Dr Constantin Lefter

Dr Sylvain Rat

M. D José Sánchez Costa

Dr María D Manrique-Juárez

Dr Carlos M Quintero

Lionel Salmon

Dr Isabelle Séguy

Dr Thierry Leichle

Liviu Nicu

Philippe Demont

Dr Aurelian Rotaru
email: rotaru@eed.usv.ro

Gábor Molnár
email: gabor.molnar@lcc-toulouse.fr

Azzedine Bousseksou
email: azzedine.bousseksou@lcc-toulouse.fr

Dr C M Manrique-Juárez

Dr L Quintero

Dr G Salmon

Dr A Molnár

Bousseksou

L Nicu Laas

Prof P Demont

Current Switching Coupled to Molecular Spin-States in Large-Area Junctions

for the investigation of the interplay between molecular spinstate switching and charge transport in SCO materials. Our approach is based on robust, well-reproducible, large-area vertical devices with nanomeric SCO spacer layers.

Obviously, the main bottleneck to our approach is the need for high quality, ultrathin, pinhole-free SCO fi lms over large electrode areas, which resist also to the deposition of the upper metallic electrode. To this aim we have chosen the [Fe(H 2 B(pz) 2 ) 2 (phen)] SCO complex 1 (H 2 B(pz) 2 = dihydrobis(pyrazolyl)borate and phen = 1,10-phenanthroline, see Figure 1 a), [ 29 ] which can be deposited on surfaces by thermal evaporation. [ 17,25,27,[30][31][32][33][34] The bulk powder of 1 displays a rather abrupt, cooperative, fi rst-order thermal spin transition around 165 K with a very narrow hysteresis, while the vacuum-deposited fi lms exhibit a very gradual (i.e., weakly cooperative) thermal spin crossover between ≈100 K (LS) and 200 K (HS). Both the powder and fi lm samples are known to exhibit also light-induced excited spin-state trapping (LIESST) phenomenon with a long-lived metastable HS state below ≈50 K. [ 30 ] Interestingly, the spin crossover properties of the fi lms are not much altered by the fi lm thickness from the micrometer scale down to the isolated molecule level. [ 33,34 ] Previous X-ray diffraction studies revealed that the fi lms are amorphous, [ 30 ] which explains probably the loss of the fi rst-order nature of the spin transition (i.e. the lack of cooperativity), the lack of signifi cant fi nite size effects as well as the remarkably smooth and homogenous fi lm morphology observed even for very small thicknesses.

The schematic representation of the device made with 1 is shown in Figure 2 a. The basic device architecture is fairly similar to the SCO devices reported by Matsuda et al. [ 14 ] Thin fi lms of 1 were fi rst grown by thermal evaporation on pre-patterned ITO (indium-tin-oxide) bottom electrodes on glass substrates. This choice of the substrate and the electrode was motivated by their optical transparency, which allows both for optical triggering and probing the spin-state changes in the devices using visible light irradiation. In the next step the upper Al electrodes were carefully deposited by thermal evaporation through a shadow mask into a crossbar confi guration (Figure 2 b). The effective junction area of each device is ≈3 mm 2 . Junctions of 1 were fabricated in a single run with three different nominal thicknesses (10, 30, and 100 nm). The fi lm thickness was monitored in situ during the deposition by a quartz crystal balance and was confi rmed also ex situ by acquiring atomic force microscopy (AFM) data and scanning electron microscopy (SEM) images of the device tranches milled by focused ion beam (FIB) etching (Figure 2 c-see also the Supporting Information.) Switchable molecular compounds are being more and more considered for the replacement of conventional inorganic semiconductor materials as the miniaturization of silicon based devices approaches fundamental technological and physical limits. [ 1 ] While early developments with molecular switches focused primarily on single molecule devices (a fi eld referred to as "molecular electronics"), [ 2 ] the recent progress of "organic electronics" [ 3 ] and "organic spintronics" [ 4 ] has paved the way for considering the use of molecular switches in more conventional devices as well, such as fi eld-effect transistors, light emitting diodes, spin valves, etc.

Transition metal complexes exhibiting spin crossover (SCO) phenomenon between the low spin (LS) and high spin (HS) electronic confi gurations [START_REF] Gütlich | Spin Crossover in Transition Metal Compounds I[END_REF][START_REF] Bousseksou | [END_REF][7] appear particularly attractive for the development of electronic and spintronic devices. Indeed, increasing interest has been focused recently on the charge transport properties of SCO compounds, including the investigation of the dielectric permittivity and electrical conductivity of bulk powders [8][9][10] and composite materials, [11][12][13] the integration of thin fi lms, [14][15][16][17] nanoparticles, [18][19][20][21] and single molecules [22][23][24][25][26][27][28] into test devices. Here, we describe a novel approach AFM studies confi rmed also the high quality of the fi lms of 1 in terms of homogeneity and continuity (see Supporting Information). Of particular interest is the thinnest, nominally 10 nm fi lm, for which AFM measurements revealed 8.5 nm thickness and 0.2 nm roughness, which was further confi rmed by X-ray refl ectometry (8.1 ± 0.2 nm) and by SEM (12 ± 3 nm). While the 10 nm fi lms are smooth and continuous, for 5 nm thickness a few pinholes could be depicted hence no junctions of this size were fabricated. Since the spin transition in 1 is accompanied by a signifi cant change in volume, the mechanical stress imposed by the "sandwich" geometry may change or even preclude the SCO in the device. To verify the occurrence of SCO we acquired optical refl ectivity spectra of the glass/ITO/SCO (100 nm)/ Al stack as a function of temperature between 300 and 5 K. This method probes the change in the refractive index associated with the spin transition. [ 35 ] As shown in Figure 1 b the refl ectance ( λ = 640 nm) of the multilayer decreases signifi cantly between ≈200 and 100 K and increases again to the initial level below 50 K. These changes of the refl ectivity signal correspond, respectively, to the wellknown thermal and photoinduced (LIESST effect) spin crossover in the fi lms of 1 , [30][31][32][33][34] providing thus unambiguous evidence that the SCO properties are preserved in the sandwich structure. The photoinduced LS → HS SCO phenomenon was also detected in the 10 and 30 nm fi lms through the refractive index changes, but due to the smaller thickness the measurement was carried out by depositing the fi lms on a surface plasmon resonance sensor (see the Supporting Information for details). I-V characteristics of the different devices were strongly nonlinear and exhibited either temperature independent or insulatinglike thermal activation behavior, which rules out the contribution of short circuit channels. These observations together with the highly reproducible device characteristics confi rm the high quality of the SCO fi lms in the junctions. The devices retained their characteristics during several tenths of thermal and bias cycles under electric fi elds up to ≈10 6 V cm -1 and could be stored in ambient air for several days refl ecting thus high robustness. However, a slow increase of the junction resistances was also detected on a weekly time scale in air.

