Brittle Mineral Foam In A Sacrificial Cladding Solution For Blast Loading Mitigation
Duarte Alcobia Braz
Thesis to obtain the Master of Science Degree in
Military Engineering
Supervisors: Lieutenant-Colonel David Lecompte
Supervisors: Lieutenant-Colonel Pedro José Da Silva Gonçalves Matias
Examination Committee
Chairperson: Prof. Dr. Paula Manuela Dos Santos Lopes Do Rego Figueiredo Supervisor: Lieutenant-Colonel David Lecompte
Members of the Committee: Prof. Dr. Hugo Bento Rebelo
Members of the Committee: Lieutenant-Colonel João Carlos Martins Rei
October 2022
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Statement
I declare that this document is an original work of my authorship and that it complies with all the requirements of the Code of Conduct and Good Practices of the University of Lisbon.
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“If you fulfil your obligations every day, you don’t need to worry about the future.”
“To learn is to die voluntarily and be born again—in great ways and small.”
Jordan Peterson
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Acknowledgments
The completion of this dissertation would not have been possible without the contribution of the most diverse institutions, to which I would like to express my deepest gratitude.
First of all, I thank Lieutenant Colonel of Military Engineering Pedro Matias, my heartfelt gratitude for the opportunity offered to me, having made himself available to establish the links that make this exchange possible.
To Lieutenant Colonel David Lecompte, my scientific advisor, for the way he received me and guided me on this project. I express my gratitude to you for your availability and support, for all your advice and for your human and technical skills committed to my work.
To Lieutenant Aldjabar Aminou, I express my heartfelt thanks for the permanent camaraderie, for the joint work and for the availability shown. Still grateful for all the help and time you gave up on your projects to actively contribute to this dissertation.
To Bachir Belkassem, my most sincere thanks for the way in which he always tried to actively participate with ideas and proposals, I also recognize mine and encourage me to reach more.
To all the elements integrated in the Department of Military and Protective Engineering of the Royal Military Academy in Belgium, a sincere thanks for your help and work.
To my parents and my sister, I leave a word of appreciation, for the support they always gave me, even with the long distance between us during the completion of this dissertation and the other adversities we went through in these months.
Finally, my deepest gratitude to Cristiana, for the time we didn't have, for her emotional support and availability throughout this journey, always being present, as far as possible, in the most difficult phases. Thank you for your patience, understanding and joy that you conveyed to me throughout the completion of this dissertation.
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Resumo
A ameaça de engenhos explosivos no campo de batalha continua a ser uma das formas mais eficazes de criar um grande número de baixas. Independentemente do alvo, estruturas ou veículos, é do interesse de países, como Portugal, que têm tropas destacadas em conflitos armados, para mitigar os seus efeitos.
No caso dos veículos, estes possuem um sistema de proteção, como a blindagem, para mitigar os efeitos dos projéteis e fragmentos inimigos. No entanto, em caso de ataque com minas ou explosivos, a sua estrutura e os seus ocupantes continuam vulneráveis. Consequentemente, surge o objetivo deste estudo: criação e avaliação de um revestimento de sacrifício. Este sistema funcionará como uma solução para mitigar a quantidade de energia que a estrutura de principal tem de suportar e assim minimizar os seus danos.
A solução de revestimento sacrificial adotada neste estudo baseia-se num núcleo de espuma mineral frágil. Este material tem a grande vantagem de ser leve, incombustível e económico. Além disso, tem também uma boa capacidade de absorção de energia. O desenvolvimento deste sistema e a avaliação do seu comportamento submetido a uma explosão ou a uma compressão dinâmica foi realizado através de campanhas experimentais e utilizando também a modelação numérica no software LS-Dyna.
Foi verificada a capacidade de absorver energia e mitigar a ação de uma explosão através de um revestimento sacrificial com espuma mineral frágil. Foi verificado que o sucesso desta solução depende não só do seu núcleo mas também do elemento anterior, a placa frontal. Além disso, a própria geometria e configuração da espuma irá influenciar a sua eficiência de absorção.
Palavras-chave: Mitigação de explosão; Revestimento sacrificial; Espuma mineral frágil; Modelação numérica.
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Abstract
The use of explosive devices on the battlefield remains one of the most effective ways to create large numbers of casualties. Regardless of the target, structures or vehicles, it is in the interest of countries, such as Portugal, which have troops deployed in armed conflicts, to mitigate its effects.
In the case of vehicles, they have a protection system, such as armour, to mitigate the effects of enemy projectiles and fragments. However, in the event of an attack using mines or explosives, its structure and occupants still remain vulnerable. Consequently, the objective of this study arises, which is the creation and evaluation of a sacrificial cladding. This system will work as a solution to mitigate the amount of energy that the supporting structure has to endure and thereby minimize the damage to that structure.
The sacrificial cladding solution adopted in this study is based on a crushable core of brittle mineral foam. This material has the great advantage of being light, non-combustible and economical. In addition, it also has a good energy absorption capacity. The development of this system and evaluation of its behaviour submitted to an explosion or a dynamic compression was carried out through experimental test campaigns and also using numerical modelling in the LS-Dyna software.
The ability to absorb energy and mitigate the action of an explosion in a sacrificial cladding with brittle mineral foam was verified. It was verified that the success of this solution depends not only on the crushable core but also on the preceding element, the front plate. In addition, the foam's own geometry and configuration will influence its absorption efficiency.
Keywords: Blast mitigation; Sacrificial cladding; Brittle mineral foam; Numerical modelling.
