Abstract
One contribution of 15 to a theme issue 'Experimental testing and modelling of brittle materials at high strain rates'.

To evaluate the vulnerability of ultra-high performance fibre-reinforced concrete (UHPFRC) infrastructure to rigid projectile penetration, over the last few years CEA-Gramat has led an experimental and numerical research programme in collaboration with French universities. During the penetration process, concrete is subjected to extreme conditions of pressure and strain rate. Plasticity mechanisms as well as dynamic tensile and/or shear damage are activated during the tunnelling phase and the cratering of the concrete target. Each mechanism has been investigated independently at the laboratory scale and the role of steel fibres especially has been analysed to understand their influence on the macroscopic behaviour. To extend the experimental results to the structural scale, penetration tests on UHPFRC slabs have been conducted by CEA-Gramat. The analysis of this dataset combined with material characterization experiments allows the role of steel fibres to be identified in the different plasticity and damage mechanisms occurring during penetration. In parallel, some improvements have been introduced into the concrete model developed by Pontiroli, Rouquand and Mazars (PRM model), especially to take into account the contribution made by the fibres in the tensile fracture process. After a primary phase of 2016 The Author(s) Published by the Royal Society. All rights reserved.
Introduction
In recent decades, the possibilities offered by ultra-high performance concrete (UHPC) have been exploited by engineers and architects to design structural elements with unusual thin shapes. This family of materials provides interesting candidates for protective structures or structural elements. Indeed, the high compressive strength and the high-energy dissipation capacity due to the presence of steel fibres (ultra-high performance fibre-reinforced concrete (UHPFRC)) has enabled the design of effective protection against blast and ballistic threats.
Introduction
Penetration tests carried out in the past 20 years have shown that increasing concrete compressive strength resulted in decreasing penetration depths or residual velocities of the projectile after the concrete slabs were perforated [1] [2] [3] . It was found also that adding steel fibres was the most effective method to reduce spalling and scabbing crater dimensions of both the impacted and rear faces of the targets [4] [5] [6] [7] .
Introduction
Twelve years ago, CEG-DGA, Gramat, France, conducted penetration tests on standard and ultra-high-strength concrete (Ductal-FM ® ) targets using a steel penetrator [8] . For impact velocities ranging from 250 to 450 m s −1 , the penetration depths were found to be 1.4 times less for a UHPFRC with a compressive strength of 200 MPa than in a standard concrete ( f c = 40 MPa). Blast experiments and ballistic impact experiments with small-calibre bullets and fragments simulating projectiles have been carried out on Ductal ® targets by Cavill et al. [9] . The ability of such concrete material to be used as protective elements has been confirmed.
Introduction
Currently, CEA-Gramat makes constant efforts to develop and to validate models that accurately describe the dynamic behaviour of concrete under extreme conditions. Numerical simulation represents a versatile tool to assess the damage and the residual load capacity of a structure subjected to blast loading or projectile impact. However, the ability of a simulation to be predictive is closely linked to the consistency of the material model. The modelling approach should take into account the main physical phenomena activated under dynamic conditions and it must be identified thanks to reliable experimental data. This is of primary importance in the case of high-amplitude loading under high strain rates, a specific loading regime associated with high-velocity impact or contact detonation. Unfortunately, there are only a few experimental data for high pressure and high strain rates available today in the literature for UHPC.
Introduction
In this work, the modelling approach developed at CEA-Gramat for standard concrete is first described. Then, several characterization experiments at the material scale are presented as well as the corresponding calibration of the Pontiroli, Rouquand and Mazars (PRM) model for UHPC. Then, the experimental results of ballistic impacts on Ductal ® are reported and compared with similar tests carried out on a standard concrete. Finally, numerical simulations of penetration and perforation tests have been conducted and compared with experimental data.
Pontiroli, Rouquand and Mazars model (a) Damage model
The PRM model has been developed to simulate the behaviour of concrete under severe loading [10] . This macroscopic model, based on the Mazars damage model [11] , includes two scalar damage variables, D t and D c , that give, respectively, the loss of stiffness under pure tensile loading and pure compressive loading (equations (2.1) and (2.2), E being the initial Young's modulus of concrete), 
Pontiroli, Rouquand and Mazars model (a) Damage model
e ft0 e (s ft , e ft ) (s ft , e ft ) Figure 1 . Principle of the PRM damage model illustrated in a cyclic one-dimensional loading.
