Mohammad Abdallah
email: mohammad.abdallah@univ-lorraine.fr

Firas Al Mahmoud

Rémi Boissière

Abdelouahab Khelil

Julien Mercier

Experimental study on strengthening of RC beams with Side Near Surface Mounted technique-CFRP bars

Keywords: CFRP rods, SNSM reinforcement, experimental, RC beam, flexural behavior. 1 Introduction

This paper presents an experimental program that was carried out for the purpose of studying the global flexural performance of reinforced concrete beams strengthened internally with Carbon Fiber Reinforced Polymer (CFRP) rods using the Side Near Surface Mounted (SNSM) technique. The CFRP rods were placed laterally, adjacent to the longitudinal steel bars inside precut grooves. The strengthening length and position of CFRP rods as well as the type of filling material were the main variables investigated in this study. Moreover, a detailed comparison between the Side Near Surface Mounted (SNSM) and the Near Surface Mounted (NSM) techniques for the strengthening of RC beams using CFRP rods was conducted as well; in order to validate and assess the effectiveness of the SNSM technique. The test results showed that using the SNSM-CFRP rods technique allowed improving in a significant way the load carrying capacity of RC beams but decreased their ductility along the deflection at maximum load. The results obtained indicated that the failure mode was influenced by the length of CFRP rods and the filling material characteristics, while the strengthening position did not have significant impact. The SNSM strengthening technique can be used as an alternative method to the NSM method, and in some cases, it may be used to prevent the nonconventional failure modes, due to degradation of the NSM strengthening system, such as the pull out of CFRP-rods or premature debonding failure. The conventional analytical model accurately predicted the strength capacity and mid-span deflection of the SNSM strengthened beams.

represent genuine handicaps in its practical execution and long-term monitoring, still exist. For instance, one can mention the pretreatment process, installation time, premature debonding failure due to interfacial stresses, and the mechanical damages resulting from accidental impacts.

In the past ten years, the Near Surface Mounted (NSM) technology is viewed as a promising alternative to the externally bonded (EB) FRP strengthening technique; this is actually a prevailing method for strengthening existing concrete structures using FRP reinforcement [START_REF] Zhang | Reinforced concrete beams strengthened in flexure with near-surface mounted (NSM) CFRP strips: Current status and research needs[END_REF]. The NSM technique has attracted worldwide attention due to the different advantages it offers: [START_REF] Lam | Strength Models for Fiber-Reinforced Plastic-Confined Concrete[END_REF] The NSM-FRP system is quite simple and does not require extensive surface preparation work. The FRP composites (rods, strips, etc) are inserted adjacent to longitudinal steel bars and embedded inside precut grooves in the beam soffit; they are bonded to concrete with epoxybased pastes or modified cement grouts [START_REF] Al-Mahmoud | Strengthening of RC members with near-surface mounted CFRP rods[END_REF]; (2) RC members strengthened with NSM-FRP reinforcement proved to be much more ductile and experienced failure at much higher level as compared to EB-FRP members [START_REF] Hassan | Investigation of bond in concrete structures strengthened with near surface mounted carbon fiber reinforced polymer strips[END_REF][START_REF] Barros | Assessing the effectiveness of embedding CFRP laminates in the near surface for structural strengthening[END_REF]; (3) Strengthening of RC components with the NSM technique offers higher bonding efficiency and better protection for the FRP reinforcement as compared to the EB strengthening technique [START_REF] Lorenzis | Near-surface mounted FRP reinforcement: an emerging technique for strengthening structures[END_REF].

Several experimental studies investigating the NSM-FRP technique have underlined its potential for strengthening reinforced concrete (RC) beams. Almahmoud et al. [START_REF] Al-Mahmoud | Strengthening of RC members with near-surface mounted CFRP rods[END_REF][START_REF] Al-Mahmoud | Failure modes and failure mechanisms of RC members strengthened by NSM CFRP composites -Analysis of pull-out failure mode[END_REF] found that the NSM method, using CFRP rods, can significantly improve the ultimate strength of RC beams by 50%, at least, over the ultimate strength of non-strengthened beam. Their experimental results also indicated that the failure mode of strengthened beam varies from degradation of the CFRP rod mounting to compressive concrete crushing and depends on the ratio of a combination of CFRP cross-section and length to the concrete compressive strength. Kreit et al. [START_REF] Kreit | Repairing corroded RC beam with near surface mounted CFRP rods[END_REF] presented a testing program for the purpose of assessing the flexural performance of corroded RC beams strengthened with CFRP rods using the NSM method. The experimental results showed that retrofitted beams with 36% loss in steel cross section displayed an ultimate capacity similar to that of an un-corroded control beam. Moreover, two types of failure modes were observed; one was due to failure of the tension corroded steel and the other was because of concrete cover peeling off at level of the additional reinforcement (CFRP). Sharaky et al. [START_REF] Sharaky | Flexural response of reinforced concrete (RC) beams strengthened with near surface mounted (NSM) fibre reinforced polymer (FRP) bars[END_REF] investigated the flexural performance of NSM strengthened RC beams with partially and fully bonded CFRP and GFRP bars. The specimens strengthened with fully bonded reinforcement showed higher stiffness and greater bearing capacity than those strengthened with partially bonded reinforcement. Also, the concrete cover separation was the dominant failure mode regarding the NSM-CFRP rods beams. On the other hand, the fully bonding NSM-GFRP rods beams failed due to either deboning at the concrete-epoxy interface or concrete splitting, depending on the number of grooves used. Furthermore, Khalifa [START_REF] Khalifa | Flexural performance of RC beams strengthened with near surface mounted CFRP strips Alexandria[END_REF] made an attempt to assess the effectiveness of NSM and EB systems with CFRP reinforcement for the purpose of improving the flexural strength of RC beams under four-point loading. The beams strengthened with NSM-CFRP strips exhibited higher ultimate loads than those strengthened with EB-CFRP sheets; the difference ranged from 12% to 18%. The tested NSM beam specimens failed due to debonding of the CFRP strips along with concrete cover peeling off. For their part, Sebastian et al. [START_REF] Sebastian | Experimental characterisation of load responses to failure of a RC frame and a NSM CFRP RC frame[END_REF] evaluated the structural performance of indeterminate RC beams internally reinforced with NSM-CFRP rods. The strengthened structure failed due to the peeling-off of CFRP bars in the positive moment region; in addition, the ultimate load carrying capacity of the strengthened beam was 37% higher than that of the control beam.

Although the above mentioned studies have indicated that NSM-FRP reinforcement can significantly enhance the flexural behavior of RC beams by increasing their ultimate capacity; this strengthening system might not be feasible in several practical cases, especially when the building is operating. In this case, widths of the beams are usually occupied by partitions as masonry walls. In other cases, the NSM technique presents some restrictions in its application : [START_REF] Lam | Strength Models for Fiber-Reinforced Plastic-Confined Concrete[END_REF] Only a limited number of FRP bars can be used in each specimen, on account of the effective groove spacing. The minimum clear spacing (SG) between grooves (Fig. 1) should be greater than twice the depth of the groove, otherwise debonding failure may occur due to stress overlapping [START_REF] Al-Mahmoud | Strengthening of RC members with near-surface mounted CFRP rods[END_REF][START_REF] Aci | Guide for the design and construction of externally bonded FRP systems for strengthening concrete structures[END_REF][START_REF] Lorenzis | Characterization of FRP Rods as Near Surface Mounted Reinforcement[END_REF]; (2) Application of the NSM technique might be difficult or even impossible for some parts of structures, particularly, over the supports. Applying anchorage of FRP rods over the supports in RC beams cannot be easily performed and could impact other structural components, such as columns and joints, during preparation and/or installation; (3) The bond efficiency of NSM-FRP rods is substantially associated with the quality of the concrete cover; for instance, in corroded beams, the bottom concrete cover is highly exposed to damage due to corrosion of the longitudinal steel bars, particularly when steel bars are arranged in one layer, and therefore, putting FRP rods in such places may reduce their expected additional resistance and drive to premature failure [START_REF] Kreit | Repairing corroded RC beam with near surface mounted CFRP rods[END_REF][START_REF] Almassri | François Mechanical behaviour of corroded RC beams strengthened by NSM CFRP rods Composites[END_REF]; hence, a need to change the strengthening position becomes essential. (4) As stated in the above-mentioned literature, application of NSM-CFRP technique for strengthening RC beams could lead to non-conventional failure modes, i.e. peeling off and pull out failures modes, due to degradation of the strengthening system, which induces a significant reduction in the ductility of beams [START_REF] Lorenzis | Near-surface mounted FRP reinforcement: an emerging technique for strengthening structures[END_REF][START_REF] Al-Mahmoud | Failure modes and failure mechanisms of RC members strengthened by NSM CFRP composites -Analysis of pull-out failure mode[END_REF][START_REF] Kreit | Repairing corroded RC beam with near surface mounted CFRP rods[END_REF][START_REF] Sharaky | Flexural response of reinforced concrete (RC) beams strengthened with near surface mounted (NSM) fibre reinforced polymer (FRP) bars[END_REF][START_REF] Khalifa | Flexural performance of RC beams strengthened with near surface mounted CFRP strips Alexandria[END_REF][START_REF] Sebastian | Experimental characterisation of load responses to failure of a RC frame and a NSM CFRP RC frame[END_REF][START_REF] De Lorenzis | Passive and active near surface mounted FRP rods for flexural strengthening of RC beams[END_REF][START_REF] De Lorenzis | Bond between near-surface mounted fiber-reinforced polymer rods and concrete in structural strengthening[END_REF][START_REF] De Lorenzis | Flexural and shear strengthening of reinforced concrete structures with near surface mounted FRP rods[END_REF]. Thus, in order to overcome the NSM limitations or/and its operational obstacles, the Side Near Surface Mounted (SNSM) technique has recently been proposed as a new alternative approach for strengthening reinforced concrete (RC) members using fiber reinforced polymer (FRP) reinforcing bars. In this proposed technique the FRP elements inserted in the vertical sides of the beam instead of the bottom side. The use of the SNSM method is quite recent and very few researchers have investigated the subject so far. Indeed, Akter et al [START_REF] Akter Hosen | Side Near Surface Mounted (SNSM) technique for flexural enhancement of RC beams[END_REF] examined the structural behavior of RC beams strengthened with CFRP and steel reinforcing bars using the SNSM technique. Although the main variable studied was the additional reinforcement ratio, the test results indicated that applying 2∅8, 2∅10 and 2∅12 SNSM-CFRP rods could increase the flexural strength of beams by up to 91%, 138% and 133% ,respectively, over the non-strengthened control beam. The impact of other variables, such as the CFRP length, CFRP position and type of filling material have still not been extensively studied yet. Therefore, the purpose of the present work is to investigate the global behavior of reinforced concrete beams strengthened with 2∅6 CFRP rods using the SNSM technique. This was carried out by studying three key parameters that are likely to affect the structural response, namely, strengthening length, strengthening position and filling material characteristics. A detailed comparison between the SNSM and NSM strengthening approaches was performed in terms of the failure and bearing capacity mechanisms. In addition, the test results obtained were compared with those of conventional analytical models; also, the computed flexural strength capacity and mid-span deflection of the strengthened beams were found to be in good agreement with the experimental ones.