Variable temperature current-voltage curves registered for devices with 10 nm SCO layer ( Figure 3 a) are clearly characteristic of tunnel junctions. In particular, the thermal variation of the I -V curves is negligible in a broad temperature range (5-293 K), which is a generic feature of tunneling conduction. The I-V curves obey the relationship I = I 0 ( V + γV 3 ) and the differential conductance d I /d V curves exhibit parabolic shape typical of tunneling junctions [ 36 ] (see Supporting Information). A closer examination of the temperature dependence of the resistance of the junction at a constant bias of 1 V reveals a slight increase (≈10%) in the thermal spin crossover range between ≈200 and 100 K (Figure 3 b). It is, however, impossible to conclude from these data on a link between the two phenomena because the reproducibility of the I ( T ) data was comparable with the magnitude of the observed (small) changes.

In order to unambiguously demonstrate the infl uence of SCO on the device characteristics we used light irradiation to alter the spin-state of the junction. We cooled the device to 5 K in dark and we irradiated the sample using a halogen lamp. Immediately the current fl owing in the device dropped by 7% from ≈4.2 to 3.9 µA ("Light ON" in Figure 3 c). Following this fi rst irradiation, the current intensity remained at the same level over successive irradiation cycles at 5 K (see OFF-ON-OFF in the blue area of Figure 3 c) for more than one night. This fi nding can be obviously correlated with the LIESST effect: light irradiation leads to the population of the HS state, which is metastable at 5 K with a long lifetime of several days. To further substantiate the relationship between the current intensity and the spin state of the device this latter was heated to 100 K. As mentioned before, above ≈50 K the metastable HS state relaxes rapidly to the LS ground state. Indeed we observed that upon heating the current intensity rises back to the same level as before the fi rst light irradiation (see yellow area in Figure 3 c). This "dark cooling-photoswitching-heating" cycle was repeated three times with the same result. The device was also irradiated by light at 100 K where no LIEEST effect occurs, and in this case no signifi cant effect on the conductivity was observed (Figure 3 d). All these results point thus to the conclusion that switching the spin-state of the junction from the LS to the HS state leads to a well reproducible decrease of the tunneling current by 7%.

Figure 4 shows the characteristics of the 30 nm device. (Very similar results were obtained with 100 nm junctions, including also junctions with a poly(3,4-ethylenedioxythiophene):polystyrene-sulfonate (PEDOT:PSS) hole-transport layer, which are shown in the Supporting Information.) As it can be expected, the increasing junction width leads to a considerable change of the charge transport characteristics. The I-V curves display a rectifying behavior, with current fl owing only under positive voltage, i.e., when the Al electrode is the cathode (Figure 4 a). We fi tted the I-V curves with the diode equation [ 37 ] exp 1

0 B I I eV nk T = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ - ⎡ ⎣ ⎢ ⎤ ⎦ ⎥
where I 0 is the reverse bias current, k B the Boltzmann constant, e the charge of electron and n the "nonideality" factor. While the fi ts are satisfactory down to 100 K, the fi tted values of n are rather high ( n = 12 at 293 K) when compared to the commonly observed ones ( n = 1-2). This deviation from the usual behavior might be an indication of transport-limited current (vide infra).

Even more important is the fi nding that the conductance of the 30 nm junction at a constant bias of 5 V is thermally activated-it increases by approximately one order of magnitude between 293 and 100 K (Figure 4 b). The Arrhenius plot of the conductivity of the device exhibits a change of slope near 200 K in a reversible and reproducible manner. It is tempting to link this phenomenon with the occurrence of thermal SCO below 200 K, but this correlation may be only fortuitous. In this case also a more obvious correlation was obtained through lightinduced spin-state switching experiments. When shining white light on the device at 5 K we observed a prompt increase of the current intensity from 250 to 872 pA, followed immediately by a decrease back to 687 pA (Figure 4 c). When the light was turned off the current level dropped back to 145 pA. In the subsequent ON/OFF cycles we observed only a reversible switching between 145 and 665 pA. The complete sequence (dark cooling and light ON/OFF cycles) and associated photocurrent phenomena were well reproduced several times-in particular the characteristic "spike" upon the fi rst irradiation was repeatedly detected. In analogy with the 10 nm junction we assign the "irreversible" current drop during the fi rst irradiation to the photoinduced spin-state trapping (LIESST) effect. One might notice that the LIESST effect in this junction is more important: when going from the LS to the HS state the current intensity drops by ≈50%. It is important to remark that at 100 K such "irreversible" photoeffect on the current was not observed in line with the fact that LIESST is not effi cient at this temperature (Figure 4 d).