Pontiroli, Rouquand and Mazars model (a) Damage model
Between these two loading states, the transition zone is defined by the couple (σ ft , ε ft ) corresponding, respectively, to crack-closure stress and crack-closure strain. If this couple is initially defined as the material parameters (σ ft0 , ε ft0 ), the evolution of σ ft is linked to the compressive damage variable D c through the following relation:
Pontiroli, Rouquand and Mazars model (a) Damage model
The evolution of damage variables is given by the following exponential forms:
Pontiroli, Rouquand and Mazars model (a) Damage model
and
Pontiroli, Rouquand and Mazars model (a) Damage model
In the compressive case, the unloading paths converge to the couple (σ fc ,ε fc ) so as to introduce permanent strains in the model.
Pontiroli, Rouquand and Mazars model (a) Damage model
In the preceding relations, ε 0t , ε 0c , A t , A c , B t and B c are material parameters and the equivalent strain is defined by equation (2.6). In this relation, X i corresponds to the positive principal strain components ε i in compression and (ε−ε ft ) i in tension,
Pontiroli, Rouquand and Mazars model (a) Damage model
For a uniaxial cyclic loading as presented in figure 1 , the mechanical behaviour is simply described by the following relation: 
Pontiroli, Rouquand and Mazars model (a) Damage model
In equation (2.10), α t plays the role of an activation factor:
Pontiroli, Rouquand and Mazars model (a) Damage model
The general three-dimensional constitutive relations relating the stress and strain tensors are given by equation (2.11), where λ 0 and μ 0 are the Lamé constants defining the initial elastic stiffness of the material. In this generalized version of the damage model, α t evolves between 0 and 1 and the actual value depends on (ε − ε ft ),
Pontiroli, Rouquand and Mazars model (a) Damage model
The PRM model is able to capture the mechanical response of concrete under quasi-static cyclic or combined loads.
Pontiroli, Rouquand and Mazars model (a) Damage model
(b) Strain-rate sensitivity of tensile strength For a dynamic regime, one has to take into account the strain-rate sensitivity of concrete. Indeed, the ultimate tensile stress reached by concrete at 100 s −1 is usually four to five times higher than its quasi-static strength [12] . Authors agreed to consider this effect as an intrinsic material phenomenon. This aspect of the behaviour is accounted for by replacing the parameter ε 0t by its dynamic equivalent ε d0t computed by the dynamic increase factor R t ,
Pontiroli, Rouquand and Mazars model (a) Damage model
This power model has already been identified for standard concrete using quasi-static direct tension experiments, spalling tests conducted with a Hopkinson bar device [12] and with ultrahigh strain-rate uniaxial deformation tests carried out with a pulsed-power generator (générateur électrique de pression intense, GEPI) [13] .
Pontiroli, Rouquand and Mazars model (a) Damage model
A problem of mesh size dependency is often observed for damage models. In order to limit this effect, the Hillerborg regularization model has been included [14] . In this method, the energy consumed by a crack to propagate through a band of finite elements is no longer dependent on the mesh size because of a fixed characteristic length L c . In fact, the method consists in modifying the softening part of the stress-strain relation in order to obtain the same fracture energy whatever the mesh size. To do so, a new variable ω is introduced,
Pontiroli, Rouquand and Mazars model (a) Damage model
where ε p = α t ε pt +(1 − α t )ε pd with ε pt and ε pd being, respectively, the peak strain in tension and compression. a(R t ) is a corrective function dependent on the dynamic increase factor R t . This variable ω is included in the damage formula (equation (2.4)) to adapt the damage kinetic, taking into account the characteristic length L c of the material and the finite-element characteristic length L e ,
Pontiroli, Rouquand and Mazars model (a) Damage model
As proposed by Hillerborg et al. [14] , the characteristic length L c is assumed to be directly linked to the fracture energy G f and the static tensile strength f t of the material under consideration, 
Pontiroli, Rouquand and Mazars model (a) Damage model
e For compressive loading at high strain rates, no dynamic increase factor has been considered because the apparent increase in compressive strength is certainly due to inertial autoconfinement or non-homogeneous lateral deformation of concrete specimens during dynamic compressive tests [15] [16] [17] .
