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Poznan University of Technology
Faculty of Civil and Environmental Engineering
Doctoral Dissertation
DYNAMIC BEHAVIOR AND FAILURE OF ALUMINUM-POLYETHYLENE SANDWICH
STRUCTURE
Thesis by Amine Bendarma
Supervisor: Prof. Tomasz Łodygowski Dr hab. inż. Tomasz Jankowiak
Poznań 2018
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Acknowledgement
This research was conducted within the Computer Aided Design division (DCAD) of Institute of Structural Engineering (ISE)
First of all, I would like to thank my supervisors. My deep gratefulness is to my thesis supervisor, Prof. Tomasz LODYGOWSKI, for having welcomed me to the research team and for having given me this thesis subject. I would like to express to him my deep gratitude for his availability and his scientific rigor.
I also would like to thank my co-director Dr.Tomasz Jankowiak for his expertise in the field of study of the dynamic behavior of materials, his instructive advice, his availability and the sympathy he has showed me for these four years of thesis. . During this journey, he introduced me to the exciting field of rapid dynamics, including impact and perforation.
My special thanks are also directed to Mr. Aziz Bouslikhane President of the International University of Agadir and to Mr. Ilias Majdouline as well for the technical and financial support I have been provided with in this thesis work.
My sincerest thanks to Alexis Rusinek, University Professor at the University of Lorraine for his great contribution in this work, particularly in the quasi-static and dynamic characterization of materials. I also want to express my gratitude to Abdellah Souleimani, Maciej Klosak and Richard Bernier for their technical assistance in manufacturing and preparing measuring devices.
I thank all the members of the Laboratory with whom human and scientific exchanges have always gone well.
It is really an honor to express my greatest satisfaction and gratitude to all the persons who contributed by all means to the development of this work.
4 I dedicate this thesis to my parents
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Contents
Acknowledgement ... 3
Abstract ... 10
Streszczenie ... 11
Nomenclature ... 12
Conversion of the measure units ... 13
General introduction ... 14
1. Chapter I : Mechanical Behavior of Materials and structures ... 20
1.1. Introduction ...20
1.2. Strain rate sensitivity of metals ...21
1.3. Temperature sensitivity of metals ...22
1.4. Metallic structure behavior under dynamic loading ...23
1.4.1. The specimen (plate) properties Influence ... 24
1.4.2. Characteristics of the projectile and its influence ... 27
1.5. Mechanical behavior of structures ...30
1.5.1. Quasi-static behavior ... 30
1.5.2. Dynamic behavior ... 31
1.6. Simulation ...33
1.6.1. Constitutive relation taking into account the strain rates and temperature 36 1.6.2. Dynamic constitutive relation ... 36
1.6.2.1. Physical constitutive relation ... 37
1.6.2.2. The empirical constitutive relation ... 40
1.6.2.3. Other constitutive relations ... 44
1.6.3. Mechanical failure criteria ... 45
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1.7. Mechanical properties of polymers ...47
1.7.1. General aspect of Polymers ... 48
1.7.2. Plastic behavior of amorphous polymers with large deformations .... 48
1.7.3. General behavior of amorphous polymers ... 49
1.7.4. The influence of temperature... 51
1.7.5. Plastic Instability ... 52
1.7.5.1. Some constitutive relation of polymers ... 54
1.7.6. Polymers plasticity criteria ... 57
1.7.6.1. Yield surface description ... 58
1.7.6.2. Tresca’s yield criterion ... 59
1.7.6.3. Huber Mises yield criterion ... 60
1.7.6.4. Modified Tresca yield criterion ... 60
1.7.6.5. Modified von Mises yield criterion ... 61
1.7.6.6. Drücker-Prager yield criterion ... 62
1.7.6.7. Raghava’s yield criterion ... 63
1.7.6.8. Criterion of Doyoyo and Wierzbicki ... 63
1.8. Conclusions ...64
2. Chapter II : Experimental conditions and techniques ... 65
2.1. Introduction ...65
2.2. Overview of the studied materials ...65
2.2.1. Aluminum AW 5005 description ... 66
2.2.2. Polyethylene low density description ... 67
2.2.3. Traction tests ... 68
2.2.4. Description of the tensile test ... 69
2.2.5. Specimen ... 69
2.3. Compression test ...70
2.3.1. Quasi-static compression ... 70
2.3.2. Description of the Hopkinson bar Device ... 72
2.4. Perforation test ...77
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2.5. Perforation using thermal chamber ...79
2.6. Conclusions ...83
3. Chapter III : Analysis and discussion of experimental results (Aluminum and polyethylene) ... 84
3.1. Introduction ...84
3.2. Characterization test results for aluminum alloy...85
3.2.1. Tensile test ... 85
3.2.2. Perforation test ... 88
3.3. Numerical analysis using different failure criterion ...90
3.3.1. Modeling and numerical simulation ... 90
3.3.2. Failure criterion model ... 92
3.3.2.1. Model I (Johnson-Cook model) ... 92
3.3.2.2. Failure modeling using optimized Model II ... 93
3.3.2.3. Failure modeling using Model III ... 94
3.4. Comparison of numerical and experimental results ...95
3.4.1. Tensile test comparisons ... 95
3.4.2. Perforation test comparison ... 97
3.4.3. Perforation test using thermal chamber ... 98
3.4.4. Parameters identification using JC model ... 102
3.5. Dynamic compression tests ...105
3.5.1. Friction effect on cylinder sheet ... 106
3.5.2. Experimental analysis results (dynamic compression test) ... 109
3.5.3. Experimental analysis results HSPB using thermal chamber ... 110
3.6. Polyethylene characterization results ...113
3.6.1. Experimental results ... 113
3.6.2. LDPE numerical simulation analysis ... 115
3.7. Conclusions ...118
8 4. Chapter IV : Alucobond structure analysis and discussion of experimental
and numerical results ... 119
4.1. Introduction ...119
4.2. Sandwich structure ...120
4.3. The constituent elements of sandwich materials ...120
4.3.1. Face sheets ... 120
4.3.2. Cores ... 120
4.3.3. The interface ... 121
4.4. Characterization test results for Alucobond structure ...121
4.4.1. Tensile test of Alucobond structure ... 121
4.4.2. Perforation test ... 123
4.4.3. Perforation test using thermal chamber ... 124
4.5. Modeling and numerical simulation ...127
4.5.1. Tensile test ... 127
4.5.2. Perforation test ... 129
4.6. Conclusions ...134
5. Chapter V : Conclusions and perspectives ... 135
References ... 138
Appendix A Shells / Solids - Finite Element Analysis ... 155
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Abstract
In this dissertation, a presentation of several experimental techniques that has been deployed to study the dynamic behavior of the Alucobond sandwich structures at small and high strain rates. The main aim of this work was to study the dynamic behavior of this structure that is composed of two completely different materials, a metallic material: AW5005 aluminum alloy, and a polymer material: LDPE low density polyethylene. Experimental, analytical and numerical studies have been carried out to analyze this behavior in details. It is therefore necessary to use new experimental techniques covering a wide range of strain rates with a temperature ranging from ambient temperature and 300 ° C.
