Long lifespan in a population of Microtus (Pitymys) duodecimcostatus

E. PARADIS and G. GUÉDON Laboratoire d'Eco-éthologie, Institut des Sciences de l 'Evolution,Case Courrier 64,Université Montpellier II,Place Eugène Bataillon,France Rodents of the family Arvicolidae (voles and lemmings) have a short lifespan in natural conditions. It is not clear whether this feature is related to cyclic fluctuations of population densities as has been observed in some populations; [START_REF] Heske | Why spacing behavior does not stabilize density in cyclic populations of microtine rodents[END_REF] state that « Microtine rodents have a lifespan of less than a year, and average survivorship may be on the order of a few weeks to a few months ». Individuals surviving through two winters are occasionally recorded in cyclic populations (Viitala 1977, Innes and[START_REF] Innes | Numbers of litters, litter size and survival in two species of microtines at two elevations[END_REF]. We report herein the existence of long-lived individuals in a population of Microtus (Pitymys) duodecimcostatus and we compare our results to the data on arvicolid lifespans available in the literature.

A population of M. duodecimcostatus has been live-trapped since 1989 (Guédon et al. 1992). In February 1989, 429 voles were marked and released; the individuals with the maximum recorded lifespan were 2 males recaptured in November 1991 (33 months). The maximum lifespan recorded among females was 27 months (one female marked in August 1990 and recaptured in November 1992). Several individuals more than two years old were regularly trapped between February 1989 and November 1992 during all seasons.

Maximum lifespan is seldom mentioned in publications on arvicolid population ecology, so few comparisons are available in spite of the abundant literature relative to capture-recapture studies of vole and lemming populations. Nevertheless, two observations can be made from Table 1. First, maximum lifespan does not exceed 2 years in natural conditions for voles, except for Microtus duodecimcostatus. Second, when data are available for both sexes, females have a higher lifespan than males, with the exception of M. duodecimcostatus. This species did not show female-biased philopatry as observed in Microtus spp. [START_REF] Boonstra | -Natal philopatry and breeding systems in voles (Microtus spp.)[END_REF].

Microtus duodecimcostatus has particular features among the Arvicolidae: a long lifespan in natural conditions, a shorter lifespan for females than for males, a small litter size (Guédon et al. 1991), and a small body size (Guédon et al. 1992). These unusual life-history traits and the fact that vole population fluctuations are almost unknown under the Mediterranean climate in Europe, underline the contribution of our current investigations on the population dynamics of this vole to a general explanation of cyclic fluctuations in arvicolid populations.

TABLE 1 .

 1 -Maximum lifespan of several arvicolid species in natural conditions.

	species	reported lifespan	references
	Microtus agrestis	20 months	Myllymäki 1977
	Microtus arvalis	< 2 years a	Frank 1957
		12 months	Dassonville 1978
	Microtus pennsylvanicus 12 months (fem.), 7.9 months (mal.)	Sheridan and Tamarin 1988
	Microtus townsendii	20.5 months (fem.), 17.7 months (mal.) Beacham 1980
	Microtus californicus	14 months (fem.), 9 months (mal.)	Cockburn and Lidicker 1983
	Microtus montanus	18 months (fem.)	Negus et al. 1986
	Microtus nivalis	13 months	LeLouarn and Janeau 1975
	Microtus guentheri	5 months	Cohen-Shlagman et al. 1984
	Microtus xanthognathus 18 months	Wolff and Lidicker 1980
	Pitymys multiplex	21 months b	Dassonville 1978
		16 months b	Salvioni 1986
	Pitymys savii	20 months b	Salvioni 1986
	Pitymys subterraneus	13 months b	Salvioni 1986
	Clethrionomys glareolus < 2 years a	Wiger 1979
	Clethrionomys gapperi	12 months	Miller and Getz 1977
	Arvicola terrestris	24 months	Saucy 1988

a : No individual surviving more than one winter was observed by the author. b : Dassonville's and Salvioni's studies were too short to estimate the maximum lifespan of these Pitymys species.