Experimental program

Test beams and materials.

A total of six RC rectangular beams, including one control beam (CB), were tested to failure by applying monotonically increasing four-point bending loads. The RC beams were designed to experience flexural failure in accordance with the ACI code [START_REF] Aci | Guide for the design and construction of externally bonded FRP systems for strengthening concrete structures[END_REF]; they were reinforced with two ordinary 12-mm-diameter deformed steel bars in the tension zone and two 6-mm-diameter ribbed bars in the compression zone. The deformed steel bars in the compression zone were utilized as hanger bars; and they were running along the shear zones. Closed 6-mm-diameter stirrups were provided against the maximum shear with a 150-mm center-to-center spacing. The concrete cover thickness in all tested beams was maintained at 20-mm for the vertical sides and 30-mm at the top and bottom sides. The beam geometry and steel reinforcement configuration are illustrated in Fig. 2.

The characteristics of concrete employed in the formulation of RC beams were determined by laboratory testing of three hardened concrete cylinder specimens (D = 160 mm and H = 320 mm). The average compressive and tensile strengths as well as the modulus of elasticity of concrete at 28 days were 37.0MPa, 3.0MPa and 30.3Gpa, respectively. However, in order to determine the mechanical properties of longitudinal steel bars, two steel bar specimens of diameter 12-mm were tested in tensile and the measured yield strength was 600MPa, and the modulus of elasticity was 210GPa.

Five beams were internally strengthened in bending with two 6-mm-diameter CFRP rods (56.5mm2) installed into two grooves, i.e. one rod in each groove, using the SNSM technique (see Fig. 2). The longitudinal grooves (12 × 12mm) were made with a special concrete saw. In addition, in order to ensure proper bonding between the filling material and concrete substrate, it was decided to apply an airbrush at high pressure to remove debris and fine particles. The grooves were filled up to two stages so to guarantee that the filling material is in-full contact with the surface of CFRP bars. Afterwards, all strengthened beams were confined at room temperature (T = 20 °C) for one week in order to obtain maximum strengthening level, depending on the filling material used, before testing.

The mechanical properties of the strengthening material were determined through axial testing of three CFRP rods by [START_REF] Al-Mahmoud | Effect of surface preconditioning on bond of carbon fiber reinforced polymer rods to concrete[END_REF]. The CFRP rods tested exhibited dissimilar response to that of the conventional steel bars, principally, the linear-elastic attitude up to the failure point (brittle failure). The ultimate tensile strength and Young's modulus of the CFRP rods used were 1875MPa and 146GPa respectively.

Two types of groove-filling materials for bonding the CFRP bars to the internal concrete surface were considered in this research, namely epoxy resin and mortar. Table 1 shows the compressive and tensile strengths of the epoxy resin and mortar at 7 days. The compressive and tensile strengths of resin were according to the manufacture, whereas the mechanical properties of mortar were determined by laboratory testing of four specimens.

Beam strengthening

Each strengthened beam was assigned a reference code that defines its strengthening scenario. For example, BC1(270-SR) means that beam BC1 is strengthened with 270-cm-long CFRP bars placed alongside the steel bars (S) and embedded in resin filling material (R).

However, the strengthening scheme of this study was designed to investigate the effectiveness of the SNSM-CFRP technique in upgrading the flexural capacity of RC beams. Therefore, the global flexural performance of the series beams BC1(270-SR), BC2(210-SR), BC3(270-SM) and BC4(210-SM), was studied with respect to the running length of CFRP rods and the type of filling material. The CFRP rods used to strengthen the above four beams were placed at the same level as the tensile steel (about 42-mm from the beam bottom surface). Beams BC1(270-SR) and BC3(270-SM) were strengthened with 270-cm-long CFRP bars, whereas 210cm-long CFRP bars were employed in strengthening beams BC4(210-SM) and BC2(210-SR). The CFRP bars were embedded in resin in the case of beams BC1(270-SR) and BC2(210-SR); however, for beams BC3(270-SM) and BC4(210-SM), these bars were inserted into mortar.

Furthermore, another RC beam, namely BC5(270-UR), was strengthened with two 6-mmdiameter CFRP rods embedded in resin and placed 20-mm higher than the longitudinal steel bars level (about 62-mm from the beam bottom surface), Fig. 2. The purpose was to assess effects of the position of CFRP rods on the flexural response of the beam. Indeed, the efficiency of the SNSM technique is substantially influenced by the quality of the concrete cover. Thus, a need to change the strengthening position in some competitive conditions might prove essential to ensure sufficient concrete cover for groove preparation and CFRP rebar installation. The flexural response of beam BC5(270-UR) was compared with that of beam BC1(270-SR). The test matrix of this study are summarized in Table 2.

Instruments and test procedure.

All beams were tested under four-point loads up to failure. The main load (P) was applied using a hydraulic actuator with capacity of 400 kN and 0.3 kN/s average loading speed. The total applied load was measured by means of an attached electrical load cell. A steel beam was utilized to distribute the total applied load into two concentrated loads. Each load (P/2) acted at a point situated at 80 cm from the adjoining support.

A vertical linear variable differential transducer (LVDT) was used for each tested beam in order to monitor the vertical mid-span deflection. Regarding the strengthened beams, four electrical strain gauges in total were used for each specimen for the purpose of monitoring the strain in the internal tensile steel bars and the adjacent CFRP rods. The strain gauges were attached to midpoint of reinforcement bars and oriented in the longitudinal direction. They were well protected with three different layers (protective paste, rubber piece and aluminum tape) after coated with polyurethane material to ensure high degree of measuring accuracy. Fig. 3 illustrates the set up and instrumentation of beams.

Results and discussion

load-deflection response

Figure 4 illustrates the curves representing the total applied load versus the mid-span deflection for the tested beams. In general, the load carrying capacities of RC beams strengthened with CFRP bars using the SNSM technique were greater than the load carrying capacity of the control beam, CB. The overall load-deflection response (P -δ) indicates that the tested beam specimens went through three distinct stages: elastic or pre-cracking stage (O-A), concrete cracking stage (A-B) and ultimate strength stage (B-C).

It should be clearly noted that in the elastic phase, all the beams displayed the same linear behavior and this is because of the flexural rigidity of beams as well as the bonding between the filling material and concrete were not yet been affected. The potential of CFRP rods to enhance the cracking strength of beams was negligible. The first turning point (A) represents the end of this phase, particularly, when concrete starts cracking.

In the concrete cracking stage, the slopes of strengthened beams were higher than the slope of control beam (A-B lines), which may be attributed to the fact that the CFRP bars improved the stiffness and yield load of the beams. As expected, the horizontal fine cracks on the leveled surface of grooves as well as the flexural cracks in the constant moment zone were observed in this stage. Note that these cracks spread and widened as the external load increased. The cracking phase ended at the second turning point (B), which represents the steel yielding load.

In the ultimate strength phase, the considerable change in slope of the (P -δ) curves as compared to the slopes in the two previous phases, was due to the intrinsic drop in the elastic modulus of tension reinforcement. It is apparent, however in this stage, that all strengthened beams presented higher load-increasing rates than that of the control beam (CB), which implies that the additional reinforcement, provided by the CFRP rods, controlled the flexural performance of the RC beams after the yielding point of tension steel was reached. At the third turning point (C), which represents the end of the ultimate strength stage, the RC beams strengthened with SNSM-CFRP rods exhibited lower ultimate mid-span deflections than that of the control beam.

Failure mods and cracking pattern

In addition to the conventional ductile flexural failure mode of the reference beam, CB (tensile steel yielding followed by concrete crushing at the beam mid-span); three different failure modes were observed in the SNSM-CFRP beams (concrete crushing in beams BC1(270-SR) and BC5(270-UR), debonding failure in beams BC3(270-SM) and BC4(210-SM) and concrete cover peeling off in beam BC2(210-SR) ), as indicated in Table 3.

Failure mode 1

The two beams BC1(270-SR) and BC5(270-UR) experienced the same failure mode although they carried different peak loads. Failure was due to crushing of the brittle compressed concrete in the constant moment zone, close to one of the two load application points, after yielding of the tension steel reinforcement, as shown in Fig. 5.

In the initial loading stage, horizontal fine cracks were apparent in the resin region. Afterwards, as the applied load was increased, these cracks gradually merged to the flexural cracks which propagated from the bottom side of the tested beams to the load application point. The flexural cracks growth in this set of beams (BC1(270-SR) and BC5(270-UR)) was influenced by the CFRP rod location. The flexural cracks in beam BC5(270-UR) were wider than those in beam BC1(270-SR), which is basically attributed to the displacement of the CFRP rods position to 20-mm above the steel bar level. The new position of the CFRP reinforcing bars in BC5(270-UR) created a new tensile stress level, which consequently caused the cracks to widen further and therefore to expanded upwards. Indeed, the crack pattern of this set confirms that the SNSM-CFRP reinforcing bars worked effectively as an additional tensile reinforcement [START_REF] Nadim Hassoun | Structural Concrete: Theory and Design[END_REF].

However, conspicuous horizontal cracks appeared in the failure sections of both beams, as a result of compressed concrete splitting. These cracks progressively extended and widened in the direction of the mid-span.

Failure mode 2

The failure mode of beams BC3(270-SM) and BC4(210-SM) was characterized by debonding between the internal surface of the groove (concrete) and mortar, as shown in Fig. 6.

The debonding failure mechanism of this set of beams was particularly influenced by the length of CFRP rods. Instantaneous debonding failure suddenly occurred in beam BC4(210-SM), immediately after yielding of the tension steel. This was very abrupt and was followed by beam failure once the concrete at the top of the beam began to crush. On the other hand, the beam BC3(270-SM) continued to deflect but did not fail even after it reached its yielding load; it continued to resist external loads until concrete started crushing in the top fibers. At this point, cracking noises could clearly and continuously be heard, due to the mortar debonding from the concrete, until the failure load was reached. Note that the failure section of beam BC3(270-SM) was observed at the mid-span, nearly the same way as for the control beam (CB); however, failure was observed close to the load application point for beam BC4(210-SM).