During this thesis, four experimental techniques were defined as quasi-static traction and compression (strain rates ranging from 10−4 𝑠−1to 10−2𝑠−1), dynamic compression (strain rates varying from 102 𝑠−1 to 103 𝑠−1) and perforation by means of a 72° conical projectile (with strain rate ranging from 10−4 1/s to 104 1/s). A Hopkinson pressure bar and a pneumatic gas gun, both equipped with a new thermal chamber, were used to conduct dynamic and perforation tests at high strain rates and temperatures. To validate this setup, experimental tests were carried out on Alucobond structures.
An analysis of the propagation of the waves made it possible to define the behavior of these materials constituting this sandwich structure, and to study the strain rate effect, the hardening and the thermal effects. The impact and perforation tests were carried out using a gas gun. The experimental results made it possible to identify the overall behavior of the structure, and to propose a constitutive relation for it.
Different failure criteria are discussed, coupling numerical and experimental analyses for a wide range of strain rates. An optimization method function is used to identify the parameters of the failure criteria of the studied aluminum alloy.
A FE 3D model has been developed to simulate the mechanical behavior of the studied materials targets subjected to impact and perforation. The results from the numerical preliminary tests were compared with the experimental results. Overall, there has been a good agreement between the numerical and experimental results, particularly in terms of ballistic curves and absorbed energy.
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Streszczenie
W pracy przedstawiono wyniki badań eksperymentalnych przeprowadzonych na panelach Alucobond dla małych i dużych prędkościach odkształceń. Głównym celem pracy była analiza dynamiczna zachowania elementu złożonego z dwóch materiałów o zasadniczo różnych charakterystykach fizycznych, tj. metalu - stopu aluminium AW5005 oraz polimeru LDPE (polietylen o niskiej gęstości). Oprócz eksperymentów przeprowadzono studium analityczne oraz numeryczne.
Zaproponowano cztery techniki eksperymentalne: quasi-statyczne rozciąganie i ściskanie (prędkości odkształceń od 10−4 𝑠−1do 10−2𝑠−1), ściskanie dynamiczne (prędkości odkształceń od 102 𝑠−1 do 103 𝑠−1) oraz perforacja za pomocą stożkowego pocisku o kącie 72° (prędkości odkształceń 10−4 1/s do 104 1/s).
Specyfika analizy wymagała użycia nowych urządzeń generujących duże prędkości odkształceń w szerokim spektrum temperatur, od temperatury pokojowej do 300 °C.
Pręt Hopkinsona został wykorzystany do analizy dynamicznego zachowania obu materiałów. Określono wzmocnienie odkształceniowe obu materiałów oraz ich wrażliwość na prędkość odkształcenia. Uwzględniono również wpływ temperatury na zachowanie materiałów. Dla stopu aluminium zdefiniowano równania konstytutywne, a także zaproponowano kryterium zniszczenia dla szerokiego spektrum prędkości odkształceń jako wynik połączonej analizy numerycznej i eksperymentalnej.
Model 3D użyty w symulacjach wykorzystujących metodę elementów skończonych pozwolił na odtworzenie zachowania analizowanych materiałów w eksperymentach dynamicznych i quasi-statycznych. Porównanie symulacji numerycznych z eksperymentami wykazało dobrą zbieżność wyników, w szczególności w teście przebijania. Również wielkości energii absorbowanej podczas perforacji były w omawianych porównaniach zgodne.
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Nomenclature
A Elongation at fracture B Material constant
c Specific heat capacity E Modulus of elasticity HV10 Hardiness, vickers
K Fitting parameter
n Strain hardening coefficient Rp0.2 0.2% proof stress
T Absolute Temperature 𝑉 Velocity
𝑚 The mass 𝜀 Equivalent strain 𝜀𝑛 void initiation strain
𝜀̇ strain rate
𝑘 Taylor-Quinney parameter 𝜆 Thermal conductivity µ Friction coefficient 𝜌 Density
𝜎 True stress 𝛾 Shear strain 𝛾̇ Shear strain rate
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Conversion of the measure units
Distance and Length:
1 m = 102 cm = 103 mm = 106 µm 1m = 3.281 foot = 39.37 inch 1 inch = 0.0254 m
1 foot = 0.3048 m Mass:
1 kg = 103 g = 2.2046 lb 1 lb = 0.4535924 kg Time:
1 s = 1000 ms = 1000000 µs Work (Energy)
1 Joule = 0.001 kJ = 0.102 KGm = 8.8507 in.lb Pressure:
1 Pa = 0.00001 bar = 0.000009869 atm = 1 N/m2 = 0.000001 N/mm2 = 0.0001 lb/in2
1 bar = 100000 Pa 1 atm = 101325 Pa 1 PSI = 6894.757 Pa 1 MPa = 145.04 PSI Abbreviations:
SHPB: Split Hopkinson Pressure Bar
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General introduction
Construction materials under rate of loading exhibit of hardening or softening that usually causes the failure and fracture phenomena. This process can be very fast for quasi-static as well as dynamic loading.
To define the behavior of materials to be used in industrial applications, it is necessary to perform several tests to understand the different loading effects of strain rates and temperatures on the material’s behavior. As shown in Fig.1, several techniques have been used to study a wide range of strain rates.
The coupling of all the data of different tests makes it possible to define the strain rate sensitivity of the material. The split Hopkinson pressure bar (SHPB) or the Taylor’s test is required in order to reach high strain rates (Fig.1), a high rate servo- hydraulic machine or the Taylor’s test might be used for lower strain rates.
The dynamic testing field covers a very wide range of conditions and is of interest to engineers from different disciplines. For example, production engineers seek to understand the problems that occur during machining, and impact problems on structures for civil engineering (Fig. 2), it is also of big interest for the aeronautics field (Fig. 3). given below some examples of such applications in increasing order of velocity:
Figure 1: Definition of the devices used to deliver the range of strain rates in mechanical testing (Julien, 2013).
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Table 1. Speed of deformation for specific physical phenomena (Łodygowski, 2011)
Type of phenomenon Velocity of deformation [𝒔−𝟏]
Creep From 10−10 to 10−5
Creep beyond the yield criteria From 10−5 to 10−1
Hot drawing From 10−1 to 101
High speed drawing From 101 to 103
Machining From 103 to 105
Drawing with use of explosion > 105
Type of phenomenon Velocity of deformation [𝒔−𝟏]
Earthquake From 10−3 to 10−1
Car crash From 10−2 to 100
Plane crash From 5.10−2 to 2.100
Hard hit From 10−0 to 5. 101
Projectile hitting From 102 to 106
Explosive loading From 106 to …
Type of phenomenon Velocity of deformation [𝒔−𝟏]
Geological movements ≈ 10−10
Creeping ≈ 10−6
One axial tension test ≈ 10−4
Drilling, rolling, drawing ≈ 100 Test with Hopkinson’s bar ≈ 103
High velocity impacts ≈ 106
For static problems the system remains elliptic whereas for dynamic ones its hyperbolic. Discussion around the type of loading, quasi-static or dynamic, should be balanced based on own experience compared with particular rates of deformation for typical processes, see Table 1. (Łodygowski, 2011).
