F Nazaret
email: nazaret@enstimac.fr

H Marzagui
email: marzagui@enstimac.fr

T Cutard
email: cutard@enstimac.fr

Influence of the mechanical behaviour specificities of damaged refractory castables on the Young's modulus determination

Keywords: Elastic modulus, Mechanical properties, Refractories

Influence of the mechanical behaviour specificities of damaged refractory castables on the Young's modulus determination

F Nazaret, H Marzagui, Thierry Cutard

Introduction

In a wide range of industrial applications, refractory castables are subjected to severe solicitations, especially from a thermomechanical point of view. Previous studies have already been performed in the field of the high temperature behaviour of refractory castables. [START_REF] Boussuge | Thermomechanical behaviour: from ceramics to refractories[END_REF][START_REF] Schacht | Thermomecanical behavior of refractories[END_REF][START_REF] Bradt | Elastic moduli, strength and fracture characteristics of refractories[END_REF][START_REF] Moore | Significance and utility of refractories test data[END_REF][START_REF] Prompt | Etude du comportement thermomécanique d'un béton réfractaire alumineux. Application à une rigole de haut fourneau[END_REF][START_REF] Robin | Comportement thermomécanique des céramiques réfractaires[END_REF][START_REF] Robin | Thermomechanical behaviour of magnesia-carbon refractories[END_REF] Nevertheless, they have been less studied than engineering ceramics [START_REF] Boussuge | Thermomechanical behaviour: from ceramics to refractories[END_REF] and there is still a need in obtaining high temperature mechanical behaviour curves and in developing constitutive laws for these materials. This is particularly true to develop numerical simulation approaches in order to improve the design and the service life duration of structures that are based on refractory castables. Bending tests are easy to perform at high temperature and are largely used to study refractory castables. But, it remains difficult to develop constitutive laws from their results. This is mainly due to the non-linear behaviour of these materials and to the dissymmetry that is classically observed between the tensile and compression behaviours. In order to be able to obtain behaviour curves under these two loading modes, high temperature tensile tests and compression tests are now under development. [START_REF] Boussuge | Thermomechanical behaviour: from ceramics to refractories[END_REF][START_REF] Nazaret | Thermomechanical behaviour of a fibre-reinforced refractory concrete: tests and FE analysis[END_REF] This paper deals with the comparison of the Young's modulus values that can be obtained from the three considered test types. Large differences can be obtained, even if the Young's modulus is normally an intrinsic property of the tested materials. Such observations have already been made by several authors who have performed two types of mechanical tests at least. [START_REF] Prompt | Etude du comportement thermomécanique d'un béton réfractaire alumineux. Application à une rigole de haut fourneau[END_REF][START_REF] Robin | Comportement thermomécanique des céramiques réfractaires[END_REF][START_REF] Robin | Thermomechanical behaviour of magnesia-carbon refractories[END_REF][START_REF] Lemaistre | Etude des propriétés thermomécaniques de divers réfractaires[END_REF][START_REF] Schmitt | Tensile behaviour of magnesia carbon refractories[END_REF] In this paper, an explanation of these Young's modulus discrepancies is proposed by considering the results of classical and of complementary original mechanical tests that have been made on two refractory castables.

As for as the paper structure is concerned, the compositions and the processing routes of the two considered materials are first given, as well as a description of the experimental devices that are used. Then, results of conventional tensile tests, four points bending tests and the compression tests are briefly presented. More attention is paid to the determination of the Young's modulus. The discrepancy phenomenon that is observed between the Young's modulus values is particularly underlined. It is then established that this phenomenon is enhanced for the materials that have been previously fired at high temperature. The cases of four points bending tests and of compression tests are presented and discussed successively. Causes of the differences between the Young's modulus values are exposed and explained.

Materials and experimental methods

Presentation of the refractory castables

Two refractory castables are considered in the present paper. One of these materials (FRRC) is a geopolymer based refractory castable that is reinforced with 1.5 vol.% of metallic fibres. It is made of a geopolymer based matrix and of cordierite aggregates. The matrix is obtained by mixing a major alumino-silicate oxide, thermal silica fume and an aqueous solution of potassium polysilicate. [START_REF] Davidovits | Geopolymers: man-made rock geosynthesis and resulting development of very early high strength cement[END_REF] Concerning aggregates, a mixture of four granulometric ranges of a cordierite grog is performed with a maximum aggregate size close to 3 mm. Metallic fibres are made of an AISI 310 stainless steel and are processed by cold drawing. They are characterized by a 0.38 mm diameter and by a 12.5 mm length. The FRRC refractory castable is shaped by mixing these different components during 5 min in a planetary mixer and by casting the mix under vibrations. The vibration amplitude is controlled to remain constant and the vibration time is of 10 min. Complete polymerisation of the matrix is obtained after an isothermal heat treatment of 12 h at 80 • C. The material is then fired at high temperature in order to stabilize the microstructure for high temperature applications. Such heat treatments are known to generate damage in the FRRC microstructure, mainly because of the differential dilatometric behaviours of the matrix, of the aggregates and of the metallic fibres. [START_REF] Cutard | Structural and mechanical properties of a refractory concrete for superplastic forming tools[END_REF][START_REF] Cailleux | Pullout of steel fibres from a refractory castable: experiment and modelling[END_REF] Three firing temperatures are considered for this material: 110, 500 and 900 • C. Examples of damage which is generated by first heat treatments are given in Figs. 1 and2. Damage mechanisms deal with aggregate/matrix decohesions, fibre/matrix decohesions and with matrix microcracking too. In the considered temperature domain, the higher the firing temperature is, the higher the damage level is.

The second material (And-LCC) is a commercial grade of a low cement andalusite based castable made of andalusite aggregates, silica fume, alpha-alumina and of a calcium aluminate cement. The samples are prepared by mixing the raw materials in a planetary mixer with a 5 wt.% water addition and during 5 min. The mix is then cast in the moulds on an amplitude controlled vibration table and during 2 min. Moulds and samples are then immediately wrapped in plastic. They are cured at room temperature during 24 h and then extracted from the moulds before a 110 • C-24 h drying step. As for the FRRC castable, high temperature heat treatments are known to develop damage in this And-LCC castable. Previous works have shown that this particularly true for firing temperatures that are in the 110-900 • C temperature range. [START_REF] Marzagui | Microstructural changes and high temperature mechanical behavior of an andalusite based low cement refractory castable[END_REF][START_REF] Marzagui | Characterisation of microstructural evolutions in refractory castables by in-situ high temperature ESEM[END_REF] As these results have shown that firing this material at 700 • C leads to an important damage state in the microstructure, this firing temperature level will be considered in the present paper for this And-LCC castable. Figs. 3 and4 enable us to observe the damage mechanisms that have occurred in this material after a 700 • C heat treatment. They deal again with aggregate/matrix decohesions and with matrix microcracking. They are mainly due to the differential dilatometric behaviour between the matrix and the aggregates. The case of the 110 • C dried And-LCC castable will be considered in that paper too.

Experimental methods

Uniaxial tensile test

Uniaxial tensile tests are performed on a MTS 810 servohydraulic universal testing machine by the way of a nonarticulated tensile system (Fig. 5). Sample extremities are glued on metallic plates that are water-cooled. At high temperature, the central part of the specimen is placed in a MTS centrate the highest stress levels in the hottest area of the specimen. Transversal dimensions of samples are limited by the furnace size. Strain measurements are performed using two high temperature extensometers that are characterized by a 12.5 mm gauge length. They are placed on two opposite faces of the specimen. Machining by grinding is performed on the two opposite faces that support the extensometers. A parallelism error lower than 0.05 mm is obtained between these two faces. A particular attention is also paid to obtain the better alignment between the sample axis and the machine loading axis in order to minimize parasitic bending. Monotonic tests and cyclic loading/unloading tests are performed for each temperature configuration with a crosshead displacement velocity of 0.04 mm/min. This tensile test configuration allows to perform tensile/compression or compression/tensile tests too.

Compression test

The tests are performed at room temperature on an IN-STRON 4467 electro-mechanical universal testing machine. Samples have a cylindrical geometry and are axially loaded between two metallic plates. They are characterized by a 30 mm diameter and a 45 mm height. As for tensile tests samples, after a diamond machining step, a parallelism error lower than 0.05 mm is obtained between the sample extremities. Two types of loading are considered: monotonic loadings and cyclic loadings-unloadings. The specimens are tested with a crosshead displacement rate of 0.1 mm/min. The load is recorded as a function of the crosshead displacement that is corrected from the machine stiffness.

