A Klepaczko

C Arias

Sauvelet

A Rusinek
email: rusinek@lpmm.univ-metz.fr

J A Rodríguez-Martínez

R Zaera

J R Klepaczko

A Arias

C Sauvelet

Experimental and numerical study on the perforation process of mild steel sheets subjected to perpendicular impact by hemispherical projectiles

Keywords: Numerical simulation, Perforation, Petalling, Ductile failure, RK model, Dynamic behavior

Experimental and numerical study on the perforation process of mild steel sheets subjected to perpendicular impact by hemispherical projectiles

T P I R C S U N A M D E T P E C C A

-Introduction

The response of materials under dynamic loading has a considerable interest. It allows for clarification of several problems in different application fields such as civil, military, aeronautical and automotive engineering. In particular, many studies on behavior of steels subjected to high strain rate concentrated in the past a large amount of efforts [START_REF] Campbell | The yield of mild steel under impact loading[END_REF][START_REF] Abramowicz | Dynamic axial crushing of square tubes[END_REF][START_REF] Abramowicz | Dynamic axial crushing of circular tubes[END_REF][START_REF] Thornton | The dynamic buckling of sheet steel[END_REF][START_REF] Klepaczko | An experimental technique for shear testing at high and very high strain rates. The case of mild steel[END_REF][START_REF] Li | Response and failure of a double-shear beam subjected to mass impact[END_REF][START_REF] Teng | Numerical prediction of fracture in the Taylor test[END_REF][START_REF] Tarigopula | Axial crushing of thin-walled high-strength steel sections[END_REF][START_REF] Rusinek | Finite element simulation of steel ring fragmentation under radial expansion[END_REF].

The extreme case of a material subjected to high strain rate solicitation is generally observed during impact or explosion. Thus, in many cases the strain rate level observed in a structure can be higher than s -1 . In addition, it is observed locally a strong temperature increase by adiabatic heating which triggers a thermal softening of material.

A special interest has been focused on the perforation resulting from an impact between non-deformable projectiles and metallic plates, [START_REF] Piekutowski | Holes produced in thin aluminium sheets by the hypervelocity impact of aluminium spheres[END_REF][START_REF] Forrestal | Penetration experiments with 6061-T6511aluminum targets and spherical-nose steel projectiles at striking velocities between 0.5 and 3.0 km/s[END_REF][START_REF] Borvik | Perforation of 12 mm thick steel plates by 20 mm diameter projectiles with flat, hemispherical and conical noses Part I: Experimental study[END_REF][START_REF] Borvik | Perforation of 12 mm thick steel plates by 20 mm diameter projectiles with flat, hemispherical and conical noses Part II: Numerical study[END_REF][START_REF] Borvik | Effect of target thickness in blunt projectile penetration of Weldox 460 E steel plates[END_REF][START_REF] Gupta | Experimental and numerical studies on the behaviour of thin aluminium plates subjected to impact by blunt -and hemispherical -nosed projectiles[END_REF][START_REF] Gupta | Effect of projectile nose shape, impact velocity and target thickness on deformation behaviour of aluminium plates[END_REF][START_REF] Arias | Numerical simulations of impact behaviour of thin steel to cylindrical, conical and hemispherical non-deformable projectiles[END_REF][START_REF] Cheshire | Perforation of sheets by pyramidal weapons such as arrowheads[END_REF]. During such kind of impact loading, petalling as a failure mode commonly appears when ogival, conical or hemispherical projectiles are applied, [START_REF] Edwards | The ballistic properties of tool steel as a potential improvised armour plate[END_REF][START_REF] Atkins | Necking and radial cracking around perforations in thin sheets at normal incidence[END_REF][START_REF] Wierzbicki | Petalling of plates under explosive and impact loading[END_REF][START_REF] Shen | A study on the failure of circular plates struck by masses. Part 1: experimental results[END_REF][START_REF] Lee | Fracture prediction of thin plates under localized impulsive loading. Part II: discing and petalling[END_REF].

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The failure mode appears to be strongly dependent on the impact velocity. Petalling can be replaced by failure mode of crack opening when impact velocity is close to the ballistic limit. In this situation a decrease of the circumferential strain slows the crack progression [START_REF] Shen | A study on the failure of circular plates struck by masses. Part 1: experimental results[END_REF]. Moreover, when the impact velocity is very high, the perforation process is governed by inertia effects and the failure mode changes from petalling to complete fragmentation of the zone affected by impact, inducing appearance of debris cloud as final stage [START_REF] Piekutowski | Holes produced in thin aluminium sheets by the hypervelocity impact of aluminium spheres[END_REF][START_REF] Piekutowski | Debris clouds generated by hypervelocity impact of cylindrical projectiles with thin aluminium plates[END_REF]. Teng et al. [START_REF] Teng | Numerical prediction of fracture in the Taylor test[END_REF] have also observed the influence of the impact velocity on the failure mode during Taylor tests performed with Weldox 460 E steel cylinders. At relatively low impact velocity no external cracks appeared in the specimen impacted. On the contrary, for high initial impact velocity formation of several radial cracks were observed propagating rapidly and causing formation of petals.

Therefore, to define properly the transition between these different failure modes appearing when hemispherical projectiles are used, it is necessary to define precisely material's behavior since all these processes are strongly depend upon strain hardening p ε , strain rate p ε & and temperature increase T responsible for thermal softening.

In order to define the behavior of steel under dynamic and complex state of stress, several constitutive relations can be found in the international literature [START_REF] Cowper | Strain hardening and strain rate effects in the impact loading of cantilever beams[END_REF][START_REF] Bodner | Constitutive equations for elastic-viscoplastic strain-hardening materials[END_REF][START_REF] Steinberg | Constitutive model for metals applicable at high strain rates[END_REF][START_REF] Nemat-Nasser | On finite deformation elastoplasticity[END_REF][START_REF] Johnson | A constitutive model and data for metals subjected to large strains, high strain rates and high temperatures[END_REF][START_REF] Zerilli | Dislocation-mechanics-based constitutive relations for material Dynamics calculations[END_REF][START_REF] Khan | Continuum Theory of Plasticity[END_REF][START_REF] Rusinek | Shear testing of sheet steel at wide range of strain rates and a constitutive relation with strain-rate and temperature dependence of the flow stress[END_REF]. A precise analysis concerning this topic can be found in the works of Liang and Khan [START_REF] Liang | A critical review of experimental results and constitutive models for BCC and FCC metals over a wide rangeof strain rates and temperatures[END_REF] and Rusinek et al. [START_REF] Rusinek | Constitutive relations in 3-D for a wide range of strain rates and temperatures -Application to mild steels[END_REF].

In present work an experimental and numerical analysis of the impact behavior of sheets of mild steel subjected to perpendicular impact by hemispherical projectiles are reported. The experiments have been carried out at the Laboratory of Physics and Mechanics of Materials (LPMM) of Metz University using a Hopkinson tube as a transducer to measure the transmitted force [START_REF] Boumaaiz | Punching by impact, case of mild steel[END_REF]. Both lubricated and dry conditions have been applied to the contact between hemispherical projectile and specimen inducing different failure modes for each case. Thus, for lubricated condition petalling process was observed, while the petalling was reduced to a circumferential notch for dry condition.

The finite element code ABAQUS/Explicit has been used to simulate the impact process. An axi-symmetric mesh configuration is commonly used to model this kind of penetration problems, mainly in order to reduce large computational time due to small element size required. However, this simplification does not allow reproducing precisely the failure mode discussed previously since petalling is not a symmetric

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process of failure, Fig. 1. In the present case, the problem was solved by 3D simulations allowing a complete analysis of the problem.

