Feng Zhu Thesis - Swinburne Research Bank | openEQUELLA
Transcript of Feng Zhu Thesis - Swinburne Research Bank | openEQUELLA
IMPULSIVE LOADING OF SANDWICH
PANELS WITH CELLULAR CORES
by
Feng Zhu B.Eng., M.Phil.
A thesis submitted for the degree of Doctor of Philosophy
Faculty of Engineering and Industrial Sciences
Swinburne University of Technology
May 2008
II
Abstract of thesis entitled
Impulsive loading of sandwich panels with
cellular cores
Submitted by
Feng Zhu
for the degree of Doctor of Philosophy
at Swinburne University of Technology
in May, 2008
Metallic sandwich panels with a cellular core such as honeycomb or metal foam have the
capability of dissipating considerable energy by large plastic deformation under quasi-static or
dynamic loading. The cellular microstructures offer the ability to undergo large plastic
deformation at nearly constant stress, and thus can absorb a large amount of kinetic energy
before collapsing to a more stable configuration or fracture. To date, research on the
performance of sandwich structures has been centred on their behaviours under quasi-static
loading and impact at a wide range of velocities, but work on their blast loading response is still
very limited. A series of analytical and computational models have been developed by previous
researchers to predict the dynamic response of a sandwich beam or circular sandwich panel.
However, no systematic studies have been reported on square sandwich panels under blast
loading.
In this research, experimental, computational and analytical investigations were conducted on a
number of peripherally clamped square metallic sandwich panels with either honeycomb or
aluminium foam cores. The experimental program was designed to investigate the effect of
various panel configurations on the structural response. Two types of experimental result were
obtained: (1) deformation/failure modes of specimen observed in the tests; and (2) quantitative
results from a ballistic pendulum with corresponding sensors.
Based on the experiments, corresponding finite element simulations have been undertaken using
commercial LS-DYNA software. In the simulation work, the explosive loading process and
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response of the sandwich panels were investigated. A parametric study was carried out to
examine the plastic deformation mechanism of the face-sheet, influence of boundary conditions,
as well as the plastic energy dissipating performance of the components of the sandwich panels.
Two analytical models have been developed in this study. The first model is a design-oriented
approximate solution, which is excellent for predicting maximum permanent deflections, but
gives no predictions of displacement-time histories. The analysis is based on an energy balance
with assumed displacement fields, where either small deflection or large deflection theory is
considered, according to the extent of panel deformation. Using the proposed analytical model,
an optimal design has been conducted for square sandwich panels of a given mass per unit area.
The second analytical model has the ability of capturing the dynamic structural response. A new
yield criterion was developed for a sandwich cross-section with different core strengths. By
adopting an energy dissipation rate balance approach with the newly developed yield surface,
upper and lower bounds of the maximum permanent deflections and response time were
obtained. Finally, comparative studies have been conducted for the analytical solutions of
monolithic plates, sandwich beams, circular and square sandwich panels.
IV
To my family
V
DECLARATION
I declare that this thesis represents my own work, except where due acknowledgement
is made, and that it has not been previously included in a thesis, dissertation or report
submitted to this university or to any other institution for a degree, diploma or other
qualification.
Signed __________________________
Feng Zhu
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ACKNOWLEDGEMENTS
I take this opportunity to thank my supervisor, Professor G. Lu for his support and supervision
in pursing this research. He has provided me with a global vision of research, strong theoretical
and technical guidance, and valuable feedback on my work. I would like to thank Professor
L.M. Zhao at Taiyuan University of Technology (TYUT), who is the co-author of my most
publications, for his valuable comments and helpful advice, especially during his sabbatical visit
at Swinburne.
I also thank Dr. Z. Wang, TYUT for his constructive suggestions for the project during his visit
at Swinburne. Besides, I benefit a lot from the discussions with Associate Professor E. Gad at
Swinburne and Professor G.N. Nurick at University of Cape Town. Their kind help are highly
appreciated. Thanks also go to the colleagues working in our research group, Mr. S.R. Guillow,
Ms. W. Hou, Dr. D. Ruan and Mr. J. Shen, for their support and friendship.
My PhD study is sponsored by Swinburne University of Technology through a scholarship, and
the research project is supported by Australian Research Council (ARC) through a discovery
grant. Their financial contributions are gratefully acknowledged. I would also like to thank the
staff members at Swinburne, Taiyuan University of Technology and North University of China
involved in this project, for their provision of the experimental facilities and technical
assistance; and thank the Victorian Partnership for Advanced Computing (VPAC), Australia, for
the access to high performance computing facilities.
Finally, I wish to express my special thanks to my parents and the other members of my family
for their support and encouragement during the course of this work.
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CONTENTS
Abstract II
Declaration V
Acknowledgements VI
Contents VII
List of tables XI
List of figures XII
List of symbols XVI
1. Introduction 1
1.1 Motivation 1
1.2 Sandwich structures 2
1.3 Blast wave and its effect 3
1.4 Methodology and workflow 5
1.5 Thesis organisation 6
2. Literature review 8
2.1 Introduction 8
2.2 Experimental investigations 8
2.2.1 Experimental facilities 8
2.2.2 Experimental observations 10
2.3 Numerical simulations 14
2.3.1 Basic formulations 15
2.3.2 Modeling blast loads 16
2.3.3 Modeling the materials of targets 17
2.3.4 Commercial FEA packages for blast loading simulations 19
2.4 Analytical modeling 20
2.4.1 Analytical models for monolithic metals 20
2.4.2 Analytical models for cellular solids 24
2.4.3 Analytical models for sandwich structures 26
2.5 Summary 28
3. Experimental investigation into the honeycomb core sandwich panels 30
3.1 Introduction 30
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3.2 Specimen 30
3.3 Experimental set-up 36
3.4 Deformation and failure patterns 40
3.4.1 Front face-sheet deformation/failure 40
3.4.2 Core deformation/failure 43
3.4.3 Back face-sheet deformation/failure 44
3.5 Pressure-time history at the central point of the front face 47
3.6 Analysis and discussion 48
3.6.1 Effect of face-sheet thickness 48
3.6.2 Effect of cell dimension of the core 49
3.6.3 Effect of charge mass 52
3.7 Summary 53
4. Experimental investigation into the aluminium foam core sandwich panels 55
4.1 Specimen 55
4.2 Results and discussion 57
4.2.1 Deformation/failure patterns 57
4.2.2 Deflection of the face-sheet 62
4.3 Summary 62
5. Numerical simulation of the honeycomb core sandwich panels 63
5.1 Introduction 63
5.2 FE model 63
5.2.1 Modeling geometry 63
5.2.2 Modeling materials 66
5.2.3 Modeling blast load 67
5.3 Simulation results and discussion 68
5.3.1 Explosion and structural response process 68
5.3.2 Deformation/failure patterns of sandwich panels 72
5.3.3 Quantitative results 76
5.4 Effect of plastic stretching and bending 77
5.4.1 Strain distribution along the x-axis 79
5.4.2 Strain distribution along the diagonal line 81
5.4.3 Analysis and discussion 84
5.5 Effect of boundary conditions 84
5.6 Summary 85
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6. Numerical simulation of the aluminium foam core sandwich panels 87
6.1 FE model 87
6.1.1 Modeling geometry 87
6.1.2 Modeling materials and blast load 88
6.2 Simulation results and discussion 91
6.2.1 Explosion and structural response process 91
6.2.2 Deformation/failure patterns 94
6.2.3 Face-sheets deflections and core crushing 95
6.3 Energy absorption 97
6.3.1 Time history of plastic dissipation 99
6.3.2 Energy partition 99
6.4 Summary 101
7. Analytical solution I – a design-oriented theoretical model 103
7.1 Introduction 103
7.2 Analytical modeling 103
7.2.1 Phase I – Front face deformation 105
7.2.2 Phase II – Core compression 105
7.2.3 Phase III – Overall bending and stretching 108
7.3 Model validation 113
7.3.1 Comparison with experiment 114
7.3.2 Comparison with the analytical model for circular plates 114
7.4 Optimal design of square plates to shock loading 115
7.4.1 Effect of side length ratio 116
7.4.2 Effect of relative density of the core 117
7.4.3 Effect of core thickness 117
7.5 Summary 118
8. Analytical solution II – a theoretical model for dynamic response 120
8.1 Introduction 120
8.2 Analytical modeling 121
8.2.1 Phase I – Front face deformation 121
8.2.2 Phase II – Core compression 122
8.2.3 Phase III – Overall bending and stretching 128
8.3 Model validation 135
8.4 Comparative studies of the analytical solutions 136
8.4.1 Effect of longitudinal strength of core after compression 136
X
8.4.2 Comparison of square monolithic and sandwich panel 138
8.4.3 Comparison among sandwich beams, circular and square sandwich panels 143
8.5 Summary 144
9. Conclusions and future work 146
9.1 Conclusions 146
9.2 Future work 150
References 152
Appendix A 159
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List of tables
Table 3-1. Sandwich panels of Group 1, where the effects of foil thickness
and face thickness are investigated 32
Table 3-2. Sandwich panels of Group 2, where the effects of cell size
and face thickness are investigated 33
Table 3-3. Sandwich panels of Group 3, where the effects of average core mass and
face thickness are investigated 34
Table 3-4. Sandwich panels of Group 4, where the effect of charge mass is investigated 35
Table 4-1. Specifications and test results of the aluminium foam core sandwich panels 56
Table 5-1. LS-DYNA material type, material property and EOS input data
for honeycomb core panels 67
Table 6-1. LS-DYNA material type, material property and EOS input data for
aluminium foam core panels 91
Table 8-1. Specifications and mechanical properties of the honeycombs
and aluminium foams 127
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List of figures
Figure 1-1. Three types of the structural damage caused by explosions 1
Figure 1-2. Typical configuration of a sandwich panel 3
Figure 1-3. Typical pressure-time history of a blast wave 4
Figure 1-4. Workflow of the project 6
Figure 2-1. Two types of ballistic pendulums 9
Figure 2-2. Some sensors used for blast tests 10
Figure 2-3. Failure modes of a beam transiting from a Mode I to a Mode III
with increasing impulsive velocity [21] 11
Figure 2-4. Using aluminium foam projectiles to simulate non-uniform shock loading [36] 13
Figure 2-5. Deflected profiles of dynamically loaded metal foam core sandwich beams [36] 13
Figure 2-6. Typical cross-section of the face-sheets and honeycomb core of a circular
sandwich plate (Mechanism II) [38] 14
Figure 2-7. Sketches of several sandwich core topologies [44] 18
Figure 2-8. Deformation patterns of cellular solids 25
Figure 2-9. Material models of cellular solids 26
Figure 3-1. Geometry and dimension of the honeycomb core specimen 31
Figure 3-2. Four-cable ballistic pendulum system 36
Figure 3-3. Sketch of the frame and clamping device 37
Figure 3-4. Two types of sensor used in the tests 38
Figure 3-5. A typical oscillation time history of the pendulum 38
Figure 3-6. Sketch of the experimental set-up 39
Figure 3-7. Indenting failure on the front face (Specimen No.: 1/8-5052-0.0015-MD-2) 41
Figure 3-8. Pitting failure on the front face (Specimen No.: 1/8-5052-0.0015-TN-1) 41
Figure 3-9. Deformation/failure map for Groups 1~3. The abscissa denotes
the specimens sorted by the cores with increasing relative densities 42
Figure 3-10. Deformation/failure map for Group 4, where all the eight panels
have identical configurations 43
Figure 3-11. Failure pattern of the honeycomb core (Specimen No.: ACG-1/4-TK-5) 45
Figure 3-12. Failure pattern of the back face (Specimen No.: ACG-1/4-TK-5) 46
Figure 3-13. Typical pressure-time history at the central point of the front face 48
Figure 3-14. Effect of face-sheet thickness. The abscissa denotes the specimens
given without any particular order 49
Figure 3-15. Effect of foil thickness. The abscissa denotes the specimens sorted
by the face-sheets with increasing thicknesses 50
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Figure 3-16. Effect of cell size. The abscissa denotes the specimens sorted
by the face-sheets with increasing thicknesses 51
Figure 3-17. Effect of the average mass of core. The abscissa denotes
the specimens sorted by the face-sheets with increasing thicknesses 52
Figure 3-18. Effect of impulse level on the panels with nominally identical configurations 53
Figure 4-1. Geometry and dimension of the aluminium foam core specimen 55
Figure 4-2. Failure patterns of the front face 58
Figure 4-3. Two types of failure in the centre of front face 59
Figure 4-4. A typical deformation/failure pattern of the back face (Specimen L-20-TK-2) 60
Figure 4-5. A typical cross-section of the specimen (Specimen L-30-TK-1) 61
Figure 5-1. Geometric model of the sandwich panel 65
Figure 5-2. Geometric model of the charge 66
Figure 5-3. A typical process of the charge detonation 69
Figure 5-4. A typical process of explosion product - structure interaction 71
Figure 5-5. A typical process of plate deformation 72
Figure 5-6. A typical profile of back face (Specimen name: ACG-1/4-TK-6) 73
Figure 5-7. Process of back face deformation and corresponding plastic hinges,
one stationary and the other traveling 74
Figure 5-8. Displacement-time history at the central points of face-sheets and core crushing
(Specimen name: ACG-1/4-TK-6) 75
Figure 5-9. Deformation patterns of honeycomb core (Specimen name: ACG-1/4-TK-6) 76
Figure 5-10. Comparison of experimental and predicated results 77
Figure 5-11. Locations of the shell elements in the two groups 78
Figure 5-12. εmid distribution for the shell elements in Group 1 79
Figure 5-13. εd distribution for the shell elements in Group 1 81
Figure 5-14. εmid distribution for the shell elements in Group 2 82
Figure 5-15. εd distribution for the shell elements in Group 2 83
Figure 5-16. Effect of boundary conditions on the time history of back face deflection and
core crushing 85
Figure 6-1. Geometric model of a sandwich panel and charge 88
Figure 6-2. Stress-strain curves for the foam core used in the simulation 89
Figure 6-3. Process of the charge detonation 92
Figure 6-4. Process of explosive-structure interaction 93
Figure 6-5. Process of plate deformation 94
Figure 6-6. Comparison of the deformation/failure patterns obtained in simulation and
experiment (Specimen L-30-TK-1) 95
Figure 6-7. Comparison of predicted and experimental deflections on the back face
(Specimen L-30-TK-1) 96
Figure 6-8. History of central point deflections and core crushing (Specimen L-30-TK-1) 97
Figure 6-9. History of plastic dissipation during plastic deformation
(Specimen L-30-TK -1) 99
Figure 6-10. Energy dissipation normalised with the total energy for Specimen No. 1 100
Figure 7-1. Schematic illustration showing the three phases in the response of
a sandwich panel subjected to the blast loads 104
Figure 7-2. Schematic illustration showing the progressive deformation mode of
cellular materials under impact loading and its simplified material model 107
Figure 7-3. Displacement field of the back face 109
Figure 7-4. Comparison between the experimental and predicted maximum deflection of
the back face of the two types of specimens 114
Figure 7-5. Comparison of the analytical predictions for circular panels and square panels 115
Figure 7-6. Comparison of the normalised maximum deflections of the rectangular plates
with various side length ratios, for three impulses 116
Figure 7-7. Dimensionless maximum deflections of a sandwich plate with various relative
densities of cores, for three impulses 117 Figure 7-8. Dimensionless maximum deflections of a sandwich plate with various
thicknesses of cores, for three impulses 118
Figure 8-1. Three phases in the response of a sandwich panel subjected to the blast loads 121
Figure 8-2. Energy absorption efficiency-strain curves and
stress-strain curves of honeycombs 125
Figure 8-3. Energy absorption efficiency-strain curves and stress-strain curves
of aluminium foams 126
Figure 8-4. Yield loci for monolithic and sandwich structures together with
their circumscribing and inscribing squares 130
Figure 8-5. Sketch of the normal stresses profile on a sandwich cross-section 132
Figure 8-6. Comparison of experimental and theoretically predicted deflections 136
Figure 8-7. Comparison of the effect of two assumptions 138
Figure 8-8. Comparison of a square solid plate and a square sandwich plate
with the same materials and mass/area 23.9 /M kg m= 140
Figure 8-9. Distribution of normalised critical impulse with respect to various thickness
ratios and core relative densities, for a square sandwich panel with
the mass/areas 23.9 /M kg m= 142
Figure 8-10. Comparison of the deflections predicted by sandwich beam and
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sandwich plates with the same materials and mass/area 23.9 /M kg m=
1 4 4
Figure A-1. Sketch of the motion of a four-cable ballistic pendulum
subjected to a shock wave 159
XV
List of symbols
Chapter 1 Is – Impulse of blast wave during the positive phase
Pa – Ambient air pressure
Ps – Peak pressure of the blast wave
R – Distance from the centre of the explosive source in meters
td – Time duration of the positive phase
W – Charge mass of TNT in kilograms
Chapter 2 A, B, C, m, n – Material constants for Johnson-Cook model
D, p – Material constants for Cowper-Symonds model
L – Side length of a square panel
Pi – Incident pressure
Pr – Reflected pressure
σdY – Dynamic yield strength
σY – Static yield strength pε – Effective plastic strain pε – Plastic strain rate
θ – Incident angle of shock wave
σzz– Stress in the transverse direction
Chapter 3 A – Working area of the PVDF film
d33 – piezoelectric constant of the PVDF film
Hc – Thickness of core
hf – Thickness of face-sheets
I – Impulse delivered onto the structure
L – Side length of a square panel
le – Cell size of hexagonal honeycomb
Q – Electric charge
XVI
m0 – Mass of core
mh – Mass of TNT charge
M – Mass per unit area
t – Nominal foil thickness of a hexagonal honeycomb
t – Thickness of the sandwich panel
w0 – Maximum deflection of back face
ρc – Mass density of the core
ρf – Material density of the face-sheets
δ – Dimensionless back face deflection
Φ – Dimensionless impulse
Chapter 4 Hc – Thickness of core
hf – Thickness of face-sheets
I – Impulse delivered onto the structure
m0 – Mass of core
mh – Mass of TNT charge
w0 – Maximum deflection of back face
Chapter 5 A, B, R1, R2, ω – Materials constants for JWL equation
P – Blast pressure
V – Detonation velocity
εmid – Middle-plane strain of the face-sheet
εlower – Lower-plane strain of the face-sheet
εupper – Upper-plane strain of the face-sheet
ρ – Explosive density
ρ0 – Explosive density at the beginning of detonation process
Chapter 6 A – Area of the plate exposed to the blast
e – Volumetric strain
Hc – Thickness of core
hf – Thickness of face-sheets
I – Impulse delivered onto the structure
lx, ly, lz – Length of a metallic foam block in x, y and z directions XVII
V – Current volume of metallic foam
V0 – original volume of metallic foam
v1 – Initial velocity of the front face of a sandwich structure
v2 – The velocity of a sandwich structure obtained after core crushing
WI – Kinetic energy of the front face of a sandwich structure before core crushing
WII – Kinetic energy of the whole sandwich structure after core crushing
ρc – Mass density of the core
ρf – Material density of the face-sheets
εx, εy , εz– Compressive strains in x, y and z directions
Chapter 7 A – Area of the plate exposed to the blast
a, b – Half side length of a rectangular plate
Dn – Johnson’s damage number
Ep – Plastic dissipation during core crushing
Hc – Thickness of core
hf – Thickness of face-sheets
ΔHc – Thickness of crushed core
cH – Final thickness of core
hf – Thickness of face-sheets
I – Impulse delivered onto the structure
L – Half side length of a square plate
lm – Length of a plastic hinge line
M – Mass per unit area
Mp – Fully plastic bending moment
t – Initial overall thickness of a sandwich structure
p – Pressure
u, v, w – Displacements in x, y and z directions
Ub – Plastic bending dissipation
Us – Plastic stretching dissipation
v1 – Initial velocity of the front face of a sandwich structure
v2 – The velocity of a sandwich structure obtained after core crushing
w0 – Maximum deflection of back face
0w – Normalised maximum deflection of back face
'0w – Maximum deflection of front face
XVIII
'0w – Normalised maximum deflection of front face
WI – Kinetic energy of the front face of a sandwich structure before core crushing
WII – Kinetic energy of the whole sandwich structure after core crushing
γxy – Shear strain
εx, εy – Normal strains
θi – Rotation angle of a plastic hinge line
*ρ – Relative density
ρc – Mass density of the core
ρf – Material density of the face-sheets
τc – Shear strength of core clσ – Longitudinal yield strength of core
cYσ – Transverse yield strength of core
fYσ – Yield strength of face material
Chapter 8
pE – Plastic dissipation per unit area during core crushing
Hc – Thickness of core
ΔHc – Thickness of crushed core
cH – Final thickness of core
hf – Thickness of face-sheets
I – Impulse per unit area
crI – Critical impulse per unit area
crI – Normalised critical impulse per unit area
L – Half side length of a square plate
lm – Length of a plastic hinge line
M – Moments per unit length
M – Mass per unit area
M0 – Fully plastic bending moment
N – Membrane forces per unit length
N0 – Fully plastic membrane force
P3 – Transverse pressure per unit area
Rn – Zhao’s response number
t – Cell wall thickness of a hexagonal cell
T – Response time
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T – Dimensionless response time
v1 – Initial velocity of the front face of a sandwich structure
v2 – The velocity of a sandwich structure obtained after core crushing
w – Transverse deflection at the central point
0W – Dimensionless maximum deflection on the back face
1W – Dimensionless maximum deflection on the front face
IW – Kinetic energy per unit area of the front face of a sandwich structure before core crushing
IIW – Kinetic energy per unit area of the whole sandwich structure after core crushing
Zn – Sandwich damage number
εc – Transverse compressive strain of cellular core
εcr – Strain at yield
εD – Densification strain
mθ – Relative angular rotation rate across a plastic hinge line
δ – Dimensionless central point deflection of a square monolithic plate
*ρ – Relative density
ρ0 – Material density
ρc – Mass density of the core
ρf – Material density of the face-sheets
σY – Yield strength cdYσ – Dynamic transverse yield strength of core
clYσ – Longitudinal yield strength of core
cYσ – Static transverse yield strength of core
fYσ – Yield strength of face material
XX
1
CHAPTER ONE
INTRODUCTION
1.1 Motivation
Today, the resistant behaviour of engineering structures under blast loading is of great interest to
both engineering communities and government agencies, due to the enhanced chance of
accidents and terrorist attacks. The high pressure and loading rate produced by explosions may
result in severe damage to the structures, e.g. structural fracture, progressive collapse and large
plastic deformation and associated ballistic penetration, as shown in Figure 1-1.
Bishop Gate, London, 1993St. Mary Axe, London, 1992
US Marine Barracks, Beirut, 1983 Murrah Building, Oklahoma City, 1995
US Navy ship, Aden, 2000 Russian armored car, Chechenia, 2000
Structural fracture
Progressive collapse
Large plastic deformation& ballistic penetration
Figure 1-1. Three types of the structural damage caused by explosions
Generally the first two types of damage take place on the large constructions made from brittle
materials such as concrete and glass; while the third type of damage usually occurs on the
2
structures with ductile metals, in which a large amount of kinetic energy is dissipated during the
large plastic deformation and failure of the structures under intense dynamic loading. Making
use of this energy absorbing feature, a large number of energy absorbers made of ductile
materials have been developed, and now they are increasingly used in a wide range of
impact/blast protective applications, such as vehicle, aircraft, ship, packaging and military
industries. Unlike conventional structures which undergo only small elastic deformation, energy
absorbers have to sustain intense impact loads, so that their deformation and failure may involve
large geometry changes, strain-hardening effects, strain-rate effects and various interactions
between different deformation modes such as bending and stretching. For these reasons, ductile
metals such as low carbon steel and aluminium alloys are most widely used materials for the
energy absorbers, while non-metallic materials, e.g. fibre-reinforced plastics and polymer foams
are also common, especially when the weight is critical [1].
A systematic investigation into the structural response of energy absorbers under shock loading
will not only help to obtain a deeper insight into the deformation and failure mechanism of these
structures, but also offer them with significant enhanced energy absorption and blast resistance
performance.
1.2 Sandwich structures
As a novel and promising energy absorber, sandwich structures have been applied in a wide
range of areas. Figure 1-2 shows a typical configuration of the sandwich plate, which consists of
two metallic face-sheets and a core made from cellular solids (e.g. honeycomb or metal foam).
The face-sheets are bonded to the core with adhesive.
3
Figure 1-2. Typical configuration of a sandwich panel
During an impact, on one hand, the kinetic energy can partially be absorbed by the bending and
stretching of the plate, which is a global response of the whole structure; and on the other hand, a
large amount of the impact kinetic energy is dissipated by the plastic collapse of sandwich cores,
which deform locally. The metallic or composite face-sheets can provide the structure with higher
bending and stretching strength, while the local indentation and damage are dominated by the
behaviour of the core material, which becomes crushed as transverse stress becomes large. Cellular
solids such as polymers, metal foams and honeycombs are excellent not only in absorbing energy
during large plastic deformation, but also have other advantages, including weight savings and ease
of manufacturing etc, hence are very suitable as core materials for sandwich structures [2, 3]. The
elastic behaviours of sandwich panels have been extensively studied and well documented in several
technical books [4-7]. But the plastic damage of the cores and the associated energy-absorbing
performance of the sandwich structures are relatively less investigated, and of current interest in
academia.
1.3 Blast wave and its effect
When an explosive charge is detonated in air, the rapidly expanding gaseous reaction products
compress the surrounding air and move it outwards with a high velocity that is initially close to
the detonation velocity of the explosive. The rapid expansion of the detonation products creates
4
a shock wave (known as blast wave) with discontinuities in pressure density, temperature and
velocity. Figure 1-3 [8] shows a typical pressure-time history for a blast wave, where ta is the
time of arrival of the blast wave, Ps is the peak pressure of the blast wave and Pa is ambient air
pressure. The discontinuous pressure rise at the shock front is shown by the jump in pressure
from Pa to Ps at time ta. Figure 1-3 also shows an approximately exponential decrease in
pressure until the pressure drops down to the pre shock level at time ta+td. The free-field
pressure-time response can be described by a modified Friedlander equation,
( ) /( ) ( )[1 ] at tas a
d
t tp t p p et
θ− −−= − − (1-1)
where td is the time duration of the positive phase and θ is the time decay constant.
Figure 1-3. Typical pressure-time history of a blast wave
Apart from Ps and ta, another significant blast wave parameter is the specific impulse of the
wave during the positive phase Is, as given by
( )a d
a
t ts
tI p t dt
+= ∫ (1-2)
where p(t) is overpressure as a function of time.
According to Cole [9], the air blast loading can be qualified based on the charge weight and
stand-off distance. Generally, the amount of charge of explosive in terms of weight is converted
to an equivalent value of TNT weight (known as TNT equivalency) by a conversion factor. In
5
other words, the TNT is employed as a reference for all explosives. Sometimes scaling laws are
used to predict the properties of blast waves resulted from large-scale explosions based on tests
on a much smaller scale. The most common form of blast scaling is Hopkinson-Cranz or
cube-root scaling [10]. It states that self-similar blast waves are produced at identical scaled
distances when two explosive charges of similar geometry and of the same explosive, but of
different sizes, are detonated in the same atmosphere. It is customary to use a scaled distance Z
as follows:
1 3
RZW
= (1-3)
where R is the distance from the centre of the explosive source in meters, and W is the charge
mass of equivalent TNT in kilograms.
In some cases, the interaction of a shock with a surface can be quite complex; hence the
geometry and the state of the incident shock are quite important when studying blast interaction
with surfaces. For example, when a shock undergoes reflection, its strength can be increased
significantly. The magnification is highly non-linear and depends upon the incident shock
strength and the angle of incidence [8].
