the MTS

machine.

144

To make the experimental set-up symmetric, one dummy load cell was made

from mild steel. Both the triaxial load cell and dummy load cell were bolted on the

surface of the loading block. Two identical specimens were placed on the two

surfaces of the triaxial load cell and dummy load cell at the same distance away from

the centre of the fixture to avoid transverse loading. To prevent slippage of the

specimen during testing, a very thin layer of double-sided sticky tape was used to fix

one end of the specimen to the surface of the loading block. Two sliding guide rods

were located on the two sides of the blocks as shown in Fig. 5.6. These two sliding

guide rods were securely fitted to one block (support block), and the other block

(loading block) could slide in the vertical direction. The sliding guide rods were to

prevent any transverse movement of one block with respect to the other. To reduce

friction between the sliding guide rods and the loading block, brass rings were

press-fitted inside two cylindrical channels in the sliding loading block. Lubricant oil

was sprayed around the sliding rods and channels to minimize friction.

The triaxial load cell (Fig. 5.7) was manufactured by Kistler (3-component force

link-type 9377C). This triaxial load cell is able to measure three force components

(Fx, Fy and Fz). The maximum load capacities of the load cell are 75 kN in the X and Y

directions and 150 kN in the Z direction. In this experimental set-up the Z direction

is the compression direction and the X direction is the shear direction. The

compression and shear forces applied to honeycombs can therefore be measured

directly by this triaxial load cell. A charge amplifier was used to output the voltage

signals from the load cell. A controller unit was connected between the computer

and charge amplifier to convert the voltage into force.

145

Figure 5.7. A photograph of Kistler triaxial load cell (3-component force link-type

9377C).

5.2. Results

5.2.1. Deformation patterns

A digital reflex camera (CANON DSLR) and a high-speed camera (FASTCAM APX

RS) were used to capture and record the deformation of honeycomb specimens in

quasi-static and dynamic tests respectively. As the two nominally identical

specimens were placed symmetrically on the fixture and crushed simultaneously

during each test, an almost identical deformation pattern was observed in each of

these two specimens.

Although loading angles varied from 15, 30 to 45, similar deformation

patterns were observed for all types of honeycombs. Moreover, no significant

difference in the collapse mechanism of honeycombs loaded in the TL and TW

planes was observed. A typical crushing process is shown diagrammatically in Fig.

146

5.8. Honeycombs were compressed in the out-of-plane (T) direction, producing

buckling (elastic or plastic buckling) of the cell walls that initiated from both the top

and bottom ends and propagated to the middle of honeycomb specimens (Fig. 5.8).

Simultaneously, honeycomb cell walls tilted due to the shear force applied at the top

and bottom surfaces of the honeycomb specimens. Due to the presence of double-

sided sticky tape between the specimen and support block, only Mode 2 proposed

by Hou et al. [111, 112] and Tounsi et al. [114, 115] was observed. When

honeycombs were loaded in the TL or TW plane, the angle between the axis of the

honeycomb and the L or W direction varied, and was no longer a right angle. The

angle between the tilted cell walls and the normal displacement direction is defined

as rotational angle, 𝛽, as shown in Fig. 5.9. The rotational angle (𝛽) was 00 before

any load was applied. Once the crushing of honeycomb occurred, the rotational

angle (𝛽) increased due to the shear force. From the images captured by the camera,

the rotational angle (𝛽) was measured manually by using a protractor.

Measurement of rotational angle (𝛽) was made in the middle part of honeycomb

specimens to avoid any edge effect.

Moreover, when loading velocity varied from 5×10-5 ms-1 to 5 ms-1, the

deformation pattern was almost the same for the same type of honeycomb loaded

in the same plane and at the same loading angle. Hou et al. [110] reported similar

deformation patterns in their combined compression-shear tests of aluminum

honeycombs at an impact velocity of 15 ms-1, which indicated that deformation

pattern did not change with loading velocity when honeycombs were under

combined compression-shear loads.

147

0 mm

3 mm

6 mm

9 mm

12 mm

15 mm

18 mm

21 mm

24 mm

27 mm

30 mm

33 mm

36 mm

39 mm

42 mm

45 mm

48 mm

51 mm

54 mm

57 mm

60 mm

Figure 5.8. Crushing process of H31 honeycomb in TL plane at 45 loading angle

under combined compression-shear load at a velocity of 5×10-3 ms-1. Displacement

indicated is vertical cross-head movement.

148

Figure 5.9. Deformation of H31 honeycomb crushed in TL plane at 45loading angle

at 5×10-3 ms-1.

Photographs of honeycomb specimens after combined compression-shear tests

at different loading angles (15, 30 and 45) and in different planes (TL and TW) at

a velocity of 5 ms-1 are shown in Fig. 5.10. With the increase of loading angle, it was

observed that rotation of cell walls (angle 𝛽) increased for all types of honeycombs

crushed in both planes. Larger angles of rotation were observed at a loading angle

of 45 compared with a loading angle of 15. For loading in both the TL and TW

planes, honeycomb cell walls in the central area deformed in a uniform pattern due

to the constraint from adjacent cell walls. However, along the four edges of the

honeycomb specimens, the cell walls deformed in an irregular pattern. Moreover,

honeycomb cell walls crushed in the TW plane were compacted more than those

crushed in the TL plane, for the same loading angles and velocities. This was due to

the double wall thickness in the W direction.

149

Figure 5.10. Photographs of three types of honeycombs tested under combined compression-shear loads at different loading angles and

in different planes.

150

5.2.2. Rotation of cell walls

The rotation of cell walls was measured from photographs taken during the

combined compression-shear tests. Rotational angles were measured at every 3 mm

displacement from 0 to 45 mm. The effect of loading velocity on the rotational angle

of honeycomb specimens is shown in Fig. 5.9 when loading in two planes (TL and

TW) and at a loading angle of 30. At 0 mm displacement, the rotational angle 𝛽 was

0, there being no deformation. After crushing was initiated, the honeycomb cell

walls were found to have rotated due to the applied shear force. It was found that

the rotational angle increased monolithically with cross-head displacement at all

loading velocities. The highest rotational angles measured were between 60 and

65 at a displacement of 45 mm. No significant influence of loading velocity on

rotational angle was observed for loading in both TL and TW planes.

Using simple kinematics and assuming uniform compression and shear strains,

the rotational angle (𝛽) and displacement (𝑑) can be estimated as 𝛽 =

arctan (𝑑 sin α

ℎ−𝑑 cos α), where 𝛼 is the loading angle and h is the original height of

honeycomb specimen. In this study, h=50 mm. and when 𝛼 = 30°, 𝛽 =

arctan (𝑑

100−√3𝑑). This calculated curve is plotted and compared with the

experimental measurement in Fig. 5.9, from which it can be seen that the

experimental measurement matches well with the estimation.

151

(a)

(b)

Figure 5.11. Effect of loading velocity on rotational angle β of honeycomb H31

loaded at 30 in: (a) TL plane; (b) TW plane.

152

Fig. 5.12 shows the effect of loading angle on rotational angle 𝛽 of cell walls for

honeycomb H31 at a loading velocity of 5×10-3 ms-1. By substituting 𝛼 = 15° and

𝛼 = 45° into 𝛽 = tan−1 (d sin α

h−d cos α), the rotational angle (𝛽) was calculated and

plotted in Fig. 10, showing good agreement with the experimental curves. It was

found that the rotational angle 𝛽 increased with loading angle when honeycomb

specimens were crushed in both the TL and TW planes. The highest and lowest

rotational angles were observed at loading angles of 45 and 15 respectively.

(a)

153

(b)

Figure 5.12. Effect of loading angle on rotational angle β of honeycomb H31 at a

velocity of 5×10-3 ms-1 in the: (a) TL plane; (b) TW plane.

5.2.3. Force-displacement curves and the effects of loading angle and plane on crushing force

As mentioned previously, combined compression-shear loads were applied at

three different loading angles (15, 30 and 45) and in two different planes (TL and

TW) for each type of honeycomb specimen. All honeycomb specimens were fully

compressed up to densification (i.e., when the load increased significantly). Quasi-

static and dynamic vertical force-displacement curves of three different types of

honeycombs at different loading angles and in different loading planes are shown in

Figs. 5.13-5.15. The vertical force-displacement curves of honeycombs H42 and H45

are similar to those of honeycomb H31, but the magnitudes of force are larger. Great

oscillations for the test results at velocity 5 ms-1 were observed. The possible

154

reasons are the resonance of the load cell as well as the vibration of specimen and

fixture subjected to dynamic loading.

