Adnan Ghribi
email: adnan.ghribi@ganil.fr

Muhammad Aburas

Yoann Baumont

Pierre-Emmanuel Bernaudin

Stéphane Bonneau

Guillaume Duteil

Robin Ferdinand

Michel Lechartier

Jean-François Leyge

Guillaume Lescalié

Yann Thivel

Arnaud Trudel

Laurent Valentin

Adrien Vassal

DØ INTERNAL DOCUMENT -NOT FOR PUBLIC DISTRIBUTION

SPIRAL 2 is a high intensity heavy ions beams accelerator project that has been going on for more than 10 years now. Countless efforts in different disciplines made it what it is today. One of the most important steps after the set up of the different equipments has been the very first full cool down of the superconducting cavities in an accelerator operation type configuration. While this has been a major achievement for the SPIRAL 2 teams, it also hi-lighted new challenges and constraints that would have to be addressed in order to have a high availability rate of the beam from the cryogenics side. This paper retraces this particular episode.

I. INTRODUCTION

The GANIL's (Grand Accélérateur National d'Ions Lourds) SPIRAL2 heavy ions accelerator [1][2][START_REF] Bertrand | 18th International Conference on Cyclotrons and their Applications-Cyclotrons 2007[END_REF][START_REF] Petit | [END_REF][5] aims at delivering some of the highest intensities of rare isotope beams on earth. The unstable beams will be produced by the ISOL(Isotope Separation On-Line) method via a converter or by direct irradiation of fissile material. The driver is designed to accelerate protons, deuterons and heavy ions at different ranges of intensities and energies. It is composed of high performance ECR sources (Electron Cyclotron Resonance sources), a RFQ(Radio-Frequency Quadrupole), and a superconducting light/heavy ion LINAC(LINear Accelerator). The latter [START_REF] Junquera | Proceedings of LINAC[END_REF][START_REF] Ferdinand | Proceedings of LINAC08[END_REF][START_REF] Bernaudin | IPAC 13-The 4th International Par-ticle Accelerator Conference[END_REF][START_REF] Ferdinand | [END_REF] is based on 26 bulk Niobium made superconducting, independently phased resonators. These resonators are assembled in two families of cryomodules that integrate passive RF, vacuum and cryogenics components needed for their operation. The first family, called type A cryomodules and developed by CEA/IRFU, is optimised for β 0 = 0.07 relative velocity and comprises 12 cryomodules (one cavity/cryomodules). The second family, called type B cryomodules and developed * Electronic address: ghribi@ganil.fr by CNRS/IN2P3/IPNO, is optimised for β 0 = 0.12 and comprises 7 cryomodules (two cavities per cryomodule). All cryomodules are interspaced with room temperature quadrupole magnets and are operated at 88.0525 MHz.

To be operated, the superconducting cavities need to be maintained at a stable temperature lower than their transition temperature with a stringent mechanical stability to avoid detuning. This is achieved thanks to a cryoplant delivering up to 1300 W cooling power at 4 K. The coldbox, at the center of the cryoplant, delivers liquid helium through a phase separator (5000L capacity LHe dewar) and a cryodistribution system to the cryomodules and manages the return gas. For every cryomodule, a valves-box allows maintaining stable LHe and stable pressure at the surface of the cavities. An overall view of the cryogenic facility is visible on figure 1. More details on the cryoplant can be found in [10].

More than 10 years after the design phase have been necessary to build and assemble all the systems and subsystems. 37 km of cables link hundreds of sensors and actuators to 37 Siemens S300/S400 programmable logic controllers. Human Machine Interface uses PCVue™ for the cryoplant and WinCC™ for the LINAC. EPICS™ is used for data acquisition in redundant distant servers. In 2015, pre-commissioning of the coldbox confirmed its capability to deliver the required refrigeration power. In 2016, a first partial cool down of two cryomodules al- lowed the pre-commissioning of mechanical, piping, vacuum and control/command systems. A year later, a second partial cool down of three cryomodules allowed the commissioning of the command, control and acquisition system [11]. The same year, all the building blocks including cryoplant and LINAC have been assembled with success.

(A) (B) (1) (3) (6) (7)
The present paper brings the development and commissioning of the cryogenic system of SPIRAL 2 a step forward with a first full cool down of the LINAC. This is also certainly a step further into maturing the system for reliable beam operation conditions. In the first section, challenges and requirements for cool down and normal operation conditions are highlighted. The second section details the cryogenic process logic that has been implemented for the cool down. Next, we show the results of the cool down phase and the normal operation phase. An opening into RF measurements of the cavities properties is also described with a possible link to cryogenics.

II. COOLING DOWN THE LINAC

A. Requirements and applied methods

Global cool down strategy

Due to the design of the cryoplant and the cryodistribution, a specific global cool down strategy has to be followed. As the cold box delivers the cooling power to cool down and maintain stable both thermal screens circuits (60 K) and liquid helium circuits, the cold box and the thermal screens are cooled down at the same time. Cool down of liquid helium circuits starts when the cold box reaches stable operation conditions and the main liquid helium dewar (5000 L capacity) has at least 1500 L of liquid helium. As there is no bypass at the edges of the LINAC, the edge cryomodules have to be cooled down in order to cool down the cryodistribution system. Once the cryodistribution is cold, the rest of the cryomodules follows in a specific order. A summary of the global cool down procedure used for 2017 cool down is shown below :

1. Purge of all systems 2. Pump the isolation vacuum and the beam vacuum 3. Helium liquefaction in the main dewar to a minimum of 1500 L.

4. Warm up the cold box 5. Cool down the cold box and at the thermal screens of CMA01 and CMB07 at the same time (the two edges of the LINAC).

6. Once the cold box is in normal operation mode, cool down the cavities of CMA01 and CMB07. This concludes the cool down of the cryo distribution system. since there is no by pass at the ends of the LINAC, cooling down the two cryomodules at the end of the LINAC is mandatory to finalise the cool down of the cryo distribution system.

7. Cool down cryomodules from the edges to the center beginning by the thermal screens then the cavities for every cryomodule. Since the heat load of one type B cryomodule is approximately twice the heat load of a type A cryomodule, for every type B cryomodule, two type A cryomodule have been cooled down at the same time.

Local cool down strategy

For every cryomodule, thermal screens are cooled down before the cavities. 2 valves ensure the cooling and control of the thermal screens and 3 valves ensure the cooling and control of the superconducting cavities. One of the three cavity valves (largest flow coefficient) is used in the beginning to cool down in a bottom to top way in order to take advantage of the cold vapours. Once, the cavity is cold and the local phase separator has a given level of liquid helium, the cool down valve is closed and the liquid helium level is regulated from the top of the phase separator using another lower flow coefficient valve. Details of the instrumentation of the cryomodules can be found in [10]. It should be noted that there are two outlet valves. The first is a warm one and is located at the outlet of a passive exchanger. When He gas is too warm ( ¿ 80 K) to be re-sent to the cold box, it sends the gas back to the inlet of the compressor station. When the gas is sufficiently cold, the warm outlet valve is closed and a second one is opened. The latter is the one that usually controls the pressure of the cavities liquid helium baths. However, during all the phases of the cool down and until the cryomodules are cold and fully thermalised, this valves is kept entirely open. It should also be noted that the passage between the two mentioned valves can be sometimes tricky depending on the programmed speed of the cool down and the heat load of the cryomodules. This means that, when not programmed carefully or during trials, the passage can lead to security interlocks activation due to overpressure. Later on, a second level security has been added with a weight added on the inlet valves opening that depends on the cavities pressure and that overtakes the programmed cool down slope when the pressure reaches a certain value and before the activation of the security interlocks.

It should be noted that, during cool down, the outlet valve that controls the pressure of the cavities liquid helium baths in normal operation modes is kept 100% open at all times.

One of the constraints of SPIRAL 2 cavities is the Q disease. This so called Q disease is a symptom of degradation of the quality factor of a cavity due to Hydrogen trapping on the inner surface [START_REF] Knobloch | AIP Conference Proceedings[END_REF]. This appears if a cavity spends more than a given time ∆(t) between two critical temperatures T1 and T2. SPIRAL 2 cavities are not heat treated and suffer from such effects. During test bench qualifications, these values have been found to be : ∆(t) = 1 hour ; T1 = 50 K and T2 = 150 K. Fortunately, the symptom is reversible as hydrogen is freed during warm up when the temperature of a cavity increases. This means that if, for some reason, a cavity spends more than 1 hour in the critical temperature range, it has to be warmed up and cooled down again. Therefore, a specific cool down procedure has been used to avoid this situation. The strategy here is to slow down the cavities temperature drop in order to allow more time for thermalisation of the mass. Previous individual cool downs allowed to determine the temperature transmitter that represents the best the thermal behaviour of the cavity. This temperature has then been set as an observer for automatically controlling the slope of the temperature drop thanks to an active proportional integral (PI) controller. Between 300 K and 250 K, the main cool down valve is opened slowly to 100%. Starting from 250 K, the PI controller on the same valve is activated with a sliding target cavity temperature from 250 K to 150 K. The slope of this variation is chosen by the operator.

Results

The procedure described in II A has been applied to oct. 2017 SPIRAL 2 LINAC cool down. It was the first full cool down of the LINAC and a good opportunity to experiment cool down methods and strategies. The resulting over all cool-down time is shown on figure 3. It begins at the start of the user controlled cavities cool down procedure and ends when the first percent of liquid helium is measured in the phase separators. As expected, the first cavities cool down time (cavities 1, 25 and 26) are higher because it matches the cool down of the liquid helium circuits of the cryodistribution. Cavities 13 and 14 (same cryomodule type B) are right in the middle of the LINAC, at the junction between the two main branches of the cryodistribution system. These cavities suffered from unequal control of thermal load distribution during cool down and not adapted cool down parameters. As a result, this deprived them from an adapted cold helium flow in a critical step of their cool down. Cavities 2, 10, 11 and 12 suffered from security mode passage. This is visible on the extended cool down time with respect to other type A cavities (the first 12 cavities) in figure 3 as well as in the slope of their temperatures during cool down in figure 2.

