IMPULSIVE LOADING OF SANDWICH PANELS WITH CELLULAR CORES

Transcription

1 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

2 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 II

3 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. III

4 To my family IV

5 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 V

6 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. VI

7 CONTENTS Abstract Declaration Acknowledgements Contents List of tables List of figures List of symbols II V VI VII XI XII XVI 1. Introduction Motivation Sandwich structures Blast wave and its effect Methodology and workflow Thesis organisation 6 2. Literature review Introduction Experimental investigations Experimental facilities Experimental observations Numerical simulations Basic formulations Modeling blast loads Modeling the materials of targets Commercial FEA packages for blast loading simulations Analytical modeling Analytical models for monolithic metals Analytical models for cellular solids Analytical models for sandwich structures Summary Experimental investigation into the honeycomb core sandwich panels Introduction 30 VII

8 3.2 Specimen Experimental set-up Deformation and failure patterns Front face-sheet deformation/failure Core deformation/failure Back face-sheet deformation/failure Pressure-time history at the central point of the front face Analysis and discussion Effect of face-sheet thickness Effect of cell dimension of the core Effect of charge mass Summary Experimental investigation into the aluminium foam core sandwich panels Specimen Results and discussion Deformation/failure patterns Deflection of the face-sheet Summary Numerical simulation of the honeycomb core sandwich panels Introduction FE model Modeling geometry Modeling materials Modeling blast load Simulation results and discussion Explosion and structural response process Deformation/failure patterns of sandwich panels Quantitative results Effect of plastic stretching and bending Strain distribution along the x-axis Strain distribution along the diagonal line Analysis and discussion Effect of boundary conditions Summary 85 VIII

9 6. Numerical simulation of the aluminium foam core sandwich panels FE model Modeling geometry Modeling materials and blast load Simulation results and discussion Explosion and structural response process Deformation/failure patterns Face-sheets deflections and core crushing Energy absorption Time history of plastic dissipation Energy partition Summary Analytical solution I a design-oriented theoretical model Introduction Analytical modeling Phase I Front face deformation Phase II Core compression Phase III Overall bending and stretching Model validation Comparison with experiment Comparison with the analytical model for circular plates Optimal design of square plates to shock loading Effect of side length ratio Effect of relative density of the core Effect of core thickness Summary Analytical solution II a theoretical model for dynamic response Introduction Analytical modeling Phase I Front face deformation Phase II Core compression Phase III Overall bending and stretching Model validation Comparative studies of the analytical solutions Effect of longitudinal strength of core after compression 136 IX

10 8.4.2 Comparison of square monolithic and sandwich panel Comparison among sandwich beams, circular and square sandwich panels Summary Conclusions and future work Conclusions Future work 150 References 152 Appendix A 159 X

11 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 XI

12 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/ MD-2) 41 Figure 3-8. Pitting failure on the front face (Specimen No.: 1/ 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 Deformation/failure map for Group 4, where all the eight panels have identical configurations 43 Figure Failure pattern of the honeycomb core (Specimen No.: ACG-1/4-TK-5) 45 Figure Failure pattern of the back face (Specimen No.: ACG-1/4-TK-5) 46 Figure Typical pressure-time history at the central point of the front face 48 Figure Effect of face-sheet thickness. The abscissa denotes the specimens given without any particular order 49 Figure Effect of foil thickness. The abscissa denotes the specimens sorted by the face-sheets with increasing thicknesses 50 XII

13 Figure Effect of cell size. The abscissa denotes the specimens sorted by the face-sheets with increasing thicknesses 51 Figure Effect of the average mass of core. The abscissa denotes the specimens sorted by the face-sheets with increasing thicknesses 52 Figure 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 Comparison of experimental and predicated results 77 Figure Locations of the shell elements in the two groups 78 Figure ε mid distribution for the shell elements in Group 1 79 Figure ε d distribution for the shell elements in Group 1 81 Figure ε mid distribution for the shell elements in Group 2 82 Figure ε d distribution for the shell elements in Group 2 83 Figure 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 XIII

14 (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 Energy dissipation normalised with the total energy for Specimen No 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 2 with the same materials and mass/area M = 3.9 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 2 the mass/areas M = 3.9 kg / m 142 Figure Comparison of the deflections predicted by sandwich beam and XIV

15 sandwich plates with the same materials and mass/area M = 3.9 kg / m Figure A-1. Sketch of the motion of a four-cable ballistic pendulum subjected to a shock wave 159 XV

16 List of symbols Chapter 1 I s Impulse of blast wave during the positive phase P a Ambient air pressure P s Peak pressure of the blast wave R Distance from the centre of the explosive source in meters t d 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 P i Incident pressure P r Reflected pressure σ dy Dynamic yield strength σ Y Static yield strength p ε Effective plastic strain p ε Plastic strain rate θ Incident angle of shock wave σ zz Stress in the transverse direction Chapter 3 A Working area of the PVDF film d 33 piezoelectric constant of the PVDF film H c Thickness of core h f Thickness of face-sheets I Impulse delivered onto the structure L Side length of a square panel l e Cell size of hexagonal honeycomb Q Electric charge XVI