(a)

(b)

155

(c)

(d)

156

(e)

(f)

Figure 5.13. Vertical force-displacement curves for honeycomb subjected to

combined compression-shear loads at a loading angle of 15: (a) H31 in TL plane;

(b) H31 in TW plane; (c) H42 in TL plane; (d) H42 in TW plane; (e) H45 in TL plane;

(f) H45 in TW plane.

157

(a)

(b)

158

(c)

(d)

159

(e)

(f)





160

(a)

(b)

161

(c)

(d)

162

(e)

(f)





163

All three types of aluminum honeycombs under combined compression-shear

loads demonstrated three phases of deformation: (1) Linear phase, where crushing

force increased linearly with displacement up to a peak crushing force; (2) Plateau

phase, where crushing force remained almost constant (for 15 loading angle); or

decreased gradually (for 30 loading angle); or decreased significantly (for 45

loading angle); (3) Densification phase, where crushing force increased significantly

due to densification of the honeycomb. For different loading velocities, similar

trends in the force-displacement were found for all types of honeycomb specimens.

The effects of loading angle and loading plane on crushing force are studied in

this section. From Fig. 5.13 it can be observed that for a 15 loading angle, the

crushing force in the plateau phase is nearly constant for honeycomb H31. From Fig.

5.14 it can be seen that at 30 loading angle, the crushing force in the plateau phase

decreases slightly with displacement. Furthermore, at an even larger loading angle

of 45 (Fig. 5.15) the crushing force in the plateau phase decreases significantly with

displacement. It was previously mentioned in Section 5.3.2 that for these tests, cell

rotational angle increased with cross-head displacement and loading angle. The

highest and the lowest rotation of cell walls were found at loading angles of 45 and

15 respectively. At a 45 loading angle, due to this cell wall rotation, the previously

vertical walls in the T direction leaned over and became easier to crush; hence the

crushing force decreased. Therefore, a significant decrease of plateau force was

observed with increasing displacement prior to densification for loading in both TL

and TW planes. At a lower loading angle of 30, a slight decrease in the plateau force

was found (Fig. 5.14).

164

Figure 5.16 shows the effect of loading angle on the force-displacement curves

of honeycomb H31, H42 and H45 for loading in both the TL and TW plane at a

loading velocity of 5×10-3 ms-1. It was observed that with an increase of loading

angle, the crushing force in the plateau region decreased.

(a)

(b)

165

(c)

(d)

166

(e)

(f)

Figure 5.16. Effect of loading angle on the vertical force-displacement curves of

different honeycombs at a loading velocity of 5×10-3 ms-1: (a) H31 loaded in the TL

plane; (b) H31 loaded in the TW plane; (c) H42 loaded in the TL plane; (d) H42

loaded in the TW plane; (e) H45 loaded in the TL plane; (f) H45 loaded in the TW

plane.

167

In all cases, the plateau force was calculated from the force-displacement curves

over a cross-head vertical displacement of 5 mm to 40 mm. For all types of

honeycomb specimen (H31, H42 and H45) it was observed that under combined

compression-shear loading at any particular angle the plateau force was slightly

lower in the TL plane than that in the TW plane. If we consider a particular loading

angle of 15 and loading velocity of 5×10-5 ms-1 for the three different types of

honeycombs, then the plateau forces in TL and TW planes for H31 honeycomb

specimens were 3.95 kN and 4.20 kN respectively. Similarly, for the same loading

conditions, the plateau forces for H42 honeycomb were 6.80 kN and 6.95 kN

respectively. For H45 honeycomb under the same loading plateau forces in the TL

and TW planes were 8.50 kN and 8.85 kN respectively. A similar trend was observed

at different loading velocities and loading angles for all types of honeycomb

specimens. This may be caused by the orientation of double ply cell walls in the

honeycomb structure. The in-depth investigation on the deformation mechanism is

required to be conducted in the future to interpret this phenomenon.

Figure 5.17 shows all force components acting on a honeycomb specimen in a

combined compression-shear test. The force mentioned above is 2𝐹𝑣 (due to two

specimens) shown in Fig. 5.17. The relations between the force components are as

follows:

𝐹ℎ = 𝐹𝑛 sin 𝜃 − 𝐹𝑠 cos 𝜃 (5.1a)

𝐹𝑣 = 𝐹𝑛 cos 𝜃 + 𝐹𝑠 sin 𝜃 (5.1b)

where, 𝐹ℎ and 𝐹𝑣 are the horizontal and vertical forces respectively, 𝐹𝑛 and 𝐹𝑠 are

the normal and shear force components, and 𝜃 is the loading angle. In the current

experimental set up, only the vertical force (𝐹𝑣) was measured, as the horizontal

168

force (𝐹ℎ) was unknown. Therefore it was not possible to separate the normal

compressive force (𝐹𝑛) and shear force (𝐹𝑠). A triaxial load cell which can measure

both 𝐹𝑣 and 𝐹ℎ simutaneously will be employed in Section 5.2.7. 𝐹𝑛 and 𝐹𝑠 will be

calculated (using Eqs. 5.1a and 5.1b) and discussed in detail then.

Figure 5.17. Sketch of the force components in a combined compression-shear test.

5.2.4. The effect of loading angle on plateau stress

Plateau stress in this study is defined as plateau force divided by the original

cross sectional area of the honeycomb specimen, 𝜎𝑝𝑙 = 𝑃 𝐴⁄ . The cross sectional

area of each specimen was 50 mm × 50 mm= 2500 mm2. Since two almost identical

specimens were crushed simultaneously in each test, the total cross sectional area

per test, A, was 5000 mm2.

169

A summary of plateau stress for each type of honeycomb tested is listed in Tables

5.2 – 5.4. It was found that with the increase of loading angle, the plateau stress of

honeycombs decreases. The possible reason is the larger rotation of cell walls at

larger loading angle as shown in Fig. 5.12. However, this requires further detailed

investigation.

Among the three types of honeycombs, for the same loading velocity and loading

angle, the largest and smallest plateau stresses were found in honeycombs H45 and

H31 respectively. This is due to the largest density of honeycomb H45 and the lowest

density of honeycomb H31 (see Table 5.1). This finding is consistent with previous

studies [22, 30] that the plateau stress of honeycombs under pure compression

increases with nominal density (kg/m3) or t/l ratio.