On the other hand, a signifi cant "reversible" photocurrent was observed both at 5 and 100 K, the junction being approximately four times more conductive in the photostationary state with respect to the "dark" state. It is important to underline, however, that the spin-transition induced drop of the current intensity is not much different in the dark and photostationary cases. We suggest therefore that the "reversible" photocurrent probably originates from light absorption in the Al electrode near the interfacial layer. Since it was not observed in the 10 nm device we tentatively attribute this phenomenon to a photoactivated charge injection through the electrode interface barrier.

While a tunneling mechanism for the description of charge transport in the 10 nm junctions is obvious, for the thicker junctions the charge transport may be either injection-limited or bulk transport-limited. A common criterion used to identify the transport mechanism in low mobility semiconductor devices is based on the analysis of the current density on thickness for a given electric fi eld. [ 38 ] In our devices a very strong dependence is observed (see Supporting Information), hence we conclude on a mostly bulk-limited mechanism, which is in line with the very low conductivity of 1 . This conclusion is also in agreement with the fact that the fi t of the I-V curves by the diode equation gave physically meaningless results. To further investigate the charge transport process in the devices we have also acquired their dielectric spectra, i.e ., the temperature and frequency f dependent complex permittivity ε *. This latter can be related to the complex conductivity σ * and dielectric modulus M * by σ * = i ωε 0 ε *, and M* = 1/ ε *, respectively. ( ε 0 is the permittivity of the free space and ω is the angular frequency.) Figure 5 shows the frequency dependence of the real part of the complex conductivity σ ′ together with the dielectric loss modulus M ″ acquired at selected temperatures for the 10 and 30 nm junctions. (See the Supporting Information for the 100 nm junction.) One can observe very similar frequency behavior between the different junctions. Each junction follows the Jonscher's power law of ac conductivity, [ 39 ] characteristic for hopping transport, σ ′( f ) = σ ac (f) = σ dc + Af s , where σ dc is the dc conductivity, s is the power law exponent and A is a constant. The frequency exponent s was found to be independent of both the temperature and the thickness of the junctions and takes a value of 1.82. A value of s > 1 means that charge carrier motion involves localized hopping between neighboring sites. In the electric modulus representation the frequency dependence of the loss modulus M ″ reveals a simple Debye peak for each junction. This loss peak can be attributed to the conductivity relaxation process and the peak maxima is related to the relaxation time by τ Μ = 1/ ω max = ε 0 ε ∞ / σ dc . [ 40 ] ( ε ∞ is the high frequency limit of ε ′.) This relaxation times range between 10 -4 and 10 -2 s for the three junctions.

Based on these data the charge transport in the different devices can be ascribed to a similar short-range hopping conductivity process. The main difference between the dielectric spectra of the junctions is observed in their temperature dependence. In the case of the 30 and 100 nm junctions both the conductivity and the relaxation frequency exhibit thermal activation with a small activation barrier (see Supporting Information), while for the 10 nm junction both quantities are temperature independent. The weak temperature dependence implies that tunneling is important, particularly for the 10 nm junction. Taking into account the signifi cant thickness of this latter junction tunneling must refer here to a multistep tunneling process by activationless hopping between localized states. [41][42][43] In the case of the 30 and 100 nm junctions the possible mechanism of the conduction can be described as a charge injection into the molecular orbitals (MOs) of 1 at one electrode interface (possibly by tunneling), followed by both diffusive tunneling and thermally activated hopping of charge carriers through the junction to the other interface, where the charge extraction occurs.

Starting from this picture of the charge transport in the different junctions we suggest that the spin-state dependence of the junction resistance is primarily related to the change of carrier hopping rates. It is well known that the hopping rate usually increases with increasing phonon frequencies and decreases with increasing distance between the localized states. [ 44 ] Since the LS to HS switching in 1 results in a signifi cant increase of the metal-ligand distances and thus a decrease of the associated vibrational frequencies we understand that this switching process must lead to a lower current in the HS state, which is indeed observed in our experiments. Of course, a number of other parameters, such as the change of the dielectric permittivity and the thickness of the junction upon the SCO, will also impact the charge transport and will have to be analyzed in detail in future work. In any case it can be suggested that the current switching in our junctions is not a direct effect of the electronic state change, but it arises mostly due to the coupling of the electronic states with the crystal lattice (volume change) and in particular with the phonon density of states.

It is also instructive to compare the work functions of Al (4.2 eV) and ITO (4.5 eV) with the relevant molecular orbital energy levels of 1 . These latter were reported from density functional theory calculations by Gopakumar et al. [ 25 ] for a molecule of 1 adsorbed on Au(111) surface. They determined the HOMO/LUMO energy levels both in the HS (-3.5/-3.9 eV) and LS (-4.7/-2.6 eV) states (see Supporting Information). From cyclic voltammetry measurements (see Supporting Information) we inferred HOMO (-4.6 eV) and LUMO (-2.5 eV) energies in the LS state in good agreement with the theoretical data, but no HS data could be obtained for the lack of appropriate solvents. Nevertheless, it is obvious that the LS to HS switching must lead to a very signifi cant decrease of the HOMO-LUMO gap. This picture would imply a substantial impact of the spin state switching on the injection barriers and thus on the current fl owing in the device. This was not observed in our experiments, presumably due to the bulk-limited nature of the charge transport in our devices. Nevertheless, other phenomena at the electrode interfaces, such as the hybridization of molecular orbitals with substrate states at the electrode interface and subsequent loss of the SCO properties, [ 26 ] must be also considered. Hence an important perspective of our work will be the design of vertical SCO junctions with appropriate properties for injection-limited charge transport in which we expect a huge effect of SCO on the current intensity. This work will require the detailed study of different SCO and electrode materials with extensive interface engineering guided by MO calculations. Since the magnetic properties of the junction change also upon the SCO (diamagnetic vs paramagnetic) perhaps even more appealing prospects appear for spintronic applications. The technical challenge is substantial, however, as it will be necessary to develop methods for the deposition of pinhole-free fi lms of these compounds between large area magnetic electrodes. Finally, taking into account the current state of the art of the SCO fi eld [START_REF] Gütlich | Spin Crossover in Transition Metal Compounds I[END_REF][START_REF] Bousseksou | [END_REF][7] we can safely claim that there is no fundamental obstacle to prepare nanometric junctions with SCO compounds exhibiting near room temperature switching properties, which is of course very relevant for any technological application.