17 m 0 Mass of core m h Mass of TNT charge M Mass per unit area t Nominal foil thickness of a hexagonal honeycomb t Thickness of the sandwich panel w 0 Maximum deflection of back face ρ c Mass density of the core ρ f Material density of the face-sheets δ Dimensionless back face deflection Φ Dimensionless impulse Chapter 4 H c Thickness of core h f Thickness of face-sheets I Impulse delivered onto the structure m 0 Mass of core m h Mass of TNT charge w 0 Maximum deflection of back face Chapter 5 A, B, R 1, R 2, ω Materials constants for JWL equation P Blast pressure V Detonation velocity ε mid Middle-plane strain of the face-sheet ε lower Lower-plane strain of the face-sheet ε upper Upper-plane strain of the face-sheet ρ Explosive density ρ 0 Explosive density at the beginning of detonation process Chapter 6 A Area of the plate exposed to the blast e Volumetric strain H c Thickness of core h f Thickness of face-sheets I Impulse delivered onto the structure l x, l y, l z Length of a metallic foam block in x, y and z directions XVII

18 V Current volume of metallic foam V 0 original volume of metallic foam v 1 Initial velocity of the front face of a sandwich structure v 2 The velocity of a sandwich structure obtained after core crushing W I Kinetic energy of the front face of a sandwich structure before core crushing W II Kinetic energy of the whole sandwich structure after core crushing ρ c Mass density of the core ρ f Material density of the face-sheets ε x, ε y, ε z 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 D n Johnson s damage number E p Plastic dissipation during core crushing H c Thickness of core h f Thickness of face-sheets ΔH c Thickness of crushed core H c Final thickness of core h f Thickness of face-sheets I Impulse delivered onto the structure L Half side length of a square plate l m Length of a plastic hinge line M Mass per unit area M p Fully plastic bending moment t Initial overall thickness of a sandwich structure p Pressure u, v, w Displacements in x, y and z directions U b Plastic bending dissipation U s Plastic stretching dissipation v 1 Initial velocity of the front face of a sandwich structure v 2 The velocity of a sandwich structure obtained after core crushing w 0 Maximum deflection of back face w 0 Normalised maximum deflection of back face ' w 0 Maximum deflection of front face XVIII

19 ' w 0 Normalised maximum deflection of front face W I Kinetic energy of the front face of a sandwich structure before core crushing W II Kinetic energy of the whole sandwich structure after core crushing γ xy Shear strain ε x, ε y Normal strains θ i Rotation angle of a plastic hinge line ρ * Relative density ρ c Mass density of the core ρ f Material density of the face-sheets τ c Shear strength of core c σ l Longitudinal yield strength of core c σ Y Transverse yield strength of core f σ Y Yield strength of face material Chapter 8 E p Plastic dissipation per unit area during core crushing H c Thickness of core ΔH c Thickness of crushed core H c Final thickness of core h f Thickness of face-sheets I Impulse per unit area I cr Critical impulse per unit area I ˆcr Normalised critical impulse per unit area L Half side length of a square plate l m Length of a plastic hinge line M Moments per unit length M Mass per unit area M 0 Fully plastic bending moment N Membrane forces per unit length N 0 Fully plastic membrane force P 3 Transverse pressure per unit area R n Zhao s response number t Cell wall thickness of a hexagonal cell T Response time XIX

20 T Dimensionless response time v 1 Initial velocity of the front face of a sandwich structure v 2 The velocity of a sandwich structure obtained after core crushing w Transverse deflection at the central point W 0 Dimensionless maximum deflection on the back face W 1 Dimensionless maximum deflection on the front face W I Kinetic energy per unit area of the front face of a sandwich structure before core crushing W II Kinetic energy per unit area of the whole sandwich structure after core crushing Z n Sandwich damage number ε c Transverse compressive strain of cellular core ε cr Strain at yield ε D Densification strain θm Relative angular rotation rate across a plastic hinge line δ Dimensionless central point deflection of a square monolithic plate ρ * Relative density ρ 0 Material density ρ c Mass density of the core ρ f Material density of the face-sheets σ Y Yield strength c σ dy Dynamic transverse yield strength of core c σ ly Longitudinal yield strength of core c σ Y Static transverse yield strength of core f σ Y Yield strength of face material XX

21 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. Structural fracture St. Mary Axe, London, 1992 Bishop Gate, London, 1993 Progressive collapse US Marine Barracks, Beirut, 1983 Murrah Building, Oklahoma City, 1995 Large plastic deformation & ballistic penetration US Navy ship, Aden, 2000 Russian armored car, Chechenia, 2000 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 1

22 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. 2

23 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 3

24 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 t a is the time of arrival of the blast wave, P s is the peak pressure of the blast wave and P a is ambient air pressure. The discontinuous pressure rise at the shock front is shown by the jump in pressure from P a to P s at time t a. Figure 1-3 also shows an approximately exponential decrease in pressure until the pressure drops down to the pre shock level at time t a +t d. The free-field pressure-time response can be described by a modified Friedlander equation, t t a pt () ( p p)[1 ] e a ( t t )/ θ = s a (1-1) td where t d 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 P s and t a, another significant blast wave parameter is the specific impulse of the wave during the positive phase I s, as given by ta+ td Is = ptdt () (1-2) ta 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 4