Pontiroli, Rouquand and Mazars model (a) Damage model
This modelling approach of damage for concrete and geomaterials is a versatile tool adequate for a wide variety of problems [9] . However, in dynamic events such as projectile impact or blast loading, the high pressure level has a strong effect on the compressive behaviour of concrete. Consequently, the PRM model has been coupled with a plasticity model and a compaction law.
(c) Plasticity and compaction modelling
The PRM damage model simulates well the dynamic response of concrete under very low confinement. However, an elastic-plastic model is more appropriate for simulating the impact of a steel projectile in a concrete slab at about 300 m s −1 . Indeed, in this case, a pressure level of several hundreds of megapascal can be observed in the vicinity of the projectile nose. Specific phenomena such as pore collapse or increase of shear strength have to be considered to model accurately the dynamic response of concrete. To do so, the PRM damage model has been coupled with a simple plasticity model proposed by Krieg [18] and Swenson & Taylor [19] .
(c) Plasticity and compaction modelling
On the one hand, in this elastic-plastic model (figure 2a), a parabolic relation describes the pressure dependency of the yield stress q,
(c) Plasticity and compaction modelling
where p is the pressure and a 0 , a 1 and a 2 are material parameters. q is the yield stress in the sense of the von Mises plasticity,
(c) Plasticity and compaction modelling
where s is the deviatoric stress tensor defined as s = σ + p.I. The increase of q with pressure can be limited by a saturation value q max . This is particularly important for standard concrete, which presents a saturation value linked to its water saturation level (η in figure 2a ) [20, 21] . On the other hand, the pore collapse phenomenon occurring at high pressure is described by a piecewise-linear compaction curve (figure 2b). The elastic behaviour becomes more and more nonlinear up to the full consolidation of concrete: at this point, the pore collapse phenomenon is achieved and the material is considered fully compacted.
(c) Plasticity and compaction modelling
These plasticity and compaction models are complementary to the PRM damage model. The fully coupled PRM model includes all these mechanisms with a perfect continuity between the compressive damage definition and the plasticity model. It has been implemented in a classical finite-element code (Abaqus/Explicit [22] ) through a VUMAT subroutine. This choice allows numerical simulations to be conducted from the material scale up to the structural response. 
Model identification for ultra-high performance fibre-reinforced concrete (a) The influence of fibres on the tensile behaviour
In order to describe the cratering mechanisms initiated by the impact of a penetrator or a contact detonation, the dynamic tensile behaviour of concrete needs to be characterized over a wide range of strain rates. At lower rates, this aspect of the mechanical behaviour is also relevant for modelling the flexural dynamic response of steel-fibre-reinforced beams [23] or the blast response of FRC slabs [24] . The strain-rate sensitivity of UHPFRC can be investigated from the quasistatic regime up to moderate dynamic rates through bending tests or direct tensile tests. Based on a Hopkinson bar device, spalling tests can be conducted to obtain data at high loading rates (approx. 100 s −1 [12] ), while for higher strain rates (greater than 1000 s −1 ) a new approach using a pulsed-power generator has been used [13] .
Model identification for ultra-high performance fibre-reinforced concrete (a) The influence of fibres on the tensile behaviour
Besides the compressive strength, the main difference between standard concrete and UHPFRC lies in the post-peak tensile behaviour. Indeed, the presence of fibres gives a pronounced ductility to UHPFRC. Disseminated in the microstructure, steel fibres play a major role during the fracture process, bridging the cracks and retaining fragments. To model their influence, the fracture energy due to fibres has been introduced directly in the evolution of the tensile damage variable D t , depending on L f the fibre length and on the volume fraction of fibres V f actually participating in the resistance to crack opening. We can rewrite equations (2.13) and (2.14) as
Model identification for ultra-high performance fibre-reinforced concrete (a) The influence of fibres on the tensile behaviour
and
Model identification for ultra-high performance fibre-reinforced concrete (a) The influence of fibres on the tensile behaviour
where a(R t , V f ) and b(V f ) are functions dependent on the dynamic increase factor R t and on the volume fraction of fibres V f . When the fibre volume is increased, the energy necessary to reach the complete fracture of UHPFRC is increased in the same way.