Regarding the cracking mechanism, the flexural cracks in beam BC4(210-SM) were narrow but more uniformly distributed than those in beam BC3(270-SM). On the other hand, the horizontal crack due to splitting of the compressed concrete was more conspicuous in beam BC3(270-SM). It is worth mentioning that the mortar-concrete interface was affected by internal longitudinal cracks, particularly at the failure sections. These cracks prevented the flexural cracks to cross the mortar in beam BC4(210-SM) or changed the flexural cracks direction in beam BC3(270-SM). This could be attributed to the debonding failure mechanism, as stated above in each beam [24].

Failure mode 3

The failure mode of beam BC2(210-SR) was certainly due to brittle concrete peeling-off in the region where maximum shear occurs; this was immediately followed by the beam failure, as shown in Fig. 7.

In the initial stage of loading, the crack pattern of beam BC2(210-SR) was almost similar to that observed in beams BC1(270-SR) and BC5(270-UR). As the applied load increased, shear cracks at the end of the additional reinforcement (CFRP bars) started to intersect the fine cracks which had previously appeared in the resin region. The intersection of the shear cracks with the minor horizontal cracks caused a significant main horizontal crack to develop and expend along the beam length at the same level as the CFRP rods. The intersection of the main horizontal crack with the bending vertical crack, which developed in the region close to the concentrated applied load, enhanced the peeling-off of the concrete cover over a significant length in the beam soffit, that is from the extremity of the CFRP bars to the region of maximum moment. In addition, one may clearly notice that visible flexural-shear cracks were formed over the failure length, particularly in the lower moment regions.

Enhancement of load carrying capacity of SNSM strengthened beams.

Table 3 summarizes the load carrying capacities of the tested beams representing the cracking load (P ), yielding load (P ) and ultimate load (P ). Fig. 8 shows the load-strain curve of the SNSM-CFRP rods. It should be noted that the CFRP load-strain curve is nearly similar to the load-deflection curve (see Fig. 4). From Fig. 8, it can be seen that the CFRP rods have not taken much load up to the first crack as the strains were almost zero; their potential to enhance the cracking load of the strengthened beams compared to the control beam was negligible. Afterwards, the CFRP strain curves have shown reasonable inclinations, which means that the additional bars have started to carry substantial loads, and thus significantly improved the yielding and ultimate loads of the SNSM-CFRP beams. The CFRP strains exhibited nearly linear curves up to tension steel yielding load, thereafter the strains were gradually increased up to the failure load.

In the subsequent sections, however, the effectiveness of the SNSM technique in improving the yielding and ultimate load carrying capacities of RC beams strengthened with CFRP rods is discussed with respect to the strengthening length, strengthening position and filling material type. The ultiame load of tested beams was determined from the load-deflection curves (Fig. 4), while the yielding load was detemined according to the strain gauge measurment (Fig. 9, ( ) ≈ 0.29%, where ( ) is the experimental yielding strain obtained from tensile test of the steel bars). It should be noted that the strain gauges installed on the steel rebar of beam BC3(270-SM) are stopped working during the loading process due to technical problem which may make the reported yielding load of this beam is not accurate.(see Fig. 9)

Effect of the CFRP strengthening length

The tension steel in beams BC1(270-SR) and BC2(210-SR) yielded at about 90kN and 94.5kN, respectively, which represents an increase of 40% and 47% over the yielding load of the control beam. Beam BC1(270-SR) failed due to crushing of brittle compressed concrete at loading of 116 kN with an increase of about 59.3% in the failure load compared to the control beam, whereas beam BC2(210-SR) failed as a result of concrete peeling-off at loading of 106.4kN with an increase of 46.2% in the failure load. The maximum tensile strain measured on CFRP rods was 0.0069mm/mm for beam BC1(270-SR) and 0.0042mm/mm for beam BC2(210-SR); these values represent respectively about 53.9% and 32.8% of the ultimate strain of CFRP bars (see Table 3).

Regarding beams BC3(270-SM) and BC4(210-SM), the yielding started, respectively, at 81.8 kN, which is 27.2% greater than that of CB, and at 92.4 kN, which is 43.7% higher than that of CB. The load at failure was 106 kN for beam BC3(270-SM), and at 94.1kN for beam BC4(210-SM). These failure loads represent an increase of 45.6% and 29.3% with respect to CB. As stated in the failure mode section, the debonding failure, between the filling material and concrete, occurred at an earlier stage in beam BC4(210-SM) than in beam BC3(270-SM). The recorded strain of CFRP bars at failure was 0.0064 mm/mm for beam BC3(270-SM) and 0.0028 mm/mm for beam BC4(210-SM); these values represent, respectively, about 50% and 21.9% of the CFRP rod ultimate strain.

Consequently, increasing length of the CFRP bars led to increase the failure load of the beam and the maximum measured strain of the SNSM-CFRP bars. It was noted that 60-cm of supplementary length of the CFRP rods helped to avoid non-conventional failure mode (peelingoff) or delayed the debonding failure, and therefore, the CFRP rods worked more efficiently as an additional tensile reinforcement. In other words, it may be stated that increasing the ratio between the strengthening length SL (distance between the end of the CFRP bar and the applied load) and the beam length BL (distance between the support and the applied load) significantly enhances the flexural response of the SNSM-CFRP beam. The ratio of beams BC1(270-SR)

and BC3(270-SM) was 0.94, whereas the ratio of beams BC2(210-SR) and BC4(210-SM) was 0.56. The failure load of BC1(270-SR) was about 9.1% grater than that of BC2(210-SR), and the failure load of BC3(270-SM) was about 12.6% grater than that of BC4(210-SM). The variation between the yielding load of beams BC1(270-SR) and BC2(210-SR), which represents about 5% relative to BC1, may either refer to experimental results scattering or influence of the SNSM strengthening length, thus further experimental studies to investigate effects of the strengthening length of SNSM-CFRP bars on performance of RC beams are necessary.

3.3.1 Effect of the filling material . The failure mode of mortar beams; i.e. BC3(270-SM) and BC4(210-SM), was characterized by debonding between the filling material and concrete substrate, whereas no signs of debonding failure were observed in the resin beams; i.e. BC1(270-SR) and BC1(270-SR), (section 3.2). These findings are in good agreement with those reported in previous studies, on NSM-CFRP beams, which indicated that using resin as a filling material forms better bonding with concrete than using mortar [START_REF] Al-Mahmoud | Strengthening of RC members with near-surface mounted CFRP rods[END_REF]. Thus, the failure loads as well as the maximum measured strain of CFRP rods in mortar beams (Table 3) were lower than those in resin beams (BC3(270-SM) vs. BC1(270-SR) and (BC4(210-SM) vs. BC2(210-SR)).

Effect of the strengthening position

As can be seen from Table 3, beam BC5(270-UR) yielded at 83.2kN before the yielding load of beam BC1(270-SR) (difference about 6.8 kN). The beam BC5(270-UR) failed due to concrete crushing, which is similar to the failure mode of BC1(270-SR), at 102.7kN. Although, this value is about 41.1% higher than that of the control beam; it is also about 11.5% lower than the failure load of beam BC1(270-SR). The maximum tensile strain measured on CFRP rods was 0.0041mm/mm and this value represents about 32% of the CFRP ultimate strain ( 40.6% lower than the maximum recorded strain of CFRP rods in BC1(270-SR)).

Consequently, the slight drop in the yield and ultimate load carrying capacities of beam BC5(270-UR) compared with beam BC1(270-SR) was due to the fact that the CFRP rods in beam BC5(270-UR) caused an additional tensile stress above the steel bars level; and this led to decrease the effective moment arm of the tensile reinforcement (CFRP and steel bars) within the beam cross section.

Ductility

Design standards require adequate ductility in order to prevent brittle failure of RC members, and therefore provide warning of impending collapse. In this study, the displacement ductility index (μ) is obtained from the load-deflection response of the beam specimens (Fig. 4), and it is calculated according to the deflection computation as follows [START_REF] Mukhopadhyaya | Optimizing structural response of beams strengthened with GFRP plates[END_REF]:

= ! " (1)
The mid-span deflection, corresponding to the beam ultimate load ! and yielding load " , and the ductility index (#) for the strengthened beams and control beam are given in Table 4.

In general, all the strengthened beams displayed less displacement ductility index ( #) compared to the non-strengthened beam; this is attributed to the increased tensile reinforcement (tension steel and CFRP bars). In fact, the ductility findings demonstrate effects of the brittle performance of CFRP rods and the SNSM strengthening approach as follows;

-Placing CFRP rods above the level of steel bars or using mortar instead of resin as a filling material proved to reduce the beam ductility. The decrease percent in μ-index of beams BC3(270-SM) and BC5(270-UR), with respect to the control beam, were about 45.1% and 33% respectively, whereas the decrease percent of BC1(270-SR) beam was 26.6%.

-The large reduction in ductility values of beams BC2(210-SR) and BC4(210-SM) was due to the insufficient strengthening length (210-cm), which led to non-conventional failure modes (peeling off or early debonding failure) as a result of degradation of the strengthening system. The percentages of decrease in the index of beams BC2(210-SR) and BC4(210-SM) were found equal to 66.6% and 76.7%, respectively, with respect to the control beam (CB). However, according to tension control design principle for reinforced concrete members, adequate ductility is achieved if the strain of steel ( $ ) at the point of concrete crushing or failure of the fiber reinforced polymer (FRP) is at least equal to 0.005 [START_REF] Aci | Building code requirements for reinforced concrete[END_REF]. Fig. 9 depicts the load-strain curves of the longitudinal tension steel in the tested beams. It can be noted that, using SNSM-CFRP rods significantly reduces tensile strains of the adjacent steel bars as compared to those in the control beam for the same applied load. The rate of increment was almost identical for all SNSM beams up to the yielding load. Afterwards, strains of the steel bars were rapidly increased but also influenced by the failure mode of the beam. It can also be found that, after the yielding load, inclinations of the load-strain curves of steel bars were lower than those of SNSM-CFRP bars for the same beam (see Fig. 8); which implies that the SNSM bars controlled the flexural behavior up to the failure load. As can be seen from Fig. 9; beams BC1(270-SR) and BC5(270-UR) meet the ductility requirement ( $ > 0.005). The measured tensile strains of the steel bars (Table 3) for beams BC1(270-SR) and BC5(270-UR) at the failure load were 0.0082-mm/mm and 0.0104-mm/mm, respectively.