1.4 Methodology and workflow
The aim of this research is to study the structural response and energy absorbing performance of
square metallic sandwich panels with cellular cores under blast loading. The whole project is
divided into three phases. In the first phase, the performance of the sandwich structures is
investigated experimentally, numerically and analytically. Then the deformation/failure modes
of the specimens obtained in Phase I are analysed, and likewise, parametric studies are carried
out to identify the influences of several key parameters on the structural response. Finally, in the
third phase, based on the analytical results in Phase II, some design guidelines are proposed,
which help to develop an optimal configuration for the sandwich panels against blast loads. The
workflow of whole project is indicated in Figure 1-4.
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Experimentalinvestigations
Numericalsimulations
Analyticalmodeling
Deformation/failuremodes analyses
Parametricstudies
Optimal designguidelines
Phase I Phase II Phase III
Figure 1-4. Workflow of the project
1.5 Thesis organisation
The rest chapters of the thesis are arranged as follows:
Chapter 2 presents a literature review on the research status of sandwich structures under blast
loading, which covers the currently available methodologies and corresponding outputs. The
methodologies include experimental investigations, numerical simulations and analytical
modeling. Due to the composite nature of sandwich structures, the literature review has a
broader scope, which is not restricted to the sandwich structures, and the responses of
monolithic metals and cellular solids are also incorporated.
In Chapters 3 and 4, blast tests on the sandwich plates with aluminium honeycomb core and
aluminium foam core are reported, respectively. The results are discussed in terms of
deformation/failure patterns observed and quantitative data, which are obtained from the tests
by means of a ballistic pendulum with corresponding sensors: including the mid-point
deflection of the face-sheet, pressure-time history at the centre of the front face, and impulse
transfer.
Based on the experimental results, corresponding finite element simulations have been
conducted. Detailed description of the models and simulation results for the two types of the
7
panels are presented in Chapters 5 and 6, respectively. In the simulation work, the loading
process of explosive and response of the sandwich panels are investigated. Besides, a parametric
study is carried out to investigate the quantitative results of interest, which are hard to be
assessed experimentally, e.g. deformation-time history, strain distributions of the face-sheets,
influence of boundary conditions and energy absorbing contributions by different components
of the sandwich panels.
Chapter 7 presents a design-oriented approximate analytical method for the performance of the
two types of sandwich panels under blast loading. This model can be used to predict maximum
stresses and deformations, but it gives no predictions of displacement-time histories. In the
analysis, either small deflection or large deflection theories are considered, according to the
extent of panel deformation. The analysis is based on an energy balance with assumed
displacement fields. Using the proposed analytical model, an optimal design has been conducted
for square sandwich panels with a given mass per unit area, and loaded by various levels of
impulse. Effect of several key design parameters, i.e. ratio of side lengths, relative density of
core, and core thickness is discussed.
Another theoretical solution is proposed in Chapter 8, to describe the dynamic response of
square sandwich panels, in which a new yield surface is developed for the sandwich
cross-section with different core strengths. By adopting an energy dissipation rate balance
approach with the newly developed yield surface, ‘upper’ and ‘lower’ bounds of the maximum
permanent deflections and response time are obtained. Finally, comparative studies are carried
out to investigate: (1) influence of the longitudinal strength of core after compression to the
analytical predictions; (2) performances of square monolith panels and a square sandwich panel
with the same mass per area; and (3) comparison of the analytical models of sandwich beams,
circular and square sandwich plates.
The findings of this research are summarised in Chapter 9, where future work is also suggested.
8
CHAPTER TWO
LITERATURE REVIEW
2.1 Introduction
In this chapter, a literature review on the research status of sandwich structures under blast
loading is presented, which covers the currently available methodologies and corresponding
outputs. Due to the composite nature of sandwich structures, the review has a broader scope,
which is not restricted to the sandwich structures, and the responses of monolithic metals and
cellular solids are also incorporated. Since sandwich structures consist of a cellular core and two
face-sheets made of monolithic materials (frequently metals), their performance would be a
combination or coupling of the behaviours of face and core materials. In other words, the
properties of sandwich structures would reflect the characteristics of both metals and porous
media. For this reason, it is essential to include monolithic and cellular materials in the review.
Like most other mechanics problems, generally there are three approaches to analyse the
behaviour of blast loaded sandwich structures, that is, experimental investigations, numerical
simulations and analytical modeling, which are reviewed in Sections 2.2, 2.3 and 2.4
respectively.
2.2 Experimental investigations
This section consists of two parts: (1) experimental facilities and (2) deformation and fracture
modes of the structures after tests, which are further classified as those for monolithic metals,
cellular solids, and sandwich structures, respectively.
2.2.1 Experimental facilities
Two types of facilities are mainly used to dynamically measure the air blast loading and its
effect: (1) pendulums, and (2) sensors.
• Pendulums
A ballistic pendulum system can be used to measure the impulse imparted to various shock
mitigation materials subjected to air blast explosion. With a charge detonated in front of the
pendulum, the blast pressure exerted on the pendulum face causes the pendulum to rotate or
translate. Based on the rotation angle or oscillation amplitude measured, the impulse transfer
can be further estimated.
In academia, Enstock and Smith [11], and Hansen et al. [12] used a two-cable pendulum which
can be applied to measure the impulse by several kilograms’ TNT. Nurick et al. [13] has used
several four-cable pendulums for small explosive loading studies for a number of years. These
two types of pendulums are shown in Figure 2-1(a) and (b) respectively.
(a) A two-cable pendulum [11] (b) A four-cable pendulum [13]
Figure 2-1. Two types of ballistic pendulums
• Sensors
According to the parameters to be measured, sensors used for blast tests include accelerometers,
displacement transducers and pressure sensors. Figure 2-2 shows several commercially
available sensors. For different specific requirements, one can choose one or more of them for a
test.
9
(a) Accelerometer (b) Displacement transducer (c) Pressure sensor
Figure 2-2. Some sensors used for blast tests
Jacinto et al. [14] used pressure sensors and accelerometers to measure the overpressure
generated by the high explosive and acceleration of unstiffened steel plates subjected to the
impact. Apart from these two sensors, Boyd [15] also used displacement transducers for his
blast experiment. Guruprasad and Mukherjee [16] conducted experiments to test the impulsive
resistance of a sacrificial structure, on which a set of potentiometers were mounted to
dynamically record the structural deformation. In the experiments by Neuberger et al. [17, 18], a
comb-like device was applied to measure the dynamic deflections of two thick armor steel
plates.
2.2.2 Experimental observations
The deformation and fracture modes observed after tests are reviewed in terms of those of
monolithic materials, cellular solids and sandwich structures.
• Failure modes of monolithic materials
Numerous failure modes of structures have been observed in blast experiments, and these
studies can be found in several review articles and books [13, 19, 20]. Menkes and Opat [21]
conducted blast experiments on clamped beams and were the first to distinguish the three
damage modes: (I) Large inelastic deformation; (II) Tearing (tensile failure) at or over the
support; and (III) Transverse shear failure at the support. Figure 2-3 shows the transition from a
Mode I to a Mode III with increasing impulsive velocity.
10
Figure 2-3. Failure modes of a beam transiting from a Mode I to a Mode III with increasing
impulsive velocity [21]
Similar modes were later observed by Teeling-Smith and Nurick [22] for fully clamped circular
plates, and Olson et al. [23] and Nurick and Shave [24] for fully clamped rectangular plates. For
Mode I, the extent of damage is described by the amount of residual deflection (Δ). The
threshold for Mode II is taken as that impulse intensity which first causes tearing. As the load
increases, Modes II and III overlap. A pure, well defined shear failure is characterized by no
significant deformation in the central section. Mode I failure of rectangular plates under blast
loading has been reported by Rudrapatna et al. [25] and Ramajeyathilagam et al. [26]. Nurick et
al. [27] experimentally studied the thinning (necking) and subsequent tearing at the boundary of
clamped circular plates subjected to uniformly loaded air blasts. Mode I was further divided as:
Mode I (no visible necking at the boundary); Mode Ia (necking around part of the boundary);
and Mode Ib (necking around the entire boundary). Mode II failure was defined as the instant
when the maximum strain reaches the failure strain obtained from the quasi-static uniaxial
tensile test. The experimental investigations for Mode II failure can be found in the literature
[22-27]. For square plates, tearing was observed to start at the middle of the boundary and
progress along the boundary towards the corners. Hence, some additions to Mode II failure were
11
12
reported [24]: Mode II*: partial tearing at the boundary; Mode IIa: complete tearing with
increasing mid-point displacement; and Mode IIb: complete tearing with decreasing mid-point
displacement. Similar failure modes have also been found for structures other than beams and
flat plates such as stiffened panels [28-30]. Mode III is characterised by insignificant flexural
deformation at most cross sections, and shear failure occurs at the supports in the early stages of
the response and generally exhibits a local response. This type of failure mode was studied by
Li and Jones for beams [31] and plates [32], and Cloete et al. [33] for centrally supported
structures. Mode III failure criteria of plastic shear sliding was adopted using a shear strain
failure criteria as proposed by Wen et al. [34] for beams. The parameters of this failure model
with respect to the circular plates have been presented by Wen and Jones [35].
• Failure modes of cellular solids
Hanssen et al. [12] used a ballistic pendulum to test the blast loading behaviour of rectangular
aluminium foam layers attached to the pendulum face with and without metallic cover panels. It
has been observed that (1) the non-covered low-density panels were all fragmented but
maintained structural integrity when a over plate was attached, (2) the degree of panel
fragmentation increased with charge mass, (3) no severe fragmentation of the high density foam
panels without cover plate took place, and (4) the front surface of the foam penal as well as the
front cover has attained an inwardly curved shape. This curvature extended in both directions of
the panel plane, i.e. a double curvature (concave shape) was obtained. The final depth of
deformation at the panel centre relative to the panel edges is termed ‘dishing’ by the authors.
• Failure modes of sandwich structures
To date, very few physical blast tests on the sandwich structures have been reported, due to high
cost and the lack of testing and measuring means.
Radford et al. [36, 37] used an aluminium foam projectile to simulate localized blast loading of
the clamped sandwich beams and circular plates, enabling the transient transverse response of
the impulsively loaded structures to be explored, as shown in Figure 2-4.
Figure 2-4. Using aluminium foam projectiles to simulate non-uniform shock loading [36]
The deflection profiles of sandwich beams with a metal foam core are shown in Figure 2-5. The
profiles of the beams are continuously curved due to the traveling plastic hinges, and significant
amounts of core crush can be observed in the central area. The test results show an acceptable
agreement with the numerical models proposed by the authors [36, 37]. The discrepancy is
mainly attributed to the fact that the foam projectiles do not provide effective impulsive loading.
In other words, the loading time of the metal foam impact is greater than that of a ‘real’ blast
load.
Figure 2-5. Deflected profiles of dynamically loaded metal foam core sandwich beams [36]
Nurick et al. [38] tested small size circular sandwich plates with a hexagonal aluminium
13
honeycomb core subjected to uniformly distributed blast loading, in which the faces were not
adhered to the core structure. The experiments identified three mechanisms of interaction
between the front and back plates, with the increase of impulse level: Mechanism I – Front and
back plates deform, with the honeycomb crushing following the plate profile in the form of a
sinusoidal shape function. Mechanism II: The rate of change of displacement with increasing
impulse for the front and back plate changes to a different linear gradient, and the honeycomb
crushing is spread over a larger area. Mechanism III: The front plate is torn, compressing the
honeycomb into a dish shape. A typical cross-section of the face-sheets and honeycomb core
(Mechanism II) is shown in Figure 2-6.
(a) Deformation mode of the face-sheets (Mechanism II)
(b) Deformation mode of the honeycomb core (Mechanism II)
Figure 2-6. Typical cross-section of the face-sheets and honeycomb core of a circular
sandwich plate (Mechanism II) [38]
2.3 Numerical simulations
Blast testing is extremely expensive and time consuming, while numerical simulations
(frequently Finite Element Analysis (FEA)), if adequately formulated and accurately realised,
help to greatly reduce the volume of laboratory and field blast tests. FEA offers the possibility to
predict distribution of stress/strain and wave propagation that are difficult to be measured
14
15
experimentally, and give the detailed process of internal structural deformation and failure
which can be hardly observed. Besides, FEA can be used to identify the influence of critical
parameters on the structural behaviour under certain conditions. Due to the highly transient and
nonlinear nature of Explosion Mechanics, the corresponding FEA often involves the dynamic
problems associated with large deformation, high pressure/temperature/strain rate, failure of
material, solid-fluid interaction etc. Finite element models solve the problems by discretising
the related equations which govern the process of explosion and consequent structural response,
and setting some initial conditions.
The first part of this section is a short introduction to the basic formulations in the impulsive
loading simulations. Then a review is presented on the main approaches to model the blast loads
and behaviours of various materials, which include monolithic metals, cellular solids as well as
sandwiches. Finally, the current commercial FEA packages for dynamic analyses are briefly
reviewed and compared.
2.3.1 Basic formulations
Generally, most of the Explosion Mechanics problems involving large deformations and
solid/liquid interactions are described by three basic formulations [39], i.e. (1) Lagrangian
methods; (2) Eulerian methods; and (3) hybrid methods.
• Lagrangian methods
In these methods, the mesh is attached with the mass particles and moves and deforms with the
material. They can handle moving boundaries or multiple materials very naturally, but perform
pooly or even fail when large deformations take place, due to the distortion of the elements.
• Eulerian methods
These methods, on the contrary, use a fixed mesh, which does not move with materials. They
are suitable to solve the problems with large deformations, but have difficulties when the
computing domain includes interactions of multiple materials or irregular surfaces.
• Hybrid methods
The hybrid methods seek a compromise between the Lagrangian methods and Eulerian methods.
A typical hybrid method is ALE (Arbitary-Lagrangian-Eulerian) method, which allows the mesh
within any material region to be continuously adjusted in predefined ways as a calculation
proceeds, thus providing a continuous and automatic rezoning capability. Therefore, it is
suitable to use an ALE approach to analyse solid and fluid motions when material strain rate is
large and significant (for example, the detonation of explosive and volume expansion of
explosion products).
2.3.2 Modeling blast loads
• Defining the pulse-time curve or velocity field directly
The idea of directly defining the pulse-time curve or velocity field on the structure is quite
straightforward and may be the easiest way to model blast loads. However, the coupling effects
of the loads and structures, such as the change of structural curvature and shock wave
reflections, are not considered. Therefore, sometimes the simulation performance of this method
is not satisfactory.
• Defining blast loads using blast pressure functions
The blast loads can be conveniently calculated using blast pressure functions such as ConWep
[39], which was developed by the US Army. The ConWep function can produce non-uniform
loads exerted on the top surface of the plates. This blast function can be used in two cases: free
air detonation of a spherical charge, and the ground surface detonation of a hemispherical
charge. The input parameters include equivalent TNT mass, type of blast (surface or air),
detonation location, and surface identification for which the pressure is applied. The pressure is
calculated based on the following equation
2 2r( ) cos (1 cos 2cos )iP P Pτ θ θ= ⋅ + ⋅ + − θ (2-1)
whereθ is the angle of incidence, defined by the tangent to the wave front and the target’s
surface; is the reflected pressure at normal incident angle; and is the incident pressure. It rP iP
16
can be seen that ConWep calculates the reflected pressure values and applies them to the
designated surfaces by taking into account the angle of incidence of the blast wave. It updates
the angle of incidence incrementally and thus account for the effect of surface rotation on the
pressure load during a blast event. The drawback of ConWep is that it cannot be used to
simulate the purely localised impulsive loads produced by explosive flakes or prisms. Some
simulation work using ConWep can be found in the literature [17, 18, 40].
• Modeling the explosive as a material
In this method, the explosive is modeled as a material. When the explosive is detonated, its
volume expands significantly and interacts with the structure. The contact force between the
expanded explosive product and structure is then calculated. The expansion of the explosive is
defined by three parameters: position of the detonation point, burn speed of the explosive and
the geometry of the explosive. The explosive materials are usually simulated by the use of the
Jones-Wilkins-Lee (JWL) high explosive equation of state, which describes the pressure of the
detonation [41].
2.3.3 Modeling the materials of targets
• Modeling monolithic materials
Two commonly used material models for metals are summarised here. The Johnson-Cook
material model is a widely used constitutive relation, which describes plasticity in metals under
strain, strain rate, and temperature conditions [42].
( )(1 ln *)(1 * )np
y A B c Tσ ε ε= + + − m (2-2)
where A, B, C, m and n are material constants; pε is effective plastic strain; *ε = pε / 0ε ,
being effective plastic strain rate, for 0ε =1s-1; T* = (T-Troom)/(Tmelt-Troom). Typical values of these
constants for a variety of materials are found in Johnson and Cook [42].
If only the strain rate effect is considered, the above model can be reduced to another well
known material model, namely the Cowper-Symonds relationship, in which the strain rate is
17
calculated for time duration from the start to the point, where the strain is nearly constant from
the equivalent plastic strain time history [20]. In the Cowper-Symonds model, the dynamic yield
stress (σdY) can be computed by
1/
1p
dY Y Dεσ σ
⎛ ⎞= +⎜⎜
⎝ ⎠⎟⎟ (2-3)
where σY and σdY are the static and dynamic yield tresses and D and p are material constants.
• Modeling cellular solids
A detailed review of constitutive models for metal foam applicable to structural impact and
shock analyses has been presented by Hanssen et al. [43]. The models have different
formulations for the yield surface, hardening rule and plastic flow rule, while fracture is not
accounted for in any of them.
• Modeling sandwich structures
In recent years a number of micro-architectured materials have been developed for uses as the
cores of sandwich structures for application in blast-resistant constructions. Some of the current
available topologies are shown in Figure 2-7: pyramidal core, diamond-celled core, corrugated
core, hexagonal honeycomb core, and square honeycomb core [44].
(a) pyramidal core, (b) diamond- celled core, (c) corrugated core, (d) hexagonal honeycomb
core, and (e) square honeycomb core
Figure 2-7. Sketches of several sandwich core topologies [44]
18
The cellular cores are assumed to be made from an elastic, perfectly-plastic solid with yield
strain εY. Their normalised transverse compressive strength nσ and longitudinal strength lσ were
predicted in Xue and Hutchinson [45]. This approach can significantly simplify the numerical
analyses for the cellular cores yet with acceptable accuracy [44, 74-77]
2.3.4 Commercial FEA packages for blast loading simulations
• LS-DYNA
LS-DYNA is a general-purpose, explicit finite element program used to analyse the nonlinear
dynamic response of three-dimensional inelastic structures. Its abundant material models, fully
automated contact analysis capability and error-checking features have enabled users to solve
many complex impact and explosion problems. The main application areas of LS-DYNA
include: Large deformation dynamics and contact simulations, crashworthiness simulation,
occupant safety systems, metal, glass, and plastics forming, multi-physics coupling, failure
analysis etc.
• MSC.Dytran
Dytran is an explicit FEA solution for analysing complex nonlinear behavior involving
permanent deformation of material properties or the interaction. Dytran combines structural,
material flow, and fluid-structure interaction (FSI) analyses in a single package, and uses a
combination of Lagrangian and Eulerian solver technology to analyse short-duration transient
events that require finer time step to ensure a more accurate solution. However, there are some
drawbacks with Dytran. For example, material models supported by Dytran are quite limited,
particularly for materials such as soils and rocks. No 2D computation is available, and thus
axisymmetric cases have to be treated as 3D problems, and time cost increases consequently.
Also the contact types are not sufficient to model complex impact problems.
• ABAQUS
ABAQUS is advanced FEA software capable of solving very complex and highly nonlinear
19
20
problems. ABAQUS product suite consists of two solvers: ABAQUS/Standard and
ABAQUS/Explicit. ABAQUS/Standard is a general-purpose solver that uses traditional implicit
integration scheme to solve finite element analyses. ABAQUS/Explicit uses explicit integration
scheme to solve highly nonlinear transient dynamic and quasi-static analyses. Each of these
solvers also comes with additional, optional modules for specific applications or requirements.
However, its performance in the simulations of explosion/impact problems is considered to be
weaker than LS-DYNA.
• AUTODYNA
AUTODYNA is a versatile explicit analysis tool for modeling the nonlinear dynamics of solids,
fluids, gases and their interaction. It uses a multi-solver approach allowing alternative numerical
techniques to be applied to the different regions of an event, where Lagrangian finite element
solvers are used to model the structural dynamics (solids, shells, beams); Eulerian finite volume
solvers are used to model the fluid and/or gas dynamics, and a mesh free partical slover (SPH)
is used to model the large deformation and fragmentation of brittle materials (ceramics,
concrete). Different solvers can be applied simultaneously to model the various regions of an
analysis and a solution is obtained by allowing these regions to interact in both space and time.
2.4 Analytical modeling
Theoretical or analytical impulsive loaded models provide valuable information for locating
damage and establishing criteria for acceptance and/or repair of structural components.
Analytical solutions that can describe deformation/damage would enable one to recognise
impact parameters. Parametric studies can then show how the failure of structures varies with
impact parameters. Furthermore, analytical solutions provide benchmark solutions for more
refined finite element analysis. In this section, the analytical models are reviewed for: (1)
monolithic metals, (2) cellular solids and (3) sandwich structures.
2.4.1 Analytical models for monolithic metals
Theoretical models for monolithic structural members have been extensively investigated,
Reviews on the relevant literature before 1990s can be found in Jones [20] and Nurick and
Martin [46], and the main points with the supplement of some recent advances are summarised
in this section. Based on the nature of the analytical models, they can be roughly classified into
three categories: (1) modal approximations, (2) rigid-plastic methods, and (3) energy solutions.
• Modal approximations
In modal approximations, the dynamic response is taken in mode form, for example, a velocity
field with separated functions for spatial and temporal variables
* *i j i j( , ) ( )W X t V X= Φ (2-4)
where is the velocity field. The function i j( , )W X t *iΦ is called the mode or mode shape; the
scalar velocity of a characteristic point is specified such that *V *i1 1− ≤ Φ ≤ . It should be noted
that the term ‘modal approximation’ here is conceptually different from that of the ‘mode
method’ commonly used in the vibration within the elastic limit.
Generally, the deformation that develops can be divided into an initial transient phase where the
pattern or location of deformation is continually changing and a modal phase where the pattern
is constant. During the transient phase the pattern of deformation steadily evolves from the
initial velocity distribution imposed at impact to a mode configuration. After attaining the
velocity distribution of a stable mode configuration, the pattern of deformation remains constant
for some period of time. In most cases a substantial part of the impact energy is dissipated in a
mode configuration during the second phase of deformation. A property of this class of problem
is that, however the motion is started, the response tends toward a modal form, and the final
stage of motion is generally in this form. The ‘minimum 0Δ ’ device is proposed for this class of
problems as a means of obtaining an approximate solution by assuming the simple mode
solution to hold for the entire motion. The amplitude of the approximating mode solution can be
chosen so that the difference between the given initial velocity and that of the mode solution is
minimised in a mean square sense.
21
22
The mode approximation technique was proposed initially for the class of rigid plastic problems
where rigid-perfectly plastic behaviour is assumed, and the dynamic loading is idealised as
impulsive, and linear small deflection form is adopted for the equations of dynamics and
kinematics [47, 48]. When large deflection effect is taken into account, the entire response
process should be treated in terms of two mode solutions valid, respectively, for small and finite
deflections [49-51]. On the other hand, some analyses for large deflections with only stretching
effect considered have been proposed [52-54]. Besides, the second-order effects in dynamic
response, such as strain rate effect and work hardening have been invesigated [50, 51]. Detailed
remarks on the mode approximation methods are in [55].
• Rigid-plastic methods
Rigid plastic methods were developed and were shown to give good agreement when the ratio
of initial kinetic energy to elastic strain energy is larger than 10, and the load duration is
sufficiently short with respect to the natural period of the structure [20]. The analysis of the
deformation is based on the assumption that the influence of elasticity is neglected, and uses a
kinematically admissible velocity field to describe the motion of a structure.
The early theoretical studies of this class started in 1950s and predicated small deflections for
beams [56] and circular plates [57, 58], in which only bending action was considered. When the
maximum deflection exceeds about twice the plate thickness, the final deflections predicated by
small deflection theory become much larger than the experimental values. This means that the
effect of membrane forces is significant to the response, and during the deformation, internal
energy dissipation occurs predominantly through the action of membrane forces on middle
surfaces strain. Jones [59] attempted to link the two distinct stages of plastic strain and describe
the behaviour of plates dynamically loaded with deflections in the range where both bending
moments and membrane forces are important, and the theory proposed predicts the experiment
with reasonable accuracy. Some other studies have been made on the strain rate effect [60],
shape of pulses [20] and dynamic transverse shear effects [61, 62] in the large deflection.
23
Compared with circular plates, much fewer investigations have been made into rectangular
plates. Small deflection analyses for rectangular plates with different boundary conditions can
be seen in [63, 64]. To solve the associated differential equations, numerical approximation
using a computational tool is needed, and no explicit solution is available. In addition, the rigid
plastic analytical solutions for fully clamped rectangular plates in large deflections by solving
kinematic differential equations have yet been reported to date.
• Energy solutions
Energy solutions are essentially design-oriented approximate analytical methods, which are
excellent for predicting peak (maximum) stresses, shears and deformations in blast loaded
structural components. These energy solutions give no predictions of displacement-time
histories, since in assessing the behaviour of a blast loaded structure it is often the case that the
calculation of final states is the principal requirement for a designer.
There are well-defined steps to the solution of a particular structure using this method. Firstly, a
mathematical representation of the deformed shape is selected for the structure which satisfies
all the necessary boundary conditions relating the displacement. Then, by operating on the
deformed shape, the curvature and then the strain of deformation is obtained, from which strain
energy can be evaluated. Next, a calculation of total kinetic energy delivered on the structure is
made, and by equating the kinetic energy acquired to the strain energy produced in the structure,
it is possible to quantify particular aspects of response such as maximum displacement,
maximum strains and maximum stresses. Energy solutions can be used for either elastic or
plastic analysis. The detailed procedure of analysing elastic and plastic beams was given in [10,
65], and the solutions for circular and rectangular plates can be seen in [10] and [66],
respectively.
Instead of using a balance of total energy dissipation, some approaches are based on the balance
of energy dissipation rate, i.e. the energy absorbed per unit time duration. In this way, it is
possible to obtain the structural response time. Jones [67] and Taya and Mura [78] applied this
24
approach to estimate the permanent transverse deflections of beams and arbitrarily shaped plates
which are subjected to large dynamic loads. The influence of finite-deflections or geometry
changes is retained in the analysis but elastic effects are disregarded. According to bending-only
theory, Yu and Chen [68] developed two membrane force factors which reflect the effect of the
stretching to formulate the governing equations for large deflections.
In theory, energy-based solutions can be used to model the beams and plates with arbitrary
geometries in both small and large deflections, and thus has a good potential of describing the
dynamic response of composite structures such as laminates or sandwiches.
2.4.2 Analytical models for cellular solids
The analytical models for the responses of cellular solids under impact/blast loading are
restricted in the one dimensional domain and they highly depend on their material constitutive
relationship and deformation mode.
There are two possible modes of cellular solids deformation, i.e., (a) homogeneous deformation
and (b) progressive collapse. A 1-D metal foam column with the two deformation modes is
shown in Figure 2-8. Under homogeneous deformation (Figure 2-8(a)), metal foam deforms
homogeneously over the entire volume of the sample. In this case, the absorbed energy per unit
volume of the foam material for a given level of deformation can be calculated as the area under
the stress–strain diagram. In the case of progressive collapse (Figure 2-8(b)), the same
deformation Δ is reached by complete densification of the portion of the foam close to the point
of load application, while the rest of the foam is assumed not to deform at all. At the end of
complete densification, the final deformation, Δmax, in case of both the modes are the same.