Three constitutive relations, Johnson-Cook (JC), Rusinek-Klepaczko (RK) and a power of strain hardening (PL) [START_REF] Klepaczko | A practical stress-strain-strain rate-temperature constitutive relation of the power form[END_REF] have been used to define the plastic behavior of material. The use of different constitutive models allows for evaluation of the influence of material's definition on the dynamic response of plates and on simulation of the failure modes. The analysis has been focused on the failure mode definition depending on dry or lubricated conditions which were applied. A wide range of impact velocities was assumed.

-Experimental setup

To analyze failure behavior of a steel sheet subjected to normal impact for a maximum velocity of s / m 100 V max 0 ≈ , an original experimental set-up has been developed [START_REF] Boumaaiz | Punching by impact, case of mild steel[END_REF][START_REF] Rusinek | Modélisation thermoviscoplastique d'une nuance de tôle d'acier aux grandes vitesses de déformation. Etude expérimentale et numérique du cisaillement, de la[END_REF][START_REF] Segreti | Experimental study on puncture of PMMA at low and high velocities, effect on the failure mode[END_REF]. This experiment is based on the RM Davies bar concept by application of Hopkinson tube, Fig. 2. With that arrangement it is possible to determine the force ) t ( F applied to the specimen during perforation process through the measurement of the transmitted elastic wave ) t ( T ε .

Moreover, black-white stripes cemented on the projectile surface enabled to record large displacements imposed onto the steel sheet specimen p δ during perforation. The details are given below. Two optical fibers transmit the light and the third one transmits the reflected light from the projectile stripes to a photodiode. A system with three optical fibers coupled with two time counters enables to determine the impact velocity V 0 at the instant of impact. The acceleration/deceleration of the projectile can be determined with precision. The projectile used in this configuration had a hemispherical shape with a diameter of mm . During the test it is possible to prepare dry or lubricated conditions using in the last case several layers of grease and Teflon foil, [START_REF] Rusinek | Modélisation thermoviscoplastique d'une nuance de tôle d'acier aux grandes vitesses de déformation. Etude expérimentale et numérique du cisaillement, de la[END_REF].

The time history of the force ) t ( F transmitted by the specimen support into the tube is defined by the following relation: The displacement imposed during perforation is obtained by decoding the signals from the photodiode in the form of maxima corresponding to the white stripes on the projectile, Fig. 3. The width of white and black stripes can be chosen by user, typical range from 0.1 mm. to 0.5 mm. This principle is based on the frequency coding of displacement [START_REF] Klepaczko | Displacement gauge with a photodiode[END_REF] The total displacement is obtained using the following equation, Eq. 2, with the time signal, Fig. 3 ( )

( ) ( ) ) t ( 4 d D E t F T 2 2 ε - π = (1)
T P I R C S U N A M D E T P E C C A ARTICLE IN PRESS -4/
∫ ς ς ε - λ = δ t 0 T 0 p d ) ( C n t ( 2
)
where n is the number of maxima measured during the process of perforation, Fig. 3, and 0 C is the elastic wave velocity in the tube. Details of this experimental technique are reported in [START_REF] Boumaaiz | Punching by impact, case of mild steel[END_REF][START_REF] Rusinek | Modélisation thermoviscoplastique d'une nuance de tôle d'acier aux grandes vitesses de déformation. Etude expérimentale et numérique du cisaillement, de la[END_REF][START_REF] Segreti | Experimental study on puncture of PMMA at low and high velocities, effect on the failure mode[END_REF] for low and high velocities using a fast hydraulic machine and a Hopkinson tube technique.

Typical experimental results in the form specimen pictures and F(t) records are shown in Fig. 4. The cases shown in Fig. 4 are for the impact velocities above the ballistic limit. The comparison shows substantial differences in specimen behavior in terms of failure using dry or lubricated conditions. In the case of dry condition a plug ejection is observed during loading, Fig. 4-c Using experimental results, a numerical study is performed to analyze constitutive relation and failure mode effects.

-Modeling of dynamic behavior of mild steel and implementation into FE code

Mild steel ES is a ferritic steel with an average grain size of m 16μ = φ

. It has a particular interest since many results are available in the international literature permitting to identify material constants in any constitutive relation -see Appendix A - [START_REF] Klepaczko | An experimental technique for shear testing at high and very high strain rates. The case of mild steel[END_REF][START_REF] Marsh | The effect of strain rate on the post-yield flow of mild steel[END_REF][START_REF] Bonorchis | The effect of welded boundaries on the response of rectangular hotrolled mild steel plates subjected to localised blast loading[END_REF]. The behavior of the mild steel is assumed as a reference since it has been tested by a number of laboratories in recent decades. Typical mild steel assumed in this study is characterized below (ES steel). The chemical composition in weight of this material is reported in Tab. 1. It must be noticed that all specimens used to perform the tests of characterisation and perforation were machined from the same plate. Different

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tests have been performed with this material in tension, shear and perforation at different strain rates, temperatures and also loading paths. As consequence of the experiments it was concluded that the material presented isotropic behaviour

To approximate the dynamic behavior of that material several constitutive relations can be used depending on the application and the required information. Thus, in this paper three constitutive relations have been applied: two phenomenological and one semi-physical. The goal was to analyze theirs effects on the prediction of the failure process during impact perforation using the same failure criterion.

-Phenomenological approach I

The first constitutive relation used in this work to analyze perforation process is a power law (PL) which takes into account strain hardening p ε , strain rate p ε & and temperature sensitivity T . The explicit formulation of the constitutive relation is defined by:

( ) ( ) ν - ε ε = ε ε σ T B ) T , , ( m p n p p p & & (3)
where B is the material constant characterizing of the stress level, n, m and ν are respectively the hardening exponent, strain rate and temperature sensitivities. Although non-linear strain rate sensitivity is considered, this constitutive equation cannot be used within a complete spectrum of strain rates and values of m must be decomposed in several ranges, Fig. 5. Thus, in the present study two different sets of constants have been used, the first one corresponding to low strain rates (PLI) and the second to high strain rates (PLII) (Appendix A). To compare FE analyses with application of this simple phenomenological approach, another phenomenological approach have been used as it is discussed in the next part of this paper.

-Phenomenological approach II

The second thermo-visco-plastic constitutive relation is the one due to Johnson-Cook (JC), frequently applied to analyze the dynamic behavior of materials [START_REF] Gupta | Experimental and numerical studies on the behaviour of thin aluminium plates subjected to impact by blunt -and hemispherical -nosed projectiles[END_REF][START_REF] Gupta | Effect of projectile nose shape, impact velocity and target thickness on deformation behaviour of aluminium plates[END_REF][START_REF] Arias | Numerical simulations of impact behaviour of thin steel to cylindrical, conical and hemispherical non-deformable projectiles[END_REF][START_REF] Borvik | Ballistic penetration of steel plates[END_REF][START_REF] Smerd | High strain rate tensile testing of automotive aluminium. alloy sheet[END_REF][START_REF] Kobayashi | Plastic flow behaviour of Inconel 718 under dynamic shear loads[END_REF]. This constitutive relation is generally pre-implemented in FE codes, including ABAQUS/Explicit. The JC constitutive relation is defined by Eq.4. The first term defines strain hardening p ε , the second strain rate sensitivity, p ε & via the constant C and the third one is related to thermal softening T , Eq.4 and 5.

T P I R C S U N A M D E T P E C C A ARTICLE IN PRESS -6/49 - [ ] ) 1 ( ) ln( C 1 ) ( B A ) T , , ( m 0 p n p p p Θ - ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ε ε + ε + = ε ε σ & & & (4) 0 m 0 T T T T - - = Θ (5)
where A and B are material constants, n is the strain hardening exponent, m is the temperature sensitivity, T 0 is the initial temperature and T m is the melting temperature.