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Table of contents
Statement ... iii
Acknowledgments ... vii
Resumo ... ix
Abstract ... xi
Table of contents ... xiii
Table of figures ... xv
Table of tables ... xvii
Table of graphs ... xix
List of abbreviations and acronyms ... xxi
List of symbols ... xxiii
Chapter 1. Introduction ... 1
1.1. Context and motivation ... 1
1.2. Objective and methodology ... 7
1.3. Dissertation outline ... 10
Chapter 2. State of art ... 13
2.2.1. Deformation mechanism of cellular materials ... 19
2.2.2. Shock absorption and constitute model of cellular materials ... 22
Chapter 3. Characterization of the brittle mineral foam ... 27
3.1. Quasi-static test ... 27
3.1.1. Introduction ... 27
3.1.2. Test results ... 29
3.2. Dynamic test ... 37
3.2.1. Introduction ... 37
3.2.2. Test set-up ... 39
3.2.3. Test results ... 41
Chapter 4. Blast Wave Attenuation Capacity ... 49
4.1. Introduction ... 49
4.2. Test set-up ... 49
4.3. Test results ... 52
Chapter 5. Numerical Model and Study ... 57
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5.1. Introduction ... 57
5.2. Consistency of units ... 57
5.3. Explosion attenuation with brittle mineral foam as SC ... 58
5.4. Modelling process ... 58
5.4.1. Geometry ... 58
5.4.2. Mesh and SPH ... 59
5.4.3. Boundary Conditions ... 60
5.4.4. Materials ... 61
5.4.5. Load ... 62
5.5. Results ... 63
Chapter 6. Conclusions and recommendations ... 67
6.1. Synthesis and main conclusions ... 67
6.2. Recommendations for the future ... 68
References ... 69 APPENDICES ... a
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Table of figures
Figure 1.1: Incidents regarding Mines and Explosives since 2021 ... 2
Figure 1.2: A truck carrying goods from a UN agency hit an explosive device in Ngoutere, killing two people ... 2
Figure 1.3: The Belgian Armoured Vehicle after IED attack ... 2
Figure 1.4: Location of terrorism events from 2002 to 2016 ... 3
Figure 1.5: Weapon type in terrorism attacks ... 3
Figure 1.6: Aerial view of the Beirut port before the explosion (left) and after the explosion (right) ... 4
Figure 1.7: Proxy map representing the likely extent of damage post explosion ... 4
Figure 1.8: Blast wave propagation ... 5
Figure 1.9: Containers used in Camp MPoko and Camp Moana where the QRF/PRT and EUTM are located, as well as other MINUSCA units ... 6
Figure 1.10: Vehicle Global Acceleration Mitigation (VGAM) ... 6
Figure 1.11: Mine-Resistant Ambush Protected (MRAP) Vehicle ... 6
Figure 1.12: (a) Schematic view of a protection layer, (b) Some examples of possible crushable materials that can be used as core component ... 8
Figure 1.13: Examples of possible crushable structures which can be used as core component ... 8
Figure 1.14: Sacrificial Cladding's philosophy ... 9
Figure 1.15: The SC system attached to a structure ... 9
Figure 1.16: Example of using SC to increase the resilience of Steel Containers ... 10
Figure 1.17: Example of using SC to increase the resilience of combat vehicles ... 10
Figure 2.1: Explosive materials classification ... 13
Figure 2.2: Typical pressure-time profile of air blast wave ... 15
Figure 2.3: Examples of a blast load on a building ... 15
Figure 2.4: Blast wave configurations depending on the dimensions of the structure and length of the shock wave ... 16
Figure 2.5: P-I diagram ... 17
Figure 2.6: Brittle mineral foam ... 19
Figure 2.7: Uniaxial compression of cellular material: (a) stress-strain curve for both single cell or specimen with strain-hardening characteristic; (b) stress-strain curve for a single cell with strain- softening characteristic; (c) macroscopic stress-strain containing collection of strain-softening cells .. 20
Figure 2.8: (a) Type I structure with strain hardening characteristics, (b) Type II structure with strain softening characteristics and (c) the quasi-static force-displacement curves for each type of structure ... 21
Figure 2.9: Schematic diagram indicating (a) the tree compression stages and (b) the η-ɛ curve ... 22
Figure 2.10: "Snowplow" compaction model ... 23
Figure 2.11: P-a compaction model ... 24
Figure 2.12: Pressure-volume space illustrating P-l curve ... 25
Figure 2.13: Illustration of a multilinear model ... 25
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Figure 2.14: Idealization of the stress-strain curve ... 26
Figure 2.15: EPPH model ... 26
Figure 3.1: Compression test Bench Zwick/Roel ... 27
Figure 3.2: Example of some of the samples used in the test campaign ... 28
Figure 3.3: Example of the confined and unconfined samples used in the test campaign ... 28
Figure 3.4: Rupture of the masking tape in R90,3 mm samples ... 29
Figure 3.5: (a) Elastic phase, (b) Crushing phase, (c) Densification phase ... 30
Figure 3.6: Deformation mechanism on some the used samples: (a) Progressive collapse, (b) Fractures ... 31
Figure 3.7: Deformation mechanism on double-layer samples ... 31
Figure 3.8: Experimental set-up of a SHPB for a high-dynamic compression test ... 37
Figure 3.9: Schematic view of the drop test setup ... 38
Figure 3.10: Schematic representation of the dynamic test set-up ... 39
Figure 3.11: (a) Pressure transducer; (b) Load cell; (c) Structure cross section ... 40
Figure 3.12: Global view of the test device ... 40
Figure 3.13: (a) Single-layer cubic sample; (b) Confined cubic samples; (c) Double-layer cubic sample ... 41
Figure 3.14: Brittle mineral foam samples after loading: (a) One-layered sample, (b) Two-layered sample where only the top layer absorbed the loading, (c) Two-layered sample where the loading attenuation was divided by the two layers ... 44
Figure 3.15: Brittle mineral foam (up) and Polyurethane foam (down) ... 46
Figure 4.1: Global view of the test set-up ... 49
Figure 4.2: Overall view of the set-up ... 50
Figure 4.3: (a) Aluminium plate with speckle pattern (b) Aluminium plate fixed (screwed) on the steel frame ... 50
Figure 4.4: Brittle mineral foam sample used (300x300x60 mm) ... 51
Figure 4.5: Pressure transducers location for load curve definition ... 51
Figure 4.6: Example of the image recorded by DIC ... 52
Figure 4.7: Example of the results in 3D obtain by DIC ... 53
Figure 5.1: Model geometry ... 58
Figure 5.2: Modelling Process ... 58
Figure 5.3: Model’s geometry (left) and the model reflected to replicate the real structure (right) ... 59
Figure 5.4: Boundary conditions ... 60
Figure 5.5: SPH Symmetry planes ... 60
Figure 5.7: Incident pressure curves measured by the four pressure transducers ... 63
Figure 5.8: EDST load curve obtained and used on the numerical model ... 63
Figure 5.9: Location of the four nodes located in the center of the control plate ... 64
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Table of tables
Table 2.1: TNT equivalent of various explosive materials ... 18
Table 3.1: Densification strain values for one-layer squared cross section samples ... 33
Table 3.2: Mechanical properties for parallelepiped samples ... 36
Table 3.3: Mechanical properties for cylindric samples ... 36
Table 3.4: Brittle mineral foam and PU foam raw results ... 46
Table 3.5: Brittle mineral foam and PU foam results with the moving average applied ... 47
Table 4.1: Summary table of the test campaign results ... 55
Table 5.1: Set of units for the numerical model ... 57
