Laurent Daniel

Ahmed Loukili

Behavior of High-Strength Fiber-Reinforced Concrete Beams under Cyclic Loading

Keywords: crack, fiber-reinforced concrete, high-strength concrete

This study investigated both the influence of longitudinal steel ratio and steel jiber length on high-strength concrete (HSC) beams' behavior under alternate cyclic bending. The evolution in both structural properties and cracking patterns was c?mpared with results from the monatonic bending test. To ~erve the mjl~e of jibers on deterioration of mechanical properties due to loadmg cycles, high-strength jiber-reinforced concrete (HSFR.C) bemn:' were tested using two jiber.lengths: 30 and 60 mm. Th1s analysis highlighted the positive effect of jibers on both the secant structural stiffness and the cracking patterns during the prepealc stage. For the postpeak stage, the ductility measurement did not reveal an! improvement. In seismic cases, however. the .measure~ of cycl1c dissipated energy is an important parameter m evaluating structural behavior. Within this framework, the positive effect of fibers ?n energy dissipation as well as on the cumulative damage capac1ty has been underscored.

INTRODUCTION

It is well-known that high-strength concrete (HSC) is a material featuring many favorable aspects, not o~y by ~e of its high compressive strength, but also through 1ts durability improvements. The mechanical tests in the literature, however, have revealed the brittleness of HSC and the low rate of increase in tensile strength. 1 • 2 These features reduced the use of HSC in earthquake-zones as a result of recommendations emphasizing the importance of obtaining a ductile material. [START_REF] Davidovici | ~me parasismique[END_REF] It is important, however, to mentio~ that, in the case of a high-strength reinforced concrete beam, the structural ductility4 of HSC can be similar to or better than that of ordinary reinfOI'Ced concrete beams if the failure mode occurs by steel failure. During an earthquake, however, structures are subjected to reverse loads, which ind~ce ~oth severe tensile damage of concrete and bond detenoration. Hence, the postpeak behavior is particularly influenced by the tensile and bond strength of the concrete. [START_REF] Shin | Flexural Ductility of Ultra-High-Strength Concrete Members[END_REF] Previous investigations have shown the effectiveness of fibers on the cracking propagation. 6 • 7 The greatest im~ve ment involved the increase of dissipated energy, charactenzed by the uniaxial compressive or tensile beha~or of a. soften~ postpeak slope. [START_REF] Hsu | Stress-Strain Behavior of Steel-Fiber High-Strength Concrete under Compression[END_REF] This focus has served to gwde certain stu?ies on structural applications and has pointed out the effecti~e ness of fibers on cracking control and on shear and bending capacities. [START_REF] Casanova | ~de fibres m6talliques: du mathiau lla structure[END_REF] The influence of fibers on the structural failure mode, however, depends on a structure's dimension, fiber type and concrete. Experimental results such as those re-' [START_REF] Bspion | De !'utilisation de fibres m~talliques conune armature compl~mentaire de poutres en b~ton arm~ soumises lla flexion[END_REF] . di dth . parted by Bspion, Devillers, and Halleux m cate e mefficiency of fibers in ductility increase, and can llf opposed to the results given by Chunxiang and Patnaikuni that revealed the good influence of fibers in ductility. These controversial results pointed out the qualified use of fibers under monotonic load, but few papers deal with the influence of fibers on structural behavior under cyclic load. The purpose of this paper is to examine the influence of fibers on cyclic beam-bending behavior (cyclic test). Two lengths of fibers are used: 30 and 60 mm. As a tradeoff between loss of workability and increase in structural strength, a reasonable level of steel fibers was chosen: 1% (by volume). To remain close to the structural dimensions used in civil engineering practice, a beam length of 2.75 m (0.11 ft) ~d a cross section of 150 x 300 mm (6 x 12 in.) are used. The mfluence. of tensile reinforcement is studied by including three different values for the tensile reinforcement ratio p.

RESEARCH SIGNIFICANCE

This study reports useful data on the app~cation. of ~gh strength fiber-reinforced concrete (HSFRC) m a se1sm1cally active zone. It has been shown that reverse loads involve severe bond deterioration. The control of macrocracks by fibers enhances not only the mechanical properties of high• strength concrete in tension and comp~ssion, but also thJ structural behavior. According to Lemmtre and Chaboche, mechanical properties of HSFRC, such as strength and ductility, are strongly influenced by length of fibers; consequen~y, this work brings some information about structural behavl<n: enhancement by fibers with respect to both the length of fibers and to the longitudinal reinforcement ratio used. These results broaden the application of fibers to reinforced concrete structures in seismically active zones.

MATERIAL PROPERTIES

• Three types of concrete were used for the purposes of this study: HSC, HSFRC with a fiber length of 30 JDID (HSFRC 30 ), and HSFRC with a fiber length of 60 JDID Laurent Daniel is a posrgradUilte student from the Laboratoire de Gtnie Civil de Nantes St-Nawire, Ecole Centrale tk Nantes, Nantes, France. His resean:h interests inchule the nwchonical improvertU!nt of new concrete (high-strength concrete and high•strength fiber-reinforr:ed concrete) and its structural application under severe enviroMII!ntal and loading conditions. (HSFRC6Q). The mixture components for these concretes are given in Table 1.

ACI rtU!mber

A portland cement with low C 3 A content was used. Undensitied silica fume was added at a proportion of 10% of cement mass. The water-binder ratio was 0.32. To avoid a loss of workability due to the inclusion of fibers, the quantity of high-range water-reducing admixture was adjusted from 0.54 to 1% dry extract of cement mass The fibers were book-ended to enhance the anchorage in the matrix. The length diameter ratio of the fibers ltd 1 served as an efficiency factor, 4 and the optimum value used in the fibers' reinforced concrete production is ltd 1 = 80. This ratio has therefore been chosen for the two types of fibers. The fiber volume was constant at 1%. Fiber characteristics are displayed in Table 2.

Compressive tests were carried out on 110 x 220 mm ( 4.3 x 8.6 in.) cylindrical specimens after 28 days using a testing machine under load control at a loading rate of 0.5 MPals. The concrete specimens were kept at 20 ac and 95% relative humidity during the setting process, and then demolded 24 h after casting. Compressive tests were performed on three specimens for each type of concrete, and results are presented in Table 3. The compressive strength of HSC was approximately 95 MPa (13.8 ksi), with a strain at peak of2.4%o. The fibers accounted for a 15% increase in compressive strength, with an average value of 112 MPa (16.2 ksi) for HSFRC 30 and 118MPa(17.1 ksi)forHSFRC 60 . The strain at peak was also increased since it reached 3.1 %o for HSFRC3o and 2. 7%o for HSFRC 60 , and highlighted the ability of HSFRC to store a higher quantity of energy in the prepeak domain than HSC.

EXPERIMENTAL PROGRAM

The general test setup and beam cross section are described in Fig. 1(a). A dynamic actuator was used to apply reverse loads on beams 2.75 m in length and 150 x 300 mm in cross section. A positive and negative loading direction was introduced by means of a threaded rod system linked to the beam. The span-depth ratio aid was chosen to obtain bending failure. To generate failure between the loading points, stirrups (diameter of 8 mm, spacing of 20 cm) were arranged outside the pure bending zone. The influence of the reinforcement on beam behavior was analyzed by testing three different values fm the tensile reinforcement ratio (0.55, 0.97 and 1.52%) using longitudinal bars 12, 16, and 20 mm in diameter, respectively. The steel was Type FeE500 (tensile yield strength: 500 :MPa).

Each support was fitted with a swiveling frame that prevented vertical displacement but allowed rotation about the z-axis (Fig. l(b)). The horizontal displacement was introduced by rollers arranged on one of the supports.

The cyclic tests were displacement controlled. A 0.5 Hz loading frequency was selected to reach the ultimate state of the beams. To measure strength deterioration at a given amplitude deflection, the loading sequence (Fig. 2) was adjusted to 30 s. All of the test features related to the beams are given in Table 3.

BOND DETERIORATION UNDER REVERSE LOADS

A cyclic loading is the worst case of loading, as each concrete layer is alternately submitted to tension and compression stresses. Concrete and steel-concrete bonds are severely damaged. The bond deterioration mechanism observed by Popov [START_REF] Popov | Bond and Anchorage of Reinforcing Bars under Cyclic Loading[END_REF] is described in Fig. 3(a) to (d) and relates to the typical cyclic load deflection curve presented in Fig. 4, which displays the following: • Before the peak load: During the positive loading (Fig. 3(a)), the difference between the steel and concrete Young's moduli generates bond stresses. Once the limit tensile stress of the concrete has been reached, cracks Dellec:llon (mm)

Fig. 4-Typical cyclic curve and envelope curve (HSFRC 3 o beam).

occur and low anelastic strain remains (Fig. 3(a)).

During unloading, the secant structural stiffness K on the load-deflection curve is regulated both by the strain of tensile bars and by the strain of the compressive zone. Once the load is close to zero, lugs and concrete are no longer connected, and compressive forces are used to close the tensile cracks (Fig. 3(b)). This mechanism involves a decrease in secant structural stiffness up until the closure of the tensile cracks and the connection between lugs and concrete (Fig. 3(c)). The secant stiffness can then be increased and new cracks generated due to compressive stresses of the bars, which may explain the pinching of the curves during the loading phases; and

(a) static test: p = 1.52% (b) cyclic test: p = 1.52%

Fig. 5-Cracking pattern of HSC beams under cyclic and monotonic loading.

• After the peak load: Bond deterioration is severe (Fig. 3(d}}, and the residual steel strain is very high. Consequently, during a reverse load. the tensile steel strain remains and prevents the closure of previous tensile cracks. The secant stiffness cannot be increased, and the pinching disappears (Fig. 4).

COMPARISON BETWEEN MONOTONIC AND CYCLIC TESTS

To measure the influence of reverse loads on structural properties, two static tests were carried out on HSC beams with p = 0.97% and p = 1.52%, respectively. The analysis focused on both the cracking pattern and the load-deflection curves.

Cracking pattern

The expansion of the cracking zone can be observed in Fig. 5(a) and (b). The reverse loads prevent the concrete from crushing, but the cracks propagate throughout the depth over a wider zone. Moreover, a greater number of horizontal cracks are located at the level of the steel bars, thereby demonstrating the severity of steel-concrete bond deterioration.