As this material shows non-linear strain rate sensitivity, Fig. 5, it is not possible to define it correctly with one set of parameters by Eq.4. Thus, the strain rate sensitivity must be defined separately in JC equation in several ranges. In present work, as previously proposed for PL, two ranges have been used, one corresponding to low strain rates (JCI), the second one for high strain rates (JCII) (Appendix A). A comparison is reported in Fig. 5. By using these two different strain rate constants in JC equation it is possible to define the strain rate behavior of the material from quasi-static to dynamic loading. The two values of the rate sensitivity C were used during numerical simulations to analyze how the value of the strain rate sensitivity can modify the results in terms of force level, residual velocity, ballistic limit and failure process.

It must be notice that recently the JC constitutive relation, Eq. 6 has been modified by introduction of non-linear terms in approximation of the rate sensitivity, [START_REF] Johnson | Strain-rate effects for high-strain ratecomputation[END_REF] to avoid the problem described previously (2 sets of constants). This, modified form is given by:

( ) [ ][ ][ ] m * C * 2 * n p T 1 ) (ln C ln C 1 B A 3 - ε + ε + ε + = σ & & (6)
where C 2 and C 3 are the new material constants allowing to define a non-linear strain rates sensitivity. However, this formulation has not been used in the present study since it is not usually pre-implemented in FE commercial codes, including ABAQUS/Explicit.

In order to complete the study a third model has been used during numerical simulations. This approach called RK constitutive relation is based on some physical aspects taking into account thermally activated processes in plasticity which are related to dislocation dynamics, for example [START_REF] Zerilli | Dislocation-mechanics-based constitutive relations for material Dynamics calculations[END_REF][START_REF] Klepaczko | Thermally activated flow and strain rate history effects for some polycrystalline FCC metals[END_REF].

-Semi-physical approach

The last model, called RK constitutive relation (Rusinek-Klepaczko) is described in detail in [START_REF] Rusinek | Shear testing of sheet steel at wide range of strain rates and a constitutive relation with strain-rate and temperature dependence of the flow stress[END_REF]. The total stress is decomposed into two parts, Eq. 7

T P I R C S U N A M D E T P E C C A ARTICLE IN PRESS -7/49 - )] T , ( ) T , , ( [ E ) T ( E ) T , , ( p * p p 0 p p ε σ + ε ε σ = ε ε σ (7)
where μ σ is the internal stress and * σ is the effective stress. In this relation the Young's modulus ) T ( E depends on temperature, Eq.8. The explicit formulation introduced by [START_REF] Klepaczko | A general approach to rate sensitivity and constitutive modeling of FCC and BCC metals, in: Impact: Effects of Fast Transient Loadings[END_REF] is given by:

⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ - θ - = ) T T 1 ( exp T T 1 E ) T ( E m * m 0 (8)
where 0 E is the Young's modulus at zero Kelvin, m T is the melting temperature and * θ is a material constant, characteristic of homologous temperature.

The explicit form of the two stress components are the following, Eqs 9-10, m is the constant allowing to define the strain rate-temperature dependency [START_REF] Klepaczko | A practical stress-strain-strain rate-temperature constitutive relation of the power form[END_REF].

) T , ( n p 0 p p p p ) )( T , ( B ) T , , ( ε μ ε + ε ε = ε ε σ T ( D 1 ) T , ( ε ε - σ = ε σ & & & ( 10
)
The modulus of plasticity and the strain hardening exponent are defined by, Eqs 11-

12: v p max m 0 p ) log( ) T T ( B ) T , ( B - ε ε = ε & & & ( 11
) ) log( ) T T ( D 1 n ) T , ( n min p m 2 0 p ε ε - = ε & & & ( 12
)
where 0 B is the material constant, v is the temperature sensitivity, 0 n is the strain where ρ is the density of material, β is the Taylor-Quinney coefficient [START_REF] Quinney | The emission of latent energy due to previous cold working when a metal is heated[END_REF] and p C is the specific heat at constant pressure.

hardening exponent at K 0 T = , 2 D
For steels, the transition between isothermal and adiabatic conditions appears generally for strain rate of the order, or higher, than 10 s -1 , [START_REF] Oussuaddi | An analysis of transition from isothermal to adiabatic deformation in the case of a tube under torsion[END_REF]. This transition is caused by a decrease of strain hardening during plastic deformation which is responsible of a thermal softening of the material. The thermal softening triggers very frequently an instability process preceding failure. Fig. 4 shows different modes of instability and failure of sheet metal specimens.

The RK model has already been successfully applied to study several processes of fast deformation such as perforation [START_REF] Rusinek | Shear testing of sheet steel at wide range of strain rates and a constitutive relation with strain-rate and temperature dependence of the flow stress[END_REF], double shear by direct impact [START_REF] Rusinek | Shear testing of sheet steel at wide range of strain rates and a constitutive relation with strain-rate and temperature dependence of the flow stress[END_REF], ring fragmentation under radial expansion [START_REF] Rusinek | Finite element simulation of steel ring fragmentation under radial expansion[END_REF], Taylor test and fast tension test [START_REF] Rusinek | Modélisation thermoviscoplastique d'une nuance de tôle d'acier aux grandes vitesses de déformation. Etude expérimentale et numérique du cisaillement, de la[END_REF][START_REF] Rusinek | Analysis of inertia and scale effects on dynamic neck formation during tension of sheet steel[END_REF].

Both, PL and RK constitutive relations have been implemented via a VUMAT in ABAQUS-Explicit using an implicit consistent algorithm developed originally by [START_REF] Zaera | An implicit consistent algorithm for the integration of thermoviscoplastic constitutive equations in adiabatic conditions and finite deformations[END_REF].

The implementation of the RK model into ABAQUS/Explicit using this algorithm is reported in [START_REF] Rusinek | Constitutive relations in 3-D for a wide range of strain rates and temperatures -Application to mild steels[END_REF] T

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-Failure criterion

In order to reproduce the perforation process it is necessary to consider a failure criterion. The use of failure criterions based on an equivalent strain level is widely extended for dynamic applications. In this work a constant value of the equivalent plastic strain at failure was assumed, as was also adopted by many authors dealing with dynamic problems [START_REF] Rusinek | Finite element simulation of steel ring fragmentation under radial expansion[END_REF][START_REF] Wood | Experimental mechanics at velocity extremes -very high strain rates[END_REF][START_REF] Teng | Evaluation of six fracture models in high velocity perforation[END_REF].

The failure strain value has been defined using several steps. The first one was to estimate the value corresponding to the Considere's criterion defined by σ = ε ∂ σ ∂ p [START_REF] Considère | L'emploi du fer de l'acier dans les constructions[END_REF]. In order to not disturb the previous solution corresponding to the plastic instability appearance, a bigger value is taken into account. Thus, the second step was to estimate the strain level corresponding to 0

p = ε ∂ σ ∂
. The gap between these two values corresponds to the development of the necking process. Finally, a coefficient is applied to this last condition as reported by [START_REF] Triantafyllidis | Onset of necking in electro-magnetically formed rings[END_REF][START_REF] Pandolfi | Finite element simulation of ring expansion and fragmentation[END_REF][START_REF] Singh | Dynamic tensile deformation and fracture of metal cylinders at high strain rates[END_REF] obtaining a final failure strain level of

1 p f = ε
. Moreover, the Considère's criterion is used since the failure mode using thin plate is mainly due to necking by tension state followed by crack propagation. Such failure criterion was also applied earlier [START_REF] Law | Use the cylindrical instability stress for blunt metal loss defects in linepipe[END_REF][START_REF] Rincón | Effect of temperature on tensile properties of an as-cast aluminum alloy A319[END_REF][START_REF] Colombo | Numerical analysis of thin-walled shaft perforation by projectile[END_REF] for dynamic tension or for ring expansion problems [START_REF] Rusinek | Analysis of inertia and scale effects on dynamic neck formation during tension of sheet steel[END_REF][START_REF] Rusinek | Finite element simulation of steel ring fragmentation under radial expansion[END_REF]. The failure strain is estimated directly from the analysis of deformation in tension predicted by a constitutive relation along with the Considère's criterion:

σ = ε σ ) d d ( ,.
A special attention has been done concerning the energy balance and its preservation during numerical simulations. It must be noticed that the number of elements deleted during the process as consequence of the failure criterion used is vastly reduced (Appendix C). The elimination of elements is restricted to the cracks propagation stage and therefore the energy balance can be considered as preserved.