(a) Homogenous deformation (b) Progressive deformation
Figure 2-8. Deformation patterns of cellular solids
Lopatnikov et al. [69] developed an analytical model to determine the energy absorption of
metal foams under the two deformation modes using the ‘elastic-plastic-rigid (E-P-R)’ material
constitutive relationship (Figure 2-9). It has been identified that the progressive collapse mode
of deformation can absorb more energy than homogeneous deformation prior to full
densification.
Reid and co-workers [70, 71] used a ‘rigid-perfectly-plastic-locking (R-P-P-L)’ model (Figure
2-9) to idealise cellular materials, where the foam is considered fully densed at the maximum
possible strain εD, and the stress level jumps from σcr to σ*. Based on the material model, the
dynamic progressive crushing behaviour of the foam under shock loading was analysed, and the
theoretical predictions compared well with the experimental data.
25
σ*
σY
εD
σ
εεM
σM
R-P-P-L Model
E-P-R Model
σcr
Metal foam
εY
Figure 2-9. Material models of cellular solids
The progressive collapse of low density cellular materials may also be reasonably idealised by
one-dimensional mass-spring models [72, 73], which should be solved using numerical
techniques.
2.4.3 Analytical models for sandwich structures
A series of analytical models have been developed by Fleck and co-workers, to predict the
dynamic response of sandwich beams and circular sandwich panels under a uniform shock
loading [44, 74] or a non-uniform one over a central patch [75]. The sandwich structures
comprise steel face-sheets and cellular solid cores, with ends fully clamped. The response to
shock loading is measured by the permanent transverse deflection at the mid-span of the
structures. In the models, a number of approximations have been made to make the problem
tractable to an analytical solution. Principally, these are
(i) the 1-D approximation of the shock events;
(ii) separation of the phases of the response into three main sequential phases:
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27
Phase I: This is actually a 1-D air-structure interaction process during the blast event,
resulting in a uniform velocity of the outer face-sheet.
Phase II: The core crushes and the velocities of the faces and core become equalised by
momentum sharing.
Phase III: This is the retardation stage at which the structure is brought to rest by plastic
bending and stretching. The problem under consideration here is turned into a
classical one for monolithic beams or plates, which has been extensively studied and
presented in the book by Jones [20].
(iii) neglect of the support reaction during the shock event and during the core compression
phases;
(iv) a highly simplified core constitutive model wherein the core is assumed to behave as an
ideally plastic locking solid with a homogeneous deformation pattern; and
(v) neglect of the effects of strain hardening.
Despite these approximations, the analysis has been shown to compare well with corresponding
numerical simulations [44, 74-76].
Based on the above three-stage procedure, Hutchinson and Xue [77] proposed a simplified
analytical model for the rectangular sandwich panels with two sides fixed using the energy
solutions. To estimate the deflection produced by the kinetic energy in Stage III, a relatively
simple estimate of the energy dissipated in bending and stretching was obtained using
approximations for the deflection that neglect details of the dynamics. The energy dissipated by
plastic deformation was sought in terms of the central deflection of the plate.
To date, no systematic investigations have been reported on the peripherally clamped
rectangular sandwich panels under blast loading, because of their non-axisymmetric geometry,
for which the principal stress directions are unknown in advance, a complete theoretical analysis
of the dynamic response is rather complicated, especially when deformation is large.
28
2.5 Summary
In this chapter, a literature review is presented on the current status of experimental
investigations, numerical simulations and analytical modeling for monolithic metals, cellular
solids and sandwich structures under shock loading.
The experimental facilities such as pendulums and sensors for monolithic structural members
can also be used for cellular solids and sandwich structures. Three failure models on the
impulsively loaded monolithic beams and panels have been distinguished: (I) large inelastic
deformation; (II) tearing (tensile failure) at or over the support; and (III) transverse shear failure
at the support. Similar tests have been conducted on the metal foam panels, and the results show
that the non-covered low-density panels were all fragmented but maintained structural integrity
when a cover plate was attached, and a dishing failure with an inwardly curved shape was
obtained on the front face. The deformation pattern of sandwich structures is characterized by
the curved face-sheets and crushed core, which is considered as the main contribution of energy
dissipation. To date, there have been no experimental studies on air blasting response of sandwich
panels, whether they are circular or rectangular/square. There is a great need for experimental data.
To simulate blast impact and corresponding structural response numerically, suitable
formulations and software packages must be chosen, which should be capable of solving the
problems involving large plastic deformation, large strain rate and fluid/solid interaction. A few
constitutive relationships are available to model the mechanical behaviours of monolithic metals
and porous media with the second-order effects (e.g. rate effect, strain hardening etc) taken into
account. For simplicity, some researchers assumed the cellular cores of the sandwich structures
to be made from an elastic, perfectly-plastic solid, to reduce the computing complexity.
Analytical modeling on the impulsively loaded structural members has been extensively studied
on monolithic metals. Generally, there are three approaches: (1) modal approximations, in which
the dynamic response is taken in mode form, i.e. a velocity field with separate functions for
29
spatial and temporal variables; (2) rigid-plastic model, which is based on the assumption that the
influence of elasticity is neglected, and uses a kinematically admissible velocity field to describe
the motion of a structure; and (3) energy solutions, in which the calculation of the maximum
deformation is made by equating the kinetic energy acquired to the strain energy produced in the
structure. The analytical models for the responses of cellular solids under impact/blast loading
are restricted in the one dimensional domain, and two deformation modes are possible: (a)
homogeneous deformation and (b) progressive collapse. The structural response of sandwich
structure is essentially a combination of the deformations of monolithic solids and porous media,
and can be divided into three phases: Phase I – Front face deformation; Phase II – Core crushing;
and Phase III – Overall bending and stretching. For simplicity, the core is assumed to behave as
an ideally plastic locking solid with a homogeneous deformation pattern, and dynamic effect
and strain hardening are neglected. However, there have been no analytical investigations
conducted so far for rectangular/square sandwich plates, due to their much more complex nature.
From the literature review, we have shown that investigations into the rectangular/square
sandwich panels with cellular core under blast loading are still very much limited and some
crucial aspects in relation to the experiments, detailed deformation mechanisms and associated
mechanics remain to be investigated systematically. This thesis attempts to resolve these issues,
which are crucial to future optimal design of such sandwich panels subject to blast loading.
30
CHAPTER THREE
EXPERIMENTAL INVESTIGATION INTO THE
HONEYCOMB CORE SANDWICH PANELS
3.1 Introduction
A large number of experiments have been conducted to test the blast resistance of square
sandwich panels with metallic face-sheets and honeycomb cores subjected to explosion, and the
experimental results are reported and discussed in this chapter. The experiment program was
designed to investigate the effects of face and core configurations and impulse levels on the
structural response. The experimental results were classified into two categories: (1)
deformation/failure modes of specimens observed in the tests, which are further grouped into
those for front face, core and back face, respectively; and (2) quantitative results, which include
the impulse on sandwich panel, permanent central point deflection of the back face and
pressure-time history at the central point of front face. Finally, a parametric study is presented to
analyse the influences of several key parameters on the performance of sandwich panels.
Further analysis of the test results is presented in subsequent chapters.
3.2 Specimen
The square specimens used consist of two face-sheets and a core of honeycomb. The face-sheets
were made of Al-2024-O aluminium alloy. Its nominal mechanical properties are as follows: E
(Young’s modulus)=73.1GPa; G (Shear modulus)=28GPa; υ (Poisson’s ratio)=0.33; and σY
(Yield stress)=75.8MPa. The HexWeb® aluminium honeycomb core comprises a square array of
normal hexagonal cells (the angle between two neighboring walls is 120°). The designation and
mechanical properties of the cells are available from [79]. A single honeycomb cell has two
critical geometrical parameters, that is, cell length le and wall thickness t, as indicated in Figure
3-1. Figure 3-1 also shows the dimensions of sandwich panels used in the tests. The side length
L and thickness of core structure Hc are constant and equal to 310mm and 12.5mm respectively.
Three different thicknesses hf are adopted for the face-sheets: TN (hf=0.5mm), MD (hf=0.8mm)
and TK (hf=1.0mm). For each test condition, two nominally identical specimens were tested.
Each specimen is denoted a unique number. For example, specimen 1/8-5052-0.0020-TK-1
indicates a sandwich panel with honeycomb core of le=1/8″ (3.18mm), made of aluminium 5052,
t=0.0020″ (0.051mm), and with a thick (TK) face sheet (hf=1.0mm) and is the first of the two
identical tests. Similarly, ACG-1/4-TK-1 stands for a sandwich panel with ACG honeycomb,
le=1/4″ (6.35mm), t=0.066mm, with a thick face sheet (hf =1.0mm), and is the first test.
L
Lhf
Hct le
Honeycomb core
Figure 3-1. Geometry and dimension of the honeycomb core specimen
All the specimens were divided into four groups as indicated in Tables 3-1 ~ 3-4. Each group was
designed to study the effect of one or two particular parameters on the structural response of the
panels. For example, the honeycomb cores in Group 1 have the same cell size (le) but increased foil
thicknesses (t), and then based on the test results the contribution of foil thickness can be identified.
Likewise, in Groups 2 and 4 the effects of cell size and mass of charge are investigated, respectively.
Group 3 is a special one, in which two cores have different cell sizes and foil thicknesses but similar
mass (and hence relative density), so that the effect of relative density of core can be analysed. It
should be noted that the two cores in this group were made of two slightly different aluminium
alloys, but have almost identical yield stress, and thus the effect of materials can be ignored. In
addition, three different face-sheet thicknesses are adopted in Groups 1-3 and thus their effect can
also be studied.
31
Table 3-1. Sandwich panels of Group 1, where the effects of foil thickness and face thickness are investigated
Name of specimen Cell size le(mm)
Nominal foil thickness t
(mm)
Face-sheets thickness hf
(mm)
Mass of core mo (g)
Cell wall material Mass of charge mh (g)
Impulse I (Ns) Back face deflection w0
(mm) 1/8-5052-0.0020-TK-1 3.18 0.051 1.0 156.1 Al-5052-H39 20 16.93 12.11/8-5052-0.0020-TK-2 3.18 0.051 1.0 159.3 Al-5052-H39 20 18.13 11.61/8-5052-0.0020-MD-1 3.18 0.051 0.8 158.2 Al-5052-H39 20 17.24 20.61/8-5052-0.0020-MD-2 3.18 0.051 0.8 155.9 Al-5052-H39 20 16.55 16.41/8-5052-0.0020-TN-1 3.18 0.051 0.5 154.9 Al-5052-H39 20 16.93 30.81/8-5052-0.0020-TN-2 3.18 0.051 0.5 156.6 Al-5052-H39 20 16.85 29.61/8-5052-0.0015-TK-1 3.18 0.038 1.0 109.3 Al-5052-H39 20 17.49 15.31/8-5052-0.0015-TK-2 3.18 0.038 1.0 111.0 Al-5052-H39 20 16.64 16.61/8-5052-0.0015-MD-1 3.18 0.038 0.8 108.7 Al-5052-H39 20 17.10 22.71/8-5052-0.0015-MD-2 3.18 0.038 0.8 107.7 Al-5052-H39 20 18.10 19.11/8-5052-0.0015-TN-1 3.18 0.038 0.5 109.2 Al-5052-H39 20 17.38 32.01/8-5052-0.0015-TN-2 3.18 0.038 0.5 108.8 Al-5052-H39 20 17.27 32.21/8-5052-0.0010-TK-1 3.18 0.025 1.0 84.7 Al-5052-H39 20 17.36 19.71/8-5052-0.0010-TK-2 3.18 0.025 1.0 84.8 Al-5052-H39 20 17.60 20.01/8-5052-0.0010-MD-1 3.18 0.025 0.8 85.4 Al-5052-H39 20 17.48 21.11/8-5052-0.0010-MD-2 3.18 0.025 0.8 85.5 Al-5052-H39 20 18.11 24.81/8-5052-0.0010-TN-1 3.18 0.025 0.5 87.6 Al-5052-H39 20 18.40 35.31/8-5052-0.0010-TN-2 3.18 0.025 0.5 85.8 Al-5052-H39 20 18.21 40.11/8-5052-0.0007-TK-1 3.18 0.018 1.0 59.0 Al-5052-H39 20 17.36 27.01/8-5052-0.0007-TK-2 3.18 0.018 1.0 58.5 Al-5052-H39 20 17.50 25.11/8-5052-0.0007-MD-1 3.18 0.018 0.8 58.6 Al-5052-H39 20 17.72 29.51/8-5052-0.0007-MD-2 3.18 0.018 0.8 57.7 Al-5052-H39 20 17.78 30.11/8-5052-0.0007-TN-1 3.18 0.018 0.5 57.9 Al-5052-H39 20 17.21 50.11/8-5052-0.0007-TN-2 3.18 0.018 0.5 58.2 Al-5052-H39 20 18.03 52.2
32
Table 3-2. Sandwich panels of Group 2, where the effects of cell size and face thickness are investigated
Name of specimen Cell size le(mm)
Nominal foil thickness t
(mm)
Face-sheets thickness hf
(mm)
Mass of core mo (g)
Cell wall material Mass of charge mh (g)
Impulse I (Ns) Back face deflection w0
(mm) 1/8-5052-0.0015-TK-1 3.18 0.038 1.0 109.3 Al-5052-H39 20 17.49 15.3 1/8-5052-0.0015-TK-2 3.18 0.038 1.0 111.0 Al-5052-H39 20 16.64 16.6 1/8-5052-0.0015-MD-1 3.18 0.038 0.8 108.7 Al-5052-H39 20 17.10 22.7 1/8-5052-0.0015-MD-2 3.18 0.038 0.8 107.7 Al-5052-H39 20 18.10 19.1 1/8-5052-0.0015-TN-1 3.18 0.038 0.5 109.2 Al-5052-H39 20 17.38 32.0 1/8-5052-0.0015-TN-2 3.18 0.038 0.5 108.8 Al-5052-H39 20 17.27 32.2
5/32-5052-0.0015-TK-1 3.97 0.038 1.0 101.3 Al-5052-H39 20 17.62 18.7 5/32-5052-0.0015-TK-2 3.97 0.038 1.0 100.6 Al-5052-H39 20 -- 18.6 5/32-5052-0.0015-MD-1 3.97 0.038 0.8 100.8 Al-5052-H39 20 18.02 25.1 5/32-5052-0.0015-MD-2 3.97 0.038 0.8 101.2 Al-5052-H39 20 -- 32.6 5/32-5052-0.0015-TN-1 3.97 0.038 0.5 101.4 Al-5052-H39 20 -- -- 5/32-5052-0.0015-TN-2 3.97 0.038 0.5 102.2 Al-5052-H39 20 17.89 41.7
33
Table 3-3. Sandwich panels of Group 3, where the effects of average core mass and face thickness are investigated
Name of specimen Cell size le(mm)
Nominal foil thickness t
(mm)
Face-sheets thickness hf
(mm)
Mass of core mo (g)
Cell wall material Mass of charge mh (g)
Impulse I (Ns) Back face deflection w0
(mm) 1/8-5052-0.0010-TK-1 3.18 0.025 1.0 84.7 Al-5052-H39 20 17.36 19.7 1/8-5052-0.0010-TK-2 3.18 0.025 1.0 84.8 Al-5052-H39 20 17.60 20.0 1/8-5052-0.0010-MD-1 3.18 0.025 0.8 85.4 Al-5052-H39 20 17.48 21.1 1/8-5052-0.0010-MD-2 3.18 0.025 0.8 85.5 Al-5052-H39 20 18.11 24.8 1/8-5052-0.0010-TN-1 3.18 0.025 0.5 87.6 Al-5052-H39 20 18.40 35.3 1/8-5052-0.0010-TN-2 3.18 0.025 0.5 85.8 Al-5052-H39 20 18.21 40.1
ACG-1/4-TK-1 6.35 0.066 1.0 86.9 Al-3104-H19 20 17.89 20.9 ACG-1/4-TK-2 6.35 0.066 1.0 85.5 Al-3104-H19 20 18.14 19.6 ACG-1/4-MD-1 6.35 0.066 0.8 85.7 Al-3104-H19 20 16.90 24.8 ACG-1/4-MD-2 6.35 0.066 0.8 85.0 Al-3104-H19 20 17.72 24.6 ACG-1/4-TN-1 6.35 0.066 0.5 86.2 Al-3104-H19 20 17.72 39.3 ACG-1/4-TN-2 6.35 0.066 0.5 86.1 Al-3104-H19 20 17.53 40.0
34
Table 3-4. Sandwich panels of Group 4, where the effect of charge mass is investigated
Name of specimen Cell size le(mm)
Nominal foil thickness t
(mm)
Face-sheets thickness hf
(mm)
Mass of core mo (g)
Cell wall material Mass of charge mh (g)
Impulse I (Ns) Back face deflection w0
(mm) ACG-1/4-TK-1 6.35 0.066 1.0 86.9 Al-3104-H19 20 17.89 20.9 ACG-1/4-TK-2 6.35 0.066 1.0 85.5 Al-3104-H19 20 18.14 19.6 ACG-1/4-TK-3 6.35 0.066 1.0 87.0 Al-3104-H19 15 15.08 17.5 ACG-1/4-TK-4 6.35 0.066 1.0 86.1 Al-3104-H19 15 14.74 17.6 ACG-1/4-TK-5 6.35 0.066 1.0 84.9 Al-3104-H19 25 18.47 22.4 ACG-1/4-TK-6 6.35 0.066 1.0 85.0 Al-3104-H19 25 21.11 21.8 ACG-1/4-TK-7 6.35 0.066 1.0 86.2 Al-3104-H19 30 22.13 25.1 ACG-1/4-TK-8 6.35 0.066 1.0 85.3 Al-3104-H19 30 22.67 26.1
35
3.3 Experimental set-up
A four-cable ballistic pendulum system was employed to measure the impulse imparted on the
blast-loaded specimen. Several similar pendulums have been used for a number of years by
Nurick and co-workers for small explosive loading studies [13, 22-24, 27, 33, 38]. Figure 3-2
shows a photograph of the pendulum set-up. When the charge (standard TNT in the present tests)
was detonated in front of the pendulum face, the impulsive load produced by explosive pushed
the pendulum to translate. Based on the oscillation amplitude recorded, the impulse exerted on
the pendulum front face can be calculated, and the effective impulse on the specimen can be
further estimated based on the exposed area of the specimen. The detailed impulse calculation
can be found in Appendix A.
Figure 3-2. Four-cable ballistic pendulum system
Each of the 310mm × 310mm sandwich panel was peripherally clamped between two
rectangular steel frames, one of which is shown in Figure 3-3(a), together with the clamping
assembly (Figure 3-3(b)).
36
(a) Sketch of the frame
(b) Sketch of the clamping device
Figure 3-3. Sketch of the frame and clamping device
The frames were clamped on the front face of the pendulum, and the charge was fixed in front
of the centre of the specimen using an iron wire with a constant stand-off distance of 200mm, as
shown in Figure 3-4. A special sensor, known as PVDF (Polyvinylidene Fluoride) pressure
gauge was mounted at the central point of specimen to record the explosion pressure-time
37
history at this point. Figure 3-4 (a) shows the gauge, which was wrapped by aluminium foils to
avoid possible damage caused by explosion heat. A laser displacement transducer
(Micro-Epsilon LD1625- 200) connected to an oscilloscope was used to measure the
translation of pendulum, as shown in Figure 3-4 (b), instead of using a recording pen as
employed by Nurick and co-workers [13, 22-24]. A typical time history recorded is shown in
Figure 3-5. Based on the magnitude of the first valley and the period of oscillation, the impulse
delivered onto the pendulum was calculated. Then according to the sizes of specimen and
clamping device and stand-off distance, the effective impulse on the specimen can be further
estimated.
(a) PVDF pressure gauge (b) Laser displacement transducer
Figure 3-4. Two types of sensor used in the tests
0 2 4 6 8 10 12 14
-100
-50
0
50
100
Dis
plac
emen
t (m
m)
Time (s) Figure 3-5. A typical oscillation time history of the pendulum
38
A sketch of the overall experimental set-up is shown in Figure 3-6. The connectors between the
I-beam and steel cables were well lubricated to reduce the damping effect to the minimum level.
The resistance of air can be neglected as the pendulum setup was very heavy (140.75kg). The
movement duration of the pendulum could be more than five minutes, as observed. The duration
of structural response is of the order of ms; while the oscillation period of pendulum is several
seconds. The deformation of sandwich plate has completed within the very beginning stage of
the pendulum’s translation, thus the moving boundary of the specimen due to swing of the
pendulum has little effect on the result in the tests.
Figure 3-6. Sketch of the experimental set-up
39
40
3.4 Deformation and failure patterns
Based on the configuration of sandwich panels, the deformation/failure of specimens observed
in the tests can be classified with respect to the front face-sheet, core and back face-sheet,
respectively. They are described in the subsequent sections.
3.4.1 Front face-sheet deformation/failure
On the front face-sheet, all the specimens show localized compression failure in the central area
and global deformation in the peripheral region. The deformation/failure modes can be
classified in terms of (1) size of plastic deformation zone; and (2) damage type at the centre.
(1) plastic deformation zone
Due to the variations of configuration and impulse level, some of the specimens exhibit a large
global deformation, while others are dominated by localized failure at the central area and their
global deformation is less evident. Therefore, the deformation/failure modes can be classified
into Mode G (global) and Mode L (localized) failures.
(2) damage type at the centre
In the central area, two types of failure: Type I (indenting) and Type P (pitting) can be observed,
and an annular band with flower-shaped deformation occurs in the immediate zone around the
centre. Type I failure is characterized by a localized large deformation without rupture damage,
while in Type P failure, the localized pit shows fracture or tearing damage on surface. Figures
3-7 and 3-8 illustrate the front face-sheets with these two failures, respectively. From the centre
to outskirts, the front face-sheet may be divided as three zones, that is, Zone 1: central localized
failure; Zone 2: flower-shaped deformation; and Zone 3: global deformation.
Figure 3-7. Indenting failure on the front face (Specimen No.: 1/8-5052-0.0015-MD-2)
Figure 3-8. Pitting failure on the front face (Specimen No.: 1/8-5052-0.0015-TN-1)
41
In order to identify the parameters which affect the deformation and failure mechanism of the front
face-sheets, deformation/failure mode maps are used, as shown in Figure 3-9 and Figure 3-10.
Figure 3-9 describes the observed deformation/failure modes of Groups 1, 2 and 3, in which the
specimens were loaded by a 20g TNT charge with a stand-off distance of 200mm. The map is in
terms of face-sheet thickness and core type characterized by its average mass (relative density).
Figure 3-10 indicates the map for Group 4, in which the panels with the same core were subjected to
different levels of shock loading. The map is plotted in terms of charge mass and specimen number.
Figure 3-9. Deformation/failure map for Groups 1~3. The abscissa denotes the specimens sorted
by the cores with increasing relative densities
42
Figure 3-10. Deformation/failure map for Group 4, where all the eight panels have identical
configurations
The maps indicate that the specimens with thicker face-sheets and denser cores, loaded by a
larger charge trend to produce a localized deformation. On the contrary, those with thinner skins
and lighter cores and subjected to lower level shocks are prone to deform globally. All of the
chargers used in the tests are cylindrical, which have the identical diameters but various lengths.
Therefore, the areas subject to all of the charges are almost the same, but larger charges release
more energy, thus producing more local deformation. As to the central damage, the occurrence
of Type I and Type P failures seems quite irregular, and no systematic trend has been observed.
This phenomenon may be due to the random and discrepant nature of blast loading, or the
explosion head produced during the chemical reaction of the explosive, and the detailed
mechanisms are open for further investigations.
3.4.2 Core deformation/failure
The deformed honeycomb core shows a progressive deformation pattern, which is the same as
that observed in the low velocity impact experiments [80, 81]. Figure 3-11 demonstrates a
typical cross-sectional view of specimen after test. From the centre to outskirts, the specimen
43
44
can be divided into three regions, according to the extent of core deformation; that is (1)
fully-folding region; (2) partially-folding region; and (3) folding-absent region, respectively.
The fully-folding region is located at the central area of the specimen, where the largest plastic
deformation occurs. Folding, interpenetrating, local tears and local separations can be observed
on the vertical edges of honeycomb. In the partially-folding region, the folding pattern is similar
but the progressive buckling only occurs on the side adjacent to the front face-sheet, and the cell
vertical walls remain nearly straight. Apart from the folding damage, in some specimens,
delamination failure between the front skin and core also occurred in regions (1) and (2). The
folding-absent region is practically the area clamped by the thick steel frames and thus no
impulse exerts on it, but shear failure is evident in the zone between regions (2) and (3).
3.4.3 Back face-sheet deformation/failure
The sandwich panels exhibit the same damage mode on the back face-sheet as that of the
monolithic square panels. In our tests, all the specimens show a typical Mode I response, which
is essentially large inelastic deformation [20, 21]. The deformation profile has the shape of a
uniform dome, moving out from the centre, transforming into a more quadrangular shape
towards the clamped edges. Plastic hinges can be observed, extending from the plate corner to
the base of the deformed dome. Typical plate profiles are shown in Figure 3-12. In some
specimens with pitting damage on the front face-sheet, a small nose occurs at the top of dome
on the back face, as indicated in Figure 3-12(b).
Figure 3-11. Failure pattern of the honeycomb core (Specimen No.: ACG-1/4-TK-5)
45
(a) A dome-like back face deformation (Specimen No.: 1/8-5052-0.0010-TN-2)
(b) A dome-like back face deformation with a small “nose” (Specimen No.: 1/8-5052-0.0015-MD-1)
Figure 3-12. Failure pattern of the back face (Specimen No.: ACG-1/4-TK-5)
46
3.5 Pressure-time history at the central point of the front face
From the direct measurement in the tests and subsequent calculation, in this research, three
types of quantitative results are mainly considered: (1) deflection of the central point on the
back face; (2) impulse imparted on the front face; and (3) history of pressure at the central point
on the front face.
The values of deflection and impulse are given in Tables 3-1 ~ 3-4, while a detailed analysis of
the effect of particular parameters is presented in Section 3.6.
The pressure-time history at the central point of front face was measured using a PVDF pressure
gauge. Made from piezoelectric polymer, this type of sensors can produce electric charge under
impact or impulsive loading, and thus offer the capability of evaluating the pressure/stress by
measuring the output voltage [82]. Based on the voltage output history recorded by an
oscilloscope, a trace of time versus stress can be further obtained. In the PVDF measurement
circuit, an equivalent capacitor was used to ensure that the piezoelectric film and amplifier
could work simultaneously. The relationship between the charge generated on the PVDF
piezoelectric film and the pressure/stress at this position is governed by the following equation:
33 zzQ d Aσ= (3-1)
where Q is the electric charge generated, which can be obtained by integration of the voltage
output over time; d33=20pC/N, being piezoelectric constant of the PVDF film; A = 6mm × 6mm,
being the working area of the film; and zzσ is stress. Figure 3-13 shows a typical resulting
curve of pressure versus time. It can be seen that the pressure on the front face of specimen
sharply increases from zero to its peak; then it decreases rapidly and finally drops down towards
zero. It is very difficult to use some sensors such as stain gauges and common accelerometers to
accurately capture the structural response subject to blasts. They are unstable and quite easy to
damage under such extreme environments. More reliable measurement means would be
considered in the future work.
47
Figure 3-13. Typical pressure-time history at the central point of the front face
3.6 Analysis and discussion
Based on the experimental results, a parametric study was conducted and the results are
presented in this section. Effect of face-sheet and core configurations, i.e. face-sheet thickness,
cell size and foil thickness of the honeycomb, and mass of charge, on the structural response of
the sandwich panels loaded by blasts is identified. The key characteristics of structural response
include (1) mechanism of deformation/failure, (2) impulse transfer, and (3) energy absorption in
plastic deformation. The mechanism of deformation/failure is considered as the most important
characteristic of structural response as all the other responses depend on it. Since personnel or
objects shielded from blast attacks are usually behind the barriers such as sandwich panels, the
back face deformation/failure of specimen is herein considered as the main response of interest.