In summary, we fabricated vertical optoelectronic devices with thin fi lms of the [Fe(H 2 B(pz) 2 ) 2 (phen)] spin crossover complex sandwiched between large area electrodes. These devices allowed us to trigger and probe the spin-state switching in the SCO layer by optical means while detecting the associated electrical resistance changes in the junctions. The stable and high current intensity in the devices provided also possibility for mechanistic studies by means of temperature-and frequency-dependent dielectric spectroscopy. The devices with 10 nm junction width displayed activationless tunneling conductivity, while the thicker (30 and 100 nm) junctions exhibited diode-like rectifying characteristics and bulk-limited thermally activated currents. We provided experimental evidence that the spin-state switching of the junctions leads to a substantial and reversible increase/decrease of the current intensity (up to 50%) in the LS/HS states and we correlated this effect with the change of carrier hopping rates upon the spin conversion. Overall these results provide very promising novel perspectives for using spin crossover compounds in nanoelectronic and spintronic devices. To this aim it will be necessary to carry out more extensive investigations on a series of different SCO compounds with systematically modifi ed MO levels and appropriate interface matching with different electrode materials.

Experimental Section

The bulk powder of complex 1 was synthetized using a procedure reported by Moliner et al. [ 29 ] with a small variation (see the Supporting Information for synthesis and characterization details). Thin fi lms of 1 were grown by thermal evaporation at 110 °C in a high vacuum chamber (10 -7 Torr) at a rate of 0.05 nm s -1 on 180 nm thick ITO electrodes. These latter were purchased from Praezisions Glas & Optik GmbH and patterned by wet etching using a conventional photolithography mask. Alternatively, for a 100 nm junction the surface of the ITO electrode was planarized by spin-coating a PEDOT:PSS hole transport layer on top of it. The upper 100 nm thick Al electrode was fi nally thermally evaporated through a shadow mask on the substrate. To avoid perforation of the SCO layer the Al fi lm was deposited at a rate of 0.1 nm s -1 and the substrate was cooled by water. The fi lms of 1 were characterized by AFM for their morphology and by optical methods to control the SCO properties (see Supporting Information for details). The junction widths were determined by SEM following FIB milling. I-V and I-T characteristics of the devices were acquired in two-probe mode using a Keithley-6430 source-meter and an Oxford Instruments OptistatCF liquid-helium cryostat. The device was irradiated by light through the cryostat windows using a 100 W halogen lamp. Dielectric spectra (1 Hz-1 MHz) of the devices and the powder sample were recorded between 100 and 250 K by means of a Novocontrol BDS 4000 broad-band dielectric spectrometer. Frequency sweeps were carried out isothermally and the applied AC voltage was 0.7 and 1 Vrms for the 10 and 30-100 nm junctions, respectively.

Figure 1 .

 1 Figure 1. Structure and properties of the spin crossover compound 1. a) Molecular structure of 1 . b) Variable temperature optical refl ectance ( λ = 640 nm) of a 100 nm thick fi lm of 1 in an ITO/ 1 /Al stack showing thermal (100-200 K) and photoinduced (<50 K) spin crossover phenomenon between the high spin and low spin states. The corresponding electronic confi gurations are also shown. The effective photoinduced population of the metastable HS state by the probe light is observed at low temperatures due to the very slow relaxation back to the ground LS state. The quantitative thermal population of the HS state at high temperatures is an entropy driven process.

Figure 2 .

 2 Figure 2. Device structure. a) Schematic representation of the ITO/SCO/Al junction. b) Photograph of a device with six crossbar junctions. The regions with ITO and SCO fi lms are highlighted by black and grey lines, respectively. Silver paste contacts on the ITO are also visible. c) SEM images of FIB milled cross-sections of the junctions with different SCO fi lm thickness (10, 30, and 100 nm).

Figure 3 .

 3 Figure 3. Electrical characteristics of a 10 nm junction. a) I -V curves registered at room temperature, 100 and 5 K at a rate of 100 mV s -1 . b) Temperature dependence of the current density in the junction registered at a rate of 5 K min -1 . c) Visible light irradiation effect on the current fl owing in the junction at 5 K through two successive ON/OFF cycles (blue area), followed by heating from 5 to 100 K in the dark (yellow area). Current intensities in the HS and LS states are indicated by dashed lines. d) Visible light irradiation effect on the current fl owing in the junction at 100 K through two successive ON/OFF cycles. The applied bias in (b-d) was 1 V.

Figure 4 .

 4 Figure 4. Electrical characteristics of a 30 nm junction. a) I -V curves registered at room temperature, 100 and 5 K at a rate of 100 mV s -1 . The inset is a zoom on the 100 and 5 K data. b) Temperature dependence of the conductivity of the junction registered at a rate of 5 K min -1 . c,d) Visible light irradiation effect on the current fl owing in the junction through three successive ON/OFF cycles at 5 K (c) and 100 K (d). Current intensities in the HS and LS states are indicated by dashed lines both in dark and in photostationary conditions. The applied bias in (b-d) was 5 V.