The minimal and maximal spacings measured in static tests are close to those obtained by Maurel [START_REF] Maurel | Contribution ll'~tude de la fissuration des membrurea tendues en ~ton arm~ de hautes performances[END_REF] on beams with a similar cross section and a shear span of 0.8 m (minimal spacing of 5 cm and maximal spacing of 12.5 cm). The longitudinal steel ratio seems to have little influence on the spacing of cracks. The cyclic tests, however, show an increase in the spacing of cracks within the shear zone, and thus, an increase in maximal spacing. This finding can be explained by the different stages of the cracking process. At first, bending cracks occur with low structural damage, and a similar cracking development to static test is denoted. Afterwards, under alternate loads, steel-concrete debonding reduces the rate of occurrence of transverse cracks wbile 1 inducing horizontal cracking.

Load-deflection curves

As a means of comparing tests under monotonic and cyclic loads, average envelope curves have been used in the cyclic' c~ses. One ex~ple of such an envelope curve is drawn in Fig. 4. A compan~on of different curves is provided in Fig. 6, and some mecharucal properties are given in Table 4.

B~fore the peak load, it has been noted that alternate loading does no~ change the secant stiffness of the HSC beams, bu~ does mduce a decrease in maximal load capacity by approxunately 10%.

Beyond the peak load, the slope of the cyclic curve drops sh~ly ~~ a~ounts for the decrease in structural ductility. This ductili_ty IS defined as the ratio of ultimate deflection to the deflection .a! the peak load l>,jl>Fmax• A study of this structural ductthty was the topic of another paper 15 Th ultimate deflection corresponds to the deflection m~asure~ at 85% of the peak load. For p = 0.97%, the cyclic loading d~reases the ductility by 36%. For p = 1.52%, data acqui-Sition was not performed beyond the postpeak, but it was thought that the loss in ductility would not be so high as a ~esult of the greater brittleness of beams exhibited by the mcre~e of tensile reinforcement under monotonic loading. Cracking degrades the steel-concrete bond. As long as the anch~rage of bars and the force redistribution in the com-p~~sive zone are effective, the structural behavior remains SlDlllar. yn1~r alternate stresses, both the cyclic softening of matenals . and the time lag of the force redistribution in the compressive zone due to crack closure lead to a steep postpeak branch.

INFLUENCE OF FIBERS ON CYCLIC BEHAVIOR

The structural tests with fiber reinforced concrete (FRC) ha~e demonstrated the influence of the steel reinforcement ratio. on ~ber efficiency . [START_REF] Kormeling | Static and Fatigue Properties of Concrete Beams Reinforced with Continuous Bars and with Fibers[END_REF] For this reason, three values of the longJ.tudinal steel ratio have been used: p = 0.55, 0.97 and 1.52%. Both the concentration and the 1/d ratio of fibe;s are also used as efficiency factors. It has therefore been decided to use a cons1a?t fiber volume of 1% with an 1/d ratio of 80. !t should be pomted out that the size of the structure is another 1Dlportant factor in regard to the structural efficiency of the ~C; however, the size effect has not been raised herein sm'7 all b~am specimens have the same height. In this paper the influence of fibers has focused on the cracking pattern' strength deterio~tion, the load-deflection envelope curves: and the cumulative damage capacity.

lnflue~ce on apparent cracking vi~e.mflu.enc~ of fibers on the apparent cracks can be fro ualized m Fig. 7(a) to (c), which have been extracted m the M-series (p = 0.97%) after the beams' failure.

B~tween .loading points-Figure 7 !Dore sinuous when fibers are added (Fig. 7(b) and (c)).lt is Important to note that this observation is made at the end of te~t. lnde~d, if fibers are often used to reduce the crack :-"Idth dunng service loads, their bridge effect allows an mcrease of the main crack width during the failure stage. The measure of the crack width, however, was not recorded due ~o the difficulties induced by both alternated loads and continuous measurement.

For the M-30 test, fibers induce a multicracking on the upper part of the beam. At the loading points, bending cracks propagate. throu~hout the beam height for both the L-ref and M-ref senes; with the long fibers, however, some bending cracks appeared but did not connect one to the other Outsi~ l~ading points-The HSC beams displa~ some characte~stic. shear and horizontal cracking, which reveals the det~~oration of the steel-concrete bond (Fig. 7(a)). With the addition of fibers, both the number and length of cracks are reduced (Fig. 7(b) and(c)). The shear cracks seem to be effectively bridged by the fibers. Moreover, the horizontal cracks. are not located on the HSFRC beams and reflect the action of fibers on cracking induced by steel-concrete de~onding. Some isolated cracks can be noted, however, which sugge~ts the random bridging effect of fibers inside the cross sections.

lnfl!-lence on load-deflection envelope curves ~Igure 8 presents cycl~~ envelope curves that resulted by taking an a~erage ?f positive and negative envelope curves.

The deflection gam at the peak load is insignificant• the curv~s were thqs drawn with respect to a nonnalized defl~on relative to the peak deflection, and correspond to the definition of sn:uctural ductility. Three stages can be distinguished: the first IS a structural behavior without any damage, as char-

FlfiiU' I)Fmax• KFmGX 11&(1)
fibers, 'JI:, MPa kN kN/mm kN mm L-ref 0

en [START_REF] Meyer | Bin werkstoffgerechtes Schll.digungsmodell und Stababschnittselement fUr Stahlbeton unter zyklisher nichtlin88111J' Beanspruchung[END_REF] 18.8 58.5 9.2 6. acterized by the secant initial stiffness Ki and the first cracking load F 1 ; the second stage is the damaged behavior up until the maximal load applied with a decreasing secant stiffness; and the last stage is the postpeak behavior that induces structural collapse. The specific values measured off of the curves are listed in Table 5.

Undamaged behavior (prepeak)-The initial secant stiffness, as shown in Table 5, is hardly increased by the presence of fibers; no real difference between HSFRC3o and HSFRC 60 can be observed. The beam's behavior is still elastic, it seems that only microcracking occurs inside the matrix. To enhance this initial stiffness, fibers would have to bridge these microcracks; consequently, the length of fibers needs to be shorter 17 (maximal length: 13 mm). With the length of fibers used in this work (that is, 30 and 60 mm), only macrocracks are bridged as the result of the improvements brought during material testing (increased strength and ductility of HSC).

When either the upper or lower concrete layer of the beam reaches the tensile stress limit, a macrocrack occurs. Should this crack not be bridged by the fibers, the tensile stress would suddenly be distributed in the tensile steel bars, and the secant structural stiffness of the beam would decrease. With HSFRC, as fiber length increases, the value ofF 1 also increases. This improvement is particularly distinct at the

Fig. 8-Load-deflection envelope curves on L-, M-, and H-series.

low reinforcement ratio (p = 0.55%) since F1 increases by 77% with HSFRC 30 and by 170% with HSFRC60. This finding highlights the efficiency of fibers to delay the macrocracks' propagation. Due to the secant stiffness being controlled by the tensile bars, however, the value of F 1 rises less rapidly when the reinforcement ratio increases.

Damaged behavior (prepeak)-In this stage, bending and shear cracks occur and propagate but also bond cracks, as detailed previously. The bridging of cracks by fibers induces an increase in secant stiffness, accentuated by the length of the fibers. This improvement is effective throughout the loading stage up to the peak load. The yielding of reinforcement occurs prior to the peak, but no significant change of stiffness is observed. Table 5 displays an increase in the peak: load of HSFRC compared with that of HSC by 47, 30, and 5% for the L-30, M-30, and H-30 tests, respectively, and an increase of 59, 40, and 18% for the L-60, M-60, and H-60 tests. It can be noted that the increase in the tensile bar ratio reduces the gain contributed by the fibers, due to a wider crack opening 120r----------------------------------- for higher reinforcement ratio. Studies focusing on the steel-concrete bond with HSC 18 have indeed shown that the crack opening is related to the steel bar diameter. For the H-30 test, the crack opening at the peak: load seems to be too wide for the 30 nun fibers, which explains the negligible gain. Nevertheless, by preventing against the propagation of cracks, both HSFRC3o and HSFRC 60 contribute to the increase in secant stiffness and thereby to the slowing rate of beam damage. At this point, it is worth mentioning that for low longitudinal reinforcement ratios, fibers can replace some of the longitudinal bars (Fig. 9). This is important since this phenomenon has not been observed with big dimension beams under monotonic bending loads. 10 Only the replacement of stirrups with fibers has been foreseen according to certain studies. 1

•ll

Collapse behavior (postpeak)-Many of the codes used to investigate structural performance during the failure stage are based on ductility measurements. Ductility has been defined as the ratio of the ultimate deflection to the deflection at the peak: load (8,j8pmax). In this paper, the ultimate deflection is assessed when the beam supports just 85% of the maximal load. This definition is based on the work of Park and Ang 19 that was then modified by Srinivas et al., 20 who considered that under cyclic and alternate loading, structures were too heavily damaged beyond this level of loading capacity. In Table 5, the ductility measurement indicates the inefficiency of the fibers. For each series, the concrete is severely degraded by tensile stresses at peak:, and the maximal load gain obtained by fibers actually induces a drop in ductility due to a severe bond deterioration between reinforcement bars and concrete.

Considering a support capacity of95% of Fmaxo the analysis conducted by Daniel and Loukili 21 indicates a similar ductility obtained with HSFRC 60 relative to HSC for the L-series and relative to HSFRC 30 for the M-series. This finding underscores the influence of long fibers with a low reinforcement ratio; however, the improvement is not substantial enough, and the small sample size only indicates some trends. More exploration should be realized to recommend the use of fibers for structural ductility-enhancement requests.

Influence on cumulative dissipated energy

Measuring ductility is not the only motivation for investigating behavior during the postpeak: phase. When earthquakes occur, energy gets injected into the structure and then has to be dissipated for safety reasons. According to this setup, the behavior of fibers inside the matrix acts as a dissipative mechanism. The measurement of dissipated energy could thus become a good efficiency index independently from structural ductility considerations.

During cyclic tests on structures, dissipative mechanisms are frequently encountered and must be distinguished to determine the action of fibers on the dissipated energy (Eq. (1)) .

In fact, a principal energy ET is injected into the structure, composed of a beam or column and supports. One component of this energy is redistributed into the soil E 1 , while the other is used b~ the structure over the elastic Ee and inelastic E domains. The first component Ee represents the energ; necessary both for beam displacement (kinematics energy Ec) and for elastic strain Ees• The latter component E includes damping energy and hysteretic energy a elastic domain inelastic domain

ET-Es = Ec + Ea (1) ~~--~ ~~--~

Ec + Ees

Structural collapse corresponds herein to the case observed when the structure is no longer able to dissipate the accumulated energy. 3 It is therefore important to increase the energy storage capability in the elastic domain and the energy dissipation in the inelastic domain. For the first aforementioned point, the use ofHSC increases structural stiffness due to the contribution of high compressive strength as well as to the improvement of the steel-concrete bond. The low tensile strength and brittle nature of concrete, however, prevent the possibility of increasing energy in the inelastic domain. Adding fibers inside the high-strength matrix limits concrete damage in the elastic domain, and results in energy dissipation in the inelastic domain due to the strain and fiber slip inside the matrix.