As described previously cavities temperatures have been monitored in order to optimise cavities temperature slopes parameters. The temperature drops of the first 12 cryomodules can be seen in figures 2. In this figure, one can distinguish four main behaviours. The first is a curve visible on all cavities. It starts at 300 K and corresponds to a radiative cool down that occurs thanks to thermal screen cool downs. The second one is a change in the slope of the previous curve. It corresponds to the beginning of the cavity cool down procedure. The third is the step-controlled cavities temperature drop down. Its slope depends on the chosen slope parameters. In figure 2, the latter is visible only for cavities 1 and 2. The fourth is an acceleration of the cool down, again visible in figure 2 for cavities 1 and 2. For most other cavities, the different parts can not be distinguished due to different reasons (beginning of cool down too fast, controlled cavities cool down too fast). For some cryomodules, the parameters entered were not adapted to the intrinsic heat load of the cryostats. This caused an increase in the outlet pressure, which caused in its turn a reduction of the cold helium flow and therefore an increase and stagnation of the cavities temperature in the critical temperature range.

The success or failure of a given cool down procedure or a parameter set is summarised in figure 4. It represents an automatic counter Qd count that starts when the cavities temperature reach 150 K and stops when it is bellow 50 K. For cryomodules 1 and 7 (cavities 1, 25 and 26), that have been cooled down at the same time as the cryodistribution, the cool down has been slowed down because of the very high thermal load of the cryodistribution compared to the cryomodules. In addition, when the temperature of the helium return is lower than 80 K, it is redirected to the cold box cold return line. For the first cryomodules, the overall return temperature is too high for the higher cold stage of the cold box. Valves at the input of the cold box, therefore, significantly reduce the return flow while the temperature is too high. The other cases for cryomodules 10, 11, 12 and 14 (cavities 15 and 16) have been explained in the previous paragraph by accidental stop of the cool down. As for cryomodule 17 (cavities 21 and 22), it suffered from an input temperature slope too high for its thermal load.

III. THE WAY TO OPERATION

A. Thermalisation

When the liquid helium level in the phase separators of the cavities is stable and regulated around a 90%, the cool down procedure is finished. However, we have to wait until all systems and subsystems are thermalised before authorising cryogenic operation in the beam configuration. To know when this can happen, we regularly measured the heat load for every cryomodule by measuring the liquid helium level decrease during a given time when outlet valve is open and inlet valve is closed. The heat load is then given by :

Q = dV dt ρC l ( 1
)
where Q is the thermal load, V the LHe volume, t the time, ρ the liquid helium density at the cryomodule measured pressure and C l the latent heat of liquid helium. For increased precision, the exact 3D volume of the phase separator have been modelled. What can be noticed from figures 5 and 6 is that the time that we are looking for strongly depends on the considered cryostat geometry, construction and constraints. For example, type A cryomodules are fast to cool down but slow to thermalise (4 to 5 days) due to the presence of a specific frequency tuning system thermally decoupled from the cold mass (see figure 5). The cryomodules are then thermalised when their frequency tuning systems temperatures reach a stable plateau (around 35 K). The type B cryomodules (see figure 6) have a much higher thermal load but are much faster to thermalise (around 1 day).

B. Cavities liquid helium bath pressures and levels control

Once the cryomodules are thermalised, there are two main constraints to authorise RF and beam operation. The first is to make sure that all cavities are submerged by liquid helium at all times. When the LHe bath pressure is stable, all cavities have a very stable temperature and therefore very stable RF properties. In order, to achieve this target, a dedicated valve is used to control liquid helium level. This valve uses a proportionalintegral (PI) controller with an integral tuned to match the corresponding cavity static heat load and a proportional term chosen to make the valve slow enough in order not to have an instantaneous effect on the pressure.

The pressure and liquid helium level of a same volume are obviously coupled. The matching valves that use independent PI controllers for liquid helium and pressure regulations therefore introduce correlations. As can be seen on figure 7, when not tuned properly, these correlations can be critical with regard to the stringent pressure stability requirements. The difficulty is enhanced here by the asymetric behaviour of the liquid helium regulation (fast to fill and slow to empty). One of the actions that has been done while keeping a simple PI controller was to limit the range of action of the liquid helium flow control valve. This proved efficient as it stabilised the pressure of the cavities with regard to correlated instabilities with liquid helium control valves positions. However, this approach is not adapted to a high variability of the thermal load.

The second is to make sure that pressure variations of the liquid helium bath do not exceed a given ∆(P ). In fact, the forces that liquid helium apply on the surface of a cavity slightly changes its shape, which changes its geometric factor. As a result, its unloaded quality factor (that we can denote Q f o here) also changes as can be seen in this equation :

Q f o = G R s ( 2
)
where G is the geometric factor and R s is the surface resistance. Depending on the bandwidth of the cavity and the margin on the amplifiers side, this can be critical as it drastically reduces the capability to accelerate particles with the required acceleration gradients. For SPIRAL 2, the limit of ∆(P ) has been set to ± 5 mbar. We then monitored every night the pressure fluctuations and counted the number of deviations from the accepted pressure range (± 5 mbar). While stabilised most of the time, the pressure showed periodic or pseudoperiodic instabilities that seemed to move from one LINAC position to another. This behaviour seemed to be more frequent for the 12 first type A cryomodules.

Figure 8 shows such behaviour for CMA05 and CMA10 and figure 9 for CMA06. Other periods were more stable and did not show any instability over ± 5 mbar (see figure 10 for example).

When looking in more details for the action of the pressure control valve during instabilities, we could see that instabilities seemed uncorrelated to the actions of the valves (see figure 11). Moreover, the maximum sampling rate of pressure data acquisition (300 ms) couldn't allow us to distinguish any reproducible behaviour with a fixed periodicity. All this pointed to the presence of hidden fast phenomena distributed all over the LINAC. During this 2017 run, first RF measurements of the cavities have been performed. This was a first cold measurement of some RF properties of the cavities in the LINAC. Due to the distance between the LINAC and the generators (power loss) and the limitations on the injected RF power in the LINAC (nuclear safety authorisation was pending), we used an external RF synthesizer to measure and characterise the transmitted power response. First, a self-oscillating closed loop circuit was used to find the resonance of the cavities. Then a low power frequency generator was used to inject the RF signal and a Yokogawa network analyser was used to analyse the output. Surprisingly, both an amplitude and a frequency modulation was noticed. These measurements have been performed on CMA07, CMA11 and CMB07. Figure 12 shows an amplitude modulation spectrum with peak to peak distance of 5.35 Hz. Other measurements consistently showed a modulation between 4 and 6 Hz. Frequencies of modulation seemed specific to the measured cryomodules as repeated measurements on the same cryomodules showed the exact same modulation frequencies. These modulations were not noticed during vertical cryostat measurements. This might mean that these fluctuations could be due to the final setup of the cryomodules in the LINAC. In a final beam configuration, the modulations would be compensated by the amplifiers power margin. However, the amplitude of these modulations are high enough to cause the system to hit its limits. Therefore, the origin of these microphonics will have to be investigated further and their effect mitigated.

The measured frequencies were too low to be a consequence of the RF test bench and seemed to be more consistent with vibrations of a mechanical nature. Several accelerometer measurements of vibrations have then been performed in the LINAC at different locations and nothing unusual has been noticed. Although few Hz seemed usually low also to be acoustic like oscillations, this could explain the fast fluctuations of the pressures in the LINAC. After countless developments and installation efforts, the SPIRAL 2 superconducting LINAC was finally cooled down for the first time in October 2017. It was also the first time that all cryogenic systems were tested in a configuration as close as possible to the operation configuration. Cooling down such a system with diphasic helium and with no by-pass at the end of the LINAC proved to be a challenge. However, trials showed that it is possible to find an optimised cool down strategy with minimal effects on the performances of the cavities. Pressure and level regulations with simple yet optimised PI controllers were for time laps as long as 12 hours. Pseudoperiodic instabilities on all cavities as well as possible hidden fast fluctuations will have to be investigated further. Another limitation of this run was that all cavities were characterised in a so called static configuration. It means that no real nor simulated RF heat load could be taken into account. Low power RF measurements of the cavities showed resonances at their design values. However, amplitude and frequency modulations of the transmitted power were detected and measured. They indicated important microphonics that could be linked to the cryogenic system and explain some of the unexplained pressure fluctuations that have been observed. Investigations of the measured microphonics were continued with a dedicated bench that allows the synchronous measurements of RF and pressure modulations. This cool down was also important to gather precious data for precise thermodynamic modelling of the SPIRAL 2 LINAC [START_REF] Vassal | [END_REF]. The next cool down will hopefully benefit of a model based pressure and liquid helium level control as well as opti-mised input cool down parameters. It will also benefit of a dedicated bench for synchronous measurements of RF and piezoelectrically measured pressure fluctuations.

FIG. 1 :

 1 FIG. 1: 3D view of the cryogenic facility of Spiral 2. (A) Cryoplant. (B) LINAC. (1) LN2 tank. (2) Compressor station. (3) LHe dewar. (4) Cold box. (5) Valves-boxes. (6) Type B double cavities cryomodules. (7) Type A single cavity cryomodules.