Model identification for ultra-high performance fibre-reinforced concrete (a) The influence of fibres on the tensile behaviour
One has to identify the real fraction of fibres V f that are playing a role during the fracture process. It is assumed that the loading direction is roughly constant during the calculation. To determine the fraction, experimental data from three-point bending tests were used. In this experimental campaign, several batches of UHPFRC (with a 2% volume of steel fibres) were characterized: the first batch was a UHPC; the second batch was a UHPFRC with fibres oriented in such a way as to give poor resistance in a bending test; the third one was a UHPFRC with fibres oriented randomly; and the last one was a UHPFRC with fibres oriented to maximize the contribution of the fibres in the bending tests. The experimental results and the corresponding numerical predictions with the PRM model are compared in figure 3 . A fairly good correlation can be seen between the UHPFRC with fibres randomly oriented and the PRM model considering 50% of the fibres contributed to the residual strength of the damaged UHPFRC. As the influence of fibres is assumed to be isotropic in the PRM model, we need to take for V f only the real fraction of fibres that are playing a role during the loading.
Model identification for ultra-high performance fibre-reinforced concrete (a) The influence of fibres on the tensile behaviour
The damage patterns D t (equation (2.4)) obtained in numerical simulations with no fibre reinforcement (figure 4a) and 2% steel fibres are plotted in figure 4 . In the simulation, while the UHPC specimen breaks up due to a single crack, the introduction of fibres tends to widen the cracking zone.
Model identification for ultra-high performance fibre-reinforced concrete (a) The influence of fibres on the tensile behaviour
At high strain rates, the tensile strength of concrete can be investigated through spalling tests. Tests have been conducted on UHPFRC with an electromagnetic machine [13, 25] (figure 5). Based on high-pulsed-power technologies, the GEPI generator is composed of 28 stages able to store 70 kJ from a charging voltage of 85 kV. The released current rises over about 500 ns to a value of 3.3 MA, converging into the centre of the electrodes where a short circuit is ensured. In this region, the intense current and the induced magnetic field generate a compressive wave applied to the internal skin of the electrodes. This compressive wave propagates through the electrodespecimen stacking until it reaches a specimen free-surface. At this point, it is reflected back as a release wave. The interaction of the incident and the reflected waves generates a tensile loading at very high strain rates (typically 10 4 s −1 with the GEPI machine). The velocity profile is measured on the UHPRFC free surface and a reference velocity signal is usually measured on the opposite face, for control purposes at the electrode-window interface as presented in figure 5b. spalling tests conducted using Hopkinson bars and GEPI spalling tests, a continuous increase of tensile strength f t versus strain rateε is observed, which can be fitted by the following power law:
Model identification for ultra-high performance fibre-reinforced concrete (a) The influence of fibres on the tensile behaviour
35 with a reference quasi-static tensile strength f t0 = 10 MPa.
Model identification for ultra-high performance fibre-reinforced concrete (a) The influence of fibres on the tensile behaviour
This dependency has been introduced into the PRM model. In figure 6a are presented the experiment-calculation correlations. The first rebound of the velocity allows the peak stress to be computed using the Novikov formula [26] . The comparison of the velocity profiles for UHPC and UHPFRC (with 2% volume steel fibres) allows the influence of the fibres to be clearly identified: the presence of fibres induces additional strength and the residual velocity for this specimen is below the signal measured for a UHPC specimen which appears more brittle. Two-dimensional plane strain numerical simulations have been carried out with the PRM coupled model (figure 6b) for different volume fraction of fibres: V f = 0%, 0.5% or 1%. For V f = 0%, the velocity profile is typical of a brittle material. Lots of similarities can be found with the velocity profile obtained on mortar specimens [13] . Increasing the volume fraction of fibres leads to a higher energy consumption during the dynamic fracture process. In this simulation, the use of V f = 1% results in an overestimation of the post-peak residual strength. This appears on figure 6b as a very plastic response (dashed line). It seems numerically that only about a quarter of the total volume fraction of fibres (i.e. 0.5%) participates in the material resistance.