The dissipated energy during a loading cycle was determined by computing the hysteretic area of the loop. The increase of dissipated energy during cycles at similar displacement amplitudes is low. It was thereby assumed that the contribution of fibers is concentrated during the first cycle of each displacement amplitude. The computation of primary dissipated energy was carried out up until total collapse (Fig. 1

0).

For L-series-Fibers increase the dissipation of energy up until collapse. The gain provided by the longest fibers, compared with the shorter ones, is observed at the end of the test for a normalized deflection of greater than 1.4. The maximal dissipated energy increases by 45% for HSFRC 60 (1490 kNmm) with respect to HSC, while for HSFRC 30 it increases by only 13%.

For M-series-Only the HSFRC 60 increases the dissipation of energy up until collapse, with an increase in ultimate dissipation (2240 kNmm) of 41 % with respect to HSC. The behavior of HSFRC 30 is similar to that of HSC, with a low increase in dissipated energy over the postpeak: zone. Furthermore, the maximal value is 26% less than HSC, which highlights the low effectiveness of shorter fibers.

For H-series-A reduction in the improvement of long fibers is noted since the maximal energy with respect to HSC increases by 28%. HSFRC 30 does not show good behavior, as the level of energy dissipation remains less than that of the HSC beam during the entire postpeak: stage.

There were not enough repetitive tests to draw firm conclusions, but some trends could be mentioned. It seems that the inclusion of fibers leads to an improvement in energy dissipation, which is particularly efficient at low longitudinal steel reinforcement ratios. As the tensile bar ratio increases, the maximal dissipation rate drops. The postpeak behavior of the HSFRC 60 beam can be considered as good for all series.

Only the L-series, however, exhibits an efficient dissipation of the HSFRC 30 beam.

STRUCTURAL DAMAGE INDEXES

Many authors have established a set of damage indexes to ascertain the residual capacitY of structures. A wide array of parameters may be used, such as number of cycles, stiffness, and ductility. When reverse loads are applied (refer to the previous section), however, the importance of energy dissipation is readily a~arent. The energy indicator proposed by Darwin and Nmai provides an assessment of the dissipative capacity compared with the elastic energy injected at peak load. This measurement has -been related to the hysteretic area of cycle i, Ei normalized to the elastic energy F max BF . The iteration on all cycles yields the total normalize~yclic energy (Bq. ( 2 KFmax and Ki = secant stiffness at the peak load and in cycle i, respectively. These last two cyclic energy indexes express numerical values without any connection to the ultimate state. Other indexes have therefore been devised to compare the energy dissipated under cyclic conditions with that dissipated under monotonic conditions, with numerical values rangip; from 0 to 100% (100% denotes structural failure). MeyeJ:-2 defined an indicator DQ based on the energy ratios in Eq. ( 4) and ( 5) Sadeghi [START_REF] Sadeghi | Simulation nummque du comportement de poteaux on ~too arm~ sous cisaillement d~vi~ altem~[END_REF] introduced a fatigue factor (Eq. ( 6)) to decrease the damage caused by cycles of similar amplitude displacement. The expression for the damage index in the positive/ negative phase of displacement is given as follows (Bq. ( 7)) ( 6) where (7) The use of this index is not straightforward for structures subjected to many cycles during the postpeak phase. Moreover, some results must be derived from tests under monotonic loading, which would increase the number of tests required overall. To overcome this constraint, a new damage index (Eq. ( 8)), based on the expression of Sadeghi's index (Eq. ( 6)), has been developed. A strain factor a.i (Eq. ( 9)) is introduced for the dissipated energy of each first cycle Epi to stand for the secant stiffness deterioration with respect to the peak load. The damage induced by successive cycles of similar displacement amplitude is attenuated by a fatigue factor A.i (Eq. ( 10)) related to the strength deterioration with respect to the first cycle. The energy summation is normalized by the total amount of energy dissipated_ by each first cycle I:.Epi• The tensile reinforcement ratio p is introduced to take into account the difference m energy behavior with increasing tensile reinforcement. The power number is computed to yield an index value close to that for the HSC beams where KFmax = secant stiffness at peak load;

~~-• = secant stiffness of primary cycles; K = maximal load of primary cycles; and 1,; = maximal load of successive cycles.

The cumulative damage capacity of the beams is computed using the index in Eq. ( 8); results are showninFig. ll(a) to (c). The HSFRC beams reveal two phases: 1) the rate of damage is reduced during the first successive cycles in comparison with the HSC beam, with this distinction being particularly strong for the lower tensile reinforcement; and 2) damage related to the loss of ductility in the beams speeds up. For p = 0.55%, however, both HSFRC 30 and HSFRC 60 successfully increase the cumulative damage capacity with values of greater than one, yet 60 mm fibers appear to reveal a faster damage growth than the others. For p = 0.97%, only the HSFRC 60 manages to enhance the cumulative damage index with respect to HSC with a value similar to that from the previous test. This fmding may suggest that 60 mm fibers are not being used up to their maximal efficiency . For p = 1.52%, the HSFRC exhibits no increase in the damage index, which underscores the inefficiency of fibers with the higher longitudinal reinforcement ratio.

CONCLUSIONS

The aim of this study has been to investigate the influence of fibers on the behavior of HSFRC beams under cyclic bending. The severe concrete damage due to alternate loading induces a loss in both maximal load capacity and ductility. This degradation suggests that the use of fibers can be efficient to prevent an early emergence of macrocracks during the prepeak stage. Fibers induce an increase in beam structural stiffness up to the peak load. For a tensile reinforcement ratio of 0.55%, HSFRC exhibits behavior similar to that of an HSC beam with a tensile reinforcement ratio of 0.97%. Nevertheless, the fibers have no influence on strength deterioration during loading cycles at a given displacement. In the Within an energy-based framework, however, and particularly so in the case of earthquake zones, the insertion of long fibers enhances the energy dissipation over both the elastic and inelastic domains for all longitudinal reinforcement ratios; with regard to the 30 mm fibers, however, enhanced energy dissipation only occurs for lower ratios. With respect to energy dissipation alone, however, 30 mm fibers may be recommended for lower reinforcement ratios. The effect on the cumulative damage capacity is positive for 60 mm fibers, with an efficiency limit for p = 1.52%. These findings have raised the possibility of using the HSFRC 60 in small-sized structures (low heights or low reinforcement ratios) in earthquake zones to increase the mechanical postpeak properties and energy dissipation in the inelastic domain.

~ 3 :Fig. 1 -

 31 Fig. 1-{a) Testing setup and beam cross section; and (b) alternate bending supports.

Fig. 2 -Fig. 3 -

 23 Fig. 2-Loading sequence.

Fig. 6 -

 6 Fig. 6-Static and cyclic tests on HSC beams.

Fig. 7 -

 7 Fig. 7-Bending and shear cracks on M-series.

  -~><;:::::::: •.~-~:::.::::.:•:::':':::~""~-.... ••••..c . ~....... Normalized deflection aJa..,.

Fig. 9 -

 9 Fig. 9-Fibers replacing reinforcement capacity ( comparison of L-30, L-60, and M-re/).

IFig. 10 -

 10 Fig. 10-Dissipated energy versus normalized deflection.

)) ( 2 )

 2 This indicator has been modified by Ehsani and Wrigh~ through introducing both the stiffness degradation and the deformation capacity (Bq. (3the peak and maximal deflection of cycle i, respectively; = dissipated energy calculated from the area of cycle i; and

  during the first half-cycle; = energy dissipated during the second half-cycle; = maximum energy dissipated under monotonic load; and ± = positive or negative deformation.

  CumuiiiUve dlllllpate energy (10 3 kN.mm) -HSC .• . HSFRC 30 c) H-series (p =1.52%)

Fig. 11 -

 11 Fig. 11-Damage index D versus cumulative dissipate energy.

=

  hysteretic energy of one cycle = first cracking load = peak load or maximal load = secant structural stiffness at peak load = liber length = strain factor = deflection at peak load

Table 1 -

 1 concrete mixture proportions

	Components, kglm 3	High-strength High-strength fiber-concrete reinforced concrete
	Cement	400	400
	Silica fume	40	40
	Filler	72.2	69
	Sand, 0 to 5 mm	722	690
	Gravels, 6 to l 0 mm	1010	966
	High-range water-reducing admixture	6	10
	Water	140	134
	Fibers	0	79

  Ahmed Loaldll is an assistant professor in the lnboratoire tk Glnie Civil tk Nantes St•Nawlre, Ecole Centrale tk Nantes, from which he received his PhD in 1996. His research interests include the early-age properties of concrete, shrinkage, creep and shrink4ge, and structural mechanics of high-strength concrete.

Table 2 -

 2 Hooked-end steel flber features

	Lengthf..t. Diameter Aspect mm D 1 ,mm ratiof.t1D 1	Shape	Tensile Modulus of strength, elasticity, MPa GPa
	30	0.38	79	__,...---..___	2000	200
	60	0.75	80	~	2000	200

Table 3 -

 3 Details of beam series

	Beam type Steel ratio p, cp mm	Loading	Concrete fcoMPa
	L-ref			HSC	97
	L-30	0.55% (12 mm)	Cyclic	HSFRC 30	110
	L-60			HSFRC60	116
	M-ref M-30	0.97% (16 mm)	Static Cyclic	HSC HSC HSFRC 30	91 95 112
	M-60			HSFRC 60	117
	H-ref H-30	1.52% (20 mm)	Monotonic Cyclic	HSC HSC HSFRC 30	97 94 114
	H-60			HSFRC60	117

I :Dynamic jade 2 : RoUcn 1iDbd by threaded rods

Table 4 -

 4 Effect of loading type on mechanical properties

	Monotonic loading

Table 5 -Results of tested beams

 5

	Vol. of fc28'	Fl,	Kt,


Antonin Delpeuch
email: antonin.delpeuch@ens.fr

Complexity of Grammar Induction for Quantum Types

Most categorical models of meaning use a functor from the syntactic category to the semantic category. When semantic information is available, the problem of grammar induction can therefore be defined as finding preimages of the semantic types under this forgetful functor, lifting the information flow from the semantic level to a valid reduction at the syntactic level. We study the complexity of grammar induction, and show that for a variety of type systems, including pivotal and compact closed categories, the grammar induction problem is NP-complete. Our approach could be extended to linguistic type systems such as autonomous or bi-closed categories.