Four points bending test

For four points bending tests, a specific device installed on a MTS 810 servo-hydraulic universal testing machine is used (Fig. 6). Several works have shown that experimental results of bending tests are very sensitive to the boundary conditions, like the loading contacts and the degrees of freedom of the contact supports. [START_REF] Quinn | Flexure strength of advanced structural ceramics: a round robin[END_REF][START_REF] Quinn | Design data for engineering ceramics: a review of the flexure test[END_REF] As a consequence, some precautions have been taken in order to minimize the error sources that have been previously identified in testing ceramics. Loading is applied with alumina parts. The two lower alumina rollers are 125 mm spaced and supported by a fixed base. The two upper rollers are 45 mm spaced and are supported by a free rotation universal joint. Samples have a 150 mm length and a 25 mm × 25 mm cross-section. The lower and upper faces of the sample are machined by grinding to obtain a parallelism error lower than 0.05 mm between these two faces. Measurement of the sample deflection is made by a linear variable displacement transducer (LVDT) coupled with a fine alumina rod that is in contact with the central point of the sample lower face. For high temperature tests, the system is entirely placed in a radiative furnace with a maximum capacity of 1600 • C. Both monotonic tests and loading/unloading cyclic tests have been performed and a constant crosshead displacement velocity of 0.02 mm/min was retained.

The material elastic strength theory is applied to calculate the Young's modulus and the nominal strength on the tension sample face. This enable us to compare four points bending curves obtained on different samples, even if sample cross sections are not exactly the same. The nominal strength, σ and the Young's modulus, E are calculated as follow:

σ = 3P(L -l) 2bh 2 (1)
E = P f 1 8bh 3 (L -l)(2L 2 + 2Ll + l 2 ) ( 2
)
where P is the applied load; b and h are, respectively the width and the thickness of the specimen; L and l are the distances between the lower and the upper loading rollers and f is the sample deflection.

Mechanical properties

Tensile test results

For the FRRC, results of tensile tests performed both on a 500 • C fired FRRC specimen and on a 900 • C fired specimen are shown in Fig. 7. Generic comments can be made to de-scribe this room temperature tensile behaviour. At the beginning of the first loading and during the first loading/unloading cycle, a linear elastic behaviour is observed. When the stress level increases, the behaviour moves to a non-linear one. This is due to microcracking damage processes that occur in the FRRC. At the stress peak, one or several macrocracks appear in the tensile test sample. The behaviour moves then to a softening part, because of the presence of fibres that bridge the macrocracks. During the application of the loading/unloading cycles, permanent strains and hysteresis loops are measured. This must be related to the energy loss by internal friction in the microcracks.

Focussing on the beginning of the tests, the Young's modulus can be calculated by a numerical interpolation of the linear part of the behaviour curves. A 11.5 GPa Young's modulus value is calculated on the strain-stress curve of the 500 • C fired FRRC and a 13.0 GPa Young's modulus value is determined on the strain-stress curve of the 900 • C fired FRRC.

For the And-LCC castable, an example of a tensile test result performed on a 700 • C fired specimen is shown in Fig. 8. As described in a previous paper, [START_REF] Marzagui | Microstructural changes and high temperature mechanical behavior of an andalusite based low cement refractory castable[END_REF] an important damage develops during the first thermal cycle at such a temperature, and particularly during the cooling stage. This thermal damage is a consequence of the differential dilatometric behaviours of the matrix and of the aggregates. The behaviour curve of Fig. 8 is characterized by a very restricted linear do-

Four points bending test results

Fig. 9 shows four points bending curves with successive loading/unloading cycles for the FRRC castable that has been fired at 500 or at 900 • C. As for tensile tests curves, a nonlinear behaviour is observed but some specificities have appeared on these four points bending curves. The first one deals with the non linear part at the early beginning of the test. Similar phenomena have been previously observed and attributed to the installation of the contacts between the specimen and the loading rollers. [START_REF] Lemaistre | Etude des propriétés thermomécaniques de divers réfractaires[END_REF] Then the behaviour moves to a linear part, on which a 8.0 GPa Young's modulus value has been calculated in the both cases of Fig. 9. The second specificity deals with the comparison of the linear domain of the first loading and of the loading/unloading loops that are performed in this domain. Calculating the Young's modulus on the first loading/unloading loop leads to a 10.5 GPa value for the 500 • C fired FRRC and to a 11.5 GPa value for the 900 • C fired one (Table 1). In both cases, a difference is observed between the Young's modulus values that are calculated on the first loading and on the first loading/unloading loop. It must be noticed that the highest values are obtained for the loading/unloading loops. In the case of a linear elastic behaviour, this second specificity is unexpected for four points bending tests. For highest stress levels, the behaviour moves then to a non linear domain indicating that damage processes occur in the materials. As for tensile tests, a peak stress is reached when macrocracking starts to occur.

A behaviour curve example is given on Fig. 10 for the And-LCC castable after a 700 • C firing. This four points bending behaviour is similar to the FRRC castable one and the same specificities are particularly observed. The early beginning of the curve exhibits a non linear part and the behaviour moves then to a linear domain. Loading/unloading cycles that have been performed in this linear domain are again characterized by a higher slope than the one of the first loading. Thus, a 8.0 GPa Young's modulus value has been determined on the linear part of the first loading against a 12.0 GPa value on the first loading/unloading loop.

Compression test results

Fig. 11 shows compression curves both for monotonic tests and for cyclic tests in the case of the FRRC castable fired at 500 and at 900 • C. For a given firing temperature, the comparison of the monotonic and of the cyclic curves allows to observed the test reproducibility and to conclude that performing cyclic tests does not significantly influence the compression behaviour. The same behaviour description can be made for these compression curves as for the four bending ones. They first starts with a non linear domain before moving to a linear one. As previously observed for tensile and four points bending tests, increasing the stress level induces another non linear domain that is due to the development of damage processes in the material. Furthermore, the two same specificities are still present and particularly the slope difference between the linear part of the first loading and the first loading/unloading loop. As a consequence, discrepancies are obtained between the Young's modulus values that can be determined from these slopes. As an example, in the case of the 900 • C fired material, a 6.5 GPa value is calculated on the first loading against a 10.0 GPa value on the first loop unloading.

Comparison between the various mechanical tests

The comparison between the behaviour curves in the tension, compression and bending configurations enable us to observe some similarities. During the first loading, a linear domain is obtained for each type of tests and the behaviour then becomes non-linear because of microcracking processes that act in the refractory castables. When the damage threshold is exceeded, evolutions of the loading/unloading loops are noticed. They deal with a decrease of their slope and with an increase of their opening. A peak stress is then observed as soon as the damage processes move to localized ones and generate one or several macrocracks. But differences are observed between the results of the three considered test types too. Some unexpected ones deal with the characteristics of the linear domain of the first loadings where the material behaviour is assumed to be linear elastic. Table 1 allows to compare the Young's modulus values that could be determined from the different behaviour curves and for different firing cycles. Results show that many factors can influence these Young's modulus values. For such materials, changing the firing temperature leads to microstructural evolutions that can modify their elastic properties and therefore their Young's modulus value. On the other hand, in the linear elastic domain, changing the loading mode or performing loading/unloading cycles should not modify the Young's modulus. Two types of discrepancies are observed in these results. The first one deals with the different Young's modulus values that are obtained for each solicitation mode, even for a given thermal history of the material. The second one deals with the higher Young's modulus values that are determined on the first loading/unloading loop compared to those that are determined on the initial loading when considering four points bending tests and compression tests. The discussion of the present paper will be devoted to this discrepancies and to their explanation.

Discussion on the Young's modulus determination

Case of the four points bending test

The bending and the compression tests will be considered separately in order to identify the mechanisms that lead to the observed Young's modulus discrepancies under these two loading modes. In the four points bending configuration, during the first loading, the contact under each loading roller is of a hertzian type. Such local effects at the roller/sample contacts are known to influence the deflection measurement and to generate non-linear behaviours for weak loadings in the case of refractory castables. [START_REF] Lemaistre | Etude des propriétés thermomécaniques de divers réfractaires[END_REF] Even if such mechanisms occur during bending tests on refractory castables, it remains difficult to explain why the Young's modulus values determined on loading/unloading loops are higher than those determined on the linear part first loading. In order to obtain more details on the effects of such a contact mode, a specific test has been developed and performed. It consists of an indentation test with a lineic contact, as described on Fig. 12. It allows to characterize the refractory castable behaviour in the contact area between the sample and the roller. Tests were carried out on 25 mm × 25 mm × 25 mm cubic samples for each consid- ered material. Loading and unloading rates are the same as those retained for the four points bending tests and the results are plotted as the variations of the applied load versus the crosshead displacement. Such a behaviour curve is given in Fig. 13 in the case of an AU4G aluminium alloy.

In agreement with the theory of Hertz applied to an elastic material, the behaviour is non linear for weak loadings and gradually tends to a linear asymptotic one when the load level increases. Loading/unloading cycles show that this behaviour is quite reversible for this metallic alloy. It is almost the case for the 110 • C dried refractory castables (Fig. 14). Nevertheless, results obtained on these dried samples, tend to indicate an increase of the slope of the unloading curves compared to the slope of the first loading as well as permanent strains are measured at low load levels. These effects are clearly enhanced when the refractory castables have been previously fired at high temperature. As an example, in the case of the FRRC castable, results indicate that the higher the firing temperature of the castable is, the higher the slope changes and the more permanent strain amplitudes are. In other words, it could be mentioned that the higher the initial damage level of the castable is, the higher the slope changes and the permanent strain amplitudes are. The same trend is observed for the And-LCC castable when comparing the behaviour curves of a 110 • C dried sample and of a 700 • C fired one.