3.6.1 Effect of face-sheet thickness
In the tests, three different face-sheet thicknesses, that is TN (hf=0.5mm), MD (hf=0.8mm) and
TK (hf=1.0mm) were tested in Groups 1, 2 and 3. Their effect on the back face deflection is
48
shown in the diagram in Figure 3-14. The diagram clearly reveals dependence of the back face
deflection on the thickness of face-sheet. Compared with the TN face-sheets, the MD
face-sheets lead to a decrease in the average deflections by 41.8%, 39.0%, 34.9%, 38.8%,
30.7% and 37.8%, respectively; while the TK face-sheets reduce the average deflections by
49.0%, 47.2%, 50.2%, 60.1%, 55.2% and 48.9%, respectively. The deflection is reduced by
increasing the face-sheet thickness and this however leads to an increase in the panel weigh.
How to make a compromise between strength and weight is one of the most important issues
that need to be considered in the design of sandwich structures
Figure 3-14. Effect of face-sheet thickness. The abscissa denotes the specimens given without
any particular order
3.6.2 Effect of cell dimension of the core
Group 1 specimens were used to test the effect of the foil thickness of the core cells. In this
group, the four honeycomb cells have the same size (le=3.18mm) but different foil thickness:
0.018mm, 0.025mm, 0.038mm and 0.051mm, respectively. It can be observed from Figure 3-15
49
that for a given face-sheet, larger foil thicknesses result in smaller back face deflections. Using
the weakest core (1/8-5052-0.0007 series) as a benchmark, the other three cores give a decrease
of average deflections by 26.4%, 37.3% and 41% for the TN face-sheets; 22.8%, 29.9% and
37.9% for the MD face-sheets; and 23.8%, 38.7% and 54.4% for the TK face-sheets,
respectively.
10
15
20
25
30
35
40
45
50
55 1/8-5052-0.0020 series 1/8-5052-0.0015 series 1/8-5052-0.0010 series 1/8-5052-0.0007 series
T N(hf = 0.5mm)
MD(hf = 0.8mm)
T K(hf = 1.0mm)
Face-sheet thickness
Deflection(mm)
Figure 3-15. Effect of foil thickness. The abscissa denotes the specimens sorted by the
face-sheets with increasing thicknesses
In Group 2, two cores with the same foil thickness (t=0.038mm) but different cell sizes (le=
3.18mm and 3.97mm, respectively) were studied. A comparison is illustrated in Figure 3-16,
which is a plot of deflection versus face-sheet thickness. As expected, the back face deflection is
larger for the specimens with a larger cell size, i.e. 5/32-5052-0.0015 series. This effect is more
significant for panels with thinner face sheet. The percentage differences in deflections of these
two panels with the face-sheet thicknesses of 0.5mm, 0.8mm and 1.0mm are 29.9%, 38.0% and
16.9%, respectively.
50
TN MD TK
15
20
25
30
35
40
45
50
55
Face-sheet thickness
(hf = 1.0mm)(hf = 0.8mm)
1/8-5052-0.0015 series 5/32-5052-0.0015 series
(hf= 0.5mm)
Deflection (mm)
Figure 3-16. Effect of cell size. The abscissa denotes the specimens sorted by the face-sheets
with increasing thicknesses
Group 3 was designed to compare the responses of two cores (1/8-5052-0.0010 series and
ACG-1/4 series) with different cell sizes and foil thicknesses, but possessing similar average
masses (i.e. 85.6g and 85.9g, respectively). The average mass of core actually reflects its
relative density. Therefore, the two cores concerned in this group have nearly the same relative
density (2.21% and 2.22% respectively). The deflections for the three different face-sheet
thicknesses are shown in Figure 3-17. It can be seen that the results of these two panels are very
close. The average differences in deflection in the cases of thin, medium and thick face-sheets
are 5.2%, 7.6% and 2.0%, respectively.
51
TN MD TK
20
25
30
35
40
45
50
55
60
1/8-5052-0.0010 series ACG-1/4 series
Deflection (mm)
Face-sheet thickness
(hf = 1.0mm)(hf = 0.8mm)(hf = 0.5mm)
Figure 3-17. Effect of the average mass of core. The abscissa denotes the specimens sorted by
the face-sheets with increasing thicknesses
Since the cell size and foil thickness are the two dimensions that determine the relative density
of core, from Figures 3-15 ~ 3-17, one can conclude that the relative density of core structure
can significantly affect the back face response of a sandwich panel subjected to impulsive loads.
By adopting honeycomb cores with higher relative density, the deflection of back face can be
reduced.
3.6.3 Effect of charge mass
In Groups 1-3, the effect of various panel configurations was studied with the charge mass kept
unchanged. In Group 4, eight nominally identical specimens (ACG-1/4-TK series) were used
and loaded by charges with four different masses: 15g, 20g, 25g and 30g, respectively, which
produce different levels of impulse. The normalised back face deflection (δ) is plotted against
normalised impulse (Φ) as shown in Figure 3-18. It is evident that, for the given panel
configuration, back face deflection increases with impulse, almost linearly.
52
Figure 3-18. Effect of impulse level on the panels with nominally identical configurations
Thus the relationship of δ and Φ can be expressed as
0
fYf
δ Φ, where Φw I
=t 2 t M 2h
ασ
= = (3-2)
where f c2t h H= + , with and being the thicknesses of face and core respectively.
fh cH
f f )(2 ccM A h Hρ ρ= + , with fρ and cρ being the densities of face and core, and A is the
exposed area. fYσ denotes the quasi-static tensile strength of face material. Fitting the data
points and taking α as 0.39, the experimental results are well predicted by Eq. (3-2).
3.7 Summary
A total of 42 experiments were conducted to test the structural response of sandwich panels
subjected to blast loads, and the experimental results are reported and discussed in this chapter.
The panels consisted of two face-sheets and a honeycomb core, which were made of aluminium
53
54
alloys. The test program consisted of four groups, each of which was designed to identify the
effect of key parameters, such as cell size and foil thickness of the honeycomb, face-sheet
thickness and mass of charge. In the tests, a four-cable ballistic pendulum system with a laser
displacement transducer was used to measure the impulse imparted on the panel, and a PVDF
pressure gauge recorded the pressure-time history at the central point of specimen’s front face.
The experimental results were classified as two categories: (1) deformation/failure modes of
specimen observed in the tests, which were further discussed for those for front face, core and
back face, respectively; and (2) quantitative results, which included the impulse on sandwich
panel, permanent central point deflection of the back face and pressure-time history at the
central point of front face. It has been shown that specimens with thicker face-sheets, a higher
density core and loaded by larger charges tend to have localized deformation on the front face,
and those with thinner skins and a sparse core and subjected to lower level shocks are prone to
deform globally. At the central area of the front face, indenting and pitting were observed on all
the specimens but their occurrence seems irregular. Folding damage took place in the
honeycomb core, with different extent of deformation at different regions. As for the back face,
all of the panels show a dome-shaped deformation.
Based on the quantitative analysis, it has also been found that the face-sheet thickness and
relative density of core structure can significantly affect the back face deformation. By adopting
thicker skins and honeycomb cores with higher relative density, the deflection of back face can
be reduced. Also, for a given panel configuration, it is evident that the back face deflection
increases with impulse, approximately linearly.
CHAPTER FOUR
EXPERIMENTAL INVESTIGATION INTO THE
ALUMINIUM FOAM CORE SANDWICH PANELS
4.1 Specimen
As the second type of specimens, aluminium foam sandwich panels have been tested using the
approach described in Chapter 3, and similarly, both the deformation/failure patterns observed
and quantitative results are analysed in the subsequent sections. The specimens consisted of two
metallic face-sheets and a core of aluminium foam, as sketched in Figure 4-1. The face-sheets
were made of aluminium alloy Al-2024-T3, which has a higher yield stress of 318.0MPa, while
the other properties are similar to those of Al-2024-O. The faces were fabricated with two
different thicknesses, i.e. 0.8mm and 1.0mm, respectively. The aluminium foam cores had two
relative densities, that is 6% (denoted L) and 10% (denoted H). The cores were cut into
300mm×300mm plates with two different thicknesses (20mm and 30mm). Specifications of the
plates are presented in Table 4-1.
hf
Hc
L
L Al foam corevn
Figure 4-1. Geometry and dimension of the aluminium foam core specimen
55
Table 4-1. Specifications and test results of the aluminium foam core sandwich panels
Specimen No.
Specimen name
Face-sheets thickness hf (mm)
Mass of core mo (g)
Relative density
(%)
Core thickness Hc (mm)
Mass of charge mh (g)
Impulse I (Ns)
Back face deflection w0 (mm)
Wrinkles at the edges
Front face tearing
1 L-20-TK-1 1.0 290 6.0 20 20 18.29 4.9 Yes No 2 L-20-TK-2 1.0 292 6.1 20 30 22.57 6.1 Yes No 3 H-20-TK-1 1.0 466 9.7 20 20 18.08 4.4 No No 4 H-20-TK-2 1.0 472 9.8 20 30 23.00 5.1 No No 5 L-30-MD-1 0.8 460 6.4 30 30 22.67 6.2 Yes No 6 L-30-MD-2 0.8 458 6.3 30 40 -- 6.3 Yes Yes 7 L-30-TK-1 1.0 461 6.4 30 30 22.32 5.6 Yes No 8 L-30-TK-2 1.0 461 6.4 30 40 25.85 7.0 Yes No 9 H-30-TK-1 1.0 728 10.1 30 30 22.36 2.4 No No
10 H-30-TK-2 1.0 714 9.9 30 40 25.55 3.9 No No
56
4.2 Results and discussion
The experimental results are listed in Table 4-1. Two types of results are presented and
discussed herein, i.e. (1) deformation/failure patterns observed in the tests, and (2) quantitative
data obtained through measurement and further calculation, which include the central point
deflection of the back face and impulse exerted onto the specimen.
4.2.1 Deformation/failure patterns
On the front face-sheet, all the specimens show localized failure on the central area and global
deformation in the peripheral region as shown in Figure 4-2(a). For the specimens with the low
density cores, a wrinkle could be observed at the edges of the front face, as shown in Figure
4-2(b), while panels with the H cores did not show any wrinkles. Face wrinkling is often the
major failure mode for thick sandwich panels with very light/weak cores due to in-plane
bending effect [83]. Wrinkling is actually a buckling mechanism, typically characterized by the
relatively short period of the buckling mode shape.
(a) Failure pattern of the front face without a wrinkle (Specimen H-30-TK-2)
57
Enlarged view
(b) Failure pattern of the front face with a wrinkle (Specimen L-30-TK-1)
Figure 4-2. Failure patterns of the front face
The observed localized failure around the central region of the specimens tested has two
patterns: (1) indenting only and (2) tearing of the front skin. The tearing damage took place for
specimen L-30-MD-2 only, and all the other nine panels show an indenting failure. Figure 4-3
illustrates these two localized failure types.
58
(a) Indenting failure at the centre (Specimen L-30-TK-2)
(b) Tearing failure at the centre (Specimen L-30-MD-2)
Figure 4-3. Two types of failure in the centre of front face
59
For deformation of the back face, all the panels in our tests show a similar pattern. The
deformation profile has the shape of a uniform dome, moving out from the centre, changing to a
more quadrangular shape towards the clamped edges. Plastic hinges are visible, extending from
the plate corner to the base of the deformed dome. A typical back face profile (Specimen
L-20-TK-2) is shown in Figure 4-4.
Figure 4-4. A typical deformation/failure pattern of the back face (Specimen L-20-TK-2)
The foam cores of the sandwich panels have mostly maintained structural integrity during blast
loading and no evident fragmentation has been observed, due to the protection of the front face.
Both the foam core and the front face-sheet have attained an inwardly curved shape (termed as
‘dishing’ [12]), and the back face has deformed outwardly. The curvature extended in all
directions of the panel plane. The core crushing damage accompanied by a cavity between the
face and the crushed foam core was observed, which is essentially a result of core fracture,
rather than debonding in the interface. Figure 4-5 shows a typical cross section taken in the
plane through the centre of a panel. The crushing/densification process of cellular media has
been studied by Reid and co-workers [70, 71]. They suggested that with the movement of
shockwave front, the cellular solids are compressed progressively and the microstructures
collapse layer by layer, and fully compacted at the densification strain εD.
60
Figure 4-5. A typical cross-section of the specimen (Specimen L-30-TK-1)
61
62
4.2.2 Deflection of the face-sheet
In this section, effect of three parameters, i.e. thickness of the face-sheet, thickness of core and
relative density of core, on the central point deflection of the back face is observed. From the
experimental results listed in Table 4-1, it is observed that
(1) The panels with thinner face-sheets exhibit a higher level of deformation, with possible
tearing failure on the front face (i.e. L-30-MD-2).
(2) Larger core thickness reduces the deflections.
(3) Higher relative densities of core result in smaller deflections.
It should be emphasised that conclusion (3) and the similar one in Chapter 3 may be just valid
for the studied problem, but may not for the sandwich panels with other configurations. If the
blast overpressure is in the same order of the plateau stress of the cellular core, back face
deflection may reduce with the decrease of core relative density because of its force limitation
capability [73].
4.3 Summary
The aluminium foam sandwich panels have been tested using the approach described in Chapter
3. Deformation/failure patterns of specimen and quantitative results have been reported and
analysed. It has been observed that the front faces show localized indentation for all the
specimens. In addition, winkling at the edges of the panels occurs for panels with a lower
density core. The back faces have a uniform quadrangular-shaped dome, moving out from the
centre to the clamped boundaries. The core crushing damage was accompanied with a cavity
between the front face and the crushed foam core. It has also been found that the panels with
dense core, both thick core and faces have small deflections.
63
CHAPTER FIVE
NUMERICAL SIMULATION OF THE HONEYCOMB
CORE SANDWICH PANELS
5.1 Introduction
Based on the experiments described in Chapter 3, corresponding finite element simulations have
been undertaken using LS-DYNA software. Detailed description of the models and simulation
results is presented. In the simulation work, the loading process of explosive and response of the
sandwich panels are investigated. The blast loading process includes both the explosion
procedure of the charge and interaction with the panel. The structural responses of sandwich
panels are studied in terms of two aspects: (1) deformation/failure patterns of the specimens;
and (2) quantitative assessment, which mainly focuses on the permanent central point deflection
of the back face of the panels. In addition, a parametric study has been carried out to examine
the contribution of plastic stretching and bending on the deformation history of the back face of
a typical sandwich panel, as well as the effect of boundary conditions.
5.2 FE model
The numerical simulations were conducted using LS-DYNA 970 software, which is a powerful
FEA tool for modeling non-linear mechanics of solids, fluids, gases and their interaction. As
LS-DYNA is based on explicit numerical methods, it is well suited for analysis of dynamic
problems associated with large deformation, low and high velocity contact/impact, ballistic
penetration and wave propagation.
5.2.1 Modeling geometry
The geometric model of the sandwich panel used in the simulations is depicted in Figure 5-1(a),
while Figure 5-1(b) shows an enlarged view of a single cell (including the corresponding
face-sheets). Since the square sandwich panel is symmetric about x-z and y-z planes, only a
quarter of the panel was modeled. Both the core and face-sheets were meshed using
Belytschko-Tasy shell element [39], which gives a high computational efficiency, and thus the
entire model comprises 17,328 shells. With the panel loaded by blasts, both face-sheets and core
would undergo large deformations, such as plastic bending, stretching and buckling. The
computational accuracy of such deformations is highly dependent on the number of elements. In
other words, the details of such deformations (particularly buckling) cannot be described
accurately by a coarse mesh. Here, an adaptive meshing approach, known as fission h-adaptivity
[39] was employed to refine the elements where large deformations take place. In an h-adaptive
method, the elements are subdivided into smaller elements wherever an indicator shows that
subdivision of the elements will provide improved accuracy. It offers the possibility to obtain a
solution of comparable accuracy using much fewer elements, and hence less computational
resources than with a fixed mesh.
(a) Geometric model of the 1/4 panel
64
(b) Geometric model of a single cell
Figure 5-1. Geometric model of the sandwich panel
The explosive charge used in the tests had a cylindrical shape. Due to the symmetric nature of
the specimen, again, only one quarter of the charge was modeled to reduce the model size.
Eight-node brick (solid) elements with the Arbitrary Lagrangian Eulerian formulation (ALE)
[39] were adopted for the explosive cylinder. Overcoming the difficulties of the traditional
Lagrangian method in large deformation analyses and the Eulerian method when dealing with
multi material interaction or moving boundaries, the ALE approach uses meshes that are
imbedded in material and deform with the material. It combines the best features of both
Lagrangian and Eulerian methods, and allows the mesh within any material region to be
continuously adjusted in predefined ways as a calculation proceeds, thus providing a continuous
and automatic rezoning capability. Therefore, it is suitable to use an ALE approach to analyse
solid and fluid motions when material strain rate is large and significant (for example, the
detonation of explosive and volume expansion of explosion products). Figure 5-2 illustrates the
geometric model of a quarter of a 20g explosive cylinder (diameter≈15mm; height≈15mm),
which consists of 17,280 solid elements. The stand-off distance is constant and equal to 200mm.
65
Figure 5-2. Geometric model of the charge
5.2.2 Modeling materials
Both the core and face-sheets of specimen used in the tests were made of aluminium alloy. In
the simulation, the mechanical behaviour of aluminium alloy was modeled with material type 3
(*MAT_PLASTIC_KINEMATIC) in LS-DYNA, which is a bi-linear elasto-plastic constitutive
relationship that contains formulations incorporating isotropic and kinetic hardening. Since
aluminium alloys show less evident strain rate effect and for simplicity, the only input
parameters of the material model are: Mass density (ρ), Young’s modulus (E), Poisson’s ratio (ν),
Yield stress (σY) and Tangent modulus (Etan).
Material type 8 (*MAT_HIGH_EXPLOSIVE_BURN) in LS-DYNA was used to describe the
material property of the TNT charge. It allows the modeling of detonation of a high explosive
by three parameters: Mass density of charge (ρM), Detonation velocity (V) and Chapman-Jouget
pressure (P). Likewise, an equation of state, named Jones-Wilkins-Lee (JWL) equation, was
used to define the explosive burn material model. This equation defines the pressure as a
function of relative volume, V*=ρ0/ρ, and internal energy per initial volume, Em0, as presented in
Eq. (5-1):
0 01 2
01 0 2 0 0
1 1R R
mP A e B e ER R
ρ ρρ ρωρ ωρ ωρ
ρ ρ
− −⎞ ⎞⎛ ⎛= − + − +⎟ ⎟⎜ ⎜
⎝ ⎝⎠ ⎠ ρ (5-1)
where P is the blast pressure, ρ is the explosive density, ρ0 is the explosive density at the
66
67
beginning of detonation process. The parameters A, B, R1, R2 and ω are material constants,
which are related to the type of explosive and can be found in most explosive handbooks.
Table 5-1 lists the values of LS-DYNA material types and mechanical properties of sandwich
panel and explosive, as well as the those of equations of state (EOS). It should be noted that the
data for face-sheets were determined through standard quasi-static tensile tests, and parameters
of the core materials and explosive were obtained from published literature [84, 85].
Table 5-1. LS-DYNA material type, material property and EOS input data for honeycomb core panels
Material Part LS-DYNA material type, material property and EOS input data
(unit = cm, g, μs) *MAT_PLASTIC_KINEMATIC RO E PR SIGY ETAN
Al-2024-O
Face sheet 2.68 0.72 0.33 7.58E-4 7.37E-3
*MAT_PLASTIC_KINEMATIC RO E PR SIGY ETAN
Al-3104-H19 [84]
Core
2.72 0.69 0.34 2.62E-3 6.90E-3 *MAT_PLASTIC_KINEMATIC RO E PR SIGY ETAN
Al-5052-H39 [84]
Core
2.68 0.70 0.33 2.65E-3 7.0E-3 *MAT_HIGH_EXPLOSIVE_BURN RO D PCJ 1.63 0.67 0.19 *EOS_JWL A B R1 R2 OMEG E0 V0
TNT [85]
Charge
3.71 3.23E-2 4.15 0.95 0.30 7.0E-2 1.0
5.2.3 Modeling blast load
Modeling the blast load on the structure, or explosive-structure interaction, can be implemented
by setting contact between them [86, 87]. In this simulation, the load imparted on the front face
of sandwich panel was defined with algorithm of *CONTACT_ERODING_SURFACE_TO_
SURFACE, which calculates the interaction between explosion product and structure. The
erosion algorithm allows for large distortion of explosion product which is caused by the
reaction of target structure, by eroding elements from its surface contacting the structure. Like
68
most of the other simulation work for the close range explosion in an open environment, due to
the large overpressure and short time duration, the influence of air is neglected.
5.3 Simulation results and discussion
The simulation results reported and discussed in this section cover the blast loading process and
deformation of the structure. Specifically, three aspects are detailed: (1) explosion and structural
response process; (2) deformation/failure patterns of sandwich panels observed; and (3) the
measured/calculated quantitative result.
5.3.1 Explosion and structural response process
Figures 5-3 ~ 5-5 illustrate a typical process of charge explosion and consequent plate response,
which was calculated by the FE model. The model shown depicts specimen ACG-1/4-TK-6
loaded with a 25g explosive. The figures illustrate three specific stages as follows:
Stage I: Expansion of the explosive from time of detonation to interaction with the plate
Stage II: Explosion product -- plate interaction
Stage III: Plate deformation under its own inertia
• Stage I (0 to32μs)
Figure 5-3 clearly shows how the explosion product (i.e. fire ball) expands. Expansion of
explosive starts at the detonation point (central point of the top surface of charge). The shock
wave created by the detonation compresses and raises the temperature of the explosive at the
detonation point of the material, initiating a chemical reaction within a small region just behind
the shock wave, known as the reaction zone. Hot gaseous detonation products are produced
from the reaction occurring in the reaction zone. LS-DYNA can capture the volume expansion
of the explosive using an EOS, although it cannot simulate chemical reactions. Figure 5-3
reveals the transient distribution of high pressure generated at the reaction zone. When the
reaction propagates through the explosive, the front of the initiation of expansion spreads
outwards from the detonation point at the detonation/burn speed of the explosive, which is
defined in the high explosive material model.
Figure 5-3. A typical process of the charge detonation
69
70
The shock wave propagation is not symmetric because the detonation point is located at the
central point of charge’s top surface, which produces a one dimensional detonation wave
propagating downwards It is at this stage I that numerical instability, such as the error of
‘out-of-range velocities’ and ‘negative volume in brick element’ may occur due to the excessive
distortion of elements. Reducing the time step scale factor is a common approach employed at
this stage of analysis to solve the problem.
• Stage II (33μs to 62μs)
At this stage, the expanded explosive interacts with the plate front surface. It can be seen from
Figure 5-4 that the explosive-plate interaction takes place from approximately t=33μs to t=62μs,
i.e. over a time period of approximately 30μs, until the contact force between explosive and
target structure almost reduces to 0. Figure 5-4 illustrates the explosive-plate interaction, and the
upward distortion of explosion products as a result of the reflection from the plate. The pressure
distribution contours on the sandwich panel are also clearly shown. At this stage, a dent failure
is first formed at the central area of sandwich front face, and then the deformation extends both
outwards and downwards with the transfer of impulse. Eroding effect takes place on a small
number of elements of the TNT charge part, due to the extremely large distortion, and thus had
little influence on the result. The purpose of erosion is to keep the computation stable. The
erosion criterion is a default strain value suggested by LS-DYNA. When the contact force
between the explosive and plate decreases to nearly zero (t=62μs), their interaction is considered
to be complete, and the high explosive model should be manually deleted from the LS-DYNA
project.
• Stage III (63μs to 2000μs)
Stage III is the final stage of the simulation process, wherein no contact between the explosive
is made with the structure, and the plate continues to deform under its own inertia. After the
deformation zone extends to the external clamped boundaries, a global dishing deformation
takes place. A slight oscillation of the plate occurs with the deformation, and the structure is
finally brought to rest by plastic bending and stretching.
Figure 5-4. A typical process of explosion product - structure interaction
71
5.3.2 Deformation/failure patterns of sandwich panels
• Deformation/failure patterns of face-sheets
All the specimens after tests show bending/stretching failure in the central area of the front face,
coupled with severe core compression, and global deformation in the peripheral region. The
transient displacement contour plots of the front face subjected to an impulsive load of 21.11Ns
are shown in Figure 5-5 (from t=0). They indicate that the front surface deforms with a dent first
developing at the centre, and this zone expands outwards. This is finally followed by a large
global plastic bending and stretching. The details of core failure can be seen in Figure 5-9 and is
discussed further in the subsequent section.
Figure 5-5. A typical process of plate deformation
72
A typical contour of back face (Specimen ACG-1/4-TK-6) obtained in the simulation is shown
in Figure 5-6, together with a photograph of the tested panel. The back face-sheets in the tests
show a typical Mode I response [20], which essentially involves a large inelastic deformation.
Plastic hinges are visible along the clamped edges. Figure 5-7 illustrates a cross-sectional view
of the deforming back face, i.e. the motion of sandwich structures after blast impact, which may
be described by the plastic hinge theory. The central portion of the structure translates with an
initial velocity vf while a segment of length ξ at each end rotates about each support. This
motion continues until the traveling hinges at the inner ends of the segments of length ξ
coalesce at the mid point of the back face. Eventually, stationary plastic hinges form at the
centre and at the ends of the structure.
Figure 5-6. A typical profile of back face (Specimen name: ACG-1/4-TK-6)
73
Figure 5-7. Process of back face deformation and corresponding plastic hinges,
one stationary and the other traveling
The displacement-time history at the central points of both face-sheets and core of Panel
ACG-1/4-TK-6 is illustrated in Figure 5-8, together with an enlarged view of 0~250μs.
Deformation of the front face starts at t=33μs, then increases gradually and reaches a plateau at
approximately t=700μs. After that, an oscillation can be observed until the structure rests. It is
clearly shown that the deflection of back face increases at a slower pace than the rate at which
the front face deforms. Core crushing commences at 33μs, and the curve goes up sharply until
about 160μs. Then the speed of crushing becomes much slower, and the curve reaches the peak
(7.85mm) at 700μs, which is the permanent core compression.
74
0 250 500 750 1000 1250 1500 1750 200002468
101214161820222426283032
Def
orm
atio
n (m
m)
Time (ms)
Front face
Back face
Core
T ime (μs)
D
efle
ctio
n (m
m)
25 50 75 100 125 150 175 200 225 2500
2
4
6
8
10
12
14
Def
orm
atio
n (m
m)
Time (μs)
Front face
Back face
Core
D
efle
ctio
n (m
m)
(enlarged review of 0~250μs)
Figure 5-8. Displacement-time history at the central points of face-sheets and core crushing
(Specimen name: ACG-1/4-TK-6)
• Deformation/failure patterns of core
Figure 5-9 shows a typical FE prediction for core deformation/failure patterns, where
progressive buckling forms on the side adjacent to the loading end, and the vertical cell walls at
the other end remain nearly straight. It can be seen that the details of failure are well captured by
the h-adaptivity algorithm mentioned in Section 5.2.1. Using this algorithm, the number of shell
elements of the FE model has been increased from initially 17,328 to 164,979. In the simulation,
75
when the total angle change of an element (in degrees) relative the surrounding elements is
greater than 5, that element would be refined, which can give the results with acceptable
accuracy.
Figure 5-9. Deformation patterns of honeycomb core (Specimen name: ACG-1/4-TK-6)
5.3.3 Quantitative results
In this section, a comparison is made between the experimental and simulation results in terms
of the most important structural response -- final permanent deformation (i.e. deflection) of the
central point of back face.