-Coefficient of friction -analytical approximations combined with experimental data

The friction can modified the failure mode as it is reported in the next part of this paper. For this reason, the friction coefficient is estimated using punching experiment in at low and intermediate velocity. For a given displacement p δ of the hemispherical punch, the contact area between the sheet and nose of the punch is limited by a value of the radial coordinate c r , Fig. 6. Along the contact zone, the sheet is assumed to adopt

T P I R C S U N A M D E T P E C C A ARTICLE IN PRESS -10/49 -
the spherical nose shape. Experimental values of c r can be obtained by imposing different displacements to the sheet. Therefore, as c r depends on the projectile displacement, it also depends on the loading time.

Assuming a constant value of the normal contact pressure p and the friction coefficient μ , the condition of equilibrium of the punch tip allows to obtain the following relation between total force F, pressure and coefficient of friction

( ) c 0 F pcos psin dS θ = θ+μ θ ∫ ( 14
)
where

2 dS 2 R sin d = π θ θ (15)
and c θ is the contact angle whose value is known through c r (for a given displacement

p δ ) ( ) ( ) ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ δ = δ θ - R r sin p c 1 p c (16)
Integration of Eq.14 leads to

( ) [ ] c c c c 2 2 cos sin sin p R F θ θ - θ μ + θ π = (17)
The ratio ξ between the two forces corresponding to dry μ F and lubricated 0 F conditions allows for elimination of the pressure dependency, Eq.18.

( )

( ) ( ) ( ) ( ) c 2 c c c c 2 0 sin cos sin sin F F θ θ θ - θ μ + θ = = ξ μ (18)
With the knowledge of the forces ratio ξ , Eq.18, it is possible to define a range for the friction coefficient μ . The obtained force ratios for different values of μ using Eq. , which corresponds to the upper limit value found Fig. 7, is used in the numerical simulations to define dry condition. In our case the dependence on the coefficient of friction with temperature and sliding velocity is not taken into account [13, 15-17, 42, 61-63]

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The plate impact problems are commonly analyzed numerically by means of axisymmetric configuration, [START_REF] Borvik | Perforation of 12 mm thick steel plates by 20 mm diameter projectiles with flat, hemispherical and conical noses Part II: Numerical study[END_REF][START_REF] Gupta | Experimental and numerical studies on the behaviour of thin aluminium plates subjected to impact by blunt -and hemispherical -nosed projectiles[END_REF][START_REF] Gupta | Effect of projectile nose shape, impact velocity and target thickness on deformation behaviour of aluminium plates[END_REF][START_REF] Arias | Numerical simulations of impact behaviour of thin steel to cylindrical, conical and hemispherical non-deformable projectiles[END_REF][START_REF] Voyiadjis | A Finite Strain Plastic-damage Model for High Velocity Impacts using Combined Viscosity and Gradient Localization Limiters: Part II -Numerical Aspects and Simulations Int[END_REF][START_REF] Gupta | Effect of projectile nose shape, impact velocity and target thickness onthe deformation behaviour of layered plates[END_REF]. This simplification permits to obtain results which encourage experimental observations in terms of ballistic limit and residual velocity for different projectile-plate experimental configuration [13, 15-17, 43, 60]. Moreover, the use of this simplification presents an advantage of increasing the mesh density without obtaining excessive computational time. However, a 3D configuration is clearly recommendable when conical, ogival or hemispherical projectiles, susceptible of inducing petalling as the final stage of perforation process, are analyzed. In order to reproduce numerically the perforation process induced by this kind of projectiles the element size is not very important as in case of cylindrical projectiles [START_REF] Borvik | Perforation of 12 mm thick steel plates by 20 mm diameter projectiles with flat, hemispherical and conical noses Part II: Numerical study[END_REF][START_REF] Teng | On the transition from adiabatic shear banding to fracture[END_REF]. Therefore, by performing an adequate mesh configuration the calculation time can be manageable. There is however a limited number of works reported in the literature where the perforation process is analyzed by 3D simulations, for example [START_REF] Colombo | Numerical analysis of thin-walled shaft perforation by projectile[END_REF][START_REF] Børvik | Experimental and numerical study on the perforation of AA6005-T6 panels[END_REF].

In the present case, the configuration used in numerical simulations is showed in Fig. 8.

Both, projectile and Hopkinson tube have been modeled as rigid bodies allowing reduction of the computational time required for simulations. On the other hand, the experimental observations have revealed an absence of erosion on the projectile surface after impact. The reason is that the thin sheet specimen has a low yield stress in comparison with the projectile oil quenched.

-Target mesh strategy

The mesh used in the present paper for the target is shown Fig. 9. The whole target has been meshed with 8-node tri-linear elements with reduced integration (C3D8R in ABAQUS notation [START_REF] Hibbitt | Abaqus User's manual[END_REF]) and eight elements along the thickness. The number of elements was 229 792 corresponding to 259 875 nodes. The elements used to mesh the zone directly affected by impact (Zone I) had a size close to The radial symmetry of the mesh avoids spurious generation of main directions for the appearance and progress of cracks. Therefore the position of the petals at the end of the

T P I R C S U N A M D E T P E C C A ARTICLE IN PRESS -12/49 -
perforation process just depends on the constitutive relation, impact velocity and friction coefficient applied. This is shown in the next part of the paper.

Using the numerical configuration shown in Fig. 8 and Fig. 9 and three constitutive relations already presented, the simulations have been carried out for a wide range of impact velocity varying from: s / m 300 V s / m 40 0 ≤ ≤ .

-Analysis and results

The experimental setup developed in LPMM allows for determination of the forcetime evolution, the displacement-time evolution of the projectile during perforation, the time history of the net displacement of the specimen axis and the ballistic limit. At the same time it is possible to compare the target shape after different boundary conditions of impact for both friction and non-friction experimental results. All these information have been analyzed and used to validate numerical results.

