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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.
A. S. M. Ayman Ashab
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:
A. S. M. Ayman Ashab
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
Figure 5.14. Vertical force-displacement curves for honeycomb subjected to combined
compression-shear loads at a loading angle of 30: (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. .................................................................................................... 159
Figure 5.15. Vertical force-displacement curves for honeycomb subjected to combined
compression-shear loads at a loading angle of 45: (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. .................................................................................................... 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 5.3. Quasi-static and dynamic experimental results for honeycomb H42 .............. 170
Table 5.4. Quasi-static and dynamic experimental results for honeycomb H45 .............. 171
Table 5.5. Normal compressive and shear results of honeycomb H31 at different loading
velocities ................................................................................................................. 188
Table 5.6. Normal compressive and shear results of honeycomb H42 at different loading
velocities ................................................................................................................. 189
Table 5.7. Normal compressive and shear results of honeycomb H45 at different loading
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
xix
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.
21
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
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)
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
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)
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
No. Researchers Velocity
(ms-1)
Loading
angle
Equipment and method
used
Type of
honeycombs
Honeycombs specification References
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]
64
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
Indentation 5×10-5 0.00
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
Compression 5×10-5 0.00
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
Indentation 5×10-5 0.00
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
Compression 5×10-5 0.00
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).
102
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
H42-Experiment [chapter 3]
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
H31-Experiment [chapter 3]
H42- FEA
H42-Experiment [chapter 3]
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
H31-Experiment [chapter 3]
H42- FEA
H42-Experiment [chapter 3]
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
H31-Experiment [chapter 3]
H42- FEA
H42-Experiment [chapter 3]
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
H31-Experiment [chapter 3]
H42- FEA
H42-Experiment [chapter 3]
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.
133
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)
Figure 5.14. Vertical force-displacement curves for honeycomb subjected to
combined compression-shear loads at a loading angle of 30: (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.
160
(a)
(b)
161
(c)
(d)
162
(e)
(f)
Figure 5.15. Vertical force-displacement curves for honeycomb subjected to
combined compression-shear loads at a loading angle of 45: (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.
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
Table 5.3. Quasi-static and dynamic experimental results for honeycomb H42
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
Table 5.4. Quasi-static and dynamic experimental results for honeycomb H45
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
Table 5.6. Normal compressive and shear results of honeycomb H42 at different
loading velocities
Exp. no Loading angle
Loading plane
Loading velocity
Normal compressive
plateau stress
Shear 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
Table 5.7. Normal compressive and shear results of honeycomb H45 at different
loading velocities
Exp. no Loading angle
Loading plane
Loading velocity
Normal compressive
plateau stress
Shear 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).
192
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.
230
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.
237
238
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