T Branza

F Deschaux-Beaume

V Velay

P Lours

A microstructural and low-cycle fatigue investigation of weld-repaired heat-resistant cast steels

Keywords:

The multi-pass weld-repair of heat-resistant cast steels is carried out using an automated shielded metal arc welding (SMAW) process, with various filler materials and pre-heating at 400 • C. Specimens weld-repaired with a filler material more resistant than the heat-resistant cast steel (over-matching) generally crack within the base metal following the tenth filling pass, whereas specimens buttered with a soft alloy prior to welding remain free of cracks.

The high temperature strain-controlled fatigue lifetime of material weld-repaired without buttering is lower than that of bulk initial material. This is due to an increase of the stress amplitude as a result of the so-called over-matching. In the case of material welded following a prior buttering, the fatigue lifetime is reduced because of the stress tri-axiality generated in the thin soft layer which prevents its plastic flow. As a consequence, it is concluded that even though buttering prevents cracking efficiently during welding, it is not acceptable as far as fatigue performance, especially lifetime, is concerned.

1.

Introduction

Heat-resistant cast austenitic stainless steels are good candidates for manufacturing high temperature components, because of their high creep strength, outstanding corrosion resistance and moderate cost compared to nickel-base superalloys. They have been used for the manufacture of pyrolysis tubes [START_REF] De Almeida | Microstructural characterization of modified 25Cr-35Ni centrifugally cast steel furnaces tubes[END_REF] or steam-reformer tubes [START_REF] Ray | Analysis of failed reformer tubes[END_REF] for the petrochemical industry since the early 1960s, and more recently for making superplastic forming (SPF) tools [START_REF] Montagnon | Application of thermo-calc to the developments of new heat-resistant alloy casting for specifics SPF-DB tooling[END_REF]. However, heat-resistant cast steels suffer from very low ductility, resulting in two major problems. First, such alloys are highly sensitive to cracking induced by thermal cycles [START_REF] Bhaumik | Failure of reformer tube of an ammonia plant[END_REF]. This type of damage is currently observed in SPF tools subject to severe thermal cycling between room temperature and 900 • C or even more (Branza et al., 2004a). Then, the welding or required weld-repair to extend the service life of those steels often provokes cracking, especially for massive bulk components [START_REF] Haro | Microstructural factors that determine the weldability of a high Cr-high Si HK 40 alloy[END_REF][START_REF] Haro | Study of weldability of a Cr-Si modify heat-resistant alloy[END_REF][START_REF] Haro | Microstructural analysis of heat-resistant welded pipes[END_REF].

It is generally assumed that a minimum tensile elongation of 4% at room temperature is required to prevent weldcracking [START_REF] Colwell | Weld cracking in modified heat-resistant castings, a microstructural investigation[END_REF]. If the material ductility is lower, the welding or weld-repair needs some precautions. A solution annealing treatment, promoting the dissolution of secondary carbides, can enhance the ductility prior to welding. However, the carbide dissolution temperature is generally higher than 1200 • C [START_REF] Wu | The eutectic carbides and creep ruptures strength of 25Cr20Ni heat-resistant steel tubes centrifugally cast with different solidification conditions[END_REF], so such a treatment is often too expensive, especially for large bulk structural parts. Another way to limit the dramatic consequences of the intrinsic material brittleness is the optimisation of the welding parameters and the use of a buttering technique [START_REF] Ebert | Fabrication of HK-40 in the field[END_REF]. It consists in depositing a soft filler metal over the surface of the steel in order to interpose a layer better able to accom- Whatever the technique used, the welding of heat-resistant cast steels has some detrimental effects on the material properties. If the microstructural changes of such steels after welding have been previously investigated [START_REF] Haro | Microstructural factors that determine the weldability of a high Cr-high Si HK 40 alloy[END_REF][START_REF] Haro | Study of weldability of a Cr-Si modify heat-resistant alloy[END_REF], the effect on the mechanical properties, and especially on the fatigue behaviour, is rarely studied.