Table 2. Load classification (Department of army, 1990)
Load classification 𝝉
𝑻 Type of the load
Quasi-static >4 Conventional testing
Quasi-static 1 Transient loading on structures
Impact < 0.25 Kinetic energy, blasts pressure
Shock < 10−6 High energy explosives
16 The study of the dynamic behavior of materials and structures is therefore a rapidly expanding field in order to improve the safety and performance of products.
Scientists working on military fields need to understand the behavior of materials, in order to design structures that are more effective in resisting the impact of a projectile or in order to design efficient ballistic missiles.
Figure 2: Asphalt road cracked and broken from earthquake (Shutterstock, 2018)
Figure 3: Impact of bird and projectile on an airplane, a) bird impact, b) projectile impact (Canada, 2007)
It is therefore a matter of meeting the scientific needs of the industry for the use of metallic materials and design of structures under dynamic loading. This especially is for better control of the constitutive relations of building materials, breakage and energy absorption for people and structures protection against natural disasters or for passive safety, for example in the construction sector.
Perforation problems are characterized by the complexity of thermomechanical processes that occur within the target structure during loading. It is therefore necessary to fully understand the relationship between the thermoviscoplastic behavior of the material, the energy absorption and the rupture mechanisms for impact and perforation loadings. This is not always easy, because it requires the development of adapted experimental and numerical means. Numerous experimental, analytical and numerical studies are now available in the international
17 literature. Works on the penetration and perforation of structures by projectiles has been carried out by researchers such as Goldsmith (Goldsmith, 1999), Corbett et al.
(Corbett, 1996), Backman et al. (Backman, 1978). (T. Borvik, 2003), the impact velocity (Arias, 2008), (Børvik, 2002), (Rusinek, 2008).
The influence of the metal plate’s thickness and the geometry of the projectile on the process were examined. The problem nowadays is the coupling of the experimental, analytical and numerical aspects in order to provide a detailed analysis of the structures behavior under rapid dynamic loading.
The ballistic behavior and resistance of aluminum sheet plates and Alucobond structures are highly dependent on the material’s behavior under dynamic loading.
The ballistic properties of the structure are strongly related to the behavior of the material and the interaction between a target structure and a projectile during the perforation process. Therefore, to find the expected curves, several works have studied many dynamic constitutive relations. For example, Johnson and Cook (Johnson, 1983) proposed a dynamic constitutive relation based on a phenomenological approach. Usually impact and perforation problems have been analyzed using this model. Verleysen et al. (Verleysen, 2011) studied the effect of strain rate on sheet metal forming behavior and described the stress-strain curves of the material using the Johnson-cook model. Erice et al. (Erice, 2014) presented a constitutive model coupled with elastoplastic damage to simulate the failure behavior of inconel plates. Rusinek and Rodrıguez-Martınez (Rusinek, 2009) provided two extensions of the original Rusinek-Klepaczko constitutive relation (Rusinek, 2001) in order to define the aluminum alloys behavior at wide ranges of strain rates and temperatures, showing a negative strain rate sensitivity. Børvik et al. (2009) studied the influence of a modified Johnson-Cook constitutive relation using steel plate perforation’s numerical simulations. Jankowiak et al. (Jankowiak, 2013) considered perforation of different configurations: mild steel and sandwich plates, Consequently, checking the effectiveness of these kinds of structures. The authors also presented the effect of strain rate sensitivity models (Johnson-Cook and Rusinek-Klepaczko) on the ballistic curve. Additionally, several effects were considered: strain hardening, yield stress and projectile mass. Based on these results, it was possible to optimize the structure and find the right plate thickness to prevent its perforation This thesis work falls within the scope of this problem: to develop a complete method for analyzing the behavior of materials ranging from characterization to validation tests and analytical models. The main objective of this thesis is to study the impact behavior of the Alucobond sandwich structure, this structure consists of two completely different materials, a metallic material: AW5005 aluminum alloy, and a polymer material: the LDPE low density polyethylene. The first step consists of making a thorough study for each material, and a complete study of the tested structure, these studies are based on two modes of quasi-static and dynamic characterization.
18 A numerical model exploiting the previously identified behavior laws (based on the characterization tests) will be developed capable of predicting the validation tests such as ballistic impact. From an editorial point of view, this work is structured around four main chapters.
The first chapter will be devoted to a complete bibliographic study: the emphasis will be mainly on the behavior under impact of metallic materials, polymers and sandwich structures. The influence of the projectile’s characteristics (shape, stiffness, density, velocity etc...) on the mechanical response of the target will be studied. The influence of the target’s properties on the perforation process will also be examined.
Then a review of different constitutive relations and failure criterion used in the finite element calculation codes to simulate the impact and perforation of metals, polymers and composite structures will be presented. The various experimental tools and methods used in this study to characterize the different studied materials will be summarized in chapter II. In the same chapter, there will be a general presentation of the studied materials. The experimental protocol of quasi-static tests (traction, compression) using a conventional machine that will also be presented. A Hopkinson pressure bar and a pneumatic gas gun, both equipped with an innovative thermal chamber, are used to conduct dynamic and perforation tests at high strain rates and temperatures ranging from room temperature to 300°C. Experimental tests were carried out on Alucobond structures in order to analyze its behavior at high strain rates and temperatures. then the technical aspects related to dynamic loading will be described, and the importance of the test specimen’s geometry. The strain rate and temperature are two very influential parameters which will be the subject of a particular study. The last part of this chapter will be devoted to the complete description of the gas gun and all its experimental measuring devices developed during this thesis and used in impact and perforation tests.
In the third chapter of this thesis, there will be a presentation, analysis and a discussion about the obtained experimental results. For the characterization tests, the influence of the strain rate and temperature on the stress level in the tested materials (Aluminum alloy AW5005 and LDPE) will be analyzed. The ballistic properties of the structure when impacted by 72 ° conical nose shape projectile are studied, different failure criterion are discussed, coupling numerical and experimental analyses for a wide range of strain rates. Optimization method functions are used to identify the parameters of the failure criteria of the studied aluminum alloy. The analytical modeling of the plastic flow of the tested materials will be carried out by means of some constitutive relations (The Johnson-Cook, Cowper symonds…). The complete identification of the retained constitutive relation parameters from the numerical studies (Tensile, compression, perforation tests) will be presented.