Based on the steel tensile strain results; the displacement ductility index μ = 3 represents the acceptable lower limit to guarantee ductile performance of the SNSM-CFRP bars strengthened beam. It should be noted that, this result corresponds to the range of μ-index, which was proposed by [START_REF] Maghsoudi | Flexural ductility of HSC members[END_REF], in order to ensure the ductile behavior of RC beams (3 ≤ μ ≤ 5). The ductile index of beams BC1(270-SR) and BC5(270-UR), obtained from Eq.( 1), are within this reasonable range.

Energy absorption

The energy absorption capacity () *+ ) of RC beam could be defined as the area enclosed by the load-deflection curve (Table3). It indicates to the energy absorbed per unit cross-sectional area of the specimens calculated at any deflection extreme point [START_REF] Gopalaratnam | On the characterization of flexural toughness in fiber reinforced concretes[END_REF]. Fig. 10 demonstrates influences of position and length of CFRP bars as well as the filling material type (resin or mortar) on the energy absorption capacity of the tested beams.

One observation was that, the strengthening length significantly influences the energy absorption capacity of strengthened beams compared to the other variables considered in this study. Compared to the control beam, 270-cm-long CFRP bars improved the energy absorption of the SNSM beam whatever the filling material used or position of the CFRP bars; the ) *+ for beams BC1(270-SR), BC3(270-SM) and BC5(270-UR) increased by 69.6%, 9.2% and 20.9%, respectively. On the other hand, the energy absorption capacity of beams BC2(210-SR) and BC4(210-SM) decreased by 41.3% and 64%, respectively compared to the control beam. This considerable reduction in the ) *+ of SNSM beams strengthened with 210-cm-long CFRP bars is principally due to their premature failure mode.

SNSM and NSM techniques: comparison and discussion.

Almahmoud et al. [START_REF] Al-Mahmoud | Strengthening of RC members with near-surface mounted CFRP rods[END_REF] carried out laboratory tests for the purpose of investigating the global behavior of reinforced concrete beams strengthened with CFRP rods using the Near Surface Mounted (NSM) technique. Their experimental work was performed on simply supported RC beams subjected to flexural loading and strengthened with CFRP rods. The study focused on changes, such as failure modes and enhancement of ultimate strengths, in the structural performance of RC beams. However, in the present work, the mechanical characteristics of the tested specimens and CFRP bars (length, section,..etc.) were cautiously selected, and the experimental setup was carefully designed in compliance with those used in the above mentioned study. This was done for the purpose of comparing the two strengthening techniques, NSM and SNSM. For this reason, the experimental results of the three NSM-CFRP beams, i.e. S-C 6 (270-R), S-C 6 (270-M) and S-C 6 (210-R), were utilized. More details about these beams can be found in [START_REF] Al-Mahmoud | Strengthening of RC members with near-surface mounted CFRP rods[END_REF].

Table 5 compares the experimental results obtained with the SNSM technique with those obtained with the NSM technique in terms of the failure mode, cracking, yielding and ultimate moments as well as the ductility index( ). Figure 11 depicts the bending moment-deflection (, --) curves of the tested SNSM beams as well as those of the NSM beams. The corresponding bending moment ,of the beam is calculated by Eq.2.

M = P 2 × a ( 2
)
Where 0 is total applied load at each increment of displacement obtained from the experimental tests and 1 is the distance between the applied point load and support, 1 =

In general, the flexural response of RC beam specimens strengthened with CFRP rods using the SNSM technique corresponded well to the flexural response of beams strengthened with CFRP rods using the NSM technique. This is also true regarding the ultimate strength capacity, specially with regard to beams BC2(210-SR) and BC3(270-SM).

Based on Fig. 11-a, Beams BC1(270-SR) and BC5(270-UR) exhibited flexural performance similar to that of beam S-C6(270-R) in the elastic and cracking phases; however, while different slopes of the moment deflection curves were observed after tensile steel yielding. This difference between slopes was attributed to the position of the CFRP rods, i.e. the SNSM-CFRP rods were closer to the neutral axis as a result of the implementation of the SNSM technique, and consequently, this led to reduce their carried tensile forces which means reduction in the increasing rate of the SNSM beam resisting moment. The maximum variation in the (, --) curves was recorded at the failure moment. Failure of beam S-C6(270-R) occurred at 53.3 kN.m; and this value is about 6.9 kN.m (14.9%) greater than that of beam BC1(270-SR), and about 12.2 kN.m (29.7%) greater than that of beam BC5(270-UR). Although the failure moments in beams BC1(270-SR) and BC5(270-UR) are smaller than the failure moment in beam S-C6(270-R) due to the position of the additional strengthening bars; the SNSM technique used in this case helped to prevent slippage of the CFRP rods out of the strengthening system. Applying the SNSM technique caused to change the failure mode from CFRP rods pull out in S-C6(270-R) to compressed concrete crushing in BC1(270-SR) and BC5(270-UR). Fig. 12 compares the failure mechanism of beams BC1(270-SR) and BC5(270-UR) with that of beam S-C6(270-R).

Figure 11-b indicates that the (, --) curves corresponding to beams BC3(270-SM) and S-C6(270-M) are almost identical. The ultimate bending moment capacity of beam BC3(270-SM) was 1.5 kN.m (3.4%) less than that of beam S-C6(270-M). This great convergence between the both strength capacities, despite different locations of CFRP bars, might suggest that to the proposed SNSM technique has the capacity to put off the debonding failure occurrence. Indeed, the S-C6(270-M) beam was mainly failed due to debonding failure at the mortar-concrete interface before concrete crushing, whereas debonding failure was occurred simultaneous with the compressed concrete crushing in the BC3(270-SM) beam. Fig. 13 illustrates the debonding failure mechanism in the both beams (BC3(270-SM) and S-C6(270-M)).

Figure 11-c indicates that the ultimate capacity of beam S-C6(210-R) was 1.4kN.m (3.3%) higher than the ultimate capacity of beam BC2(210-SR). The S-C6(210-R) and BC2(210-SR) beams were both failed due to concrete peeling off at the end of the CFRP rods. This result confirms that, the peeling off failure is attributed more to the strengthening length than to the strengthening technique used (NSM or SMSM). The strengthening length (SL) to the beam length (BL) ratio of this set of beams ( = 0.56) is consistent with the expected peeling off failure ratio suggested in [START_REF] Al-Mahmoud | Failure modes and failure mechanisms of RC members strengthened by NSM CFRP composites -Analysis of pull-out failure mode[END_REF]. However, Fig. 14 illustrates the peeling off failure mechanism of beams S-C6(210-R) and BC2(210-SR).

Regarding the displacement ductility, it is worth noting that beams strengthened with CFRP rods using the SNSM technique proved to be more ductile as compared to the NSM beams, except for BC2(210-SR) for which the ductility index remained the same as that of beam S-C6(210-R). The overall improvement in ductility of SNSM beams is most likely referred to the failure mode and the interaction between the additional reinforcement location and the tension steel deformation. Fig. 15 gives a comparison between the ductility indices ( ) of SNSM and NSM beams.

Analytical model-Application of conventional prediction techniques

Prediction of deflection response ((δ) 567 )

The American Concrete Institute ACI 318-08 [START_REF] Aci | Building code requirements for reinforced concrete[END_REF] recommendations were employed in calculating the short-term deflection of reinforced concrete beams strengthened with CFRP rods using the SNSM technique under service loads. The mid-span deflections of beams could be computed as a function of the bending moment (M=(P/2).a) as expressed in Eq.( 3). Hence, the center deflections corresponding to the cracking, yielding and ultimate moment of beams were obtained in this study according to the sequence presented in Fig. 16.

(δ) 567 = , 24 E I ;; (3L = -4a = ) (3)
Where, E is the elastic modulus of concrete (E = 4700?@ A B ) and I ;; is the effective moment of inertia of the entire beam. The effective moment of inertia suggested by Branson was adopted and can be determined as follows [START_REF] Aci | Building code requirements for reinforced concrete[END_REF][START_REF] Branson | Instantaneous and Time-Dependent Deflections of Simple and Continuous Reinforced Concrete Beams[END_REF]:

C ;; = ( , AD , * ) E C F + H1 -I , AD , * J E K C AD ≤ C F (4)
Where, , * is the maximum moment calculated at the point of deflection of the beam, C F is the gross moment of inertia, C AD is the cracked moment of inertia and , AD is the cracking moment.

The moment of inertia, C AD of the cracked section is computed by assuming that the material has a linear elastic behavior, while neglecting the concrete below the neutral axis, as depicted in Fig. 17. Therefore, the position of the neutral axis (x) and the moment of inertia of the cracked section were computed using Eq.( 5) and Eq.( 6), respectively.

L = -MNO PQ N R O R S ∓ U MNO P + N R O R S V + W X V MN R O R Y R + NO P Y P S X (5)
Z [\ = XL ] ] + NO ^(Y P -L) V + N R O R _Y R -L`V (6)
Where, a = , b is the width of the cross section, f is the area of tension steel, f ; is the area of SNSM CFRP bars, g $ is the effective depth of steel bars and g ; is the effective depth of the CFRP rods.

In fact, applying Eqs.( 5) and ( 6), for deflection calculations (Eq.( 3)), when , * ≫ , AD (i.e. I ;; equals to I AD ) was found to underestimate the deflection in the SNSM-CFRP beams. For this reason, the moment of inertia, C AD was computed again while neglecting the elastic modulus of the steel reinforcement in the stage of _i ! -i " ` only, Fig. 16. This slight modification could be justified by following:

-The additional resistance of the strengthened beams to the external applied load in the ultimate strength phase (from the steel yielding load to failure load) was basically induced due to the presence of CFRP rods [START_REF] Al-Mahmoud | Strengthening of RC members with near-surface mounted CFRP rods[END_REF]. This can be seen in Fig. 4 (section 3.1) where all the strengthened beams showed load increasing rates greater than that of the control beam.

-The SNSM reinforcing bars start controlling the propagation of cracks in the strengthened beams once the tensile stress in the steel reinforcement exceeded the yielding point [START_REF] Akter Hosen | Side Near Surface Mounted (SNSM) technique for flexural enhancement of RC beams[END_REF].

5.2 Prediction of cracking moment(, AD )

The predicted cracking moment (, AD = @ j C jD k + l ) of the SNSM-CFRP beams was calculated by assuming that concrete fibers in the tension zone had reached their maximum tensile stress (m = @ j ). Where f o is the concrete tensile strength, I o is the un-cracked moment of inertia of the transformed section (see appendix A) and y q is the distance between the most tensile concrete fiber and the neutral axis before cracking.