It is however well known that the welding of steels can greatly change their fatigue behaviour. One of the main parameters affecting the fatigue strength is the mis-matching ratio, generally defined as the ratio M between the yield strengths of the filler metal and the base metal (Ravi et al., 2004a(Ravi et al., ,b,c, 2006)). Under-matched joints (M < 1) are generally used for repair-welding [START_REF] Ravi | Influences of post weld heat treatment on fatigue life prediction of strength mis-matched HSLA steel welds[END_REF] or for the assembling of brittle materials, so as to concentrate the major part of the plastic strain in the welding zone during welding. On the other hand, over-matching is recommended for ductile materials. In such a way, the welding zone is less sensitive to the presence of flaws (porosities, inclusions) inherent to the welding process [START_REF] Vojvodic Tuma | Analysis of the unstable fracture behaviour of a high strength low alloy steel weldment[END_REF].

Numerous investigations on the effect of the mis-matching on the toughness [START_REF] Angamuthu | Investigation of dynamic fracture toughness (JId) behaviour of strength mis-matched Q & T steel weldments using instrumented Charpy impact testing[END_REF] or the fatigue properties [START_REF] Zhang | Influence of weld mis-matching on fatigue crack growth behaviors of electron beam welded joints[END_REF] of welded alloys have been carried out. Generally speaking, cracks preferentially propagate within the softest zone (welding zone or heat affected zone of the base metal according to the mis-matching), at room or high temperature [START_REF] Saxena | Role of nonlinear fracture mechanics in assessing fracture and crack growth in welds[END_REF]. The fatigue-crack growth-resistance of the welding zone has also been observed to be enhanced for over-matching, and cracks then deviate in the base metal [START_REF] Anand | Fatigue behaviour of tailor (laser)-welded blanks for automotive applications[END_REF]. These results can be understood considering that the plastically strained zone, which is the driving force for crack propagation, is larger at the crack-tip in a soft material than in a hard material. An unsymmetrical crack-tip plastic zone is then formed when the cracks are located in the welding zone close to the interface with the base metal and the deviation occurs in the side where the plastic zone is larger [START_REF] Kim | Numerical analyses of strength mis-match effect on local stresses for ideally plastic materials[END_REF].

Another important parameter affecting the fatigue strength of welded structures is the presence of flaws and residual stresses. The fracture of over-matched welded samples under cyclic tensile loading in the welding zone has been attributed to the presence of flaws and unfavourable stress state in this region [START_REF] Borrego | Fatigue behaviour of laser repairing welded joints[END_REF].

The mis-matching effect can be also responsible for a heterogeneous distribution of stresses and strains in the mechanically loaded welded samples, as reported by [START_REF] Boronski | Cyclic material properties distribution in laser-welded joints[END_REF], which generates stress concentrations. In the case of high temperature constant isothermal loading, stress concentrations occur due to the difference of creep strain rate in the base metal and the welding zone [START_REF] Chellapandi | Influence of mis-match of weld and base material creep properties on elevated temperature design of pressure vessels and piping[END_REF].

The present article focuses on the weld-repair of a heat-resistant cast austenitic stainless steel used for the manufacturing of SPF tools. Multi-pass weld-repair tests are carried out with various operating conditions and the weld-repaired specimens are investigated. Both the microstructural features and the low-cycle fatigue behaviour are analysed.

Materials

Base material

The base material studied is a heat-resistant cast austenitic stainless steel corresponding to the grade GX30NiCr3924 according to the EN 10027 European Norm. The chemical composition of the steel is given in Table 1. The high carbon content (0.31 wt.%) promotes the precipitation of a rather large amount of carbides during the solidification (primary carbides) or during ageing treatments (secondary carbides), which enhances the creep strength.

The mechanical properties of the base material are given in Table 2. The room temperature yield strength is rather low, but it only decreases slightly with temperature remaining rather high at 500 • C. Note that the elongation to rupture is extremely low in the whole temperature range from room temperature to 500 • C.

Filler materials

Two following types of filler materials, chosen for their specific properties, are used in this study:

• soft filler materials for the buttering, in order to form a ductile layer between the base metal and the filling metal allowing the release of thermal stresses through plastic strain;

• heat-resistant filler materials for the filling of the V-groove, to maintain the high temperature properties of the base material.