76
A plot of the experimental values versus the predicted values of all the specimens is shown in
Figure 5-10. The data points are very close to the line of perfect match, thus representing a
reasonable correlation between the experimental and predicted results.
0 5 10 15 20 25 30 35 40 45 50 550
5
10
15
20
25
30
35
40
45
50
55Ex
perim
enta
l def
lect
ion
(mm
)
Predicted deflection (mm)
11
Figure 5-10. Comparison of experimental and predicated results
5.4 Effect of plastic stretching and bending
In order to better understand the back face deformation mechanism, a study was further carried
out on a typical panel ACG-1/4-TK-6, in which the contributions of plastic stretching and
bending were analysed in detail. The plate under stretching essentially exhibits a membrane
deformation behaviour. The level of membrane deformation can be indicated by the
middle-plane strain of the plate εmid, while the bending states can be identified by calculating the
difference of the values of the in-plane normal strains at the lower- and upper-surfaces of the
back face, i.e. εd = εlower - εupper, which indicates the curvature.
The distributions of εmid and εd have been investigated both temporally and spatially. The 1/4
meshed back face was placed in a 2D Cartesian coordinate system, with the symmetric centre of
the plate placed at the point of origin. For comparison purposes, two groups of shell elements
were selected, the first group is located along the x axis and the second group is located along a
77
diagonal passing through the origin. The exact locations of the two groups of shells are shown
in Figure 5-11, with the elements re-numbered for the purpose of presentation only.
(a) Shell elements in Group 1
(b) Shell elements in Group 2
Figure 5-11. Locations of the shell elements in the two groups
78
5.4.1 Strain distribution along the x axis
The development of the middle-plane strains in x and y directions for the shells on the x axis, i.e.
εmidx and εmidy, is shown in Figure 5-12. The figure clearly reveals that both εmidx and εmidy
increase with time, and the strains progress from the clamped end to the centre. When t=800μs,
the middle-plane strains at the centre reach the maximum values, i.e. 3.2% and 3.1%,
respectively, while the strains near the edge remain small. Therefore, one can conclude that the
highest level membrane deformation occurs at the plate centre, and almost no stretching takes
place near the boundary.
(a) εmidx distribution for the shell elements in Group 1
(b) εmidy distribution for the shell elements in Group 1
Figure 5-12. εmid distribution for the shell elements in Group 1
79
The bending states of the plate are indicated by εdx and εdy in Figure 5-13. It can be observed that,
in the x direction, bending deformation propagates like a wave from the fully supported end to
the plate centre. At the time of 200μs, bending first takes place near the boundary, and when the
structure comes back to rest (t=800μs), the maximum residual bending deformation is near the
centre; while the bending deformation originally in the boundary area decreases to a very small
value and eventually becomes negative. In the y direction, no bending occurs near the boundary,
while in the middle area, εdy goes up with time, and larger deformations take place at the
locations closer to the centre. At the final stage of the structural response, the maximum value of
εdy occurs at the central region.
(a) εdx distribution for the shell elements in Group 1
80
(b) εdy distribution for the shell elements in Group 1
Figure 5-13. εd distribution for the shell elements in Group 1
5.4.2 Strain distribution along the diagonal line
The progressions of εmid and εd of the shell elements along the diagonal line are illustrated in
Figure 5-14 and Figure 5-15 respectively.
(a) εmidx distribution for the shell elements in Group 2
81
(b) εmidy distribution for the shell elements in Group 2
Figure 5-14. εmid distribution for the shell elements in Group 2
(a) εdx distribution for the shell elements in Group 2
82
(b) εdy distribution for the shell elements in Group 2
Figure 5-15. εd distribution for the shell elements in Group 2
The transient distributions of εmidx and εmidy in Figure 5-14 exhibit a similar pattern to those
shown in Figure 5-12. Compared with the shell elements along the x axis, the shells along the
diagonal line show more consistent transient distribution between the x and y directions. The
stretching states of these elements are symmetric in both directions.
As to bending states, Figures 5-15 and 5-13 show some similarity in the original and final
distributions of εd, at the regions adjacent to the centre and locations near the clamped edge, in
both directions. However, compared with the elements along the x axis, the trends of the
bending deformation propagation along the diagonal line are less regular. One possible reason is
that, during large plastic deformation of a square plate, plastic hinges travel along the diagonal
lines from the clamped edges to the plate centre [20, 24, 26], and thus the progression of εd is
not exactly in the x or y directions. It has been shown in the Figure 5-15 that the distributions of
εdx and εdy are not monotonic in both spatial and temporal domains, and very sensitive to
position and time. Due to the limit of the element sizes and shapes, it is very difficult to set
elements exactly on the diagonal line, and thus discrepancy occurs between εdx and εdy of the
83
84
selected elements.
5.4.3 Analysis and discussion
In summary, from the FE study on a typical panel, it is concluded that
(1) The stretching deformation increases with time, and propagates from the boundary to the
centre.
(2) Maximum stretching deformation occurs at the centre, and it reduces with the increase of
the distance from the centre. No stretching takes place near the edge.
(3) Bending deformation has a traveled pattern, from the boundary to the centre.
(4) The maximum permanent bending deformation takes place near the central area, and the
final bending near the edge is almost zero.
(5) The permanent peak value of εd is less than 13% of the permanent maximum εmid. Therefore,
in this case, stretching has a much more significant contribution to the final shape of the
back face, and stretching/membrane deformation can be considered as the main effect in the
back face deformation mechanism. The numerical simulation has confirmed the dominating
effect of membrane force in the large plastic deformation of plates, which was theoretically
analysed by Jones [20], and Symonds and Wierzbicki [52].
5.5 Effect of boundary conditions
Boundary conditions can significantly affect the deflections of impulsive loaded structural
members made from monolithic materials [20]. However, no such investigations have been
made on the sandwich beams or plates. In this study, a numerical simulation was conducted to
examine the effect of two boundary conditions: fully clamped versus simply supported. In the
simply supported case, the plate has the same geometry with that shown in Figure 5-1(a), and
the nodes on the back face at the boundary of previously clamped and opening regions are
allowed to rotate with respective to x or y axes but restricted in translations, and all of the other
nodes are set free to move. Once again, the result of Specimen ACG-1/4-TK-6 is presented here,
as shown in Figure 5-16.
Figure 5-16. Effect of boundary conditions on the time history of back face deflection and core crushing
In the figure, it is found that simply supported boundary increases the back face deflection by
about 20%. However, the cores have very similar compressions in the two cases.
5.6 Summary
Based on the experiments in Chapter 3, this chapter presents a corresponding numerical
simulation study using software LS-DYNA.
In the simulation, both the face-sheet and core were modeled using shell elements and bi-linear
elasto-plastic constitutive relationship. To improve the computational accuracy of local large
plastic deformation, an adaptive meshing approach, known as fission h-adaptivity was
employed. This approach is capable of refining the elements where large deformations take
place. The TNT charge was meshed into solid elements with the ALE formulation. Its
mechanical behaviour is governed by a high explosive material model incorporating the JWL
85
86
equation of state. The interaction between explosion products and structure was modeled with
an erosion contact algorithm, which enables failed elements to be eliminated.
The process of charge explosion and plate response was simulated with three stages, that is,
Stage I - Expansion of the explosive from time of detonation to interaction with the plate; Stage
II - Explosive plate interaction; and Stage III - Plate deformation under its own inertia. The FE
model predicted similar deformation/failure patterns to those observed experimentally for both
face-sheets and core structure. Likewise, the simulation results demonstrate a good agreement
with the measured data obtained from the tests, which mainly include the permanent deflection
of the central point of back face-sheet.
A study was conduced to analyse the contribution of plastic stretching and bending on the
deformation history of a typical sandwich panel back face, as well as the effect of boundary
conditions. The results show that both the stretching and bending deformations progress from
the clamped boundaries to the centre, and in the present case, stretching has a much more
significant contribution to the final shape. Simply supported boundaries increase the back face
deflections but have no effect on core crushing. The simulation study provides an insight into
the process of the blast loading process and the deformation mechanism of the panels, and
therefore can be used as a valuable tool to accurately predict structural response of sandwich
panels under impulsive loading.
CHAPTER SIX
NUMERICAL SIMULATION OF THE ALUMINIUM
FOAM CORE SANDWICH PANELS
6.1 FE model
Using the approach described in Chapter 5, the numerical simulation for the second type of
specimens, aluminium foam core sandwich panels, is reported in this chapter. The FE model is
quite similar to that of the honeycomb core panels except the component of foam core.
6.1.1 Modeling geometry
The geometric model of 1/4 sandwich panel is indicated in Figure 6-1(a). The face-sheets were
meshed using the Belytschko-Tasy shell elements, and the entire model comprises 6,050 shells.
The foam core was meshed into the eight-node brick (solid) elements, and consists of 90,750
brick elements. Figure 6-2(b) illustrates the geometric model of the 1/4 explosive cylinder,
which consists of 12,000 solid elements.
(a) Geometry model of a sandwich panel
87
(b) Geometry model of a charge (enlarged view)
Figure 6-1. Geometric model of a sandwich panel and charge
6.1.2 Modeling materials and blast load
The face-sheets of specimens used in the tests were made of aluminium alloy, which was
modeled with the material type 3 (*MAT_PLASTIC_KINEMATIC) in LS-DYNA.
The material type 63 (*MAT_CRUSHABLE_FOAM) in LS-DYNA was used to model the
aluminum foams. This is a very simple material model, which allows for a description of the
foam behavior through the input of a stress versus volumetric strain curve. The stress versus
strain behaviour is depicted in Figure 6-2(a), which shows an unloading from point a to the
tension stress cutoff at b then unloading to point c and finally reloading to point d. The input
parameters required by this material model are: a material ID, density, Young’s modulus,
Poisson’s ratio, a load curve ID, tensile stress cutoff and damping coefficient [39]. In this model,
the foam is assumed isotropic and crushed one-dimensionally with a Poisson’s ratio that is
essentially zero. The model transforms the stresses into the principal stress space where the
yielding function is defined, and yielding is governed by the largest principal stress. The
principal stresses σ1, σ2, σ3 are compared with the yield stress in compression and tension Yc and
Yt, respectively. If the actual stress component is compressive, then the stress has to be
compared with a yield stress from a given volumetric strain-hardening function specified by the
user, Yc=Yc0+H(ev). On the contrary, when the considered principal stress component is tensile,
the comparison with the yield surface is made with regard to a constant tensile cutoff stress
88
Yt=Yt0. Hence, the hardening function in tension is similar to that of an elastic, perfectly plastic
material [43]. Model 63 assumes that the Young’s modulus of the foam is constant. The
stress-strain curves for the two aluminium foams (6% and 10%) used in this study were from
uniaxial compression tests, and are shown in Figure 6-2(b).
Stre
ss
Volumetric strain
a
bc
d
(a) Schematic representation of a stress-strain curve for the material model 63
(b) Experimental stress-strain curves for the two foams
Figure 6-2. Stress-strain curves for the foam core used in the simulation
The stress versus volumetric strain curve is generated for the foam by conversion of the stress
versus percent crush distance. The volumetric strain e is defined as
change in volumeoriginal volume
e = (6-1)
89
The original volume of a foam block is given by V0=lxlylz, where lx, ly and lz are the side lengths
of the block in three dimensions respectively. Then the current volume is
(1 ) (1 ) (1 )x x y y z zV l l lε ε ε= + + + (1 )x y z x y z x y x z y z x y zl l l ε ε ε ε ε ε ε ε ε ε ε ε= + + + + + + + (6-2)
whereε denotes the engineering strain, and the foam is assumed to be crushed in Z direction. It
is also assumed that the expansion of the foam under a compressive load can be negligible. The
only change in the volume of the foam is due to the change in the crushed depth, i.e. xε = yε =0
This is a reasonable assumption based on the behaviour of the foam as observed in static and
dynamic testing. Then the expression of V can be rewritten as
0(1 ) (1 )x y z z zV l l l Vε ε= + = + (6-3)
Therefore, in this simple case, the volumetric strain is equal to the compressive engineering
strain, or the change in the depth of the block divided by the original depth of the block.
Since delamination cracks occur in the foam core along a path adjacent to the front face-sheet,
the foam core was subdivided such that a thin layer of elements was presented at the interface.
The delamination of the foam core was modeled by removing the thin foam interface elements
from the mesh, using the material erosion capability of LS-DYNA. Maximum tensile strain
(MTS) and maximum shear strain (MSS) were used to define the failure criteria, i.e. any
element that has tensile strain greater than MTS or shear strain greater than MSS will fail and be
removed from further calculation. Here, it is taken that MTS=0.2% and MSS=0.3% [88].
Table 6-1 lists the LS-DYNA material types and mechanical properties of sandwich panel,
explosive, as well as the parameters of equations of state (EOS). The data for face-sheets and
core were determined through tensile/compression tests and parameters of explosive were
obtained from published literature. Similar to the FE model for the honeycomb core panels,
material type 8 (*MAT_HIGH_ EXPLOSIVE_BURN) in LS-DYNA was used to describe the
material property of the TNT charge. In the simulation, the load imparted on the front face of
sandwich panel was defined with algorithm of *CONTACT_ERODING_SURFACE_TO_
SURFACE, which calculates the interaction between explosion product and structure.
90
91
Table 6-1. LS-DYNA material type, material property and EOS input data
for aluminium foam core panels
Material Part LS-DYNA material type, material property and EOS input data
(unit = cm, g, μs) *MAT_PLASTIC_KINEMATIC RO E PR SIGY ETAN
Al-2024-T3
Face sheet
2.68 0.72 0.33 3.18E-3 7.37E-3 *MAT_CRUSHABLE_FOAM RO E PR LCID TSC DAMP
Aluminium foam (6%)
Core
0.16 7.27E-4 0.0 Figure 6-2(b) 2.18E-5 0.1 *MAT_ CRUSHABLE_FOAM RO E PR LCID TSC DAMP
Aluminium foam (10%)
Core
0.27 1.55E-3 0.0 Figure 6-2(b) 4.66E-5 0.1
*MAT_HIGH_EXPLOSIVE_BURN RO D PCJ 1.63 0.67 0.19 *EOS_JWL A B R1 R2 OMEG E0 V0
TNT [85]
Charge
3.71 3.23E-2 4.15 0.95 0.30 7.0E-2 1.0
6.2 Simulation results and discussion
The simulation results are reported and discussed in this section, which include three aspects: (1)
explosion and structural response process; (2) failure patterns of the sandwich panels observed;
and (3) the measured/calculated quantitative result.
6.2.1 Explosion and structural response process
Similar to the model of the honeycomb core panels, three stages can be distinguished for an
entire process in the simulation of aluminium foam core specimens: Stage I – Expansion of the
explosive from time of detonation to interaction with the plate (0~35μs); Stage II –
Explosive-plate interaction (36μs~70μs); and Stage III – Plate deformation under its own inertia
(71μs~5000μs), which are illustrated in Figures 6-3 ~ 6-5, for Specimen L-30-TK-1 loaded with
a 30g explosive.
Figure 6-3. Process of the charge detonation
92
Figure 6-4. Process of explosive-structure interaction
93
Figure 6-5. Process of plate deformation
Figure 6-5 clearly reveals the whole process of the panel deformation (from t=0), in which a
dent failure is first formed at the central area of sandwich front face, and then deformation
extends both outwards and downwards with the transfer of impulse. Likewise, with the
development of denting, the thin foam layer adjacent to the front face begins to fail, and
delamination occurs between the front face and core. After the deformation zone extends to the
external clamped boundaries, a global dishing deformation takes place. A slight oscillation of
the plate occurs with the deformation, and the structure is finally brought to rest by plastic
bending and stretching.
946.2.2 Deformation/failure patterns
A typical contour of deformation/failure pattern obtained in the simulation is shown in Figure
6-6, together with a photograph of a tested specimen. It can be seen that the details of the
deformation/failure have been well captured by the simulation. Both face-sheets in the FE
model show a typical Mode I response [20], which is essentially a large inelastic deformation,
with a denting deformation on the front face and a quadrangular-shaped convexity on the back
side. A cavity occurs between the front face and foam core, due to the failure of the thin foam
layer adjacent to the front skin. Foam densification can also be observed clearly.
Figure 6-6. Comparison of the deformation/failure patterns obtained in simulation and
experiment (Specimen L-30-TK-1)
6.2.3 Face-sheets deflections and core crushing
A comparison is made between the experiment and simulation results in terms of the final
permanent deformation (i.e. deflection) of the central point of back face. A plot of the
experimental values versus the predicted values of all the specimens is shown in Figure 6-7. The
95
data points are very close to the line of perfect match, thus representing a reasonable correlation
between the experimental and predicted results.
0 1 2 3 4 5 6 7 80
1
2
3
4
5
6
7
8
Expe
rimen
tal d
efle
ctio
n (m
m)
Predicted deflection (mm)
11
Figure 6-7. Comparison of predicted and experimental deflections on the back face
(Specimen L-30-TK-1)
A typical displacement-time history of the central points of both face-sheets and front surface of
the core is illustrated in Figure 6-8(a). In order to clearly show the details of deformation
initiation at the beginning stage, the curves beyond t=900μs were cut off. It can be observed
from the figure that the deformation of the front face and top surface of the core starts at t=36μs,
when the explosion product contacts with the plate, Approximately 55 microseconds later (i.e. t
≈ 90μs), the back face begins to deform, and its deflection increases at a slower pace than the
rate at which the front face and front surface of the core deforms. Almost at the same time,
delamination between the front face and core takes place, due to the failure of thin foam layer in
the interface. After that, the front face-sheet keeps deforming under inertia, at a much slower
rate, and reaches its peak at t ≈ 180μs. On the other hand, the deformation of core and back
faces continues, until the deflections reach their respective maximum values at 820μs. Figure
6-8(b) shows the history of core crushing at the central point. At 110μs, the core stops crushing,
i.e. the thickness of core would not change any more, but it still moves downwards under inertia,
together with the back face.
96
(a) History of central point deflections
(b) History of core crushing
Figure 6-8. History of central point deflections and core crushing (Specimen L-30-TK-1)
6.3 Energy absorption
A parametric study has been conducted to investigate the energy absorbing behaviour of the
blast loaded square sandwich panels, which include the time history of plastic dissipation in the
97
face-sheets and core, as well as partition of the plastic energy absorbed by the different
components of the panels; effect of panel configurations is also analysed.
During the interaction between the explosion product and structure, the explosion energy is
transferred to the sandwich panel, and then dissipated by the panel as it deforms. The initial
energy transferred to the structure (ET) is essentially the sum of kinetic (EK) and internal energy
(EI, also known as deformation energy ED). The kinetic energy would reduce with time, while
the internal energy of the system would increase. Given the impulse delivered on the front face
(I), with the impulse transmission, analytically, the front face’s initial velocity can be written as
1f f
IvA hρ
= (6-4)
where A is the exposed area, and fρ and fh are the material density and thickness of
face-sheets, respectively. The corresponding kinetic energy of the front face is calculated by Eq.
(6-2), which is the total energy of the structure obtained from the blast load.
2
I 2 f f
IWA hρ
= (6-5)
After core crushing, the whole structure would have an identical velocity, and the kinetic energy
at that instant can be calculated by
2
II 2 (2 )f f c c
IWA h Hρ ρ
=+
(6-6)
where cρ and are the mass density and thickness of the core, respectively. This part of
energy would be dissipated by plastic bending and stretching of the panel. The above three
equations will be used again in the next chapters. Stages of front face deformation and core
curshing may be coupled due to different structural configurations, material properties and
boundary or loading conditions. The discussion in this issue is beyond the scope of this research.
But in general cases, the whole structure can be assumed to have the identical velocities after
core crushing, as suggested in Refs. [44, 45, 74-77].
cH
98
6.3.1 Time history of plastic dissipation
Figure 6-9 presents a typical time history of the internal energy in each component of a panel
(Specimen L-30-TK-1) during plastic deformation, i.e. front face, back face and core, and the
small amount of energy reduction during the thin layer foam failure in the interface is neglected.
The figure shows that in the early stage of the response, lasting until approximately 120μs, the
front face sheet flies into the core, resulting in core crushing and significant energy dissipation.
After that, the foam core compression almost ceases. From the figure it can be seen that the
large deformation of front face and core compression result in significant energy dissipation and
core compaction constitutes a major contribution, which is 75% of the total dissipation. Much
less energy is absorbed by the back face, as its deformation is maintained at a low level. More
discussion is given in the next section.
in the core
in the front face
in the back face
0 100 200 300 400 500 600 700 800 900 1000 1100 12000.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Plas
tic e
nerg
y di
ssip
atio
n ra
tio
Time (μs)
Figure 6-9. History of plastic dissipation during plastic deformation
(Specimen L-30-TK -1)
6.3.2 Energy partition
The partition of the energy absorbed by different parts of the panels during deformation is
indicated in a stack bar diagram in Figure 6-10. The numbers, designations and specifications of
the specimens can be seen in Table 4-1. Using the plastic energy absorption in Specimen No. 1
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as a benchmark, the plastic dissipations by the other nine panels are expressed in a normalised
form with the total energy absorbed by the first panel. Their energy dissipation is compared and
analysed in terms of (1) impulse level, (2) relative density of core, (3) face-sheet thickness and
(4) core thickness.
1
2
3
4
5
6
7
8
9
10
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Core Front face Back face
Normalised energy
Spe
cim
en N
o.
Figure 6-10. Energy dissipation normalised with the total energy for Specimen No. 1
• Effect of impulse level
In order to study the performance of the panels at different levels of blast loading, all the ten
panels are divided into five groups, i.e. Specimens 1 & 2, 3 & 4, 5 & 6, 7 & 8 and 9 & 10, and
in each group, the two panels have identical configurations but loaded by charges with different
masses. Increasing impulse levels by 23.4%~27.0% (for Specimens 1 & 2 and 3 & 4) and
14.3%~15.8% (for the rest) leads to a rise of total internal energy dissipation in the panels. The
increases in internal energy in each group are 53.8%, 60.4%, 37.8%, 41.5%, and 34.8%,
respectively, which are close to the results obtained from Eq. (6-2) that the total energy input
(WI) is proportional to the square of the total impulse input (I2).
• Effect of face-sheet thickness
100
101
Four specimens have been selected and grouped as two pairs (i.e. Specimens 5 & 7 and 6 & 8)
to investigate the effect of face-sheet thickness on their energy absorbing performance. It is
evident that at two levels of impulse, compared with the panels with thicker face-sheets (1mm)
the internal energy in those with thinner faces (0.8mm) increases significantly, i.e. by 31.6% and
28.0% respectively. Eq. (6-2) indicates that the 0.8mm skin would lead to a 25% increase in the
total energy, which is close to the simulation result obtained. Therefore, it is concluded that a
sandwich panel with thinner face-sheets can improve its energy absorbing capability. However,
when under large blast loading, tearing damage may take place on the thinner front face (e.g.
Specimen 6 (L-30-MD-2)).
• Effect of relative density of core
Effect of relative density of core has been analysed by taking eight panels, which are divided
into four groups: Specimens 1 & 3, 2 & 4, 7 & 9 and 8 &10, respectively. Specimens 1, 2, 7 and
9 have low density cores (6%) while the cores in the other panels are of high density (10%). The
simulation result shows that all the four groups exhibit a similar trend. The total internal energy
for the panels with different core densities in each group is very close, but the contribution of
core in Specimens 3, 4, 9 and 10 increases by 7.0%, 8.0%, 8.0% and 5.9% respectively,
compared with in Specimens 1, 2, 7 and 8. Therefore one can conclude that the portion of
energy absorption by the core can be increased by increasing its density.
• Effect of core thickness
Four panels have been grouped as Specimens 2 & 7 and 4 & 9. Each group has a single core
thickness, i.e. 200mm and 300mm respectively. The simulation result shows that the total
dissipations by the four panels are very similar. Compared with Specimens 2 and 4, in Panels 7
and 9, the percentages of the dissipation by the back faces, reduce from 6.3% to 1.3% and 3.9%
to 0.9%, respectively. This is because in the panels with a thicker core, back faces have smaller
deflections, and thus less energy is dissipated.
6.4 Summary
102
A numerical simulation study has been conducted for the aluminium foam core sandwich panels
using LS-DYNA software, and the results are reported and discussed in this chapter.
In the simulation, a crushable foam constitutive relationship has been used to model the material
property of aluminium foam. A thin layer of foam have been set with a failure criterion in the
interface of front face and core to simulate the delamination crack by removing the failed
elements. The TNT charge has been meshed using solid elements with the ALE formulation. Its
mechanical behaviour is governed by a high explosive material model incorporating the JWL
equation of state. The process of charge explosion and plate response was simulated with three
stages, that is, Stage I -- Expansion of the explosive from time of detonation to interaction with
the plate; Stage II -- Explosive plate interaction; and Stage III -- Plate deformation under its own
inertia. The FE model predicts similar deformation/failure patterns as observed experimentally
for both face-sheets and core structure. Likewise, the simulation results demonstrate a
reasonable agreement with the measured data obtained in the experiment. Finally, a parametric
study was conduced to analyse the energy absorption in each part during plastic deformation. It
is concluded that the foam core constitutes a major contribution to energy dissipation; thinner
face-sheets can raise the total internal energy; while denser and thicker core can increase its
portion of energy dissipation.
103
CHAPTER SEVEN
ANALYTICAL SOLUTION I – A DESIGN-ORIENTED
THEORETICAL MODEL
7.1 Introduction
This chapter presents a design-oriented approximate analytical method for the performance of
the two types of sandwich panels under blast loading. Since in assessing the behaviour of a blast
loaded structure it is often the case that the calculation of final states is the principal requirement
for a designer, a simple model is developed to predict maximum deflections in square sandwich
panels under blast loading, but gives no predictions of displacement-time histories, In analytical
modeling, the deformation is divided into three phases, corresponding to the front face
deformation, core crushing and overall structural bending and stretching, respectively. The
response in the last phase is considered using small deflection and large deflection theories,
respectively, based on the extent of panel deformation. The analysis is based on an energy
balance with assumed displacement fields, which are simplified to reduce the calculation cost
but give acceptable results. Using the proposed analytical model, an optimal design has been
conducted for square sandwich panels with a given mass per unit area, and loaded by various
levels of impulse. Effect of several key design parameters, i.e. ratio of side lengths, relative
density of core, and core thickness is discussed.
7.2 Analytical modeling
According to the theoretical analyses for the blast loaded response of sandwich beams or
circular sandwich plates by Fleck and co-workers [44, 74], the whole structural deformation
process can be split into three phases (Figure 7-1):
Phase I – The blast impulse (I) is transmitted to the front face of sandwich structure, and the
front face is assumed to have instantly obtained a velocity v1 while the rest of the
structure is stationary.
Phase II – The core is compressed while the back face remains undeformed.
Phase III – The back face starts to deform and the component parts of the plate obtain an
identical velocity v2, and finally the structure is brought to rest by plastic bending
and stretching.
Explosive
Phase I
Front facedeformation
Phase II
Core crushing
Phase III
Overall bending& stretching
Al foam corevcH
v1
Support
cHΔ
v2
ytycH
c0 Hw +Δ
0w
Face-sheets
ytycH
Figure 7-1. Schematic illustration showing the three phases in the response of a sandwich panel
subjected to the blast loads
The justification for splitting the analysis into three distinct phases is the observation from the
FE analyses [44, 74] that the time periods for the three stages differ significantly: 0.1ms for the
104
primary shock, 0.4ms for the core crush and 25ms for the overall response. Deformation process
of the square or rectangular sandwich plates has the same phases. As to the structural
deformation in Phase III, the problem under consideration is effectively the same as a classical
one for monolithic plates. To date, such studies have been centred on sandwich beams and
circular sandwich panels, and no theoretical analyses for square plates are available, due to their
more complex nature. In this phase, the residual kinetic energy of the structure (WII) is totally
dissipated by plastic bending and stretching. It has been suggested that if the maximum back
face deflection of the sandwich structure is greater than its original panel thickness ( 2 cfh H+ ),
stretching plays a key role in the deformation mechanism and bending effect can be ignored; on
the other hand, in small deflection cases, bending dominates and the effect of stretching is
negligible [77]. In our tests, all of the aluminium foam core specimens show small deflections,
while 40 of the totally 42 honeycomb core panels exhibit large deflections. Therefore, the
analysis in Phase III is separated as two categories: (1) small deflection analysis, to be used for
aluminium foam core specimens, and (2) large deflection analysis for honeycomb core panels,
which are discussed in detail in Section 7.2.3.