Table 5.2. Quasi-static and dynamic experimental results for honeycomb H31

Test No. Loading

Angle

Loading

Plane

Loading

Velocity

Plateau

Stress

Dissipated

Energy

ms-1 MPa J

H31-TL-1 15 TL 5×10-5 0.79 160

H31-TL-3 15 TL 5×10-4 0.8 162

H31-TL-5 15 TL 5×10-3 0.81 164

H31-TL-7 15 TL 5×10-1 0.86 172

H31-TL-9 15 TL 5 0.89 180

H31-TW-2 15 TW 5×10-5 0.84 171

H31-TW-4 15 TW 5×10-4 0.89 181

H31-TW-6 15 TW 5×10-3 0.91 185

H31-TW-8 15 TW 5×10-1 0.94 191

H31-TW-10 15 TW 5 0.99 201

H31-TL-11 30 TL 5×10-5 0.66 137

H31-TL-13 30 TL 5×10-4 0.68 139

170

H31-TL-15 30 TL 5×10-3 0.70 141

H31-TL-17 30 TL 5×10-1 0.78 161

H31-TL-19 30 TL 5 0.82 165

H31-TW-12 30 TW 5×10-5 0.71 146

H31-TW-14 30 TW 5×10-4 0.76 156

H31-TW-16 30 TW 5×10-3 0.77 158

H31-TW-18 30 TW 5×10-1 0.8 164

H31-TW-20 30 TW 5 0.87 179

H31-TL-21 45 TL 5×10-5 0.46 100

H31-TL-23 45 TL 5×10-4 0.48 104

H31-TL-25 45 TL 5×10-3 0.5 108

H31-TW-22 45 TW 5×10-5 0.47 102

H31-TW-24 45 TW 5×10-4 0.49 107

H31-TW-26 45 TW 5×10-3 0.52 113


Test No. Loading

Angle

Loading

Plane

Loading

Velocity

Plateau

Stress

Dissipated

Energy

ms-1 MPa J

H42-TL-1 15 TL 5×10-5 1.36 277

H42-TL-3 15 TL 5×10-4 1.4 285

H42-TL-5 15 TL 5×10-3 1.47 292

H42-TL-7 15 TL 5×10-1 1.51 301

H42-TL-9 15 TL 5 1.56 312

H42-TW-2 15 TW 5×10-5 1.39 281

H42-TW-4 15 TW 5×10-4 1.47 294

H42-TW-6 15 TW 5×10-3 1.5 300

H42-TW-8 15 TW 5×10-1 1.54 311

H42-TW-10 15 TW 5 1.58 320

171

H42-TL-11 30 TL 5×10-5 1.17 239

H42-TL-13 30 TL 5×10-4 1.26 257

H42-TL-15 30 TL 5×10-3 1.28 261

H42-TL-17 30 TL 5×10-1 1.35 275

H42-TL-19 30 TL 5 1.47 291

H42-TW-12 30 TW 5×10-5 1.21 247

H42-TW-14 30 TW 5×10-4 1.32 269

H42-TW-16 30 TW 5×10-3 1.36 277

H42-TW-18 30 TW 5×10-1 1.42 289

H42-TW-20 30 TW 5 1.49 301

H42-TL-21 45 TL 5×10-5 0.96 190

H42-TL-23 45 TL 5×10-4 0.98 194

H42-TL-25 45 TL 5×10-3 1.02 201

H42-TW-22 45 TW 5×10-5 1.05 208

H42-TW-24 45 TW 5×10-4 1.11 210

H42-TW-26 45 TW 5×10-3 1.16 214


Test No. Loading

Angle

Loading

Plane

Loading

Velocity

Plateau

Stress

Dissipated

Energy

ms-1 MPa J

H45-TL-1 15 TL 5×10-5 1.7 341

H45-TL-3 15 TL 5×10-4 1.73 347

H45-TL-5 15 TL 5×10-3 1.8 359

H45-TL-7 15 TL 5×10-1 1.85 369

H45-TL-9 15 TL 5 1.89 378

H45-TW-2 15 TW 5×10-5 1.77 351

H45-TW-4 15 TW 5×10-4 1.82 361

H45-TW-6 15 TW 5×10-3 1.85 371

172

H45-TW-8 15 TW 5×10-1 1.91 376

H45-TW-10 15 TW 5 1.96 388

H45-TL-11 30 TL 5×10-5 1.42 290

H45-TL-13 30 TL 5×10-4 1.44 294

H45-TL-15 30 TL 5×10-3 1.45 296

H45-TL-17 30 TL 5×10-1 1.69 345

H45-TL-19 30 TL 5 1.72 344

H45-TW-12 30 TW 5×110-5 1.51 302

H45-TW-14 30 TW 5×10-4 1.54 312

H45-TW-16 30 TW 5×10-3 1.59 320

H45-TW-18 30 TW 5×10-1 1.76 352

H45-TW-20 30 TW 5 1.81 362

H45-TL-21 45 TL 5×10-5 1.08 218

H45-TL-23 45 TL 5×10-4 1.1 221

H45-TL-25 45 TL 5×10-3 1.13 228

H45-TW-22 45 TW 5×10-5 1.17 234

H45-TW-24 45 TW 5×10-4 1.17 234

H45-TW-26 45 TW 5×10-3 1.19 238

Plateau stress ratio is defined as the ratio between plateau stress at different

loading angles of 15, 30 and 45 and the plateau stress at 0, for a loading velocity

of 5×10-5 ms-1; i.e., plateau stress ratio = 𝜎𝑝𝑙 𝜃

𝜎𝑝𝑙 0 . Figure 5.18 shows the effect of

loading angle on this plateau stress ratio. A best-fitted line is also plotted in Fig. 5.18

for the different honeycombs (H31, H42 and H45). The relationship between plateau

stress ratio, 𝜎𝑝𝑙 𝜃

𝜎𝑝𝑙 0 and loading angle, θ (rad) is described by Eq. (5.2):

𝜎𝑝𝑙 𝜃

𝜎𝑝𝑙 0 = 1 − 0.95𝜃0.51 (5.2)

173

Figure 5.18. Effect of loading angle on plateau stress ratio for different honeycombs.

5.2.5. The effect of loading velocity on the plateau stress

The effect of loading velocity on plateau stress for different types of honeycombs

is shown in Fig. 5.19. The plateau stress was found to increase exponentially with

loading velocity for all loading angles (15, 30 and 45) in both the TL and TW

planes.

174

(a)

(b)

175

(c)

(d)

176

(e)

(f)

Figure 5.19. Effect of loading velocity on plateau stress at different loading angles

and in different planes: (a) H31 in TL plane; (b) H31 in TW plane; (c) H42 in TL

plane; (d) H42 in TW plane; (e) H45 in TL plane; (f) H45 in TW plane.

177

The Normalized plateau stress ratio is defined as the ratio between plateau

stresses at different loading velocities to the plateau stress at a loading velocity of

5×10-5 ms-1, 𝜎𝑑

𝜎𝑞𝑠 , where 𝜎𝑑 and 𝜎𝑞𝑠 are the dynamic and quasi-static plateau stresses

respectively. Figure 5.20 shows the effect of loading velocity on the normalized

plateau stress ratio at different loading angles for honeycomb H31, H42 and H45

loaded in both TL and TW plane. At all loading angles it was found that the

normalized plateau stress ratio increased with loading velocity. Moreover, the

enhancement was found to increase with loading angle. The highest and lowest

enhancements were observed at loading angles of 45 and 0 respectively.

(a)

178

(b)

(c)

179

(d)

(e)

180

(f)

Figure 5.20. Effect of loading velocity on normalized plateau stress ratio at different

loading angles for honeycombs: (a) H31 in TL plane; (b) H31 in TW plane; (c) H42

in TL plane; (d) H42 in TW plane; (e) H45 in TL plane; (f) H45 in TW plane.

5.2.6. Energy dissipation under combined compression-shear load

Energy absorbed by the honeycomb specimens was calculated from the force-

displacement curves. The total energy dissipated by all types of honeycomb

specimens was calculated for a vertical displacement of 0 mm to 40 mm (prior to

the densification phase). A summary of total dissipated energy for each type of

honeycomb at different loading velocities, loading angles and in different planes is

listed in Tables 5.2–5.4. Similar to our observations for plateau stress, the total

dissipated energy increased with loading velocity at all loading angles for all types

of honeycombs.

181

Specific energy is defined as the total dissipated energy divided by the mass of

the honeycomb specimen. All test specimens were nominally of the same size and

average mass for each type of honeycomb. The average mass for honeycombs H31,

H42 and H45 were 5.6 g, 9.6 g and 10.051 g, respectively. Figure 5.21 shows the

influence of loading velocity on the specific energy for the three types of

honeycombs at different loading angles and in different planes. Similar to our

finding regarding plateau stress, it was found that a power law relationship exists

between specific energy and loading velocity for all types of honeycombs tested. At

any particular loading velocity, the largest specific energy was found in honeycomb

H45, which had the highest nominal density, whereas the lowest specific energy was

found in honeycomb H31, which had the smallest nominal density among the three

types of honeycomb studied.

(a)

182

(b)

(c)

183

(d)

(e)

184

(f)

Figure 5.21. Effect of loading velocity on specific energy at different loading angles

and in different planes (a) H31 in TL plane; (b) H31 in TW plane; (c) H42 in TL plane;

(d) H42 in TW plane; (e) H45 in TL plane; (f) H45 in TW plane.

5.2.7. Measurement of normal compression and shear forces using a triaxial load cell

In this section the results of the compression and shear forces measured by a

triaxial load cell are discussed. Three types of honeycombs (H31, H42 and H45)

were loaded at 15 loading angle and different low loading velocities (5×10-5, 5×10-

4 and 5×10-3 ms-1). Figure 5.22 shows the force-displacement curves of different

honeycombs loaded in both the TL and TW planes. This is the first time that the

normal compression and shear forces were directly measured. Moreover, from the

measured compression and shear forces, the vertical force could be calculated by

using Eq. 5.1(b). On the other hand, the vertical force was measured by the uniaxial

load cell fitted with the MTS machine. When comparing the calculated and measured

vertical force, nearly identical curves were found.