FIG. 2 :FIG. 3 :

 23 FIG.2: CMA cavities temperature drop. The time axis is normalised so that all temperature appear begin their slope at the same time

2 FIG. 4 :

 24 FIG.4: Q disease counter for all cryomodules. Green bars correspond to counters within the requirements. Red bars are for counters above requirements.

FIG. 5 :FIG. 6 :

 56 FIG.5: Bars : evolution of the measured thermal load Q of the type A cryomodules (first 12 cavities) as a function of the elapsed time since the end of the cool down procedure. Circles : Evolution of the temperature of the frequency tuning system.

FIG. 7 :

 7 FIG. 7: Pressure and liquid helium level regulations (top plot) with matching valves (bottom plot) for CMA05. Data of 2017-11-20.

FIG. 8 :

 8 FIG. 8: Pressure fluctuations (gray) and ± 5 mbar deviation counter (red). Data of : 2017-11-27 -start 20:00.

FIG. 9 :FIG. 10 :

 910 FIG. 9: Pressure fluctuations (gray) and ± 5 mbar deviation counter (red). Data of : 2017-11-28 -start 20:00.

  FIG. 11: 2017-11-22 -CMA01

  FIG.12: Transmitted power modulation measurement of cavity number 25 (CMB07 cavity 1). Cryomodule was cold, thermalised and regulated both in liquid helium level and pressure. Pressure of liquid helium bath, as measured, was stable within ± 5 mbar. Modulation frequency was measured to be 5.35 Hz


Mohamed Saidi
email: mohamed.saidi@univ-lyon1.fr

Aron Gabor
email: aron.gabor@univ-lyon1.fr

Iterative analytical modelling of the global behaviour of textile-reinforced cementitious matrix composites subjected to tensile loading

Keywords: Textile-reinforced mortar TRM, cracking behaviour, analytical modelling, distributed fibre-optic sensing (DFOS), global and local mechanical behaviour

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Textile-reinforced cementitious matrix composites (TRCMCs) are formed by embedding one or several layers of textile reinforcement in a cementitious matrix. This composite has high mechanical tensile strength [START_REF] Promis | Failure behavior of E-glass fiber-and fabric-reinforced IPC composites under tension and compression loading[END_REF], making it an effective alternative to polymer matrix composites for reinforcing and repairing civil engineering structures [START_REF] Ferrara | Tensile strength of flax fabrics to be used as reinforcement in cement-based composites: experimental tests under different environmental exposures[END_REF][START_REF] Escrig | Experimental and analytical study of reinforced concrete beams shear strengthened with different types of textile-reinforced mortar[END_REF][START_REF] Krevaikas | Experimental study on carbon fiber textile reinforced mortar system as a means for confinement of masonry columns[END_REF][START_REF] Askouni | Textile Reinforced Mortar-to-masonry bond: Experimental investigation of bond-critical parameters[END_REF][START_REF] Santis | Tensile behaviour of mortar-based composites for externally bonded reinforcement systems[END_REF]. However, due to the brittleness of the cementitious matrix, cracks appear at relatively low tensile stresses [START_REF] Carozzi | Mechanical properties and debonding strength of Fabric Reinforced Cementitious Matrix (FRCM) systems for masonry strengthening[END_REF][START_REF] D'antino | Mechanical characterization of textile reinforced inorganic-matrix composites[END_REF],

which influences the mechanical behaviour of the TRCMC. According to the literature [START_REF] Carozzi | Mechanical properties and debonding strength of Fabric Reinforced Cementitious Matrix (FRCM) systems for masonry strengthening[END_REF][START_REF] D'antino | Mechanical characterization of textile reinforced inorganic-matrix composites[END_REF][START_REF] Larrinaga | Experimental and numerical modeling of basalt textile reinforced mortar behavior under uniaxial tensile stress[END_REF][START_REF] Mercedes | Mechanical performance of vegetal fabric reinforced cementitious matrix (FRCM) composites[END_REF][START_REF] Ascione | A qualification method for externally bonded Fibre Reinforced Cementitious Matrix (FRCM) strengthening systems[END_REF], this mechanical behaviour is distinguished by the following three zones:

• Pre-cracking zone: It is characterised by a quasi-linear behaviour expressed by a constant Young's modulus E1 as shown in Fig. 1-a. This Young's modulus is mainly influenced by the mechanical properties of the matrix, and is not very sensitive to those of the textile reinforcement [START_REF] Colombo | Erratum to: Textile Reinforced Concrete: experimental investigation on design parameters[END_REF]. This zone is limited by a strain ε1 corresponding to a stress σ1 that is related to the ultimate stress of the matrix.

• Crack propagation zone: It begins after the ultimate stress of the matrix is reached, as manifested by the appearance of the first crack on the TRCMC (see Fig. 1-a). At the time of each crack, a drop in the stress applied to the TRCMC, σ(N)drop, is recorded (Fig. 1-b). The applied stress then increases as the applied displacement on the TRCMC increases during the tensile test, with an equivalent Young's modulus Eeq(N) that gradually decreases as the test progresses, giving the stress-strain curve a curved shape (Fig. 1-b). A new crack appears at a stress σNc that is generally greater than those of the previous cracks. These successive cracks appear on the TRCMC in random order [START_REF] Du | Experimental study on basalt textile reinforced concrete under uniaxial tensile loading[END_REF][START_REF] Caggegi | Experimental study on effective bond length of basalt textile reinforced mortar strengthening system: Contributions of digital image correlation[END_REF], with a final spacing that decreases as the reinforcement ratio increases [START_REF] Colombo | Erratum to: Textile Reinforced Concrete: experimental investigation on design parameters[END_REF]. The end of this zone is characterised by a stress σ2 that corresponds to a strain ε2.

• Post-cracking zone: It is characterised by an equivalent Young's modulus E3 that is generally lower than E1 (Fig. 1-a). This area is mainly driven by the mechanical characteristics of the textile reinforcement [START_REF] Colombo | Erratum to: Textile Reinforced Concrete: experimental investigation on design parameters[END_REF]. It is limited by the ultimate stress σ3 that corresponds to the ultimate strain ε3. Two modes of failure can occur: [START_REF] Promis | Failure behavior of E-glass fiber-and fabric-reinforced IPC composites under tension and compression loading[END_REF] slippage of the textile in the matrix, particularly between the plates that transmit the load from the test machine to the specimen, or (2) the failure of the textile reinforcement. The tensile behaviour of TRCMC is generally influenced by several mechanical and geometrical parameters of its components, as well as the boundary conditions of the specimens. The structure and density of the textile [START_REF] Ferrara | Experimental Study on the Adhesion of Basalt Textile Reinforced Mortars (TRM) to Clay Brick Masonry: The Influence of Textile Density[END_REF], the durability and ageing of the reinforcement [START_REF] Signorini | Mechanical performance and crack pattern analysis of aged Carbon Fabric Cementitious Matrix (CFRCM) composites[END_REF] considerably influence the mechanical response of TRCMC. The behaviour of the textile at the core of the TRCMC, whether monolithic or telescopic [START_REF] Contamine | Contribution à l'étude du comportement mécanique de composites textile-mortier : application à la réparation et/ou renforcement de poutres en béton armé vis-à-vis de l'effort tranchant[END_REF], also plays a significant role in the micromechanical interactions of TRCMC. Indeed, the type of coating used for the textile reinforcement [START_REF] Signorini | Sustainable mineral coating of alkali-resistant glass fibres in textile-reinforced mortar composites for structural purposes[END_REF][START_REF] Ferrara | Influence of an Impregnation Treatment on the Morphology and Mechanical Behaviour of Flax Yarns Embedded in Hydraulic Lime Mortar[END_REF][START_REF] Donnini | Mechanical properties of FRCM using carbon fabrics with different coating treatments[END_REF] as well as the degree of impregnation of this coating [START_REF] Messori | Mechanical performance of epoxy coated AR-glass fabric Textile Reinforced Mortar: Influence of coating thickness and formulation[END_REF] impact the textile/matrix bond, the ultimate strength of the TRCMC as well as the failure mode. As well, the type of matrix and the shape of the specimens play an important role [START_REF] Colombo | Erratum to: Textile Reinforced Concrete: experimental investigation on design parameters[END_REF][START_REF] Brameshuber | State-of-the-Art report of RILEM Technical Committee TC 201-TRC "Textile Reinforced Concrete[END_REF]. The boundary conditions and the experimental setup used affect the load transfer mode from the tensile machine to the TRCTM specimens [START_REF] Hartig | Influence of experimental setups on the apparent uniaxial tensile load-bearing capacity of Textile Reinforced Concrete specimens[END_REF][START_REF] Arboleda | Testing Procedures for the Uniaxial Tensile Characterization of Fabric-Reinforced Cementitious Matrix Composites[END_REF][START_REF] Caggegi | Experimental analysis on tensile and bond properties of PBO and aramid fabric reinforced cementitious matrix for strengthening masonry structures[END_REF]. Several experimental works have shown that the shear stress at the textile/matrix interface near the load-transmitting jaws is influenced by the mechanism used (such as clamps, hydraulic grips, clevis joints, etc.), which impacts the ultimate strength of the TRCMC.