-Validation and influence of constitutive relations

The first step is to compare the value of ballistic limit bl V predicted by numerical simulations with the value obtained during the experiments. Experiments showed that in the case of dry conditions the ballistic limit s / m 45 V dry bl = -is slightly higher that in the case of lubricated contact surface s / m 40 V lub bl = -. The relation between the ballistic limit and the residual velocity of projectile corresponding to both conditions is well reproduced by the numerical simulations using all the constitutive relations considered, see Appendix B -. In addition, the value of ballistic limit predicted by numerical simulation remains close to the experimental observations inside the interval: , the numerical estimation of the fracture time can not be validated due to absence of experimental data. The evolution of failure time with the impact velocity for all the constitutive relations used, presents the classical parabolic profile of perforation. This is in agreement with the experimental and numerical data reported in [START_REF] Borvik | Perforation of 12 mm thick steel plates by 20 mm diameter projectiles with flat, hemispherical and conical noses Part II: Numerical study[END_REF][START_REF] Arias | Numerical simulations of impact behaviour of thin steel to cylindrical, conical and hemispherical non-deformable projectiles[END_REF][START_REF] Rusinek | Influence of conical projectile diameter on perpendicular impact of thin steel plate[END_REF][START_REF] Voyiadjis | A Finite Strain Plastic-damage Model for High Velocity Impacts using Combined Viscosity and Gradient Localization Limiters: Part II -Numerical Aspects and Simulations Int[END_REF].

exp bl num bl exp bl V 1 . 1 V V 9 . 0 - - - ⋅ ≤ ≤ ⋅
Concerning the target shape after impact in the case of lubricated conditions and for impact velocity s / m 40 V 0 = appearance of four symmetric petals is observed for all constitutive relations Fig. 11. Although for JCII one can observe a secondary crack that has been arrested before reaching the rear side of the plate, Fig. 11. Anyhow, the failure mode predicted for all cases is petalling with absence of plug ejection. The number and the disposition of the petals are in agreement with the experimental observations, Fig. 4b. Albeit, some differences in the residual velocity can be observed, Fig. 11.

On the contrary to the failure mode for s / m 40 V 0 = , in the case of s / m 300 V 0 = and lubricated conditions the number of simulated radial cracks changed substantially depending on the constitutive relation, Fig. 12. Notice that the higher number of radial cracks corresponds to the case with the higher residual velocity, JCI Fig. 12. The relevance of this agreement will be analyzed ahead in this paper. In addition, using PLII model the failure of the plate is produced in the zone corresponding to the contact: plate -Hopkinson tube. This kind of failure has not been observed during experiments, although it is true that for this impact velocity experimental data does not exist.

Anyhow, the failure mode predicted is common to all constitutive relations. That is ejection of a plug as the final stage of the perforation process.

It is shown in Fig. 13 for the case of dry conditions and s / m 100 V 0 = that for all constitutive relations the failure mode is in agreement with the experimental observations, Fig. 4-a-c. That is a plug ejection as the final stage of perforation and therefore reduction of petalling. Nevertheless, the number of radial cracks changes with the constitutive relation applied. Again, the number of radial cracks predicted numerically is larger when the residual velocity is higher, JCI, Fig. 13.

T P I R C S U N A M D E T P E C C A ARTICLE IN PRESS -14/49 -
. 0 = ε & s -1 , 01 . 0 = ε & s -1 , 10 = ε & s -1 and 130 = ε & s -1 . Also a
comparison with experimental curves is reported. In the case of low strain rates JCI, PLI and RK fit correctly the experimental results Fig. 14 a-b, however JCII and PLII are not able to define the behavior at low strain rate since they are fitted for strain rates larger than 10 s -1 , Fig. 14 c-d. For this reason at high strain rates JCII and PLII produce better results with experiments. However, it is observed that RK constitutive relation is able to reproduce the material behavior in the whole spectrum of strain rates shown in In order to demonstrate the importance of defining correctly the material behavior, and particularly the effect of underestimating the flow stress, an extreme case has been taken into account. This extreme case consists of subtracting the term corresponding to the strain rate sensitivity from the JC formulation. The main goal is to investigate the relevance of the strain rate on the perforation process and at the same time the effect of the decrease of the flow stress of the target material.

As result of this analysis -see Appendix Dit is observed how a decrease of the rate sensitivity of the flow stress induces a reduction of the plastic field during perforation. In that case the inertia effects becoming relevant and it favors the appearance of a greater number of radial cracks. This explains the previously reported relation between the number of radial cracks and residual velocity. As it is analyzed in the next part, this effect is equivalent to an increase in the impact velocity.

In view of the complexity of failure dynamics demonstrated by the perforation problem and of the strong dependence of simulations on the definition of the material behavior, it is necessary the use of a constitutive relation capable to predict the response of the material with a good precision within a wide range of load conditions. Therefore relying on the good results that RK constitutive relation has offered for the problem treated in the present work as well as for other problems that involve the appearance of plastic instabilities [START_REF] Rusinek | Finite element simulation of steel ring fragmentation under radial expansion[END_REF][START_REF] Rusinek | Analysis of inertia and scale effects on dynamic neck formation during tension of sheet steel[END_REF], this constitutive relation was chosen to perform a more

T P I R C S U N A M D E T P E C C A ARTICLE IN PRESS -15/49 -
complex analysis of the perforation depending on the impact velocity and on the conditions of friction. Simulations performed under those assumptions are reported in the following part of the paper.

Analysis of the perforation process

In addition to the numerical simulations dealing with the role of the constitutive relation in analyzing the target behavior in the perforation process, in the next part of the paper a detailed analysis is offered showing the effects of impact velocity and friction on the failure mode. In this part of the numerical study only RK constitutive relation was applied.

Effect of friction

In present case, different kinds of failure mode are found during impact of steel sheets depending mainly on the friction coefficient μ and the impact velocity. Thus, for ) ejection of a plug is observed as the final stage of perforation process for the whole range of impact velocities considered.

Nevertheless, in the case of lubricated conditions ( 0 = μ ), the plug ejection appears only when the ballistic limit is considerably exceeded, that is for s / m 100 V 0 ≥ , Fig. 16d. This effect has been observed and studied previously in tension and perforation. This last is due to the trapping of plastic deformation corresponding to the Critical Impact

T P I R C S U N A M D E T P E C C A ARTICLE IN PRESS -16/49 -
Velocity appearance for this material as reported in [START_REF] Rusinek | A numerical study on the wave propagation in tensile and perforation test[END_REF][START_REF] Rusinek | A viscoplastic modeling of sheet metal in the range of large deformation and at low and high strain rates: Application to perforation Zeitschrift fur angewandte mathematik und mechanik[END_REF]. When that impact velocity is reached, the hole enlargement process becomes reduced and the target failure is induced by circumferential necking, it causes a plug ejection. Thus, for a high impact velocity, the differences concerning the failure mode between dry and lubricated condition become reduced.

Anyway, the friction effect is observed in the whole considered range of impact . Moreover, the constant value used for the friction coefficient based on the assumption of a constant pressure along the contact zone projectile-plate is supported by the results obtained from the numerical simulations as reported in Fig. 18.

It is observed using this measurement technique (force obtained by knowledge of projectile deceleration history) that inertia effect appears at the beginning of loading, Fig. 19. When the inertia effect is dissipated, a quasi-static loading curve is obtained, as for example when the force is measured along the Hopkinson tube, as reported in [START_REF] Rusinek | Modélisation thermoviscoplastique d'une nuance de tôle d'acier aux grandes vitesses de déformation. Etude expérimentale et numérique du cisaillement, de la[END_REF].

However, even if the force history is different, the energies absorbed in the target are very close. A complete analysis is reported in [START_REF] Rusinek | Modélisation thermoviscoplastique d'une nuance de tôle d'acier aux grandes vitesses de déformation. Etude expérimentale et numérique du cisaillement, de la[END_REF] concerning the effect of point measurement during experiment. In the present case, as the tube is defined as rigid, the force is obtained via time deceleration of the projectile.

Moreover, the friction level can also be observed for any impact velocity by measuring the plug size. This is possible since the tangential stress that appears by the friction amplifies the process of necking and therefore favours plug ejection. In the case of 26 . 0 = μ the plug diameter is always larger than in the case of 0 = μ . The largest difference appears for impact velocities close to the ballistic limit. In that case for 0 = μ and s / m 70 V 0 ≤ there is no plug ejection, Fig. 20.