19 There will be a discussion about the relevance of analytical models to describing the intrinsic behavior of a metallic material and Alucobond structure in terms of stress / strain curve. Emphasis will then be placed on the plate perforation test which makes it possible to reach extreme stress levels in terms of deformation and strain rates.
Then the complete ballistic curves of different kinds of studied materials as well as the absorbed energy will be analyzed. The results of these perforation tests will subsequently validate the constitutive relation that will be identified later in another section.
Finally, in the fourth chapter of this thesis, the behavior of Alucobond composite structure (composed of two material types) will be identified, polymers and metals undergoing a ballistic impact, the quasi-static tensile tests will be performed for four different strain rates, i.e. 0.0001, 0.001, 0.01 and 0.03𝑠−1. The quasi-static uniaxial tensile tests of Alucobond were performed using a conventional hydraulic machine.
Moreover, in dynamic testing the used strain rates range for compression and perforation are between 104 𝑠−1. ≤𝜀̇ ≤105 𝑠−1 at different temperatures ranging from room temperature to 300°C. in the same chapter a description of the numerical model that’s used to simulate the tests in quasi-static and dynamic modes was done using FE code “Abaqus”. The Johnson-Cook constitutive relation initially Implemented in the FE Abaqus Explicit code will be used to simulate the perforation and compression tests under adiabatic conditions.
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Chapter I Mechanical Behavior of Materials and structures
1.1. Introduction
The understanding of the mechanical behavior of a material subjected to high dynamic loads are found necessary to optimize the design of structures intended to undergo extreme loading conditions. To this end, numerous models aim to reproduce the observed phenomena by linking the various mechanical parameters (stress, strain rate, deformation, temperature, energy, etc.). These models, called behavior laws or constitutive relation, have a major application in computational codes for their role in predicting the thermomechanical behavior of materials. This makes it possible to simulate the mechanical response of structures under extreme loading such as impact and perforation.
The concept of this chapter consists in drawing up the state of the art of the behavior of metallic, polymers and composite materials subjected to a ballistic impact. Various parameters influence this behavior: the geometry and mechanical properties of the projectile, the impact velocity, the thermomechanical behavior of the target and the angle of penetration.
For this, a systematic analyze the influence of the various parameters mentioned above on the process of perforation of metal plates will presented. A summary of the principal constitutive relation of predicting the mechanical behavior of solid materials subjected to high dynamic stresses, these laws of behavior are the ones that will be implemented in the FE code calculations to simulate impact and perforation.
A brief description of some breakthrough criteria that are available in the literature (Wierzbicki, 2005) will be cited. In addition, the different numerical methods for simulating the perforation test will be described below.
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1.2. Strain rate sensitivity of metals
Numerous experimental studies highlight the variation of the flow stress of metallic materials in accordance with the strain rate deformation. The curves in Figure 4 below represent the results of the torsion tests carried out by Costin et al (Costin, 1979), using the Kolsky bar device, on tubular specimens of cold-rolled steel.
These results show that the behavior of the material is sensitive to the strain rate.
Indeed, the flow stress increases with a certain strain rate level. This figure also shows that for the dynamic loading test where 𝛾̇= 500 𝑠−1', the stress begins to decrease from a certain level of deformation. This decrease is explained by a thermal softening due to the heating of the material and its sensitivity to temperature Zener- Hollomon (Zener, 1944).
Figure 4: Effect of strain rate on the behavior of a thin 1018CRS steel tube in a torsion test, Costin et al (Costin, 1979)
Other studies highlighted the influence of strain rate on metal flow stress, to mention a few ; the work of Zener-Hollomon (Zener, 1944), Klepaczko (Klepaczko, 1969), Lindholm-Yeakley (Lindholm, 1968), Sensenye et al. (Senseny, 1976), Duffy (Duffy, 1980) and Regazzoni and Montheillet (Regazzoni, 1984), The experimental observations that are found in the literature often show differences between the effects of velocity on cubic-centered metals (CC) and face-centered cubic metals (FCC) .
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Figure 5: The influence of the strain rates on the yield stress of metallic materials with CC structure (Rosenfield, 1966)
This figure shows the existence of three strain rate domains. The first domain corresponds to slow loadings, the stress is barely sensitive or even insensitive to the strain rate. This zone is marked by the predominance of athermic deformation mechanisms. The second domain corresponds to a zone in which the variation of the stress as a function of the strain rate is almost linear. In this zone, the thermoactive mechanisms are predominant. The third domain corresponds to very fast loadings, the flow stress is strongly influenced by the strain rate. Probably, the viscous drag mechanisms are predominant and the metal behaves like a viscous fluid in this third and last zone.
1.3. Temperature sensitivity of metals
Much of the plastic deformation energy of a metal structure subjected to a given loading state dissolves as thermal energy. This can result in very high local elevations of temperature and, consequently, a decrease in the strength of the material. Indeed, several studies show a sensitivity of the flow stress to temperature (Zener-Hollornon (Zener, 1944), Eleiche-Campbel (Eleiche, 1976), Eleiche-Duffy (Eleiche, 1975), Chiem-Klepaczko (Klepaczko, 1986) among others). The curves in Fig. 6 below show the results of the torsion tests carried out at a strain rate of 103𝑠−1 by Eleiche and Campbell on mild steel. These results show that the flow stress is sensitive to temperature. This sensitivity, which is due to the evolution of the microstructure with temperature, depends on the nature of the deformation mechanisms.
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Figure 6: Temperature dependent of follow stress of mild steel at high strain rate (Eleiche, 1976)
1.4. Metallic structure behavior under dynamic loading
The perforation of plates by projectile impact is a complex process often revealing several phenomena such as elastic and plastic deformations, the effects of strain rate, thermal softening, crack formation, the adiabatic shear band, the formation of Plugs and petals, and even sparks. The most important parameters affecting the ballistic capacity of a target plate appear to be the projectile (geometry, density and hardness), the intrinsic properties of the perforated plate (hardness / strength, ductility, microstructure and thickness) such as impact velocity, impact angle and the state of the projectile / plate contact.
Before diving to the rest of this section, it is important to define the following expressions (Cailleau, n.d.):
Penetration
Interaction of a projectile with a target that leads to the formation or absence of a blind whole.
The perforation
Interaction of a projectile with a target, which leads to a craterization emerging from the target, with or without expulsion of elements (plug) from the target.
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Ballistic limit, VB
The ballistic limit of a target is the average of the highest impact velocity (projectile) that the target can withstand without being completely perforated and the lowest velocity allowing complete perforation of the target.
Residual velocity, VR
Residual velocity refers to the velocity of the projectile after complete perforation of the target (without ricochet).
Please noted that experimentally, the concepts of ballistic limit and residual velocity are difficult to implement with precision.
The test benches are often equipped with a compressed gas gun (air, nitrogen, etc.), a support for fixing the plate targets, and devices for measuring speeds.