In the case of four point bending tests performed on samples that have been damaged during a firing thermal cycle, the measurement of the beam deflection is influenced by local permanent strains that appear at the roller/sample contacts since low load levels. That is the reason why important similarities are observed between the load-displacement curves of lineic indentation tests and those of four point bending tests. Thus, the measured deflection values do not only reflect the elastic bending of the beam. As a consequence, the Young's modulus values that are calculated from the linear part of the first loading of the four points bending curves are only apparent ones and they are lower than those determined on the tensile test curves. In previous works, Schmitt et al. [START_REF] Schmitt | Tensile behaviour of magnesia carbon refractories[END_REF] have measured a 25.2 GPa Young's modulus value during tensile tests performed on a magnesia-carbon refractory. For the same material, they have determined a 18.6 GPa value when considering the load/deflection curve of three points bending tests. It can be considered that the same trend is observed than for the materials considered in the present paper. Moreover, it is interesting to notice that these authors have determined the Young's modulus from the measurements of strain gauges too. These gauges were glued on the three points bending beam and on the face that is mainly subjected to tension stresses. In that case, the obtained Young's modulus value is of 24.0 GPa and is almost equal to the Young's modulus determined on direct tensile test curves. This result confirms that for refractory castables, the Young's modulus values that are determined on load/deflection curves are underestimated because the beam deflection measurements are influenced by the local phenomena that occur at roller/sample contacts. In the case of high temperature bending tests, the deflection measurement is the easiest way to obtain data on the strain state of the beam and is often performed. Nevertheless, load/deflection bending curves must be carefully considered in the case of damaged materials. For a given refractory castable, they can give qualitative data on the high temperature behaviour and quantitative ones dealing with the hot modulus of rupture or of other specific strengths. But dealing with the Young's modulus determination and with strain calculations, they can be highly influenced by local phenomena that act at the roller/sample contact.

Case of the compression test

In the case of compression tests, the same discrepancies are observed:

-the Young's modulus values calculated on the linear part of the first loading are low compared with tensile test ones, -the Young's modulus values calculated on the loading/unloading loops are higher than those calculated on the linear part of the first loading.

As for bending tests and in order to obtain more details on the sources of such discrepancies, specific compression tests have been developed and performed too.

First they deal with successive interrupted compression tests that have been carried out on the FRRC castable. They consist of applying a load on a compression sample up to a given stress level that is just below the damage threshold. Load is then totally withdrawn by removing the contact between the loading plates and the sample. The same sequence is run four times on the same sample. Results of such successive compression tests are shown on Fig. 15 for the FRRC castable, both after a 110 • C drying and after a 500 • C firing. Considering only the loading curves, results show that the slope of the linear part gets steeper as the sequence number increases. This is particularly true between the two first sequences. Young's modulus values have been calculated by linear interpolation on these linear parts and are given in Table 2. Between the two first loading curves, a 24% increase of the Young's modulus is measured for the 110 • C dried sample whereas a 130% increase is measured for the 500 • C fired sample. Thus, the firing cycle appears as enhancing drastically the amplitude of this Young's modulus increase. This could be related to the initial damage state of the material that is higher in the case of the 500 • C fired sample. It appears that the higher the initial damage level is, the higher the Young's modulus increase is. These successive compression tests show that the difficulty to determine the Young's modulus is enhanced for fired refractory castables.

A second type of compression tests was carried out by gluing the two sample extremities on the loading plates. This configuration allows to minimize interface effects between the sample and the plates and to apply a tensile loading on the sample too. A behaviour curve example is given in Fig. 16A for a FRRC castable fired at 500 • C and is compared with the behaviour curve of a classical compression test (non-glued sample). In the glued configuration, a linear behaviour is observed in tension and in compression since the beginning of the test. The Young's modulus remains constant between the tension and compression domains and is equal to 11.5 GPa. This indicates that the Young's modulus is not intrinsically different under these two loading modes. Furthermore, it can be noticed that this Young's modulus value is the same as the one obtained during direct tensile tests (Table 1).

Up to a stress level close to -15 MPa, the behaviour curve remains linear. Beyond this threshold, the compression behaviour curve moves to an non-linear domain indicating that damage processes occur in the castable. It is now interest- ing to focus on the comparison between this behaviour curve and the one of a classical compression test (Fig. 16A andB). Fig. 16B allows to see that adding an offset to the strain values of the glued configuration has made it possible to perfectly superpose the high stress domain of the resulting curve with the linear domain of the classical compression test curve. This indicates that damage processes occur in this domain and that the 6.0 GPa Young's modulus value that has been calculated on the classical compression test (Table 1) must be considered as an apparent one. Moreover, concerning the discrepancies between the beginning of these two curves, it can be assumed that local effects can act at the sample/loading plate contacts, as previously observed for four points bending tests. Indeed, in spite of the machining of the sample extremities and the adjustment of the parallelism between the loading plates, there is no certainty that two perfect plane to plane contacts are obtained between the sample extremities and the loading plates.

In order to characterize the influence of such local effects, a third type of compression test has been performed. It consists of introducing by machining a parallelism error between the two sample extremities. During that test, strains have been determined both by the measurements of the crosshead displacement and of the two extensometers that have been previously described. The extensometers have been placed opposite to each other in the middle part of the sample and their gauge length is fifteen times lower than the sample height. Results of this test are shown on Fig. 17. At the beginning of the test, the two behaviour curves obtained from the extensometers are highly divergent. This indicates a tensile state on one side of the sample and a compression state on the opposite one. This is clearly due to the geometric defect that has been voluntarily introduced. It appears that it is necessary to reach a stress level close to -4 MPa in order to place the whole sample in a compression state. Furthermore, it should be noticed from the loading/unloading cycles that the extensometers, that are located far away from the contact surfaces, do not record permanent strains. On the other hand, the behaviour curve obtained from the crosshead displacement exhibits permanent strains. Similar observations had already been made by Prompt [START_REF] Prompt | Etude du comportement thermomécanique d'un béton réfractaire alumineux. Application à une rigole de haut fourneau[END_REF] for compression tests performed on an alumina based refractory castable. Such behaviour discrepancies between local and global strain measurements indicate that many local phenomena occur at the sample/plate contacts and generate the appearance of permanent strains on the global compression curves. As a consequence, the linear domain that is observed on these curves during the first loading is not only due to elastic strains of the sample. As for bending tests, the Young's modulus values that can be calculated in this linear domain are apparent ones.

In the case of compression tests at high temperature, the measurement of the crosshead displacement or of the distance between the two loading plates are the easiest ways to obtain data on the strain state of the sample and is often performed. Nevertheless, the resulting behaviour curves must be carefully considered in the case of damaged materials. As for deflection measurements in bending, local phenomena are active and the global displacement values that can be measured are not only due to the elastic behaviour of the compression sample, even if the apparent damage threshold has not been exceeded. Fig. 17. Result of a compression test with a parallelism defect on the sample: strains have been calculated from the measurements of two extensometers and of the crosshead displacement (FRRC after firing at 500 • C).

Conclusions

As a conclusion, it is really difficult to determine Young's modulus values of refractory castables by the way of compression or bending tests. That can be done by using strain gauges in the case of room temperature tests but this technique cannot be used at high temperature. As high temperature extensometers are not easy to be used in such test configurations, behaviour curves are often obtained from the measurement of global strains calculated from the beam deflection in bending and from the distance between plates in compression. In bending tests, results have shown that local strain effects take place at the roller/sample contacts and influence the deflection measurement. In compression tests, it is necessary to be aware of the influence of similar contact phenomena that occur at the sample/plate interfaces even after correction of the device stiffness. As a consequence, it is necessary to be extremely careful in the exploitation of behaviour curves obtained under such conditions.

Especially in the case of fired refractory castables, the obtained tensile, compression and bending curves indicate specific behaviours for this type of materials. As already known for such materials, firing heat treatments that are performed in the intermediate temperature range can lead to the development of damage by decohesion and microcracking in the bulk material. For a given mechanical test type, changing the initial damage state will of course modify the behaviour curve and will affect the Young's modulus value. Moreover, it appears from the results of this paper that this damage state plays a major role on the local mechanical behaviour of such refractory castables, particularly in the loading contact areas. Increasing the initial damage state highly increases the amplitude of these local phenomena. Several mechanisms can be evoked to explain them. On the one hand, local micromechanical mechanisms of microcrack closing can influence the behaviour when the loading contact occurs. On the other hand, in the case of the bending tests, a triaxial stress field appears locally under each loading roller. In the case of the lineic indentation test, applying a load of 1 kN leads to an hydrostatic stress close to 138 MPa according to the Hertz theory. [START_REF] Johnson | Contact Mechanics[END_REF] Even if the assumptions of the Hertz theory are not perfectly respected, since the castable behaviour does not remain linear-elastic, this calculation allows to estimate the very high stress levels that are locally generated. Moreover, under such a triaxial compression stress field, it can be assumed that the damaged castables can be compacted. [START_REF] Burlion | Compaction and tensile damage in concrete: constitutive modelling and application to dynamics[END_REF] Furthermore, it can be considered that the firing heat treatment will lead to a decrease of the compaction stress thresholds too.