The global and local mechanical behaviour of the TRCMC has been the subject of several experimental, analytical' and numerical studies. This behaviour was modelled by ACK [START_REF] Aveston | Theory of multiple fracture of fibrous composites[END_REF] and Cuypers [START_REF] Cuypers | Stochastic matrix-cracking model for textile reinforced cementitious composites under tensile loading[END_REF], assuming that the law of mixtures is applicable in the first zone. ACK model [START_REF] Aveston | Theory of multiple fracture of fibrous composites[END_REF] assumes that the Young's modulus of the second zone is zero, while the Cuypers model [START_REF] Cuypers | Stochastic matrix-cracking model for textile reinforced cementitious composites under tensile loading[END_REF] proposes a progressive evolution of the crack propagation zone based on the Weibull distribution and the Widom statistical study [START_REF] Widom | Random Sequential Addition of Hard Spheres to a Volume[END_REF]. Both models also consider that the appearance of each crack causes damage over a distance on both sides of the crack that is called the load transfer length. For the third zone, the textile/matrix interface is assumed to be completely damaged, which means that the textile bears all the load applied to the TRCMC.

These models are combined with statistical studies [START_REF] Ahn | Strain and hysteresis by stochastic matrix cracking in ceramic matrix composites[END_REF] that correlate the load transfer length with the final crack spacing, which has improved the ACK model.

These models in literature do not reproduce the exact behaviour of the crack propagation zone. Indeed, several mechanical interactions are not taken into account or reproduced in these models, such as the stress drop during cracking, the changes in the local and global stiffnesses of the TRCMC as the cracks propagate, the evolution of the textile/matrix interface damage, reflected by the increase in the load transfer lengths, etc. In addition, Cuypers' model uses statistical characteristics as input data for the model [START_REF] Cuypers | Stochastic matrix-cracking model for textile reinforced cementitious composites under tensile loading[END_REF]. These characteristics require a series of preliminary tests in order to statistically characterise the behaviour of the matrix, a contrario to this new model developed in this paper which uses the mechanical and geometrical characteristics of the TRCMC as input data in order to obtain the global behaviour of these composites with all the micromechanical interactions at the textile/matrix interface, such as the load transfer length, the shear stress at the textile/matrix interface, the change in local stiffness due to cracking and its influence on the global stiffness. This allowed to reproduce and predict the cracking behaviour of the TRCMC by calculating the stress drops at each cracking instant and deducing the stiffness of the TRCMC during the tensile test (Fig. These models in literature are based on both experimental observations and mechanical analyses. The experimental part employs traditional measurement techniques such as strain gauges, linear variable differential transformers, digital image correlation, etc. [START_REF] Caggegi | Experimental study on effective bond length of basalt textile reinforced mortar strengthening system: Contributions of digital image correlation[END_REF][START_REF] Tran | Experimental and analytical analysis of the effect of fibre treatment on the thermomechanical behaviour of continuous carbon textile subjected to simultaneous elevated temperature and uniaxial tensile loadings[END_REF][START_REF] Bilotta | Use of DIC technique for investigating the behaviour of FRCM materials for strengthening masonry elements[END_REF][START_REF] Bilotta | Experimental tests on FRCM strengthening systems for tuff masonry elements[END_REF],

2-a and b).

allowing the strain or displacement field on the surface of the TRCMCs to be measured. The internal mechanical characteristics between the textile and the matrix are then deduced by combining continuum and fracture mechanics with the experimental results [START_REF] Mercedes | Mechanical performance of vegetal fabric reinforced cementitious matrix (FRCM) composites[END_REF][START_REF] Colombi | Analytical assessment of the stress-transfer mechanism in FRCM composites[END_REF][START_REF] D'antino | Estimation of a matrix-fiber interface cohesive material law in FRCM-concrete joints[END_REF].

Numerical studies have been conducted [START_REF] Larrinaga | Experimental and numerical modeling of basalt textile reinforced mortar behavior under uniaxial tensile stress[END_REF][START_REF] Donnini | Tensile behaviour of glass FRCM systems with fabrics' overlap: Experimental results and numerical modeling[END_REF] to reproduce the global and local behaviour of the TRCMC and to observe the internal interactions and the propagation of damage at the textile/matrix interface.

To verify the validity of these assumptions of the developed analytical models, to better understand the internal behaviour' and to measure the micromechanical parameters that control the behaviour of the TRCMC, the authors carried out experimental tests using distributed optical fibres as strain sensors [START_REF] Saidi | Experimental analysis of the tensile behaviour of textile reinforced cementitious matrix composites using distributed fibre optic sensing (DFOS) technology[END_REF]. The reliability of the results of this measurement technique used in a cracking material such as TRCMCs has been analysed and validated [START_REF] Saidi | Use of distributed optical fibre as a strain sensor in textile reinforced cementitious matrix composites[END_REF]. The experimental results confirm that the law of mixtures can be applied in the first zone. In the second zone, each crack develops two load transfer lengths. The latter length defines the final crack spacing, influences the distribution of shear stress at the textile/matrix interface' and locally changes the Young's modulus of the composite. In the third zone, the hypothesis that the textile takes up all the applied load was validated again.

(a) (b)

In this work, based on these experimental and analytical studies in the literature, and a series of DFOS instrumented tests, an analytical model of the global behaviour of the TRCMC was developed. Nine configurations were tested: two types of textiles, two types of matrices' and three reinforcement ratios. The model was developed with a step-by-step process, zone by zone, using an iterative method with an imposed displacement. This allowed the algorithm to reproduce the global behaviour of the TRCMC with the details of the crack propagation zone. This model was compared with both the experimental results and the ACK model. After its validation, a parametric study was carried out that allowed the effect of some geometrical and mechanical parameters of the matrix and textile on the global mechanical response of the TRCMC to be determined.

Experimental work

The experimental part of the research is now briefly described. Detailed information on the experimental test (devices, optical fibre, backscattering principle, geometrical and metrological characteristics of the sensors, etc.) and on the specimen preparation with optical fibre (insertion of the fibre, casting of the matrix, etc.) is presented in [START_REF] Saidi | Experimental analysis of the tensile behaviour of textile reinforced cementitious matrix composites using distributed fibre optic sensing (DFOS) technology[END_REF][START_REF] Saidi | Use of distributed optical fibre as a strain sensor in textile reinforced cementitious matrix composites[END_REF].

2.1.Materials

The studied composite is composed of a cementitious matrix reinforced by one or several textile layers. Nine configurations were studied: two types of self-placing matrices, two types of textile reinforcement' and three reinforcement ratios. The following subsections describe the mechanical properties of these components.

Matrices

For practical reasons of specimen preparation, and given the fragility of the optical fibre, two ettringitic and self-placing cementitious matrices are used. Mechanical characterisation tests of the two matrices are performed in the laboratory by following the detailed procedure described in [START_REF]Methods of testing cement-Part 1: Determination of strength, European Committee for standardization[END_REF], and the results are summarised in Table 1. The w/c ratios are 0.33 for matrix M1 and 0.24 for matrix M2. The M1 matrix consists of an ettringitic binder (24%), white cement (40%), slag and gypsum (10%), admixtures (3%) and water (33%). The M2 matrix consists of cement, silica sand, specific additives (76%) and water (24%). Both matrices exhibit brittle elastic mechanical behaviour under tension. following the detailed procedure described in [START_REF] Saidi | Experimental analysis of the tensile behaviour of textile reinforced cementitious matrix composites using distributed fibre optic sensing (DFOS) technology[END_REF], and the results are summarised in Table 2.

Geometrical properties, such as the cross-sectional area and the perimeter of the yarns in the weft direction of the textile, are apparent characteristics measured in the laboratory using a sliding calliper. The cross-sectional areas of the two textiles are assumed to be of elliptical shape. All geometric parameters related to the textile are calculated using the apparent crosssectional area and perimeter. These two textile reinforcements are coated with resin and exhibit monolithic behaviour. The applied tensile load is supported by the yarns in the weft direction for both the used textiles.

TRCMC specimens

The investigated TRCMC specimens have a rectangular parallelepipedic shape, with dimensions of approximately 600 × 50 × 10 mm 3 (±1 mm) (length × width × thickness) as shown in Fig. 3. This shape is based on experimental studies reported in the literature [START_REF] Caggegi | Experimental study on effective bond length of basalt textile reinforced mortar strengthening system: Contributions of digital image correlation[END_REF][START_REF] Contamine | Contribution to direct tensile testing of textile reinforced concrete (TRC) composites[END_REF] and approximate the dimensions required to strengthen engineering structures.

These composites were equipped with optical fibres to quantify their internal strain and deduce the global and local mechanical behaviour. Fig. 3 shows the details of the position of the optical fibre. For more details, see [START_REF] Saidi | Experimental analysis of the tensile behaviour of textile reinforced cementitious matrix composites using distributed fibre optic sensing (DFOS) technology[END_REF][START_REF] Saidi | Use of distributed optical fibre as a strain sensor in textile reinforced cementitious matrix composites[END_REF]. The preparation of the specimens involves placing the optical fibre at six different positions as illustrated in Fig. 3. For each reinforcement layer, a 20 cm segment of optical fibre is glued to the textile' and another segment is simply positioned on the textile, which allows measurement of the behaviour of the textile and the matrix during the test. Other optical fibre segments are embedded directly into the matrix as described in [START_REF] Saidi | Experimental analysis of the tensile behaviour of textile reinforced cementitious matrix composites using distributed fibre optic sensing (DFOS) technology[END_REF][START_REF] Saidi | Use of distributed optical fibre as a strain sensor in textile reinforced cementitious matrix composites[END_REF].

At both ends of each specimen, two aluminium plates are glued as shown in Fig. 3. During the tensile test, these two plates are clamped by external plates not connected to the tensile machine in order to increase the textile/matrix bond at the aluminium plates. The load transfer from the tensile machine to the TRCMC is done by two ball-joint loading heads, which reduce the effect of eventual parasitic bending. More details of the boundary conditions of the TRCMCs tested are presented by the authors in a previous paper [START_REF] Saidi | Experimental analysis of the tensile behaviour of textile reinforced cementitious matrix composites using distributed fibre optic sensing (DFOS) technology[END_REF].