Another possibility to quantify the friction effect, which is directly related with the plug size, is the displacement of projectile at failure time. The results obtained from the numerical simulations reveal that this parameter is larger when 0 = μ . This statement is

T P I R C S U N A M D E T P E C C A ARTICLE IN PRESS -17/49 -
true for the complete range of impact velocities considered, Fig. 20. The displacement of the projectile at failure quantifies the larger deformation of the target in the case of 0 = μ .

Influence of impact velocity

Influence of impact velocity, 0 V , on the failure mode considering both conditions of friction, 0 = μ and 26 . 0 = μ , can be easily observed. In both friction conditions, the number of radial cracks increases with impact velocity. This phenomenon is induced by the increase of the circumferential strain level responsible of the crack initiation and progression, [START_REF] Wierzbicki | Petalling of plates under explosive and impact loading[END_REF]. This is caused by the increase of the kinetic energy transferred to the material of the target when the impact velocity increases. Although the number of cracks is larger in case of high impact velocity, the failure mode induced by necking that appears in this situation reduces the size of the petals.

In addition, the initial impact velocity affects also the force level during perforation.

The maximum force is reached at the maximum impact velocity considered, s / m 300 V 0 = , and next it decreases with impact velocity, Fig. 21. This effect is caused by the strain rate sensitivity of the material target. At high impact velocity the mean strain rate increases and the target material is subjected to a strong process of strain hardening, which in turn increases the mean force. This process is limited at low impact velocity where the force level is reduced.

In order to analyze the phenomena involved in the process, an energy balance is proposed, defined by Eq. 19.

tp f p p W W W K + + = Δ (19)
Where p K Δ is the kinetic energy lost by the projectile, p W is the plastic work, f W is the friction energy and tp W is the kinetic energy transferred into the plate. Globally, it can be observed how at low impact velocity the process is governed by the plastic work, however, when the impact velocity increases the inertia effects become predominant due to larger amount of kinetic energy transferred into the plate, concentrated mainly in the kinetic energy of the plug, Fig. 22.

However some differences appear between lubricated and dry conditions. In the case of 0 = μ , and the impact velocity close to the ballistic limit, the plastic work represents As previously discussed, the plug size is larger for dry conditions.

T P I R C S U N A M D E T P E C C A
In addition, the increase of the inertia effect with impact velocity can be measured by the gap projectile -plate which appears when a certain impact velocity is exceeded, s / m 150 V 0 ≥ , Fig. 23. This generates a hole expansion process, [START_REF] Lee | Cavitation and Mushrooming in Attack of Thick Targets by Deforming Rods[END_REF].

It is observed after Fig. 22 how a complete petalling (lubricated condition for

s / m 100 V V 0 bl ≤ ≤
) is restricted in perforation processes completely governed by plastic work. In that case, an absence of plug ejection is found.

Analysis of petalling process

In order to study the petalling process, several studies based on analytical developments can be found in the international literature explaining the mechanics responsible for such behavior. Traditionally, the petalling process in the analytical models available in the literature is approximated as a simple hole enlargement, [START_REF] Taylor | The formation of enlargement of circular holes in thin plastic plates[END_REF][START_REF] Zaid | Mechanics of high speed projectile perforation[END_REF][START_REF] Paul | Normal perforation of a thin plate by truncated projectiles[END_REF][START_REF] Johnson | Hole anging and punching of circular plates with conically headed cylindrical punches[END_REF].

However, the theories proposed in [START_REF] Landkof | Petalling of thin, metallic plates during penetration by cylindroconical projectiles[END_REF] and [START_REF] Wierzbicki | Petalling of plates under explosive and impact loading[END_REF] have approached the process by means of a more rigorous treatment.

In [START_REF] Wierzbicki | Petalling of plates under explosive and impact loading[END_REF] it is reported that the appearance of four or five symmetric petals corresponds to a minimum for total rate of energy dissipation. The experiments performed in the present work for lubricated condition also satisfied the previous observations. At impact velocity close to the ballistic limit the number of petals appearing is four. In this situation the energy needed by the projectile to perforate the plate is minimum, Fig. 24.

This fact is well defined by the numerical simulations performed with application of RK constitutive relation, Fig. 24. In such case the damage induced by the projectile in the sheet steel is concentrated in the dome of the target by avoiding necking and therefore by plug ejection, Fig. 25. This induces generation of several cracks in the zone, four of them quickly progress inducing the formation of four petals symmetrically disposed, Fig. 25. The fast progression of the cracks triggers an increase of the circumferential strain induced by the projectile advances.

T P I R C S U N A M D E T P E C C A ARTICLE IN PRESS -19/49 -
The circumferential plastic strain in adiabatic conditions strongly increases the temperature at the bottom of the crack. This induces a local thermal softening, commonly responsible of instabilities appearance in dynamic problems, [START_REF] Rusinek | Finite element simulation of steel ring fragmentation under radial expansion[END_REF][START_REF] Rusinek | Analysis of inertia and scale effects on dynamic neck formation during tension of sheet steel[END_REF][START_REF] Batra | Instability strain and shear band spacing in simple tensile/ compressive deformations of thermoviscoplastic materials[END_REF][START_REF] Batra | Effect of viscoplastic relations on the instability strain, shear band initiation strain, the strain corresponding to the minimum shear band spacing, and the band width in a thermoviscoplastic material[END_REF][START_REF] Alos | Analytical and numerical analysis of sheet metal instability sing a stress based criterion[END_REF][START_REF] Kuroda | Ductility of interstitial-free steel under high strain rate ension: Experiments and macroscopic modelling with a physically-based consideration[END_REF][START_REF] Rusinek | A viscoplastic modeling of sheet metal in the range of large deformation and at low and high strain rates: Application to perforation[END_REF][START_REF] Rusinek | Analysis of thermovisco-plastic behaviour of six high strength steels[END_REF][START_REF] Rusinek | A numerical study on the wave propagation in tensile and perforation test[END_REF],

which favours the fast progression of the cracks and diminishing the amount of energy absorbed by the plate during the perforation.

Three different stages of the perforation process have been analyzed in order to quantify the local gradient of temperature in the proximity of the cracks, Fig. However in the case of stage II the strong temperature gradient coupled with the high level of circumferential strain caused by the advances of the projectile induces the progression of the crack with just a small amount of energy absorbed by the plate. In the case of stage III, although the gradient of temperature is comparable to stage II, the circumferential strain is not large enough to induce the crack progression and cracks are finally arrested close to the rear side of the target.

In the case of mm 1 x ≥ , the temperature decay is progressive until reaching K 293 T ≈ which corresponds to a zone with absence of plastic deformation. The relevance of the gradient of temperature during the cracks progression demonstrates that the constitutive model used to define the behaviour of the plate in perforation problems, susceptible of inducing plastic instabilities such as petalling, is crucial.

Conclusions

The failure process of steel sheets when subjected to normal impact by hemispherical projectiles was examined. Experiments have been carried out using an for a complete approximation of the non-linear strain rate sensitivity. However, in the cases of JC and PL constitutive equations the strain rate sensitivity must be approximated in several parts. Therefore, RK relation is chosen to carry out the numerical analysis of the perforation process. Since it is well known that impact events are strongly coupled with strain hardening, strain rate sensitivity and adiabatic temperature increase, a precise approximation of material behaviour is crucial.

T P I R C S U N A M D E T P E C C A ARTICLE IN PRESS - 20
In addition, the friction effect on the failure mode has been analyzed. Close to the ballistic limit velocity, appreciable differences appear on the failure mode depending on the conditions of friction applied between the specimen and the projectile. Using dry conditions the failure process appears in the form of circumferential necking inducing plug ejection as final stage with reduced petalling. Whereas in the case of lubricated conditions the longer hole enlargement process is observed which induces a complete petal formation. At high impact velocities the differences between both conditions of friction are reduced and the plug ejection due to necking process is the common failure mode observed.