Table 5.2: Mechanical properties of the steel frame and campling frame ... 61
Table 5.3: Mechanical properties of the aluminium plate ... 62
Table 5.4: Mechanical properties of the brittle mineral foam ... 62
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Table of graphs
Graph 3.1: Stress-strain curves for different parallelepiped samples ... 30
Graph 3.2: Example where the determination of the onset densification strain required an valuation from Method D, E and B. ... 32
Graph 3.3: Plateau Stage and Densification stage for 60mm thickness, considering a square and circular cross section samples ... 34
Graph 3.4: Plateau Stage and Densification stage for 60mm thickness confined, considering the considering a square and circular cross section samples ... 35
Graph 3.5: Energy absorption efficiency for a double-layer sample ... 35
Graph 3.6: Plateau Stage and Densification stage for 60mm thickness, considering only the squared cross section (two layers of parallelipiped samples) ... 36
Graph 3.7: Reflected pressure measured at end of the EDST and the transmitted force to the high mass structure for the one-layer samples ... 42
Graph 3.8: Sacrificial cladding philosophy applied to the One-layer Test 1 sample ... 43
Graph 3.9: One-layered results for Test 3 ... 43
Graph 3.10: Two-layered samples results ... 44
Graph 3.11: One-layered confined samples results ... 45
Graph 4.1: Displacement of the aluminium plate´s centre as a function of time with and without the mineral foam as a SC ... 53
Graph 4.2: Displacement results with different front plate/skin solutions besides aluminium ... 54
Graph 4.3: Difference in displacement when adopting aluminium plates (2 and 0.5 mm) to the SC solution ... 55
Graph 5.1: Center plate z displacement with no foam ... 64
Graph 5.2: Center plate z displacement with foam (no front plate) ... 65
Graph 5.3: Center plate z displacement with the sacrificial cladding solution of foam and 0,5mm aluminium front plate ... 65
Graph 5.4: Center plate z displacement with the sacrificial cladding solution of foam and 2mm aluminium front plate ... 66
Graph 5.5: Center plate z displacement overall result comparison ... 66
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List of abbreviations and acronyms
3D Three dimensional
ANFO Ammonium nitrate/Fuel oil
CAR Central African Republic
CL-20 Hexanitrohexaazaisowurtzitane DIC Digital image correlation EDST Explosive driven shock tube e.g. exempli gratia (“for example”) EPPH Elastic-perfectly-plastic-hardening etc. et cetera (“and others”)
FE Finite elements
FEM Finite elements method
HMX Cyclotetramethylene-tetranitramine i.e. id est (“this is”)
IED Improvised explosive device
ISAF International Segurance Assistance Force
KFOR Kosovo Force
LAV Light armoured vehicles
LED Light-emitting diode
LSTC Livermore Software Technology Corporation
LV Light vehicle
MINUSCA United Nations Multidimensional Integrated Stabilization Mission in Central African Republic
MINUSMA United Nations Multidimensional Integrated Stabilization Mission in Mali MONUA United Nations Observer Mission in Angola
MRAP Mine-Resistant Ambush Protected NATO North Atlantic Treaty Organization NTM-I NATO Training Mission Iraq PETN Pentaerythritol tetranitrate PMMA Poly(methyl methacrylate)
PU Polyurethane
RDX Cyclotrimethylenetrinitramine
R-LHP-L Rigid-linearly hardening plastic-locking RPPL Rigid-perfectly plastic locking
SC Sacrificial cladding
SDOF Single degree of freedom
SHPB Split Hopkinson pressure bar
SNMG 1 Standing NATO Maritime Group One
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SPH Smoothed particle hydrodynamics
TNT Trinitrotoluene
UN United Nations Organization
UNAVEM III United Nations Angola Verification Mission in Angola UNOGIL United Nations Observation Group in Lebanon UNOMOZ United Nations Angola Operations in Mozambique UNPROFOR United Nations Protection Force
VGAM Vehicle Global Accelerations Mitigation
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List of symbols
Latin alphabet
A Wave decay coefficient
â Adjustment coefficient
Ab Cross-section area
As Cross section of the sample
cb Stress pulse of the bar
D Stand-off distance
E, Eb Young’s Modulus
Hd,HE Detonation energy of the explosive
Hd,TNT Detonation energy of TNT
I Impulse
Is Length of the sample
is Specific wave impulse
Me,TNT TNT equivalent weight
MHE Weight of the explosive
n Homogeneity parameter
P Local pressure
P0 Ambient pressure
Pc Critical pressure at densification state Pe Hydrodynamic pressure at the elastic limit
PH Pressure from Hugoniot curve
P*H Rankine-Hugoniot pressure
PS Local pressure of the shock wave
PS0 Peak incident pressure
𝑟̇ Plastic strain rate
𝑟̇∗ Normalized damage-equivalent plastic strain rate
t time
T Temperature
t0 Positive phase duration
tA Time of arrival
T* Homologous temperature
Tm Melting Temperature
Tr Room temperature
V Volume
V0 Initial volume
V*0 Initial expanded form volume
𝑤 Angular frequency
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W Weight of the explosive
Y Local elastic resistance
Greek alphabet
a Compaction
ae Compaction at the elastic limit
e Nominal compressive strain
ec0 Pre-collapse strain
ed Densification strain
ed0 Onset densification
ei Incident strain
er Tensile strain
et Compressive strain
ė" User-defined reference strain
G0 Mie-Grüneisen
h Energy absorption efficiency
l Irreversible state variable
s Nominal compressive stress
sc0 Pre-collapse stress
spl Plateau stress
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Chapter 1. Introduction
1.1. Context and motivation
The deployment of troops by Portugal in different world conflicts is a consequence of its part in some international military and non-military organizations such as NATO (North Atlantic Treaty Organization) and UN (United Nations). The Portuguese Armed Forces marked their presence on many military operations under the aegis of the Atlantic Alliance: in Iraq, under NATO Training Mission Iraq (NTM-I); in Kosovo, under Kosovo Force (KFOR), in Afghanistan, under International Segurance Assistance Force (ISAF), in the Mediterranean Sea, under Active Endeavour, and on Standing NATO Maritime Group One (SNMG 1) [1]. More recently, Portuguese troops were sent to Romania under Tailored Forward Presence Mission, regarding the instability on the East of Europe [2].
Regarding UN’s mission, Portugal has a strong presence in peace mission, where the focus of this are activities that fall within the concepts of conflict prevention, peacemaking, peacekeeping, peace enforcement and peace consolidation provided for in the UN Capstone Doctrine. The first Portuguese appearance in peace mission by UN was in 1958, under the United Nations Observation Group in Lebanon (UNOGIL). After this intervention, many have follow: in former Yugoslavia, under United Nations Protection Force (UNPROFOR), in Mozambique, under United Nations Angola Operations in Mozambique (UNOMOZ), in Angola, under United Nations Angola Verification Mission (UNAVEM III) and United Nations Observer Mission in Angola (MONUA), and till present day in Mali, under United Nations Multidimensional Integrated Stabilization Mission in Mali (MINUSMA) [3], an in Central African Republic (CAR), under United Nations Multidimensional Integrated Stabilization Mission in Central African Republic (MINUSCA) [4].