25 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: R Z = (1-3) W 13 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

26 Phase I Phase II Phase III Experimental investigations Numerical simulations Analytical modeling Deformation/failure modes analyses Parametric studies Optimal design guidelines 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 6

27 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. 7

28 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 Experimental facilities Two types of facilities are mainly used to dynamically measure the air blast loading and its 8

29 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

30 (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 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

31 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

32 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

33 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

34 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

35 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 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. 15

36 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) 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 P = P + P i + θ (2-1) 2 2 ( τ ) r cos θ (1 cos θ 2cos ) whereθ is the angle of incidence, defined by the tangent to the wave front and the target s surface; P is the reflected pressure at normal incident angle; and P is the incident pressure. It r i 16

37 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] 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]. n p m σ y = ( A+ Bε )(1+ cln ε*)(1 T* ) (2-2) where A, B, C, m and n are material constants; p ε is effective plastic strain; ε * = ε p / ε 0, being effective plastic strain rate, forε 0 =1s -1 ; T* = (T-T room )/(T melt -T room ). 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

38 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 (σ dy ) can be computed by 1/ p ε σdy σ = Y 1+ D (2-3) where σ Y and σ dy are the static and dynamic yield tresses and D and p are material constants. Modeling cellular solids A detailed review of constitutive models for metal foam applicable to structural impact and shock analyses has been presented by Hanssen et al. [43]. The models have different formulations for the yield surface, hardening rule and plastic flow rule, while fracture is not accounted for in any of them. Modeling sandwich structures In recent years a number of micro-architectured materials have been developed for uses as the cores of sandwich structures for application in blast-resistant constructions. Some of the current available topologies are shown in Figure 2-7: pyramidal core, diamond-celled core, corrugated core, hexagonal honeycomb core, and square honeycomb core [44]. (a) pyramidal core, (b) diamond- celled core, (c) corrugated core, (d) hexagonal honeycomb core, and (e) square honeycomb core Figure 2-7. Sketches of several sandwich core topologies [44] 18

39 The cellular cores are assumed to be made from an elastic, perfectly-plastic solid with yield strain ε Y. Their normalised transverse compressive strengthσ and longitudinal strengthσ 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] n l Commercial FEA packages for blast loading simulations LS-DYNA LS-DYNA is a general-purpose, explicit finite element program used to analyse the nonlinear dynamic response of three-dimensional inelastic structures. Its abundant material models, fully automated contact analysis capability and error-checking features have enabled users to solve many complex impact and explosion problems. The main application areas of LS-DYNA include: Large deformation dynamics and contact simulations, crashworthiness simulation, occupant safety systems, metal, glass, and plastics forming, multi-physics coupling, failure analysis etc. MSC.Dytran Dytran is an explicit FEA solution for analysing complex nonlinear behavior involving permanent deformation of material properties or the interaction. Dytran combines structural, material flow, and fluid-structure interaction (FSI) analyses in a single package, and uses a combination of Lagrangian and Eulerian solver technology to analyse short-duration transient events that require finer time step to ensure a more accurate solution. However, there are some drawbacks with Dytran. For example, material models supported by Dytran are quite limited, particularly for materials such as soils and rocks. No 2D computation is available, and thus axisymmetric cases have to be treated as 3D problems, and time cost increases consequently. Also the contact types are not sufficient to model complex impact problems. ABAQUS ABAQUS is advanced FEA software capable of solving very complex and highly nonlinear 19

40 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 Analytical models for monolithic metals 20

41 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 W ( X, t) = V Φ ( X ) (2-4) * * i j i j where i( j, ) is the velocity field. The function * i W X t Φ is called the mode or mode shape; the scalar velocity * V of a characteristic point is specified such that 1 Φ 1. It should be noted * i 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 Δ 0 device is proposed for this class of problems as a means of obtaining an approximate solution by assuming the simple mode solution to hold for the entire motion. The amplitude of the approximating mode solution can be chosen so that the difference between the given initial velocity and that of the mode solution is minimised in a mean square sense. 21

42 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. 22

43 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 23

44 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 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, Δ max, in case of both the modes are the same. 24

45 (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 ε D, and the stress level jumps from σ cr 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

46 σ R-P-P-L Model E-P-R Model σ * Metal foam σ M σ cr σ Y ε Y ε D ε M ε 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 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) (ii) the 1-D approximation of the shock events; separation of the phases of the response into three main sequential phases: 26

47 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. 27

48 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 28

49 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. 29

50 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 σ Y (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 l e and wall thickness t, as indicated in Figure 30