Introduction

Overview

Category theoretic approaches to linguistics are flourishing. They provide a convenient abstract framework for both syntax and semantics [START_REF] Coecke | Lambek vs. Lambek: Functorial vector space semantics and string diagrams for Lambek calculus[END_REF], and these insights enable some progress on natural language processing tasks [START_REF] Grefenstette | Experimental Support for a Categorical Compositional Distributional Model of Meaning[END_REF]. This framework is flexible, because it allows for different types of grammars, such as the Syntactic Calculus of Lambek [START_REF] Lambek | Deductive systems and categories[END_REF] or Compact Bilinear Logic [START_REF] Preller | Free Compact 2-categories[END_REF], also known as pregroups [START_REF] Lambek | Pregroup Grammars and Chomsky's Earliest Examples[END_REF]. It also allows for different kinds of compositional semantics, which can be distributional [START_REF] Coecke | Lambek vs. Lambek: Functorial vector space semantics and string diagrams for Lambek calculus[END_REF], Montagovian and extensional [START_REF] Preller | Category theoretical semantics for pregroup grammars[END_REF], Montagovian and intensional [START_REF] Delpeuch | From Natural Language to RDF Graphs with Pregroups[END_REF], or even hybrid models [START_REF] Preller | Semantic Vector Models and Functional Models for Pregroup Grammars[END_REF]. But whatever the syntax or the semantics are, these approaches rely on a functor from the syntactic category to the semantic category to give meaning to a sentence.

S F ----------→ C syntax semantics
We propose to study the complexity of lifting the information flow at the semantic level to a valid expression at the syntactic level. In a quantum setting, this could correspond to representing a family of quantum circuits as (planar) string diagrams, for instance. In a linguistic framework, this is the task of grammar induction. Given a set of example sentences belonging to a language, the problem is to infer a grammar of this language. Originally motivated by the study of language acquisition by children [START_REF] Pinker | Language Learnability and Language Development, With New Commentary by the Author[END_REF], this task has been widely investigated in the field of formal languages [START_REF] De | A bibliographical study of grammatical inference[END_REF]. If the example sentences are just raw strings, the problem is known to be intractable for most expressive classes of grammars [START_REF] Gold | Language identification in the limit[END_REF].

Hence variations have been introduced, one of them consisting in adding some semantic information about the words in the example sentences. In a categorical framework, words are given syntactic types, which are objects in a monoidal category. The semantic type of a word is the image of this syntactic type under a monoidal functor to the semantic category. The categories we will use are defined in Section 2.2 and are summarised in figure 1. Our results focus on the lower part of our hierarchy of categories, which consists in quantum structures, whereas the linguistic type systems are higher up in the hierarchy. Since the grammatical correctness of a sentence is witnessed by an arrow from the product of its syntactic types to S (the type of a sentence), the problem of grammar induction can be seen as lifting an arrow from the semantic category to the syntactic category, as we will see in Section 4.1.

Bi-closed

Symmetric closed Autonomous

It turns out that many instances of this problem are hard, in the sense of computational complexity theory. This is mainly because we require that the syntactic type assigned to each word remains consistent among all the example sentences. This creates global constraints which restrict the solutions of the inference problem. In Section 4, we use this fact to reduce NP-complete problems to our grammar learning problem.

An example

Suppose we use a compact closed category for the semantics and a pivotal category for the syntax. We have to infer the possible syntactic types t i based on their images F(t i ), where F is the canonical monoidal functor from the free pivotal category to the free compact closed category on a given set of generators. In the following expressions, the tensor product ⊗ is implicit.

Syntax t 1 t 2 t 3 t 4 → S (1) Semantics ABC B * A * A C * A * A A * S → S (2)
There are many different arrows of the required domain and codomain at the semantic level. One of them is

A B C B * A * A C * A * A A * S.
As the only difference between a free compact closed category and a free pivotal category is the symmetry, the problem bends down to finding a permutation of the basic types of each t i such that the type reduction holds at the syntactic level. In other words, we have to find a diagrammatic reduction without crossing, such as this one:

C B A A * B * A A * C * A A * S
In this particular example, one can see that it is necessary that C occurs before B in t 1 . We can add a second sentence:

Syntax t 1 t 5 t 6 t 7 → S (3) Semantics ABC B * C * C A * C * C C * S → S ( 4
)
This examples forces A to occur before B in t 1 . Hence every solution of the learning problem made of these two sentences will be such that C and A occur before B in t 1 . In Section 4, this technique enables us to reduce the problem of betweenness [START_REF] Guttmann | Variations on an ordering theme with constraints[END_REF] to our grammar lifting problem. This problem is known to be NP-complete.

A grammar hierarchy

Monoidal categories as type systems

We define how monoidal categories can be used as type systems. Both the syntactic and the semantic categories will be seen as type systems in our induction problem. Definition 1. A type system (C , S) is a strict monoidal category C with a distinguished object S in C .

When the object S is clear from the context, the type system is simply noted C . The objects of this category will be used to denote types. We require the category to be monoidal, so that we can define the sentence type as the product of the types of its words. The distinguished object will play the role of the type for a grammatical sentence. The arrows in the category play the role of reductions: A reduces to B when C (A, B) is not empty.

The type systems we will consider are monoidal categories with some additional structure (which will be detailed in section 2.2), and freely generated by a basic category, whose objects are called basic types and morphisms are understood as subtyping relations: there is a morphism between two basic types A and B when A is a subtype of B. Definition 2. A lexicon l over a set of words W and a type system (C , S) is a function l : W → C . Although it is interesting to consider the case where multiple types can be assigned to a single word, the previous definition restricts our lexicons to one type per word. We restruct ourselves to rigid grammars, according to the terminology of [START_REF] Béchet | Learnability of Pregroup Grammars[END_REF]. Definition 3. A sequence of words w 1 , . . . , w n ∈ W is grammatical for a lexicon l when C (l(w 1 ) ⊗ • • • ⊗ l(w n ), S) is not empty. In this definition, S is the distinguished type of the underlying type system. Definition 4. A functor of type systems from

(C 1 , S 1 ) to (C 2 , S 2 ) is a functor of monoidal categories F : C 1 → C 2 such that F(S 1 ) = S 2 .
From this definition, the following property follows immediately: Proposition 1. Let F : T 1 → T 2 be a functor of type systems. If a sentence w 1 , . . . , w n is grammatical for the lexicon L 1 over T 1 , then it is grammatical for the lexicon F • L 1 over T 2 . This property expresses that if a sentence is correct at the syntactic level, then there is a valid reduction at the semantic level.

Various structures in monoidal categories

We now move on to the definition of the categories involved in the hierarchy of figure 1. Definition 5. A bi-closed category is a monoidal category in which for all object B, the functor _ ⊗ B has a right adjoint _/B and the functor B ⊗ _ has a right adjoint B\_. In other words, this means that for every pair of objects A, B, we have morphisms eval l A,B : B ⊗ (B\A) → A and eval r A,B : (A/B) ⊗ B → A satisfying some coherence equations, and similarly some morphisms A → (A ⊗ B)/B and A → B\(B ⊗ A). Type systems built on bi-closed categories correspond to grammars defined in the Syntactic Calculus of Lambek. Definition 6. An autonomous category 1 is a monoidal category where for each object A, there are two objects, the left (A l ) and right (A r ) adjoints, equipped with four morphisms ε l A :

A l ⊗ A → 1, ε r A : A ⊗ A r → 1, η l A : 1 → A ⊗ A l and η r A : 1 → A r ⊗ A satisfying the following equalities : (ε r A ⊗ 1 A ) • (1 A ⊗ η r A ) = 1 A (ε l A ⊗ 1 A l ) • (1 A l ⊗ η l A ) = 1 A l (1 A ⊗ ε l A ) • (η l A ⊗ 1 A ) = 1 A (1 A r ⊗ ε r A ) • (η r A ⊗ 1 A r ) = 1 A r
Type systems built on a free autonomous category define pregroup grammars. For instance, let n be the type of a noun phrase and s be the distinguished type of a sentence. If we give the type n to the words Mary and John, and the type n r ⊗ s ⊗ n l to loves, the sentence Mary loves John has the type n ⊗ n r ⊗ s ⊗ n l ⊗ n. This type reduces to s through the morphism [START_REF] Lambek | Pregroup Grammars and Chomsky's Earliest Examples[END_REF] for a linguistic presentation of pregroup grammars and [START_REF] Preller | Free Compact 2-categories[END_REF] for the links with category theory.

(ε r n ⊗ 1 s ⊗ ε l n ) : n ⊗ n r ⊗ s ⊗ n l ⊗ n → s See
The distinction between n l and n r is important at a syntactical level to reject ill-formed sentences. For instance, we can give the type s r ⊗ s to adverbs placed at the end of a sentence. If s l = s r , then the type s r ⊗ s = s l ⊗ s reduces to 1 through ε l s , hence the adverb can be written at any place in the sentence, which does not reflect the usual rules of grammar. As one can show that for any object n, n rl ≃ n ≃ n lr , the iterated adjoints of a type n are of the form . . . , n lll , n ll , n l , n, n r , n rr , n rrr . . . so we can write n ll = n -2 , n l = n -1 , n = n 0 , n r = n 1 , n rr = n 2 , and so on.

However, it makes sense to drop the distinction between left and right adjoints at the semantic level: in terms of flow of information, an adjoint is just something that can consume a resource, no matter whether it comes from the left or the right side. Definition 7. A pivotal category is an autonomous category with a monoidal natural isomorphism between A r and A l . We set A * = A l . Pivotal categories correspond to groups, in the sense that in a free pivotal category, two objects have an arrow between them if and only if they are equal in the corresponding free group (where * plays the role of the inverse, hence * will be sometimes noted -1 ).

The canonical morphism between the free pregroup and the free group is defined by

h : t e 1 1 ⊗ • • • ⊗ t e n n → t (-1) e 1 1 ⊗ • • • ⊗ t (-1) en n
where t e 1 1 ⊗ • • • ⊗ t e n n is the canonical form of a pregroup element.

Definition 8. A compact closed category is an autonomous category which is symmetric, i.e. for each objects A and B there is a monoidal natural isomorphism s A,B

: A ⊗ B → B ⊗ A such that s -1 A,B = s B,A .
For instance, the category of finite-dimensional vector spaces is compact closed. One can wonder why we introduced the isomorphism A l ≃ A r before adding the symmetries s A,B . The following fact explains our choice. Proposition 2. Compact closed categories are pivotal. This property is well known (it is stated in [START_REF] Coecke | Lambek vs. Lambek: Functorial vector space semantics and string diagrams for Lambek calculus[END_REF], and implicitly in [START_REF] Selinger | A survey of graphical languages for monoidal categories[END_REF]) but I have never seen a proof of it.