2.3.Experimental device

Optical data acquisition

The optical fibre data is acquired with the ODiSI-B LUNA (Optical Distributed Sensor Interrogator) shown in Fig. 4. The measurement is based on the Rayleigh backscattering principle [START_REF] Soller | High resolution optical frequency domain reflectometry for characterization of components and assemblies[END_REF]. The used optical fibre has a diameter of 250 µm and a spatial resolution of 2.6 mm over its entire length (2 m).

Tensile testing machine

TRCMC specimens are tested using a Zwick universal testing machine as shown in Fig. 4.

It has a maximum capacity of 65 kN and is equipped with sensors which allow the measurement of the displacement and the applied load with high accuracy, 0.06% mm and 0.2%, respectively. The test is controlled in displacement with a velocity of 0.1 mm/minute by following the detailed procedure described in [START_REF] Saidi | Adaptation of the strain measurement in textile reinforced cementitious matrix composites by distributed optical fibre and 2D digital image correlation[END_REF] with the necessary recalibration of the optical fibre at the appearance of each crack. Two ball-joint loading heads are used to clamp the TRCMC specimens to the Zwick machine to avoid eventual parasitic bending effects due to the geometric imperfections of the specimens (Fig. 4).

Fig. 4: Experimental protocol and measurement technique

Model construction

In this section, the model is built while comparing it to the experimental results for each of the three zones of the TRCMC's mechanical behaviour. The model is developed zone by zone, based on the experimental results of the DFOS tests, the results reported in the literature' and the micromechanical parameters that control the behaviour of TRCMCs.

Global behaviour

The experimental results for the global behaviour of the tested TRCMCs are grouped in Fig. 5, where the three zones can be clearly identified. The stress-strain curve of the global behaviour is obtained by multiplying the strain recorded at each measurement point of the optical fibre by the elementary length of the spatial resolution (2.6 mm), thus obtaining the displacement at each point. Next, the sum of these displacements, which represents the total displacement, was calculated and divided by the length of the segment glued to the textile (20 cm). Stress is the ratio of the force applied to the TRCMC to its cross-sectional area. The curves presented on Fig. 5 are mean values. The repeatability of the results is detailed by the authors in a previous paper [START_REF] Saidi | Experimental analysis of the tensile behaviour of textile reinforced cementitious matrix composites using distributed fibre optic sensing (DFOS) technology[END_REF]. The results in Fig. 5 show a difference in the slopes of the first and third zones, as well as in the number of cracks and the ultimate stress. For the following sections, the model is constructed and compared with the experimental results reported in this work, as well as with the results reported by other studies in the literature.

First zone of mechanical behaviour of TRCMC: pre-cracking zone

Based on the experimental results, the Young's modulus value of the first zone E1 is measured and compared to the Young's modulus calculated by the law of mixtures.

Note that a comparison of the results for glass-reinforced TRCMCs and those based on the law of mixtures is published in [START_REF] Saidi | Experimental analysis of the tensile behaviour of textile reinforced cementitious matrix composites using distributed fibre optic sensing (DFOS) technology[END_REF]. These results showed that the law of mixtures is applicable in the first zone of the mechanical tensile behaviour of the TRCMC.

For greater certainty, a comparison is also made with the results for carbon-textilereinforced TRCMCs, presented in Table 3, which complements the study described in [START_REF] Saidi | Experimental analysis of the tensile behaviour of textile reinforced cementitious matrix composites using distributed fibre optic sensing (DFOS) technology[END_REF] with the addition of the effect of the textile type. The experimental value of E1 is calculated from the origin to the transition point from the first to the second zone (E1 = σ1/ɛ1). From these results, it can be concluded that the law of mixtures applies before cracks appear in the TRCMC. Thus, Fig. 6 corroborates the hypothesis of the law of mixtures by showing that the matrix strain is almost identical to the textile strain.

(a ) (b) (c)

3.3.Second zone: crack propagation zone

The crack propagation zone begins after the ultimate stress of the matrix is reached and the first crack appears on the composite. At the time of cracking, a drop in the stress is recorded; then' as the stress increases again, a second crack appears, a stress drop is recorded, etc.

To model this behaviour, it is necessary to estimate the drop in the stress due to cracking and the change in Young's modulus after each crack. Experimental results for the matrix strain distribution along the specimen shown in Fig. 7 and other results presented in a previous paper [START_REF] Saidi | Experimental analysis of the tensile behaviour of textile reinforced cementitious matrix composites using distributed fibre optic sensing (DFOS) technology[END_REF] show that after the first crack appears, the matrix strain is zero at the crack position, and gradually increases along the load transfer length δo, after which it becomes constant and reaches its nominal value. The repeatability of results is presented and analysed by the authors in a previous paper [START_REF] Saidi | Experimental analysis of the tensile behaviour of textile reinforced cementitious matrix composites using distributed fibre optic sensing (DFOS) technology[END_REF]. This means that over the area covered by the

Estimation of the equivalent Young's modulus Eδo of the 'δo -crack-δo' area

The assessment of the equivalent Young's modulus of the area covered by the two load transfer lengths of a given crack is based on the following assumptions:

• The Young's modulus of the matrix in tension Em and that of textile Ef are assumed to be constant at all points of the TRCMC (Fig. 8).

• Since the textile reinforcement is composed of continuous fibres along the TRCMC, the local strain of the TRCMC ɛc(x) is attributed to the local strain of the textile reinforcement ɛf(x), while working with the stress applied to the composite σc.

• At the crack position, the mortar does not contribute to the global stiffness, which reduces the stiffness of the TRCMC at this point (Ef.Vf). Moving away from the crack position, the matrix strain increases (Fig. 7), and as a result the TRCMC By subdividing the area between the crack position and the load transfer length δo into small infinitesimal elements of length dx, the stiffness of each element Kdx is obtained by applying Eq. 2 as follows:

Kdx = × ( )

Eq. 3

These elements are assembled serially, and therefore the equivalent stiffness Kδo can be calculated as follows:

Kδo = × Eq. 4 = × = ∑ × ( ) = × ( ) = ×( × × ( ))
Eq. 5

From this integral (Eq. 5), we obtain

= ×( ) Ln ( × ( ) × ( ) × )
Eq. 6

We obtain the expression for Eδo as follows:

Eδo = ×( ) ( × (!) × ("# ) × )
Eq. 7

Estimation of the stress drop during cracking

To estimate the stress drop at the occurrence of the first crack, it is necessary to calculate the equivalent Young's modulus of the TRCMC at that time. To accomplish this, the equivalent stiffness Keq of the entire specimen is calculated after the first crack occurs. It is considered that the composite is composed of three parts in series (Fig. 10):

• Two areas not covered by the two load transfer lengths of the first crack, L1 and L2, and Young's modulus E1

• The area 'δo -crack-δo' of length 2δo and Young's modulus Eδo.

× $%(") = + " + ' = ( × + " × " + ' × "
Eq. 8

with Ltot = 2δo + L1 + L2 Eq. 9

From Eq. 8 and Eq. 9, we obtain the expression for the equivalent Young's modulus of the TRCMC at the appearance of the first crack Eeq(1) as follows:

Eeq(1) = Ltot × " × ( × " × ( & & ( )
Eq. 10

Because the test is controlled in displacement with a relatively low velocity (0.1 mm/min),

and the crack appears instantly, the global strain of the composite εc just before and just after the cracking can be considered to be the same. Therefore, and by applying Hook's law (σc = Ec × εc), the stress drop during cracking is due to the drop in the global Young's modulus of the composite. It can be estimated as follows:

ε1 = σ1 / E1 = σ * / Eeq(1)
Eq. 11

where σ * is the stress applied to the composite just after the appearance of the crack.

Therefore, from Eq. 11, the stress drop at the first crack σ(1)drop is expressed as follows:

σ As shown in Fig. 9, the stress drop at the first crack decreases parabolically as the reinforcement ratio increases. At relatively high reinforcement ratios (over 6%), the force drop tends to be zero. This explains and corroborates the results in the literature [START_REF] Kang | Influence of reinforcement ratio on tension stiffening of reinforced engineered cementitious composites[END_REF][START_REF] Tlaiji | Développement et caractérisation du comportement thermomécanique des matériaux composites TRC[END_REF] of a high reinforcement ratio for this type of composite. Indeed, experimental studies have reported the absence of the crack propagation zone for TRCMCs with relatively high reinforcement ratios [START_REF] Tlaiji | Développement et caractérisation du comportement thermomécanique des matériaux composites TRC[END_REF], but by using digital image correlation it was found that cracks form, which means that the crack propagation zone exists but is not visible on the curve of the global behaviour of this material due to this nearly zero drop in the applied force.

Estimation of the global Young's modulus after each cracking

In subsection 3.3.2, the equivalent Young's modulus formula of the TRCMC at the time of occurrence of the first crack was demonstrated (Eq. 10). After the first crack, the applied force increases again until another crack appears; then the force drops, increases again, etc., until the beginning of the third zone.

When the stress increases again after a crack appears, the load transfer lengths of the previous cracks increase. This increase can be estimated by using the Cuypers model [START_REF] Cuypers | Stochastic matrix-cracking model for textile reinforced cementitious composites under tensile loading[END_REF] as follows:

δ + (, -) = / 0 1 0 ×2 3 4 5 0 6×7 5 0 ×(7 ×5 0 7 ×4 5 0 6) , - Eq. 13

Therefore, the increase in δo must be considered in Eq. 10. This means that the equivalent slope of the TRCMC when the applied stress increases again after cracking is not constant and decreases as the applied stress increases. This explains the curved shape of the stress-strain curve of the global behaviour of the TRCMC between two successive force drops (Fig. 1-b).