Influence of impact velocity on the failure mode considering both conditions of friction has been evaluated. In both cases the number of radial cracks increases with impact velocity. This phenomenon is induced by the increase of the circumferential strain level responsible of crack initiation and progression.

To evaluate the contribution of the particular energy components involved in the process, an energy balance was carried out. From this analysis a conclusion can be drawn that the inertia effect increases as the impact velocity does, mainly due to the kinetic energy increase with the plug velocity. On the contrary, plastic work mainly governs the process for impact velocities close to the ballistic limit. For lubricated condition, plastic work represents almost % 100 of the energy absorbed by the plate.

When these conditions of friction and impact velocity are satisfied (ballistic limit velocity and lubricated condition) a complete petalling process is observed. The Rusinek-Klepaczko (Low and high strain rates)

T P I R C S U N A M D E T P E C C A T P I R C S U N A M D E T P E C C A ARTICLE IN PRESS -22/
B 0 [MPa] 591.6 n 0 [-] 0.285 0 ε [-] 1.8*10 -2 D 1 [-] 0.48 ν [-] 0.2 * 0 σ [MPa] 406.3 m [-] 2.8 D 2 [-] 0.19 E 0 [GPa] 212 * ϑ [-] 0.59 T m [K] 1600 max ε & (s -1 ) 10 7 min ε & (s -1 ) 10 -5
C p (Jkg -1 K -1 ) 470

β [-] 0.9 ρ (kgm -3 ) 7800 α (K -1 ) 10 -5
Power law, set I (Low strain rates) Power law, set II (High strain rates)

K [MPa] 1598 1598 n [-] 0.149 0.149 m [-] 0.02 0.062 ν [-] 0.2 0.2 T P I R C S U N A M D E T P E C C A ARTICLE IN PRESS -23/
P I R C S U N A M D E T P E C C A T P I R C S U N A M D E T P E C C A T P I R C S U N A M D E T P E C C A T P I R C S U N A M D E T P E C C A T P I R C S U N A M D E T P E C C A T P I R C S U N A M D E T P E C C A ARTICLE IN PRESS -31/49 -

Figure captions

49 -

 49 where E is the Young's modulus of the tube, D and d are respectively the external and internal diameters of the Hopkinson tube (

18 are shown in Fig. 7 .

 7 Comparing analytical predictions with experimental results, the value of μ is close to 26

Fig. 9 .

 9 Fig.[START_REF] Rusinek | Finite element simulation of steel ring fragmentation under radial expansion[END_REF]. In order to not increase excessively the computational time, the element size has been increased in the zone not affected by the impact, Fig.9. (Zone II). This optimized mesh has been obtained by a convergence study making use of different mesh densities.

Fig. 14 shows

 14 Fig. 14 shows stress vs. strain curves predicted by each constitutive relation and the comparison with experimental data for mild steel ES. These curves reproduce the true stress versus true plastic strain relations defined by Eqs.3, 4 and 7 for K 300 T = and different strain rates 001 . 0 = ε & s -1 , 01 . 0 = ε & s -1 , 10 = ε & s -1 and 130 = ε & s -1 . Also a

Fig. 14 .

 14 Fig.14. These differences in the predicted material behavior modify considerably the response of plate during perforation. The different behavior predicted by each constitutive relation will affect the absorption of energy consumed in the process of perforation. It is therefore possible to introduce some modifications in the failure mode of the target, the main goal of analysis in the present work.

For

  of the projectile is stuck to the steel sheet inducing a circumferential failure by necking. The diameter of the circumferential failure is close to projectile diameter. It follows by initiation of small radial cracks, Fig.4a, c. The process of failure observed for lubricated case is slightly different. For the last case ( 0 ≈ μ ), due to radial sliding of the steel sheet along the projectile nose a small plug ejection appears close to the dome of the sheet steel, stage A, Fig. 4-a-c and Fig. 15. In that case the process of hole enlargement is longer allowing to initiate more small cracks, stage B, responsible of petal formation as ultimate stage by crack propagation, stage C. Both situations are faithfully reproduced during numerical simulations in agreement with the experimental observations, Fig. 4 using the RK constitutive relation, Fig. 16.

  velocity by measuring the force time history. The force level is slightly larger in case of the friction effect. The difference in the maximum force level between the numerical values obtained for 26 impact velocity, Fig.17.This fact is in agreement with the experimental observations and also with the hypothesis assumed to obtain the analytical prediction of the friction coefficient used for dry conditions, 26 . 0 = μ

  . The first stage (Stage I) corresponds to the early development and progression of cracks, the second one (Stage II) represents the progression of cracks and the third one (Stage III) leads to the crack arrest. It is observed that the temperature quickly decreases as one moves away from the crack. The temperature distribution is represented by the parabolic profile, Fig. 26. Thus, Fig. 26 shows a gradient of temperature in the zone close to the crack: mm 1 x ≤ (x = distance from the crack), for stage II and stage III, being slightly minor in case of stage I. The reason of this difference is that for stage I the zone beside the crack is not fully plastic and an external contribution of energy represented by the force induced by the advance of the projectile is needed to propagate the crack.

/ 49 -

 49 original set-up developed in LPMM of Metz University based on the R.M. Davies bar concept by application of Hopkinson tube. The tests were conducted covering a wide range of initial impact velocities. Lubricated and dry conditions were applied between specimen and steel sheet making possible the analysis concerning the influence of friction during perforation process. Numerical calculations have been made by application of 3D analysis using ABAQUS/Explicit FE code and considering three different constitutive relations, JC, PL and RK. The last constitutive equation allows

Fig C- 1 .

 1 Fig C-1. Elements eroded during numerical simulations for both, lubricated and dry conditions for the case V 0 = 50 m/s and V 0 = 100 m/s

Fig. 1 .

 1 Fig. 1. Perforation process by 3D numerical approach, visualization of petalling phenomenon.

Figure 2 Fig. 2 .

 22 Figure 2

Figure 3 Fig. 3 .Fig. 4 .: 5 .

 3345 Figure 3

Figure 6 Fig. 6 .Fig. 7 .

 667 Figure 6

Fig. 8 .Figure 9 Fig. 9 .Fig. 10 .Fig. 11 Fig. 12 Fig. 13 Fig. 14 .

 8991011121314 Fig.8. Configuration applied in the present work

  Figure 15

Fig. 15 .

 15 Fig.[START_REF] Gupta | Experimental and numerical studies on the behaviour of thin aluminium plates subjected to impact by blunt -and hemispherical -nosed projectiles[END_REF]. Schematic representation of petal formation during perforation of sheet steel using lubricated conditions, V e corresponding to expansion velocity along radial direction and V r is the residual velocity.