The motivation for this master thesis relays on the last two missions taking place in the African continent. Landmines, anti-vehicle mines and improvised explosive devices (IEDs) are extremely widespread in Africa [5]. In July 2020, the suspected use of anti-vehicle mines was first reported in the country since the UN peacekeeping mission MINUSCA was established in 2014. One of the suspected devices damaged a MINUSCA tank near the border with Cameroon. As shown in Figure 1.1, the number of attacks and incidents with mines and explosives have increased, especially since April 2021. In more recent times, two trucks contracted by a United Nations agency transporting construction materials from Bocaranga to Bozoum hit an anti-tank mine in Ngoutéré, 40 km from Bocaranga in December 2021 and March 2022 (Figure 1.2) [6], [7].
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Figure 1.1: Incidents regarding Mines and Explosives since 2021
In the beginning of 2020, two Belgian soldiers currently serving in Mali were injured when their vehicle drove over an improvised explosive device (Figure 1.3) [8]. Fortunately, the Portuguese forces, till the making of this thesis, didn’t suffer any attacks or casualties by this kind of devices, but as the dangerous increases it will be not a question of if, but when it will it happen.
Figure 1.2: A truck carrying goods from a UN agency hit an explosive device in Ngoutere, killing two people [7]
Figure 1.3: The Belgian Armoured Vehicle after IED attack
Military structures such as barracks, combat vehicles and civilian buildings of great importance are the most common targets in these terrorist attacks. For this reason, it’s fair to say that these terrible events are likely to occur in routine times or in theatres of operations (Figure 1.4).
BANGUI BOUAR
BOZOUM BOSSANGOA
BERBERATI
NOLA
MBAIKI
KAGA-BANDORO
SIBUT
NDELE
BAMBARI
MOBAYE
BANGASSOU BRIA
OBO BIRAO
KOUKI
MARKOUNDA
ALINDAO MOYENNE SIDO
YALINGA SAM-OUANDJA
BAKOUMA BATANGAFO
KAMBA KOTA
CARNOT
General area of the location of explosions (2021)
General area of the location of explosions (2022
3 Figure 1.4: Location of terrorism events from 2002 to 2016 [9]
In 2018, more than 9600 terrorist attacks were perpetrated worldwide, killing more than 22000 people [10]. The number of terrorist attacks has increased from 833 to 9600 in a decade with a proportionate number of victims [11]. As is showed in Figure 1.5 most of the mechanism used to carry out these attacks are mainly explosives, which can guarantee a big amount of damage on structures and human lives.
Figure 1.5: Weapon type in terrorism attacks [12]
Apart from terrorism attacks, explosions can be originated by accidental events. On August 4, 2020, a massive chemical detonation occurred in the Port of Beirut (Figure 1.6 and Figure 1.7), Lebanon.
An uncontrolled fire in an adjacent warehouse ignited around 2750 tons of Ammonium Nitrate. The blast supersonic pressure and heat wave claimed the lives of 220 people and injured more than 6500 instantaneously, with severe damage to the nearby dense residential and commercial areas [13]. The accident was considered to be the largest of its kind and the most severe anthropological disaster of the decade [14].
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Figure 1.6: Aerial view of the Beirut port before the explosion (left) and after the explosion (right) [14]
Figure 1.7: Proxy map representing the likely extent of damage post explosion [14]
As we can see from the explosive and mine attacks in Africa, the accidental explosion in Beirut and the global terrorism situation, there are threats both in military and civilian environments. For this reason, it remains a challenge for engineers and government authorities to minimize and develop more efficient solutions to protect structures from explosions.
The concept of protection is hardly absolute. Some trade-offs can be made so that the protection provided versus the cost of the potential loss is optimal. Some losses may be negligible compared to the loss of human lives, for example. In order to determine a level of protection, it is essential to carry out a risk assessment which combines: the type of threat, the probability and the consequences of an attack or an accident. In addition, it is important to clearly identify the behaviour of both civil and military structures (train stations, airports, combat vehicles, etc.) subjected to blast loads [15].
Currently the solution to protect structures and the human lives inside them from the effects of explosions, depending on the structure and the threat can be active and passive.
An active system is defined as a system which manually or automatically activates before the structure or element is overloaded or deformed excessively to protect that structure or element from
5 failure [16]. This protection is assured by resorting to fluids, gases or additional mechanical or structural elements to resist the applied load or at least a portion of it, prestressing critical elements or by transferring the blast loads directly to the foundation [16]. The systems are often activated electronically, using sensors for example. The disadvantage of these systems is that they are not as economical and reliable as the structural system designed to withstand a bending load [11], [16].
Regarding passive solutions, these do not need to be activate since they are intrinsic to the structure. One common solution of this type is making sure that it is possible increase the distance between the structure and the origin of the explosion, avoiding the higher pressure (Figure 1.8).
However, this solution cannot always be effective because of the limitation of space, mainly on an urban environment [11]. For structural engineering, researchers have come up with different possibilities and solutions of this type such as thicker concrete cover layers, extra armour by steel plates or also external reinforcement ([17]–[21]) and more are being developed.
Figure 1.8: Blast wave propagation (adapted from [19], [31])
In a military sense, these types of solutions can be applied in armoured vehicles or steel container used for many purposes in the field for instance, in the form or amour or shielding. The Portuguese troops deployed in CAR are equipped with light vehicles (LV) and light armoured vehicles (LAV). The latter offers protection against the weapons of the enemy, but not as much when it comes to explosives such as landmines or IEDs. Nowadays there are many ways to mitigate the damage from this kind of attacks on these vehicles, but there are some disadvantages to this. In the case of the containers which are used as barracks, offices or to storage important equipment and other materials (Figure 1.9) there are many kinds of defence mechanism installed. With this solution it would be possible to use these containers more often on the field without being necessary to build walls or fences in front.
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Figure 1.9: Containers used in Camp MPoko and Camp Moana where the QRF/PRT and EUTM are located, as well as other MINUSCA units
Passive solutions are almost universally based on: V-shaped hulls, high ground clearance, hight weight, hight centre of gravity and modular structures [22]–[24]. In the case of active solutions, there are mechanisms such as the Vehicle Global Accelerations Mitigation (VGAM) (Figure 1.10), which will directly counteract the lifting forces caused by the mine blast under the vehicle with motors on the roof (Figure 1.10) [23], avoiding mainly the overturning but still leaving the vehicle somewhat vulnerable to the blast wave effect.