Model identification for ultra-high performance fibre-reinforced concrete (a) The influence of fibres on the tensile behaviour
(b) Confined behaviour of ultra-high performance fibre-reinforced concrete
Model identification for ultra-high performance fibre-reinforced concrete (a) The influence of fibres on the tensile behaviour
To identify the parameters describing the confined behaviour of UHPFRC, triaxial experiments were carried out with a high-pressure cell. The UHPC specimens were small cylinders (40 mm height and 20 mm in diameter). Tests were conducted on specimens without fibre reinforcement for two reasons: firstly, the fibres complicate the measurement of strain by means of gauges during triaxal tests and, secondly, it has been shown previously [27] that the steel fibres have only a negligible influence on similar UHPFRC compositions.
Model identification for ultra-high performance fibre-reinforced concrete (a) The influence of fibres on the tensile behaviour
The increase in deviatoric strength with confining pressure has been fitted from tests conducted with an initial fluid pressure of 200, 300, 400 and 600 MPa. Moreover, the compaction curve has been extracted from a test carried out with a pure hydrostatic loading. However, this technique was limited to a maximum pressure of about 1 GPa, which is quiet low considering the high levels of confinement during a fragment penetration or a contact detonation. Consequently, plate impact experiments were conducted to complete the dynamic characterization up to 6 GPa axial stress. of a UHPC plate on a tantalum target plate at 470 and 750 m s −1 . The experimental method, the main results and the parameters identified from this dataset are given in [28] . The experimentcalculation comparisons illustrate the capacities of the PRM model to reproduce the dynamic behaviour of concrete under high pressures.
Ballistic impact on ultra-high performance fibre-reinforced concrete slabs (a) Experimental configuration and main results
The predictive capabilities of the PRM coupled model have to be assessed by comparing numerical results with impact experiments. Several impact tests (no obliquity) were performed on Ductal-FM ® (with 2% volume steel fibre) concrete slabs with a 100 mm calibre gas gun. In a concrete formulation, the maximum aggregate size of sand is about 0.5 mm and the high-strength steel straight fibres have a length of 13 mm and a diameter of 0.2 mm. The target section was 1.5 × 1.5 m with a thickness of either 20 or 80 cm. The projectile used was 52 mm in diameter and 300 mm in total length. Its nose was a tangent ogive with a calibre-radius head ratio of 5.77. The penetrator, made of 35NCD16 steel, had an internal cavity filled with an autonomous acceleration recorder (figure 8a). The total mass of the projectile was 2.6 kg. One may observe that the penetrator remained nearly undeformed after an impact at 233 m s −1 even though the nose of the projectile was slightly eroded (figure 8b). Several impact tests were performed on thick UHPFRC targets with impact velocities ranging from 233 to 440 m s −1 . The increase in penetration depth with velocity is plotted in figure 9 where the experimental results gathered on Ductal-FM ® are compared with the corresponding data obtained on standard concrete with the same steel projectile. For a given velocity in the considered range, the penetration depth in standard concrete is typically 30% higher than the value measured experimentally for UHPFRC (figure 9).
Ballistic impact on ultra-high performance fibre-reinforced concrete slabs (a) Experimental configuration and main results
Front craters formed during two penetration tests are shown in figure 10 . The front and rear craters obtained after a perforation test conducted on a thin UHPFRC target (200 mm) can also be observed in figure 10 . For each of them, the crater is irregular and many fragments are kept in place by the steel fibres. The average diameters of the front and rear craters have been estimated. The main experimental parameters and the main results of these three reference tests are shown in table 1. Again, significant differences can be seen between UHPFRC and standard concrete: in standard concrete targets, the crater size is about two times larger than that measured in Ductal-FM ® targets with the same impact configuration.