Dynamic study laboratories working on the impact and perforation there are equipped with these gas launchers to impact targets at relatively high impact velocities (up to 180 m / s).
Depending on the study to be carried out, and the physical quantities to be measured, the test bench is equipped with appropriate devices in order to make reliable measurements.
1.4.1. The specimen (plate) properties Influence
Any type of manufactured products or natural materials (rubber, wood, glass ...) can be a target to be impacted. However, the targets used in surveys that are subject to the statutes thereafter almost exclusively aluminum and polymer grades. The metal plates are macroscopically considered homogeneous and isotropic.
Researchers have carried out many studies in order to make it possible to estimate the ballistic limit. In general, the ballistic limit of a structure is the highest velocity of the projectile that the structure can support without being fully perforated.
The precise definitions of this parameter vary according to the interpretation of the term "perforation", more details are available in (Goldsmith, 1978).
A target can break by several mechanisms. Zukas et al (Zukas, 1990) identifies five types of possible metal plate damage:
The cracking and fragmentation during which part of the fragile plate breaks up into several small pieces, this is the case in particular of high hardness plates and ceramics.
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The ductile rupture in which a part of the specimen energy is absorbed by the plate by deformation.
The localize adiabatic shear, which leads to the ejection of a plug, the deformation is strongly localized in the zone of impact and the plate absorbs little energy compared to the ductile rupture.
The chipping caused by the reflection of the shock wave on the rear surface of the plate.
The erosion of the material under the impact of hollow-charge projectile.
By varying, several parameters in order to analyze their influence on the mechanical behavior of the metal plate of the recent experimental work on the perforation have been carried out. Gupta et al. (Gupta, 2007) studied the influence of the thickness of the plate on the deformation of aluminum sheets impacted by blunt, conical and hemispherical projectiles with a diameter of 19 mm. The range of explored velocities varies from 20 m / s to 140 m / s. The ballistic curves showing the residual velocity as a function of the initial velocity of impact are shown in Fig.
7. The authors found that as the thickness of the target plate increases, it provides more resistance to the projectile. The smallest ballistic velocity of projectiles is observed in the case of impact on 0.5 mm thick plates. The ballistic limit increases with the thickness of the target. It is also observed that in the case of blunt and conical projectiles, the energy required to perforate the target plate is almost similar for a given thickness, but in the case of the hemispherical projectile, the plate absorbs more energy. The authors also compared the obtained results for monolithic plates and plates in a sandwich configuration (several thin plates of the same thickness put in contact side by side) with an equivalent total thickness. They found that the ballistic limit increases with an increase in the thickness of the target plate, whether in a monolithic or sandwich configuration
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Figure 7: Ballistic curve for different thicknesses of aluminum sheet. a- Blunt projectile; b- Conical projectile; c- Hemispherical projectile. (Gupta, 2007).
Jankowiak et al. (Jankowiak, 2013) Analyzed the influence of the elasticity limit of the material on the ballistic curve. For this, the authors studied the impact behavior of several metallic materials using a conical projectile. The ballistic curves of the various materials are shown in Fig.8.
A linear relationship has been established between the elastic limit and the ballistic boundary of the material. The authors have shown that the ballistic limit increases with the yield stress of the material. The influence of hardening of the material on the ballistic limit was also examined. It has been shown that, at the equivalent elastic limit, materials with a high strain hardening have a higher ballistic limit than materials with low hardening (Jankowiak, 2013).
27
Figure 8: (a) RK constitutive relation fitted to various materials; (b) Ballistic curve predictions for the materials studied (Jankowiak, 2013).
1.4.2. Characteristics of the projectile and its influence
The geometry of the projectile and more particularly the shape of the impacting end has a significant influence on the rupture mode and the ballistic limit of the plate to be punched. In 1978, Wilkins (Wilkins, 1978) presented some types of target fracture (thin plates) that depend on the nose shape of the projectile. The majority of the studied projectiles in the literature have a cylindrical shape and are differentiated by the geometry of the impacting end, the best known of which are the hemispherical, conical, Blunt, conical ends and sometimes a combination of two of these shapes.
The most commonly used materials for making these impactors are very often metals, including many iron bases ranging from mild steel to the most complex form of alloy steel, with a hardness up to 64 HRC (Rm ≃ 2500 MPa).
28
Figure 10: Experimental ballistic curve depending on the shape of the projectile, thickness plate t= 1 mm (Jankowiak, 2013)
Borvik et al. (Børvik, 2002) examined the behavior of 12 mm thick "Weldox 460E" steel plates, subjected to ballistic impact using blunt, hemispherical and conical projectiles. The explored range of velocities varied from 150 to 500 m.𝑠−1. They found from the experiments that flat projectiles are more efficient penetrators than hemispherical or conical projectiles at low impact velocities. However, at high impact velocities, the conical projectile requires less energy to perforate the target.
Ipson and Recht (Ipson, 1977) found that the blunt projectile penetrates the target plate more effectively than the conical projectile. Corran et al. (Corran, 1983) studied the effect of the shape of the end of the projectile on the penetration of steel and aluminum alloy plates. They used flat and conical projectiles 12.5 mm in diameter.
Figure 9: Experimentally observed failure patterns for different kinds of projectile, 𝑉0141 m/s and dry condition. (a)Hemispherical ;( b) conical; and(c) blunt. (Jankowiak, 2013)
29 The mass of the projectiles varies from 15 to 100 g and the impact velocity from 50 to 250 𝑚. 𝑠−1. It has been observed that the critical impact energy depends on the radius of the projectile. Recht and Ipson (Ipson, 1963) proposed an analytical law in the 1960s to model the ballistic curve; the model is based on the consideration of momentum and energy balance and is widely used in the literature.
𝑉𝑅= 𝑚𝑝
𝑚𝑝𝑙+ 𝑚𝑝(𝑉0𝑘− 𝑉𝐵𝑘)1⁄𝐾 Eq. I.1 Where 𝑚𝑝 the mass of the projectile and 𝑚𝑝𝑙 is the mass of the plug, 𝑉𝐵 is the ballistic limit and 𝑉0 is the initial velocity, 𝑘 is the ballistic curve shape parameter.
Iqbal et al. (Gupta, 2008) analyzed the influence of the projectile’s shape on the energy absorption capacity of perforated aluminum plates by blunt, conical and hemispherical projectiles. They have shown that the aluminum plate absorbs more energy when it is impacted by a hemispherical projectile followed by the flat and conical projectiles respectively.
Gupta et al. (Gupta, 2007) studied the effect of projectile shape, impact velocity and target thickness on the deformation behavior of aluminum plates impacted by blunt, conical and hemispherical projectiles. They found that for plates of 1mm thickness, the ballistic speed limit is greater for the hemispherical projectile followed by blunt and conical projectiles. Different modes of plate rupture were observed depending on the shape of the projectile. For the flat projectile, the rupture of the target occurs by shearing and ejection of a circular plug of diameter equal to that of the projectile. The perforation by the conical projectile causes the formation of petals in the plate. The hemispherical projectile causes a rupture of the target material by traction with reduction of the thickness of the plate in the contact zone and then ejection of a plug.