7.2.1 Phase I – Front face deformation
The impulse delivered onto the sandwich structure (I) is assumed to have a uniform distribution
over the front face. With the impulse transmission, the front face has an initial velocity
1f f
IvA hρ
= (7-1)
where A is exposed area of the panel. ρf and hf are material density and thickness of the faces,
respectively. The corresponding kinetic energy of the front face is obtained by 2
I 2 f f
IWA hρ
= (7-2)
7.2.2 Phase II – Core compression
At the end of this stage, the front and back faces as well as the core structure all have an
identical velocity:
2 (2 )f f c c
IvA h Hρ ρ
=+
(7-3)
105
where ρc is mass density of core material, and Hc is core thickness.
Correspondingly, the kinetic energy of the entire structure at the end of Phase II is written as 2
II 2 (2 )f f c c
IWA h Hρ ρ
=+
(7-4)
Hence, the energy absorption in core compression is:
p IE W W= − II (7-5)
Or
I II
I
12
W WW
μμ
− +=
+ (7-6)
with /c c f fH hμ ρ ρ= being the ratio of core mass and face mass.
Two different scenarios of crushing behaviour of cellular materials have been distinguished [69],
i.e., (a) homogeneous deformation and (b) progressive collapse. Under homogeneous
deformation, cellular medium deforms homogeneously over the entire volume of the sample. In
this case, the absorbed energy per unit volume of the foam material for a given level of
deformation can be calculated as the area under the stress–strain diagram. In the case of
progressive collapse, on the other hand, the same deformation is reached by complete
densification of the portion of the cellular material adjacent to the location where the load
applies, while the rest of the cellular solid is assumed undeformed. At the end of complete
densification, the final deformations in both the cases are the same, when the elasticity and
strain hardening are disregarded. In literature [44, 74], crushing of cellular core of sandwich
structures is assumed to have the homogeneous deformation mode.
Tan et al. [71, 89] reported shock effect on porous media, and they suggested that there exists a
critical velocity (108m/s for the small cells and 42m/s for the large cells), beyond which the
cellular solids have the progressive collapse mode. In our research, all of the cores exhibit this
type of deformation, which has been confirmed by both the observation after tests and
numerical simulations. A 1-D metal foam column with the progressive collapse mode is shown
106
in Figure 7-2(a), in which the final thickness cH is reached by complete densification of the
portion close to the point of load application, while the rest of the core does not deform at all. A
‘rigid-perfectly-plastic-locking (R-P-P-L)’ model (Figure 7-2(b)) is used to idealise cellular
materials, where the core is considered fully densed at the densification strain εD, and the stress
level jumps from cYσ to *c
Yσ [71, 89], which can be determined by
21* cC
D
cY Y
vρε
σ σ += (7-7)
,ccYσ ρ
,ccYσ ρ
**, ccYσ ρ
cH
cHΔ
*l
'cH
cH
(a) Progressive deformation mode of a cellular material under impact loading
cYσ
*cYσ
Dε
Stress
Strain
Uniaxial compression testR-P-P-L model
(b) A typical stress-strain curve for cellular material, which is idealised into a R-P-P-L model
Figure 7-2. Schematic illustration showing the progressive deformation mode of cellular
materials under impact loading and its simplified material model
The profile of the front face and front surface of core at the end of this stage is approximated by
107
the following shape function:
c c( , ) cos cos2 2
x yw x y Ha b
π π= Δ (7-8)
where a and b are half side lengths of the panel, and ; for square plates, a = b. a b≥
Then the energy dissipation during core crushing can be obtained by
2p 0 044 cos cos2 2 c
CY
b ac
A Hy CY
xE H dxdya b πππσ σ Δ= Δ =∫ ∫ (7-9)
where the value of CYσ is estimated using the formulae given in [44], that is,
3/ 20.3( *)CY ρ Yσσ = (7-10)
for metal foams, where *ρ is relative density and Yσ is the yield stress of the solid, and
*CY ρ Yσσ = (7-11)
for honeycombs. Then
2 2 2
c
2p I II
2( ) 1
4 4 2 8c c cY Y Y f f
E W W IH A A A hπ π μ πσ σ μ σ ρ
− +Δ = = = ⋅+
(7-12)
7.2.3 Phase III – Overall bending and stretching
In this phase, two scenarios of panel deformation of both front and back faces are considered:
small deflections and large deflections. For simplification, the initial flat plate is considered. To
make the analytical model more general, a rectangular plate, rather than a square one, is
considered here.
• Small deflection analysis
In the small deflection analysis, bending is the main effect and stretching can be neglected. All
the remaining kinetic energy at the end of Phase II is assumed to dissipate by plastic
deformation at the hinge lines generated within the front and back faces, with the contribution
from the core neglected [77].
A sketch of the displacement field of the panel back face is shown in Figure 7-3, where lines AB, BD, CD and AC correspond to fully clamped edges. In addition, five hinge-lines are needed for
108
the plate to become a mechanism (i.e. EF, AE, CE, FB and FD). Since the plate deflection is small, the length of the hinge-lines can be considered equal to the projection on the undeformed plate ABDC. ξ0 is a constant and 0<ξ0<=1. When ξ0=1, the rectangular plate reduces to a square plate.
2b
2a
w0
w0
aξ0
ϕ
A J I B
H E G F
CK
D
xy
Figure 7-3. Displacement field of the back face
In the analysis, the face material satisfies von Mises yielding criterion. The plastic energy
dissipation ( bU ) depends on the length of the plastic hinge-lines, their angle of rotation, and
fully plastic bending moment per unit length (Mp). Due to the symmetric nature of the problem,
only a quarter of the plate is considered here. In AIGH, the angle ϕ is determined using upper
bound theorem [90] by
2tan (3 )ϕ η= + −η (7-13)
where ab
η = .
The rotation angle of the plastic hinge-line is given by
1 00
cos sin(w
a b)
ϕ ϕθ
ξ= + (On AE) (7-14a)
2 0 /w bθ = (On AI) (7-14b)
109
3 0 /w bθ = (On EG) (7-14c)
4 0 /w a 0θ ξ= (On AH) (7-14d)
Then the bending dissipation of the whole structure ( ) can be obtained by bU
p 1 m p 2 m p 3 m p 4 m p 024 ( )3 AE AI EG AHb M dl M dl M dl M dl M w RU θ θ θ θ= × + + + =∫ ∫ ∫ ∫ (7-15)
where p ( )f
Y f f cHM h hσ += ; 2
0 020 0
cos 1 1( sin ) (2 )4[ ]R
ϕη ϕ ξ η ξ
ξ η η= + + + − +
ξ. In the present
small deflection cases, the front face deflection and core crushing are much less than the panel
thickness, and thus for simplicity, Mp can be calculated based on back face and deformed core,
which can significantly reduce the computational complexity but give acceptable accuracy.
Equating Eq. (7-4) and Eq. (7-15), we have
II bW U= (7-16)
i.e. 2
p 02 (2 )f f c c
I M w RA h Hρ ρ
=+
(7-17)
Then 2
0p2 (2 )f f c c
IwA h H Mρ ρ
=+ R
(7-18)
Johnson [91] defined a dimensionless number, namely damage number, which can be presented
in the following form
2
2n f 2f Y f
IDA hρ σ
= (7-19)
Then Eq. (7-18) can be normalised and expressed in terms of nD as
00 2 (2 )(
n
c f
w hDAwt Rt h H hρ
= = ⋅+ − )
(7-20)
110
where 2 f ct h H= + being the initial overall thickness of the sandwich panel. /f ch h H= and
/c fρ ρ ρ= .
Taking account of core compression, the normalised maximum deflection at the front face is
then given by '
' 0 00
cw w Hw
t t tΔ
= += (7-21)
As an interest, if it is assumed that the maximum deformation is achieved by a constant
quasi-static pressure P, by equating the work done by load to the total strain energy dissipated in
the structure, the limit pressure can be obtained using the following equation:
00 04 cos cos2 2
b abP yxw dxda b y Uππ =∫ ∫ (7-22)
• Large deflection analysis
Following conventional large deflection analysis [92] of a rectangular plate, under a uniformly
distributed impulsive loading, its final profile is assumed to have the shape governed by Eqs.
(7-23a) and (7-23b) for the back and front faces, respectively.
0
0
0
sin cos2
sin cos2
cos cos2 2
back
back
back
x yu ua bx yv va b
x yw wa b
π π
π π
π π
⎧ =⎪⎪⎪ =⎨⎪⎪ =⎪⎩
(7-23a)
0
0
0
sin cos2
sin cos2
( )cos cos2 2
front
front
front c
x yu ua bx yv va b
x yw w Ha b
π π
π π
π π
⎧ =⎪⎪⎪ =⎨⎪⎪
= + Δ⎪⎩
(7-23b)
with uback, vback and wback being displacements of the back face in x, y and z directions,
respectively and similarly, ufront, vfront and wfront for the front face. u0, v0 and w0 are the maximum
displacements (corresponding to the plate centre) in x, y and z directions.
Here, compared with w, the magnitudes of u and v are very small and will be neglected in the
following calculation [46]. The in-plane strain components of the back face, front face and core
111
can be calculated by 2
2
12
12
back backx
back backy
back back backxy
wx
wy
w wx y
ε
ε
γ
⎧ ∂⎛ ⎞=⎪ ⎜ ⎟∂⎝ ⎠⎪⎪
⎛ ⎞∂⎪ =⎨ ⎜ ⎟∂⎝ ⎠⎪⎪ ⎛ ⎞∂ ∂⎛ ⎞⎪ = ⎜ ⎟⎜ ⎟⎪ ∂ ∂⎝ ⎠⎝ ⎠⎩
(7-24a)
2
2
12
12
frontfrontx
frontfronty
front frontfrontxy
wx
wy
w wx y
ε
ε
γ
⎧ ∂ ⎞⎛=⎪ ⎟⎜ ∂⎪ ⎝ ⎠
⎪∂ ⎞⎛⎪ =⎨ ⎟⎜ ∂⎝ ⎠⎪
⎪ ∂ ∂⎞ ⎞⎛ ⎛⎪ = ⎟ ⎟⎜ ⎜∂ ∂⎪ ⎝ ⎝⎠ ⎠⎩
(7-24b)
22
2 2
14
14
12
frontcore backx
frontcore backy
front frontcore back backxy
wwx x
w wy y
w ww wx y x y
ε
ε
γ
⎧ ⎡ ⎤∂⎛ ⎞∂⎛ ⎞⎪ ⎢ ⎥= + ⎜ ⎟⎜ ⎟∂ ∂⎪ ⎝ ⎠⎢ ⎥⎝ ⎠⎣ ⎦⎪⎪ ⎡ ⎤∂⎛ ⎞ ⎛ ⎞∂⎪ ⎢ ⎥= +⎨ ⎜ ⎟ ⎜ ⎟∂ ∂⎢ ⎥⎝ ⎠⎝ ⎠⎪ ⎣ ⎦⎪
⎡ ⎤∂ ∂⎛ ⎞⎛⎛ ⎞∂ ∂⎪ ⎛ ⎞= +⎢ ⎥⎜ ⎟⎜⎜ ⎟⎜ ⎟⎪ ∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠ ⎝ ⎠⎝⎣ ⎦⎪⎩
⎞⎟⎠
(7-24c)
with backxε , back
yε and being strain components of the back face, backxyγ core
xε , coreyε and core
xyγ being
strain components of the core, and frontxε , front
yε and frontxyγ being strain components of the front face,
respectively.
Then the energy dissipated during plastic stretching ( sU ) can be expressed as
0 0 0 0
0 0
4 [ ( ) ] 4 [ ( ) ]3
4 [ ( ) ]
f ff fY Y
Y Y
cl
b a b aback back back front front fronts f x y xy f x y xy
b a core core c corec x y xy
U h dxdy h dxdy
H dxdy
σ σσ ε ε γ σ ε ε γ
σ ε ε τ γ
= × + + + × + +
+ × + +
∫ ∫ ∫ ∫
∫ ∫3 (7-25)
where clσ and cτ are in-plane tensile stress and shear stress of the core, which have been
discussed in [2]. Here, compared with face-sheets, the contribution of core in stretching is
relatively small, and in-plane stretching of the thin dense layer can be reasonably neglected. In
the present case, since the in-plane tensile strength of the hexagonal cells and aluminium foam
is very small, their contribution to the stretching dissipation is ignored [74]. Then Eq. (7-25) can
be re-written as
112
0 0 0 0
220 0
4 [ ( ) ] 4 [ ( ) ]3
[ ( ) ]
f ff fY Y
Y Y
b a b aback back back front front fronts f x y xy f x y xy
fc Y f
U h dxdy h dxdy
C w w H h
σ σσ ε ε γ σ ε ε γ
σ
= × + + + × + +
= + + Δ
∫ ∫ ∫ ∫ 3 (7-26)
where 2 1( )(
8 3b aCa b
π= + + ) .
Equating the kinetic energy and stretching dissipation UIIW s gives
2 2II 0 0[ ( ) ] f
Ys cW U C w w H hσ= = + + Δ f (7-27)
For simplicity, Eq. (7-27) can be re-written as
2
0 01 2 3 0K w K w K+ − = (7-28)
where 1 2 fY fK C hσ= ; 2 2 f
Y f cK C h Hσ= Δ ; 2
23 2 (2 )
fY f c
f f c c
IK CA h H
σρ ρ
= −+
h HΔ .
Solving Eq. (7-28), the maximum deflection of the back face is obtained by
22 2 1
01
4
23K K K K
wK
− − += (7-29)
Similarly, Eq. (7-29) is normalised and expressed in terms of Dn as
200 2 (2 )
12
c nc
w H AhDw
t C ht t ρΔ
= + − Δ+
= H (7-30)
The normalised maximum deflection at the front face is then given by '
' 0 00
cw w Hw
t t tΔ
= += (7-31)
7.3 Model validation
In this section, the above analytical model is validated by comparing its predictions with the
experimental data. Results from the previous analytical models for circular sandwich plates are
also included.
113
7.3.1 Comparison with experiment
Figures 7-4 shows the comparison between the normalised theoretically predicted back face
deflections and the experimental results, for both foam core and honeycomb core panels. In the
figure, it can be seen that the data points are concentrated around a straight line of a slope equal
to 1, thus representing a reasonable correlation between the experimental and predicted results.
In our tests, all of the aluminium foam core specimens show small deflections, while 40 of the
totally 42 honeycomb core panels exhibit large deflections.
-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
Def
lect
ion/
Initi
al th
ickn
ess r
atio
(Exp
erim
enta
l res
ult)
Deflection/Initial thickness ratio (Predicted result)
Aluminium foamcore panels
Honey combcore panels
Figure 7-4. Comparison between the experimental and predicted maximum deflection of
the back face of the two types of specimens
7.3.2 Comparison with the analytical model for circular plates
As an interest, the present model for square plates is compared with that for a circular plate
proposed by Qiu et al. [74]. In their analytical model the structural response was discussed for
small deflection analysis and large deflection analysis, and the exact yield locus was
approximated by either inscribing or circumscribing squares, which simplified the subsequent
calculation and gave upper and lower bounds of the maximum deflection, respectively.
The analytical predictions for circular and square sandwich panels are shown in Figure 7-5. The
114
model for circular panels is from [74] (here denoted as ‘the QDF model’ for convenience), while
that described in Section 7-2 (denoted as ‘the present model’) is used for the square panels. In
this comparison, the diameter of the circular panels is set equal to the side length of the square
panels, and all the other parameters (i.e. impulse, face and core thickness and material properties)
are identical. The comparison is made by plotting normalised deflections against normalised
impulses, which can be written as
/fY f
IIAM σ ρ
= (7-32)
where 2 f f c cM hρ ρ= + H being the mass of the plate per unit area. It is clearly shown in
Figure 7-5 that the QFD model gives similar results with the present model. With the increase of
impulse, the QFD model leads to more rapid increase in deflection than the present one.
Large deflections
Small deflections
/ /Y fI AM σ ρ
0/
wt
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5square - test
square - analytical prediction (von Mises yield locus)
circular - analytical prediction (circumscribing yield locus)
circular - analytical prediction (inscribing yield locus)
Figure 7-5. Comparison of the analytical predictions for circular panels and square panels
7.4 Optimal design of square plates to shock loading
The objective of the present optimal design is to minimize the permanent maximum deflections
of a sandwich plate for a given mass, exposed area and blast impulse [45, 77, 93]. It should be
115
emphasised that in this research, a ‘local’ optimal design of the panel configuration, rather than
a ‘global’ one is sought. The optimization is limited to a relatively narrow scope. Using the
analytical model proposed in Section 7.2, this section presents an optimal design for a square
sandwich panel with a given mass and exposed area, loaded by various levels of impulse. The
design variables include (1) ratio of side lengths, (2) relative density of core, and (3) core
thickness.
7.4.1 Effect of side length ratio
In this section, ten fully clamped rectangular honeycomb core sandwich plates with various side
length ratios are compared and plotted in Figure 7-6. The side length ratio (a/b) on the
horizontal axis varies from 1 to 10, and the vertical axis denotes the normalised permanent
maximum deflection of the face sheet. All the panels have an identical exposed area (0.0625m2)
and mass per unit area (11.18kg/m2) (i.e. core thickness is 16.67mm, the relative density of core
is 0.03 and face thickness is 1.84mm), and loaded by three values of identical impulses (32Ns,
40Ns and 48Ns). The analytical prediction shows that the square panel (a/b=1) exhibits the
largest deflections on both faces, and the deflections decrease monotonically with an increasing
in a/b. But it is found that the side length ratio has little effect on the core crushing behaviour.
1 2 3 4 5 6 7 8 9 10
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
a/b
Max
imum
def
lect
ion/
thic
knes
s rat
io
Front face, 32Ns Back face, 32Ns Front face, 40Ns Back face, 40Ns Front face, 48Ns Back face, 48Ns
Figure 7-6. Comparison of the normalised maximum deflections of the rectangular plates
with various side length ratios, for three impulses
7.4.2 Effect of relative density of the core 116
Now consider a square panel with a constant dimensionless mass per unit
area /( ) 0.0334fM Lρ = , where L is the half side length of the square exposed area; and the
value of the dimensionless core thickness ( /cH L ) is fixed at 0.134. A search for the optimal
relative density of the honeycomb core is shown in Figure 7-7, which is plotted by
dimensionless maximum face deflection (ratio of maximum deflection and half side length)
against the relative density of the core ( *ρ ). It is concluded from the figure that, a relative
density of 0.03 may be considered as the optimal value, at which the back face experiences the
smallest maximum deflections. Also it is noted that weaker cores can significantly increase the
core compression.
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
Front face, 32Ns Back face, 32Ns Front face, 40Ns Back face, 40Ns Front face, 48Ns Back face, 48Ns
ρ∗
Max
imum
def
lect
ion/
half
side
leng
th ra
tio
*ρ
Figure 7-7. Dimensionless maximum deflections of a sandwich plate with various relative
densities of cores, for three impulses
7.4.3 Effect of core thickness
With the same value of mass per unit area as above and fixing the value of relative density as
0.03, the optimal core thickness for a square panel is searched in Figure 7-8, by plotting the
ratio of maximum deflection and half side length against dimensionless core thickness ( /cH L ).
The result shows that a dimensionless core thickness of approximately 0.5 gives the best
performance, and thicker cores yield larger compressions.
117
0.00 0.09 0.18 0.27 0.36 0.45 0.54 0.63 0.720.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
Hc/l
Max
imum
def
lect
ion/
half
side
leng
th ra
tio
/cH L
Figure 7-8. Dimensionless maximum deflections of a sandwich plate with various thicknesses of
cores, for three impulses
7.5 Summary
This chapter proposes a design-oriented analytical model to describe the structural response of
the square sandwich panels tested in Chapters 3 and 4, and based on the theoretical model, an
attempt is made to optimise the configuration of the sandwich panels.
The complete deformation process is split into three phases. In Phase I: the blast impulse is
transmitted to the front face of the sandwich structure and, as a result, the front face would
attain an initial velocity while the rest of the structure is stationary. In Phase II: the core is
compressed while the back face is stationary; and in Phase III: the back face starts to deform
and finally the structure is brought to rest by plastic bending and stretching. In Phases I and II,
based on momentum and energy conservation, and idealising the cellular core as a
rigid-perfectly-plastic-locking material, the energy dissipated during core crushing and the
compressive strain of core structure were calculated, and the residual kinetic energy at the end
of Phase II was further obtained. The analysis in Phase III was either for small deflection or for
large deflection case, according to the degree of panel deformation. In the small deflection
118
119
analysis, bending is the main energy dissipation mechanism and stretching can be neglected; the
kinetic energy is assumed to be dissipated solely at the plastic hinge lines generated. In the large
deflection analysis, on the other hand, stretching plays a key role in the deformation mechanism
and bending effect can be ignored. The residual kinetic energy is dissipated in the continuous
deformation fields. In both cases, the contribution of core in the last phase can be disregarded.
By equating the kinetic energy acquired to the plastic strain energy produced in the structure,
the permanent maximum deflections of the face-sheets were obtained. The analytical model was
validated by comparing the predictions with the experimental data as well as the theoretical
calculations based on the analytical model for circular sandwich plates.
Using the present model, an optimisation was conducted for minimal permanent maximum
deflection of square sandwich panels for a given mass/per unit area and loaded by several levels
of impulse. The design variables include (1) ratio of the two side lengths, (2) relative density of
core, and (3) core thickness.
120
CHAPTER EIGHT
ANALYTICAL SOLUTION II – A THEORETICAL MODEL
FOR DYNAMIC RESPONSE
8.1 Introduction
This chapter presents a new analytical model, which can capture the dynamic response, i.e. not
only the final profile, but also the structural response time, and a new yield criterion for the
sandwich panel is proposed by considering the core strength. The deformation process is
assumed to have three phases, which is similar to the procedure described in Chapter 7, that is,
the front face deformation, core crushing and overall structural bending and stretching,
respectively. However, In Phase III, both the front face and back face are assumed to have the
identical profile, as shown in Figure 8-1. It is extremely difficult to mathematically describe the
final profiles of the sandwich plates, especially the top faces with pitting failure. Here, a
simplified equation has been used to approximate the permanent deflections. Rate sensitivity of
the cellular cores in the out-of-plane direction is considered, but its longitudinal strength is
assumed unaffected by compression. By adopting an energy dissipation rate balance approach
with the newly developed yield surface, ‘upper’ and ‘lower’ bounds of the maximum permanent
deflections and response time are obtained. Finally, comparative studies are carried out to
investigate: (1) influence of the longitudinal strength of core after compression to the analytical
predictions; (2) performances of square monolith panels and a square sandwich panel with the
same mass per unit area; and (3) analytical models of sandwich beams and circular and square
sandwich plates.
Al foam corev
Face-sheets
Explosive
Support
cH
cH
cHPhase I
Front facedeformation
Phase II
Core crushing
Phase III
Overall bending& stretching
v1
v2
Figure 8-1. Three phases in the response of a sandwich panel subjected to the blast loads
8.2 Analytical modeling
8.2.1 Phase I – Front face deformation
The impulse delivered onto the sandwich structure is assumed to have a uniform distribution
over the front face. With the impulse transmission, the front face has an initial velocity
1f f
Ivhρ
= (8-1)
where I is the impulse per unit area. ρf and hf are material density and thickness of the faces,
respectively.
Based on momentum conservation, the kinetic energy per unit area of the front face is obtained
by 2
I 2 f f
IWhρ
= (8-2)
121
8.2.2 Phase II – Core compression
At the end of this stage, the front and back faces as well as the core structure would all have an
identical velocity:
2 2 f f c
Ivh Hρ ρ
=+ c
(8-3)
where ρc is mass density of core material, and Hc is core thickness.
Correspondingly, the kinetic energy per unit area of the sandwich structure at the end of Stage II
can be written as 2
II 2(2 )f f c c
IWh Hρ ρ
=+
(8-4)
Hence, the energy absorption per unit area during core compression is:
p I IIE W W= − (8-5)
Or
I II
I
12
W WW
μμ
− +=
+ (8-6)
with /c c f fH hμ ρ ρ= being the ratio of core mass and face mass.
The cellular core is assumed to have the progressive deformation pattern as described in Chapter
7. The plateau stress or crushing strength in the out-of plane direction of the cores ( cYσ ) can be
roughly approximated using some equations in the simplified forms, with the dynamic effect
disregarded [44, 74]. In this chapter, instead of using the empirical approximate methods, an
energy-based approach is proposed to calculate the effective dynamic transverse plateau stress,
through the stress-strain curves obtained from the standard uniaxial compression tests, and rate
sensitivity is taken into account.
Define energy absorption efficiency ( )aεη as the energy absorbed up to a given nominal
122
strain aε normalised by the corresponding stress value ( )cσ ε [94]:
( )( )
( )a
a
c
ca
d
εε
ε
εσ
εεσεη
=
∫= 0 (8-7)
Densification strain Dε is the strain value corresponding to the stationary point in the
efficiency-strain curve where the efficiency is a global maximum, i.e.
( ) 0== Ddd
εεεεη
(8-8)
To remove the recoverable energy at the stage of elasticity, Eq. (8-7) is modified as
( )( )
( )cr
a
a
c
ac
dε
ε
ε ε
σ ε εη ε
σ ε=
=∫
(8-9)
where crε is the strain at yield corresponding to the start of the plateau regime.
Quasi-static uniaxial compression tests were carried out for the honeycombs and aluminium
foams to obtain their densification strains. The two materials were cut into 75×75×12.5mm and
120×120×30mm plates, respectively. Figures 8-2 and 8-3 show the curves plotted by their
energy absorption efficiency against compressive strain, together with the original compressive
stress-strain relationships. By taking the maximum value of ( )aεη , the corresponding
densification strain Dε can be easily obtained, which are smaller than the strain in theory [2],
i.e. 1 1.4 *Dε ρ= − , with *ρ being relative density.