Proof. Let φ A and ψ A be the following morphisms :

A l A l A r A r A r A A l A φ A = A A l A r A l A A r A l ψ A = We have ψ A • φ A = 1 A l and φ A • ψ A = 1 A r .
By symmetry, let us show the first equality only.

= = = = =
Moreover, one can check with similar techniques that this isomorphism is monoidal and natural.

Definition 9. A self-dual compact closed category is a compact closed category with a family of isomorphisms h

A : A → A * .
Self-dual compact closed categories have been studied in detail by Selinger in [START_REF] Selinger | Autonomous categories in which A ≃ A *[END_REF]. The definition we adopt here corresponds to his first option, namely self-duality without coherence. As a finite-dimensional vector space is isomorphic to its dual, the category of finite-dimensional vector spaces is self-dual. This category has been widely used as the underlying semantic category for models of meaning, such as in [START_REF] Coecke | Lambek vs. Lambek: Functorial vector space semantics and string diagrams for Lambek calculus[END_REF], [START_REF] Preller | Semantic Vector Models and Functional Models for Pregroup Grammars[END_REF] or [START_REF] Delpeuch | From Natural Language to RDF Graphs with Pregroups[END_REF]. The objects in this category have also been used in [START_REF] Béchet | Learnability of Pregroup Grammars[END_REF] as semantic types in a learning task. However, they did not introduce a whole typing system at the semantic level, as they had no notion of reduction on semantic types.

We have introduced the commutativity first and then the isomorphism between A and A * . It is possible to swap these properties, although it requires to be more careful: Definition 10. A free self-dual pivotal category is the free pivotal category generated by a category C where for each object A ∈ C , A ≃ A * . A self-dual pivotal category models a rewriting system where any two identical adjacent letters cancel.

It is important to notice that we require that A ≃ A * only for basic objects. If this were true for all objects, then as noted by Selinger [START_REF] Selinger | A survey of graphical languages for monoidal categories[END_REF], we would get the following isomorphism

A ⊗ B ≃ (A ⊗ B) * ≃ B * ⊗ A * ≃ B ⊗ A
. This isomorphism is not a symmetry in general but would have the same effects on our type system.

A widespread category for semantic types in the linguistic literature is the free symmetric monoidal closed category. It has been used, among others, in [START_REF] Buszkowski | A note on the Lambek-van Benthem calculus[END_REF] and [START_REF] Dudau-Sofronie | A learnable class of Classical Categorial Grammars from typed examples[END_REF].

Definition 11. A symmetric closed category is a symmetric bi-closed category. For all objects A and B, B\A ≃ A/B, so we note A|B

= A/B.
The objects of this category can be thought of simple types for the simply-typed λ -calculus with pairs. The object A|B plays the role of the type B → A and we have a morphism eval A,B : (A|B) ⊗ B → A satisfying the required coherence conditions.

Functional types

Restricting the set of possible types

Not all types are likely to be used in a type-logical grammar. We expect types to be functional, i.e. to be built using only abstractions, the operations \ and /.

For instance, the type n ⊗ s ⊗ n belongs to the free pregroup generated by n and s, but cannot be constructed by iterated abstractions. The type n r ⊗ s ⊗ n l however can be constructed as n\(s/n) or (n\s)/n. Definition 12. Let L be the free bi-closed monoidal category. The set P ⊂ Ob(L) is the closure by / and \ of the set of basic types. Given a type system (C , S) and a bi-closed functor F : L → C the set of functional types in C is F(P).

Restricting our search of types to this form of type reduces our search space. This restriction makes sense because these types are more likely to be relevant from a linguistic point of view. For instance, [START_REF] Lambek | Pregroup Grammars and Chomsky's Earliest Examples[END_REF] builds a fairly advanced grammar of English and he uses only functional types in his grammar, while not mentioning this constraint at all.

Properties of functional types

The generative power of pregroup grammars is not reduced when we require functional types: the proof given in [START_REF] Buszkowski | Pregroup grammars and context-free grammars[END_REF] that every ε-free context free grammar is weakly equivalent to a pregroup grammar uses only functional types.

For group grammars (i.e. type systems built on pivotal categories), restricting the assignments to functional types does not harm the expressiveness either, as it is enough to multiply by a -1 a the types that are not functional to get an equivalent grammar with functional types only. This remark will be made clear by the following proposition, which characterises functional types in pivotal categories. Proposition 3. In a pivotal category, functional types are exactly those which are either

• basic types (generators of the free autonomous category), or

• products of basic types with exponents t e 1 1 ⊗ • • • ⊗ t e n n , where at least one e i is -1 and at least one e i is +1.

Proof. By induction on a functional Lambek type t, let us show that F(t) satisfies the characterization above. If t = a, a basic type, then F(t) = a, falling into the first option. If t = u/v, then F(t) = F(u)F (v) -1 . By induction, there is a basic type occurring with a +1 exponent in F(u), so it occurs again with the same exponent in F(t). Similarly, there is a basic type occurring with a +1 exponent in F(v), so it occurs with a -1 exponent in F(t).

Conversely, let us show by induction on the length of a group type t = t e 1 1 ⊗ • • • ⊗ t e n n satisfying the characterization that it is the image of a functional Lambek type. If n = 1, then t = a where a is a basic type, so F(a) = t. If n > 1, there are several cases:

• e n = -1 and e 1 , . . . , e n-1 satisfies the characterization. Then by induction we can find a functional Lambek type u such that

F(u) = t e 1 1 ⊗ • • • ⊗ t e n-1
n-1 and hence F(u/t n ) = F(u)F(t n ) -1 = t. • e n = -1 and e 1 , . . . , e n-1 = +1 : then (-e 2 ), . . . , (-e n ) satisfies the characterization and hence there is a functional u such that F(u

) -1 = t e 2 2 ⊗ • • • ⊗ t e n n , hence F(t 1 /u) = t.
• if e n = +1 and (-e 1 ), . . . , (-e n-1 ) satisfies the characterization. Then by induction we can find a functional Lambek type u such that F(u)

-1 = t e 1 1 ⊗ • • •⊗t e n-1 n-1 and hence F(u\t n ) = F(u) -1 F(t n ) = t • if e n = +1
and e 1 , . . . , e n-1 = -1: then e 2 , . . . , e n satisfies the characterization and hence there is a functional u such that F(u) = t e 2 2 ⊗ . . .t e n n , hence F(t 1 \u) = F(t 1 ) -1 F(u) = t. This completes the proof.

Corollary 1.

In a compact closed category, the characterization of functional types is the same. In a self-dual compact closed category, every type but 1 is functional.

Complexity of the grammar induction problem 4.1 Definition of the problem

We study the complexity of learning syntactic types based on positive samples (i.e. a set of grammatical sentences) with semantic types. Each word occurrence in the samples comes with a semantic type. The nature of the syntactic and semantic types depends on the problem.

Definition 13.

A training sample for a type system (C , S) and a finite set of variables V is a finite set of sentences, where each sentence is a finite sequence of the form (v 1 ,t 1 ), . . . , (v n ,t n ), where v i ∈ V and t i ∈ C is functional, and such that all the sentences are grammatical for their respective type assignment.

Note that we do not require that a variable is always paired with a single type. The type of the word can depend on the context in which it appears.

In the following sections, we study the complexity of inducing a grammar, given a finite training sample. First we give a definition of the problem. Definition 14. Let (C , S) be the syntactic type system, (C ′ , S ′ ) be the semantic type system and F be a morphism from (C , S) to (C ′ , S ′ ). We call grammar induction the problem of, given a training sample T for the type system (C ′ , S ′ ), find a lexicon h :

V × Ob(C ′ ) → C such that for all pair (v i ,t i ) ∈ T, F(h(v i ,t i )) ≃ t i and h(v i ,t i ) is functional and for all sentence (v 1 ,t 1 ), . . . , (v n ,t n ) ∈ T, C (h(v 1 ,t 1 ) ⊗ • • • ⊗ h(v n ,t n ), S)

is not empty

In other words, the problem is to find functional syntactic types that are compatible with the semantic types and all the sentences are grammatical at the syntactic level. Note that we require that each pair (variable, semantic type) is associated to an unique syntactic type, following [START_REF] Béchet | Learnability of Pregroup Grammars[END_REF]. Without this restriction, the problem is trivial as the syntactic types can be chosen independently for each sentence.

Learning pivotal categories from compact closed categories Theorem 1. Type inference from a compact closed category to a pivotal category is NP-complete.

Proof. We give a reduction of the betweenness problem [START_REF] Guttmann | Variations on an ordering theme with constraints[END_REF] to our grammar induction problem. The betweenness problem is as follows. Given a finite set A and a set of triples C ⊂ A 3 , the problem is to find a total ordering of A such that for each (a, b, c) ∈ C, either a < b < c or c < b < a. This problem is NP-complete [START_REF] Guttmann | Variations on an ordering theme with constraints[END_REF].

The compact closed category we will consider contains the objects a for each a ∈ A and d a,b,c for each (a, b, c) ∈ C, with the following reduction between basic types: a → d a,b,c and c → d a,b,c . We set y = ∏ x∈A x. The preimage of this type will define the total order satisfying the constraints induced by the sentences. For each triple (a, b, c) ∈ C, we define the following compact closed types:

w = ∏ x∈A\{a,b,c} x c 1 = d -1 a,b,c w -1 w c 2 = b -1 w -1 w
and add the following sentence to the training sample:

(Y, y)(W a,b,c,1 , c 1 )(W a,b,c,2 , c 2 )(W a,b,c,3 , c 1 )(W a,b,c,4 , w -1 )
where the W are words chosen to be different from any word previously seen. This reduction is polynomial. Let us show that this grammar induction problem has a solution if and only if the corresponding betweenness problem has a solution. If there is a total ordering < of A satisfying the constraints, let A = {x 1 , . . . , x n } where x 1 < • • • < x n . One can check that with the following preimages, the sample is grammatical in the pivotal category:

• The type of Y becomes y ′ = ∏ n i=1 x i . • For each (a, b, c) ∈ C, let p,q,r and s be such that

y ′ = p • a • q • b • r • c • s or y ′ = p • c • q • b • r • a • s
(where p, q, r and s are possibly equal to 1).

y ′ reduces to p • d a,b,c • q • b • r • d a,b,c • s.
P possible type assignment for the W is:

W a,b,c,1 : s -1 d -1 a,b,c spqr(pqr) -1 W a,b,c,2 : (rs) -1 b -1 rs(pq)(pq) -1 W a,b,c,3 : (qrs) -1 d -1 a,b,c (qrs)pp -1
One can check that the image of this assignment is equal to the assignment from the training sample and that it makes the sentences grammatical in the pivotal category.