185

(a)

(b)

186

(c)

(d)

187

(e)

(f)

Figure 5.22. Force-displacement curves of honeycombs subjected to combined

compression-shear load at 15° loading angle: (a) H31 in the TL plane; (b) H31 in the

TW plane; (c) H42 in the TL plane; (d) H42 in the TW plane; (e) H45 in the TL plane;

(f) H45 in the TW plane.

188

The normal compressive and shear plateau stresses of different honeycombs at

different quasi-static loading velocities are listed in Tables 5.5-5.7. In Section 5.3.5

it has been discussed that with the increase of loading velocity, the vertical plateau

stress increased. Similar phenomena were also found in both normal compressive

and shear plateau stresses which increased with loading velocity for all the

honeycombs. Among the three, Honeycomb H45 has the greatest nominal density

(72.09 kg/m3) and t/l ratio, while honeycomb H31 has the smallest nominal density

(49.66 kg/m3) and t/l ratio. It is known that plateau stress increases with nominal

density and t/l ratio [22, 30]. Due to this, the largest and smallest plateau stresses

(normal compressive and shear) were found in honeycombs H45 and H31

respectively at a particular loading velocity.

Table 5.5. Normal compressive and shear results of honeycomb H31 at different

loading velocities

Exp. no Loading angle

Loading plane

Loading velocity

Normal compressive

plateau stress

Shear plateau stress

ms-1 MPa MPa H31-T-1 15 TL 5×10-5 0.79 0.11

H31-T-2 15 TW 5×10-5 0.82 0.12

H31-T-3 15 TL 5×10-4 0.80 0.10

H31-T-4 15 TW 5×10-4 0.91 0.13

H31-T-5 15 TL 5×10-3 0.86 0.12

H31-T-6 15 TW 5×10-3 0.94 0.15

189


loading velocities


Loading plane

Loading velocity

Normal compressive

plateau stress


ms-1 MPa MPa H42-T-1 15 TL 5×10-5 1.44 0.18

H42-T-2 15 TW 5×10-5 1.53 0.22

H42-T-3 15 TL 5×10-4 1.46 0.18

H42-T-4 15 TW 5×10-4 1.54 0.20

H42-T-5 15 TL 5×10-3 1.50 0.22

H42-T-6 15 TW 5×10-3 1.56 0.23


loading velocities


Loading plane

Loading velocity

Normal compressive

plateau stress


ms-1 MPa MPa

H45-T-1 15 TL 5×10-5 1.79 0.17

H45-T-2 15 TW 5×10-5 1.88 0.22

H45-T-3 15 TL 5×10-4 1.84 0.19

H45-T-4 15 TW 5×10-4 1.94 0.26

H45-T-5 15 TL 5×10-3 1.86 0.21

H45-T-6 15 TW 5×10-3 1.93 0.26

5.3. Summary

In this chapter, the mechanical response of three different types of aluminum

honeycombs under combined compression-shear loads has been investigated

experimentally. The three aluminum honeycombs tested had different cell sizes and

wall thicknesses. Both quasi-static and dynamic combined compression-shear loads

have been applied at different loading velocities from 5×10-5 to 5 ms-1. The load has

been applied at three different loading angles of 15, 30 and 45 by employing three

sets of specially designed fixtures.

190

During the crushing process, compression and shear occurred simultaneously in

the honeycomb specimens. As the specimens were compressed, elastic or plastic

buckling of the honeycomb cell walls has occurred, starting from both ends of the

specimen. Shear has occurred through rotation of the cell walls. The rotational angle

of the honeycomb cell walls has been measured and found to increase with vertical

displacement of the machine‘s cross-head. No significant effect of loading velocity

has been observed on cell rotation angle. However, the rotation angle has been

found to increase with increasing loading angle.

It has been found that the shape of the force-displacement curves is similar for

all honeycombs crushed at the same loading angle. In the plateau region, the

crushing force remained almost constant for a loading angle of 15°; the force

decreased slightly for a loading angle of 30°; and the force decreased significantly at

a loading angle of 45°. For the same loading velocity and loading angle, the plateau

force has been found to be slightly larger in the TW plane than in the TL plane.

The plateau stresses of the aluminum honeycombs tested has been found to

decrease with loading angle and increase with loading velocity. An empirical

formula has been proposed to describe the relationship between plateau stress and

loading angle. Both the plateau stress ratio, 𝜎𝑝𝑙 𝜃

𝜎𝑝𝑙 0 and the normalized plateau stress

ratio have been found to increase with both loading angle and loading velocity. For

the same loading plane, both plateau stress and energy dissipation increased with

loading angle and loading velocity.

Furthermore by employing a triaxial load cell in the experimental set-up, normal

compressive and shear force components of the applied combined load have been

measured directly at the 15° loading angle. It has been found that both the normal

191

compressive and shear stresses increased with loading velocity. Due to the

limitation of the experimental set-up (i.e., the fixtures for 30 and 45° are too large to

be housed in the testing machines), it has not been possible to conduct similar tests

at loading angles of 30 and 45° and higher loading velocities. These will be further

studied by using finite element analysis in the subsequent chapter (Chapter 6).

193

Chapter 6. Numerical simulation of aluminum honeycomb

subjected to combined compression-shear loads

The previous chapter (Chapter 5) discussed the experimental work of aluminum

honeycombs subjected to combined compressive-shear loads. In this chapter, a

numerical simulation of honeycombs subjected to combined compression-shear

will be carried out using ANSYS LS-DYNA. The finite element (FE) models of

honeycombs will be verified by the experimental results in terms of deformation

mode, rotation of cell wall and plateau stress at different loading angles. Verified FE

models will be used to calculate the compressive and shear stresses of honeycombs

at various loading angles and loading velocities. Crushing envelopes of honeycombs

will thereafter be proposed. The effects of honeycomb cell wall to edge length ratio

(𝑡/𝑙) and loading velocity will be discussed as well.

6.1. Finite element modelling

In this chapter, two full scale FE models of aluminum hexagonal honeycomb

were developed using ANSYS LS-DYNA [131]. The cell size and cell wall thickness of

two types of honeycombs simulated, H31 and H42, are the same as those in the

previous experiments (Chapter 5) and shown in Table 6.1.

194

Table 6.1. Specification of aluminum honeycombs simulated [20]

Type Material description* Cell

size, D

Single cell wall

thickness, t

Cell wall thickness

to edge length

ratio, t/l

mm mm

H31 3.1-3/16-5052-.001N 4.763 0.0254 0.00924

H42 4.2-3/8-5052-.003N 9.525 0.0762 0.0139

*In the material description, 3.1 and 4.2 are the nominal densities in pounds per

cubic foot, 3/16, 3/8 are the cell size in inches, 5052 is the aluminum alloy grade,

0.001 or 0.003 is the nominal foil thickness in inches and N denotes non-perforated

cell walls. Data were provided by the manufacturer.

The dimensions of honeycombs simulated are the same as the specimens used

in the experiments (50 mm × 50 mm × 50 mm). Similar to the experimental

specimen, each cell of the simulated honeycomb consists of four single walls and two

double walls. In the physical honeycomb specimens, adhesive was used to bond two

single walls together to form the double walls. However, in the FE models, double

walls were set-up by doubling the thickness of single walls without any adhesive. A

typical honeycomb model developed is shown in Fig. 6.1.

To mesh the honeycombs, quadrilateral mapped elements with a size of 0.7 mm

were selected, which was the same as that used in the simulation of compression

and indentation honeycombs in Chapter 4. A convergence test was conducted and

results confirmed this element size. Belytschko-Tsay Shell 163 elements with five

integration points were employed to simulate the honeycomb cell walls for high

computational efficiency. Two sets of real constants for thickness were defined for

195

the single walls and double walls respectively. The detailed material properties used

in the FE model for aluminum honeycombs are listed in Table 6.2.

Figure 6.1. A finite element model of honeycomb (H31).

The support and loading blocks were simulated using rigid bodies. The detailed

material properties used in the FE models for rigid bodies are listed in Table 6.2. All

contacts in the FE models were defined as SURFACE_TO_SURFACE to avoid any

penetration.