51 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 H c are constant and equal to 310mm and 12.5mm respectively. Three different thicknesses h f are adopted for the face-sheets: TN (h f =0.5mm), MD (h f =0.8mm) and TK (h f =1.0mm). For each test condition, two nominally identical specimens were tested. Each specimen is denoted a unique number. For example, specimen 1/ TK-1 indicates a sandwich panel with honeycomb core of l e =1/8 (3.18mm), made of aluminium 5052, t= (0.051mm), and with a thick (TK) face sheet (h f =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, l e =1/4 (6.35mm), t=0.066mm, with a thick face sheet (h f =1.0mm), and is the first test. h f H c t l e Honeycomb core L L 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 (l e ) 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

52 Name of specimen Table 3-1. Sandwich panels of Group 1, where the effects of foil thickness and face thickness are investigated Cell size l e (mm) Nominal foil thickness t (mm) Face-sheets thickness h f (mm) Mass of core m o (g) Cell wall material Mass of charge m h (g) Impulse I (Ns) Back face deflection w 0 (mm) 1/ TK Al-5052-H / TK Al-5052-H / MD Al-5052-H / MD Al-5052-H / TN Al-5052-H / TN Al-5052-H / TK Al-5052-H / TK Al-5052-H / MD Al-5052-H / MD Al-5052-H / TN Al-5052-H / TN Al-5052-H / TK Al-5052-H / TK Al-5052-H / MD Al-5052-H / MD Al-5052-H / TN Al-5052-H / TN Al-5052-H / TK Al-5052-H / TK Al-5052-H / MD Al-5052-H / MD Al-5052-H / TN Al-5052-H / TN Al-5052-H

53 Name of specimen Table 3-2. Sandwich panels of Group 2, where the effects of cell size and face thickness are investigated Cell size l e (mm) Nominal foil thickness t (mm) Face-sheets thickness h f (mm) Mass of core m o (g) Cell wall material Mass of charge m h (g) Impulse I (Ns) Back face deflection w 0 (mm) 1/ TK Al-5052-H / TK Al-5052-H / MD Al-5052-H / MD Al-5052-H / TN Al-5052-H / TN Al-5052-H / TK Al-5052-H / TK Al-5052-H / MD Al-5052-H / MD Al-5052-H / TN Al-5052-H / TN Al-5052-H

54 Name of specimen Table 3-3. Sandwich panels of Group 3, where the effects of average core mass and face thickness are investigated Cell size l e (mm) Nominal foil thickness t (mm) Face-sheets thickness h f (mm) Mass of core m o (g) Cell wall material Mass of charge m h (g) Impulse I (Ns) Back face deflection w 0 (mm) 1/ TK Al-5052-H / TK Al-5052-H / MD Al-5052-H / MD Al-5052-H / TN Al-5052-H / TN Al-5052-H ACG-1/4-TK Al-3104-H ACG-1/4-TK Al-3104-H ACG-1/4-MD Al-3104-H ACG-1/4-MD Al-3104-H ACG-1/4-TN Al-3104-H ACG-1/4-TN Al-3104-H

55 Name of specimen Cell size l e (mm) Table 3-4. Sandwich panels of Group 4, where the effect of charge mass is investigated Nominal foil thickness t (mm) Face-sheets thickness h f (mm) Mass of core m o (g) Cell wall material Mass of charge m h (g) Impulse I (Ns) Back face deflection w 0 (mm) ACG-1/4-TK Al-3104-H ACG-1/4-TK Al-3104-H ACG-1/4-TK Al-3104-H ACG-1/4-TK Al-3104-H ACG-1/4-TK Al-3104-H ACG-1/4-TK Al-3104-H ACG-1/4-TK Al-3104-H ACG-1/4-TK Al-3104-H

56 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

57 (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

58 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 LD ) 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 100 Displacement (mm) Time (s) Figure 3-5. A typical oscillation time history of the pendulum 38

59 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

60 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 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. 40

61 Figure 3-7. Indenting failure on the front face (Specimen No.: 1/ MD-2) Figure 3-8. Pitting failure on the front face (Specimen No.: 1/ TN-1) 41

62 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 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

63 Figure 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 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

64 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) 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 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). 44

65 Figure Failure pattern of the honeycomb core (Specimen No.: ACG-1/4-TK-5) 45

66 (a) A dome-like back face deformation (Specimen No.: 1/ TN-2) (b) A dome-like back face deformation with a small nose (Specimen No.: 1/ MD-1) Figure Failure pattern of the back face (Specimen No.: ACG-1/4-TK-5) 46

67 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: Q= d33aσ zz (3-1) where Q is the electric charge generated, which can be obtained by integration of the voltage output over time; d 33 =20pC/N, being piezoelectric constant of the PVDF film; A = 6mm 6mm, being the working area of the film; and σ zz 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

68 Figure 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 Effect of face-sheet thickness In the tests, three different face-sheet thicknesses, that is TN (h f =0.5mm), MD (h f =0.8mm) and TK (h f =1.0mm) were tested in Groups 1, 2 and 3. Their effect on the back face deflection is 48

69 shown in the diagram in Figure 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 Effect of face-sheet thickness. The abscissa denotes the specimens given without any particular order 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 (l e =3.18mm) but different foil thickness: 0.018mm, 0.025mm, 0.038mm and 0.051mm, respectively. It can be observed from Figure