123
(a) 1/8-5052-0.0020 core
(b) 1/8-5052-0.0015 core
(c) 1/8-5052-0.0010 core
124
(d) 1/8-5052-0.0007 core
(e) 5/32-5052-0.0015 core
(f) ACG-1/4 core
Figure 8-2. Energy absorption efficiency-strain curves and stress-strain curves of honeycombs
125
(a) 6% foam core
(b) 10% foam core
Figure 8-3. Energy absorption efficiency-strain curves and stress-strain curves
of aluminium foams
Then static transverse plateau stress of the cellular core can be calculated by
C coreY
D
Eσε
= (8-10)
where is the energy absorption per unit area of the core in crush. The static plateau stresses
for the honeycombs and aluminium foams are listed in Table 8-1.
coreE
126
Table 8-1. Specifications and mechanical properties of the honeycombs and aluminium foams
Core type Material
yield
strength
σY (MPa)
Cell
size le
(mm)
Foil
thickness
t (mm)
Static
transverse
plateau stress
cYσ (MPa)
Dynamic
transverse
plateau stress
cdYσ (MPa)
Longitudinal
strength
clYσ (MPa)
1/8-5052-0.0020 core 265 3.18 0.051 3.32 8.97 0.14 1/8-5052-0.0015 core 265 3.18 0.038 3.13 8.45 0.08 1/8-5052-0.0010 core 265 3.18 0.025 1.79 4.83 0.03 1/8-5052-0.0007 core 265 3.18 0.018 0.95 2.56 0.02 5/32-5052-0.0015 core 265 3.97 0.038 2.42 6.53 0.05 ACG-1/4 core 260 6.35 0.066 1.58 4.27 0.06 6% foam core 268 - - 2.06 2.06 1.18 10% foam core 268 - - 5.39 5.39 2.54
Aluminium honeycombs compressed in the out-of-plane direction (normal to the hexagonal
cells) has a softening quasi-static response after a peak load, characteristic of a Type II structure
[95]. It has been found that under dynamic loading, the crush strength increases significantly
[96, 97]. Wierzbicki [98] theoretically studied the progressive crushing behaviour of aluminium
honeycombs, and it has been found that the dynamic crush strength of hexagonal honeycombs
made of Al-5052 and Al-3104 can be approximately calculated based on their static crush
strength as
2.7CYdYCσ σ= (8-11)
The calculated dynamic plateau stresses are also listed in Table 8-1. The metal foam used in the
experiment did not exhibit evident strain rate effect in the previous impact tests.
Adopting the dynamic plateau stresses, the energy dissipation per unit area during core crushing
can be obtained by
pCdY cE Hσ= Δ (8-12)
Then
127
p I II Ic
21 12 2 2c c c c
dY dY dY dY f f
E W W W IH hμ μ
σ σ μ σ μ σ ρ− + +Δ = = = ⋅ = ⋅
+ + (8-13)
It should be emphasised that if the resultant compressive strain cε is beyond the densification
strain Dε , then cε is set equal to Dε , to neglect the additional dissipation mechanisms
required to conserve energy [44, 74].
8.2.3 Phase III – Overall bending and stretching
A theoretical model for monolithic rectangular plates is extended for sandwich panels by
adopting the proper yield loci, where the core strength is considered. The basic formulations,
yield loci and solution are described in the subsequent sections. It is noted that the slight shear
effect in the early stage of this phase is ignored.
(1) Basic formulations
Jones [67] proposed an approximate theoretical procedure to predict the permanent transverse
deflections of solid rectangular plates subjected to large dynamic loads. The influences of
finite-deflections or geometry changes are retained in the analysis but elastic effects are
disregarded. In the model, it is assumed that the shape of the displacement field due to dynamic
loads which produce finite-deflections is the same as the velocity profile developed for the
corresponding static collapse load and keeps constant. The analysis is based on a balance of
plastic dissipation rate, which is governed by the following equation:
( ) ( )31
( ) ' 'm
p
mA l Am
'p Mw wdA w w dl w w dA=
− = − + −∑∫ ∫ ∫N M M N (8-14)
where M and p3 are mass per unit surface area of a plate and transverse pressure; w is the
transverse deflection at the mid-point and A is the total exposed area; M and N are moments and
membrane forces per unit length, respectively; lm and p are length and number of plastic
hinge-lines, respectively.
128
The first term on the right hand side of Eq. (8-14) gives the internal energy dissipated at
traveling plastic “hinges” while the remaining term is the energy dissipated in continuous
deformation fields. Jones assumed that the initial energy transferred to the structure is all
dissipated by the plastic hinge lines, and thus the second term of Eq. (8-14) vanishes. In the
present case, the structure obtains a uniform velocity field after the application of a pressure
pulse with a neglegible short period, such that p3 vanishes in the equation as well. Since a
square plate is divided into four rigid regions separated by eight straight line hinges (finally
located at the four clamped edges and four diagonal lines), m-th hinge line with length lm, the
energy equilibrium reduces to the following equation: 8
1( ) ( ) m mA
mMw wdA w lθ
=
= −∑∫ N M (8-15)
where mθ denotes the relative angular rotation rate across a hinge-line.
It is convenient to define
( mD Nw M )θ= − (8-16)
which is the internal energy dissipation per unit length of a hinge. Clearly, the dissipation
function D will depend on the type of supports around the boundary of a plate and on the yield
condition which is selected. If a square yield condition is employed [67], the corresponding
dissipation relation for a clamped monolithic plate is
0
41 m
wD M
Hθ= +⎛ ⎞
⎜ ⎟⎝ ⎠
(8-17)
with M0 being circumferential plastic bending moment and H is thickness. Based on Eqs. (8-14)
and (8-16), and solving the associated nonlinear differential equation approximately, the
maximum permanent deflection of a monolithic square plate was obtained.
(2) Yield loci for sandwich structures
Now the model by Jones [67] has been extended for the square sandwich panels by applying
proper yield loci. In Jones’ model, the exact maximum yield curve can be described by the
following equation:
129
2
0 0
1M NM N
+⎛ ⎞⎜ ⎟⎝ ⎠
= (8-18)
where N0 is circumferential plastic membrane force, and
20 / 4YM Hσ= (8-19a)
0 YN Hσ= (8-19b)
where Yσ is yield stress.
For simplicity, the exact yield locus of the plate is approximated by either inscribing or
circumscribing squares as sketched in Figure 8-4.
0MM
0NN1
1
-1
-1
•
•
•
• •
•
•
•
•
•
•
••
•
•
•• •0.5
0.5
-0.5
-0.5
0.618
0.618
-0.618
-0.618
•
•
•
•
•
•
ζ
ζ
ζ−
ζ−
h2+σσ
Circumscribing yield locus for all of the three cases
Inscribing yield loci
Exact yield locus with core strength considered
Yield locus with core strength disregarded (Qiu et al. [74])
Exact yield locus for monolithic plates (Jones [67])
•
Figure 8-4. Yield loci for monolithic and sandwich structures together with
their circumscribing and inscribing squares
Based on the yield surface described above, Qiu et al. [74] proposed a yield criterion for
sandwich structures with thin and strong face-sheets and a thick and weak core, which is
130
governed by
0 0
1M NM N
+ = (8-20)
where M0 and N0 are given by
02( )f c
Y f c f lY cM h H h Hσ σ= + + / 4 (8-21a)
0 2clY c Y fN Hσ σ= + f h (8-21b)
where (1 )c c cH H ε= − ; clYσ is longitudinal strength of the core, which can be calculated using
the empirical equations given in [2]:
22 ( )3
clY Y
tl
σ σ= (8-22)
for hexagonal honeycombs, where t and l are cell wall thickness and minor cell diameter,
respectively, and
3/ 20.3( *)clYσ ρ= (8-23)
for metal foams. The longitudinal strengths of the two types of core are listed in Table 8-1.
The analysis can be further simplified by approximating the above yield locus by either
inscribing or circumscribing squares, as indicated in Figure 8-4. In this paper, a new yield
criterion is developed for the sandwich structures, where the effect of core strength is
considered, and it is assumed that the cellular cores have the same properties for tension and
compression. Figure 8-5 shows the distribution of the normal stresses on a sandwich cross
section, subjected to a bending moment M and a membrane force N simultaneously. Based on
the magnitude of N, the stress distribution is divided into two regimes, i.e. 0
02
NN h
σσ
≤ ≤+
(Figure 8-5(a)), and 0
12
Nh N
σσ
≤ ≤+
(Figure 8-5(b)), where c flY Yσ σ σ= , f ch h H= . Then the
profile of the stresses can be described by the combination of a symmetric component with
respect to the central axis (stretching effect) and an antisymmetric component (bending effect).
131
/ 2cHξ= +
σ
Total stress σm+σn Bending σm Stretching σn
fYσ σ
fYσ σ
cH
fh
cH
fh
fh
/ 2cHξ
fhclYσ
clYσ
clYσ
/ 2cHξ
(a) 0
02
NN h
σσ
≤ ≤+
fhξ
= +
Total stress σm+σn Bending σm Stretching σn
σfYσ σf
Yσ σ
fh
clYσ c
lYσ
fYσ
fhξ
cH cH
(b) 0
12
Nh N
σσ
≤ ≤+
Figure 8-5. Sketch of the normal stresses profile on a sandwich cross-section
According to Figure 8-5, circumferential bending moment M and membrane force N in both two
cases can be calculated by
0 2NN h
ξσσ
=+
, 2
0
14 (1 )
MM h h
σξσ
= −+ +
when 0
02
NN h
σσ
≤ ≤+
(8-24a)
hh
NN
221
0 +−=σξ
, ( )[ ]
( ) σξξ
++−+
=hh
hhMM
142422
0
when 12 0
≤≤+ N
Nhσ
σ (8-24b)
whereξ is a constant and 0 1ξ≤ ≤ .
132
Eliminatingξ , the corresponding yield locus is expressed as
( )( ) 114
22
02
2
0
=⎟⎟⎠
⎞⎜⎜⎝
⎛++
++
NN
hhh
MM
σσσ
when hN
N2
00 +≤≤σσ
(8-25a)
( ) ( ) ( )( ) 014
2112 22
0
0
=++
+−⎥⎦
⎤⎢⎣
⎡−++⎟⎟
⎠
⎞⎜⎜⎝
⎛
+σ
σσ
hh
hhNN
MM
when 12 0
≤≤+ N
Nhσ
σ (8-25b)
where M0 and N0 can be determined using Eqs. 8-21(a) and 8-21(b), respectively.
The sketch of the curve governed by Eq. (8-25) can also be seen in Figure 8-4.
If 1c flY Yσ σ σ= = , the above locus reduces to the yield criterion for a solid monolithic plate, i.e.
Eq. (8-18); while if 1σ and 1f ch h H= , then Eq. (8-25) reduces to the locus for the
sandwich structures with thin, strong faces and a thick, weak core, i.e. Eq. (8-20).
For simplicity, the exact yield locus can also be approximated by either circumscribing or
inscribing squares, as illustrated in Figure 8-4. The circumscribing square locus can be
described as
0M M= (8-26a)
0N N= (8-26b)
Likewise, the inscribing square locus is governed by
0MM ζ= (8-27a)
0NN ζ= (8-27b)
where ( )
( )
1 2 2
1
22 3 2 2 2
1 4 1, 8 1 0
2
4, 8 1 0
2
sh h
s
s s sh h
σ
ζ
σ
+ −+ − ≤
=+ −
+ − >
⎧⎪⎪⎨⎪⎪⎩
in which ( )( ) ,14
22
2
1 σσσ
+++
=hhhs
( )( )( )2 2
1 3 81
2
hs
h
σ σ
σ
− += +
+,
( )( )3
2 1 21
2
hs
hσ
+= −
+.
133
(3) Solution
Taking the circumscribing square, the corresponding dissipation function is
( )( )0
4 21
(4 1 ) mc
w hD M
h h H
σθ
σ
+= +
+ +
⎛ ⎞⎜⎜⎝ ⎠
⎟⎟ (8-28)
It is assumed that the shape of the displacement field for the dynamic case is of the same pattern
as the velocity profile used by Jones [67] to give an upper bound to the collapse load of the
corresponding static problem. Thus, the displacement field of the square plate is given by
0( ) ( )(1 )(1 )x yw t W tL L
= − − (8-29)
where L is half side length; is the maximum deflection. 0W
Substituting Eqs. (8-28) and (8-29) into Eq. (8-15) and solving the associated nonlinear
differential equation approximately, which satisfies the initial conditions
(0) 0w = (8-30a)
2(0)w = v (8-30b)
and final conditions at t = T
( ) 0w T = (8-31)
the normalised maximum central deflection of the back face and structural response time T can
be obtained by
1 2
212
2 1 3
00
2ˆ 13
nWc
LZW αα
α α α= + 1−
⎡ ⎤⎛ ⎞= ⎢ ⎥⎜ ⎟
⎝ ⎠⎢ ⎥⎣ ⎦ (8-32)
1 2
13 22
2 1
2tan
6 3f n
Y f
TT
LZα α
σ ρα α α
−=⎛
= ⎜⎝ ⎠3
⎞⎟ (8-33)
where ˆ /cc H L= ; ˆ /(1 )ch h ε= − ; 1ˆ4 (1 )h hα σ= + + ; 2
ˆ2hα σ= + ; ( )3ˆ2 1 chα ρ ε= + −⎡ ⎤⎣ ⎦ , with
c fρ ρ ρ= and ρc and ρf being mass density of core and density of face-sheet material,
respectively. It can be found that 1α , 2α and 3α actually reflect the effect of plastic bending,
134
stretching and dimensionless mass, respectively. nZ is herein defined as ‘sandwich damage
number’, which can be used to assess the dynamic plastic response of sandwich structures, and
2
22
1ˆn f
Y f c
IZ
H cσ ρ= (8-34)
Then the maximum deflection of the front face is given as
'1 0
cHL
W W Δ= + (8-35)
Similarly, applying the inscribing square gives
1 2
212
2 1 30
2ˆ 13
ncZW αα
α ζ α α= +
⎡ ⎤⎛ ⎞⎢⎜ ⎟⎝ ⎠⎢ ⎥⎣ ⎦
1− ⎥ (8-36)
1 2
13 22
2 1
2tan
6 3nT
Zα αα ζ ζ α α
−=⎛⎜⎝ ⎠3
⎞⎟ (8-37)
8.3 Model validation
The theoretically predicated normalised back face deflections ( 0 /W L ) from Eqs. (8-32) and
(8-36) are compared with the experimental data in Figure 8-6. It is clearly shown in the figure
that the data points are concentrated around a straight line of a slope equal to 1, thus
representing a reasonable correlation between the experimental and predicted results. In this
model, an UPPER BOUND was actually given when assuming a kinetically admissible velocity
field [Eq.(8-29)]. The inscribed and circumscribed yield surfaces then gave the upper bound and
lower bound of this UPPER BOUND solution. Therefore, the over-prediction in Figure 8-6 can
be easily explained.
135
0.0 0.1 0.2 0.3 0.4 0.5 0.6
0.0
0.1
0.2
0.3
0.4
0.5
0.6
circumscribing yield locusinscribing yield locus
Theo
retic
al p
redi
cted
def
lect
ion
Experimental deflection
Aluminiumfoam core
panels
Honey combcore panels
Figure 8-6. Comparison of experimental and theoretically predicted deflections
8.4 Comparative studies of the analytical solutions
In the subsequent sections (8.4.1 ~ 8.4.3), comparative studies are carried out on the analytical
solutions of monolithic and sandwich structures. In Section 8.4.1, an assessment is made on the
influence of the longitudinal strength of core after compression to the analytical solution
described in Section 8.2. Then the performances of square monolithic panels and a square
sandwich panel with the same mass per unit area are compared in Section 8.4.2. Finally, a
comparison among the analytical models of sandwich beams, circular and square sandwich
plates is presented in Section 8.4.3.
8.4.1 Effect of the core strength after compression
The yield criterion employed in the present model assumes that the core has a progressive
deformation pattern, and the longitudinal strength of core clYσ is unaffected by core
compression. On the other hand, in the model by Qiu et al. for circular sandwich plates [74], it
was assumed the core deforms uniformly and the plastic membrane force N0 is unaffected by the
136
degree of core compression. Thus the average yield longitudinal strength 'clYσ of the
compressed core, was in effect assumed to increase with the transverse compression and can be
reasonably estimated by
' /(1c clY lY cσ σ )ε= − (8-38)
Then the plastic bending moment '0M and plastic membrane force of the sandwich
structure with the core after compression are given by
'0N
'0 ( ) 'f c
Y f c f lY cM h H h Hσ σ= + + 2 / 4 (8-39a)
'0 2'c
lY c Y fN Hσ σ= + f h (8-39b)
Substituting '0M and into the yield criterion gives the result '
0N
1 2''' 21
0 ' '2 '2 1 3
2ˆ ' 1 13
ncZW αα
α α α+ −
⎡ ⎤⎛ ⎞= ⎢⎜ ⎟
⎝ ⎠⎢ ⎥⎣ ⎦⎥ (8-40)
1 2
1' '3 2
' '22 1
2' tan
6 3nT
Zα αα α α
− ⎛= ⎜⎝ ⎠
'3
⎞⎟ (8-41)
for the circumscribing square locus, and
1 2''' 21
0 ' '2 '2 1 3
2ˆ ' 13
ncZW αα
α ζ α α1+ −
⎡ ⎤⎛ ⎞= ⎢⎜ ⎟
⎝ ⎠⎢ ⎥⎣ ⎦⎥ (8-42)
1 2
1' '3 2' '22 1
2' tan
6 3nT
Zα αα ζ α αζ
− ⎛= ⎜⎝ ⎠
'3
⎞⎟ (8-43)
for the inscribing square locus, and 'ˆ /cc H L= ; '1 4 (1 )h hα σ= + + ; '
2ˆ/(1 ) 2c hα σ ε= − + ;
( )'3
ˆ2 1 chα ρ= + −⎡⎣ ε ⎤⎦ .
The theoretical predictions rising from the two different assumptions are compared and the
result is illustrated in Figure 8-7. For convenience, they are named as
Assumption 1: Longitudinal strength of core is unaffected by compression
137
Assumption 2: Plastic membrane force is unaffected by compression
The comparison is made by plotting normalised deflections against normalised impulses, which
can be written as
ˆ/f
Y f
II
M σ ρ= (8-44)
where 2 f f c cM hρ ρ= + H being the mass of the plate per unit area. The figure reveals that the
predictions based on the two different assumptions are very similar. Thus one can further
conclude that the longitudinal strength of core after compression has little effect on the final
deflections of the honeycomb and aluminium foam core sandwich panels in Phase III.
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55 testAssumption 1 - circumsbribing locusAssumption 1 - inscribing locus Assumption 2 - circumsbribing locusAssumption 2 - inscribing locus
/( / )fY fI M σ ρ
0W
Aluminiumfoam core
panels
Honey combcore panels
Figure 8-7. Comparison of the effect of two assumptions
8.4.2 Comparison of square monolithic and sandwich panels
Performances are compared for a square solid plate and sandwich plate with the same material
and same mass per unit area. The configuration of the sandwich panel is as follows: fh =0.6mm,
138
fρ =2600kg/m3, fYσ =150MPa, L =125mm, =10.0mm, cH 78.0cρ = kg/m3 and c
Yσ =4.5MPa.
It should be emphasized that for simplicity, both the face and core are assumed to have the same
base material, and strain rate effect and elasticity are disregarded. Besides, the core is modeled
as a compressive isotropic material with equal longitudinal and normal strength. Then the
corresponding solid panel has density 0ρ =2600kg/m3, yield strength Yσ =150MPa and
thickness H=1.50mm.
Since the present theoretical analysis for square sandwich panels is based on Jones’ work [67], it
is interesting to make a comparison between the analytical predictions from the present model
and Jones’ model for monolithic plates. In Jones’ model, the central dimensionless deflection (δ)
of a square plate can be expressed as
0 213 n
LL Hw Rδ = = + −1 (8-45)
where 22
2nY
I LRH Hρσ
⎛ ⎞= ⎜ ⎟
⎝ ⎠ , which was termed as ‘the response number’ by Zhao [99].
Figure 8-8 presents the comparison result for solid and sandwich plates from Eq. (8-45) and Eqs.
(8-32) and (8-35), with only circumscribing yield locus used. The figure is plotted for
normalised maximum deflection against normalised impulse per unit area. It can be found that
the model for sandwich panels leads to more rapid increase in deflection than the model for
solid panels. Also there exists a critical impulse value, within which the sandwich structures
have superior blast resistance, while if the impulse is larger than the critical value, solid plates
produce smaller deflections. For example, the critical impulse for the back face deflection of the
sandwich panel with circumscribing yield surface is approximately 0.49.
139
0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
Max
imum
def
lect
ion/
half
side
leng
th ra
tio
Solid plate Sandwich plate (back face) Sandwich plate (front face)
/( / )f fI M σ ρ
Critical impulsepredicted by Eq. (8-55)
Critical impulse predicted by Eqs. (8-32), (8-35) and (8-45)
Figure 8-8. Comparison of a square solid plate and a square sandwich plate
with the same materials and mass/area 23.9 /M kg m=
Here a criterion is derived to determine the critical impulse, which would be helpful to the
primary design of blast resistant sandwich panels. Assuming that nZ is not very large (in the
reasonable scope), Eq. (8-32) can be simplified using Taylor expansion:
1 22
2 1 30
1 3
1 2ˆ 1 12 3
ˆ3
nc nZ cZW α αα α α α α
+ × −⎡ ⎤
= ⎢ ⎥⎣ ⎦
= (8-46)
Only the first two terms of Taylor series are kept here, which retains reasonable accuracy and
the error introduced is less than 10%. Similarly, Eq. (8-45) can be simplified as
31 2(1 1)2 3
nn
L LRH H
Rδ = + × − = (8-47)
Then the ratio of deflections for sandwich panel back face and monolithic plate is given by
03
31 3
1
c
W HHδ α α
= ⋅ (8-48)
140
Since the two plates have an identical mass per unit area, we have
02c c f fH h Hρ ρ+ = ρ (8-49)
which can be re-written as
2c
HH
hρ + = (8-50)
Substituting Eq. (8-50) into Eq. (8-48), and equating the two deflections, we have
{ }0
2( 2 ) 1(1 ) 4 [(1 ) ] (1 )c c c
W hh h
ρδ ε ε σ ε
+=
− − + + −= (8-51)
or
2( 4 )(1 ) (1 ) ( 2 )c ch h hρ ε ε ρ+ − + − − + =2 0
( 8 - 5 2 )
Solving Eq. (8-52), the compressive strain of core is given by
11cε = − Ψ (8-53)
where 2 2
1[ 4(4 )(2 ) ]
2(4 )h h h
hρ ρ
ρ+ + + −
Ψ =+
1/ 2 h .
According to Eq. (8-13), cε is determined by
2
2 2c
c cc dY f
H IH h
εσ ρ
Δ= = Ψ ⋅
f cH (8-54)
where 2 2hh
ρρ
+Ψ =
+.
Equating Eq. (8-53) and Eq. (8-54), the critical impulse crI can be found by
1/ 21
2
2(1 )[ ccr dY f f cI σ ρ−Ψ=
Ψ]h H (8-55)
141
When the impulse applied is smaller than crI , the sandwich structures have superior blast
resistance (with a smaller back face deflection) than solid structures; otherwise solid structures
would have a better performance. Using Eq. (8-55), the normalised critical impulse ( ) was
predicted and plotted in Figure 8-8, which is approximately equal to 0.47 and very close to the
value obtained from the analytical models, i.e. Eqs. (8-32) and (8-45).
ˆcrI
According to Eq. (8-55), it is clear that for a sandwich panel with a given mass per unit area,
different combinations of h and ρ correspond to different values of . As an example, a
square sandwich panel with the mass per unit area
crI
23.9 /M kg m= is considered here. The
distribution of with respect to various thickness ratios and core relative densities is plotted
as a 3D surface and shown in Figure 8-9. It can be seen that higher
crI
ρ and lower h give
smaller , while lowercrI ρ and higher h yield larger . crI
ρh
crI
0.100.08
0.06
0.04
0.02
0.00
0.25
0.50
0.75
1.00
0.020.03
0.040.05
0.060.07
Figure 8-9. Distribution of normalised critical impulse with respect to various thickness ratios
and core relative densities, for a square sandwich panel with the mass/areas
23.9 /M kg m= 142
8.4.3 Comparison among sandwich beams, circular and square sandwich panels
As an interest, the present model for square sandwich plates is compared with the existing
models for beams [44] and circular plates [74]. Consider a sandwich beam and a circular plate
with half span l and radius R , respectively, which are equal to the half side length ( L ) of a
square sandwich plate. The other dimensions, e.g. core and face thickness and material
properties are all the same as those of the model described in Section 8.4.2. The structures are
subjected to impulses with an identical magnitude per unit area.
Assuming the longitudinal core strength keeps constant during compression, the analytical
solution for sandwich beams with the circumscribing square yield surface can be expressed as a
function of sandwich damage number Zn as
1 2
212
2 1 30
8ˆ 12 3
ncZW αα
α α α1+ −
⎡ ⎤⎛ ⎞= ⎢⎜ ⎟
⎝ ⎠⎢ ⎥⎣ ⎦⎥ (8-56)
21/ 2 2
1 3 1 312 321 3 22 2
21 3
241 1 tan3 32 3 4
n
n
n n
T
ZZ
Z Z
αα α α αα α
α α αα αα α
−=
⎡ ⎤⎢ ⎥⎡ ⎤⎛ ⎞ ⎢ ⎥⎢ ⎥+ − +⎜ ⎟ ⎢ ⎥⎢ ⎥⎝ ⎠⎣ ⎦ +⎢ ⎥⎢ ⎥⎣ ⎦
(8-57)
For the inscribing square yield surface,
1 2
212
2 1 30
8ˆ 12 3
ncZW αα
α ζ α α1+ −
⎡ ⎤⎛ ⎞= ⎢⎜ ⎟
⎝ ⎠⎢ ⎥⎣ ⎦⎥ (8-58)
21/ 2 2
1 3 1 312 321 3 22 2
21 3
241 1 tan
3 32 3 4
n
n
n n
T
ZZ
Z Z
αα α α αα α
ζ α α α ζα αζα α
−=
⎡ ⎤⎢ ⎥⎡ ⎤⎛ ⎞ ⎢ ⎥⎢ ⎥+ − +⎜ ⎟ ⎢ ⎥⎢ ⎥⎝ ⎠⎣ ⎦ +⎢ ⎥⎢ ⎥⎣ ⎦
(8-59)
As to the analytical model for circular panels, it is interesting to find that it has completely the
same expression with the present formula of square panels when it is assumed that the
longitudinal strength of core is unchanged during compression, i.e. Eqs. (8-32) and (8-33) for
the circumscribing yield locus, and Eqs. (8-36) and (8-37) for the inscribing yield locus,
143
respectively.
The result of comparison is plotted in Figure 8-10 for the maximum deflection. It can be clearly
seen that the model of sandwich beam produces larger deflections than the plate models. The
theoretical back face deflections of a beam are 1.4 and 1.35 times that of a plate at Zn=1.0, for
inscribing and circumscribing yield surface, respectively. It is reasonable, since the beam is
equivalent to a square panel with only two opposite sides fixed, which gives a weaker constraint
than the panels fully clamped along all the four edges.
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16 beam - circumscribing yield locus beam - inscribing yield locus plate - circumscribing yield locus plate - inscribing yield locus
Max
imum
def
lect
ion/
half
side
leng
th ra
tio
Zn
Figure 8-10. Comparison of the deflections predicted by sandwich beam and sandwich plates
with the same materials and mass/area 23.9 /M kg m=
8.5 Summary
This chapter proposes a theoretical model to predict the dynamic response of square sandwich
panels with cellular cores, and a comparative study is made among the proposed model and
analytical solutions of monolithic structures, sandwich beams and circular sandwich panels.
144
145
In the model, the complete deformation process is split into three phases, as described in
Chapter 7. However, both the front face and back face are assumed to have identical profiles.
The cellular core is assumed to have a progressive deformation mode in crushing with the
longitudinal core strength unaffected by compression, and an energy-based approach is
proposed to calculate the effective dynamic transverse plateau stress, through the stress-strain
curves obtained from the standard uniaxial compression tests with the strain rate effect
considered. A new yield surface is developed for the sandwich cross-section with different core
strengths. By adopting an energy dissipation rate balance approach with new yield surface, the
‘upper’ and ‘lower’ bounds of the maximum permanent deflections and response time
corresponding to the inscribing and circumscribing yield loci are obtained.
Finally, comparative studies are carried out to investigate: (1) influence of the change of core
in-plane strength during compression to the analytical predictions; (2) performances of square
monolith panels and a square sandwich panel with the same mass per unit area; and (3)
analytical models of sandwich beams, circular and square sandwich plates. It has been found
that that the change of longitudinal strength of core after compression has little effect on the
final deflection of the structure in Phase III. There exists a critical impulse value, within which
the sandwich structures have superior blast resistance than solid structures, and a formula is
derived to estimate the critical impulse. Finally, the analytical models for circular and square
sandwich panels have been found to produce the same formulae, which predict smaller
deflections than the sandwich beam model.