Forrestal et al. (Warren, 2009) and Chen et al. (Chen, 2003) proposed new analytical models to describe ballistic curves. However, these two analytical models are valid only for the study of the perforation of ductile metallic plates by rigid pointed projectiles contrary to the model of Recht and Ipson, which adapts to all forms of projectiles.
This review of the projectile shape studies shows the importance of the shape of the impacting end of the projectile on the perforation process.
30
1.5. Mechanical behavior of structures
1.5.1. Quasi-static behaviorA large number of experimental, numerical and analytical cellular materials quasi-static behavior studies are reported in the literature (reviewed in (Gibson, 1999), (Zhao, 2004)). In general, a typical stress versus quasi-static strain curve of a cellular material has a weak elastic zone at the beginning of the loading, then a near- perfect plastic plateau, which corresponds to the successive collapse of the skeleton walls until the densification zone. For this strain, the stress rapidly increases and approaches in an asymptotic manner the Young's modulus of the material constituting the skeleton as indicated in the Fg.11. (Elnasri, 2006). The behavior of cellular materials under quasi-static loading can also be estimated from skeletal base material data, and relative density from a micromechanical analysis (Gibson, 1999);
(Zhao, 2004).The geometric microstructure of cellular materials can be characterized by the image tomography technique (Maire, 2003), allowing to deduce the behavior of the foam (Fazekas, 2002).
Figure 11: Typical stress-strain curve of a cellular material under compression quasi-static loading.
(Zhao, 2005)
31 1.5.2. Dynamic behavior
At low impact velocities (< 40 m / s), the tests found in the literature seem to show a low sensitivity of the behavior to the velocity for certain sandwich structure and cellular materials. It can be attributed to the effect of micro-inertia, or the velocity sensitivity of the base material (Klinworth, 1988), (Kenny, 1996), (Mukai, 1999);
(Zhao, 1998); (Deshpande, 2000), (Zhao, 2004), (Zhao, 2005).
Figure 12: Strength enhancement under impact loading (Zhao, 2005).
Fig.12 shows the sensitivity of aluminum honeycombs (5052, 5056) of different types and densities, in accordance to the impact velocity under compression that is applied along the axis of the hexagonal cells. An increase in stress of about 15% is observed. It should be noted that the behavior of the honeycomb impacted along the slice axis is identical to the quasi-static case (Zhao, 1998).
32
Figure 13: Strength enhancement under impact loading for foams. (Zhao, 2005).
Fig.13 shows the velocity sensitivity of IFAM and Cymat type foams.
Unlike honeycomb, foams are usually isotropic and heterogeneous.
The density of the tested IFAM foam is about 620 Kg / 𝑚3 and that of Cymat is about 250 kg / 𝑚3. An increase of 18 % in stress is observed for the IFAM foam while no increase was observed for the Cymat foam.
Figure 14: Strength enhancement under impact loading for hollow spheres (Zhao, 2005).
33 Fig.14 shows the velocity sensitivity of the hollow spheres agglomerates of two basic materials (steel, diameters 3 and 1.5 mm, and nickel, diameter 2.5 mm). There is an increase in the average impact stress of 57% for steel spheres and 17% for nickel spheres.
1.6. Simulation
The Abaqus code is a finite element code whose Lagrangian formulation is used in this work. Two versions of the code are available for the users, to choose according to the application:
1. The first, called standard version, uses the Newmark scheme (Abaqus, 2014) as a time-integration algorithm. Since this scheme is implicit, it leads to a system of nonlinear kinematic equations whose resolution requires the use of iterative methods such as the Newton-Raphson method (Abaqus, 2014) or, where possible, one of the equivalent methods to reduce the calculation time (Abaqus, 2014). This version is more used to simulate applications involving a quasi static loading of the material.
2. The second version, called explicit version, uses as its name suggests an explicit integration scheme over time. This scheme, which corresponds to the fine-centered difference method (Abaqus, 2014), leads to a system of linear equations that requires no iterative process to be resolved The stability condition of this method is related to the time increment, which must be below a certain critical value. This value is determined by the time that the elastic dilation waves take to travel the length of the smallest element in the used mesh :
∆𝑡 ≤ ∆𝑡𝑐𝑟 = min (𝐿𝑒
𝐶0) Eq. I.2
Where 𝐿𝑒 being the shortest length characteristic of the element, and 𝐶0 the velocity of the elastic expansion waves. This version is suitable for simulating dynamic loading applications.
Examples of the use of numerical techniques to analyze the general problem of penetration and ballistic impact perforation are reported in the work of Backman and Goldsmith (Goldsmith, 1978), Jonas and Zukas (Zukas, 1978) , Wilkins (Wilkins, 1978) and Gupta et al. (Gupta, 2007). The latter also contain extensive bibliographies on the subject.
34
Figure 15: Schematic representation of tensile test data in stress-displacement space for elastic- plastic materials (Abaqus, 2014)
For example In Fig. 15 the progressive damage model is used in this work for the aluminum alloy (Bendarma, 2017). The description includes the elastic part with 𝐸0 (part a-b), the plasticity range (b-c). The damage initiation with JC criterion can be expressed by Eq. I23 (c). Along the line (c-e), the damage variable evolution grows from 0 to a maximum degradation ratio 𝐷𝑚𝑎𝑥 (d) therefore, the stiffness of the material is degraded and reduced to (1 − 𝐷) 𝐸0 where D is the damage variable and E0 is the initial Young modulus. The damage evolution is described by the mesh- independent measurements (displacement at failure and damage energy dissipation) in the model. A linear evolution damage rule is used by defining a value of displacement at failure 𝑢𝑓 (e). Thus, the maximum stiffness degradation as well as the maximum damage have been taken finally as failure criterion (d). After reaching the failure criterion, the element is deleted from the mesh in simulation.
Gupta et al. (Gupta, 2007) performed a numerical analysis of the perforation problem using the ABAQUS Explicit finite element code. They created an axially symmetrical geometric model of the projectile and the target plate in the preprocessing module code. The target plate was modeled as a deformable body and the projectile as a rigid body with a single reference node to assign mass and initial velocity. The effect of friction between the projectile and the target is neglected. The prediction of the residual velocities and the target plates rupture modes are in adequacy with the corresponding experimental results (Fig. 16).
35
Figure 16: Comparison of the numerical and experimental ballistic curves. a-Blunt projectile; b- Conical projectile; C-Hemispheric projectile. (Gupta, 2007)
Several constitutive relationships have been proposed to model the plastic flow of metallic materials under impact loading for use in numerical simulation. In these constitutive models, the thermomechanical behavior of a material is represented by a mathematical expression linking the quantities such as stress, strain, strain rate, temperature and structure. A description of some of the used behavior models in the literature will later be presented.