Prediction of yielding moment(, )

In the same way of cracking moment, the predicted yielding moment (, = @ C AD ak l ) was calculated by assuming that the tension steel reinforcement had achieved the elastic behavior (m = @ ). Where @ is the elastic limit stress of the tensile steel and (k = g $ -r) is the distance between the steel bars and the neutral axis.

Prediction of ultimate moment(, s )

The ultimate moment of SNSM strengthened beam is assessed by the force equilibrium conditions and the stress-strain relation of concrete in compression and steel reinforcement in tension [START_REF] Aci | Guide for the design and construction of externally bonded FRP systems for strengthening concrete structures[END_REF], as presented in (appendix B). The assumption includes: (1) Conventional flexural failure of strengthened beam; i.e. crushing of the compressed concrete after yielding of the tension steel reinforcement. [START_REF] Li | An experimental and numerical study of the effects of thickness and length of CFRP on performance of repaired reinforced concrete beams[END_REF] Perfect bond between the internal surface of concrete and the filling material (resin or mortar). (3) The tensile strength of concrete is totally neglected. (4) The stress-strain relation of the concrete and steel suggested by ACI-318-08 is adopted. . . 5.5 Performance of the analytical model.

Figure 18 compares the calculated deflections of the tested RC beams at the cracking, yielding and ultimate levels with the experimental results. Overall, the analytical model gave quite a good estimation of the experiment deflections in beams strengthened with SNSM-CFRP bars. The ratio of the computed ultimate deflection to the experimental deflection ((δ) ( tu utov7) /(δ) (vwxv 65vyotu). ) are approximately 0.87, 0.88, 1.15, 1.24 and 1.06 for beams BC1(270-SR), BC2(210-SR), BC3(270-SM), BC4(210-SM) and BC5(270-UR), respectively.

Moreover, Fig. 19 shows a comparison between the experimental and the predicted bending moment-deflection relationships for the SNSM-CFRP beams. The bending moments obtained from the above analytical model and the experimental values were found to be in good agreement. The ratio of the , s (A*zAsz*j {) to the , s ( D|} ~j*z) are 1.03, 1.12, 1.13, 1.27 and 1.07 for beams BC1(270-SR), BC2(210-SR), BC3(270-SM), BC4(210-SM) and BC5(270-UR), respectively. The significant overestimated ultimate values in beam BC4(210-SM) (1.24 for deflection and 1.27 for moment) were primarily due to the early debonding failure which is not considered in the current analytical model. Further analytical studies to include such failures are strongly recommended.

Conclusion:

The present study aimed to analyze the global flexural response of RC beams strengthened with CFRP rods using the SNSM technique. The experimental results obtained and the studies conducted allowed drawing the following conclusions:

The Side Near Surface Mounted (SNSM) technique helped to improve the global flexural performance of RC beams whatever the strengthening scenario used. The load-bearing capacity of the beams BC1(270-SR), BC2(210-SR), BC3(270-SM),BC4(210-SM) and BC5(270-UR) were greater than that of the control beam by 59.3%, 46.2%, 45.6%, 29.3% and 41.1%, respectively. CFRP rods placed laterally adjacent to the steel bars and embedded in resin formed better resisting action to the flexural bending than those embedded in mortar or placed upper than the main tension steel. In BC1(270-SR) beam, the flexural strength increased over 59% compared to the control beam without large loss in ductility. The length of CFRP rods was found to have a considerable influence on the failure mode and energy absorption capacity. Two strengthened beams, i.e. BC2(210-SR) and BC4(210-SM), out of five beams showed premature failure due to the insufficient strengthening length (210 cm). The ratio of the strengthening length SL to the beam length BL of the both beams was about 0.56. However, using mortar caused the failure mode to change from peeling-off failure in BC2(210-SR) to early debonding failure in BC4(210-SM).

The strengthening position did not display significant impact on the failure mode. Beams BC1(270-SR) and BC5(270-UR) were both failed due to concrete crushing. However, the failure mode of beam BC3(270-SM) was debonding failure at the interface concrete-mortar because of the mortar filling material. According to the experimental data provided by this investigation; the Side Near Surface Mounted (SNSM) strengthening system can be used as an alternative to the Near Surface Mounted (NSM) system; it can in some cases help to avoid the non-conventional failure modes (pull-out of CFRP rods or early debonding failure). Furthermore, the SNSM-CFRP beams showed higher ductility behavior as compared to the NSM-CFRP beams. The computed flexural strength capacity and mid-span deflection of the strengthened beams, using the conventional prediction models, showed excellent agreement with the experimental results. (2) U: above than steel level.

Tables

( Debonding at the mortar-concrete interface followed by compressed concrete crushing. 3) and a = 800mm.

3) SL/BL: is the ratio between strengthening length SL and beam length BL, where SL is the distance between the end of the CFRP bar and the applied load and BL is the distance between the support and the applied load

(2) & # (yielding moment) = ) # * ; where ) # is the yielding load of the beam obtained from experimental test (Table 3) and a = 800mm.

(3) & (ultimate moment) = ) * ; where ) is the ultimate load of the beam obtained from experimental test (Table 3) and a = 800mm.

(4) ! (ductility index): (Table 4)

*According to the adjusted value (Table 3)

FiguresFig. 1 .Fig. 2 .Fig. 3 .Fig. 4 .Fig. 5 .Fig. 6 .Fig. 7 .Fig. 8 .Fig. 9 .FigFig. 11 .Fig. 12 .Fig. 13 .Fig. 14 .Fig. 15 .Fig. 16 .Fig. 18 .

 12345678911121314151618 Figures

Fig. 19 .

 19 Fig. 19. Comparison between experimental and theoretical moment-deflection curves

( 1 )( 3 )

 13 Strain of CFRP: the recorded strain in CFRP rods at failure load of a strengthened beam. (2) : Efficiency ratio of the CFRP rods = ( Strain of CFRP (1) /) . : Ultimate tensile strain of CFRP rod obtained from axial testing of three specimens = ./ . Strain of steel: recorded strain in steel bars at failure load of a strengthened beam. (4) : Energy absorption capacity. *Adjusted value, determined by considering 5% variation from BC4

9 ( 1 )

 91 & '( (cracking moment) = ) '( * , where ) '( is the cracking load of the beam obtained from experimental test (Table

Table 1 Characteristics of the filling material

 1

	Material	Compressive strength (MPa)	Tensile strength (MPa)
	Epoxy resin	83	29.5
	Mortar	74.6	6.2

Table 2 Strengthening details of the test beams.

 2

	Beam	Number of CFRP rod	length (cm) Strengthening	Material Filling	Position Strengthening	SL/BL (3)
	CB	----	----	----	----	----
	BC1(270-SR)	2∅6	270	Resin	S (1)	0.94
	BC2(210-SR)	2∅6	210	Resin	S	0.56
	BC3(270-SM)	2∅6	270	Mortar	S	0.94
	BC4(210-SM)	2∅6	210	Mortar	S	0.56
	BC5(270-UR)	2∅6	270	Resin	U (2) (+20mm)	0.94

(1) S: on the same level of the steel bars.

Table 3 Experimental results of the test beams

 3

	Beam	Failure Mode	Cracking load, P (kN)	Yielding Load, P (kN)	Failure Load, P (kN)	Strain of CFRP (1) (mm/mm)	(2)	Strain of steel (3) (mm/mm)
	CB	Conventional flexural failure	17.1	64.3	72.8	----	-----	0.0193	2702.5
	BC1(270-SR)	1	18.3	90	116.0	0.0069	0.54	0.0082	4583.8
	BC2(210-SR)	3	17.3	94.5	106.4	0.0042	0.33	0.0043	1586.1
	BC3(270-SM)	2	18.0	81.8 (88)*	106.0	0.0064	0.50	0.0035	2951.2
	BC4(210-SM)	2	18.8	92.4	94.1	0.0028	0.22	0.0033	972.0
	BC5(270-UR)	1	17.8	83.2	102.7	0.0041	0.32	0.0104	3267.9

Table 4 Mid-span deflection and ductility of the tested beams

 4

	Beam	δ (1)	δ (2)	! (3)
		(mm)	(mm)

Table 5 Comparison between experimental results of the SNSM and NSM techniques

 5

	SNSM technique

Acknowledgement

The authors acknowledge the financial support of the Freyssinet Company, France.

Appendix (A): Calculation of the transformed moment of inertia.

The moment of inertia of the transformed section of the beam strengthened with SNSM-CFRP rods was calculated based on Eq. (A.1).

Where y y is the neutral axis depth of the transformed section; it was calculated by equating the sum of areas above and below the neutral axis as given in Eq. (A.2):

Appendix (B): Calculation of the failure moment.

The ultimate resisting moment (, s ) of the SNSM-CFRP bars strengthened beam was assessed by applying the equilibrium conditions (C= • $ + • ; ) and the stress-strain correlation requirements, as presented in Fig. B1, where C, • $ and • ; are the compressive force, tensile steel force and tensile CFRP force, respectively.

The neutral axis location in a beam section where failure occurs can be computed as given below:

The ultimate compressive strain As in concrete was assumed equal to 0.003, based on the compressive test results; this value is similar to the one suggested by the ACI 318-08 Building Code. However, the maximum strain of CFRP rods, corresponding to the failure load, was evaluated using similar triangles as follows:

Therefore, the ultimate moment of the SNSM-CFRP section can be expressed using the following formula: , s = f @ Ig $ - †" 2 J + ; f ; ) ; (g ; - †" 2 ) (B.7)


Pierre-Henri Chavanis

Giuseppe Alberti

Gravitational phase transitions and instabilities of self-gravitating fermions in general relativity

Keywords: numbers: 04.40.Dg, 05.70.-a, 05.70.Fh, 95.30.Sf, 95.35.+d

We discuss the occurrence of gravitational phase transitions and instabilities in a gas of selfgravitating fermions within the framework of general relativity. In the classical (nondegenerate) limit, the system undergoes a gravitational collapse at low energies E < Ec and low temperatures T < Tc. This is called "gravothermal catastrophe" in the microcanonical ensemble and "isothermal collapse" in the canonical ensemble. When quantum mechanics is taken into account and when the particle number is below the Oppenheimer-Volkoff limit (N < NOV), complete gravitational collapse is prevented by the Pauli exclusion principle. In that case, the Fermi gas undergoes a gravitational phase transition from a gaseous phase to a condensed phase. The condensed phase represents a compact object like a white dwarf, a neutron star, or a dark matter fermion ball. When N > NOV, there can be a subsequent gravitational collapse below a lower critical energy E < E ′′ c or a lower critical temperature T < T ′ c leading presumably to the formation of a black hole. The evolution of the system is different in the microcanonical and canonical ensembles. In the microcanonical ensemble, the system takes a "core-halo" structure. The core consists in a compact quantum object or a black hole while the hot halo is expelled at large distances. This is reminiscent of the red giant structure of low-mass stars or the implosion-explosion of massive stars (supernova). In the canonical ensemble, the system collapses as a whole towards a compact object or a black hole. This is reminiscent of the implosion of supermassive stars (hypernova).