Fig. 1 .

 1 Fig. 1. Observation of damage in the FRRC castable after drying at 110 • C (scanning electron microscopy).

Fig. 2 .

 2 Fig. 2. Observation of damage in the FRRC castable after firing at 500 • C (scanning electron microscopy).

Fig. 3 .

 3 Fig. 3. Observation of damage in the And-LCC castable after firing at 700 • C (scanning electron microscopy).

Fig. 4 .Fig. 5 .

 45 Fig. 4. Observation of damage in the And-LCC castable after firing at 700 • C (scanning electron microscopy).653 resistive furnace with two heating zones and a maximum capacity of 1400 • C. The specimen geometry has been defined to allow tests in the 20-900 • C temperature range. A reduced cross-section of 30 mm × 25 mm allows to con-

Fig. 6 .

 6 Fig. 6. Four points bending device in the room temperature configuration: zoom on the active parts.

Fig. 7 .

 7 Fig. 7. Tensile test behaviour curves of the FRRC castable after firing at 500 • C (A) and at 900 • C (B).

Fig. 8 .

 8 Fig. 8. Tensile test behaviour curve of the And-LCC castable after firing at 700 • C.

Fig. 9 .

 9 Fig. 9. Four point bending behaviour curves of the FRRC refractory castable after firing at 500 • C (A) and at 900 • C (B).

Fig. 10 .

 10 Fig. 10. Four point bending behaviour curve of the And-LCC castable and after firing at 700 • C.

Fig. 11 .

 11 Fig. 11. Monotonic and cyclic compression behaviour curves of the FRRC castable after firing at 500 • C (A) and at 900 • C (B).

Fig. 12 .

 12 Fig. 12. Principle of the lineic indentation test.

Fig. 13 .

 13 Fig. 13. Lineic indentation test: behaviour curve of an AU4G aluminium alloy.

Fig. 14 .

 14 Fig. 14. Lineic indentation test: behaviour curves of the FRRC castable (A), of the And-LCC castable (B), for different firing temperatures.

Fig. 15 .

 15 Fig. 15. Loading behaviour curves of successive compression tests on the FRRC castable after a 110 • C drying (A) and after a 500 • C firing (B).

Fig. 16 .

 16 Fig. 16. Comparison between the results of a classical compression test (without glue at the sample extremities) and the results of a compression test with glue at the sample/plate interface (FRRC after firing at 500 • C).

Table 2

 2 Young's modulus values (in GPa) calculated from the four loadings of successive interrupted compression tests

		FRRC sample	FRRC sample
		fired at 110 • C	fired at 500 • C
	First loading	20.0	7 .0
	Second loading	25.0	1 6.0
	Third loading	25.0	1 6.5
	Fourth loading	25.0	1 7.0

Acknowledgements

Authors gratefully acknowledge ACB, Airbus-France, EDF, Hurel-Hispano, Pyromeral Systems, Vesuvius, TRB and the French Ministry for Industry for their technical and financial supports.


Sergei Soloviev
email: soloviev@irit.fr

Studies of Hilbert's ε-operator in the USSR

Keywords: Hilbert's epsilon-operator, history of logic, research in the USSR

The aim of this paper is to give a short survey of the studies that concern the notion of Hilbert's ε-operator and its applications by the researchers in the USSR and continuation of their works abroad in post-soviet time.

The aim of this paper is to give a short survey of the studies that concern the notion of Hilbert's ε-operator and its applications by the researchers in the USSR. These works mostly belong to the domain of mathematical logics, but interactions with philosophical logics and mathematical linguistics cannot be ignored, all three domains being quite active.

The paper is based on a systematic historical investigation and my personal recollections, that helped to organize the search for information. I knew personally Grigori Mints, Albert Dragalin and Vladimir Smirnov mentioned below. Mints was the adviser of my graduate work, and I met Dragalin and Smirnov at various conferences. It is possible that my investigation is not completely exhaustive, but I recall whose names were mentioned then in connection with the ε-operator, and this knowledge was confirmed when I did look for references. I am reasonably sure that there were no other authors who did significant study of the ε-operator in the USSR.

My personal recollections were used also to reconstruct to some extent the atmosphere of Soviet times, at least as far as the relationship with the international science is concerned.

Three main aspects will be addressed.

• How the interest for ε-operator had arisen, in particular, the external sources of logical research in the USSR.

• Principal works by soviet researchers who studied ε-operator and continuation of their works in post-soviet time. There are three main names: A. G. Dragalin (1941Dragalin ( -1998)), G. E. Mints (1939Mints ( -2014) ) and V. A. Smirnov (1931Smirnov ( -1996)).

• Influence of these works on the research worldwide in their own time, and how they influence contemporary research.

Two volumes of German edition of "Grundlagen der Mathematik" by Hilbert and Bernays [START_REF] Hilbert | Grundlagen der Mathematik. I[END_REF] were known to leading researchers in the USSR. Russian translation was done in 1968/70 and published in 1979/82 by "Nauka" (transl. from German by N. M. Nagorny). The full English translation still does not exist.

In general, the translation of scientific literature into Russian at this period was extremely active, and this fact mostly answers the question about sources. One may mention the publication of Kleene's "Introduction to Metamathematics" in 1957 (translated by Essenin-Volpin), the translation of A. Robinson's "Introduction to Model Theory and to the Metamathematics of Algebra" (1967), the translation of selected foundational papers in proof theory published as "Mathematical theory of logical deduction" (edited by A. V. Idelson and G.E. Mints) in 1967. In 1973 Kleene's "Mathematical Logic" (translated by Yu. Gastev and edited by G. Mints) was published. In 1981 G. Kreisel's selected papers in proof theory translated by Gastev and Mints [START_REF] Kreisel | Issledovaniya po theorii dokazatelstv[END_REF] were published. For scientific literature at this time the interval between publication of an original and its Russian translation was often 5-7 years. The number of copies was usually at least several thousands, e.g., 7800 for [START_REF] Kreisel | Issledovaniya po theorii dokazatelstv[END_REF]. Partly (but only partly) this may be explained by the fact that the USSR did not sign most of the international copyright agreements1 .

What else was accessible? Just as an illustration, one may mention that the library of Steklov Mathematical Institute included almost all "Mathematische Annalen" until June 1941 (with Gentzen's foundational paper of 1936, and Ackermann's paper of 1940), and again since 1949. It got also "Dissertationes Mathematicae" (published by Polish Academy), all issues 1953 -1989 (including such papers as [START_REF] Kreisel | Formally self-referential propositions for cut-free analysis and related systems[END_REF] by Kreisel and Takeuti)... Personal contacts also should not be underestimated. For example, many prominent logicians attended the International Congress of Mathematicians in Moscow (1966), among them Tarski, Church, Kleene, Curry, Schütte, Feferman, Cohen.

An one-hour talk was given by Schütte, 30 minutes talks by Cohen, Ershov (from Novosibirsk), joint talk by Shanin (the head of the logic group at Leningrad Branch of Steklov Mathematical Institute), Tseitin, and Zaslavski. Junior logicians from Leningrad (Maslov, Matiyasevich, Mints, Orevkov, Slissenko) participated in ICM with 15 minutes talks and had many opportunities to discuss logical problems with western colleagues. Tarski mentions [START_REF] Fefereman | Life and Logic[END_REF] his dinner at "Praha" restaurant with Mal'cev (Maltsev), Markov, Shanin, Ershov, Kleene, Curry, Chang, Feferman. After the ICM Tarski visited Leningrad and gave a talk. Earlier, in 1965, both Moscow and Leningrad were visited by John McCarthy. G. Mints, whose works on the ε-symbol we shall consider below, was greatly influenced by G. Kreisel. Mints considered Kreisel (who did several important works concerning the ε-symbol himself) as one of his teachers. Kreisel visited the USSR in 1976, but actively communicated with Mints before (cf. [START_REF] Kreisel | The use of abstract language in elementary metamathematics: some pedagogic examples[END_REF]).

I cannot go too much into detail in this short paper, but would like to recommend the book [START_REF]Beyond the Divide. Entangled Histories of Cold War Europe[END_REF] for a more general picture of the East-West scientific and cultural exchanges during Cold War times.

Another illustration of active interaction of soviet researchers with worldwide research community is a series of biannual Finnish-Soviet Logic Conferences that started in 1976. "Somewhere around 1975 J. Hintikka and V.A. Smirnov have agreed to hold Finnish-Soviet Conference on logic" [START_REF] Karpenko | The History of Finnish-Soviet Logic Colloquium[END_REF] 2 .