The chemical compositions of all filler materials are given in Table 1. Two different buttering alloys (referred to as BA) are used. They both have an austenitic structure, so as to be metallurgically compatible with the base metal, and a low yield strength at room temperature and high temperature, to promote easy stress release. The first one, Safonte (SAF) is supplied in the form of shielded electrodes with 2.5 mm core diameter, and the second one, Invar (Imphy Alloys) in the form of 2-mm diameter welding wires.

Two different filling materials (referred to as FM), supplied in the form of shielded electrodes with 2.5 mm core diameter, are used (Table 1). The first one, designated by the commercial reference 2133Mn (B ölher Thyssen), shows Ni and Cr contents rather close to those of the base material, with lower C content, and higher Mn and Nb contents. The second one, designated by 6222Mo (B öhler Thyssen), has a chemical composition corresponding to a 625 alloy (UNS Designation NO6625). It is a typical filler alloy currently employed for the welding of heat-resistant austenitic steels or nickel-base alloys [START_REF] Bohler Thyssen | Welding guide[END_REF].

Experimental

Welding process

The welding samples consisted of bulk 400 mm × 160 mm × 120 mm parallelepiped blocks, representative of industrial SPF tools. A 20-mm deep V-groove (60 • ) was machined on the two larger sides of the specimens, corresponding to a typical preparation prior to the weld-repair of a cracked component.

An automatic shielded metal arc welding (SMAW) process, allowing the control of both the welding rate and the arc voltage, with a Lincoln PW 455 STT power source, was used for the filling with 2133Mn and 6222Mo, and for the buttering with Safonte. Buttering with Invar was carried out using a manual gas tungsten arc welding (GTAW) process, with a Kemppi Master 2850 power source, in DC electrode negative (DCEN) mode. The welding parameters, i.e. current, voltage and welding rate were optimised for each couple base metal/filler metal (Table 3).

Prior to welding, the base metal samples were systematically pre-heated uniformly at 400 • C using heat resistances, in order to reduce the thermal gradients and stresses during welding. A delay of 5-10 min was respected between two passes, in order to make the temperature field uniform after each pass. In this way, the interpass temperature is below 430 • C.

The weld-repair tests were done without buttering using the two filling materials, with a partial filling of the V-groove (10 or 16 passes), or a complete filling (40 passes). Except for the first three passes, where the arc tension was slightly higher, the same energy was used for all the weld passes of one assembly (Table 3). Weld-repair tests with prior buttering with Safonte or Invar, and a complete filling with 6222Mo were also carried out.

Microstructural investigations

Following welding, microstructural investigations were carried out on cross-sections of the welding specimens using a Philips XL30 scanning electron microscope (SEM) with an EDAX energy dispersive spectrometer (EDS). In order to quantify the extent of cracking for each welded sample, the total crack length was measured by image analysis (Aphelion software) from low magnification (30×) SEM micrographs taken over the whole cracked zone. Three cross-sections were analysed for each welded sample. Vickers (500 g) micro-hardness was also measured on the different zones of the cross-sections. Vickers microhardness profiles were obtained along lines crossing the base material-fusion zone interface, and if present the buttering layer, with a spacing of 100 m. However, in order to maintain a 300 m distance between two indents, to prevent the interaction between the plastic zones of two neighbouring indents, the indents were made by "zigzagging" along the profile line.

Fatigue tests

The main objective of the fatigue tests performed in this study was to evaluate the effect of weld-repair on the in-service lifetime of SPF tools, which are subject to thermal and mechanical cycles. In addition, some indications concerning the behaviour of the materials during welding, especially the propensity to cracking according to the type of filler material used, were also expected from these tests.