Figure 5 .

 5 Figure 5. Frequency dependence of the real part of the ac conductivity and that of the loss modulus at selected temperatures for a 10 (a) and 30 nm (b) junction.

Acknowledgements

This work was funded by the French RENATECH network and a joint ANR-UEFISCDI project (9RO-FR/01.02.2013, ANR-12-IS07-0003-01). C.L. thanks the French Ministry of Foreign Affairs for an Eiffel scholarship. S.R. thanks the French Education and Research Ministry for a MESR fellowship. M.D.M.-J. thanks the CONACYT for a PhD grant. J.S.C. thanks the IEF Marie-Curie research program (NanoSCOpe 328078).

Supporting Information

Supporting Information is available from the Wiley Online Library or from the author.


Thanh Song Phan

Pierre Rossi
email: pierre.rossi@ifsttar.fr

Jean-Louis Tailhan

Numerical modelling of the concrete/rebar bond

Keywords: Reinforced Concrete, Flat Steel, Concrete/Rebar Interface, Cracking

The simulation of the behaviour of the rebar-concrete bond is of primary importance for an accurate description of cracking processes in reinforced concrete structures and an improved prediction of their durability. In this paper, the methodology used to identify the mechanical behaviour of a rebar-concrete bond in the case of a particular steel reinforcement is first mentioned. The methodology consists in simulating the probabilistic mechanical behaviour of RC tie-beams (170 x 10 x 10 cm), subjected to tension, using a probabilistic approach for the mechanical behaviour of the concrete and a deterministic model for the concrete/rebar bond.

The tie-beams are reinforced by a flat steel rebar with a rectangular cross section (25 x 3.5 mm). This approach at macro-scale level is compared to another one at micro-scale level in which notches/indentations of the flat steel rebar are explicitly taken into account in the simulation, the bond behaviour being considered through the local concrete cracking around the rebar.

Introduction

In the process of developing a new technological solution of reinforcement for RC structures, it is of great importance to understand and identify the main physical mechanisms involved.

Numerical simulation tools can then be very useful to demonstrate, for example, the mechanical relevancy of the solution. A French construction company has developed in recent years, a new type of flat steel reinforcement with notches/indentations (Figure 1). It can be used in structural elements and shows some advantages:

• reduction of the total thickness of the structure by reducing the thickness of the steel,

• decrease in the quantity of steel by using tendrils replacing the anchorages of steel round rebars,

• increase in relative surface of adhesion,

• easy implementation.

Mechanically speaking, the main difference between this new type of steel reinforcement and the traditional one (cylindrical rebar) concerns the interface behaviour between the flat steel and the concrete. So, the modelling of the interface behaviour is very important for a relevant analysis of the behaviour of the structure reinforced with this type of steel reinforcement.

In this work, it has been chosen, as a starting point, to model a tie-beam reinforced by flat steel and subjected to axial tension. The study is mainly focused on the mechanical effect of the concrete/rebar bond on the cracking process of the tie-beams. The behaviour of the concrete in tension is represented by a probabilistic cracking model [START_REF] Rossi | A probabilistic model for material behaviour analysis and appraisement of concrete structures[END_REF][START_REF] Rossi | Scale effects on concrete in tension[END_REF], which takes into account the heterogeneity of the material and its inherent scale effect and gives an explicit representation of the cracks in the concrete.

Concerning the concrete/rebar bond, three strategies of modelling are considered:

1. In the first one, called macro-scale level strategy, the concrete/rebar bond is represented by standard interface elements. Their behaviour is described by a simple deterministic damage model which macroscopically takes into account the main physical phenomena of the interface through only two parameters: cohesion and slip (i.e. tangential relative displacement between steel and concrete).

An inverse analysis approach is used to identify the parameters of this type of behaviour. The numerical results are compared with those of the experimental ones [START_REF] Augustin | Report the results of experimental tests on RC tie-beams[END_REF]. The comparison is made in terms of global response (load vs. relative displacement), as well as local information (cracks opening, cracks spacing and number of cracks).

2. In the second one, called mix-scale strategy level, two manners of representing the concrete/rebar bond are considered in the same numerical simulation:

• In the central part of the steel rebar, on a length of 30 cm, the notches/indentations of the flat steel rebar are explicitly taken into account in the simulation and no interface elements are used.

• In the rest of the rebar, the interface is modelled as previously by interface elements.

Concerning the part of the rebar where the notches/indentations are explicitly modelled, the concrete/rebar interface behaviour is taken into account only by the concrete cracking around the rebar (the bond between the rebar and the concrete being considered as perfect) while, in the rest of the rebar, the values of the parameters related to the interface elements are the same than those previously determined.

3. In the last one, called micro-scale level strategy, the notches/indentations are explicitly modelled along the totality of the rebar (without any interface elements and with a perfect bond between the concrete and the rebar).

The purpose of this paper and of the use of the proposed multi-scale strategy of modelling is to evaluate if:

• the cracking behaviour of reinforced concrete is correctly modelled by using a very simple macroscopic model of the concrete/rebar bond if the modelling of concrete cracking is correctly done (physically based) ?

• it is possible, in a same numerical analysis, to get, at the same time, global information related to macro-cracks around the rebar and very local one related to micro-cracks situated in very small zone around this ribbed rebar ?

• the concrete/steel bond behaviour is mainly due to the micro-cracking of the concrete around the ribbed rebar (due to the presence of notches/indentations) ?