36 1.6.1. Constitutive relation taking into account the strain rates and temperature
In the context of studying materials and metallic structures properties, one speaks of dynamic regime for strain rates greater than 1𝑠−1. The dynamic regime is known to be slow for strain rates of 1 to 10𝑠−1 it is then close to the quasi-static case.
When the strain rate is between 10 and 1000𝑠−1 one speaks of an average dynamic regime, for example, in the automotive industry, strain rate reached are of the order of 100 𝑠−1for an impact at 60 km / h (see Cunat (Cunat, 2000)). In this type of medium velocity test, the impact velocities are at most 100 m / s. The time scale of these phenomena is in milliseconds. Beyond 1000𝑠−1, it is the fast dynamics regime.
The time scale is then of the order of microseconds. This is the case of ballistics and explosions, these applications are mainly in the military domain and the test velocities are then of the order of km / s. The focal point of this work is the study of the phenomena between 10 and 10000 𝑠−1 .
There are two major families of constitutive relations: phenomenological or empirical models (eg, Johnson-Cook (Johnson, 1983), Cowper-Symonds (Symonds, 1957) or Zhao (Zhao, 1997)) and physical Zerilli-Armstrong (Armstrong, 1987), Bodner and Partom (Partom, 1975) or Rusinek and Klepaczko (Klepaczko, 2001)), which take into account microscopic phenomena such as grain size, crystal structure or dislocation structure.
In case the strain rate is taken into account in the evaluation of the elastic limit, the plasticity criterion must be modified accordingly. Langrand et al. (Langrand, 1999) have studied various types of laws including the strain rate. They divided these laws into three categories:
The empirical constitutive relations, obtained by an experimental procedure whose formulation is more or less complex, the constitutive relations of viscous hardening of additive type and the constitutive relations of viscous hardening of the multiplicative type.
1.6.2. Dynamic constitutive relations
In this section, the constitutive relation used specifically to describe a fast phenomenon has been focused on, whose strain rates are typically in the order of 100 to 1000𝑠−1. The models that apply to large plastic deformations and high strain rates will be described. These models widely used for numerical simulation in different fields for example: military applications, machining processes, aeronautics, automotive and shipbuilding. Physical and empirical models are the simplest and most used because they are based on experimental observations.
37 1.6.2.1. Physical constitutive relation
Physical models use more physical considerations, in particular by making assumptions about the nature of the deformation mechanisms involved. Among the most elaborate are those of de Rusinek-Klepaczko (Klepaczko, 1988) and Zerilli- Armstrong (Armstrong, 1987).
a- Rusinek-Klepaczko’s model
The model proposed by Rusinek-Klepaczko (RK) for BCC microstructure is based partly on the theory of dislocations and its phenomenological formulation adopts the additive decomposition of the total stress (Klepaczko, 1987):
𝜎(ԑ, 𝜀̇, 𝑇) =𝐸(𝑇)
𝐸0 [𝜎µ(ԑ, 𝜀̇, 𝑇) + 𝜎∗(𝜀̇, 𝑇)] Eq. I.4 Where 𝜎µ represents the internal stress, 𝜎∗ is the effective stress and 𝐸0 is the Young's modulus at 𝑇 = 0 K. The first term is directly related to the hardening of the material and the second term defines the effects of thermal activation (instantaneous sensitivity to deformation rate and temperature).
The multiplier factor that is in front of the sum of the stresses, E (T) /𝐸0, defines the evolution of the Young modulus with the temperature (Klepaczko, 1988):
𝐸(𝑇) = 𝐸0(1 − 𝑇
𝑇𝑚𝑒𝑥𝑝 [𝜃∗(1 −𝑇𝑚
𝑇]) 𝑇 > 0 Eq. I.5 Where 𝑇𝑚 and 𝜃∗denote respectively the melting temperature and the homologated temperature characteristic of the material. This expression makes it possible to define the thermal softening as a function of the crystal lattice of the material (Rusinek, 2009). This factor is important in dynamics for the study of wave propagation.
The internal component of the flow stress is defined by the following equation:
𝜎𝜇(ԑ, 𝜀̇, 𝑇) = 𝐵(𝜀̇, 𝑇)(ԑ0+ ԑ)𝑛(𝜀̅ ̇,𝑇) Eq. I.6
Where 𝐵(𝜀̇, 𝑇) is the modulus of plasticity which is dependent on the strain rate and temperature, 𝑛(𝜀̇, 𝑇) is the coefficient of hardening, it depends on the strain rate and temperature, ԑ0 is the deformation value corresponding to the elastic limit for a specific deformation velocity and temperature.
T. Børvik, O. S. Hopperstad and T. Berstad (Børvik, 1999), propose the following formulations to describe the plasticity modulus and coefficient of hardening:
38 𝐵(𝜀̇, 𝑇) = 𝐵0((𝑇
𝑇𝑚) log (𝜀̇𝑚𝑎𝑥 𝜀̇ ))
−𝑣
𝑇 > 0 𝒏(𝜀̇, 𝑇) = 𝑛0〈1 − 𝐷2(𝑇
𝑇𝑚) 𝑙𝑜𝑔 ( 𝜀̇
𝜀̇𝑚𝑖𝑛)〉
Eq. I.7
With 𝐵0 a material parameter, 𝑣 is the sensitivity to temperature 𝑛0 is the Coefficient of hardening at 𝑇 = 0 K, 𝐷2 is a parameter related to strain hardening, 𝜀̇𝑚𝑖𝑛 and 𝜀̇𝑚𝑎𝑥 are the limits of the model in terms of deformation velocity.
Rusinek and J. R. Klepaczko, (Klepaczko, 2001) deduced the following expression, (Børvik, 1999). This formulation takes into account the link between the deformation velocity and temperature through the Arrhenius equation:
𝝈∗(𝜀̇, 𝑇) = 𝜎0∗〈1 − 𝐷1(𝑇
𝑇𝑚) 𝑙𝑜𝑔 (𝜀̇𝑚𝑎𝑥
𝜀̇ )〉𝑚∗ Eq. I.8
Where 𝜎0∗ is the effective stress at 𝑇 = 0 K (it is related to the mechanical threshold stress MTS), 𝐷1 is a constant of the material, 𝑚∗ translates the sensitivity to the strain rate (Klepaczko, 1987) .
Under adiabatic loading conditions, the law of behavior is combined with the heat equation (Atkins, 1998), allowing to take into account the thermal softening due to adiabatic heating (without heat transfer, k=0):
∆𝑇(ԑ, 𝜎) = 𝛽
𝜌𝐶𝑝∫ 𝜎(ԑ, 𝜀̇, 𝑇)𝑑ԑ
𝛿𝑚𝑎𝑥 0
Eq. I.9
where 𝛽 denotes the Quinney-Taylor coefficient, 𝜌 is the density of the material and 𝐶𝑝 is the specific heat of the material. The transition from isothermal to adiabatic conditions occurs for 𝜀̅̇𝑝=10 𝑠−1in the case of steels.