70 that for a given face-sheet, larger foil thicknesses result in smaller back face deflections. Using the weakest core (1/ 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. Deflection (mm) / series 1/ series 1/ series 1/ series TN (h f = 0.5mm) MD (h f = 0.8mm) TK (h f = 1.0mm) Face-sheet thickness Figure 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 (l e = 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/ 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

71 Deflection (mm) / series 5/ series Face-sheet thickness TN MD TK (h f = 0.5mm) (h f = 0.8mm) (h f = 1.0mm) Figure 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/ 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 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

72 Deflection (mm) / series ACG-1/4 series Face-sheet thickness TN MD TK (h f = 0.5mm) (h f = 0.8mm) (h f = 1.0mm) Figure 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 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 It is evident that, for the given panel configuration, back face deflection increases with impulse, almost linearly. 52

73 Figure Effect of impulse level on the panels with nominally identical configurations Thus the relationship of δ and Φ can be expressed as w I = t 2t M 2hf σ 0 δ = = α Φ, where Φ f Y (3-2) where t = 2h + H, with h and H being the thicknesses of face and core respectively. f c f c M = A h + H, with ρf and (2 ρ ρ ) f f c c ρc being the densities of face and core, and A is the exposed area. f σ denotes the quasi-static tensile strength of face material. Fitting the data Y 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

74 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. 54

75 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. h f H c Al foam corevn L L Figure 4-1. Geometry and dimension of the aluminium foam core specimen 55

76 Table 4-1. Specifications and test results of the aluminium foam core sandwich panels Specimen No. Specimen name Face-sheets thickness h f (mm) Mass of core m o (g) Relative density Core thickness Mass of charge m h (g) Impulse I (Ns) Back face deflection Wrinkles at the Front face tearing (%) H c (mm) w 0 (mm) edges 1 L-20-TK Yes No 2 L-20-TK Yes No 3 H-20-TK No No 4 H-20-TK No No 5 L-30-MD Yes No 6 L-30-MD Yes Yes 7 L-30-TK Yes No 8 L-30-TK Yes No 9 H-30-TK No No 10 H-30-TK No No 56

77 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 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

78 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

79 (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

80 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 ε D. 60

81 Figure 4-5. A typical cross-section of the specimen (Specimen L-30-TK-1) 61

82 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. 62

83 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 Modeling geometry The geometric model of the sandwich panel used in the simulations is depicted in Figure 5-1(a), 63

84 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

85 (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

86 Figure 5-2. Geometric model of the charge 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 (σ Y ) and Tangent modulus (E tan ). 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 (ρ M ), 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 * =ρ 0 /ρ, and internal energy per initial volume, E m0, as presented in Eq. (5-1): ρ 0 0 ωρ R1 R2 ρ ωρ ρ ωρ P= A 1 e + B 1 e + E m R1ρ0 R2ρ0 ρ0 ρ 0 (5-1) where P is the blast pressure, ρ is the explosive density, ρ 0 is the explosive density at the 66

87 beginning of detonation process. The parameters A, B, R 1, R 2 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 Al-2024-O Face RO E PR SIGY ETAN sheet E E-3 *MAT_PLASTIC_KINEMATIC Al-3104-H19 Core RO E PR SIGY ETAN [84] E E-3 *MAT_PLASTIC_KINEMATIC Al-5052-H39 Core RO E PR SIGY ETAN [84] E-3 7.0E-3 *MAT_HIGH_EXPLOSIVE_BURN RO D PCJ TNT [85] Charge *EOS_JWL A B R1 R2 OMEG E0 V E E 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 67

88 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 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 68

89 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

90 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. 70

91 Figure 5-4. A typical process of explosion product - structure interaction 71

92 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

93 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 v f 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

94 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

95 Deformation Deflection (mm) Time (ms) (μs) Front face Back face Core Deformation Deflection (mm) Front face Core 4 Back face Time (μs) (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 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

96 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) 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

97 A plot of the experimental values versus the predicted values of all the specimens is shown in Figure The data points are very close to the line of perfect match, thus representing a reasonable correlation between the experimental and predicted results Experimental deflection (mm) Predicted deflection (mm) Figure 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 ε mid, 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. ε d = ε lower - ε upper, which indicates the curvature. The distributions of ε mid and ε d 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

98 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 Locations of the shell elements in the two groups 78

99 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. ε midx and ε midy, is shown in Figure The figure clearly reveals that both ε midx and ε midy 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) ε midx distribution for the shell elements in Group 1 (b) ε midy distribution for the shell elements in Group 1 Figure ε mid distribution for the shell elements in Group 1 79

100 The bending states of the plate are indicated by ε dx and ε dy in Figure 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, ε dy 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 ε dy occurs at the central region. (a) ε dx distribution for the shell elements in Group 1 80

101 (b) ε dy distribution for the shell elements in Group 1 Figure ε d distribution for the shell elements in Group Strain distribution along the diagonal line The progressions of ε mid and ε d of the shell elements along the diagonal line are illustrated in Figure 5-14 and Figure 5-15 respectively. (a) ε midx distribution for the shell elements in Group 2 81