Previous works have shown that the main damage factor of SPF tools is related to the thermal cycling imposed to the dies [START_REF] Baleix | Surface and image analysis of oxides grown and spalled on heat resistant cast steels exposed to thermal cycles[END_REF][START_REF] Baleix | Oxidation and oxide spallation of heat resistant cast steels for superplastic forming dies[END_REF]. A strain-controlled fatigue test was selected in order to simulate the mechanical strain resulting from the thermal strain gradient due to thermal cycles. Strain-controlled fatigue is also convenient to investigate the behaviour of samples during multi-pass welding, each welding pass generating a rapid thermal cycle. The strain amplitudes were selected to be as representative as possible of a typical strain cycle of an industrial SPF operation [START_REF] Baleix | Surface and image analysis of oxides grown and spalled on heat resistant cast steels exposed to thermal cycles[END_REF][START_REF] Baleix | Oxidation and oxide spallation of heat resistant cast steels for superplastic forming dies[END_REF], which is in the same order of magnitude as the strain level induced by a welding pass. Three strain amplitudes, 0.4, 0.6 and 0.8%, resulting in a slight plastic strain at each cycle, were chosen.

Low-cycle fatigue tests were carried out using a 250 kN electro-hydraulic fatigue machine (MTS) and specimens were heated in a resistance furnace. The strain in the samples was controlled with a water-cooled 21-mm gauge length extensometer. Because of the high temperature achieved for specimen testing, special grips in Udimet 720 superalloy were used. Two test temperatures, typical of the SPF process, i.e. 750 and 920 • C, were selected. It is also in this temperature range that the mechanical resistance of the heat-resistant cast steels decreases somewhat. The strain rate was 10 -4 s -1 for all tests.

The fatigue specimens (150 mm length and 12 mm diameter) were machined from the welded blocks free of cracks, in order to obtain an unsymmetrical sample containing the weldroot (Fig. 1). In such a way, it is possible to observe structural effects due to the heterogeneous and unsymmetrical characteristics of a weld-repaired component. Two fatigue samples were prepared for each condition. Before machining, the weldrepaired blocks were annealed 20 h at 950 • C with slow heating and cooling rates (5 • C/min), to promote residual stress relief. This phenomenon of stress release is supposed to occur on SPF tools during their first in-service heating after weld-repair. In addition, this treatment prevents the samples from distorting during machining. Due to the great thermal stability of the base material, this annealing treatment has no effect on the grain size, but should have a slight effect on the precipitation and growth of secondary carbides (Branza, 2005).

4.

Results of weld-repairs

Microstructural observations

SEM micrographs in Fig. 2 show the base material microstructure. It is a typical dendritic structure (Fig. 2a), with coarse primary arms (about 200 m width). EDS analysis reveals the presence of two different phases in the interdendritic regions (Fig. 2b), a major chromium-rich phase, in dark contrast, and a minor niobium-rich phase, in bright contrast. This indicates the formation of two types of primary carbides (possibly Cr 23 C 6 and Nb 23 C 6 according to EDS analysis) at the end of the solidification, in the interdendritic regions. A fine precipitation of secondary carbides is also observed in the dendrite core (Fig. 2).

When the number of passes is above or equal to 16, the cross-sections of weld-repaired samples without buttering reveal the presence of cracks in the base metal, close to the fusion zone. Fig. 3 shows the evolution of the total crack length measured by image analysis versus the number of passes for the two different filling materials. The first cracks appear between the tenth and the sixteenth passes. Following this threshold, initial cracks tend to extend and/or new cracks initiate and propagate until the filling of the V-groove is completed. This indicates a progressive increase of the maximum stress with the increase of the number of passes, which could reach the failure strength of the material, or a cyclic damage phenomenon induced by the successive heating and cooling, the 5-10-min delay between two passes allowing the cooling of the welded area. Note that cracking is much more important with the filling material 2133Mn, than with 6222Mo.

The cracks formed during welding are always located in the base metal, very close to the fusion zone (Fig. 4), and propagate across the carbide-rich interdendritic regions. On the other hand, prior buttering is shown to prevent weld-induced cracking completely (Fig. 5).