196

Table 6.2. Material properties of aluminum honeycombs and blocks used in the

finite element analysis [38]

Description Aluminum

honeycomb

Loading and

support blocks

Element type SHELL 163 SOLID 164

Material model Bilinear

kinematic hardening

material

Rigid material

Mass Density [ρ] 2680 kg/m3 7800 kg/m3

Young’s Modulus

[E]

69 GPa 207 GPa

Poisson’s ratio [υ] 0.33 0.34

Tangent modulus

[Etan]

690 MPa -

Yield Stress [σys] 292 MPa -

Translational

constraints

All degrees of

freedom of one node

at the corner of the

bottom surface are

fixed

All displacements for

support block and, Z and X

displacements for loading block

are fixed

Rotational

constraints

No rotational

constraint applied

All rotations for both blocks

are fixed

The finite element models of honeycombs subjected to combined compression-

shear loads at three different loading angles are shown in Fig. 6.2. Being the same as

the experimental set-up, two identical honeycomb specimens were located on top

of the support block where the loading block moved downward to crush the

honeycomb specimen against the support block.

197

(a)

(b)

198

(c)

Figure 6.2. Finite element models of honeycombs subjected to combined

compression-shear loads at three different loading angles: (a) 15; (b) 30; (c) 45.

6.2. Validation of the FE models

6.2.1. Deformation model

Figure 6.3 shows the comparison between experimental and numerical results

of honeycombs under combined compression-shear loads in the TL plane at two

different loading angles of 15 and 30 and a velocity of 5ms-1. The experimental

images were captured by a high-speed camera (FASTCAM APX RS) while the FEA

images were taken from the animation using LS-PrePost software. Almost identical

deformation models were observed in both the experiment and FEA. When

honeycombs were loaded under the combined compression-shear load, progressive

199

buckling (elastic or plastic buckling) of the cell walls was observed. The buckling of

the cell walls initiated from both the top and bottom ends of a honeycomb specimen

and propagated to the middle of the specimen. A similar deformation mechanism

was also observed when honeycombs were loaded in the TW plane. No influence of

loading angle on deformation model was observed. Hou et al. [110] also found that

the deformation mode of honeycombs under dynamic combined compression-shear

loads did not change with loading angles.

(a)

(b)

200

(c)

(d)

Figure 6.3. Comparison between experimental and simulated results of

deformation model of honeycomb (H31): (a) experimental result at loading angle

15; (b) simulated result at loading angle 15; (c) experimental result at loading

angle 30; (d) simulated result at loading angle 30.

6.2.2. Rotation of cell walls

Under the combined compression-shear loads at different loading angles,

honeycomb cell walls rotate due to the applied shear force. Figure 6.4 shows the

rotation of honeycomb cell walls when loaded at 15 and 5 ms-1 in both the

experiment and FEA. The rotational angle, 𝛽, is defined as the change in angle of the

cell walls (Figs. 6.4 a and b).

201

(a)

(b)

Figure 6.4. Rotation of honeycomb cell walls subjected to combined compression-

shear load at 15 and 5 ms-1: (a) experiment; (b) FEA.

The rotational angle, 𝛽, was measured from the images captured by the high-

speed camera in experiments and by LS-PrePost in FEA. Measurements were taken

at every 3 mm vertical cross-head displacement (see Fig. 6.2c), initiating from 0 mm

to 45 mm (prior to densification). A comparison between experimental and

simulated results of cell wall rotation of honeycomb H31 at 15 and 30 loading

angles in the TL plane are shown in Fig. 6.5. A similar trend in the cell wall rotation

was observed at both loading angles. At 0 mm displacement (no deformation), the

rotational angle, 𝛽 was 0. After the combined compression-shear crush initiated,

due to the contribution of the shear force, the honeycomb cell walls rotated. It was

202

found that with the increase of displacement the rotational angle increased

monolithically. Cell walls of honeycombs rotated more when loaded at 30 loading

angle than at the 15 loading angle.

(a)

(b)

Figure 6.5. Comparison between experimental and simulated rotational angle-

displacement of honeycomb H31 loaded at a velocity of 5 ms-1: (a) at 15 loading

angle; (b) at 30 loading angle.

203

6.2.3. Force- Displacement curves

The comparison between experimental and simulated force-displacement

curves of honeycomb H31 in the TL and TW planes at 15 and 30 loading angles

and a velocity of 5 ms-1 are shown in Figs. 6.6 -6.7. Simulated force-displacement

curves of honeycomb H31 matched well with the corresponding experimental

results. Good agreement was also found for simulated force-displacement curves of

honeycomb H42 at both loading angles and planes.

(a)

204

(b)

(c)

205

(d)

Figure 6.6. Comparison between experimental and simulated force-displacement

curves of different honeycombs loaded at 15 loading angle and a velocity of 5 ms-1:

(a) H31-TL plane; (b) H31-TW plane; (c) H42-TL plane; (d) H42-TW plane.

(a)

206

(b)

(c)

207

(d)

Figure 6.7. Comparison between experimental and simulated force-displacement

curves of different honeycombs loaded at 30 loading angle and a velocity of 5 ms-1:

(a) H31-TL plane; (b) H31-TW plane; (c) H42-TL plane; (d) H42-TW plane.

6.2.4. Plateau stress

The cross-sectional area of honeycomb specimens is identical in both

experiment and finite element analysis, which is 50 mm × 50 mm= 2500 mm2. Since

the two specimens were crushed simultaneously, the total cross-sectional area is

5000 mm2. The plateau force is defined as the average force between 5 mm to 40

mm vertical displacement. The plateau stress is calculated as the ratio of plateau

force to the total cross-sectional area of honeycombs, 𝜎𝑝𝑙 = 𝑃 𝐴⁄ . The plateau

stresses of honeycombs calculated from both experiments and FEA are summarized

in Table 6.3. A good correlation was found between experimental and FEA results

in terms of plateau stress. For two different types of honeycombs (H31 and H42) the

simulated vertical plateau stresses were found to be slightly lower than the

208

corresponding experimental values for honeycombs H31 and H42 in both loading

angles. One possible reason is the imperfect alignment in cell walls and in the

geometry of cells in the physical honeycomb specimens. The other possible reason

is the adhesive in the double walls of honeycombs. In physical honeycomb

specimens, adhesive was used to bond two single walls together to form the double

walls. However, in FE models, double walls were simulated by doubling the

thickness of single walls without any adhesive. The possible de-bonding was ignored

in FEA. The maximum difference between experimental and FEA result measured

approximately 7.8 % in plateau stress.

Table 6.3. Comparison between experimental and simulated results of honeycombs

H31 and H42 at 5 ms-1 loading velocity

Honeycomb Loading

angle

Loading

plane

Vertical

plateau

Stress

(Experiment)

Vertical

plateau

stress

(FEA)

Difference

MPa MPa %

H31 15 TL 0.89 0.82 7.86

H31 15 TW 0.99 0.93 6.06

H31 30 TL 0.82 0.77 6.10

H31 30 TW 0.87 0.81 6.89

H42 15 TL 1.56 1.45 7.05

H42 15 TW 1.58 1.47 6.96

H42 30 TL 1.47 1.36 7.48

H42 30 TW 1.49 1.40 6.04

209

Based on the above comparisons, it is concluded that the proposed FE models

are valid and can be used to conduct the following parametric study of honeycombs

subjected to combined compression-shear loads.

6.3. Results and discussions

6.3.1. Force distribution

In the previous experimental study (chapter 5) the vertical force was measured

as the total force applied to crush each set of honeycomb specimens (two specimens

in each set). Since the dimensions of the two specimens in each set were identical

and the fixture was symmetric, it was assumed that each honeycomb specimen

carried half of the total force. The current FEA confirmed this assumption. Figure 6.8

shows the vertical force-displacement curves of each honeycomb (H31) in a set

loaded at 5 ms-1 loading velocity and 15, 30 and 45 loading angles in both the TL

and TW planes. Due to the symmetrical setting of the two honeycomb specimens,

the force applied to each specimen is identical and is half of the total vertical force.

Similar force distributions in each honeycomb specimen were also observed for

honeycomb H42 at all loading angles, plane orientations and loading velocities.

210

(a)

(b)

211

(c)

(d)

212

(e)

(f)

Figure 6.8. Vertical force-displacement curves of honeycomb H31 at different

loading angles and a velocity of 5 ms-1: (a) 15 loading angle in the TL plane; (b) 15

loading angle in the TW plane; (c) 30 loading angle in the TL plane; (d) 30 loading

angle in the TW plane; (e) 45 loading angle in the TL plane; (f) 45 loading angle in

the TW plane.