102 (b) ε midy distribution for the shell elements in Group 2 Figure ε mid distribution for the shell elements in Group 2 (a) ε dx distribution for the shell elements in Group 2 82

103 (b) ε dy distribution for the shell elements in Group 2 Figure ε d distribution for the shell elements in Group 2 The transient distributions of ε midx and ε midy in Figure 5-14 exhibit a similar pattern to those shown in Figure 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 ε d, 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 ε d is not exactly in the x or y directions. It has been shown in the Figure 5-15 that the distributions of ε dx and ε dy 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 ε dx and ε dy of the 83

104 selected elements 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 ε d is less than 13% of the permanent maximum ε mid. 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, 84

105 as shown in Figure Figure 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

106 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. 86

107 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 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

108 (b) Geometry model of a charge (enlarged view) Figure 6-1. Geometric model of a sandwich panel and charge 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 σ 1, σ 2, σ 3 are compared with the yield stress in compression and tension Y c and Y t, 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, Y c =Y 0 c +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

109 Y t =Y 0 t. 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). Stress d a c b Volumetric strain (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 volume e = (6-1) original volume 89

110 The original volume of a foam block is given by V 0 =l x l y l z, where l x, l y and l z are the side lengths of the block in three dimensions respectively. Then the current volume is V = l (1 + ε ) l (1 + ε ) l (1 + ε ) = lll(1 + ε + ε + ε + εε + εε + εε + εεε) (6-2) x x y y z z x y z x y z x y x z y z x y z 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. ε x = ε y =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 V = l l l (1 + ε ) = V (1 + ε ) (6-3) x y z z 0 z 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

111 Table 6-1. LS-DYNA material type, material property and EOS input data for aluminium foam core panels Material Part Face LS-DYNA material type, material property and EOS input data (unit = cm, g, μs) *MAT_PLASTIC_KINEMATIC Al-2024-T3 sheet RO E PR SIGY ETAN E E-3 Aluminium *MAT_CRUSHABLE_FOAM foam Core RO E PR LCID TSC DAMP (6%) E Figure 6-2(b) 2.18E Aluminium *MAT_ CRUSHABLE_FOAM foam Core RO E PR LCID TSC DAMP (10%) E Figure 6-2(b) 4.66E TNT [85] Charge *MAT_HIGH_EXPLOSIVE_BURN RO D PCJ *EOS_JWL A B R1 R2 OMEG E0 V E E 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 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. 91

112 Figure 6-3. Process of the charge detonation 92

113 Figure 6-4. Process of explosive-structure interaction 93

114 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 Deformation/failure patterns 94

115 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) 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

116 data points are very close to the line of perfect match, thus representing a reasonable correlation between the experimental and predicted results. 8 7 Experimental deflection (mm) Predicted deflection (mm) 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

117 (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

118 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 (E T ) is essentially the sum of kinetic (E K ) and internal energy (E I, also known as deformation energy E D ). 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 I v = 1 Aρ h (6-4) f f where A is the exposed area, and ρ f and hf 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 W = (6-5) I 2Aρ f hf After core crushing, the whole structure would have an identical velocity, and the kinetic energy at that instant can be calculated by W II 2 I = 2 A(2 ρ h + H ρ ) f f c c (6-6) where ρ c and H c 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]. 98

119 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. 0.8 Plastic energy dissipation ratio in the core in the front face in the back face Time (μs) Figure 6-9. History of plastic dissipation during plastic deformation (Specimen L-30-TK -1) 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 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

120 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 Specimen No Core Front face Back face Normalised energy Figure 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 (W I ) is proportional to the square of the total impulse input (I 2 ). Effect of face-sheet thickness 100

121 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 101

122 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. 102

123 CHAPTER SEVEN ANALYTICAL SOLUTION I A DESIGN-ORIENTED THEORETICAL MODEL 7.1 Introduction This chapter presents a design-oriented approximate analytical method for the performance of the two types of sandwich panels under blast loading. Since in assessing the behaviour of a blast loaded structure it is often the case that the calculation of final states is the principal requirement for a designer, a simple model is developed to predict maximum deflections in square sandwich panels under blast loading, but gives no predictions of displacement-time histories, In analytical modeling, the deformation is divided into three phases, corresponding to the front face deformation, core crushing and overall structural bending and stretching, respectively. The response in the last phase is considered using small deflection and large deflection theories, respectively, based on the extent of panel deformation. The analysis is based on an energy balance with assumed displacement fields, which are simplified to reduce the calculation cost but give acceptable results. Using the proposed analytical model, an optimal design has been conducted for square sandwich panels with a given mass per unit area, and loaded by various levels of impulse. Effect of several key design parameters, i.e. ratio of side lengths, relative density of core, and core thickness is discussed. 7.2 Analytical modeling According to the theoretical analyses for the blast loaded response of sandwich beams or circular sandwich plates by Fleck and co-workers [44, 74], the whole structural deformation process can be split into three phases (Figure 7-1): 103