Presentation of the concrete cracking model

The behaviour of concrete is represented by a probabilistic cracking model developed at the IFSTTAR 2 (formerly LCPC 3 ) by Rossi [START_REF] Rossi | A probabilistic model for material behaviour analysis and appraisement of concrete structures[END_REF] and recently improved by Tailhan et al. [4]. This model has the particularity to take into account two major characteristics of concrete: heterogeneity on the one hand and its sensitivity to scale effects, on the other hand [START_REF] Rossi | Scale effects on concrete in tension[END_REF]. The physical basis of the model (presented in detail in the past [START_REF] Rossi | A probabilistic model for material behaviour analysis and appraisement of concrete structures[END_REF][START_REF] Rossi | Scale effects on concrete in tension[END_REF]) can be summarized as following:

• The heterogeneity of the concrete is due to its composition. The local mechanical characteristics (Young's modulus E b , tensile strength f t ) are randomly distributed.

• Scale effects are also a consequence of the heterogeneity of the material. The mechanical response of the material directly depends on the volume of material stressed.

• The cracking process is controlled by the presence of defects in the cement paste, by the heterogeneity of the material and by the development of tensile stress gradients.

In terms of numerical modelling (in the frame of finite elements method), these physical evidences can be taken into account as following: • The tensile strength is initially randomly distributed on all elements of the mesh using a probability distribution whose characteristics depend on the volume of the finite element/volume of the largest aggregate ratio. So, for a given concrete, the distribution functions related to a given mesh are known [START_REF] Rossi | A probabilistic model for material behaviour analysis and appraisement of concrete structures[END_REF][START_REF] Tailhan | From local to global probabilistic modelling of concrete cracking[END_REF].

• A crack is represented by the fact that the stiffness of a volume element becomes equal to zero. That happens when the principal tensile stress at the centre of the element reaches the tensile strength (which is randomly distributed in the mesh) attributed to this element.

• The cracking propagation in concrete is, in this model, the result of the creation of elementary "holes" that appear randomly.

The distribution function of the concrete tensile strength used in the numerical calculation is determined by just knowing the volume of the finite element/volume the largest aggregate ratio and the concrete average compressive strength [START_REF] Rossi | A probabilistic model for material behaviour analysis and appraisement of concrete structures[END_REF][START_REF] Tailhan | From local to global probabilistic modelling of concrete cracking[END_REF]. .

Presentation of the concrete/rebar bond modelling

Many studies have been performed to simulate the behaviour of RC structures taking into account the behaviour of the concrete/rebar bond (models based on the plasticity theory [START_REF] Cox | Implementation of a plasticity bond model for reinforced concrete[END_REF] or the damage theory [START_REF] Ragueneau | Thermodynamic-based interface model for cohesive brittle materials : Application to bond slip in RC structures[END_REF][START_REF] Willam | Interface damage model for thermomechanical degradation of heterogeneous materials[END_REF]. But very few of them were focused on the role of this interface in the cracking process of the concrete.

In this section, a simple and robust interface model is presented. It takes into account the nonlinear behaviour of the concrete/rebar bond in the frame of damage mechanics. It can also takes into account physical phenomena of concrete/rebar bond such as interface sliding, cracks appearance and degradation process. The interface zone is represented by interface elements that connect the concrete to the rebar. Their role is:

• to ensure the displacement continuity between the concrete and the steel before the slip of the interface and before the concrete cracking and consequently to ensure the transfer of load (and therefore of stresses) between steel and concrete.

• to represent the macroscopic mechanical effect of the rebar at the ribs level (which are not explicitly represented in the mesh).

• to simulate a local failure between steel and concrete along the rebar, if a shear cracking occurs and leads to a loss of the local adhesion.

• to simulate the local friction between the concrete and the steel after the interface failure.

The model is implemented in 2D and 3D. It considers the concrete/rebar bond as a material zone that progressively degrades in shear (the tensile failure is neglected). During this degradation, and before the total failure of the interface, stresses are considered to be still transmitted to the concrete.

A very simple approach is chosen, based on a damage model which maintains a constant level of stress when the critical shear has been reached (figure 2). When the relative tangential displacement between the concrete and the rebar exceeds a critical value, the interface element is declared broken [START_REF] Rossi | Comportement dynamique des bétons : du matériau la structure[END_REF]. After the failure, friction behaviour of Mohr-Coulomb's type is considered. After the failure, when δ t > δ t cri and d = 1, a friction behaviour is considered and a Mohr-Coulomb type criterion is applied (Figure 3). In that case, an associated flow rule g is also used:

g = |τ| -σ n tanψ (4) tanϕ = tanψ
Where τ = f (τ 1 , τ 2 ) is the resultant tangential stress, ψ the dilatancy angle and φ the friction angle. Without any further information concerning φ and ψ and considering that the failure essentially occurs in the concrete surrounding the steel, a value of 30• is retained (value obtained from Rossi [START_REF] Rossi | Comportement dynamique des bétons : du matériau la structure[END_REF]).

In this study, only the values of the maximum shear stress, C, and of the tangential critical relative displacement, δ t cri , have to be determined. This identification is made by a numerical inverse analysis based on a comparison with experimental results. In consequence, this determination is available only for a given rebar geometry and a given concrete.

Modelling of the tie-beam test -Use of a flat steel rebar

An example of the ribs of the flat steel rebar studied in the frame of tie-beam tests is presented in figure 1.

Presentation of the tie-beam test

The experimental tests on tie-beams were conducted in the laboratory of Polytech Clermont-Ferrand (Blaise Pascal University of Clermont-Ferrand, France). In these tests, the reinforced concrete tie-beams was subjected to pure tension. The dimensions of the concrete prismatic specimens were: 170 × 10 × 10 cm. The reinforcement was a flat steel with a rectangular cross section of 25 × 3.5 mm. The steel rebar was centred in the middle of the tie-beam. To minimize edge effects in the concrete during the test, "no friction zones" of 10 cm length were considered at each extremity (figure 4).