The proceedings of the first Soviet-Finnish Logic Conference included 6 papers by soviet participants. Among other contributions let us mention the papers by S. Feferman, J. Hintikka, J. Ketonen, G. Kreisel, D. Prawitz, R. Statman, D. Van Dalen. As Karpenko writes [START_REF] Karpenko | The History of Finnish-Soviet Logic Colloquium[END_REF], the second Finnish-Soviet Logic Conference was held in Moscow, at the Institute of Philosophy, in 1979. The first Finnish-Soviet-Polish Logic Conference at Polanica-Zdrój was held in 1981. The series of Finnish-Soviet Logic Conferences continued (Helsinki, 1983;Telavi, Georgia, 1985;Helsinki, 1987;Moscow, 1989). It continued even after the fall of the USSR (until 1997, and restarted in 2012) 3 .

Among early works on the ε-operator one may cite V. A. Smirnov [START_REF] Smirnov | Elimination des termes ε dans la logique intuitioniste[END_REF] (from Institute of Philosophy), and G. E. Mints [START_REF] Mints | Heyting Predicate Calculus with Epsilon Symbol (Russian). Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im[END_REF] (from Leningrad Branch of Mathematical Institute). In Smirnov's paper [START_REF] Smirnov | Elimination des termes ε dans la logique intuitioniste[END_REF] (published in French) Hilbert, Bernays, Quine, 2 Finland played a special role in this interaction. It was a liberal democracy, but it was not a NATO member and had a special relationship with the USSR because of the conditions of the Agreement of Friendship, Cooperation, and Mutual Assistance (1948). In times of the Cold War it was considered as a kind of "neutral ground". For example, in 1975 it was the venue of the Conference on Security and Co-operation in Europe.

3 I guess that the interruption was connected with the changes in the finance of scientific research.

Russell, Sloupiétzski4 were mentioned, but there is no bibliography. Mints [START_REF] Mints | Heyting Predicate Calculus with Epsilon Symbol (Russian). Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im[END_REF] cited several publications, in particular [START_REF] Dragalin | ) Istoriia i Metodologiia Estestvennykh Nauk[END_REF], [START_REF] Leisenring | Mathematical logic and Hilbert's ε-symbol[END_REF], [START_REF] Leivant | Existential instantiation in a system of natural deduction for intuitionistic arithmetics[END_REF], [START_REF] Maehara | A General Theory of Completeness Proofs[END_REF], [START_REF] Shirai | Intuitionistic Predicate Calculus with ε-symbol[END_REF]. From this brief outline it is clear that a large variety of sources was accessible, at least to researchers working in academic institutions. It is interesting to notice the role of Japanese and Polish sources, in addition to German, English and American. We may agree that the most important contributions of the USSR researchers concerning the ε-symbol belong to mathematical logics. This is true also for continuation of this line of research after the fall of the USSR. As B. H. Slater writes: "A good deal of technical work has been done, as a result of such proofs, to create epsilon extensions of Intuitionistic Logic which are conservative." [START_REF] Slater | Hilbert's Epsilon Calculus and its Successors[END_REF] It is more difficult to agree with Slater that this work is now mostly of academic interest. The research on ε in the post-soviet time included the studies of ε-substitutions for analysis [START_REF] Mints | Epsilon substitution method for elementary analysis[END_REF] and other theories [START_REF] Arai | Epsilon substitution method for theories of jump hierarchies[END_REF], [START_REF] Mints | Epsilon-substitution method for the ramified language and ∆ 1 1comprehension rule[END_REF] (see the discussion in Stanford Encyclopedia of Philosophy [START_REF] Avigad | The Epsilon Calculus[END_REF]).

As said above, the first paper on the ε-operator by a soviet researcher was the paper [START_REF] Smirnov | Elimination des termes ε dans la logique intuitioniste[END_REF] published in "Revue Internationale de Philosophie" in 1971. In 1974 two more mathematical papers were published: the paper [START_REF] Dragalin | ) Istoriia i Metodologiia Estestvennykh Nauk[END_REF] by Albert Dragalin and [START_REF] Mints | Heyting Predicate Calculus with Epsilon Symbol (Russian). Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im[END_REF] by Grigori Mints.

Here is a brief outline of the main points of Mints' paper (which takes into account the two others):

• It is known, that adding the ε-axiom A[t] → A[εxA] to Heyting's (intuitionistic) predicate calculus HA gives a non-conservative extension -for example, the formula ∃x(¬P x → ¬P b ∧ ¬P a) becomes derivable5 .

• We know (says Mints) two conservative ε-extensions of HA. In one of them, due to V. A. Smirnov, a rather limiting constraint is imposed on the notion of proof. We shall use another formulation, due to A. G. Dragalin; in this formulation functions defined by ε-expressions, are seen as partially defined.

• A. G. Dragalin...uses model-theoretical methods; we shall use proof-theoretical methods, that may be extended to stronger systems, and obtain a supplementary theorem about cut-elimination for proofs of arbitrary formulas, not only ε-free.

• These results admit natural extension to Intuitionistic Predicate Calculus with decidable equality (i.e., with supplementary axiom ∀x∀y(x = y ∨ ¬x = y) and to Heyting's arithmetic with free functional variables and a choice principle in the form Γ → ∀x∃yA ∀xA y [f (x)], Γ → C Γ → C where f does not occur in Γ, C, A. An extension to Heyting's arithmetic with bound variables of higher types and with a respective choice principle requires new ideas.

• The system studied in the paper is denoted HP C ε . It is obtained from Heyting Predicate Calculus HP C with two sorts of variables (free and bound), with functional symbols but without equality by addition of the following term formation rule: for a formula A, a free and x bound variable εxA a [x] is a term. The system HP C ε has the same postulates as Hentzen's LJ (except modified ∃ →), but the definition of proof is different.

• An occurrence V of some sequent in a tree-form figure (of deduction) is called meaningful 6 if for every quasiterm εxA in V , there is in the antecedent of V or in some sequent lying below V a member

(1) ∀α 1 ...∀α n ∃xA
where α 1 , ..., α n is the full list of free variables of quasiformula ∃xA. The ( 1) is denoted !εxA.

• The figure (of deduction) that is built from axioms A → A using deduction rules is called meaningful if all sequents in it are meaningful.

• There are three theorems: the cut elimination theorem for HP C ε , the conservativity theorem for HP C ε w.r.t. HP C and the theorem that the following rule

∀xA y [f (x)], Γ → C ∀x∃yA, Γ → C is admissible in HP C ε and HP C for function symbols f that do not occur in A, Γ, C.
This work by Mints keeps its relevance for modern research. A surprising fact is that later he might underestimate its relevance himself. Bruno Woltzenlogel Paleo met Mints in 2012 and discussed his current work about epsilonization (in collaboraion with Giselle Reis). As he wrote to the author, concerning [START_REF] Mints | Heyting Predicate Calculus with Epsilon Symbol (Russian). Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im[END_REF]: "the work you pointed to us is much more relevant than the work Grigori Mints recommended. He must have forgotten." 7 Mints recommended [START_REF] Mints | Intuitionistic Existential Instantiation and Epsilon Symbol[END_REF].

Next paper by V. A. Smirnov [START_REF] Smirnov | Theory of Quantification and ε-calculi[END_REF], presented initially as a talk at the First Soviet-Finnish Conference, was published in 1979. In this paper Smirnov recalls the history of his intuitionistic system of natural deduction with the ε-symbol (a conservative extension of the system without ε). Dragalin's technique is used to obtain a similar result for the system of natural deduction with identity. At the same conference was presented also the talk by Yu. Gladkich "Singular Terms, Existence and Truth: Some Remarks on a First Order Logic of Existence" published later in the same proceedings as [START_REF] Smirnov | Theory of Quantification and ε-calculi[END_REF].

In 1982 and 1989 Mints published two papers [START_REF] Mints | A simplified consistency proof for arithmetic[END_REF], [START_REF] Mints | Epsilon substitution method for the theory of hereditarily finite sets (Russian, English Summary)[END_REF].

• Zbl 0523.03043 [START_REF] Mints | A simplified consistency proof for arithmetic[END_REF]: The paper contains a simplified proof of Ackermann's theorem about the consistency of Peano Arithmetic. The proof is based on Hilbert's idea to apply so called ε-substitutions to systems of arithmetical formulas (...). A careful analysis of the behaviour of the above substitutions and (...) give a proof of the convergence of a suitable system of ε-substitutions. Hence Ackermann's theorem follows. (E. Adamowicz)

• Zbl 0677.03040 [START_REF] Mints | Epsilon substitution method for the theory of hereditarily finite sets (Russian, English Summary)[END_REF]: The Hilbert epsilon-substitution method is extended to some formalizations of the theory of hereditarily finite sets. Applying the methodology developed in an earlier paper (...) the convergence (...) for the theory of hereditarily finite sets is established which generalizes the Ackermann theorem of the convergence of the epsilon-substitution method for first-order arithmetic. (B.R. Boricić)

After the end of the USSR, G. Mints (with some co-authors of the next generation) was, no doubt the most active (among the researchers formed in the USSR) to pursue the studies of ε. V. A. Smirnov also continued to work in this direction [START_REF] Smirnov | Logika i Komputer[END_REF], [START_REF] Smirnov | Cut elimination in ε-calculus // Book of abstracts[END_REF], [START_REF] Smirnov | Proof search in intuitionistic natural deduction with ε-symbol and existence predicate[END_REF] 8 .