146
CHAPTER NINE
CONCLUSIONS AND FUTURE WORK
9.1 Conclusions
This thesis presents a systematic study (including experimental, computational and analytical
investigations) on the resistant behaviour and energy absorbing performance of square metallic
sandwich panels with honeycomb core and aluminium foam core under blast loading.
The honeycomb core panels consisted of two face-sheets and a honeycomb core, which were
made of aluminium alloys. The test program was divided into four groups, each of which was
designed to identify the effect of several key parameters, such as cell size and foil thickness of
the honeycomb, face-sheet thickness and mass of charge. In the tests, a four-cable ballistic
pendulum system with a laser displacement transducer was used to measure the impulse
imparted on the panel, and a PVDF pressure gauge recorded the pressure-time history at the
central point of specimen’s front face. Two types of experimental results were obtained: (1)
deformation/failure modes of specimen observed in the tests, which were further separated as
those for front face, core and back face, respectively; and (2) quantitative results, which include
the impulse on sandwich panel, permanent central point deflection of the back face and
pressure-time history at the mid-point of front face.
It has been shown that specimens with thicker face-sheets, a higher density core and loaded by
larger charges tend to have localized deformation on the front face, and those with thinner skins
and a sparse core and subjected to lower level shocks are prone to deform globally. At the
central area of the front face, indenting and pitting were observed on all the specimens but their
occurrence seems irregular. Folding damage took place in the honeycomb core, with different
extent of deformation at different regions. As for the back face, all of the panels showed a
147
dome-shaped deformation. Based on the quantitative analysis, it has also been found that the
face-sheet thickness and relative density of core structure can significantly affect the back face
deformation. By adopting thicker skins and honeycomb cores with higher relative density, the
deflection of back face can be reduced. Also, for a given panel configuration, it is evident that
the back face deflection increases with impulse, approximately linearly.
The aluminium foam sandwich panels have been tested using the same approach.
Deformation/failure patterns of specimen and quantitative results have been reported and
analysed. It has been observed that the front faces show localized indentation for all the
specimens. In addition, winkling at the edges of the panels occurs for panels with a lower
density core. The back faces have a uniform quadrangular-shaped dome, spreading from the
centre to the clamped boundaries. The core crushing damage was accompanied with a cavity
between the front face and the crushed foam core. It has also been found that the panels with
dense core, both thick core and faces have small deflections.
Based on the experiments, a corresponding numerical simulation study has been conducted
using software LS-DYNA. In the simulation for honeycomb core panels, both the face-sheet and
core were modeled using shell elements and bi-linear elasto-plastic constitutive relationship. To
improve the computational accuracy of local large plastic deformation, an adaptive meshing
approach, known as fission h-adaptivity was employed. This approach is capable of refining the
elements where large deformations take place. The TNT charge was meshed into solid elements
with the ALE formulation. Its mechanical behavior is governed by a high explosive material
model incorporating the JWL equation of state. The interaction between explosion products and
structure was modeled with an erosion contact algorithm which enables failed elements of
explosion products to be eliminated. The process of charge explosion and plate response was
simulated with three stages, that is, Stage I - expansion of the explosive from the time of
detonation to the beginning of the interaction with the plate; Stage II – explosive-plate
interaction; and Stage III - plate deformation under its own inertia. The FE model predicted
similar deformation/failure patterns to those observed experimentally for both face-sheets and
148
core structure. Likewise, the simulation results demonstrate a good agreement with the
measured quantitative data obtained from the tests, which mainly include the permanent
deflection of the central point of back face-sheet. A parametric study was conduced on a typical
panel to analyse the contribution of plastic stretching and bending on the deformation history of
the sandwich panel back face, as well as the effect of boundary conditions. The results show that
both the stretching and bending deformations progress from the clamped boundaries to the
centre, and in the present case, stretching has a much more significant contribution to the final
shape. Changing from simply supported boundaries to fully fixed edges increases the back face
deflections but have no effect on core crushing.
In the simulation for aluminium foam core panels, a crushable foam constitutive relationship
has been used to model the material property of aluminium foam. A thin layer of foam was set
with a failure criterion in the interface of front face and core to simulate the delamination crack
by removing the failed elements. The blast loading process and structural response are quite
similar to those of the honeycomb core panels. A study was conduced to analyse the energy
absorption in each part during plastic deformation. It is concluded that the foam core constitutes
a major contribution to energy dissipation, thinner face-sheets can raise the total internal energy;
while denser and thicker core can increase its portion of energy dissipation.
Two analytical solutions have been proposed to describe the structural responses of the two
types of sandwich panels under blast loading. Both analyses were based on the assumption that
the complete deformation process could be split into three phases. In Phase I: the blast impulse
is transmitted to the front face of the sandwich structure and, as a result, the front face would
attain an initial velocity while the rest of the structure is stationary. In Phase II: the core is
compressed while the back face is stationary; and in Phase III: the back face starts to deform
and finally the structure is brought to rest by plastic bending and stretching. The first solution is
a design-oriented approximate analytical method, which is excellent for predicting permanent
deformations, but gives no response time. In Phases I and II, based on momentum and energy
conservation, and idealising the cellular core as a rigid-perfectly-plastic-locking material, the
149
energy dissipated during core crushing and the compressive strain of core structure were
calculated, and the residual kinetic energy at the end of Phase II was further obtained. The
analysis in Phase III is either for small deflection or for large deflection case, according to the
extent of panel deformation. In the small deflection analysis, bending is the main energy
dissipation mechanism and stretching can be neglected; the kinetic energy is assumed to be
dissipated solely at the plastic hinge lines generated. In the large deflection analysis, on the
other hand, stretching plays a key role in the deformation mechanism and bending effect can be
ignored. The residual kinetic energy is dissipated in the continuous deformation fields. In both
cases, the contribution of core in the last phase can be disregarded. By equating the kinetic
energy acquired to the plastic strain energy produced in the structure, the permanent maximum
deflections of the face-sheets were obtained. The analytical model was validated by comparing
the predictions with the experimental data as well as the theoretical calculations based on the
analytical model for circular sandwich plates. Using the model, an optimisation was conducted
for minimal permanent maximum deflection of square sandwich panels for a given mass per
unit area and loaded by several levels of impulse. The design variables included (1) ratio of the
two side lengths, (2) relative density of core, and (3) core thickness.
The second analytical model can capture the dynamic response, i.e. not only the final profile,
but also the total response time. The cellular core was assumed to have a progressive
deformation mode in crushing with the longitudinal core strength unaffected by compression,
and an energy-based approach was proposed to calculate the effective dynamic transverse
plateau stress, through the stress-strain curves obtained from the standard uniaxial compression
tests with the strain rate effect considered. A new yield surface was developed for the sandwich
cross-section with different core strengths. By adopting an energy dissipation rate balance
approach and newly developed yield surface, the upper and lower bounds of the maximum
permanent deflections and response time were obtained. Finally, comparative studies were
carried out to investigate: (1) influence of the change for core in-plane strength after
compression to the analytical predictions; (2) performances of square monolith panels and a
square sandwich panel with the same mass per unit area; and (3) analytical models of sandwich
150
beams, and circular and square sandwich plates. It has been found that that the longitudinal
strength of core after compression has little effect on the plastic stretching in Phase III. There
exists a critical impulse value, within which the sandwich structures have superior blast
resistance than solid structures; and a criterion was derived to estimate the critical impulse.
Finally, the analytical models for circular and square sandwich panels have been found to
produce the same formula, which predicts smaller deflections than the sandwich beam model.
9.2 Future work
In this section, some future work is recommended:
• Experimental and computational studies
In the present study, no response time history was recorded, due to the limitation of
measurement means. This issue would be addressed in the future work. Besides, more physical
tests and numerical simulations should be carried out on the sandwich panels with various
configurations, for instance, rectangular panels with different side length ratios and the two
faces with unequal thicknesses. Also, the blast resistant performances of several novel core
topologies should be studied: square honeycombs, corrugated lattices, and octet, tetrahedral,
pyramidal or Kagome trusses, which can offer the lightweight structures with high strength and
stiffness. Besides, a wider range of impulse levels should be applied to identify the failure
criteria of the panels, and various damage modes, e.g. tearing, fracture and heating effect.
• Analytical modeling
Both of the current analytical models disregard the displacement-time variations, as the
structural response of the sandwich plate in phase III is actually a procedure of plastic hinge
lines’ forming and traveling, which are not considered in the current analytical models. In the
second solution, the deformation mode is assumed constant. To trace the time history
displacement, the exact yield locus rather than inscribing and circumscribing approximations
must be adopted, which leads to huge difficulty to solve the differential equations. This issue
151
would be considered in the future work. In the previous analyses, the total energy of the
structure obtained from the blast load was assumed to be all dissipated during core crushing and
subsequent overall plastic bending and stretching, and the energy loss due to front face-core
delamination, shear at the supports and vibratory motion was neglected. This additional
dissipation therefore needs to be considered in the future work. Also both of the two analytical
solutions were based on the assumption that the complete deformation process is split into three
phases, which can significantly simplify the problem. However, the rational of this assumption
might be still arguable, and therefore, the coupling effects between Phases I and II and Phases II
and III should be investigated in detail. Besides, more detailed optimisation should be
considered.
References
[1] Lu G and Yu TX. Energy absorption of structures and materials. Cambridge: Woodhead
Publishing Ltd.; 2003.
[2] Gibson LJ and Ashby MF. Cellular solids: structure and properties, 2nd edition. Cambridge:
Cambridge University Press; 1997.
[3] Ashby MF, Evans AG, Fleck NA, Gibson LJ, Hutchinson JW and Wadley HNG. Metal
foams: a design guide. Oxford: Butterworth-Heinemann, 2000.
[4] Allen HG. Analysis and design of structural sandwich panels. Oxford: Pergamon Press,
1969.
[5] Plantema FJ. Sandwich construction. New York: Wiley Ltd., 1966.
[6] Vinson JR. The behavior of sandwich structures of isotropic and composite materials.
Lancaster: Technomic Publishing Co., 1999.
[7] Zenkert D. An introduction to sandwich construction. Warley: Emas Press, 1995.
[8] Kinney G and Graham K. Explosive shocks in air, 2nd edition. Berlin: Springer-Verlag, 1985.
[9] Cole RH. Underwater explosions. New York: Dover, 1948.
[10] Baker W, Cox P, Westine P, Kulesz J and Strehlow R. Explosion hazards and evaluation.
New York: Elsevier, 1983.
[11] Enstock LK, and Smith PD. Measurement of impulse from the close-in explosion of
doped charges using a pendulum. International Journal of Impact Engineering, 2007;
34(3): 487-494.
[12] Hanssen AG., Enstock L, and Langseth M. Close-range blast loading of aluminium foam
panels. International Journal of Impact Engineering, 2002; 27: 593-618.
[13] Nurick GN and Martin JB. Deformation of thin plates subjected to impulsive loading – a
review, Part II: Experimental studies. International Journal of Impact Engineering, 1989;
8(2): 171-186.
[14] Jacinto AC, Ambrosini RD and Danesi RF. Experimental and computational analysis of
plates under air blast loading. International Journal of Impact Engineering, 2001; 25: 927-
947.
[15] Boyd SD. Acceleration of a plate subject to explosive blast loading – trial results. Internal
Report DSTO-TN-0270, Department of Defense, Australia, 2002.
[16] Guruprasad S and Mukherjee A. Layered sacrificial claddings under blast loading Part II –
experimental studies. International Journal of Impact Engineering, 2000; 24: 975-984.
[17] Neuberger A, Peles S and Rittel D. Scaling the response of circular plates subjected to
large and close-range spherical explosions. Part I: Air-blast loading. International Journal
of Impact Engineering, 2007; 34(5): 859-873.
152
[18] Neuberger A, Peles S and Rittel D. Scaling the response of circular plates subjected to
large and close-range spherical explosions. Part II: Buried loading. International Journal
of Impact Engineering, 2007; 34(5): 874-882.
[19] Jones N. Recent studies on the dynamic plastic behaviour of structures. Applied
Mechanics Review, 1989; 42(4): 95-115.
[20] Jones N. Structural impact. Cambridge: Cambridge University Press, 1989.
[21] Menkes SB and Opat HJ. Tearing and shear failures in explosively loaded clamped beams.
Experimental Mechanics, 1973; 13: 480-486.
[22] Teeling-Smith RG and Nurick GN. The deformation and tearing of thin circular plates
subjected to impulsive loads. International Journal of Impact Engineering, 1991; 11(1):
77-91.
[23] Olson MD, Nurick GN, Fagnan JR and Levin A. Deformation and rupture of blast loaded
square plates – predictions and experiments. International Journal of Impact Engineering,
1993; 13(2): 279-291.
[24] Nurick GN and Shave GC. The deformation and tearing of thin square plates subjected to
impulsive loads – an experimental study. International Journal of Impact Engineering,
1996; 18(1): 99-116.
[25] Rudrapatna NS, Vaziri R and Olson MD. Deformation and failure of blast-loaded square
plates. International Journal of Impact Engineering, 1999; 22: 449-467.
[26] Ramajeyathilagam K, Vendhan CP and Bhujanga Rao V. Non-linear transient dynamic
response of rectangular plates under shock loading. International Journal of Impact
Engineering, 2000; 24: 999-1015.
[27] Nurick GN, Gelman ME and Marshall NS. Tearing of blast loaded plates with clamped
boundary conditions. International Journal of Impact Engineering, 1996; 18(7-8): 803-827.
[28] Rudrapatna NS, Vaziri R and Olson MD. Deformation and failure of blast-loaded
stiffened plates. International Journal of Impact Engineering, 2000; 24: 457-474.
[29] Yuen SCK and Nurick GN. Experimental and numerical studies on the response of
quadrangular stiffened plates. Part I: subjected to uniform blast load. International Journal
of Impact Engineering, 2005; 31: 55-83.
[30] Langdon GS, Yuen SCK and Nurick GN. Experimental and numerical studies on the
response of quadrangular stiffened plates. Part II: localised blast loading, International
Journal of Impact Engineering, 2005; 31: 85-111.
[31] Li QM and Jones N. Shear and adiabatic shear failures in an impulsively loaded fully
clamped beam. International Journal of Impact Engineering, 1999; 22: 589-607.
[32] Li QM and Jones N. Formation of shear localization in structural elements under
transverse dynamic loads. International Journal of Solids and Structures, 2000; 37: 6683-
6704.
153
[33] Cloete TJ, Nurick GN and Palmer RN. The deformation and shear failure of peripherally
clamped centrally supported blast loaded circular plates. International Journal of Impact
Engineering, 2005; 32: 92-117.
[34] Wen HM, Yu TX and Reddy TY. Failure maps of clamped beams under impulsive loading.
Mechanics Based Design of Structures and Machines, 1995; 23: 453-472.
[35] Wen HM and Jones N. Low velocity perforation of punch-impact loaded metal plates.
Journal of Pressure Vessel Technology, ASME, 1996; 118(2): 181-187.
[36] Radford DD, Fleck NA and Deshpande VS. The response of clamped sandwich beams
subject to shock loading. International Journal of Impact Engineering, 2006; 32: 968-987.
[37] Radford DD, McShane GJ, Deshpande CS and Fleck NA. The response of clamped
sandwich plates with metallic foam cores to simulate blast loading. International Journal
of Solids and Structures, 2006; 43: 2243-2259.
[38] Nurick GN, Langdon GS, Chi Y and Jacob N. Behaviour of sandwich panels subjected to
intense air blast: Part I – experiments. In: Proceedings of 6th International Conference on
Composite Science and Technology, Durban, South Africa, 2007.
[39] Hallquist JO. LS-DYNA theoretical manual. Livermore: Livermore Software Technology
Co.; 1998.
[40] Yen CF, Skaggs R and Cheeseman BA. Modeling of shock mitigation sandwich structures
for blast protection, In: Proceedings of the 3rd First International Conference on Structural
Stability and Dynamics, Kissimmee, FL, USA, 2005.
[41] Yun SR and Zhao HY. Explosion Mechanics. Beijing: National Defence Industry Press;
2005. (In Chinese)
[42] Johnson GR and Cook WH. A constitutive model and data for metals subjected to large
strains, high strain rates and high temperatures. In: Proceedings of the 7th International
Symposium on Ballistics, Hague, The Netherlands, 1983.
[43] Hanssen AG, Hopperstad OS, Langseth M and Ilstad H. Validation of constitutive models
applicable to aluminium foams. International Journal of Mechanical Sciences, 2002; 44:
359-406.
[44] Fleck NA and Deshpande VS. The resistance of clamped sandwich beams to shock
loading. Journal of Applied Mechanics, ASME, 2004; 71: 1-16.
[45] Xue Z and Hutchinson JW. A comparative study of blast-resistant metal sandwich plates.
International Journal of Impact Engineering, 2004; 30: 1283-1305.
[46] Nurick GN and Martin JB. Deformation of thin plates subjected to impulsive loading — a
review, Part I: Theoretical considerations. International Journal of Impact Engineering,
1989; 8(2): 159-170.
[47] Martin JB and Symonds PS. Mode approximation for impulsively loaded rigid-plastic
structures. Journal of the Engineering Mechanics Division Proceedings, ASCE, 1966; 92:
154
43-46.
[48] Strong WJ and Yu TX. Dynamic models for structural plasticity. Berlin: Springer-Verlag,
1993.
[49] Symonds PS. Elastic, finite deflection and strain rate effects in a mode approximation
technique for plastic deformation of pulse loaded structures. Journal of Mechanical
Engineering Science, 1980; 22(4): 189-197.
[50] Symonds PS and Chon CT. Finite viscoplastic deflections of an impulsively loaded plate
by the mode approximation technique. Journal of the Mechanics and Physics of Solids,
1979; 27(2): 115-133.
[51] Chon CT and Symonds PS. Large dynamic plastic deflection of plates by mode method.
Journal of the Engineering Mechanics Division Proceedings, ASCE, 1977; 103(1): 169-
187.
[52] Symonds PS and Wierzbicki T. Membrane mode solutions for impulsively loaded circular
plates. Journal of Applied Mechanics, ASME, 1979; 46(1): 58-64.
[53] Wierzbicki T and Nurick GN. Large deformation of thin plates under localized impulsive
loading. International Journal of Impact Engineering, 1996; 18(7-8): 899-918.
[54] Nurick GN, Pearce HT and Martin JB. Predictions of transverse deflections and in-plane
strains in impulsively loaded thin plates. International Journal of Mechanical Sciences,
1987; 29(6): 435-442.
[55] Huang ZP and Wang R. The mode approximation technique in dynamic plastic response
of structures. Advances in Mechanics, 1985; 15(1): 1-20. (In Chinese).
[56] Lee EH and Symonds PS. Large deformation of beams under transverse impact. Journal
of Applied Mechanics, ASME, 1952; 19: 308-314.
[57] Hopkins HG and Prager W. On the dynamics of plastic circular plates, Journal of Applied
Mathematics and Mechanics, 1954; 5: 317-330.
[58] Wang AJ. The permanent deflection of a plastic plate under blast loading. Journal of
Applied Mechanics, ASME, 1955; 22: 375-376.
[59] Jones N. Impulsive loading of a simply supported circular rigid-plastic plate. Journal of
Applied Mechanics, ASME, 1968; 3: 59-65.
[60] Wierzbicki T and Florence AL. A theoretical and experimental investigation of
impulsively loaded clamped circular viscoplastic plates. International Journal of Solids
and Structures, 1970; 6: 553-568.
[61] Shen WQ and Jones N. Dynamic response and failure of fully clamped circular plates
under impulsive loading. International Journal of Impact Engineering, 1993; 13: 259-278.
[62] Li QM and Jones N. Blast loading of fully clamped circular plates with transverse shear
effects. International Journal of Solids and Structures, 1994; 31: 1861-1876.
[63] Cox AD and Morland LM. Dynamic plastic deformations of simply-supported square
155
plates. Journal of Mechanics and Physics of Solids, 1959; 229-241.
[64] Komarov KL and Nemirovskii YuV. Dynamic behaviour of rigid-plastic rectangular plates.
International Applied Mechanics, ASME, 1985; 21(7): 683-690.
[65] Smith PD and Hetherington JG. Blast and ballistic loading of structures. Oxford:
Butterworth-Heinemann, 1994.
[66] Duffey TA. The large deflection dynamic response of clamped circular plates subject to
explosive loading. Sandia Laboratories Research Report, No. SC-RR-67-532, 1967.
[67] Jones N. A theoretical study of the dynamic plastic behaviour of beams and plates with
finite-deflections. International Journal of Solids and Structures, 1971; 7: 1007-1029.
[68] Yu TX and Chen FL. The large deflection dynamic plastic response of rectangular plates.
International Journal of Impact Engineering, 1992; 12(4): 603-616.
[69] Lopatnikov SL, Gama BA, Gillespie Jr. JW. Modeling the progressive collapse behavior
of metal foams. International Journal of Impact Engineering, 2007; 34: 587-595.
[70] Reid SR and Peng C. Dynamic uniaxial crushing of wood. International Journal of Impact
Engineering, 1997; 19(5-6): 531-570.
[71] Tan PJ, Reid SR, Harrigan JJ, Zou Z and Li S. Dynamic compressive strength properties
of aluminium foams. Part II – ‘shock’ theory and comparison with experimental data and
numerical models. Journal of Mechanics and Physics of solids, 2005; 53: 2206-2230.
[72] Shim VPW and Yap KY. Modeling impact deformation of foam-plate sandwich systems.
International Journal of Impact Engineering, 1997; 19: 615-636.
[73] Li QM and Meng H. Attenuation or enhancement – a one-dimensional analysis on shock
transmission in the solid phase of a cellular material. International Journal of Impact
Engineering, 2003; 27: 1049-1065.
[74] Qiu X, Deshpande VS and Fleck NA. Dynamic response of a clamped circular sandwich
plate subject to shock loading. Journal of Applied Mechanics, ASME, 2004; 71: 637-645.
[75] Qiu X, Deshpande VS and Fleck NA. Impulsive loading of clamped monolithic and
sandwich beam over a central patch. Journal of the Mechanics and Physics of Solids,
2005; 53: 1015-1046.
[76] Qiu X, Deshpande VS and Fleck NA. Finite element analysis of the dynamic response of
clamped sandwich beams subject to shock loading. European Journal of Mechanics,
A/Solids, 2003; 22: 801-814.
[77] Hutchinson JW and Xue Z. Metal sandwich plates optimized for pressure impulses.
International Journal of Mechanical Sciences, 2005; 47: 545-569.
[78] Taya M and Mura T. Dynamic plastic behaviour of structures under impact loading
investigated by the extended Hamilton’s principle. International Journal of Solids and
Structures, 1974; 10: 197-209.
[79] Hexcel Co.. HexWeb® Honeycomb energy absorption system design data, 2005.
156
[80] Wu E and Jiang WS. Axial crush of metallic honeycomb. International Journal of Impact
Engineering, 1999; 19(5-6): 439-456.
[81] Yamashita M and Gotoh M. Impact behaviour of honeycomb structures with various cell
specifications – numerical simulation and experiment. International Journal of Impact
Engineering, 2005; 32: 618-630.
[82] Bauer F. Advances to piezoelectric PVDF shock compression sensors. In: Proceedings of
the10th International Symposium on Electrets, Greece, 1999.
[83] Grenestedt JL and Danielsson M. Elastic-plastic wrinkling of sandwich panels with
layered cores, Journal of Applied Mechanics, ASME, 2005; 72: 276-281.
[84] http://www.matweb.com
[85] Meyer R, Köhler J and Homburg A. Explosives, 5th edition. Weinheim: Wiley- VCH, 2002.
[86] Grobbelaar WP and Nurick GN. An investigation of structures subjected to blast loads
incorporating an equation of state to model the material behaviour of the explosive. In:
Proceedings of 7th International Symposium on Structural Failure and Plasticity
(IMPLAST2000), Melbourne, Australia, 2000; p.185-194.
[87] Mahoi S. Influence of shape of solid explosives on the deformation of circular steel plates
– Experimental and numerical investigations. Ph.D. thesis, University of Cape Town,
Cape Town, South Africa, 2006.
[88] Sriram R, Vaidya UK and Kim JE. Blast impact response of aluminum foam sandwich
composites, Journal of Material Science, 2006; 4023-4039.
[89] Tan PJ, Reid SR, Harrigan JJ, Zou Z and Li S. Dynamic compressive strength properties
of aluminium foams. Part I – experimental data and observations. Journal of Mechanics
and Physics of Solids, 2005; 53: 2174-2205.
[90] Johnson W and Mellor PB. Engineering plasticity. Chichester: Ellis Horwood, 1986.
[91] Johnson W. Impact strength of materials. London: Edward Arnold, 1972.
[92] Timoshenko S. Theory of plates and shells. New York: McGraw-Hill, 1959.
[93] Xue Z and Hutchinson JW. Preliminary assessment of sandwich plates subject to blast
loads. International Journal of Mechanical Sciences, 2003; 45: 687-705.
[94] Li QM, Magkiriadies I and Harrigan JJ. Compressive strain at the onset of densification
of cellular solids. Journal of Cellular Plastics, 2006; 42: 371-392.
[95] Cote F, Deshpande VS, Fleck NA and Evans AG. The out-of-plane compressive behaviour
of metallic honeycomb. Materials Science & Engineering A, 2004; 380: 272-280.
[96] Goldsmith W and Louie DL. Axial perforation of aluminium honeycombs by projectiles.
International Journal of Solids and Structures, 1995; 32(8/9): 1017-1046.
[97] Zhao H and Gary G. Crushing behaviour of aluminium honeycombs under impact loading.
International Journal of Impact Engineering, 1998; 21(10): 827-836.
[98] Wierzbicki T. Crushing analysis of metal honeycombs. International Journal of Impact
157
Engineering, 1983; 1(2): 157-174.
[99] Zhao YP. Suggestion of a new dimensionless number for dynamic plastic response of
beams and plates. Archive of Applied Mechanics, 1998; 68: 524-538.
158
159
Appendix A
Impulse calculation Figure A-1 shows the motion of a four-cable ballistic pendulum. After the application of the
shock wave, the pendulum reaches the maximum displacement x1 at t=1/T, and when t=3/4T, its
maximum displacement in the opposite direction is x2, with T being the period of the oscillation
of the pendulum.
Figure A-1. Sketch of the motion of a four-cable ballistic pendulum subjected to a shock wave
The oscillation of the pendulum is governed by the following equation: 2
2 0d x dx MM C gxdt dt R
+ + = (A-1)
where M is the total mass; x is the horizontal displacement; C is the damping coefficient; and R
is the length of cable. The solution for the horizontal displacement is given by
0 sin( )t xx e tβ ωω
−= (A-2)
where β=C/M and ω=2π/T. Substituting t=T/4 and t=3T/4 into Eq. (A-2) yields
41 0 2
T Tx e xβ
π−
= (A-3)
34
2 0 2
T Tx e xβ
π−
= (A-4)
x1x2
θ
R R
Shock wave
160
Then we have
1 2
2
Tx ex
β
= (A-5)
Solving β gives
1
2
2ln xx
Tβ = (A-6)
If the values of x1 and x2 are known, then β can be obtained, and the initial velocity of the
pendulum can be further calculated using Eq. (A-7):
40 1
2 T
x x eT
βπ= (A-7)
Then the impulse is estimated by
0I Mx= (A-8)
In our tests, 0.03β ≈ , M=140.75kg, T ≈ 4.2s, R=4.38m. Generally, the rotation angle (θ ) should
be greater than 5º. In the present case, θ is approximately 1.5º, which can ensure the high
accuracy of measurement.