213

From Figs. 6.8(a)-6.8(b) it can be seen that at the 15 loading angle the vertical

force in the plateau region is nearly constant. From Figs. 6.8(c)-6.8(d) it can be

observed that when honeycombs are loaded at the 30 loading angle, the vertical

force in the plateau region decreases slightly. Furthermore, at the 45 loading angle

(Figs. 6.8e-6.8f) the vertical force in the plateau region decreases significantly with

displacement. This trend was also observed in experiments as mentioned in Chapter

5.

6.3.2. Vertical and horizontal force

In the experimental investigation (15, 30 and 45 loading angle) only the

vertical forces were measured by the uniaxial load cell in the machine, while the

horizontal force was unknown. The current finite element analysis facilitates the

measurement of both the horizontal and vertical forces applied to honeycombs

under combined compression-shear loads. The force-displacement curves of a

single H31 honeycomb under combined compression-shear loads in the TL plane at

5 ms-1 and three different loading angles of 15, 30and 45 are shown in Fig. 6.9. It

is observed that with the increase in loading angle, the vertical compressive force

decreases (see Fig. 6.9a) and horizontal shear force increases (see Fig. 6.9b). A

similar trend in the horizontal and vertical force-displacement curves was also

observed for honeycomb H31 loaded in the TW plane and honeycomb H42 in both

the TL and TW planes.

214

(a)

(b)

Figure 6.9. Force-displacement curves of a single H31 honeycomb under combined

compression-shear loads in the TL plane at 5 ms-1: (a) vertical force; (b) horizontal

force.

215

6.3.3. Normal compressive and shear stresses

As mentioned in the previous section, the current finite element analysis

facilitates the measurement of both the horizontal and vertical forces of

honeycombs subjected to combined compression-shear loads. From the vertical and

horizontal forces, the normal compressive force and shear force can be calculated

according to the following equations:

𝐹𝑣 = 𝐹𝑛 cos 𝜃 + 𝐹𝑠 sin 𝜃 (6.1a)

𝐹ℎ = 𝐹𝑛 sin 𝜃 − 𝐹𝑠 cos 𝜃 (6.1b)

where, 𝐹ℎ, 𝐹𝑣 , 𝐹𝑠 and 𝐹𝑛 are the horizontal, vertical, shear and normal

compressive forces respectively, and 𝜃 is the loading angle (Fig. 6.2b).

The normal compressive and shear plateau forces are defined as the average

forces from the 5 mm to 40 mm normal compressive displacement. Using Eqs. 6.1(a)

and (b), the normal compressive and shear plateau stresses are calculated and listed

in Tables 6.4 (a) and (b) for honeycombs H31 and H42 respectively. Both the normal

compressive and shear plateau stresses in the TW plane were slightly higher than

those in the TL plane for the same type of honeycomb at the same loading angle and

loading velocity.

For the same loading angle and loading velocity, both normal compressive and

shear stresses of honeycomb H42 are larger than those of honeycomb H31 due to

the larger t/l ratio (i.e., higher density) of honeycomb H42.

216

Table 6.4. (a) Normal compressive and shear plateau forces and stresses of

honeycomb H31 at 5 ms-1

FEA

no.

Loading

angle

Loading

plane

Normal

compressive

plateau

Force

Shear

plateau

force

Normal

compressive

plateau

stress

Shear

plateau

stress

kN kN MPa MPa

H31-1 15 TL 2.35 0.33 0.94 0.13

H31-2 15 TW 2.40 0.38 0.96 0.15

H31-3 30 TL 2.15 0.53 0.86 0.21

H31-4 30 TW 2.28 0.55 0.91 0.22

H31-5 45 TL 1.88 0.63 0.75 0.25

H31-6 45 TW 2.03 0.73 0.81 0.29

Table 6.4. (b) Normal compressive and shear plateau forces and stresses of

honeycomb H42 at 5 ms-1

FEA

no.

Loading

angle

Loading

plane

Normal

compressive

plateau

Force

Shear

plateau

force

Normal

compressive

plateau

stress

Shear

plateau

stress

kN kN MPa MPa

H42-1 15 TL 3.48 0.53 1.39 0.21

H42-2 15 TW 3.65 0.63 1.46 0.25

H42-3 30 TL 3.25 0.81 1.30 0.32

H42-4 30 TW 3.43 0.95 1.37 0.38

H42-5 45 TL 2.88 0.98 1.15 0.39

H42-6 45 TW 3.03 1.05 1.21 0.42

217

6.3.4. Crushing envelopes

Different loading angles between 0 (pure compression) to 90 (pure shear) are

applied in this section to measure the contribution of normal compressive stress

and shear stress in the combined compression-shear loads. Figure 6.10 shows the

normal compressive plateau stress-shear plateau stress of honeycombs H31 when

loaded at 5 ms-1 in both TL and TW plane. The elliptical curves characterized the

normal compressive and shear plateau stresses of honeycombs subjected to

combined compression-shear loads. At the maximum value of normal compressive

plateau stress the shear plateau stress is found minimum (𝜎𝑠 = 0), which indicates

pure compression of honeycombs. Similar to this at the maxinum value of shear

plateau stress the normal compressive plateau stress is found minimum (𝜎𝑛 = 0),

which indicates pure shear of honeycombs.

(a)

218

(b)

Figure 6.10. Crushing envelopes of honeycombs H31 in the normal stress-shear

stress coordinate system when they are subjected to combined compression-shear

loads at 5 ms-1: (a) TL plane; (b) TW plane.

6.3.5. Effect of t/l ratio

The effect of 𝑡 𝑙⁄ ratio on the normal compressive and shear plateau stress of

honeycombs is studied in this section. Different loading angles from 0 to 90 are

employed in this extensive study. The cell edge length was kept the same as 2.75

mm and the cell wall thickness varied from 0.0178 mm to 0.0762 mm. The

corresponding 𝑡 𝑙⁄ ratios ranged from 0.00647 to 0.02771. The normal compressive

and shear plateau stresses for honeycombs with different 𝑡 𝑙⁄ ratios subjected to

combined compression-shear loads in the TL plane at 5 ms-1 are listed in Table 6.5.

Figure 6.11 shows the effect of 𝑡 𝑙⁄ ratio on the plateau stress (normal compressive

and shear) of honeycombs at different loading angles, at a loading velocity of 5 ms-1

and in the TL plane.

219

Table 6.5. Normal compressive and shear plateau stresses of honeycombs with different 𝑡 𝑙⁄ ratios at a loading velocity of 5 ms-1

FEA

no.