I. INTRODUCTION

The study of phase transitions is an important problem in physics. Some examples include solid-liquid-gas phase transitions, superconducting and superfluid transitions, Bose-Einstein condensation, liquid-glass phase transition in polymers, liquid crystal phases, Kosterlitz-Thouless transition etc. Self-gravitating systems also undergo phase transitions but they are special because of the unshielded long-range attractive nature of the gravitational interaction [1][2][3]. This leads to unusual phenomena such as quasistationary states, negative specific heats, ensembles inequivalence, long-lived metastable states, and gravitational collapse. The statistical mechanics of systems with long-range interactions is also an important topic in statistical mechanics [START_REF]Dynamics and thermodynamics of systems with long range interactions[END_REF][5][START_REF] Campa | Physics of long-range interacting systems[END_REF] because of its peculiarities and its various applications in astrophysics, plasma physics and hydrodynamics.

In this Letter, we consider the statistical mechanics of self-gravitating fermions in general relativity. This is a fundamental problem that combines general relativity, quantum mechanics and statistical mechanics. In addition, the thermodynamics of the self-gravitating Fermi gas can have application in relation to the physics of white dwarfs, neutron stars and dark matter halos made of massive neutrinos. It may also be related, as we shall see, to the supernova and hypernova phenomena. It represents therefore a topic of considerable interest both from a fundamental and an astrophysical point of view.

The statistical mechanics of self-gravitating systems dates back to the works of Antonov [START_REF] Antonov | [END_REF] and Lynden-Bell and Wood [8] who considered nonrelativistic classical stellar systems such as globular clusters. They enclosed the gas of stars within a spherical box of radius R in order to prevent its evaporation. They showed the absence of a statistical equilibrium state below a critical energy E c = -0.335GM 2 /R in the microcanonical ensemble (MCE) or below a critical temperature T c = GM m/2.52k B R in the canonical ensemble (CE) (the critical temperature T c was found earlier by Emden [9]). In that case, the system collapses. The gravitational collapse in the MCE is called "gravothermal catastrophe" [8]. In the case of globular clusters, it leads ultimately to the formation of a binary star surrounded by a hot halo [10] (this structure has an infinite entropy S → +∞ at fixed energy [11]). The gravitational collapse in the CE is called "isothermal collapse" [12]. In the case of self-gravitating Brownian particles it leads to the formation of a Dirac peak containing all the mass [13] (this structure has an infinite free energy F → -∞ [11]).

We may expect that quantum mechanics will prevent complete gravitational collapse on account of the Pauli exclusion principle (for fermions) giving rise to an additional quantum pressure. This is indeed the case for a nonrelativistic system of self-gravitating fermions as shown by Fowler [14], Stoner [15] and Chandrasekhar [16] at T = 0 in the context of white dwarf stars. The statistical mechanics of self-gravitating fermions at finite temperature in the nonrelativistic limit was developed by Hertel and Thirring [START_REF] Hertel | Thermodynamic Instability of a System of Gravitating Fermions[END_REF], Baumgartner [START_REF] Baumgartner | [END_REF], Messer [19], Bilic and Viollier [20] and Chavanis [3,21]. They showed that a gas of fermions experiences a gravitational phase transition from a gaseous phase to a condensed phase when the energy or the temperature passes below a critical value. In that case, an equilibrium state (gaseous or condensed) exists for all accessible values of temperature and energy. However, the situation is expected to change for the general relativistic Fermi gas because we already know from the work of Oppenheimer and Volkoff [22] on neutron stars that, at T = 0, there is no equilibrium state if the particle number N overcomes the Oppenheimer-Volkoff limit N OV = 0.39853 ( c/G) 3/2 m -3 .1 In this Letter, we report the results of our recent investigations concerning the statistical mechanics of selfgravitating fermions at finite temperature in the framework of general relativity. They complete the previous works of Bilic and Viollier [27,28] and Roupas [29][30][31]. The complete study being extremely rich, all the details are given in a series of exhaustive papers [32][33][34][35]. These papers also contain a detailed history of the subject with a long list of references. In this Letter, we emphasize the main results of our study and refer to our companion papers for more details and additional results.

II. BASIC EQUATIONS

We consider a spherically symmetric self-gravitating system in general relativity described by a metric of the form ds 2 = e ν c 2 dt 2 -r 2 (dθ 2 + sin 2 θdϕ 2 ) -e λ dr 2 where ν and λ are functions of r only. The statistical equilibrium state of the system is obtained by maximizing the entropy S = -k B ∫ C(f )e λ(r)/2 4πr 2 drdp at fixed mass-energy M c 2 = ∫ f E(p)4πr 2 drdp and particle number N = ∫ f e λ(r)/2 4πr 2 drdp [27,29,34]. Here f (r, p) is the distribution function (DF) and e λ(r)/2 4πr 2 dr is the proper volume element involving the metric coefficient e λ(r)/2 = [1 -2GM (r)/rc 2 ] -1/2 where M (r)c 2 is the mass-energy contained within the sphere of radius r. For fermions, S is the Fermi-Dirac entropy with

C(f ) = f ln(f /f m ) + (f m -f ) ln(1 -f /f m )
where f m = 2/h 3 . In the nondegenerate (classical) limit f ≪ f m , the Fermi-Dirac entropy reduces to the Boltzmann entropy with

C(f ) = f [ln(f /f m ) -1]. The extremization problem leads to the Fermi-Dirac DF f (r, p) = 2 h 3 1 1 + e -α e E(p)/k B T (r) , ( 1
)
where

E(p) = √ p 2 c 2 + m 2 c 4
is the energy of a particle and T (r) is the local temperature. In general relativity, the temperature is spatially inhomogeneous even at statistical equilibrium (Tolman's effect) [36]. The extremization of entropy at fixed mass-energy and particle number also yields the Tolman-Oppenheimer-Volkoff (TOV) equations [22,36]

dM dr = ϵ c 2 4πr 2 , ( 2
)
1 T dT dr = 1 ϵ + P dP dr = - 1 c 2 GM (r) r 2 + 4πG c 2 P r 1 -2GM (r) rc 2 , ( 3
)
expressing the condition of hydrostatic equilibrium, and the Tolman-Klein [36,37] relations T (r) = T ∞ e -ν(r)/2 and µ(r) = µ ∞ e -ν(r)/2 relating the local temperature T (r) and the local chemical potential µ(r) to the metric coefficient ν(r) (T ∞ and µ ∞ are the temperature and the chemical potential measured by an observer at infinity). Their ratio α = µ(r)/k B T (r) = µ ∞ /k B T ∞ is constant since they are redshifted in the same manner.

From the DF (1), we can express the particle number density n(r) = ∫ f dp, the energy density ϵ(r) = ∫ f E dp and the pressure P (r) = (1/3) ∫ f pE ′ (p) dp in terms of T (r) and α. We can then obtain T (r) by solving the TOV equations ( 2) and (3) with the boundary conditions T (0) = T 0 ≥ 0 and M (0) = 0, and determining α by the constraint N (R) = N . The mass and the Tolman (global) temperature are then given by M = M (R) and

T ∞ = T (R) √ 1 -2GM/Rc 2 .
Finally, we can plot the caloric curve T ∞ (M c 2 ) for a given value of N by varying T 0 from 0 to +∞. Instead of the mass-energy M c 2 it is better to use the binding energy E = (M -N m)c 2 which reduces to the Newtonian energy E = E kin + W in the limit c → +∞. Finally, we introduce the dimensionless energy Λ = -ER/GN 2 m 2 and the dimensionless inverse

temperature η = β ∞ GN m 2 /R with β ∞ = 1/k B T ∞ .

III. CLASSICAL SYSTEMS

The caloric curve η(Λ) of a classical self-gravitating gas in general relativity depends on a single parameter ν = GN m/Rc 2 called the compactness parameter. It is plotted in Fig. 1 for ν = 0.1. It has the form of a double spiral parametrized by the energy density contrast R = ϵ(0)/ϵ(R) [30,32]. The density contrast R is minimum at the "center" of the caloric curve and increases along the series of equilibria in the directions of the spirals.

The cold spiral (on the right) is a relativistic generalization of the caloric curve obtained by Katz [38] for the nonrelativistic classical self-gravitating gas in Newtonian gravity. It corresponds to weakly relativistic configurations (except when ν is large). It exhibits a minimum energy in the MCE at E c and a minimum temperature in the CE at T c below which the system undergoes a gravitational collapse as in the nonrelativistic case. The hot spiral is a purely general relativistic result. It is similar (but not identical [35]) to the caloric curve obtained by Chavanis [39] for the self-gravitating black-body radiation. It corresponds to strongly relativistic configurations. It exhibits a maximum energy in the MCE at E max and a maximum temperature in the CE at T max above which the system undergoes a gravitational collapse leading presumably to the formation of a black hole. 2 In the CE the series of equilibria is stable on the main branch between η min and η c . According to the Poincaré criterion [38,40], it becomes unstable at the first turning points of temperature η min and η c where the specific heat is infinite passing from positive to negative values. A new mode of instability is lost at each subsequent turning point of temperature as the spirals rotate clockwise. In the MCE the series of equilibria is stable on the main branch between Λ min and Λ c . According to the Poincaré criterion [38,40], it becomes unstable at the first turning points of energy Λ min and Λ c where the specific heat vanishes passing from negative to positive values. 3 A new mode of instability is lost at each subsequent turning point of energy as the spirals rotate clockwise. There are two regions of ensembles inequivalence, one on each spiral, between the turning points of temperature and energy, i.e., in the first region of negative specific heat.