If we consider that there are N cracks on the TRCMC, and we want to calculate the equivalent Young's modulus Eeq(N) after this number of cracks, we proceed to a stiffness analysis as in subsection 3.3.1. As illustrated in Fig. 10, for N cracks, there will be N 'δocrack-δo' areas with length 2δo and Young's modulus Eδo, and a total equivalent length area of Ltot-2×N×δo. Calculating the global stiffness as an element in series using Eq. 15 and taking into consideration the overlaps of the δo denoted by α in subsection 3.3.4 (Eq. 16), we have

Eeq(N) = Ltot × " × ( × 8 × 9 × × " × ( & & ( × 8 × 9 × )
Eq. 14 consideration with the increase in the applied load on the TRCMC.

Criterion for stopping cracks

After a number of cracks on the TRCMC, the third zone begins, and no further cracks appear. To determine the number of cracks, an experimental study [START_REF] Saidi | Experimental analysis of the tensile behaviour of textile reinforced cementitious matrix composites using distributed fibre optic sensing (DFOS) technology[END_REF] shows that the third zone begins after the entire length of the TRCMC specimen is covered by the load transfer lengths of the cracks that appeared and that the final crack spacing 'X' lies between δo and 2 δo.

δo < X ≤ 2 δo Eq. 15

This means that the average final crack spacing is strongly related to the load transfer length as follows: X = α × δo, with 1< α ≤ 2 Eq. 16 From the results in Table 4, by introducing a coefficient β as the ratio of σ1 to σ2, it can be empirically concluded that :

σ2 = (1+β) × σ1 with 20% ≤ β ≤ 40% Eq. 20
Given the complexity of calculating the exact values of the stresses at the time of the appearance of cracks σNc, and based on the observation that generally σ1 < σ2, we consider that the stresses σNc increase linearly with the increase in the number of cracks. We conclude from Eq. 20 that for a given N-order crack :

σNc = (1+ < < & & × β) × σ1 Eq. 21
The stress drop σ(N)drop at the N-order crack can, therefore, be calculated based on Eq. [START_REF] Du | Experimental study on basalt textile reinforced concrete under uniaxial tensile loading[END_REF] and Eq. 21 as follows:

σ(N)drop = σNc × (1-Eeq(N) / Eeq(N-1))
Eq. 22

Critical reinforcement ratio

When a crack forms, the textile takes up the full load applied to the composite at the crack position. This implies that if the load applied to the textile exceeds its ultimate strength, it breaks before reaching the third zone. For this purpose, a critical reinforcement ratio must be applied to resist the stress σ2 and reach the third zone. It can be calculated as follows:

σ2/Vf critical < σuf Eq. 23

From Eq. 20 and Eq. 23, and by applying the law of mixtures, σ2 is expressed as follows:

σ2 = (1+β) × σ1 = (1+β) × (Vf × σf + (1-Vf) × σm) Eq. 24
where σf = (Ef / Em) × σm Eq. 25

From Eq. 23, Eq. 24, and Eq. 25, we can deduce that

Vf critical = ( =) × > × > 0? × ( =) × > ×( 0 )
Eq. 26

In this model, at each crack occurrence, a test is conducted for the ultimate strength of the textile as follows:

σNc / Vf < σuf Eq. 27

If Eq. 27 is satisfied, the calculation will continue. Otherwise, the conclusion is that the textile has reached its ultimate stress' because the critical reinforcement ratio was not exceeded.

3.4.Third zone: post-cracking zone

To characterise the behaviour of the post-cracking zone, it is sufficient to determine the slope E3, the stress σ3, and the strain ε3 at the failure of the composite.

Assessment of the Young's modulus E3

The third zone begins after the entire length of the specimen is covered by the load transfer lengths of the cracks that have appeared. Several studies [START_REF] Carozzi | Mechanical properties and debonding strength of Fabric Reinforced Cementitious Matrix (FRCM) systems for masonry strengthening[END_REF][START_REF] Aveston | Theory of multiple fracture of fibrous composites[END_REF][START_REF] Cuypers | Stochastic matrix-cracking model for textile reinforced cementitious composites under tensile loading[END_REF][START_REF] Ahn | Strain and hysteresis by stochastic matrix cracking in ceramic matrix composites[END_REF][START_REF] Larrinaga | Non-linear analytical model of composites based on basalt textile reinforced mortar under uniaxial tension[END_REF] consider that in the third zone the entire interface between the matrix and the textile is damaged. This means that the textile takes up all the load applied in the post-cracking zone as indicated in [START_REF] Saidi | Experimental analysis of the tensile behaviour of textile reinforced cementitious matrix composites using distributed fibre optic sensing (DFOS) technology[END_REF]. From this fact, we deduce that E3 = Ef × Vf Eq. 28

To verify Eq. 28, a comparison is shown in Table 5 between the Young's modulus E3 and the Young's modulus of the textile alone.

Table 5. Comparison of the mechanical behaviour of the textile at the core of the TRCMC with that of the textile alone: glass textile [START_REF] Saidi | Experimental analysis of the tensile behaviour of textile reinforced cementitious matrix composites using distributed fibre optic sensing (DFOS) technology[END_REF] and carbon textile

✓ ✓ ✓ ✓ ✓ ✓ - ✓ ✓

TRCMC specimen failure

Failure of the TRCMC specimen can occur in two cases:

• Textile failure occurs when its ultimate stress is reached:

σc/ Vf = σuf Eq. 29

• Slippage of the textile in the matrix in the area covered by the aluminium plates that transmit the load from the testing machine to the specimen.

To estimate the stress at which the textile slippage begins, pull-out tests were carried out for the used textiles and matrices, allowing the ultimate shear stress τpo to be determined for the three configurations tested: glass-M1, glass-M2, and carbon-M2. The experimental results of the pull-out tests are presented in details by the authors in a previous paper [START_REF] Saidi | Experimental analysis and analytical modelling of the textile/matrix interface shear stress in textile reinforced cementitious matrix composites[END_REF]. These pull-out tests allow to characterise the textile/matrix bond behaviour by quantifying the shear stress generated by the contact between the two TRCMC components at the interface level.

This stress is multiplied by the lateral surface of the textile yarns in the longitudinal direction which are covered by the aluminium plates that transmit the tensile force of the Zwick machine to the TRCMC specimens, and the result is compared to the experimental slippage force of the textile in Table 6:

Fslippage-theo = τpo × Pf × Lanc × Nyarns Eq. 30 Table 6 shows that the slippage force of the textile reinforcement can be estimated from a pull-out test. It is generally less than 0% to 35% of the theoretical value of the slippage force Fslippage-theo. By introducing a coefficient γ, the ratio of the experimental to the theoretical slippage force of the textile reinforcement, we find:

Fslippage-exp = Fslippage-theo × γ ; with 0.65 ≤ γ ≤ 1.00 Eq. 32

Note that the boundary conditions of TRCMC in tension influence the mode of failure and ultimate strength of these composites. Indeed, the mode of load transmission from the tensile machine to the specimen (such as clamps, hydraulic grips, clevis joints, etc.) influences the resistance of the textile/matrix interface in the regions nearby of these transmission systems [START_REF] Hartig | Influence of experimental setups on the apparent uniaxial tensile load-bearing capacity of Textile Reinforced Concrete specimens[END_REF][START_REF] Arboleda | Testing Procedures for the Uniaxial Tensile Characterization of Fabric-Reinforced Cementitious Matrix Composites[END_REF][START_REF] Caggegi | Experimental analysis on tensile and bond properties of PBO and aramid fabric reinforced cementitious matrix for strengthening masonry structures[END_REF]. For this reason, the authors have defined an interval of 0.65 ≤ γ ≤ 1, and have left the choice to the user to calibrate the coefficient γ according to the load transmission mode used.

To estimate the slippage stress, Eq. 32 is rewritten as follows:

σ3 × Sc = γ ×τpo × Pf × Lanc × Nyarns Eq. 33

If Eq. 33 is divided by Nyarns and Sf, it can be shown that the slippage stress σslip is

σslip = γ × Vf × τpo × Lanc × Pf / Sf Eq. 34
Thus, in the third zone, while the stress applied to the composite is lower than the slippage stress σslip (Eq. 34), the calculation will continue-until textile failure occurs. Otherwise, slippage will occur, upon which the calculation is terminated.

3.5.Recap

Based on the model construction steps in the previous subsections, an algorithm is developed (see Fig. 11) by grouping all the calculation steps for the three typical zones of TRCMC mechanical behaviour in tensile loading.

3.6.Model validation

To validate the developed model, it was implemented in MATLAB [49]. A comparison between the developed model, the ACK model, and the experimental results is shown in Fig.

12-Fig. 20. The values of the parameters α, β, and ɣ in this comparison are 1.337, 0.4, and 0.7, respectively.