The papers of this period are easily accessible, so there is less need for a detailed presentation.

G. Mints did work on versions of ε-substitution method in various systems, problems of completeness, termination and cut elimination [START_REF] Mints | Hilbert's Substitution Method and Gentzen-type Systems[END_REF], [START_REF] Mints | Epsilon substitution method for elementary analysis[END_REF], [START_REF] Mints | Strong termination for the epsilon substitution method[END_REF], [START_REF] Mints | Skolem and the epsilon substitution method for predicate logic[END_REF], [START_REF] Mints | Strong termination for the epsilon substitution method for predicate logic[END_REF], [START_REF] Mints | Epsilon-substitution method for the ramified language and ∆ 1 1comprehension rule[END_REF], [START_REF] Mints | A termination proof for epsilon substitution using partial derivations[END_REF], [START_REF] Mints | Completeness of indexed epsilon-calculus[END_REF], [START_REF] Mints | Cut Elimination for a Simple Formulation of PAepsilon[END_REF], [START_REF] Mints | Cut elimination for a simple formulation of epsilon calculus[END_REF], [START_REF] Mints | Intuitionistic Existential Instantiation and Epsilon Symbol[END_REF], [START_REF] Mints | Epsilon substitution for first-and second-order predicate logic[END_REF]. Among his collaborators were W. Buchholz, D. Sarenac, S. Tupailo. Some of these papers concerned other aspects of ε-methods, such as their relationship to Kripke semantics. Here is the abstract of [START_REF] Mints | Intuitionistic Existential Instantiation and Epsilon Symbol[END_REF]:

• A natural deduction system for intuitionistic predicate logic with the existential instantiation rule presented here uses Hilbert's ε-symbol. It is conservative over intuitionistic predicate logic. We provide a completeness proof for a suitable Kripke semantics, sketch an approach to a normalization proof, survey related work and state some open problems. Our system extends intuitionistic systems with the ε-symbol due to A. Dragalin and Sh. Maehara.

In the paper with Sarenac [START_REF] Mints | Completeness of indexed epsilon-calculus[END_REF] Mints studied context-dependent descriptions and so called "salience hierarchy". Here is the abstract:

• Epsilon terms indexed by contexts were used by K. von Heusinger 9 to represent definite and indefinite noun phrases as well as some other constructs of natural language. We provide a language and a complete first order system allowing to formalize basic aspects of this representation. The main axiom says that for any finite collection S 1 , ..., S k of distinct definable sets and elements a 1 , ..., a k of these sets there exists a choice function assigning a i to S i for all i ≤ k. We prove soundness and completeness theorems for this system S ε i .

The method of the proof proposed in [START_REF] Mints | Completeness of indexed epsilon-calculus[END_REF] is based on a modification of the completeness proof for first-order predicate calculus given by Henkin [START_REF] Henkin | The completeness of the first-order functional calculus[END_REF]. Recently a critical analysis of [START_REF] Mints | Completeness of indexed epsilon-calculus[END_REF] was presented in a talk by Hans Leiss10 . His criticism concerns two important points: one technical (there is a gap in the proof), and one conceptual (whether the first-order theory proposed by Mints and Sarenac is an adequate representation of the second-order theory of von Heusinger).

It is clear, however, that independently of the results of this controversy [START_REF] Mints | Completeness of indexed epsilon-calculus[END_REF] turns out to be a very stimulating work.

In his talk at Montpellier conference mentioned above Hans Leiss proposed what he himself described as a non-conclusive counterexample, and tried to "fill the gap" in the Mints-Sarenac proof (to modify the construction of term model).

An attempt to reflect a second-order theory in a first-order system also looks promising even if the details need to be fixed. One may hope that this theme will see new interesting developments.

It reminds the story of another proof proposed by G. Mints that was included in [START_REF] Troelstra | Basic Proof Theory[END_REF]. One may read there (p. 281): "Our treatment follows Mints [1992d], with a correction in proposition 8.4.12. (The correction was formulated after exchanges between Mints, Solovjov and the authors.)" It is meant here a "short proof" of so called Coherence Theorem for Cartesian Closed Categories. First proof that was considerably longer was published by the author of this article and A. Babaev in 1979. G. Mints suggested a much shorter proof in 1992.

Conclusion. If I wanted to make a "lesson" from the history of studies of the ε-symbol outlined above, I would stress the role of continuity and interaction. Indeed, the papers by Smirnov, Mints and Dragalin could hardly appear in 1970s without continuity of the development of logic inside and outside the USSR, and strong collaboration between researchers on an international level.

The role of these factors remains vital even after the end of the USSR, as it is clearly seen from the examples above.

A curious fact, mentioned in[START_REF] Kaataja | Expert Groups Closing the Divide: Estonian-Finnish Computing Cooperation since the 1960s[END_REF], is that some Finnish universities, for example, the University of Turku, purchased in 1960s and 1970s Russian translations of American or West-European research due to financial reasons, since they were much cheaper than the original editions.

In fact it has to be Jerzy Słupecki, who published a book with Ludwik Borkowski on logic and set theory; its russian edition appeared in 1965.

Here HA may be a misprint, since below Mints speaks about Heyting's arithmetic, and calls Heyting's predicate calculus HP C.

"Osmyslennyi" in Russian.

E-mail to the author, 6 Oct. 2015.

It must be noticed that in the USSR and post-soviet Russia mathematical logic interacted very little with philosophical logic and philosophy of language. The V. A. Smirnov's works had considerable influence on philosophical community.

in his paper[START_REF] Heusinger | Reference and Salience[END_REF]

At the same conference where the initial version of this work was presented, Epsilon 2015. Hilbert's epsilon and tau in logic, informatics and linguistics, Montpellier, LIRMM, 10-12 June 2015.

Bruno Woltzenlogel Paleo, and Hans Leiss for fruitful discussions. Thanks also are due to the anonimous referees. This work was partially supported by the Goverment of the Russian Federation Grant 074-U01.


F Nazaret
email: nazaret@aurock.fr

T Cutard

O Barrau
email: cbarrau@aurock.fr

Sizing of refractory castable gas-burner using thermomechanical simulations

Keywords: Damage, refractory castables, simulation

Damage is a crucial characteristic of refractory castables and has to be considered to simulate correctly the behaviour of refractory structures. But, damage modelling by finite element simulations remains difficult. Indeed, the use of a continuum damage model with softening leads to strain localization phenomena. Numerical results depend on the mesh. Several numerical methods allow solving this meshing dependence by introducing an internal length in the material constitutive laws. In this paper, a regularization method has been applied with the damage plasticity model, considering a scalar value for damage. This model enables to take into account permanent strains due to plasticity and damage before and after the peak stress in tension and compression. Thermomechanical simulations are performed with this model to predict damage in a gas-burner. The damage level is evaluated after a thermal simulation generating high temperature gradients. Interests to take into account damage in the refractory structures are discussed. Sensitivity of results to material properties is studied. This work gives an example of using thermomechanical simulations to improve the design of refractory castable structures and to help in the material choice.

Introduction

Refractory castables present a damage behaviour up to a transition temperature domain located in the 800°C to 1100°C temperature range. This transition temperature domain depends on the material composition. Beyond this domain, the mechanical behaviour becomes mainly viscoplastic. The present work deals with the damage modelling for temperature values lower than this transition temperature. Considering a such non-linear behaviour is crucial to well predict the high temperature behaviour of complex refractory structures by finite element methods.

Damage behaviour is difficult to reproduce because it generates a strain localization phenomenon and a rigidity loss. During experiments, crack initiation appears after a microcraking developpement in the refractory matrix. Softening behaviour after the peak stress is generated by aggregates cohesion and is characteristic of quasi brittle materials. This mechanism is at the origin of the size effect in quasi-brittle materials [START_REF] Bazant | Fracture and size effect in concrete and other quasi-brittle materials[END_REF]. This property allows refractory structures working above the initiation stress level while a stress redistribution phenomenon appears. So, refractory structures are able to support higher load levels compared to structures based on brittle materials. Moreover, damage decreases rigidity and limits stress levels in the structure. That is really important under gradient thermal solicitations.

This paper deals with the study of the behaviour of a gas burner refractory part. This part is submitted to severe solicitations. Damage model is described first. Then, a sensitivity study is performed to evaluate influence of non linear behaviour before the peak stress during simulations of tensile and bending tests. Then, damage modeling is applied to the gas burner part. Results are discussed through damage fields.