Figure 1.10: Vehicle Global Acceleration Mitigation (VGAM) [23]
Figure 1.11: Mine-Resistant Ambush Protected (MRAP) Vehicle [25]
The Mine-Resistant Ambush Protected (MRAP) family of vehicles (Figure 1.11) provides soldiers with highly survivable multisession platforms capable of mitigating improvised explosive devices, rocket-propelled grenades, explosively formed penetrators, underbody mines and small arms fire threats [25]. But these vehicles are tall, creating several major disadvantages, such as: instability which might lead to a poor handling and roll-over accidents, consequence of a mine or IED attack, difficult access from ground level, large target, more difficult recovery; increased weight due to larger surface area to protect and finally, the most crucial, it requires a very heavy fuel consumption, that will not guarantee a high mobility and combat capability on the battlefield or due to the structure of the vehicle itself [24].
7 Currently, LV and LAV vehicles operating in an armed conflict zone, such as CAR, meet the relevant requirements for ballistic, anti-fragmentation, anti-mine protection and IED protection. These requirements are defined on NATO’s and institutes cooperating with NATO documents where constitute the basis for protection of crew members [24]. Most of the solutions available nowadays are based on ceramic shielding, whose main purpose is to protect military personnel and vehicles from projectiles and fragments generated by mines or IED explosions and from enemy firearms, and on composite (or sandwich) armour, this is a multitude of layers of different materials which combined give those materials a better performance then the base materials, the important factor here is the weight of these materials ([11], [22], [26], [27]).
1.2. Objective and methodology
As it was showed above, in present days solutions are being developed regarding structural protection against explosives in military vehicles, but the most common disadvantage presented is the weight of this solution and the energy absorption efficiency related to its decrease.
Ceramics and polymer have the advantage of being lighter than reinforced concrete or steel plates. In order to present a proper ballistic and blast effects protection manufactures require a multi- material assembly process, which lead to a very high cost [11], [28].
It is a possibility to develop a composite armour that can garantee more efficiency, remaining very resistant and very light. However, the types of armour in the market are very expensive when compared to steel or concrete as solutions for armouring. For this reason, it is relevant to develop a less expensive solution, that will ensure the protection of the structures and human lives.
Light weight protection solutions have been a common target of research in current years ([11], [21], [28]–[32]). The protection layer in these solutions is composed of crushable core sandwiched between two sheets called front and rear plate (Figure 1.12 (a)). The face sheets are often made of thin metallic plates or composite laminates. The front plate which serves on the one hand as protection for the core and to uniformly distribute the blast wave on the core, acting act as a piston that crushes the core material. The crushable core can be made of cellular material (Figure 1.12 (b)) or of crushable structures with regular or irregular topologies (Figure 1.13) [21], [30].
Cellular materials are lightweight, incombustible and inexpensive when compared to other types of solutions. Under compression, this material presents a good capacity to absorb energy (either due to a shock wave or from an impact) by plastic deformation or densification [11], [21], [29], [30]. Foams can be open or closed cells [28], [33].
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Figure 1.12: (a) Schematic view of a protection layer, (b) Some examples of possible crushable materials that can be used as core component [21]
Figure 1.13: Examples of possible crushable structures which can be used as core component [28]
The philosophy behind the sacrificial cladding (SC) (Figure 1.14) is to attenuate the accelerations and the large damages resulting from an explosion by converting high pressures on a short time period to lower pressures (below the maximal strength of the structure) on longer time period [11], [29], [34]. Moreover, with lightweight material the self-weight of the structure will not increase significantly [35].
2D 3D
Honeycombs Prismatic Truss Textile
Hexagonal
Square
Triangular
Triangular
Diamond
Navtruss
Tetrahedral
Pyramidal
3D kagome
Diamond textile
Diamond Collinear
Square Textile
Front Plate
Crushable Core
Rear Plate
(a) (b)
Alporas Cymat
Polyurethane Expanded polystyrene
9 Figure 1.14: Sacrificial Cladding's philosophy [11], [35]
This dissertation focuses on the characterization and evaluation of mineral foams as a sacrificial cladding against a blast loading. With this work, the intention is to contribute to the development of knowledge in this area, namely in the most efficient way of dissipating energy by this type of foams. The SC system is planned to be attached to a supporting structure, shown in Figure 1.15. In this solution it is intended to improve the protection of structures, building (Figure 1.16) and vehicles (Figure 1.17), against the effects of blast loading from explosives by means of a sacrificial cladding (Figure 1.17).
Figure 1.15: The SC system attached to a structure [11], [35]
10
Figure 1.16: Example of using SC to increase the resilience of Steel Containers
Figure 1.17: Example of using SC to increase the resilience of combat vehicles
1.3. Dissertation outline
This master's dissertation is divided into six chapters. The first chapter presents and explains the context, motivation, objective and methodology of this work.
The second chapter explains the concept of an explosion and how its loading action is characterised. Furthermore, in the same chapter, some properties and mechanical characteristics of cellular materials are explained, with emphasis on the brittle mineral foam, which is the target of this study.
In chapters three and four, the experimental campaigns carried out in this thesis are presented.
The first campaigns, in chapter three, were performed in order to characterize and evaluate the mechanical properties of the brittle mineral foam. Then, in chapter four, the tests carried out were aimed at evaluating the reduction of deformation in a structure with the use of a brittle mineral foam based SC.
Afterwards, in chapter five, a numerical model of the above-mentioned experimental campaign was created and its values validated by the experimental ones.
Sacrificial Cladding
Mine
11 Finally, in chapter six, the conclusions of this dissertation and the proposals for future works are presented.
12
13
Chapter 2. State of art
2.1. Explosion and blast loading characterization
The word “explosion” is generally used to describe events associated with a loud noise and sudden disruption of the objects at the site of its occurrence and around it [36], as well as a large-scale, rapid and sudden release of energy and production of gas [20], [21], [36]. This energy comes in form of light, heat, sound and shock waves. These waves propagate through the structure in a very short duration and lead to the collapse of the structure [37].
Explosive materials can be classified according to their physical state as solid, liquids or gases [38]. We can also classify them based on their sensitivity to ignition as secondary or primary explosives.
The last ones can be easily detonated by simple ignition from a spark, flame or impact. Secondary explosives when detonated create blast (shock) waves which can result in widespread damage to the surroundings [20], [36].
Figure 2.1: Explosive materials classification(adapted from [21], [36])
The energy or heat released by the chemical reaction that occurs during the combustion of propellant or detonation of an explosive is called the “heat of combustion” or “heat of detonation”, the latter is used for explosives. This value describes the heat caused by the reaction of the explosive itself yielding the detonation products [39].
It is important to establish the difference between the energy and the heat of detonation, avoiding confusion since these two are used interchangeably. The heat of detonation is determined in a closed calorimetric tank and does not account for the energy available from the highly compressed gas products. On the other hand, the detonation energy is used to refer to the calculated detonation energy of an explosive without the presence of air [39].