It has to be noted that the stable equilibrium states are in fact metastable as there is no global maximum of 2 Below Tc the system collapses because it is too cold and the thermal pressure cannot balance the gravitational attraction. By contrast, above Emax the system collapses because it is too hot and feels "the weight of heat" [36] (energy is mass so that it gravitates). 3 Stable equilibrium states may have a negative specific heat in the MCE (isolated systems with fixed energy) while this is not possible in the CE (systems in contact with a heat bath with fixed temperature) [1][2][3][START_REF]Dynamics and thermodynamics of systems with long range interactions[END_REF][5][START_REF] Campa | Physics of long-range interacting systems[END_REF].

entropy at fixed energy or global minimum of free energy for classical self-gravitating systems [START_REF] Antonov | [END_REF]8]. However, these metastable states have a tremendously long lifetime, scaling as e N , so they are stable in practice [41]. 4The evolution of the caloric curve with ν is described in detail in [30,32]. As ν increases, the cold and hot spirals approach each other, merge at ν ′ S = 0.128, form a loop above ν S = 0.1415, reduce to a point at ν max = 0.1764, and finally disappear. The limit ν → 0 depends on the normalization that we use to plot the caloric curve. If we use the normalized variables Λ = -ER/GN 2 m 2 and η = β ∞ GN m 2 /R appropriate to the nonrelativistic limit we find that the hot spiral is rejected at infinity and we obtain a limit curve corresponding to the caloric curve of the nonrelativistic classical self-gravitating gas. It has the form of a (cold) spiral with turning points at Λ c = 0.335 and η c = 2.52 [START_REF] Antonov | [END_REF][8][9]38]. Alternatively, if we use the normalized variables M = GM/Rc 2 and B = β ∞ Rc 4 /GN appropriate to the ultrarelativistic limit we find that the cold spiral is rejected at infinity and we obtain a limit curve corresponding to the caloric curve of the ultrarelativistic classical self-gravitating gas [35]. It has the form of a (hot) spiral with turning points at M max = 0.24632 (like for the self-gravitating black-body radiation [39,42]) and B min = 17.809 (different from the self-gravitating black-body radiation [35]).

Remark-Let us assume here that the previous results apply to star clusters (they may also apply to gaseous stars as discussed in Sec. V in which case the following discussion would be different). In Newtonian gravity, it can be shown that all the isotropic DFs of the form f = f (ϵ) with f ′ (ϵ) < 0, where ϵ = v 2 /2 + Φ(r) is the energy of a star by unit of mass, are dynamically stable with respect to the Vlasov-Poisson equations (which describe collisionless stellar systems), even those that are thermodynamically unstable [43]. On the other hand, in general relativity, Ipser [44] has shown that dynamical stability with respect to the Vlasov-Einstein equations coincides with thermodynamical stability so that the equilibrium states located after the turning points of energy E c and E max are both thermodynamically and dynamically unstable. To solve this apparent paradox, it is expected that the growth rate of the dynamical instability decreases as the relativity level decreases. From these considerations, one expects that the collapse at the turning point E c of the cold spiral (weakly relativistic configurations) is mainly a thermodynamical -collisional -instability (gravothermal catastrophe) that takes place on a long, secular, timescale while the collapse at the turning point E max of the hot spiral (strongly relativistic configurations) is mainly a dynamical -collisionless -instability that takes place on a short, dynamical, timescale.

IV. QUANTUM SYSTEMS

The caloric curve T ∞ (E) of a quantum gas of fermions in general relativity depends on two parameters, the box radius R and the particle number N (throughout the paper, the radius is measured in terms of R * = ( 3 /m 4 Gc) 1/2 = 0.11447 R OV and the particle number is measured in terms of N * = ( 3 c 3 /m 6 G 3 ) 1/2 = 2.5092 N OV ) [33].

A. N -shape structure

We first consider a radius in the range R CCP = 12.0 < R < R MCP = 92.0 so that the caloric curve may display canonical, but not microcanonical, phase transitions [33].

For N ≪ N OV = 0.39853 we are in the nonrelativistic limit considered in [3,[START_REF] Hertel | Thermodynamic Instability of a System of Gravitating Fermions[END_REF]20,21]. When N < N CCP (R), where N CCP (R) ≃ 2125/R 3 corresponds to a canonical critical point, the caloric curve is monotonic like in Fig. 14 of [3] for µ = 10 and there is no phase transition. When N CCP (R) < N < N OV the caloric curve has an Nshape structure similar to Fig. 23 of [3]. In the MCE, the whole series of equilibria is stable. It displays a region of negative specific heat C = dE/dT ∞ < 0 between T c and T * . In the CE, the region of negative specific heat is replaced by a phase transition from the gaseous phase to the condensed phase. The condensed phase corresponds to a fermion ball at T = 0 (similar to a white dwarf or a neutron star) with only a tiny atmosphere. This structure has a very low energy. In principle, we expect a first order phase transition to take place at the transition temperature T t at which the two phases have the same free energy. 5 It would be marked by a discontinuity of energy. However, this first order phase transition does not take place in practice because the metastable gaseous states with T c < T < T t (which are local but not global minima of free energy) have tremendously long lifetimes, scaling as e N [41]. Not only are stable in practice, but they are also more physical than strictly stable equilibrium states (global minima of free energy) which can be reached only if the system spontaneously crosses a huge barrier of free energy by forming a "critical droplet" (maximum of free energy) represented here by a fermion ball (quantum core), a very unlikely process (rare event) in the CE. As a result, the physical phase transition does not take place at T t but rather at T c (spinodal point), the temperature at which the metastable gaseous branch disappears. At that point, the system undergoes a gravitational collapse (isothermal collapse) and becomes denser and denser until quantum effects come into play and stabilize the system. Once in the condensed phase, if 5 The canonical transition temperature Tt can also be determined by a method analogous to that of Maxwell's plateau in the discussion of the van der Waals gas. the temperature increases, there is an explosion, reverse to the collapse, from the condensed phase to the gaseous phase at T * (spinodal point). We can in this manner generate an hysteresis cycle in the CE [3].

When N < N OV , there is an equilibrium state (gaseous or condensed) for any accessible value of energy E ≥ E min or temperature T ≥ 0. Quantum mechanics (Pauli's exclusion principle) prevents gravitational collapse of classical self-gravitating systems below E c or below T c . However, as N approaches N OV , general relativistic effects come into play. For N OV < N ≪ N max , the caloric curve has the form of Fig. 2. In the MCE, there is no equilibrium state below a minimum energy E ′ c . In that case the system is expected to collapse towards a black hole. The series of equilibria is stable until E ′ c and becomes unstable afterwards. The entropy versus energy curve is plotted in Fig. 3. We can see the

convex intruder d 2 S/dE 2 = dβ ∞ /dE = -1/k B T 2
∞ C > 0 associated with the region of negative specific heat on the caloric curve, the minimum energy E ′ c at which the curve S(E) presents a spike (because δE = 0 implies δS = 0 since δS = β ∞ δE), and the branch of unstable states with lower entropy than the stable states with the same energy. In the CE, the evolution is more interesting. As we reduce the temperature, starting from the gaseous phase, the system first undergoes a phase transition at T c towards the condensed phase (fermion ball), followed by a catastrophic collapse at T ′ c towards a black hole. The series of equilibria is stable until T c , becomes unstable between T c and T * , is stable again between T * and T ′ c and becomes unstable again afterwards. The free energy versus temperature curve is plotted in Fig. 4. We can see the signal of a first order phase transition at T t marked by the discontinuity of the first derivative d(β ∞ F )/dβ ∞ = E of the free energy (this first order phase transition does not take place in practice as we have explained) and the two spinodal points T c and T * at which the curve F (T ∞ ) presents a spike (because δT ∞ = 0 implies δF = 0 since δ(β ∞ F ) = Eδβ ∞ ) and where zeroth order phase transitions (associated with the discontinuity of the free energy) take place. We also see the minimum temperature T ′ c below which the system collapses towards a black hole (the curve F (T ∞ ) also presents a spike there). For large values of N , the caloric curve approaches the classical caloric curve of Fig. 1.

Remark -The equilibrium states in the region of negative specific heat have a core-halo structure with a quantum core (fermion ball) surrounded by an isothermal atmosphere. They are unstable in the CE (saddle points of free energy) and represent a "critical droplet" or a free energy "barrier" that the system must cross in order to trigger a phase transition from the gaseous phase to the condensed phase (a rare event as we have seen). By contrast, these core-halo solutions are stable in the MCE (maxima of entropy at fixed energy). They may describe DM halos with a quantum core (representing a large bulge) and an isothermal atmosphere as suggested in [45,46].

B. Z-shape structure

We now consider a radius in the range R > R MCP = 92.0 so that the caloric curve may display both canonical and microcanonical phase transitions [33]. The description of the CE is the same as above so we focus on the MCE.

For N ≪ N OV we are in the nonrelativistic limit considered in [3]. When N < N MCP (R), where N MCP (R) ≃ 2.20 × 10 6 /R 3 corresponds to a microcanonical critical point, the situation is the same as above. When N MCP (R) < N < N OV the caloric curve has a Z-shape structure similar to Fig. 15

of [3] (dinosaur's neck).

There is a phase transition from the gaseous phase to the condensed phase. The condensed phase corresponds to a fermion ball at T = 0 (similar to a white dwarf or a neutron star) surrounded by a hot and massive atmosphere. 6 As before, the physical phase transition does not take place at the transition energy E t where the two phases have the same entropy but rather at E c (spinodal point), the energy at which the metastable gaseous branch disappears and the system collapses (gravothermal catastrophe). Once in the condensed phase, if the energy increases, there is an explosion, reverse to the collapse, from the condensed phase to the gaseous phase at E * (spinodal point). We can in this manner generate an hysteresis cycle in the MCE [3].

For N OV < N ≪ N max , the caloric curve has the form of Fig. 5. As we reduce the energy, starting from the gaseous phase, the system first undergoes a phase transition at E c towards the condensed phase (fermion ball), followed by a catastrophic collapse at E ′′ c towards a black hole. The series of equilibria is stable until E c , becomes unstable between E c and E * , is stable again between E * and E ′′ c and becomes unstable again afterwards. The entropy versus energy curve is plotted in Fig. 6. We can see the signal of a first order phase transition at E t marked by the discontinuity of the first derivative dS/dE = β ∞ of the entropy (this first order phase transition does not take place in practice for the reason explained above) and the two spinodal points E c and E * , at which the curve S(E) presents a spike and where zeroth order phase transitions (associated with the discontinuity of the entropy) take place. We also see the minimum energy E ′′ c be- 6 In the condensed phase the system has a core-halo structure. The core is condensed and behaves as a completely degenerate Fermi gas at T = 0 (because k B T ≪ µ(r)). For physically relevant parameters we find that the condensed core contains about 1/4 of the total mass [33]. It has a very negative potential energy.

Since the total energy is conserved the halo must have a very high kinetic energy, i.e., a very high temperature.

-1.2 -0.6 0 0.6 1.2 low which the system collapses towards a black hole (the curve S(E) also presents a spike there). For large values of N , the caloric curve approaches the classical caloric curve of Fig. 1.