Figure 4 schematizes the experimental device and figure 5 shows a general view.

To measure the elongations of the concrete and the steel, six displacement sensors (LVDTs) were placed on the specimen (Figure 4). These LVDTs were fixed on a common undeformable support independent of the specimen.

Two sensors (LVDTs 5 and 6) were located on the steel rebar (on both extremities of the tiebeam).

Four other sensors (LVDTs 1, 2, 3 and 4) on each face of concrete at the top and bottom of the tie-beam were also placed.

As a consequence, each couple of LVDTs provided the measurement of a relative displacement on the base length of 170 cm or 150 cm respectively. The six displacement sensors and the two load sensors were connected to a central acquisition system where the information was automatically recorded.

The tests were run with a load rate corresponding to a quasi-static condition so without any influence on results. During this loading procedure, steps were realized to measure crack opening evolution at the surface of the specimens.

During this experimental study, 9 specimens were tested.

Concrete and steel reinforcement

The concrete used in these tests is a C40/50 one, and its mix design formula is reported in table 1.

To determine the material characteristics of the concrete, compressive tests were performed on standardized cylindrical specimens (160 mm in diameter and 320 mm high). It must be noticed that the compressive tests were performed at the same age as the tie-beam tests, and under the same conditions of preservation.

Table 2 gives the material characteristics of the concrete and the flat steel rebar, considered in the numerical simulations.

Strategy of analysis of the test

A strategy of analysis of the tests results is proposed to get a consistent method to analyse the experimental tests and the calculations.

Determination of the mean crack opening

• At each loading step, the cracks opening is measured on each side of the specimen by using a fissurometer.

• A crack is counted if it is seen across an entire width of one side of the specimen, i.e. two adjacent edges are open. The opening of this crack is equal to the average of the openings related to the adjacent edges. For example, in figure 6, Face 1 presents two opened adjacent edges (w 1 ≥ 0 and w 2 ≥ 0); a crack is then counted, having a width given by:

2 2 1 w w w + = (5)
and Face 2 presents only one opened (w 2 ≥ 0); then no crack is counted on this face. For this kind of test, and according to Eurocode 2 [START_REF]Calcul des structures en béton et Document d'Application Nationale[END_REF] in the case of RC structures, a crack is considered only if its opening w is equal to or higher than 300 μm (value which corresponds to prejudicial crack opening for service limit states).

• For each face, the average of all cracks openings (if cracks are counted on this face) is computed.

• The final mean value of the cracks opening of the tie-beam is given by the average of the preceding mean values (if it exists) obtained for each face.

Determination of the number of cracks and cracks spacing

The same strategy is adopted.

Note that this results analysis strategy can be performed on the 3D experimental test configuration as well as on the ones of the 2D and 3D modelling. Although this strategy only considers cracks of openings ≥ 300 μm, the model is able to describe smaller crack openings.

These smaller cracks are not taken into account in this analysis.

Parametric study-Determination of the parameters values of the interface model

In the following work, only 2D numerical simulations are considered (plane stresses conditions).

This work consists, in a first step, to identify the parameters of the behaviour of the concrete/rebar bond by an inverse analysis approach. The numerical results are compared with the experimental ones.

The mesh adopted for the 2D numerical simulations is given in figure 7.

Linear triangular elements are used for both concrete and steel reinforcement, whereas linear interface elements are used to represent the steel/concrete bond. The probabilistic cracking model and the interface model, described above, are applied on concrete and steel/concrete bond respectively. The steel behaviour remains elastic as far as only the service limit state is considered.

The damage model parameters (the cohesion C and the critical tangential relative displacement δ t cri ) describing the behaviour of the steel/concrete bond have to be determined (it is the objective of the inverse approach). The objective of these calculations is to determine the (C, δ t cri ) couple of values which leads to the better fitting of the experimental results. So, the inverse approach proposed is a simple fitting one.

The material characteristics used in the numerical simulation are presented in table 2 ( except for C and δ tcri , all the others material parameters are experimentally determined).

Numerical results and comparison with experiments -Macro-scale level

The numerical results of the parametric study have been analyzed in the same way than those related to the experiments and as detailed in chapter 4.3. The comparison has been made at a global level (load vs. displacement curves) and also at a local level (analysis of the cracking: number of cracks, cracks openings and cracks spacing vs. load). It is recalled that only cracks widths openings greater than or equal to 300 μm are considered. It appears that the numerical approach is efficient to represent the global response as well as local information (about cracks) of the behaviour of the tie-beam in tension (this despite the rough assumption of a 2D simulation of a 3D experiment).

Numerical results and comparison with experiments -Mix-scale level

As presented in the introduction, the concrete/rebar bond can be represented following two manners in the same numerical simulation: in the central part of the steel rebar, on a length of 30 cm, the notches/indentations of the flat steel rebar are explicitly taken into account in the simulation while in the rest of the rebar, the interface is modelled as previously by interface elements.

In the part of the rebar where the notches/indentations are explicitly modelled, the concrete/rebar interface behaviour is taken into account only by the concrete cracking around the rebar (probabilistic cracking model presented in chapter 2).

It is important to precise that the parameters values of the interface behaviour are those determined in the chapter 4.5 (C = 10 MPa and δ t cri = 10 μm).

In figure 10 is presented the mesh used for this "double scale approach" simulation (2D simulation) and an example of cracking profile obtained with the numerical simulations.