Conversely, if there exists a pivotal type assignment, then as the type b does not occur in the types assigned to W a,b,c,1 and W a,b,c,3 , there is an a or a c on the right side of the occurrence of b, and similarly on the left side. But as there cannot be two occurrences of the same basic type in y ′ , we have either

a < b < c or c < b < a.
Hence the problem is NP-hard. As one can check a solution in polynomial time, the problem is NP-complete.

Learning self-dual pivotal categories from self-dual compact closed categories

Similarly, the previous proof can also be carried when a ≃ a -1 , giving the following theorem: Theorem 2. Type inference from a self-dual compact closed category to a self-dual pivotal category is NP-complete.

Learning compact closed categories from self-dual compact closed categories

The problem of grammar induction from a self-dual compact closed category to a compact closed category bends down to assigning exponents to the types. It can be reduced to an integer linear programming problem where we are interested in nonnegative solutions only. This problem is NP-complete and we will show that the grammar induction problem itself is actually NP-complete. Theorem 3. Type inference from a self-dual compact closed category to a compact closed category is NP-complete.

We give a polynomial reduction from 3-SAT to the problem of learning symmetric pivotal types from self-adjoint (symmetric pivotal) types. As 3-SAT is NP-complete [START_REF] Richard | Reducibility among combinatorial problems[END_REF], and the learning problem is in NP, this will complete the proof. Let φ = c 1 ∧ • • • ∧ c n be a conjunction of 3-clauses. We adopt an approach similar to the strategy of [START_REF] Shay | Viterbi training for PCFGs: Hardness results and competitiveness of uniform initialization[END_REF]:

(i) We replace each positive occurrence of a variable x i by a new variable x i,1 and each negative occurrence by a different variable x i,0 . We add to the training sample one sentence, ensuring that all the clauses are satisfied (lemma 1).

(ii) We add one sentence for each pair of variables x i,0 , x i,1 , ensuring that they are assigned opposite truth values (lemma 2).

The full training example will hence encode satisfiability, completing the proof.

Let c = x i 1 ,b 1 ∨ x i 2 ,b 2 ∨ x i 3 ,b 3
be a 3-clause, where x i,1 stands for x i and x i,0 for ¬x i . For each i and b we define a self-adjoint type

v i,b = z i,b y i,b y i,b . Let t(c) = dz i 1 ,b 1 z i 2 ,b 2 z i 3 ,
b 3 be a self-adjoint type. Our idea is that we will force d to have a -1 exponent in the corresponding group type, and hence this group type will be functional if and only if one of the z a,b occurs with a +1 exponent. As a clause is true when at least one of the literals it contains is true, this will encode satisfiability.

(S, s) n ∏ i=1 ((C i ,t(c i ))(D i , d)) p ∏ i=1 1 ∏ b=0 (X b i , v i,b ) n i,b
are exactly those for which x 1,0 , x 1,1 , x 2,0 , x 2,1 , . . . , x p,0 , x p,1 are assigned values making all the clauses true.

Proof. Let us show first that if φ is satisfiable, then the grammar induction problem has a solution. Let x 1 = a 1 , . . . , x p = a p be a satisfying boolean assignment. We give X b i the group type z e i,b y -1 i,b y i,b , where e = -1 if b = a i and e = 1 otherwise. This type is functional.

We give [START_REF] Buszkowski | Pregroup grammars and context-free grammars[END_REF] where e k = 1 if b i,k = a i k and e k = -1 otherwise. As the clause c i is satisfied, there is at least one k ∈ {1, 2, 3} such that b i,k = a i k , hence the type is functional.

C i the type c -1 z e 1 i 1 ,b i,1 z e 2 i 2 ,b i,2 z e 3 i 3 ,b i,
Let us show that the sentence is grammatical. As the exponent of z i,b in the type assigned to a clause only depends on a i , there are n i,b occurrences of z i,b , with the same exponent, in ∏ n j=1 (c j ,t j ). By construction, the exponent of z i,b is inversed in the type assigned to x b i , and there are n i,b such occurrences in the sentence. Hence all the z i,b cancel. The type d assigned to D i cancels with d -1 in the type assigned to t(c i ), and the y i,b cancel as well. Hence only s remains: the sentence is grammatical.

Conversely, suppose there are functional group types r j preimage of t(c j ) and w i,b preimage of v i,b such that the sentences are grammatical at the syntactic level. Note that the types of the words S and D i are basic types, so the only functional syntactic types compatible with the learning problem are these basic types. As the pregroup type d occurs with exponent +1 n times in the product due to the words D j , all the occurences of d in the r j have the exponent -1, otherwise they would not cancel. For each r j , it is functional so one of the z i,b has exponent +1. For each x b i , z i,b occurs n i,b times with the same exponent, thanks to w i,b , and n i,b other times in the clauses, so the exponent assigned to z i,b is the same in every r j . Lemma 2. The assignments making the following sentence grammatical

(S, s)(X 0 i , v i,0 )(Z i , z i,0 z i,1 )(X 1 i , v i,1 )
are exactly those for which the z i,0 and z i,1 in v i,0 and v i,1 get opposite exponents.

Proof. One can check that the following assignment is valid, for e = ±1:

s • z e i,0 y -1 i,0 y i,0 • z -e i,0 z e i,1 • z -e i,1 y -1 i,1 y i,1 → s
Conversely, as the type assigned to Z i has to be functional, z i,0 and z i,1 get opposite exponents in any solution of the grammar inference problem.

Learning pivotal categories from self-dual pivotal categories

The construction of Theorem 3 can be adapted to work in non-commutative structures, hence the following theorem: Theorem 4. Type inference from self-dual pivotal categories to pivotal categories is NP-complete.

Composing complexity results

Suppose we know the complexity of the grammar induction problem between C 1 and C 2 , and between C 2 and C 3 . What can be said about grammar induction between C 1 and C 3 ?

C 1 C 2 C 3 ?
Given a syntactic category C and a semantic category C ′ , we introduce the notion of exact samples.

Definition 15. A training sample is said exact for some syntactic type t when it contains a word-type pair (w, F(t)) such that for all solutions h of this training sample, h(w, F(t)) = t.

We say that a grammar induction problem has exact samples when there exists exact samples for each syntactic type t.

In other words, a grammar induction problem has exact samples when we can build sentences forcing the preimage of a particular type. Proof. Take an exact sample S for t from C 2 to C 3 . For each type t ′ occurring in S, take exact samples for t ′ from C 1 to C 2 . Concatenate these samples with the image of S under the functor from C 2 to C 1 .

Future work

A number of questions remain open. Being able to classify the complexity of the inference problem in the higher half of the hierarchy would enable us to give complexity results on the problems studied in [START_REF] Béchet | Learnability of Pregroup Grammars[END_REF] and [START_REF] Dudau-Sofronie | From Logic to Grammars via Types[END_REF].

Another issue is the expressivity of the classes of grammars defined by the categories in the lower half of the hierarchy. These grammars generate sub-classes of the context-free grammars, but it would be interesting to relate these sub-classes to known classes from the field of formal languages.

One could also use this framework to study inference problems in which the structure of a parse is known, but the types are unknown. This notion of learning with structural examples has been studied for the syntactic calculus [START_REF] Kanazawa | Identification in the limit of categorial grammars[END_REF].

Self

  functors, dashed lines are complexity results and dotted lines are existing algorithms.

Figure 1 :

 1 Figure 1: A hierarchy of type systems

Lemma 1 .

 1 For each literal x b i , i ∈ {1, . . . p}, b ∈ {0, 1}, let n i,b be the number of occurrences of x i,b in φ . The assignments making the following sentence grammatical

Lemma 3 .

 3 If the grammar induction problem has exact samples, then for all finite set of syntactic types T = {t 1 , . . . ,t n } there exists a training sample which is exact for t 1 , . . . ,t n .Proof. Take an exact training samples for each element of T . Make these training samples disjoint by ensuring that they use different words. The concatenation of these training samples satisfies the property claimed.

Lemma 4 .

 4 If grammar induction from C 1 to C 2 and from C 2 to C 3 has exact samples, then so does grammar induction from C 1 to C 3 .

Proposition 4 .

 4 If grammar induction from C 1 to C 2 has (polynomial) exact samples and grammar induction from C 2 to C 3 is NP-complete, then grammar induction from C 1 to C 3 is NP-complete. Proof. Take an instance of SAT. It can be represented as an equivalent training sample from C 2 to C 3 . Take the image of this training sample by the functor from C 2 to C 1 and force this new sample to have the original preimages in C 2 by adding an exact sample. This problem has a solution if and only if the instance of SAT is satisfiable.

Lemma 5 .Corollary 2 .

 52 The following grammar induction problems have exact samples:• from self-dual compact closed to self-dual pivotal • from self-dual compact closed to compact closed • from self-dual pivotal to pivotal• from compact closed to pivotal Proof. Use the same techniques as the ones we developped for our reductions. Grammar induction from a self-dual compact closed category to a pivotal category is NP-complete.Proof. Combine Lemma 5 and Theorem 1 with Proposition 4.

Some authors use the name compact closed category instead, but this term has been used for both symmetric and planar categories. As we want to insist on the fact that these categories are not symmetric (contrary to some other ones in this article), we follow the terminology of[START_REF] Selinger | A survey of graphical languages for monoidal categories[END_REF].

Acknowledgements

This work has been supported by the Quantum Group in the Department of Computer Science of the University of Oxford. I am grateful to Thomas Bourgeat, Amar Hadzihasanovic, Anne Preller and Isabelle Tellier for their help, and the reviewers for their useful comments. Special thanks go to Jamie Vicary who helped me a lot by supervising and reviewing my work.


L Augustins
email: louis.augustins@safranmbd.com

F Hild

R Billardon

S Boudevin

Experimental and numerical analysis of thermal striping in automotive brake discs

Keywords: Thermal striping, automotive brake discs, shielding process. NOMENCLATURE

In the present study, thermal striping development on friction bands of brake discs is investigated through an experimental and numerical analysis. A test consisting of a series of several hundred severe brakings was carried out on a specific bench at PSA Peugeot Citroën. The experimental observations of the crack network evolution and a numerical analysis of a brake disc with a single crack helped to propose a macroscopic criterion capable of predicting the criticality of the network, namely, its ability to lead to the complete failure of the structure. In addition, a numerical analysis of the crack shielding process is developed whose results are compared with experimental findings.