Loading

angle

Loading

plane

Cell edge

length, l

Cell wall

thickness, t

t/l

ratio

Normal compressive

Plateau stress

Shear plateau

stress

mm mm MPa MPa

TL-1 0 TL 2.75 0.0178 0.00647 0.89 0

TL-2 0 TL 2.75 0.0254 0.00924 0.99 0

TL-3 0 TL 2.75 0.0381 0.01385 2.01 0

TL-4 0 TL 2.75 0.0508 0.01847 3.17 0

TL-5 0 TL 2.75 0.0635 0.02309 4.18 0

TL-6 0 TL 2.75 0.0762 0.02771 5.24 0

TL-7 15 TL 2.75 0.0178 0.00647 0.75 0.1

TL-8 15 TL 2.75 0.0254 0.00924 0.94 0.13

TL-9 15 TL 2.75 0.0381 0.01385 1.81 0.21

TL-10 15 TL 2.75 0.0508 0.01847 2.86 0.36

TL-11 15 TL 2.75 0.0635 0.02309 3.76 0.65

TL-12 15 TL 2.75 0.0762 0.02771 4.88 0.86

TL-13 30 TL 2.75 0.0178 0.00647 0.65 0.16

220

TL-14 30 TL 2.75 0.0254 0.00924 0.86 0.21

TL-15 30 TL 2.75 0.0381 0.01385 1.34 0.58

TL-16 30 TL 2.75 0.0508 0.01847 2.23 0.83

TL-17 30 TL 2.75 0.0635 0.02309 2.96 1.28

TL-18 30 TL 2.75 0.0762 0.02771 3.84 1.62

TL-19 45 TL 2.75 0.0178 0.00647 0.51 0.20

TL-20 45 TL 2.75 0.0254 0.00924 0.7 0.25

TL-21 45 TL 2.75 0.0381 0.01385 1.05 0.69

TL-22 45 TL 2.75 0.0508 0.01847 1.65 1.05

TL-23 45 TL 2.75 0.0635 0.02309 2.09 1.71

TL-24 45 TL 2.75 0.0762 0.02771 3.17 2.08

TL-25 60 TL 2.75 0.0178 0.00647 0.27 0.23

TL-26 60 TL 2.75 0.0254 0.00924 0.33 0.42

TL-27 60 TL 2.75 0.0381 0.01385 0.65 0.88

TL-28 60 TL 2.75 0.0508 0.01847 1.07 1.33

TL-29 60 TL 2.75 0.0635 0.02309 1.48 2

TL-30 60 TL 2.75 0.0762 0.02771 1.99 2.68

221

TL-31 75 TL 2.75 0.0178 0.00647 0.19 0.35

TL-32 75 TL 2.75 0.0254 0.00924 0.18 0.45

TL-33 75 TL 2.75 0.0381 0.01385 0.35 0.96

TL-34 75 TL 2.75 0.0508 0.01847 0.57 1.55

TL-35 75 TL 2.75 0.0635 0.02309 0.85 2.14

TL-36 75 TL 2.75 0.0762 0.02771 1.15 2.84

TL-37 90 TL 2.75 0.0178 0.00647 0 0.26

TL-38 90 TL 2.75 0.0254 0.00924 0 0.52

TL-39 90 TL 2.75 0.0381 0.01385 0 1.15

TL-40 90 TL 2.75 0.0508 0.01847 0 1.77

TL-41 90 TL 2.75 0.0635 0.02309 0 2.46

TL-42 90 TL 2.75 0.0762 0.02771 0 3.12

222

Figure 6.11. Effect of 𝑡 𝑙⁄ ratio in the normal compressive plateau stress-shear

plateau stress curves of honeycombs at a loading velocity of 5 ms-1 in the TL plane.

Elliptical curves are found for different 𝑡 𝑙⁄ ratios which follow a similar trend.

The equation of the ellipse is derived from the curves (see Fig 6.11) for the different

𝑡 𝑙⁄ ratio as follows:

(𝜎

𝜎𝑜)

2

+ (𝜏

𝜏𝑜)

2

= 1 (6.2)

where, 𝜎 and 𝜏 are the normal compressive and shear plateau stresses at

different loading angles, and 𝜎𝑜 and 𝜏𝑜 are the pure compression and shear plateau

stresses. For different 𝑡 𝑙⁄ ratios, the pure compression and shear plateau stresses

are also identified from the best-fitted curves and are listed in Table 6.6.

223

Table 6.6. Pure compressive and shear plateau stresses of honeycombs for different

𝑡 𝑙⁄ ratios at a loading velocity of 5 ms-1

t/l ratio Pure compressive stress, 𝝈𝒐

Pure shear stress, 𝝉𝒐

MPa MPa 0.00647 0.87 0.25

0.00924 0.97 0.46

0.01385 1.89 0.95

0.01847 2.97 1.48

0.02309 3.97 2.13

0.02771 5.02 2.87

Figure 6.12 shows the effect of 𝑡 𝑙⁄ ratio on the pure compressive and shear

plateau stresses of honeycombs at a loading velocity of 5 ms-1 in the TL plane. Both

the plateau stresses (pure compressive and shear) were found to increase

exponentially with 𝑡 𝑙⁄ ratio.

(a)

224

(b)

Figure 6.12. The effect of t/l ratio on the plateau stresses of honeycombs at a

loading velocity of 5 ms-1 in the TL plane: (a) pure compressive plateau stress; (b)

shear plateau stress.

From Fig. 6.12 empirical formulae are derived for both the pure compressive and

shear plateau stresses in terms of 𝑡 𝑙⁄ ratio as follows:

𝜎𝑜 = 677.86(𝑡/𝑙)1.41 (6.3a)

𝜏𝑜 = 1008.11(𝑡/𝑙)1.59 (6.3b)

Please note that 𝜎𝑜 is slightly different from that discussed in chapter 4 for pure

compression. In chapter 4 the plateau stress for pure compression is calculated from

the FE analysis; however, in this section the pure compressive plateau stress is

identified from the fitted curves. Hou et al. [112], Mohr and Doyoyo [107] also used

225

curve-fitting to identify 𝜎𝑜 and 𝜏𝑜 for the combined compression-shear analysis of

honeycombs.

6.3.6. Effect of loading velocity

In the previous experimental study, honeycombs were crushed at low and

intermediate loading velocities (5 × 10−5– 5 ms-1) for the loading angles of 15 and

30. Due to the limitation of testing facilities (fixture for 45 was too large for the

high-speed Instron machine), experiments at 45 loading angle were only conducted

at low velocities (5 × 10−5 – 5× 10−3 ms-1). In this finite element analysis, different

loading angles from 0 and 90 and loading velocities from 5 to 55 ms-1 were applied

to honeycombs with corresponding strain rates from 100 and 1100 s-1. Figure 6.13

shows the effect of strain rate on the plateau stresses (normal compressive and

shear) of honeycomb H31. Elliptical curves are found for the different strain rates

for honeycomb H31 which also follows the equation of an ellipse (see Eq. 6.2). For

different strain rates, the pure compressive and shear plateau stresses are identified

from the best-fitted curves and listed in Table 6.7.

226

Figure 6.13. Effect of strain rate in the normal compressive plateau stress-shear

plateau stress curves of honeycombs at a loading velocity of 5 ms-1 in the TL plane.

Table 6.7. Pure compressive and shear plateau stresses of honeycombs for different

strain rates

Strain rate Pure compressive stress, 𝝈𝒐

Pure shear stress, 𝝉𝒐

s-1 MPa MPa 100 0.97 0.46

300 1.28 0.57

500 1.54 0.61

700 1.69 0.68

900 1.83 0.74

1100 1.98 0.87

Figure 6.14 shows the effect of strain rate on the pure compressive and shear

plateau stresses of honeycombs loaded in the TL plane. Both the plateau stresses

(pure compressive and shear) were found to increase exponentially with strain rate.

227

(a)

(b)

Figure 6.14. The effect of strain rate on the plateau stresses of honeycombs at a

loading velocity of 5 ms-1 in the TL plane: (a) pure compressive plateau stress; (b)

shear plateau stress.

228

Empirical formulae are derived for both the pure compressive and shear plateau

stresses in terms of 𝑡 𝑙⁄ ratio and strain rate as follows:

𝜎𝑜 = 277.92(𝑡/𝑙)1.41(1 + 0.14𝜀̇0.46) (6.4a)

𝜏𝑜 = 735.92(𝑡/𝑙)1.59(1 + 0.008𝜀̇0.56) (6.4b)

6.4. Summary

In this finite element analysis, the mechanical response of aluminum

honeycombs under combined compression-shear loads has been investigated using

ANSYS LS-DYNA. A good agreement between the numerical and experimental

results has been found in terms of deformation mode, rotation of cell walls, force-

displacement curves and plateau stress.

The verified FE model facilitates the measurement of horizontal force, which was

not obtained in the previous experimental study. Furthermore, the FE models

enable the calculation of the normal compressive and shear forces acting on

honeycombs.

The normal compressive and shear plateau forces and stresses have been

calculated from the vertical and horizontal plateau forces. The FEA results show that

the normal compressive plateau stress decreases with loading angle, while shear

plateau stress increases with loading angle. Elliptical stress envelopes have been

found between the normal compressive stress and shear plateau stress.

The elliptical stress envelopes for different 𝑡 𝑙⁄ ratios and strain rates have been

studied and an equation of the ellipse is derived from the stress envelopes. The pure

compressive and shear plateau stresses have been identified from the curve-fitting

in both cases. It has been found that both plateau stresses (pure compressive and

229

shear) increase exponentially with 𝑡 𝑙⁄ ratio and strain rate. Empirical formulae have

been derived to describe the relationship between plateau stresses, 𝑡 𝑙⁄ ratio and

strain rate.

231

Chapter 7. Conclusions and recommendations for future

work

In this chapter, the findings of the present research work are summarized.

Recommendations for potential future work are proposed.