Fig. 17

 17 Figure 16

Fig. 18 Fig. 19 .Figure 20 V 0 =Fig. 20

 181920020 Fig.17Force-time history comparison between lubricated and dry conditions for several initial impact velocities using RK model, (a) s / m 40 V 0 = (b) s / m 75 V 0 = , (c) s / m 150 V 0 = Figure 18

Fig. 21 .Fig. 22 Fig. 23 .Fig. 24 GapFig. 25 .GenerationFig. 26 (

 212223242526 Figure 21

  for all cases. It must be noticed however that PL and RK are the constitutive relations which predict the closer value of the ballistic limit ballistic limit the numerical values obtained for the failure time are close to the experimental measurements for all the constitutive relations used, except the case of

					ARTICLE IN PRESS
	PLII for which the obtained values are out of	10	%	range in comparison to the
	experimental data, Fig.10. It must be noticed that for numerical simulations
	corresponding to	V 0 ≥	100	m	/	s
	in comparison with experimental measurements. The results obtained from the simulations with the JC constitutive relation are not so precise. It must be notice that using JCI the numerical simulations corresponding to impact velocities under s / m 50 P I R C S U N A M D E T P E C V T C A

0 ≤ are prematurely ended due to the appearance of a numerical problem avoiding complete time of calculation. Later, the failure time predicted by the numerical simulations is compared with the value obtained during experiments. As reported for the -13/49 -

49 - -Appendix A -Parameters of the mild steel for different constitutive relations Johnson-Cook, set

  I (Low strain rates) Johnson-Cook, set II (High strain rates)

	A [MPa]	57.27	57.27
	B [MPa]	479.93	79.641
	n [-]	0.316	0.316
	C [-]	0.0362	0.37848
	0 ε & [s -1 ]	10 -3	10 -3
	m [-]	0.28	0.28
	T 0 [K]	300	300

49 - -Appendix B -Relevant results of the numerical simulations

	JCI
	Impact	Residual	Work, W	Plastic	Failure	Plug	Friction
	velocity, V 0 (m/s)	velocity, V 0 (m/s)	(J)	work, W p (J)	time, t f (μs)	diameter, Φ p (mm)	coefficient μ [-]
	300	295,9	188,1	96,0	28	16,6	0.26
		296,3	169,8	115,7	28	9,0	0
	200	196,4	109,8	68,9	35	15,6	0.26
		195,5	137,0	104,0	49	11,8	0
	100	93,8	92,3	74,9	104	11,1	0.26
		94,3	85,1	82,0	118	9,9	0
	75	67,2	84,9	72,8	135	10,2	0.26
		67,8	78,5	77,2	144	7,8	0
	50	-	-	-	-	-	0.26
		-	-	-	-	-	0
	40	-	-	-	-	-	0.26
		-	-	-	-	-	0
	JCII
	Impact	Residual	Work, W	Plastic	Failure	Plug	Friction
	velocity, V 0 (m/s)	velocity, V 0 (m/s)	(J)	work, W p (J)	time, t f (μs)	diameter, Φ p (mm)	coefficient μ [-]
	300	295,9	188,1	107,6	28	16,5	0.26
		295,9	188,1	129,9	28	12,0	0
	200	196,0	121,9	85,7	35	13,1	0.26
		195,8	128,0	109,7	50	13,0	0
	150	145,2	109,1	91,0	72	10,8	0.26
		145,2	109,1	103,1	80	9,9	0
	100	93,0	103,7	87,9	91	8,7	0.26
		93,6	94,5	92,6	105	6,9	0
	75	65,3	104,2	88,1	128	7,8	0.26
		66,5	92,6	88,3	128	4,8	0
	50	34,1	102,5	85,9	207	8,1	0.26
		37,7	82,7	82,0	184	0,0	0
	40	17,5	99,6	83,5	272	8,8	0.26
		23,1	82,1	81,3	238	0,0	0
	RK
	Impact	Residual	Work,	Plastic	Failure
	velocity, V 0	velocity, V 0	W (J)	work,	time,
	(m/s)	(m/s)		W p (J)	t f (

μs) Plug diameter, Φ p (mm) Friction coefficient μ [-]

				ARTICLE IN PRESS
	PLI
	Impact	Residual	Work, W	Plastic	Failure	Plug	Friction
	velocity, V 0 (m/s)	velocity, V 0 (m/s)		(J)	work, W p (J)	time, t f (μs)	diameter, Φ p (mm)	coefficient μ [-]
	300	295,8		192,7	104,7	28	17,1	0.26
		295,7		197,2	128,2	30	14,8	0
	200	196,1		118,9	87,4	40	14,8	0.26
	100 75 50 40 Impact PLII velocity, V 0 (m/s) 300 200 100 75 50	194,9 92,7 93,0 65,0 66,0 38,6 37,0 17,7 20,5 Residual velocity, V 0 (m/s) 293,8 293,3 194,4 191,2 88,0 89,6 59,0 61,6 22,1	155,0 108,3 104,0 107,8 97,7 77,8 86,9 99,0 90,8 Work, W (J) 283,5 306,0 170,0 265,0 173,7 151,8 165,1 140,7 154,9	116,9 90,7 100,2 91,7 95,3 85,0 91,0 84,1 89,4 Plastic work, W p (J) 219,4 219,5 155,2 177,9 149,2 148,4 A 115 54 110 154 168 220 240 310 320 Failure S 14,5 10,5 8,8 12,5 10,4 10,5 -10,2 R T 0 0.26 0 0.26 P 0 I 0.26 0 0.26 -0 C Friction time, t f (μs) (mm) coefficient μ [-] 42 13,9 0.26 U 42 12,0 0 50 13,0 0.26 N 60 -0 130 10,8 0.26 124 7,5 0 140,3 165 11,4 0.26 139,0 170 5,8 0 M 133,5 270 -0.26
	40	28,0 0 0		E 62,8 123,2 D 130,6 114,0 123,2 121,0	255 --	---	0 0.26 0
		T P
	295,5 295,2 194,2 193,6 E C 143,1 143,3 90,0 91,2 C 300 200 150 100 A	206,3 220,0 175,7 194,0 157,9 151,3 146,3 129,5	139,0 169,6 140,6 159,7 132,5 144,0 125,0 126,8	28 32 50 55 70 84 108 120	15,9 13,5 12,9 12,3 11,4 10,2 9,8 6,9	0.26 0 0.26 0 0.26 0 0.26 0
	80	67,4		142,7	123,0	145	9,9	0.26
		69,3		123,0	120,5	150	5,7	0
	75	61,5		141,9	121,2	165	10	0.26
		64,0		120,3	118,5	165	5,6	0
	50	27,8		133,0	112,9	250	10,5	0.26
		32,1		113,1	111,7	250	0	0
	40	0		123,2	107,9	360	-	0.26
		13,7		108,7	107,5	320	0	0
								-24/49 -

Table 1

 1 Chemical composition of the mild steel ES (% of wt)

	Mn	Al	Cr	C	Ni	S	Cu	Si	P	N	Ti
	0.203 0.054	0.041	0.03	0.018	0.011	0.009	0.009	0.008 0.0063 0.002

& & &

& & &

Acknowledgements -1

Prof. J.R. Klepaczko, December, 2006 Janusz Roman Klepaczko passed away on August 15, 2008 at the age of 73. Graduated from Warsaw University of Technology in 1959, began the research work in 1960 at IPPT -Institute of Fundamental Technological Research, Polish Academy of Sciences, Warsaw, Poland and continued it until 1984, becoming full professor in 1983. Since 1985 he was working in LPMM (Laboratory of Physic and Mechanic of Materials), Paul Verlaine University of Metz, France, where he was founder of the experimental laboratory. He was well known in the field of dynamic behaviour of materials; he was the author of over 200 publications and supervised 30 doctors in several research centres around the world. Janusz was involved in research until the end of his life. He was a great researcher and had a passion for Science. He was for us a source of motivation and inspiration. We pay our tribute to him for his teaching and contribution in Science.

Acknowledgements -2

The researchers of the University Carlos III of Madrid are indebted to the Spanish Ministry of Education (project DPI2005-06769), and to the Region of Madrid (project CCG06-UC3M/DPI-0796) for the financial support that allowed to perform a part of the numerical simulations. The researchers from the Metz University (Laboratory of Physics and Mechanics of Materials) acknowledgement some support by CNRS-France.

number and disposition of petals numerically reproduced is in agreement with experimental observations and analytical predictions.

-Appendix D-Influence of strain rate sensitivity on the plastic flow

The explicit formulation of the JC constitutive relation without taking into account the term concerning the strain rate sensitivity is defined as follows:

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