From these results, it can be concluded that the developed model acceptably reproduces the global behaviour of the TRCMCs with the details of the three zones. Thus, the developed model is a significant improvement over the ACK model in the crack propagation zone. We observe that the strain at the end of the second zone, ε2, is overestimated by the ACK model in the case of one textile reinforcement layer, and it approaches the experimental results when the reinforcement ratio is increased, as is the case for two and three textile reinforcement layers. The ACK model estimates that the strain ɛ2 is equal to : ɛ2 = [1+0.666 Eq. 35

From Eq. 35, we can see that ɛ2 has an increasing trend with the decrease of the reinforcement ratio Vf, which explains the overestimation of this strain with the decrease of the number of reinforcement layers. A second comparison is made in terms of the number of cracks (Table 7) and strains ɛ1 and ɛ2 (Table 8). The number of cracks depends on the value of the parameter α (the ratio of the final crack spacing to the load transfer length), which lies between 1 and 2. The authors chose the value of 1.337 in accordance with the study conducted by Cuypers and Wastiels [START_REF] Cuypers | Stochastic matrix-cracking model for textile reinforced cementitious composites under tensile loading[END_REF]. The results of the comparison are presented in Table 7. This comparison shows that in general the number of experimental cracks lies between Ltot/δo and Ltot/(2δo), equivalent to α = 1 and α = 2, respectively. Thus, the value of α = 1.337 gives a value for the number of cracks that is close to the experimental results, which validates the proposed model. Regarding the comparison of the strains ɛ1 and ɛ2 presented in Table 8, the difference between the experimental results and the developed model is between 0% and 15%. This discrepancy can be justified by several effects, in particular the uncertainties expressed by the standard deviations, related to the textiles, matrix, reinforcement ratio, shape of the specimen, etc (see

(a) (b) (a) (b)
Table 1 and Table 2). Thus, the coefficient α influences not only the number of cracks, but also the strain ɛ2, which means that by varying this coefficient, the discrepancy between the model and the experimental results can be reduced.

Parametric study

To study the influence of the geometrical and mechanical parameters of the TRCMC on its global behaviour, a parametric study is carried out. In this study, three parameters are chosen: the total length, the shear stress at the textile/matrix interface' and the reinforcement ratio.

The strain at the end of the second zone, ε2, the ultimate strain, ε3, and the number of cracks are chosen for the parametric study. This choice is made under the assumption that the first zone does not depend on the above-mentioned geometric parameters apart from the reinforcement ratio, and its variation is predictable, unlike the second and third zones. This parametric study is conducted for the M1-G-1L configuration, varying one of the abovementioned parameters each time.

4.1.Effect of the total length of the specimen Ltot

A study of the effect of the total length of the TRCMC specimens is conducted to verify if a minimum length exists for the specimen length in the tensile tests. Thus, it establishes whether or not studies in the literature are comparable if the lengths of the specimens are not the same.

A step of 50 mm is chosen, with Ltot being varied from 50 mm to 1000 mm. The results are presented in Fig. 23. Fig. 23-a shows that ε2 and ε3 evolve almost parabolically as the total length increases, with an increasing trend for ε2 and a decreasing trend for ε3. Thus, from Ltot = 200 mm, both strains tend towards a fixed value. Therefore, it can be deduced that the specimen length must conform to a minimum length to minimise the variation in overall behaviour and ensure good repeatability. According to the authors, this length can be related to the load transfer length δo, which is 30 mm in this configuration. This total length must be at least 7δo, which will allow at least three cracks along the specimen.

In terms of cracks, because the spacing between cracks is related to the load transfer length δo and the latter is not influenced by the length of the specimen, it can be predicted that the number of cracks is proportional to the total length and evolves linearly as Ltot increases, as

shown in Fig. 23-b.

Note that Ltot influences the force drop at each crack. The latter decreases as Ltot increases.

4.2.Effect of shear stress at the textile/matrix interface τpo

A study of the effect of the shear stress at the textile/matrix interface is carried out to verify the influence of the bond between the cementitious matrix and the textile reinforcement at the interface.

A step of 0.5 MPa is chosen, with τpo varying from 0.50 MPa to 20 MPa. The results are presented in Fig. 22. Fig. 22-a shows that ε2 first increases almost linearly up to τpo = 2.5

MPa, and then decreases almost linearly as τpo increases, while ε3 exhibits a decreasing parabolic trend as τpo increases.

In terms of cracks, the load transfer length δo decreases as τpo increases, which means that the spacing between the cracks decreases. This explains the increase in the number of cracks as τpo increases, as shown in Fig. 22-b.

4.3.Effect of the reinforcement ratio Vf

A study of the effect of the reinforcement ratio is carried out to understand the observations reported in the literature on the influence of this parameter. Note that a variation in the reinforcement ratio implies a variation in the cross-sectional area Sf of the longitudinal yarns of the textile reinforcement, and therefore this parametric study is also valid for the parameter Sf.

A step of 0.25 % is chosen, with Vf being varied from 0.25 % to 10 %. The results are presented in Fig. 23. Fig. 23-a shows that ε2 increases almost linearly to 0.75%, after which it decreases almost linearly as Vf increases. For ε3, no results are obtained for Vf < 0.75%, after which this deformation exhibits an increasing parabolic trend as Vf increases. This behaviour of ε2 and ε3 observed at Vf = 0.75% is related to the critical reinforcement ratio explained in subsection 3.3.6, which is 0.8% for the M1-G-1L configuration. The decreasing trend of ε2 c explains the observations reported in the literature regarding the shrinkage of the second zone until it almost disappeared as the reinforcement ratio increased [START_REF] Larrinaga | Experimental and numerical modeling of basalt textile reinforced mortar behavior under uniaxial tensile stress[END_REF].

In terms of cracks, the load transfer length δo decreases as Vf increases, which implies that the spacing between the cracks decreases. This explains the increase in the number of cracks as Vf increases, as shown in Fig. 23-b andc. For reinforcement ratios below 5%, the ACK model diverges (Fig. 24). For reinforcement ratios that are too low (less than 0.5%), the ACK model yields ε2 values that exceed the ultimate textile strain, which is not consistent with the developed model. From Vf = 5%, both models give almost the same value (Fig. 24). • In the pre-cracking zone, a comparison of the experimental results with the law of mixtures confirmed the validity of this law before the appearance of cracks on the TRCMC. This law was used for the analytical modelling of this zone.

• The developed model is based on the load transfer length, interface shear stress, and the change in the local Young's modulus during the cracking of the TRCMC, which is expressed in terms of serial stiffnesses. This allowed estimation of the drop in the force applied to the composite during cracking in the second zone, as well as the global change in Young's modulus during the mechanical test.

• In the post-cracking zone, the experimental results for the textile reinforcement at the core of the composite and for the textile alone were compared. This confirmed the hypothesis that the textile takes up the total applied load to the TRCMC in the post-cracking zone. This provided the basis for the analytical modelling of this zone.

• The developed model allows one to take into consideration the eventual slippage of the textile in the matrix between the plates that serve as load transmission equipment from the tensile machine to the specimen.

• Due to manual manufacturing, the random nature of geometric defects' and the possible heterogeneity of the TRCMC components, the modelling and mechanical approximation of the stresses at cracking is complicated. The use of an empirical approach allows estimation of these parameters.

• The developed model is compared with the experimental results. It is found that the model reproduces the global behaviour of the TRCMC with details of cracking, force drop, Young's modulus change' and the number of cracks. Thus, the developed model is validated.

• A comparison between the developed model and the ACK model highlights the contribution of this model in relation to the available literature, particularly in the crack propagation zone. Unlike the ACK model, it shows good agreement in terms of the strain at the end of the second zone for low reinforcement ratios.

• A parametric study is performed by varying the total length of the TRCMC specimens, the shear stress at the textile/matrix interface' and the reinforcement rate. This allows visualisation of the sensitivity of the global mechanical response of the TRCMC to these parameters.

• As an outlook, the authors propose to investigate the shrinkage behaviour during the hardening process of the matrix and quantify its effect on the micromechanical parameters that drive the cracking behaviour of TRCMC.

• This developed model is based on the experimental results for a TRCMC reinforced by a coated textile that exhibits a monolithic tensile behaviour. It would be interesting to study TRCMCs with an uncoated textile, vegetal textiles, etc. to improve this model by considering the mechanical aspects at the textile/matrix interface for this type of textile' which exhibits telescopic behaviour.

σuf

  Ultimate stress of the textile reinforcement σ1 Stress corresponding to the first crack in the composite σ2 Stress corresponding to the last crack in the composite σ3 Ultimate stress of the composite

Fig. 1 :

 1 Fig. 1: Mechanical tensile behaviour of the TRCMC: (a) overall view; (b) close-up of the crack propagation zone

Fig. 2 :

 2 Fig. 2: Analytical models of the tensile behaviour: (a) ACK model [26]; (b) Cuypers model [27].

Fig. 3 :

 3 Fig. 3: Diagram of the TRCMC specimen, showing the optical fibre positions

Fig. 5 :

 5 Fig. 5: Global behaviour of the nine tested TRCMC configurations: (a) M1 matrix reinforced with glass textile [36]; (b) M2 matrix reinforced with glass textile [36]; (c) M2 matrix reinforced with carbon textile

Fig. 6 :

 6 Fig. 6: Experimental results for the M2-C-1L configuration of textile and matrix strain distribution along the specimen before cracking

Fig. 7 :

 7 Fig. 7: Experimental results for the M2-C-1L configuration of the matrix strain distribution along the specimen during crack propagation : (a) before cracking ; (b) first crack ; (c) second crack ; (d) third crack

1 Fig. 8 :

 18 Fig. 8: Hypothesis for calculating the equivalent Young's modulus along the load transfer length To calculate the equivalent Young's modulus, a stiffness analysis is performed. The stiffness Ki of an element of length Li, Young's modulus Ei, and section Si is expressed as

12 Fig. 9 :

 129 Fig. 9: Evolution of the stress drop at the appearance of the first crack as a function of the reinforcement ratio: comparison of the model with experimental results

Fig. 10 :

 10 Fig. 10: Illustrative diagram of the principle of equivalent serial stiffness in the crack propagation phase of the TRCMC's mechanical behaviour

Fig. 11 :

 11 Fig. 11: Algorithm representing the calculation steps of the TRCMC global mechanical behaviour model

Fig. 12 :Fig. 13 :Fig. 14 :Fig. 15 :Fig. 16 :Fig. 17 :Fig. 18 :Fig. 19 :Fig. 20 :

 121314151617181920 Fig. 12: Comparison of experimental results for the global behaviour of the M1-G-1L configuration with the developed model and the ACK model: (a) global view; (b) close-up of the crack propagation zone

Fig. 21 :

 21 Fig. 21: Parametric study of the effect of the total length of the specimen on the global behaviour of the TRCMC: (a) evolution of strains ε2 and ε3 ; (b) evolution of the number of cracks

Fig. 22 :

 22 Fig. 22: Parametric study of the effect of shear stress at the interface on the global behaviour of the TRCMC: (a) evolution of strain ε2 and ε3; (b) evolution of the number of cracks

Fig. 23 :

 23 Fig. 23: Parametric study of the effect of the reinforcement ratio on the global behaviour of the TRCMC: (a) evolution of strains ε2 and ε3 ; (b) evolution of the number of cracks; (c) global behaviour

Fig. 24 :

 24 Fig. 24: Evolution of the strain ε2 as the reinforcement ratio increases: comparison of the developed model with the ACK model

Table 1 .