Damage Modeling

Formulation. Below the transition temperature domain, the mechanical behaviour is divided in two parts. Firstly, the behaviour is linear elastic. When the stress level increases, the behaviour moves to a non-linear one due to diffuse damage processes by microcracking. At higher stress levels, damage localizes and one or several macrocracks initiate. After the peak stress, the behaviour curve exhibits a softening part. The behaviour between compression and tension is asymmetric. The consideration of this phenomenon is taken in account through the yield surface and damage criteria.

To reproduce such as behaviour, a damage plasticity model for quasi brittle materials implemented in the Abaqus ® software has been used [START_REF] Lubliner | A plastic-damage model for concrete[END_REF][START_REF] Hillerborg | Application of the fictitious crack model to different types of materials[END_REF]. A strain rate decomposition is assumed for the rate-independent model :

pl el ε ε ε ! ! ! + = ( 1 )
Where ε! is the total strain rate, el ε! is the elastic part of the strain rate and pl ε! is plastic part of the strain rate. Stress-strain relations are governed by scalar damaged plasticity :

( ) ( ) ( ) pl el pl el d ε ε ε ε - = - - = : : 1 0 D D σ ( 2 )
where el 0 D is the initial (undamaged) elastic stiffness of the material;

( ) el el d 0 1 D D - =
is the degraded elastic stiffness; and d is the scalar stiffness degradation variable, which can take values in the range from zero (undamaged material) to one (fully damaged material). Damage associated with the failure mechanisms of the concrete (cracking and crushing) therefore results in a decrease of the elastic stiffness. Within the context of the scalar-damage theory, the stiffness degradation is isotropic and characterized by a single degradation variable, d. The Cauchy stress is related to the effective stress σ through the scalar degradation equation:

( )σ σ d - = 1 ( 3 )
Nevertheless, when damage occurs, the effective stress is more representative than the Cauchy stress because it is the effective stress area that is resisting the external loads. It is, therefore, convenient to formulate the plasticity problem in terms of the effective stress. As discussed later, the evolution of the degradation variable is governed by a set of hardening variables, pl

      = pl c pl t pl ε ε ε ~ ; ( ) pl pl pl ε ε ε ! ! ⋅ = , ~σ h ( 4 )
Microcracking and crushing in the concrete are represented by increasing values of the hardening variables. These variables control the evolution of the yield surface and the degradation of the elastic stiffness. They are directly related to the dissipated fracture energy required to generate micro-cracks too.

The yield function,

( ) pl F ε σ ,
, represents a surface in the effective stress space, which determines the state of failure or damage. The yield function enables to reproduce asymmetric behaviour between tension and compression (Fig. 2). Variables p and q correspond respectively to hydrostatic pressure and to the deviatoric part of the stress tensor.

Plastic flow is governed by a flow potential G according to the flow rule:

( )

σ σ ∂ ∂ = G pl λ ε ! ! ( 5 )
where λ ! is the nonnegative plastic multiplier. The plastic potential is defined in the effective stress space. The specific form of the flow potential for the concrete damaged plasticity model is discussed later in this section. The model uses nonassociated plasticity, therefore requiring the solution of nonsymmetric equations. Details of uniaxial cyclic conditions and of multiaxial conditions are described in the Abaqus ® handbook [START_REF] Lubliner | A plastic-damage model for concrete[END_REF].

The use of a continuum damage model with softening leads to strain localization phenomena. Results depend on the mesh size. A viscoplastic regularization is helpful to obtain the convergence of the model. Several numerical methods allow solving such a meshing dependence problem. They are based on the introduction of an internal length in material constitutive laws. The regularization method with a dependence of the dissipated energy by cracking on the characteristic element length has been chosen [START_REF] Hillerborg | Application of the fictitious crack model to different types of materials[END_REF]. This needs to introduce the post peak behavior as a stress-displacement curve. This curve can correspond to the crack opening measured during a tensile test on a notched sample. The description of the softening behaviour by a stress-displacement curve allows obtaining quite stable simulations even if a severe damage occurs.

Sensitivity study. It's well known that softening behaviour influences highly the macroscopic behaviour during bending tests [START_REF] Nazaret | Caractérisation et modélisation d'un béton réfractaire renforcé de fibres métalliques[END_REF]. This phenomenon is notably very marked for fiber reinforced material [START_REF] Olesen | Fictitious crack propagation in fiber-reinforced concrete beams[END_REF]. But with some damage models as smeared crack model [START_REF] Rots | Crack models for concrete : discrete or smeared? Fixed, multidirectional or rotating?[END_REF] or Mazars model [8], it's not possible to reproduce non-linear behaviour before the peak stress. Nevertheless, this behaviour domain due to microcraking in the matrix is present for a lot of refractory castables. This domain is often larger at high temperature. To study the influence of this domain on the bending tests, two lists of material parameters determined arbitrarly have been tested. The tensile behaviour established with both sets of material property is represented on the Fig. 3. The softening part is strictly similar for the both data in terms of permanent strains and damage. But for the set numerated 1, none permanent strain is considered before the peak stress. Both curves correspond to classical behaviour in tension of refractory castables. Damage increases quickly after crack initiation.

Four points bending tests have been simulated with these material parameters in 2D modelling imposing a displacement of the upper supports. The behaviour is considered elastic close to contact with support to avoid local effects [START_REF] Nazaret | Influence of the mechanical behaviour specificities of damaged refractory castables on the Young's modulus determination[END_REF]. The load-deflection curve is represented on the Fig. 4. In this figure, stress is calculated using elastic approach. This method enables to compare responses of the both sets of material parameters. For the curve without damage during pre-peak domain, it can be observed that the maximal strength corresponds to the tress level of initiation crack in tension. While the damage is introduced before the peak stress the apparent strength is increased by 80%. At the beginning of the damage apparition at 1.5 MPa, the damage repartition is really diffuse and the strain localization occurs later (Fig. 6). The damage plasticity model allows reproducing non-linear behaviour which is often neglected. This domain has a strong influence on the macroscopic behaviour during bending tests. Now, the influence of both sets of material data is going to be used on an industrial refractory part.

Industrial Cases

Description. This part deals with several simulations performed on a gas burner with a diameter of 400mm. The structure is made of two refractory castables part. Four injectors are located in the lower refractory block. The assembly is disposed in a steel case (Fig. 7). This structure is submitted to severe thermal gradients induced by convection and heat flux. Convection is imposed by air temperature on all exterior faces. Heat fluxes concern areas close to injectors and in the chimney. A temperature level of 1350°C is imposed on the bottom face. The temperature field is calculated from all these thermal boundary conditions by finite element analysis for a steady-state. All parts are tied. Thermal gradients are in the 32°C to 1360°C temperature range (Fig. 8). Damage modeling. The thermal field previously calculated has been used to perform simulations using the damage plasticity model. During this thermomechanical simulation, stress level remains low. This is due to crack initiation and propagation phenomena. Damage initiates around holes and propagate up the bottom face and the external faces of the refractory castable part. Without damage before the peak stress, cracks propagate quickly and the damage level reaches the high value of 0.8 for the four main cracks (Fig. 9).

With the second material parameters integrating damage before peak stress, the final value of damage is lower (Fig. 10). Cracks propagate for only two holes. The structure supports larger strains. With these simulations on a complex structure, the important rule of non-linear before the peak stress has been underlined.

Conclusion

A damage plasticity model has been used to model damage behaviour in refractory castable. This model implemented in Abaqus ® software allows integration of permanent strains before and after the peak stress. This capability gives the possibility to take in account non-linear behaviour classically observed experimentally before the peak stress. This data is often neglected by damage model. Four points bending tests have been simulated with two material parameters sets with and none damage before peak stress. These simulations have shown that the maximal load supported by the sample is increased while the non-linear behaviour before the peak stress is considered. This tendency is reinforced by simulations performed on a complex part in refractory castable submitted to several thermal gradients. By considering damage before the peak stress, the crack propagation is less marked and damage level is lower. So, in order to correcly size refractory structure, it's crucial to integrate damage behavior to take in account stress redistribution and to calculate realistic stress levels. The integration of non linear behaviour is also an important issue to not underestimate load supported by a structure.

  tension and compression are characterized independently by two hardening variables, pl t ε ~and pl c ε ~. They are respectively referred to as equivalent plastic strains in tension and in compression. The evolution of the hardening variables can be expressed as follow :

Fig. 1 :Fig. 2 :

 12 Fig. 1 : Response of concrete to uniaxial loading in tension (a) and in compression (b)

Fig. 3 :Fig. 4 :Fig. 5 :Fig. 6 :

 3456 Fig. 3 : Stress-strain curves in tension with material parameters sets

Fig. 7 :Fig. 8 :

 78 Fig. 7 : Design of the gas burner. The studied refractory part is bottom.

Fig. 9 : 1 Fig. 10 :

 9110 Fig. 9 : Damage field in th gas burner with material parameters set 1 Fig. 10 : Damage field in th gas burner with material parameters set 2

Acknowledgment

Authors gratefully acknowledge Loïc LeBars and Optimise society for their technical supports.