Λ 0 0.7 1.4 2.1 2.8 η N = 1.3 R = 600 Λ c Λ'' c Λ * Λ t N OV < N < N f E C C''

V. ASTROPHYSICAL APPLICATIONS

We can use our model to draw a physical scenario of stellar evolution (of course in a very simplified setting) complementing the results of [47,48]. Let us consider a gaseous star initially with a high energy and a high temperature. We consider two types of evolution, a canonical one correponding to Fig. 2 and a microcanonical one corresponding to Fig. 5. As the star radiates light into space, its temperature T (t) (resp. energy E(t)) progressively decreases. As a result, the star follows the series of equilibria on the gaseous branch until it reaches the minimum temperature T c (resp. minimum energy E c ) at which point it becomes unstable and collapses. The collapse (due to a loss of hydrostatic equilibrium) continues until quantum mechanics comes into play. Then, if its mass is not too high, the star settles on a quiescent equilibrium state, a compact object similar to a white dwarf or a neutron star. The instability at the critical point corresponds to a saddle-center bifurcation. The evolution of the radius R(t) of the star on the gaseous branch can be described by a Painlevé I equation of the form R = -ct -bR 2 which explains the slow-fast transition in the life and death of a star [47,48]. The following discussion depends whether we work in the CE or in the MCE.

In the CE it can be shown that the perturbation δρ(r) that triggers the instability at the critical temperature T c has a core structure (one node) [12,47] while the velocity perturbation δv(r) has an implosive structure (no node) with δv < 0 (implosion) [47]. Therefore, the star is expected to collapse as a whole and form a compact object (white dwarf or neutron star) containing all the mass [3,33,47]. This phase transition is reminiscent of the hypernova phenomenon for supermassive stars (above 40 M ⊙ ) which shows very intense and directive gamma ray bursts, but no explosion of matter (or a very faint one). When N is large enough (N N OV ) there is no equilibrium state anymore, and the star collapses towards a black hole.

In the MCE it can be shown that the perturbation δρ(r) that triggers the instability at the critical energy E c has a core-halo structure (two nodes) [1,48] while the velocity perturbation δv(r) has an implosive-explosive structure (one node) with δv < 0 in the core (implosion) and δv > 0 in the halo (explosion) [48]. Therefore, the star is expected to split into a collapsing core, leading ultimately to a compact object (white dwarf or neutron star) containg a finite fraction (∼ 1/4) of the total initial mass [33], and an explosive (hot) halo expanding at large distances [3,33,48]. This phase transition is reminiscent of the red giant structure of stars with low or intermediate mass (roughly 0.3 -8 M ⊙ ) in a late phase of stellar evolution before the white dwarf stage. The implosion of the core and the explosion of the halo is also similar to the supernova explosion of massive stars with mass in the range of 8 -40 M ⊙ resulting in the formation of a neutron star. When N is large enough (N 4N OV ) there is no equilibrium state anymore, and the core collapses towards a black hole.

Similar results apply to stellar systems (globular clusters, galactic nuclei,...) and dark matter halos with, however, some differences. For these systems, only the MCE makes sense. At the critical energy E c , they undergo the gravothermal catastrophe [8] and separate into a collapsing core and a halo. They are in hydrostatic equilibrium but their evolution is induced by the temperature gradient between the core and the halo and by the fact that the core has a negative specific heat C c < 0 [8,49]. Therefore, by losing heat, the core grows hotter, contracts, loses heat again to the profit of the halo, and evolves away from equilibrium in an unstoppable process (thermal runaway). In the case of classical objects like globular clusters, core collapse leads to a binary star surrounded by a hot halo [10]. In the case of fermionic dark matter halos where quantum mechanics becomes important at small scales, core collapse may lead to a "fermion ball" (representing a quantum core or a large bulge) + the expulsion of a hot envelope. 7 The gravitational energy released by the collapse of the core heats up the envelope. In the box model, the atmosphere is held by the walls of the box. Without the box, the atmosphere is expelled at large distances (see Figs. 38 and 41 of [50] for an illustration in the context of the fermionic King model). Therefore, in the MCE the remnant is a fermion ball. Alternatively, in the case of galactic nuclei (star clusters) [51][52][53] and self-interacting dark matter [54] the collapsing core during the gravothermal catastrophe may become relativistic and finally experience a dynamical instability of general relativistic origin leading to a supermassive black hole of the right size (10 6 ≤ M/M ⊙ ≤ 10 9 ) to explain quasars and active galactic nuclei (AGNs). In that case, the halo is not crucially affected by the collapse of the core and maintains its initial (isothermal) structure. The timescale governing the phase transition of these different systems (stars and stellar systems) is very different. For supernovae where the energy is carried quickly by neutrinos they are fast, a few days, but for globular clusters, galactic nuclei, and dark matter halos they are very slow (secular), of the order of the age of the Universe.

In summary, we suggest that the microcanonial phase transition occurring in the self-gravitating Fermi gas may be related to the onset of red giant structure or to the supernova phenomenon. In these spectacular events, the collapse of the core of the system (resulting ultimately in the formation of a white dwarf, a neutron star, or a black hole) is accompanied by the explosion and the expulsion of a hot envelope. Newtonian gravity is sufficient to describe the white dwarf and the planetary nebula that follow the red giant stage while general relativity is necessary to describe neutron stars or black holes formed from supernova explosion. Similarly, dark matter halos made of fermions may have a core-halo structure where the core may be a fermion ball (bulge) or a black hole depending on the mechanism at work.

Remark -It is interesting to compare our results with previous results obtained in the same context. Lynden-Bell and Wood [8], considering a classical self-gravitating gas in the MCE, found the emergence of a core-halo structure and related it to the onset of red giants. Hertel and Thirring [START_REF] Hertel | Thermodynamic Instability of a System of Gravitating Fermions[END_REF] argued that, through the electromagnetic radiation, a star is in heat contact with the rest of the Universe which acts as heat bath. As a result they worked in the CE and mentioned the analogy between the canonical phase transition and the formation of supernovae. However, this analogy may not be fully correct because the phase transition that they obtained just corresponds to an implosion. This is because they worked in the CE and considered relatively small systems while the phase transition leading to an implosion-explosion phenomenon, associated with a core-halo structure, occurs in the MCE for larger systems [33,48]. Chandrasekhar, in the last sentence of [55], suggested that the supernova phenomenon may result from the inability of a star of mass greater than M Chandra to settle down to the final state of complete degeneracy without getting rid of the excess mass. He assumed that sufficient mass is ejected so that the remnant has a mass M core < M Chandra . He did not anticipate the formation of a black hole when M core > M Chandra . At a qualitative level, our scenario encompasses and generalizes these original ideas.

VI. CONCLUSION

In this Letter, we have presented the main results of our recent investigations concerning the statistical mechanics of box-confined self-gravitating fermions at finite temperature in the framework of general relativity [32][33][34][35]. We have highlighted a situation of physical interest, namely the canonical and microcanonical phase transitions of self-gravitating fermions slightly above the OV limit. In that case, the system first displays a transition from a gaseous phase to a condensed phase (white dwarf, neutron star, fermion ball) followed by a gravitational collapse towards a black hole, as the temperature or the energy of the system is reduced. Our study completes the results of Bilic and Viollier [28] who studied the canonical phase transition of self-gravitating fermions below the OV limit. It also complements the study of Roupas [31] who considered self-gravitating fermions under different boundary conditions appropriate to neutron stars (by imposing a fixed density on the boundary of the system instead of a fixed size) and studied the stability of the system as a function of the temperature. Additional results and details of our study can be found in our companion papers [32][33][34][35]. In particular, we have obtained in Ref. [33] the complete phase diagram of the general relativistic Fermi gas in the (N, R) plane determining the canonical and microcanonical critical points of this system, thereby generalizing the Newtonian results of [3]. This phase diagram was useful, for example, to interprete the results of Roupas and Chavanis [56] who studied phase transitions in the general relativistic Fermi gas using different control parameters, GN m/Rc 2 and R/R OV , instead of N/N OV and R/R OV .

We have also discussed astrophysical applications of our thermodynamical approach. In particular, we have argued that the CE may be relevant to describe the life and death of supermassive stars which collapse (implode) without exploding (hypernova phenomenon) while the MCE may be relevant to describe the life and death of less massive stars which present a more complex evolution marked by the collapse (implosion) of the core and the explosion of the halo (supernova phenomenon) [47,48]. This is an interesting consequence of ensembles inequivalence for systems with long-range interactions [START_REF] Campa | Physics of long-range interacting systems[END_REF]. Other interesting applications of the fermionic model in relation to dark matter halos made of massive neutrinos are given in [50,[57][58][59].

FIG. 1 :

 1 FIG.1: Caloric curve of the general relativistic classical selfgravitating gas for ν = 0.1. It presents a double spiral. The system collapses at low energies and low temperatures as in the case of a nonrelativistic classical self-gravitating gas in Newtonian gravity (cold spiral). It also collapses (towards a black hole) at high energies and high temperatures as in the case of the self-gravitating black-body radiation (hot spiral).

FIG. 2 :

 2 FIG. 2: Caloric curve for NOV < N ≪ Nmax (specifically R = 50 and N = 0.399). The arrows refer to the CE: (C) collapse at Tc from the gaseous phase to the condensed phase (fermion ball); (C') collapse at T ′ c from the condensed phase (fermion ball) to a black hole; (E) explosion at T * from the condensed phase (fermion ball) to the gaseous phase.

FIG. 3 :

 3 FIG. 3: Entropy per fermion as a function of the normalized energy for NOV < N ≪ Nmax (specifically R = 50 and N = 0.399). (S): stable; (U): unstable in the MCE.

FIG. 4 :

 4 FIG. 4: Normalized free energy as a function of the normalized inverse temperature for NOV < N ≪ Nmax (specifically R = 50 and N = 0.399). (S): stable; (M): metastable; (U): unstable in the CE.

FIG. 5 :FIG. 6 :

 56 FIG. 5: Caloric curve for NOV < N ≪ Nmax (specifically R = 600 and N = 1.3). The arrows refer to the MCE: (C) collapse at Ec from the gaseous phase to the condensed phase (fermion ball); (C") collapse at E ′′ c from the condensed phase (fermion ball) to a black hole; (E) explosion at E * from the condensed phase (fermion ball) to the gaseous phase.

This is the general relativistic extension of the Anderson-Stoner-Chandrasekhar-Landau[23][24][25][26] maximum mass of special relativistic Newtonian white dwarf stars.

Only a large random fluctuation can drive the system out of a local maximum of entropy (or local minimum of free energy). This is a rare event whose probability scales as e -N . Accordingly, the time scale for this phenomenon is exponentially large.

It is also possible that the halo is left undisturbed during the gravothermal catastrophe. In that case, the system acquires a core-halo structure with a quantum core (fermion ball) surrounded by an isothermal halo.