The detonation of a condensed high explosive generates hot gases under pressures up to 30 GPa and temperatures of about 3000-4000°C. These gases will push out the volume which they occupy and, in consequence, a layer of compressed air (blast wave) forms in front of this gas volume containing most of the energy released by the explosion generating an increase above the ambient atmospheric
High explosives Primary explosives
Military explosives Industrial explosives Secondary explosives
• TNT
• RDX
• HMX
• PETN
• CL-20
• PbAzide
• PbStyphnate
• HgFulminate
• Tetrazene • ANFO
• Gelignites/dynamites
• Slurries/emulsions
14
pressure [20], [40]. As the blast wave moves outwards from the source, the air pressure at the wave front decreases (Figure 1.8).
The observed characteristics of air blast waves are found to be affected by the physical properties of the explosion source [20]. A blast wave is characterized by instantaneous increase in pressure from ambient atmospheric pressure (P0) to a peak incident overpressure (PS0). The peak incident overpressure decays exponentially with time and returns back to ambient air pressure in time t0, which is known as positive phase duration. This is followed by a negative pressure wave with duration, t0-, which is approximately 2 to 5 time of the positive phase in duration. In most reinforced structure designs, this negative phase is ignored for being small [20], [41]. The blast wave profile is described by modified Friedlander’s equation as follows for the detonation of a spherical charge detonation in free air [41], [42]:
𝑃#(𝑡) = 𝑃#")1 − 𝑡
𝑡", 𝑒$%&''!( (2.1)
PS(t) – pressure at time, t (kPa);
PS0 – peak incident pressure (kPa);
t0 – positive phase duration (ms);
A – wave decay coefficient.
The incident peak overpressure PS0 is amplified by a reflection factor as the shock wave encounters an object or a structure in its path [20], for example, for a hemispherical detonation, a factor of 1.8 is multiplied to take into account the reflection from the ground [41], [42].
The specific wave impulse, iS, is equal to the area under the pressure-time curve from the moment of arrival, tA, to the end of the positive phase and is given by the expression [40]:
𝑖#= / 𝑃#
'")'!
'" (𝑡)𝑑𝑡
(2.2) The typical pressure profile of a blast wave in time for an explosion in air is given in Figure 2.2.
15 Figure 2.2: Typical pressure-time profile of air blast wave [15], [20], [21], [37], [40]–[42]
The interaction between a structure and the blast wave is dependent on the characteristics of the explosive (explosive material, released energy and weight of explosive), the radius of the blast wave and the location relative to the structure, the dimension of the targeted structure, the positive phase duration and the natural period (T) of the structure [40], [42].
Charges situated extremely close to a target structure impose a highly impulsive, high intensity pressure load over a localized region of the structure. On the other hand, charges situated further away produce a lower-intensity, longer-duration uniform pressure distribution over the entire structures.
Eventually, the entire structure is engulfed in the shock wave, with reflection and diffraction effects creating focusing and shadow zones in a complex pattern around the structure (Figure 2.3). During the negative phase, the weakened structure may be subjected to impact by debris that may cause additional damage.
Figure 2.3: Examples of a blast load on a building [43]
16
Due to the complexity of the structure-wave interaction, by accepting certain assumptions about structural response and loads, a simplified analysis can be used. To do this, the structure is idealized as a system with a single degree of freedom (SDOF) and a relation is established between the duration of the positive phase and the natural period of the structure. Depending on the relation between the last two, there are three possible loading cases [11], where 𝑤=2π/T:
𝑤𝑡"< 0.4 5'0(1-23)!(+,-.')6 (impulse loading) (2.3)
0.4 < 𝑤𝑡"< 40 5'0!≈ 16 (dynamic loading) (2.4) 40 < 𝑤𝑡"5'0(+,-.')!(1-23)6 (quasi-static loading) (2.5) Moreover, the interaction of the blast waves with the structure is depending upon the radius of the pressure wave and the dimension of the structure [43]. For this reason, there can be identified three different configurations of blast wave-structure interaction [21], [42], [43]:
• Large blast wave and large structure (Figure 2.4 (a)): the structure will be completely engulfed by the blast wave. In addition to the higher crushing force, some translational forces will be generated;
• Large blast wave and small structure (Figure 2.4 (b)): the entire structure will be engulfed and crushed by the blast wave. The over-pressure acting on several parts of structure will be more or less the same. A significant magnitude of drag force will be generated on the structures;
• Small blast wave and large structure (Figure 2.4 (c)): the loading on the structures will not be uniform. The response of individual elements of the structure should be analysed separately.
Figure 2.4: Blast wave configurations depending on the dimensions of the structure and length of the shock wave [34]
(a)
(c) (b)
17 As for the distance between the origin of the explosion and the targeted structure, known as the stand-off distance [44], there are two distinguished cases:
• Far to intermediate field blast (global action): the stand-off distance is greater than the characteristic dimension of the structure. Thus, the non-uniformity and arrival time difference of the blast load applied on the structure is not significant due to its small value, so it can be neglected. In these cases, the blast wave can be neglected [42]. The peak overpressure is related to the weight (W) and stand-off distance (D) of the explosive charge by:
𝑃#"≈ 53.9 ;𝑊".55 𝐷 >
6.65
(2.6)
• Near field blast (local action): typically, the stand-off distance is within 10 to 20 times of the charge radius. This type of explosion corresponds to an impulsive loading regime of the structure [11], [44].
In order to quantify the domain of survivability of a structure related to a combination of pressure and momentum a P-I diagram can be established, as is shown in Figure 2.5. The objective of this diagram is to summarize the response of a structure associated with a blast wave with a certain over- pressure peak during a certain duration [42].
Figure 2.5: P-I diagram
Different explosives with equal mass will differ in the energy produced after the explosion. The energy released depends on the chemical composition of the explosive itself. The universal TNT explosive implies that it is convenient to compare the effects of fuel-air or high explosive detonation to TNT. This comparison can be conducted using the TNT equivalency, which represents the mass of TNT
Impulse loading
Dynamic loading
Safe
Unsafe
• Effects due to stress: flaking
• Near field blast wave
• Blunt trauma effect
• Internal transmission of strong accelerations
Quasi-static loading
• Strain rate dependence
• Intermediate field blast wave
• Effects due to stress: deformations
• Far field blast wave
Pressure (P)
Impulse (I)
18
that would result in an explosion of the same energy level as the unit of the explosive under consideration. The concept of TNT equivalency is given by proportional ratio of the TNT mass to explosive mass, which produces a blast wave equal magnitude (or impulse pressure) at the same radial distance for each charge, which assumes the scaling laws of Sachs and Hopkinson [39]. In the case of high explosives, the mass of TNT can be calculated with the following expression:
𝑀7,090=𝐻:,;<× 𝑀;<
𝐻:,090 (2.7)
where Me,TNT is the TNT equivalent weight (kg), MHE is the weight of the explosive considered (kg), Hd,HE is the detonation energy of the explosive considered (MJ kg-1) and Hd,TNT is the detonation energy of TNT (MJ kg-1) [38], [39].