INTRODUCTION

Thermal fatigue [START_REF] Serpa | What is thermal fatigue ?[END_REF] is characterized by high temperature gradients that prevent free expansion and induce fatigue cracks. In such cases, a crack network appears, often called thermal striping or heat checking. This type of damage is commonly encountered in industrial structures such as molds for metal forming [START_REF] Oudin | Thermo-mechanical fatigue of hot work tool steels[END_REF][START_REF] Persson | Thermal fatigue cracking of surface engineered hot work tool steels[END_REF][START_REF] Srivastava | Computer modeling and prediction of thermal fatigue ckacking in die-casting tooling[END_REF] , automotive [START_REF] Vallet | Etude de la fissuration d'un disque de frein à partir de l'analyse de son comportement thermomécanique[END_REF] and railway vehicles [START_REF] Degallaix | Fatigue thermique de disques de frein ferroviaires[END_REF] brake discs, or pipes of the cooling systems in nuclear power plants [START_REF] Seyedi | Numerical modeling of crack propagation and shielding effects in a striping network[END_REF][START_REF] Malésys | A probabilistic model to predict the formation and propagation of ckack networks in thermal fatigue[END_REF] .

The crack network appears more or less rapidly, depending on the stress level, but once initiated, crack propagation can be very slow before the failure of the structure or may even stop under certain conditions related to the stress field shape [START_REF] Seyedi | Numerical modeling of crack propagation and shielding effects in a striping network[END_REF] .

The present work focuses on an experimental and numerical analysis of a crack network located on the friction bands of an automotive brake disc made of flake graphite (gray) cast iron [START_REF] Augustins | Constitutive model for flake graphite cast iron automotive brake discs. From macroscopic multiscale models to a 1D rheological description[END_REF][START_REF] Augustins | Constitutive model for flake graphite cast iron automotive brake discs. Induced anisotropic damage model under complex loadings[END_REF] . On bench tests and even in particular real conditions, a crack network in the form of radial cracks may appear after a low number of brakings without necessarily propagating toward the outer or inner radii or resulting in complete disc failure. Therefore being able to estimate crack initiation through classical low cycle fatigue criteria is not sufficient for an efficient design. It is necessary to estimate the criticality of the network, namely, its likelyhood to result in through thickness radial cracks on the friction ring [START_REF] Bagnoli | Thermal fatigue cracks of fire fighting vehicles gray iron brake discs[END_REF] .

In order to study the evolution of the network, a test campaign on a dynamometric bench, which is appropriate for simulating any braking condition, was carried out at PSA Peugeot Citroën. Based on the analysis of the results, a numerical approach is proposed to assess the criticality of the network.

THEORETICAL APPROACHES

Early studies devoted to the analysis of thermal striping were developed in the field of metal forming, where molds are subjected to alternate heating and cooling conditions. Cracks are mostly treated independently, estimating initiation from dissipative mechanical variables -such as plastic strain amplitude -and simulating mode I propagation with Linear Elastic (LEFM) or Non-Linear (NLFM) Fracture Mechanics laws 2 . Authors generally agree that an increase in stress level results in an intensification of the crack network density and a decrease in the distance between cracks [START_REF] Le Roux | Role of heat-flux density and mechanical loading on the microscopic heat-checking of high temperature tool steels under thermal fatigue experiments[END_REF] .

More recent studies [START_REF] Maillot | Thermal fatigue crack networks parameters and stability: an experimental study[END_REF][START_REF] Seyedi | Numerical modeling of crack propagation and shielding effects in a striping network[END_REF][START_REF] Malésys | A probabilistic model to predict the formation and propagation of ckack networks in thermal fatigue[END_REF][START_REF] Rupil | Identification and probabilistic modeling of mesocrack initiations in 304L stainless steel[END_REF] are connected with an incident that occurred in 1998 in the Civaux nuclear power plant [START_REF] Stephan | Evaluation of the risk of damage in mixing zones: EDF R&D[END_REF] where observations revealed the presence of thermal striping on the internal part of pipes of the cooling system. Their purpose was to improve the understanding of the failure and to develop efficient tools to simulate the evolution and criticality of a crack network. Based on Seyedi et al. work [START_REF] Seyedi | Numerical modeling of crack propagation and shielding effects in a striping network[END_REF] , Malésys et al. [START_REF] Malésys | A probabilistic model to predict the formation and propagation of ckack networks in thermal fatigue[END_REF] proposed a probabilistic model suitable for describing initiation and cyclic change of crack networks using two main assumptions:

 Continuous formation of cracks, where microcracks develop into mesocracks in the course of time but not all at the same time;

 Crack obscuration or shielding, which characterizes the presence of stress relaxation zones around the cracks. Consequently, the latter ones cannot propagate any farther.

The shielding phenomenon is of critical importance for understanding the partition between arrested and propagating cracks, and the model -using LEFM -gives access to the information about the possible saturation of the crack network and describes its density.

In the case of automotive brake discs, the density or the saturation of the crack network is not the major issue. The design must avoid through thickness cracks that would result in a complete brake disc failure. Considering a single crack and using LEFM or NLFM to simulate propagation could only give access to qualitative results because of the strong nonlinear behavior of the material [START_REF] Augustins | Constitutive model for flake graphite cast iron automotive brake discs. From macroscopic multiscale models to a 1D rheological description[END_REF][START_REF] Augustins | Constitutive model for flake graphite cast iron automotive brake discs. Induced anisotropic damage model under complex loadings[END_REF] for the given loadings in this zone. It is therefore necessary to define engineering indicators that enable the criticality of the crack network to be assessed, namely, its ability to propagate or stop in the radial and through-thickness directions.

EXPERIMENTAL CAMPAIGN

Experimental set-up

The disc used in this study is a large ventilated model 330 mm in diameter (denoted ) mounted on a heavy vehicle. Each friction band is 10 mm deep. A series of severe brakings (i.e. deceleration from to at ) was carried out on the dynamometric bench shown in Figure 1, which is composed of an electrical engine and weights representing the vehicle inertia on its rotation axle. The caliper is a fixed part and the deceleration can reach 1.2 . Every 100 brakings, the crack network is revealed with dye penetrant and pictures are taken. Brake pads are changed every 600 brakings. In order to accurately simulate the thermal behavior of the disc during braking, thermocouples were utilized (Figure 2a) on a disc on which several brakings were carried out in the desired configuration. Temperature records are shown in Figures 2b and2c. On the external friction band, temperature measurements are essentially steady for successive brakings (Figure 2b) with changes of a few tens of degrees near the outer radius. Conversely, on the internal friction band, temperature measurements are not steady (Figure 2c) and fluctuations can reach 100 °C and the maximum temperature is not measured on the same thermocouple for successive brakings. This phenomenon is probably due to a high geometrical concentration of heat flux and hot band phenomena [START_REF] Barber | Thermoelastic instabilities in the sliding of conforming solids[END_REF][START_REF] Cristol | Coupling between thermal localisation and friction mechanisms: heat accumulation effect[END_REF] , which disrupt the stationary response. Because of this uncertainty about the heat flux distribution on the internal friction band, observations and thermal simulations focus on the external friction band, for which correlations between numerical models and experimental results are more relevant.

Crack network evolution

Figures 3 and4 show pictures taken after dye penetration on the external friction band for 200 to 500 brakings and for 600 to 2,400 brakings, respectively. A crack network development is observed after 200 to 400 brakings in the area located between radius (near the inner radius) and

(near the outer radius). From 500 brakings onward (Figure 3(d)), tends to move toward the inner radius and a second crack network appears in this region, superimposed on the first one. After 600 brakings (Figure 4), this second network becomes more visible. This phenomenon can be explained by nonuniform wear of the friction pad. Figure 5 shows the wear profile of the external friction pad after 300 and 600 brakings. The incoming torque area corresponds to that which is first in contact with a point on the disc during a wheel rotation. The z-axis variable relates to the difference between the thickness measured at a particular point and the minimum thickness measured on the whole surface. The lower the value, the higher the wear, namely, a 0-value relates to the area where wear is highest. After 300 brakings (Figure 5(a)), wear is much higher on the outer radius than on the inner one, which is consistent with the flux distribution and the damaged area. After 600 brakings (Figure 5(b)), wear is more uniformly distributed, which is meaning that the flux presents a similar distribution in the radial direction. As a result, a flux appears near the inner radius and gives rise to a second crack network in that area. As friction pads are replaced every 600 brakings, it is reasonable to assume that the phenomenon will repeat itself, namely, a heat flux concentrated in the outer radius area, which mainly acts on the first network and then becomes more uniform and less intense. This phenomenon is illustrated in Figure 6a.

Between 1,200 and 2,400 brakings, the crack network does not evolve significantly in the radial direction, i.e. and remain unchanged. The crack density is also stable but the diffuse network of the early cycles changes to a more organized one where long cracks propagate and stop the shorter ones. This shielding process is clearly observed in Figure 6b and 6c after 3,200 brakings. The test was finally interrupted after 3.200 brakings when visual observations showed that cracks no longer evolved in the radial direction or only very slowly. Non-destructive depth measurements were carried out on several long cracks with a crack depth meter RMG 4015 [START_REF] Deutsch | RMG 4015 Crack Depth Measurement[END_REF] using the principle of potential drop measurements between both sides of the crack. Data relating to the longest crack on the external friction band are shown in Figure 7. In view of the semielliptical profile observed on all measurements, the crack depth is approximated with an elliptical shape.

THERMAL STRIPING SIMULATION

A precise thermo-mechanical simulation of the experimental braking configuration is carried out with the commercial FE code Abaqus. As described in the previous section, the thermal flux gradually evolves during the cycles. However, as temperatures are only recorded for the first cycles, and taking into account the fact that the heat flux is more concentrated and more intense during the first 300 brakings of brake pads life, the simulation of the heat flux is restricted to this time interval. Only an angular sector of the disc is used for the simulations. Adiabatic, cyclic and symmetric boundary conditions are applied to the lateral faces for the thermal and mechanical simulations, respectively. The mesh is composed of 10-node tetrahedra as shown in Figure 8.