Note that, in this mesh, both parts (refined and not refined) are interlinked via the mesh nodes, ensuring therefore the continuity of the displacements. In addition, since the probabilistic cracking model takes into account volume effects, the strength of each concrete element directly depends on its own volume, it avoids therefore any mesh sensitivity of the results.

Figure 11 summarizes the results obtained in terms of mean curves.

These figures 10 and 11 allows the following comments:

• The cracking process (especially if cracks with openings greater than or equal to 300 μm are considered) obtained by combining, in the same simulation, two scales of modelling (macro-scale with interface elements and micro-scale by explicitly taking into account the rebar ribs) is similar to the one obtained by testing (Figure 11).

This positive result indicates that this two scales approach (in a same simulation) is fair and that the concrete/rebar bond behaviour can mainly be considered as the consequence of concrete micro-cracking around the ribbed rebar.

• In the part of the rebar where the notches/indentations of the flat steel rebar are explicitly taken into account, it is found that the micro-cracking zone surrounding the rebar has a thickness similar to the rebar one (Figure 10).

Numerical results and comparison with experiments -Micro-scale level

At this level, the notches/indentations are explicitly modelled along the entire rebar.

Therefore, the mesh is made by taking the same pattern than the one in the central part of figure 10, but repeating it other the entire length of the tie beam. As already detailed in the preceding section, the probabilistic concrete cracking behavior is used here for describing all cracking processes around the steel rebar: deterioration of the steel/concrete bond and also cracking in concrete.

As for the other levels of simulation (chapter 4.5 and 4.6), an example of cracking profile is presented in the figure 12. The results in terms of mean curves are summarized in figure 13.

These figures 12 and 13 allow the following comments:

• Concerning the thickness of the micro-cracked zone surrounding the rebar, the simulation at the micro-scale level gives similar information than at the mix-scale level: this thickness is around the same that the rebar thickness (Figure 12).

• The results (in term of cracks opening and spacing) obtained at this micro-scale level (Figure 13) are very acceptable if it is considered that the 2D modelling of the notches/indentations along the flat rebar is very simplistic compared with the reality.

Conclusions

This work is related to the numerical modelling (finite elements method) of the concrete/rebar interface. Three strategies of modelling have been considered to model this interface:

• A Macro-scale level: the concrete/rebar bond (the rebar is a flat steel one with notches/indentations) is represented by standard interface elements.

• A Mix-scale level: the notches/indentations of the rebar are explicitly taken into account in the central part of the rebar while the interface is modelled by interface elements in the rest of the rebar.

• A Micro-scale level: the notches/indentations are explicitly modelled along the entire rebar.

A macroscopic discrete probabilistic cracking model is chosen to describe the concrete cracking process around the rebar.

At the macro-scale level, the values of the parameters of the interface element behaviour (the maximum shear stress, C, and the tangential critical relative displacement, δ t cri ) are determined through an inverse analysis approach. This one consists in a parametric analysis which permits to determine the values of these parameters fitting well experimental results.

These experimental results are obtained in the frame of tie-beams reinforced by a flat steel rebar (with ribs) and subjected to tension.

This first part of the work and of the paper shows a good agreement between experience and modelling both when the global behaviour of the tie-beam and its cracking process (crack openings, number of cracks and cracks' spacing) are concerned. It means that the use of an interface element (having a deterministic macroscopic behaviour) with a discrete probabilistic cracking model is physically and quantitatively relevant to model the cracking behaviour of reinforced concrete.

The mix-scale and micro-scale levels numerical simulations lead to the following conclusions:

• To model the concrete/rebar bond (without notches/indentations) by an interface element having damage behaviour is equivalent to model micro-cracking of concrete around the rebar (with notches/indentations) if the objective is to get information about macro-cracks around this rebar.

The concrete/rebar bond behaviour can mainly be considered as the consequence of concrete micro-cracking around the ribbed rebar (it means due to the presence of notches/indentations).

• This micro-cracking concerns a zone with a thickness similar to the rebar one.

It is important to underline that all the results (values of the interface behaviour parameters)

and conclusions are valid and relevant for the types of concrete and rebar studied.

List of figures

2 IFSTTAR-

 2 The French institute of science and technology for transport, development and networks 3 LCPC -French Public Works Laboratory

  Fig

Figure 8

 8 Figure 8 presents an example of the 2D numerical cracking profile of the tie-beam, showing,

Figure 9

 9 Figure 9 summarizes this comparison in terms of mean curves.

Fig. 1 .Fig. 2 .Fig. 3 .Fig. 4 .Fig. 5 .Fig. 6 .Fig. 7 .Fig. 8 .Fig. 9 .Fig. 10 .Fig. 11 .

 1234567891011 Fig.1. Flat steel rebar -MATIERE ®

FiguresFig. 1

 1 Figures

  Fig. 2

FigureFig. 3

 3 Figure

Fig. 4 3Fig. 5

 45 Fig. 4

Fig. 6 4Fig. 7

 67 Fig. 6

Fig. 8

 8 Fig. 8

Fig. 9 6Fig. 10

 910 Fig. 9

Fig. 11

 11 Fig. 11

Fig.12 9 Fig. 13

 913 Fig.12

  The parametric study is realized by considering different cohesion C (6 MPa, 10 MPa, 15 MPa, 20 MPa, 25 MPa, 30 MPa) and critical tangential relative displacement δ t cri (6 μm, 10 μm, 15 μm, 20 μm, 25 μm, 30 μm) values of the concrete/rebar bond behaviour in total of 36 configurations. In order to compare numerical simulations to experiment, 9 calculations per configuration have been performed by Monte-Carlo method (each simulation corresponding to one random distribution of mechanical properties in the concrete). A total of 324 calculations have been performed for this parametric study.

List of tables