Table 2.1 presents a few explosive materials, with the corresponding density, specific energy and the equivalent TNT. Since the comparison can be different for pressure or impulse, the average equivalent TNT factors must be used [45].
Table 2.1: TNT equivalent of various explosive materials [40], [41], [45]
Equivalent TNT Density
ρ
Specific energy Es
Specific energy
Pressure Impulse
Explosive [kg/m3] [kJ/kg] [/] [/] [/]
TNT 1630 4520 1 1 1
RDX 1840 5360 1.185 1.14 1.09
PETN 1700 5800 1.283 1.27 1.11
M112 1650 5360 1.185 1.37 1.19
2.2. Brittle Mineral Foam
In nature it is possible to encounter many raw materials composed of unique cell structures, such as wood and coral. This intrinsic property enables them to possess an ultra-low density and multi- functionality. For this reason, humans have always been motivated to recreate these properties by producing similar synthetic materials in order to meet the continuing demands of weight reduction and function integration in engineering structures [46]. These synthetic materials include cellular materials.
Numerous material and structural components have been investigated over the years for their suitability as impact energy absorbers. The recent developments of less expensive foams manufacturing techniques have led to an increase of interest in their use for applications in energy absorbing systems.
Man-made cellular material has distinctive characteristics that include an excellent strength to weight ratio and the capacity to undergo large deformation at nearly constant load (nominal stress) [47].
By the above, it his understandable that there is a big importance in having a detailed knowledge of the mechanical behaviour of cellular material, mainly foams. The properties of the latter depend on
19 the structures of the cells but also on the base material, from which the foam was produced. The most important characteristics of a foam rely on its relative density, the degree to which its cells are open or closed and the anisotropic shape ratios R12 and R13. The later will cause the foams to have different resistance values for different directions. The cell walls or struts will also influence the behaviour of the foam, the properties related to them are the density of the material, Young’s Modulus, the elastic limit and the fracture resistance [11], [46].
The brittle mineral foam studied in this thesis has a closed-cell structure and is composed of chalk, sand, cement and water. The mixture of these components is stabilized under pressure of 12 bar and temperature of about 190°C in the autoclave [29]. The chemical composition in mass percentage of the brittle mineral foam studied is: 70-80% hydrated calcium silicate, less of 2% quartz, 15-20% of calcite and 3-8% of gypsum and anhydrite. It has a density of about 110kg/m3 and, since it is manufactured as lightweight panels, it can be used to add resilience to an existing structure [11]. The technical sheet of this brittle mineral foam is shown in Appendix A.
Figure 2.6: Brittle mineral foam 2.2.1. Deformation mechanism of cellular materials
The compressive stress-strain curve of a cellular solid can often be simplified into three regimes or stages: the linear elastic phase (pre-collapse stage), the plateau stage and the densification stage [33]. These three regimes differ for different types of cellular materials (Figure 2.7). For many types of cellular solids, the plateau stage starts from the pre-collapse strain, ɛc0, or pre-collapse stress, σc0, representing the initiation of the new deformation mechanism of the cell wall or the cell wall failure, and ends at a critical strain, ɛd0, representing the onset densification. With further compression, more cell walls come into contact with each other. The cellular material is completely compacted when the strain reaches a complete densification strain, ɛd, causing a steep increase in the slope of the stress-strain curve [47], [48].
20
Figure 2.7: Uniaxial compression of cellular material: (a) stress-strain curve for both single cell or specimen with strain-hardening characteristic; (b) stress-strain curve for a single cell with strain- softening characteristic; (c) macroscopic stress-strain containing collection of strain-softening cells (adapted from [48])
For consistency, the definition of either the pre-collapse stress, the plateau stress and the densification strain must be defined and standardised. The pre-collapse stress is defined as the first peak stress or the stress at which the tangent modulus is the first minimum. It is possible to also determine an offset yield stress but this may vary with strain and requires extra effort for a more accurate measurement [46]. The pre-collapse strain is defined as the strain corresponding to the pre-collapse stress.
The densification strain corresponds to different levels of interaction between cells, and therefore to different points on a compressive stress-strain curve. Li et al [48] shows the relevance of establishing a distinction between “onset strain of densification” and “densification strain”, where in many publications the densification strain is used instead of the onset strain of densification, which is more accurate. The first one was defined by Gibson and Ashby [49] to define as limit strain where “the opposing walls of the cells crush together and the cell wall material itself is compressed”, this is, the complete densification of the material. In this thesis, the densification strain will be defined as the “onset strain of densification” for which there are several methods to determine it:
• Method A: the onset strain of densification is defined by the intersection of the tangents to the stress stage and the densification stage [50];
• Method B: the onset strain of densification is defined as the strain at the last local minimum before the stress rises steeply [48];
• Method C: the onset strain of densification is defined as the strain at which the slope of the tangent is equal to that of the elastic regime [50].
21 Since the results obtain by these methods may introduce errors when used on cellular material, an energy parameter can be used to determine the optimal energy absorption of the foam. This absorption efficiency is defined by [48]:
𝜂(ɛ) = 1
𝜎(ɛ)/ 𝜎(ɛ)𝑑ɛ
ɛ
" (2.8)
where σ and ɛ are the nominal compressive stress and strain, respectively.
Furthermore, thin structures under compression can be divided into two types: Type I (Figure 2.8 (a)) and Type II (Figure 2.8 (b)) [46], [48]. Type I structure have a stable load-deflection plateau and are frequently featured by a slight hardening in the plateau stage. The load-deflection relation for a Type II structure is unstable, which leads to a strong softening feature in the plateau stage [48].
Figure 2.8: (a) Type I structure with strain hardening characteristics, (b) Type II structure with strain softening characteristics [48] and (c) the quasi-static force-displacement curves for each type of
structure [35]
When subjected to large deformations, the plateau stage ends when the interactions between structural walls start to play an important role. A cellular solid, mainly the brittle mineral foam targeted in this thesis, is a collection of regular and irregular thin-walled cells and its macroscopic compressive behaviour is determined by the individual compressive behaviour of each cell and the possible interactions between the ones surrounding it. For this reason, Li et al [48] presents three distinguished behaviours for cellular solids that can occur during the plateau stage: hardening (Figure 2.7 (a)), softening (Figure 2.7 (b)) and perfectly-plastic (Figure 2.7 (c)).