Ballistic impact on ultra-high performance fibre-reinforced concrete slabs (a) Experimental configuration and main results
(b) Numerical simulation using the Pontiroli, Rouquand and Mazars coupled model
Ballistic impact on ultra-high performance fibre-reinforced concrete slabs (a) Experimental configuration and main results
The PRM coupled model was implemented in ABAQUS/explicit through a VUMAT subroutine. This finite-element code is used to evaluate the consistency of the modelling approach by means of penetration and perforation experiments in UHPFRC blocks. The numerical simulations were run with two-dimensional axisymmetry and the projectile was assumed to be a rigid body. figure 12 ). For these two configurations, the damage patterns D t with no fibres or 1% fibres contributing to resist the penetration force are plotted; the black arrow indicates the average crater diameter determined experimentally. The steel fibres cause a noticeable decrease in the diameter of the damage zone. Considering the inherent uncertainties of impact experiments, the PRM coupled model provides a fairly good estimation of the crater width. The axial deceleration curves (figures 11c and 12c) and the penetration depth versus time of the projectile (figures 11d and 12d) confirm the performance of the modelling approach. The phase of constant deceleration, observed on experimental and numerical signals, corresponds to the tunnelling process during pure penetration [29] . It can be remarked that, at low velocity, the PRM model with fibres overestimates the penetration resistance. The orientation of the steel fibres near the free surface during the cast can explain this effect. For the penetration test performed at a higher velocity (440 m s −1 ), a good correlation is obtained assuming that 50% of the fibres (i.e. 1% fibre ratio) actually contribute to the tensile response of the material.
Ballistic impact on ultra-high performance fibre-reinforced concrete slabs (a) Experimental configuration and main results
The last numerical simulation concerns a perforation test conducted on a 0.2 m thick UHPFRC block. In this experiment, the projectile perforates the concrete slab at an impact velocity of 455 m s −1 and the measured residual velocity was 300 m s −1 . The computations were run without any fibre reinforcement or with V f = 1%.
Ballistic impact on ultra-high performance fibre-reinforced concrete slabs (a) Experimental configuration and main results
If no fibre reinforcement is assumed (figure 13a), the numerical result underestimates the penetration resistance of the concrete slab and the residual velocity of the projectile is 11% higher than the experimental value. With 50% of 'active fibres', the crater sizes in the front side and in the rear face of the concrete slab determined by numerical simulation appear consistent with the average values obtained experimentally (black arrows, figure 13 ). Again, with V f = 1%, the PRM coupled model provides an accurate estimated residual velocity of the projectile as well as front and rear crater sizes caused by the hard impact.
Conclusion
The design of new protective structures to guard against blast loads or high-velocity impact using UHPFRC can be undertaken through numerical simulations. However, the constitutive and damage models have to be accurate and validated with reliable data. Thus, considering the relatively recent knowledge concerning the mechanical response of this class of materials, extensive mechanical characterization is still necessary. In this work, quasi-static and dynamic mechanical tests were conducted on UHPC and UHPFRC to identify the main characteristics of the UHPFRC response in tension and in confined compression. Then, a new material parameter dataset was calibrated for the fully coupled PRM model. This sophisticated concrete model consists in a phenomenological modelling approach including the main mechanisms activated under high strain rate and high confining pressure, coupling damage to plasticity and compaction. The influence of steel fibres disseminated in the concrete has been included in the model: the damage kinetic has been modified to improve the description of the influence of the fibres on the dynamic tensile fracture of UHPFRC.
Conclusion
The modelling approach was validated step by step using characterization data. The final evaluation of the PRM model consists in simulating penetration problems in UHPFRC targets. Penetration depth or deceleration of the projectile in the concrete block allowed the numerical predictions to be compared qualitatively with experimental results. Taking into account all the mechanisms previously studied independently, the PRM model allows an overall good correlation with experimental data. In the three impact tests, the width of the craters provoked by the impact was fairly well estimated by computations for 1% active fibres. However, the PRM model is based on an isotropic formulation and the fibre ratio actually contributing to the material resistance needs to be more accurately determined.

{'paper_id': '31521023', 'title': 'Ultra-high performance fibre-reinforced concrete under impact: experimental analysis of the mechanical response in extreme conditions and modelling using the Pontiroli, Rouquand and Mazars model', 's2_url': 'https://api.semanticscholar.org/CorpusID:31521023', 'source': 'S2ORC'}