7.1. Conclusions

This thesis presents comprehensive experimental and numerical studies on

three types of aluminum honeycombs to investigate their mechanical response

when subjected to different types of loadings (indentation, compression and

combined compression-shear). The findings of this research work are summarized

as follows:

The experimental indentation and compression tests on three types of

aluminum honeycombs (differing in cell size and 𝑡 𝑙⁄ ratio or relative density) in the

out-of-plane direction have been conducted at different low and intermediate

constant loading velocities from 5×10-5 ms-1 to 5 ms-1. An MTS and a high-speed

INSTRON machine have been employed to conduct such tests. In compression,

progressive plastic buckling of the cell walls has been observed from both the upper

and lower interfaces of the honeycomb specimens. In indentation, it was difficult to

observe experimentally the crushing mechanism of the specimens because the

indenter penetrated into the middle portion of the honeycomb specimens which

was surrounded by the un-deformed cells. Moreover, irregular tearing of the cell

walls has been observed in specimens post-test. The effects of loading velocity or

strain rate on the plateau stress and total dissipated energy has been investigated

for both out-of-plane indentation and compression. In both cases (indentation and

232

compression) the plateau stress and total dissipated energy have been found to

increase with strain rate for all types of honeycombs. The largest magnitude of

plateau stress and total dissipated energy has been found for the honeycomb which

had the greatest nominal density (72.09 kg/m3). Similarly, the smallest magnitude

of plateau stress and total dissipated energy has been found for the honeycomb

which had the smallest nominal density (49.66 kg/m3). In indentation tests, the

tearing of the cell walls occurred along the four edges of the indenter. The tearing

energy has been calculated from the difference between the total dissipated energy

in indentation and compression tests. Similar to the plateau stress and total

dissipated energy, tearing energy has also been found to increase with strain rate.

The magnitude of tearing energy has been the largest for the honeycomb which has

the largest 𝑡 𝑙⁄ ratio. Furthermore, empirical formulae have been proposed to

describe the relationship between the tearing energy per unit fracture area and

strain rates for different honeycombs.

Based on the experimental out-of-plane indentation and compression tests,

finite element analysis (FEA) has been conducted using ANSYS LS-DYNA software.

In simulation, honeycombs have been modelled using bilinear kinematic hardening

material models. For high computation efficiency, Belytschko-Tsay Shell 163

elements with five integration points were employed for honeycomb cell walls. The

optimum element size has been identified by mesh convergence tests. The

honeycomb models have been verified by the experimental results presented in

Chapter 3 in terms of deformation modes, stress-strain curve and total energy

dissipation. The aim of the FEA was to study the effects of 𝑡/𝑙 ratio (0.00462 to

0.05542) and strain rate (1 × 102 to 1 × 104 s-1) on the plateau stress, total

233

dissipated energy and tearing energy of honeycombs extensively. The 𝑡/𝑙 ratio has

great influence on the plateau stress in both the indentation and compression of

honeycombs. The indentation and compression plateau stresses have been found to

increase with 𝑡 𝑙⁄ ratio by power laws with exponents of 1.36 and 1.47 respectively

(Eqs. 4.2a and 4.2b). In both the indentation and compression of aluminum

honeycombs, significant enhancement in the plateau stress, dissipated energy and

tearing energy has been observed at high strain rates. A generic formula has been

developed to describe the relationship between tearing energy per unit fracture

area and relative density for both aluminum honeycombs and foams under quasi-

static loading condition (Eq. 4.3, foam data were from other researchers).

Furthermore, from the parametric study of honeycombs an empirical formula has

been proposed for the tearing energy per unit fracture area in relation to the strain

rate and relative density of honeycombs (Eq. 4.4).

Quasi-static and dynamic combined compression-shear tests of aluminum

honeycombs have been conducted using MTS and high-speed INSTRON machines.

Different loading angles (15, 30 and 45) and loading velocities (5×10-5 ms-1 to 5

ms-1) have been employed in this experimental study. Specially designed fixtures

have been employed for different loading angles where the compression and shear

crushing process occurred simultaneously without the effect of any transverse load

on the loading machine. During crushing, at all loading angles, it has been observed

that elastic or plastic buckling of cell walls occurred from both interfaces of the

honeycomb specimens. The rotation of cell walls has been observed for different

loading angles and it was found that the rotational angle increased with loading

angle when honeycomb specimens had been loaded in both the TL and TW planes.

234

The effect of loading angles on the force-displacement curves has been discussed. In

the plateau region of the force-displacement curves it has been noticed that the

crushing force changes with loading angle. At the 15° loading angle, the crushing

force remains nearly constant; at the 30° loading angle, the crushing force decreases

slightly; and at the 45° loading angle, the crushing force decreases significantly. In

all cases, the crushing force in the plateau region has been found to be slightly larger

in the TW plane than in the TL plane, at any particular loading velocity. Subjected to

combined compression-shear load, the plateau stress and dissipated energy have

been found to increase with loading velocity and decrease with loading angle. The

highest magnitude of the plateau stress and dissipated energy has been found for

the honeycomb which has greatest nominal density. An empirical formula between

plateau stress and loading angle has been proposed (Eq. 5.2). Furthermore, the

normal compressive and shear force components of the applied combined

compression-shear load at the 15° loading angle have been measured directly by

employing a triaxial load cell in the MTS machine. At this particular loading angle,

the contribution of normal compressive force in the crushing of honeycombs

subjected to combined compression-shear load has been found to be much larger

than that of the shear force.

In the experimental combined compression-shear tests of honeycombs, it was

very challenging to conduct tests on a variety of aluminum honeycombs at a wide

range of dynamic loading velocities and also to employ a triaxial load cell at larger

loading angles, owing to the limitation of experimental facilities. For a

comprehensive study, ANSYS LS-DYNA software has been used to simulate the

dynamic crushing of honeycombs with different t/l ratios at a wide range of

235

velocities or strain rates (100 to 1100 s-1) and loading angles (0° to 90°). Finite

element models of honeycombs have been developed and verified by the

experimental results. The finite element model facilitates the measurement of

horizontal forces applied and therefore the calculation of normal compressive and

shear stresses acting on the honeycombs at different loading angles. An elliptical

stress envelope has been found in the normal compressive plateau stress and shear

plateau stress space. The elliptical stress envelope enlarges with the 𝑡 𝑙⁄ ratio and

strain rate. An equation of the ellipse has been derived (Eq. 6.2) and empirical

formulae (Eqs. 6.4a and 6.4b) have been developed to reflect the effects of 𝑡 𝑙⁄ ratio

and strain rate on the ellipses.

7.2. Recommendations for future work

In the experimental parts of this thesis (Chapters 3 and 5), various low and

intermediate loading velocities (up to 5 ms‐1) were employed in both indentation

and combined compression‐shear tests. Due to the limitation of experimental

facilities, higher loading velocities could not be achieved in both types of tests.

Although higher constant loading velocities were achieved in the following finite

element analyses (Chapters 4 and 6) for both type of tests (Indentation, combined

compression‐shear), further experimental work at higher velocities is expected to

be conducted.

Beside this, only three types of hexagonal honeycombs with different cell size

and wall thicknesses were experimentally studied in this thesis. Other types of

aluminum honeycombs with different cell shapes, such as square, rectangular,

circular and triangular, different cell size and wall thicknesses are expected to be

studied in future.

236

In indentation, only square shape indenter was employed for indentation

analysis in both experimental and finite element analyses. The influence of various

shapes of indenter on the plateau stress, dissipated energy and tearing energy are

expected to be investigated.

In combined compression‐shear experiments, different loading angles were

employed in different quasi‐static loading velocities using uniaxial load cell to

measure vertical forces. Only at 15° loading angle it was possible to measure normal

compressive and shear force components of the applied combined compression‐

shear load directly using a triaxial load cell. Due to the limitation of experimental

setup it was difficult to employ larger loading angles to crush different honeycombs

and to measure all the force components using the triaxial load cell. Although finite

element modelling was employed to measure force components in both the vertical

and horizontal directions at different loading angles and velocities, further

experimental investigation are required to be carried out to measure the

compressive normal and shear forces directly..

No in‐depth theoretical analysis was conducted in this thesis due to the tight time

constraint. Theoretical work is expected to be conducted to unveil the deformation

mechanisms of aluminum honeycombs subjected to indentation as well as combined

compression and shear loadings.

238

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