 1 Mechanical properties of the two cementitious matrices M1 and M2

	Component Parameter	σum Tension (MPa)	εum Tension (µε)	Em Tension (MPa)
		Average	3.5	300	10700
	Matrix M1	Standard deviation	0.2	27	316
		Average	4.5	400	14000
	Matrix M2	Standard deviation	0.36	34	492
	2.1.2. Textile reinforcement
	Two textile reinforcements are used and studied in this work. The first is an alkali-resistant
	(AR) glass-based reinforcement, and the second is a carbon-based reinforcement, with
	respective inter-axis opening meshes of 5 × 5 mm² and 17 × 17 mm². Their mechanical and
	geometrical properties are obtained by standard characterisation tests in the laboratory by

Table 2 .

 2 Mechanical and geometrical properties of the two used textile reinforcements

			Mechanical properties	Geometrical properties	Surface and linear weight
	Textile	Value	σf (MPa)	εf (%)	Ef (MPa)	Sf (mm²) Pf (mm)	Textile weight (g/m²)	Weft direction (Tex)	Warp direction (Tex)
		Average	520	1.5	35000	2.2	7	525	2400	272
	AR glass	Standard deviation	48	0.12	3180	0.04	0.3	-	-	-
		Average	518	0.65	105000	2.98	11	520	3200	3200
	Carbon	Standard deviation	34	0.09	6820	0.05	0.32	-	-	-

Table 3 .

 3 Verification of applicability of the law of mixtures in pre-cracking zone for carbon-

	textile-reinforced TRCMCs
	Matrix		M2
	Carbon reinforcement	1 Layer	2 Layers	3 Layers
	Reinforcement ratio Vf (%)	1.75 ± 0.12	3.5 ± 0.24	5.2 ± 0.36
	Em matrix (MPa)		14000 ± 492
	Ef textile (MPa)		105000 ± 4820
	Experimental value of E1 (MPa)	14600 ± 330 16500 ± 416 19400 ± 562
	Law of mixtures (MPa) E1 = Vf × Ef + (1-Vf) × Em	15592 ± 677 17185 ± 862 18732 ± 1044
	Average difference (%)	2.37	3.35	2.02
	The uncertainty interval of E1 exp. intersects with that of E1 law of mixtures?	✓	✓	✓

Table 4 .

 4 Ratio of the end of the first zone σ1 to the end of the second zone σ2

			σ1 (MPa) σ2 (MPa)	(σ2 -σ1)/σ1 (%)
		M1-G-1L	3.44	4.67	35.76
		M1-G-2L	4.05	4.88	20.49
		M1-G-3L	3.94	4.98	26.40
	Exp. Results	M2-G-1L M2-G-2L	4.77 4.08	6.02 5.2	26.21 27.45
		M2-G-3L	4.62	6.43	39.18
		M2-C-1L	4.78	6.35	32.85
		M2-C-2L	5.48	6.8	24.09
		M2-C-3L	5.87	7.86	33.90
	Nguyen et al [46]	3.5	4.7	34.29
	Caggegi et al [47]	3.16	4.1	29.75
		Maximum value		39.18
		Minimum value		20.49

Table 6 .

 6 Comparison of experimental and theoretical results for the textile slippage force

		Textile			Glass				Carbon
	Designation	M1-1L M1-2L M1-3L M2-1L M2-2L M2-3L M2-1L M2-2L M2-3L
		Number of yarns Nyarns	6	12	18	6	12	18	3	6	9
		Pf (mm)	7	7	7	7	7	7	11	11	11
		Aluminium
	Theoretical results	bonding length Lanc (mm) Shear stress τpo (MPa)	70 1.4	70 1.4	70 1.4	70 1.87	70 1.87	70 1.87	70 2.2	70 2.2	70 2.2
		Theoretical
		slippage force	4116	8232 12348 5497.8 10996 16493	5082	10164 15246
		Fslippage-theo (N)
		Stress σ3 (MPa)	5.17	12.55 15.09 10.32 15.34 18.67	6.35	14.78 22.42
	Exp. results	Experimental slippage force	2843.5 6902.5 8299.5 5676	8437 10269 3492.5	8129 12331
		Fslippage-exp (N)
	Difference (%)	30.92 16.15 32.79	3.24	23.27 37.74	31.28	20.02 19.12

Table 7 .

 7 Comparison of experimental crack numbers with those obtained by the analytical

	model
	TRCMC Configuration		M1-Glass			M2-Glass			M2-Carbon
	Number of reinforcement layers	1 L	2 L	3 L	1 L	2 L	3 L	1 L	2 L	3 L
	Experimental final crack number	11	14	25	9	12	35	10	27	28
	Model results	α = 1.337 α = 1 α = 2	8 11 5	16 21 11	24 32 16	8 11 5	16 21 11	24 32 16	8 11 5	16 21 11	24 32 16

Table 8 .

 8 Comparison of the model to the experimental results of strains ɛ1 and ɛ2

	TRCMC	Strain ɛ1 (μm/m) Exp. Results Model Difference (%) Exp. Results Model Difference (%) Strain ɛ2 (μm/m)
	M1-G-1L	304	337	10.86	1455	1489	2.34
	M1-G-2L	311	327	5.14	1204	1246	3.49
	M1-G-3L	301	327	8.64	1288	1124	12.73
	M2-G-1L	320	321	0.31	1546	1401	9.38
	M2-G-2L	381	322	15.49	1212	1320	8.91
	M2-G-3L	321	332	3.43	1095	1182	7.95
	M2-C-1L	318	324	1.89	1227	1248	1.71
	M2-C-2L	321	324	0.93	1122	1020	9.09
	M2-C-3L	316	324	2.53	916	918	0.22

20

We know that if we have Ntot cracks after which the third zone begins, we will then have Ntot-1 crack spacings. So, from Eq. 16, we have X = Ltot / (Ntot-1) = α × δo, with 1< α ≤ 2

Eq. 17

Hence' the total number of cracks is Ntot =

In the literature, statistical studies [START_REF] Cuypers | Stochastic matrix-cracking model for textile reinforced cementitious composites under tensile loading[END_REF][START_REF] Ahn | Strain and hysteresis by stochastic matrix cracking in ceramic matrix composites[END_REF] have found that α =1.337. Nevertheless, in our model, we prefer to let the user choose a value while ensuring that Eq. 16 is satisfied. As a result, the total number of cracks becomes Ntot = Integer portion (

This allows the creation of a loop to define the crack stopping criterion: while the number of cracks is lower than Ntot, another crack must appear, etc.

Stresses of appearance of cracks in the second zone

When the TRCMC is subjected to tensile loading, the cementitious matrix strain increases until it reaches its maximum strain at one point, at which the first crack appears [START_REF] Saidi | Experimental analysis of the tensile behaviour of textile reinforced cementitious matrix composites using distributed fibre optic sensing (DFOS) technology[END_REF]. Then, because of the change in the global Young's modulus, the load applied to the composite decreases, and therefore the local matrix strain also decreases [START_REF] Saidi | Experimental analysis of the tensile behaviour of textile reinforced cementitious matrix composites using distributed fibre optic sensing (DFOS) technology[END_REF]. When the stress applied to the TRCMC increases again, the cementitious matrix strain increases and reaches its maximum at a point not located in the 'δo -crack-δo' zone of the first crack. At this point, the second crack appears.

On the other hand, in terms of stress, several studies [START_REF] Contamine | Contribution to direct tensile testing of textile reinforced concrete (TRC) composites[END_REF][START_REF] Tran | Mesoscale experimental investigation of thermomechanical behaviour of the carbon textile reinforced refractory concrete under simultaneous mechanical loading and elevated temperature[END_REF][START_REF] Larrinaga | Non-linear analytical model of composites based on basalt textile reinforced mortar under uniaxial tension[END_REF] have noted that the cracking stress σNc is not constant, but rather increases; i.e. the cracking stress of a crack N is greater than that of the crack N-1, etc. This can be explained by the geometrical defects in the specimen, which results in a non-uniform cross-sectional area of the composite, and by variation in the mechanical properties (Young's modulus, ultimate strain) of the matrix along the specimen due to curing and hardening, in addition to possible local heterogeneities in the matrix and textile. These parameters cannot be quantified, estimated' or modelled, and therefore the use of an empirical approach is necessary in this case.

The cracking stress of the first crack σ1 and that of the end of the second zone σ2 were compared for our experimental results as well as for results reported in the literature. This comparison is presented in Table 4.