Joe Lorkowski

Olga Kosheleva
email: olgak@

Vladik Kreinovich
email: vladik@utep.edu

Sergei Soloviev'
email: sergei.soloviev@irit.fr

How Design Quality Improves with Increasing Computational Abilities: General Formulas and Case Study of Aircraft Fuel Efficiency

Keywords: design quality, computational abilities, air craft fuel efficiency

It is known that the problems of optimal design are NP-hard -meaning that, in general, a feasible algo rithm can only produce close-to-optimal designs. The more computations we perform, the better design we can produce. In this paper, we theoretically derive quantitative formulas describing how the design qual ities improves with the increasing computational abil ities. We then empirically confirm the resulting the oretical formula by applying it to the problem of air craft fuel efficiency.

Formulation of the Problem

Design objective is to produce an optimal design. Start ing from 1980s, computers have become ubiquitous in en gineering design; see, e.g., [1][START_REF]International Council on Systems Engineering (INCOSE)[END_REF][START_REF] Laplante | Requirements Engineering for Software and Sys tems[END_REF][START_REF] Madsen | Engineering Drawing and De sign[END_REF]. An important break through in computer-aided design was Boeing 777, the first commercial airplane which was designed exclusively by using computers; see, e.g., [START_REF] Sabbagh | Twenty-First-Century Jet: The Making and Market ing of the Boeing 777[END_REF].

The main objective of a computer-aided design is to come up with a design which optimizes the corresponding objective function -e.g., fuel efficiency of an aircraft.

Optimization is, in general, NP-hard. The correspond ing optimization problems are non-linear, and non-linear optimization problems are, in general, NP-hard; see, e.g., [START_REF] Kreinovich | Computational Complexity and Feasibility of Data Processing and Interval Com putations[END_REF]7]. This means that -under the belief of most computer scientists that P,,eNP -a feasible algorithm can not always find the exact optimum; see, e.g., [START_REF] Garey | Computers and Intractability: A Guide to the Theory of NP-Completeness[END_REF]9]. In general, we can only find an approximate optimum.

Problem. The more computations we perform, the better the design. It is desirable to come up with a quantitative description of how increasing computational abilities im-prove the design quality.

Analysis of the Problem and the Derivation of the Resulting Formula

Because of NP-hardness, more computations simply means more test cases. In principle, each design opti mization problem can be solved by exhaustive searchwe can try all possible combinations of parameters, and see which combination leads to the optimal design. This approach may work if we have a small number of parame ters, then we can indeed try all possible combinations. If, on average, we have C possible values of each parameter, then:

• we need to compare C test cases when we have only one parameter,

• we need C 2 test cases when we have two parameters,

• and we need C 3 test cases when we have three pa rameters.

In general, when we have d parameters, we need to an alyze C d test cases. For large systems (e.g., for an air craft), we have thousands of possible parameters, and for d "' 10 3 , the exponential value c a exceeds the lifetime of the Universe. As a result, for realistic d, instead of the exhaustive search of all possible combinations of param eters, we can only test some combinations.

NP-hardness means, crudely speaking, that we cannot expect optimization algorithms to be significantly faster than this exponential time C d . This means that, in effect, all possible optimization algorithm boil down to trying many possible test cases.

When compntational abilities increase, we can test more cases. From this viewpoint, increasing computa tional abilities mean that we can test more cases. Thus, How can we describe this in quantitative terms? by fi nd in a better creasing the design. scope of our search, we will hopefully How to describe quality of an individual design. Since we cannot do significantly better than with a simple search, the resulting search is not well directed, we can not meaningfully predict whether the next test case will be better or worse -because if we could, we would be able to significantly decrease the search time.

The quality of the next test case -i.e., in precise terms, the value of the objective function corresponding to the next test case -cannot be predicted and is, in this sense, a random variable.

Many different factors affect the quality of each indi vidual design. It is known that, under reasouable condi tions, the distribution of the resulting effect of several iu dependent random factors is close to Gaussian; this fact is known as the Central Limit Theorem; see, e.g., [10]. Thus, we can conclude that the quality of a (randomly selected) individual design is normally distributed, with some mean µ and standard deviation cr.

What if we test n possible designs. After computation, we select the design with the largest value of the objective function. Let n denote the number of designs that our pro gram tests. If x; denotes the quality of the i-th design, then the resulting quality is equal to x = max(x, , ... ,x n )-We know that the variables x; are independent and identically normally distributed with some mean µ and standard de viation µ. What is the resulting probability distribution for the quality x? What is the expected value of this qual ity?

To answer this question, let us first reduce this ques tion to its simplest case of a standard normal distribution, with µ = 0 and cr = I. It is know that a general nor mally distributed random variable x; can be represented as x; = µ + cr • y;. Since adding µ and multiplying by a positive constant cr > 0 does not change which of the val ues are larger and which are smaller, we have

x = max(x1 1 ••• ,x n) = max(µ+ cr-y,, ... ,µ + <5 • Y n ) = µ + cr-y, def ( ) where y = max YI,•••,Y n .
For large n, the max-central limit theorem [START_REF] Beirlant | Statistics of Extremes: Theory and Applications[END_REF][START_REF] De Haan | Extreme Value Theory: An Introduc tion[END_REF][13][START_REF] Gumbel | Statistics of Extremes[END_REF] (also known as Fisher-Tippet-Gnedenko Theorem) says that the cumulative distributive function F (y) for y is approxi mately equal to F( y ) ""FEv (y::n), where: FEv(Y ) � exp ( -exp ( -y )) is known as the Gumbel distribution, µn �<I>-l (1-�), <5n �<I>-1 (r-�-e-1 )-<I>-1 (1-�), and <I>-1 (t) is the inverse function to the cumulative distri bution function <I>(y) of the standard normal distribution (with mean 0 and standard deviation 1). In other words, the distribution of the random variable y is approximately equal to the distribution of the variable � + crn • !; , where !; is distributed according to the Gumbel distribution.

It is known that the mean of the Gumbel distribution is equal to the Euler's constant y"" 0.5772. Thus, the mean value mn of y is equal to µn + y-crn. For large n, we get asymptotically

m n ~ Y• ✓21n(n),
hence the mean value en of x = µ + cr •xis asymptotically equal to en~µ+ cr-y-✓21n(n ) .

Resulting formula. When we test n different cases to fi nd the optimal design, the quality en of the resulting design increases with n as en~µ+ cr-y-✓21n(n).

Case Study of Aircraft Fuel Efficiency Confirms the Theoretical Formula

Case study: brief description. As a case study, let us take the fuel effi ciency of commercial aircraft; see, e.g., [START_REF] Lee | Historical and future trends in aircraft performance, cost and emissions[END_REF][START_REF] Peeters | Fuel Efficiency of Commercuial Aircraft: An Overview of Historical and Future Trends[END_REF][17]. It is known that the average energy effi ciency E changes with time T as E =exp(a+b-ln ( T)) = C-T b , for b"" 0.5.

How to apply our theoretical formula to this case?

The above theoretical formulas describes how the qual ity changes with the number of computational steps n. In the case study, we know how it changes with time T. So, to compare these two formulas, we need to know how the number of computational steps which can be applied to solve the design problem changes with time T. In other words, we need to know how the computer's computa tional speed -i.e., the number of computational steps that a computer can perform in a fi xed time period -changes with time T.

This dependence follows the known Moore's law, ac cording to which the computational speed grows expo nentially with time T: n"" exp(c • T) for some constant c. Crudely speaking, the computational speed doubles every two years; [START_REF] Hennessy | Computer Architecture: A Quantitative Approach[END_REF][START_REF] Moore | Cramnting More Components onto Integrated Cir cuits[END_REF].

Applying the theoretical formula to this case study. When n ""exp(c • T), we have ln(n) ~ T. Thus, the de pendence en~µ+ cr-y-✓21n(n) of quality q � en on time takes the form q""a+b•VT. This is exactly the empirical dependence that we actually observe.

Thus, the empirical data confirm the above theoretical formula.

Comment. It is important to be cautious when testing the formula. For example, in a seemingly similar case of cars, the driving force for their fuel efficiency is not computer design but rather federal and state regulations which pre scribe what fuel efficiency should be. Because of this, for cars, the dependence of fuel efficiency on time T is deter mined by the political will and is, thus, not as regular as for the aircraft.

Acknowledgements This work was supported in part by the National Science Foun dation grants HRD-0734825 and HRD-1242122 (Cyber-ShARE Center of Excellence) and DUE-0926721, and by Government of the Russian Federation Grant 074-U0l. The authors are thank ful to the organizers of the International Interdisciplinary Confer ence Philosophy, Mathematics, Linguistics: Aspects of Interac tion 2014 PhML'20!4 (St. Petersburg, Russia, April 21-25, 2014) -this inter-disciplinary conference brought the authors together and started this research, and also to the organizers and partici pants of the International Symposium on Management Engineer ing ISME'2014 (Kitakyushu, Japan, July 27-30, 2014), where these results were first presented. The authors are also thankful to the anonymous referees for valuable suggestions.

Name:

Olga Kosheleva Affiliation: University