124 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 v 1 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 v 2, and finally the structure is brought to rest by plastic bending and stretching. Face-sheets Explosive Phase I v 1 Front face deformation Al foam corev H c Support Phase II Core crushing ΔH c Hyty c v 2 Phase III Overall bending & stretching w 0 +ΔH c Hyty c w 0 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

125 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 (W II ) 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 ( 2h f + H ), c 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 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 I v = 1 Aρ h (7-1) f f where A is exposed area of the panel. ρ f and h f are material density and thickness of the faces, respectively. The corresponding kinetic energy of the front face is obtained by 2 I W = (7-2) I 2Aρ f hf 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: I v2 = A(2 ρ h + ρ H ) f f c c (7-3) 105

126 where ρ c is mass density of core material, and H c is core thickness. Correspondingly, the kinetic energy of the entire structure at the end of Phase II is written as 2 I WII = 2 A(2 ρ h + ρ H ) f f c c (7-4) Hence, the energy absorption in core compression is: E = W W II (7-5) p I Or WI WII 1+ μ = W 2 + μ I (7-6) with μ = ρ H / ρ h being the ratio of core mass and face mass. c c f f 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

127 in Figure 7-2(a), in which the final thickness H c 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 ε D, and the stress level jumps from c* σ to σ [71, 89], which can be determined by c Y Y ρ v σ = σ + (7-7) 2 c* C c 1 Y Y ε D ΔH c H c σ Y c, ρ c σ ρ c* * Y, c * l σ Y c, ρ c ' H c H c (a) Progressive deformation mode of a cellular material under impact loading Stress c* σ Y Uniaxial compression test R-P-P-L model σ c Y ε D Strain (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

128 the following shape function: π x π y wc( x, y) =Δ Hccos cos (7-8) 2a 2b where a and b are half side lengths of the panel, and a b; for square plates, a = b. Then the energy dissipation during core crushing can be obtained by π x π y 4A Ep = 4σ Δ H cos cos dxdy = 2a 2b π b a C C Y 2 c 0 0 c σy ΔH (7-9) where the value of C σ Y is estimated using the formulae given in [44], that is, σ = σ (7-10) C 3/2 Y 0.3( ρ*) Y for metal foams, where ρ * is relative density andσ Y is the yield stress of the solid, and C σ = * σ (7-11) Y ρ Y for honeycombs. Then π E π ( W W ) 1+ μ π I Δ Hc = = = p I II c c 2 c AσY AσY + μ A σρ Y f hf (7-12) 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

129 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. ξ 0 is a constant and 0<ξ 0 <=1. When ξ 0 =1, the rectangular plate reduces to a square plate. aξ 0 A J I B w 0 ϕ 2b H E G F x y C 2a K D w 0 Figure 7-3. Displacement field of the back face In the analysis, the face material satisfies von Mises yielding criterion. The plastic energy dissipation ( U b ) depends on the length of the plastic hinge-lines, their angle of rotation, and fully plastic bending moment per unit length (M p ). 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 2 tan ϕ (3 η ) where = + η (7-13) a η =. b The rotation angle of the plastic hinge-line is given by cosϕ sinϕ θ = w ( + ) 1 0 (On AE) (7-14a) a ξ b 0 θ = b (On AI) (7-14b) w /

130 θ = b (On EG) (7-14c) w / 3 0 θ w / a 4 = 0 ξ0 (On AH) (7-14d) Then the bending dissipation of the whole structure ( 2 Ub = 4 ( M pθ1dlm + M pθ2dlm + M pθ3dlm + M pθ4dlm) = M AE AI EG AH pw0r 3 U b ) can be obtained by (7-15) f where Mp = σ h ( h + H ) ; Y f f c cosϕ 1 1 R = 4[ ( + η sin ϕ ) + ξ + η (2 ξ ) + ]. In the present ξ η η ξ small deflection cases, the front face deflection and core crushing are much less than the panel thickness, and thus for simplicity, M p 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 W II i.e. = U (7-16) b 2 I = M pwr 0 2 A(2 ρ h + ρ H ) f f c c (7-17) Then 2 I w0 = 2 A(2 ρ h + ρ H ) M R f f c c p (7-18) Johnson [91] defined a dimensionless number, namely damage number, which can be presented in the following form D 2 = I (7-19) n 2 f 2 A ρσ f Y hf Then Eq. (7-18) can be normalised and expressed in terms of D n as w w A hdn t 2 Rt (2 h + ρ )( H h ) 0 0 = = c f (7-20) 110

131 where t = 2h f + H c being the initial overall thickness of the sandwich panel. h = h / H and ρ = ρ / ρ. c f f c Taking account of core compression, the normalised maximum deflection at the front face is then given by ' ' w w ΔH 0 0 w = = + c (7-21) 0 t t t 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: π x π y 4 cos cos 2a 2b b a P w dxdy= Ub (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. u v w back back back πx πy = u0 sin cos a 2 b πx πy = v0 sin cos 2a b π x π y = w0 cos cos 2a 2b πx πy ufront = u0 sin cos a 2 b πx πy vfront = v0 sin cos 2a b π x π y wfront = ( w0 +ΔHc)cos cos 2a 2b (7-23a) (7-23b) with u back, v back and w back being displacements of the back face in x, y and z directions, respectively and similarly, u front, v front and w front for the front face. u 0, v 0 and w 0 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