Rusinek and Martinez (Rusinek, 2009) studied the behavior of aluminum alloys at wide ranges of strain rate and temperature to describe the behavior two aluminum alloys (AA 5083-H116 and AA 7075).
In the following plot, Fig.17, experimental data are compared with analytical predictions of both models for two different strain rate levels at room temperature.
In the case of 𝜀̅̇𝑝 =3.95 𝑠−1 the extended RK model fit properly the experimental data. The Modified JC model overestimates the flow stress and the strain hardening of the material. In the case of 𝜀̅̇𝑝=1313 𝑠−1, both models offer predictions close to the experimental data and the thermal softening due to adiabatic heating is well defined.
39
Figure 17: Description of the flow stress evolution along plastic strain using extended RK and MJC models and comparison with experiments at room temperature (Clausen, 2004). (a) 3.95 𝑠−1, (b) 1313 𝑠−1.
In this work the first step was evaluating the predictions extended RK model of the for different strain rate levels. It is reported in Fig.18 a satisfactory agreement between the model and the experiments from quasi-static loading to high strain rate 0.001𝑠−1 ≤ 𝜀̅̇𝑝 ≤ 2529 𝑠−1.The difference only takes place after saturation stress stage d𝜎̅/d𝜀̅̇𝑝= 0 which corresponds to non-homogeneous behavior (Rusinek, 2009).
Figure 18: Description of the flow stress evolution along with plastic strain using extended RK model and comparison with experiments at room temperature (El-Magd, 2006). (a) 0.001s−1, (b) 1s−1, (c) 10s−1, (d) 2529s−1.
40 b- The Zerilli-Armstrong model
In 1987 Zerilli and Armstrong made a complete demonstration of a physical model that is presented in (Armstrong, 1987).The relationship is expressed differently for materials with a centered (T. Borvik, 2003) and a face-centered (Arias, 2008) cubic structure (Eq.I.10 for FCC microstructure and Eq.I.11 for BCC microstructure):
𝜎 = 𝜎0+ 𝜎1𝜀𝑛+ 𝐶1exp(−𝐶3𝑇 + 𝐶4𝑇𝑙𝑛(𝜀̇)) + 𝑘𝜀𝑑−1/2 Eq. I.10
𝜎 = 𝜎0+ 𝐶2𝜀1/2exp(−𝐶3𝑇 + 𝐶4𝑇𝑙𝑛(𝜀̇)) + 𝑘𝜀𝑑−1/2 Eq. I.11
where 𝜎0 designates the yield stress, 𝜎1 and n are strain-hardening parameters: 𝐶1 , 𝐶2 , 𝐶3 , 𝐶4 are model coefficients, and 𝑘𝜀 is the average grain diameter of the material.
These equations make it possible to take into account, in a coupled manner, the effect of the work hardening, the strain rate and the temperature on the flow stress of the material.
However, they neglect the effect of the temperature and the rate of deformation on the coefficient of hardening n. Indeed, it has been observed experimentally (Klepaczko, 1988), (Klepaczko, 2009) that the coefficient of hardening decreases when the strain rate of the material is increased.
A comparison between the Zerelli-Amstrong and JC model will be presented and discussed in the following section.
1.6.2.2. The empirical constitutive relation
The empirical models are directly related to the analysis and exploitation of experimental data.
a. The model of Klopp, Clifton and Shawki
Klopp, Clifton and Shawki integrates the influence of temperature in their model (Klopp, 1985) apart from the influence of the deformation velocity:
𝜎 = 𝑘𝜀𝑛𝜀̇𝑚𝑇−𝑣 Eq. I.12
where 𝐾 is a constant of the material, 𝑛 is the coefficient of hardening, 𝑚 is the coefficient of sensitivity to the strain rate and 𝑣 is the thermal softening Parameter.
These constants will be the easier to determine sins the carried out tests allow decoupling the three variables independently.
41 b. The Lindholm model
This model (Lindholm, 1964) connects the stress 𝜎 to the deformation 𝜀 and to the strain rate 𝜀̇ by the following relation:
𝜎 = 𝜎0+ 𝜎1𝜀𝑛+ (𝜎2𝜀 + 𝜎3)ln ( 𝜀̇) Eq. I.13 Where 𝜎0 represents the elastic limit, 𝜎1 is a material parameter, 𝜎2 and 𝜎3 are related to the material’s strain rate sensitivity, and n corresponds to the coefficient of hardening. This constitutive relation involves the strain rate, but not the temperature.
It therefore does not make it possible to reproduce the behavior of the material under the conditions of stress in temperature. Therefore, it cannot be used for dynamic process simulation due to thermal softening.
c. The Johnson-Cook model
Johnson and Cook propose an empirical law (Johnson, 1983), (Johnson, 1985) based on experimental results and designed for rapid implementation in calculation codes. This model is based on that of Ludwik (Ludwik, 1909) and includes the influences of deformation velocity; strain hardening and temperature (see fig.19):
𝜎 = (𝐴 + 𝐵𝜀𝑝𝑙𝑛) (1 + 𝐶 𝑙𝑛𝜀̅̇𝑝
𝜀̇0) (1 − 𝑇∗𝑚) Eq. I.14 Where 𝐴 is the yield stress, 𝐵 and 𝑛 are the strain hardening coefficients, 𝐶 is the strain rate sensitivity coefficient, 𝜀̇0 is strain rate reference value and 𝑚 is the temperature sensitivity parameter. In this work, isothermal conditions are assumed.
Therefore, the last term of the JC model related to the non-dimensional temperature where 𝑇∗ is the homologous temperature.
In the context of this work’s applications, a model has been chosen, it is an empirical model from Johnson-Cook to describe the behavior of the studied materials, the parameters will be presented in chapter III of this thesis.
Using this model, it is necessary to associate a failure criterion of the structure in order to model the mechanical behavior of structures subjected to a ballistic impact.
42
Figure 19: True stress-strain relationships using JC model for different materials. (Johnson, 1985)
In order to characterize the constitutive response of AA5754, AA5182 and AA6111 at high strain rates, (Smerd, 2005) performed a tensile experiments tests at strain rates between 600 𝑠−1 and 1500 𝑠−1, and at temperatures between ambient and 300°C using (TSHB) tensile split Hopkinson bar, these experimental data was fit to the Johnson-Cook and Zerilli-Armstrong constitutive models for all three alloys. The resulting fits were analyzed by numerically simulating the tensile experiments conducted using a finite element approach. Of the two models, the Zerilli- Armstrong constitutive model was more accurate in predicting the flow stress of these materials at the strain rates and temperatures considered.
