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

III

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

VI

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.

VII

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

VIII

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

IX

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


7.2.1 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


8.2.1 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

XI

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

XII

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

XIII

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

XIV

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



m0 – Mass of core

mh – Mass of TNT charge


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




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



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



ΔHc – Thickness of crushed core

cH – Final thickness of core



L – Half side length of a square plate

lm – Length of a plastic hinge line


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




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



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


ΔHc – Thickness of crushed core

cH – Final thickness of core


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


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

XIX

T – Dimensionless response time



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



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.

6

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]

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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

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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.

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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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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



(mm)


(mm)

Mass of core mo (g)



(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



(mm)


(mm)

Mass of core mo (g)



(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



(mm)


(mm)

Mass of core mo (g)



(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


(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)


(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

99

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.


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


8.2.1 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


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


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.

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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.

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