132 can be calculated by 2 back 1 wback ε x = 2 x 2 back 1 wback ε y = 2 y back wback w back γ xy = x y (7-24a) 2 front 1 w front ε x = 2 x 2 front 1 w front ε y = 2 y w front front wfront γ xy = x y (7-24b) 2 2 core 1 w w back front ε x = + 4 x x 2 2 core 1 wfront w back ε y = + 4 y y core 1 w w back wback front w γ xy = + 2 x y x y front (7-24c) with back x back back ε, ε and being strain components of the back face, y γ xy core ε, ε core x y and γ core xy being strain components of the core, and front front ε x, ε respectively. y and γ front xy being strain components of the front face, Then the energy dissipated during plastic stretching ( U s ) can be expressed as f f b a b a f back back σy back f front front σy front U s = hf 4 [ σ ( ε ) ] 4 [ ( ) ] Y Y 0 0 x + εy + γ xy dxdy + hf σ ε 0 0 x εy γ xy dxdy (7-25) + H 4 [ ( + ) + ] dxdy where b a c core core c core c σ ε l 0 0 x εy τ γ xy c σ l and c τ 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

133 f f b a b a f back back σy back f front front σy front U s = hf 4 [ σ ( ε ) ] 4 [ ( ) ] Y Y 0 0 x + εy + γ xy dxdy + hf σ ε 0 0 x + εy + γ xy dxdy 3 3 (7-26) f = Cw [ + ( w +ΔH) ] σ h c Y f where 2 π 1 b a C = ( + )( + ). 8 3 a b Equating the kinetic energy WII and stretching dissipation Us gives f W U C[ w ( w H ) ] σ h = = + +Δ f (7-27) 2 2 II s 0 0 c Y For simplicity, Eq. (7-27) can be re-written as Kw + K w K = (7-28) f f where K1 = 2Cσ h ; K2 = 2Cσ h Δ H ; Y f Y f c K 3 2 = I Cσ 2 A(2 ρ h + ρ H ) h f f c c Δ H. f 2 Y f c Solving Eq. (7-28), the maximum deflection of the back face is obtained by 2 K K + 4K K w = (7-29) 0 2K 1 Similarly, Eq. (7-29) is normalised and expressed in terms of D n as w w ΔH 1 AhD t 2 t 2t C(2 h + ρ ) 0 c n 2 = = + ΔH 0 c (7-30) The normalised maximum deflection at the front face is then given by ' ' w w ΔH 0 0 w = = + c (7-31) 0 t t t 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

134 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. Deflection/Initial thickness ratio (Experimental result) Aluminium foam core panels Honeycomb core panels Deflection/Initial thickness ratio (Predicted result) Figure 7-4. Comparison between the experimental and predicted maximum deflection of the back face of the two types of specimens 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

135 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 I I = (7-32) f AM σ / ρ Y f where M = 2ρ f h f + ρ c H c 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 square - test square - analytical prediction (von Mises yield locus) circular - analytical prediction (circumscribing yield locus) w / t circular - analytical prediction (inscribing yield locus) Large deflections Small deflections I / AM σ / ρ Y f 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

136 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 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 (0.0625m 2 ) and mass per unit area (11.18kg/m 2 ) (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. 5.5 Maximum deflection/thickness ratio Front face, 32Ns Back face, 32Ns Front face, 40Ns Back face, 40Ns Front face, 48Ns Back face, 48Ns a/b Figure 7-6. Comparison of the normalised maximum deflections of the rectangular plates with various side length ratios, for three impulses Effect of relative density of the core 116

137 Now consider a square panel with a constant dimensionless mass per unit area M /( ρ L) = , where L is the half side length of the square exposed area; and the f value of the dimensionless core thickness ( Hc / L ) is fixed at 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 Maximum deflection/half side length ratio Front face, 32Ns Back face, 32Ns Front face, 40Ns Back face, 40Ns Front face, 48Ns Back face, 48Ns ρ * ρ Figure 7-7. Dimensionless maximum deflections of a sandwich plate with various relative densities of cores, for three impulses 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 ( Hc / L ). The result shows that a dimensionless core thickness of approximately 0.5 gives the best performance, and thicker cores yield larger compressions. 117

138 0.60 Maximum deflection/half side length ratio H / L ch c /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

139 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. 119

140 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. 120

141 Face-sheets Explosive Phase I Front face deformation v 1 Support Al foam corev H c Phase II Hc Core crushing v 2 Phase III Overall bending & stretching Hc Figure 8-1. Three phases in the response of a sandwich panel subjected to the blast loads 8.2 Analytical modeling 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 I v = 1 ρ h (8-1) f f where I is the impulse per unit area. ρ f and h f 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 W = I h (8-2) 2ρ f f 121