Consequently, the method to determine the onset strain of densification by the energy parameter needs to take into account the correspondent behaviour (Figure 2.9). For that one of the following methods [48]:
• Method D (for hardening/softening behaviour): the global maximum of the efficiency curve is used to determine the densification strain:
𝑑𝜂(ɛ) 𝑑ɛ F
ɛ>ɛ#! = 0 (2.9)
Local oscillations in the efficiency curve are smoothed;
22
• Method E (for perfectly plastic behaviour): the last local maximum of the efficiency curve is used to determine the densification strain.
The plateau stress can be determined by the energy equivalence in the plateau stage, by:
𝜎?1=∫ɛɛ#!𝜎(ɛ)𝑑ɛ
$!
ɛ:"− ɛ@" (2.10)
where ɛc0 is the strain at yield (pre-collapse strain) and ɛd0 the onset strain densification corresponding to the start and end of the plateau stage, respectively.
Figure 2.9: Schematic diagram indicating (a) the tree compression stages and (b) the η-ɛ curve [51]
It is of fundamental importance to know closely the elastic modulus approaches to that of the constituent materials as the foam is compressed towards the density of the constituent material. This is needed when the foam is used under extreme loading conditions, such as intensive blast loading or high-speed impact, which can introduce very large deformations [52].
2.2.2. Shock absorption and constitute model of cellular materials
The energy absorption mechanism of a specific material may manifest itself in different ways such as by the change of its state (for example the change from a solid state to liquid or gaseous state), deformations (elastic or plastic), cracking and other types of visual damage, thermal transfer, compaction, etc [11].
The latter is related to cellular and porous materials. When shock occurs, the nominal stress- strain relation is unable to describe the constitutive behaviour of cellular materials and the nominal strain-rate should not be used to characterise the dynamic behaviour, since the axial inertia leads to a significant difference between stresses and strains across the shock front [46]. By cell-based simulations and test campaign there have been compared by [46], demonstrated that there is a difference between stresses measured at the loading and supporting ends, as well as deformations in crushed and uncrushed areas. Therefore, it is possible to establish stress and strain under shock throughout a sample and measure the constitutive stress-strain model, by determining the physical quantities locally.
(a) (
b
23 Furthermore, to better understand the response of this material under shock through a numerical modelling, constitutive models have been developed. The models presented below can be implemented and used in Finite Element (FE) software, like the one used in this thesis LS-Dyna, so it is possible to recreate this materials’ behaviour.
A very simple way to do it can be by using the “Snowplow” compaction mode (Figure 2.10). This model was introduced in the beginning of 1960 and takes the basic assumption that there is no resistance. The behaviour of the material is descripted as a fluid and a hydrodynamic material, according to its Hugoniot curve which is characteristic to the material subjected to a shock. From the relations between conservation of mass, momentum and energy, historically expressed by Rankine-Hugoniot, the follow equations can be obtained [53], [54]:
𝑃;∗(𝑉) = 𝑃;(𝑉) 1 −G"
2 J1 −𝑉 𝑉"K 1 −G"
2 L𝑉"∗ 𝑉"− 𝑉
𝑉"M (2.11)
where V0 and V*0 are respectively the initial specimen volumes of the compacted material and its expanded form. G0 is the Mie-Grüneisen coefficient of the material.
Figure 2.10: "Snowplow" compaction model
In case of heavier shocks, this model is representative of the phenomenon of dynamic compaction of the porous material, which is contrary to what happens with weaker shocks [53].
Apart from this model there is also a pressure-compaction model, P-a, proposed by Herrmann, in 1969, for ductile materials (Figure 2.11). It is based on a continuous Hugoniot curve which represents three regions where the material is fully compacted, uncompacted and elastic. In 1994, Maxwell proposed two possible expressions to describe the compaction dynamics of a porous material [53]:
a= 1 − (a7− 1)𝑒$â(B$B%) (2.12)
a=a7− 2(a7− 1)𝑃 − 𝑃7
𝑃@− 𝑃7+ (a7− 1) L𝑃 − 𝑃7 𝑃@− 𝑃7M
C
(2.13)
24
where Pe and ae are respectively the hydrodynamic pressure and the compaction at the elastic limit of the material. Pc is the critical pressure at which the material is completely compacted (densification state) and â is an adjustment coefficient.
Figure 2.11: P-a compaction model [53], [54]
According to the P-a compaction model there are different states of compression and decompression [11]:
1. The initial state of elastic compression which results from the elastic crushing of the cells of the material;
2. The state of plastic deformation from Pe at the beginning of which the permanent change of the initial volume begins;
3. Dynamic compaction takes place for different pressures along the Rayleigh lines until reaching PC beyond which densification begins;
4. Release from a partially compacted state proceeds elastically with final volume smaller than the volume at the onset of release due to localized densifications.
A similar compaction model to this is a P-l model (Figure 2.12), where it essentially differs on the latter in the sense that this one is a heterogeneous material compaction model, which allows to take into account the air contained in the pores of the porous material during the modelling. For the porous material is considered to exist two different physical states: either uncompacted or completely unpacked state. An irreversible state variable, l, describes the mass fraction of the compacted material [54] where monotonically increases in function of the pressure, according to:
l= 1 − 𝑒$&BD(
&
(2.14) where Y is the local elastic resistance, P the local pressure and n the homogeneity parameter.
Pressure
α αp
P
1
Rayleigh’s Lines Plastic compression
Hugoniot solid
Elastic compression
αe
Pc
Pe Elastic state
25 Figure 2.12: Pressure-volume space illustrating P-l curve [54]
For a more complete approach, by considering the stress-strain curve of the crushed material in a multilinear way there is the Multilinear model (Figure 2.13). The variants of this model are both bilinear and trilinear models.
Figure 2.13: Illustration of a multilinear model
A compressive model for shock wave in cellular material called rigid-perfectly plastic-locking (RPPL) idealization was presented by Reid et al [55], which does not take into account the strain rate.
There are only two material parameters used to define the material properties: the plateau stress and the densification strain (initially defined as locking strain) [56].
To consider the strain rate of the material a rigid-linearly hardening plastic-locking (R-LHP-L) is used. According to this model, there are three important parameters: the elastic limit, the hardening modulus, E1, and the densification strain. If the hardening modulus tends to zero, the R-LHP-L model
Constraint [GPa]
26
tends to the RPPL model (Figure 2.14). In reality, in a quasistatic regime there is always a hardening part [56]. Both of these two models do not take into account the densification state, so for that the elastic- perfectly-plastic-hardening model (EPPH) should be used (Figure 2.15). According to this the deformation of a cellular material is the average response of the cells considered separately [57].
Figure 2.14: Idealization of the stress-strain curve
Figure 2.15: EPPH model Constraint
Deformation at
densification Deformation
RPPL R-LHP-L E₁ Plateau constraint
Elasticity limit