Thermal loading

A test bench with inertia spinning with an initial velocity is considered, on which braking with constant deceleration (i.e. the speed decreases linearly with time during braking) is carried out several times. Thus, if is the braking duration, the angular velocity at time reads (1)

The kinetic energy is then deduced at time

(2)

The assumption is that all the kinetic energy is converted into heat and that 95% of the latter is transmitted to the disc, while the remaining 5% is transmitted to the linings. The global heat flux transmitted to the disc is then equal to the derivative of the kinetic energy with respect to time, weighted by a factor 0.95. The expression of the global heat flux as a function of time then reads (3) where et are the inner and outer friction surface radii. Assuming a uniform pressure between the friction pad and the disc, the heat flux per unit area is expressed as a linear function of the radius and the angular velocity (4) with a constant.

Equations ( 3) and (4) then give (5) which leads to the expression of the flux on each of the friction band (6) This flux can be tuned to correlate thermocouple measurements with simulated data by modifying the inner and outer radii and the value of , keeping the same global flux . then becomes an affine function . Using this type of expression, simulations were carried out to obtain a good correlation with thermocouple measurements on the external friction band (Figure 9). Even if experimental data are not perfectly described (with a maximum error of a few tens of degrees), the orders of magnitude and geometrical localization are faithfully simulated. Given the sources of variability of the temperatures field on the friction bands, this result is deemed satisfactory.

This simulated transient field fits the disc thermal behavior only when new friction padsor friction pads whose wear is fairly loware taken into consideration, namely, high temperatures are located near the outer radius. It follows that the secondary crack network will presumably not be predicted by the simulation. In the following, the analysis therefore focuses only the primary network near the outer radius.

Mechanical field analysis

Damage mechanisms and stress paths on a brake disc friction band are comprehensively described in Ref. [10]. During braking, the strong through thickness thermal gradient induces high levels of radial and mostly hoop (i.e. orthoradial) compressive stresses, denoted in a cylindrical basis. Upon cooling, residual tensile stresses following plastic compressive strains appear. Thanks to the thermal simulation and a suitable constitutive model, 9,10 complete thermomechanical simulations of the braking phases are performed. The residual (positive) hoop stress field in the friction band depth is presented in Figure 10a. It is superimposed to the experimentally observed crack profile, whose shape is similar to the stress contours.

Discussion

The experimental observations show that the crack network has not evolved between 2,400 and 3,200 brakings in the radial direction. In the thickness direction, the crack driving force is the thermal gradient, which becomes very low. It is assumed that the crack does not propagate farther in that direction. The residual hoop stresses can therefore, as a first approximation, be viewed as a boundary beyond which the main cracks of the network do not propagate. In order to support this assumption, a simulation was carried out for a more severe test where a through thickness crack near the outer radius is observed after a few hundred brakings. The residual hoop stresses (Figure 10b) show that the expected boundary reaches the outer radius. This boundary seems to be a relevant indicator of the criticality of the network but strongly depends on the thermal simulation of the braking, which is still a major issue in thermomechanical design of brake discs as many sources of instabilities exist (e.g. thermoelastic instabilities, hot spots, disc and friction band wear).

NUMERICAL ANALYSIS OF THE SHIELDING PROCESS

The previous section proposed an approach that enables the main crack growth in the radial direction to be predicted. An attempt is made in this section to justify the experimental network morphology in the hoop direction. Figure 6b shows an organized and relatively regular crack network where short cracks have been stopped by the shielding zones of the longer ones, according to the mechanism described earlier. [START_REF] Seyedi | Numerical modeling of crack propagation and shielding effects in a striping network[END_REF][START_REF] Malésys | A probabilistic model to predict the formation and propagation of ckack networks in thermal fatigue[END_REF] Since the total number of main radial cracks can be visually estimated, it is possible to define the average angular spacing in degrees between two main cracks. This quantity relates to the shielding or obscuration angle of the main cracks and thus to the network pitch. The optical assessment of the number of main cracks depends on the observer and on what is considered to be a main crack, and is necessarily subjective. Bounds are nevertheless defined and the obscuration zone is finally estimated such that . The objective is to predict with a numerical approach and to compare it with the experimental observations. As a first step, it is necessary to identify an area around a crack where hoop stresses are relaxed.

The previous braking simulation must be carried out on an FE model accounting for the crack reported in Figure 7. In order to avoid too long re-meshing and CPU time, a simplified model of the external friction band is used instead of the complete brake disc model. The angular sector is larger than the previous model (i.e. 18° instead of 10°) to avoid side effects and to contain the whole relaxed zone (Figure 11). Details concerning the representativity of the numerical model from a thermal and mechanical point of view are described in Appendix 1. The residual tensile stress is then plotted as a function of the angle along several radii. For this purpose, post-processing paths are defined on the simplified model (Figure 11). The change of on the central path (half-way between the radial ends of the crack) is plotted in Figure 12. The stress gradually drops toward near the crack, resulting in a relaxed zone, which is characterized by an angle . An upper bound is defined, which corresponds to the angular distance between the values from where the stress begins to differ from its asymptotic value . An arbitrary offset needs to be defined in that case. A lower bound is thus preferred. The latter is defined using an idealization of the profile against with a binary representation, namely, either the stress value is equal or it is null (dotted lines in Figure 12a). The angle is then defined by equivalence between the integral of the real stress along the path and that of the idealized representation (7) which graphically means that the area of the surface under the dotted lines is equal to that under the solid lines. A triangular shape is a possible alternative, which is adopted for the study of brittle matrix composites cracking [START_REF] Curtin | Exact theory of fiber fragmentation in single-filament composite[END_REF] . The estimation of would have been slightly higher. The idealized relaxed zone is plotted in Figure 12b, which is approximated along the radius with a second order polynomial function, superimposed on the residual hoop stress field.

With the chosen idealization, once a crack has formed and propagates, there is an angular domain of width   relax / 2 about the crack surface (Figure 12a) in which no new initiation and propagation is possible since the hoop stresses are fully relaxed. With such an assumption corresponds to the minimum angular separation between two macrocracks, i.e., the corresponding relaxation zones should not overlap, Figure 13a). At saturation of the cracking process the maximum angular separation between two macrocracks is equal to (Figure 13b). Consequently, the minimum separation is not equivalent to the average angular separation. For applications related to particle adsorption, Widom [START_REF] Widom | Random sequential addition of hard spheres to a volume[END_REF] studied the distribution of -long rigid bars randomly distributed along a line. The author proposed a statistical analysis that is suitable for reaching the average distance between the center of each bar defined by . Curtin [START_REF] Curtin | Exact theory of fiber fragmentation in single-filament composite[END_REF] successfully used an identical approach to predict the average debond length between two cracks in a fragmented monofilament composite in the limit of an infinite Weibull modulus. Analogously, the mean angular separation (or angular pitch) is for the crack network. For each considered path (Figure 11) is calculated and the results are reported in Figure 14. The maximum angular separation is 4°, which is to be compared with the visual estimate . There is a good agreement between theoretical values and experimental results, reflecting the consistency of the approach for addressing the development of thermal striping.

CONCLUSION

The purpose of this paper was to study the propagation of a thermal fatigue crack network on the friction band of an automotive brake disc. The experimental set-up made it possible to carry out more than 3,000 severe consecutive brakings. Observations with dye penetrant revealed a stabilized network that does not lead to the complete failure of the disc.

Numerical thermomechanical simulations of the experimental braking were performed and showed that the residual hoop stresses coincide with the main radial stabilized crack profile. It was concluded that this field represents a boundary beyond which the crack network does not further evolve in the radial and depth directions.

In addition, a numerical analysis on a single crack brake disc model was carried out in order to justify, from a numerical point of view, the observed network shape and its associated angular pattern. Based on the proposed assumptions (i.e. stress relaxation induced by propagating cracks), the latter is satisfactorily predicted by the numerical model.

=

  maximum vehicle velocity g = gravity acceleration = angular velocity = initial angular velocity = kinetic energy = braking duration = global heat flux = flux per unit area = inner radius of the friction band = outer radius of the friction band = lower bound radius to which the heat flux is applied = upper bound radius to which the heat flux is applied = obscuration angle = extrusion angle of the model = residual hoop tensile stresses on the friction band = maximum residual hoop tensile stresses on the friction band = relaxation angle

Fig. 1 :

 1 Fig. 1: Dynamometric test bench. (a) Schematic view; (b) assembly area

Fig. 2 :

 2 Fig. 2: (a) Thermocouple implementation and measured temperature history; (b) External friction band; (c) Internal friction band

Fig. 3 :Fig. 4 :

 34 Fig. 3: Pictures of the external friction band from 200 to 500 brakings

Fig. 5 :Fig. 6 :

 56 Fig. 5: External friction pad wear profile after two different numbers of brakings

Fig. 7 :

 7 Fig. 7: Depth measurement of the longest crack on the external friction band and its elliptical approximation. (a) Location of the considred section. (b) Measured crack front

Fig. 8 :

 8 Fig. 8: Mesh of an angular sector of brake disc

Fig. 9 :

 9 Fig. 9: Correlation between thermal simulation (Sim) and thermocouple measurements on the external friction band

Fig. 10 :

 10 Fig. 10: (a) Superposition of the experimental crack profile (black solid line) and the simulated residual hoop stresses (in MPa) upon cooling. (a) For the considered test; (b) For a more severe test

Fig. 11 :

 11 Fig. 11: Simplified model of the external friction band with a crack and associated postprocessing paths for the analysis of the relaxation of the hoop stresses

Fig. 12 :

 12 Fig. 12: (a) profile on the central path and its idealized representation (dashed lines). (b) field and idealized relaxation zone (black solid lines)

Fig. 13 :

 13 Fig. 13: Schematic illustration of anglular separations between two neighboring macrocracks (red dots). The stress relaxation zones correspond to the gray shaded areas. Minimum (a) and maximum (b) separation at saturation of macrocracks to avoid any overlap of relaxation zones

Fig. 14 :

 14 Fig. 14: Obscuration zone angle

Acknowledgements

This work was carried out within the Executive Doctorate in Industry framework, and was supported by PSA Peugeot Citroën.

APPENDIX 1

From a heat transfer point of view, merely modeling the external friction ring and applying the same loads and boundary conditions as in the case of the complete model has very little influence on the final response. The only difference lies in the fact that heat is not transmitted through the fin and this results in a slight increase in temperature on the underside of the friction ring. The temperature field at hottest braking time is shown in Figure 15 for the complete and the simplified model with no crack. The same mechanical boundary conditions as those applied to the complete disc are applied to the lateral sides of the simplified model. Displacement in the direction corresponding to the thickness of the friction ring is clamped on its underside. The residual (tensile) hoop stress field for the simplified model withno crack is shown in Figure 15c and presents a profile similar to that obtained with the complete model.