Micro-hardness tests

Fig. 6 shows the Vickers hardness profiles obtained over the cross-sections of weld-repaired samples, with or without buttering. The hardness profile in the base metal is widely scattered because of the heterogeneous microstructure of the cast steel, constituted of a quite soft austenitic matrix surrounded by interdendritic hard carbide-rich regions. The hardness of the welding zone for the two filling materials is higher than that of the base metal, suggesting an overmatching at room temperature. So this supposed higher yield strength of the filling materials at room temperature promotes the stress increase during cooling after welding and, as a consequence, the resulting cracking of the base metal. However, the difference in room temperature hardness between the base and the filling materials is not sufficient to explain the propensity to cracking satisfactorily, because the hardness of the welding zone with 6222Mo is higher than with 2133Mn, whereas the total crack length in the base material is lower with 6222Mo (Fig. 3). Note that due to the great thermal stability of heat-resistant cast austenitic stainless steels, no heat affected zone (HAZ) is observed, so the hardness, as well as the microstructure, of the base material is not changed close to the fusion zone.

For the weld-repaired specimens achieved with buttering, the hardness of the buttering layer strongly depends on the buttering alloy. With Invar, the buttered layer hardness is slightly lower than the mean hardness of the base material, whereas with Safonte, hardness is higher than that of the filling zone (with 6222Mo). The beneficial effect of buttering with Invar is then attributed to the under-matching effect at room temperature. But in the case of Safonte, the strong overmatching observed at room temperature does not explain the absence of cracks. However, due to the large temperature variations and thermal gradients observed during welding, it is clear that the mechanical behaviour of the materials at high temperature should have a great impact on the stress increase in the base material. It is then important to obtain data concerning the high temperature behaviour of the materials.

Fatigue tests

Two types of result are expected from the low-cycle fatigue tests carried out on homogeneous fatigue samples (without weld) and on weld-repaired samples, with or without buttering. First, the number of cycles before failure is an indication on the lifetime of repaired components which can be compared to bulk material. Secondly, the comparison of the stress amplitudes, for a fixed strain range (supposed to correspond to mechanical strain resulting from a thermal gradient during SPF cycle or multi-pass welding), of welded and unwelded samples can assess the beneficial or detrimental effect of a filler material.

Cyclic behaviour of the base metal and of the welded samples

At 750 • C, the stress amplitude ( /2) for a fixed strain range ( ε) of the base material (without weld) increases during the first cycles, indicating a hardening effect. At 920 • C, stabilisation is reached as soon as the first cycle is completed. However, wide scattering in stress range, reaching 20% for tests at 920 • C, is observed for samples tested in the same conditions. These differences are too important to be explained only by a certain scattering of the test parameters, so it can be assumed that they are due to the large grain size of the base metal (Fig. 1), which makes the plastic behaviour of the samples very sensitive to the grain orientation. Fig. 7 compares the evolution of the stress amplitude during the first 10 cycles of a fatigue test, respectively performed on homogeneous (base material) and welded samples without buttering, for a maximum strain amplitude of 0.4%. For the same strain, the welded samples, either filled with 2133Mn or 6222Mo, sustain higher stress range than the homogeneous material. This trend is confirmed at 750 • C for all fatigue tests whatever the maximum imposed strain is (Fig. 8). In contrast, at 920 • C the stress range for a fixed strain is just slightly higher than in homogeneous base material, in samples welded with 6222Mo, whereas rather similar stress amplitudes are observed in samples welded with 2133Mn (Fig. 8). Note that at 750 • C, the stress is higher, for the same strain, with 2133Mn filler metal, whereas at 920 • C, the stress is higher with 6222Mo.

Fig. 9 compares the stress amplitude sustained at various temperatures by the specimens without welding, the specimens welded with 6222Mo and the specimens welded following prior buttering with Invar. Despite the small number of fatigue tests carried out on samples welded with buttering (only three fatigue test conditions, with two samples per condition), it seems that buttering has no significant effect on the material behaviour.

Fatigue lifetime and localisation of the fracture

If the behaviour of the buttered and non-buttered samples during the first 10 cycles is rather similar, the localisation of fracture shows some significant differences. When the samples are not buttered, fracture generally occurs in the base metal, close to the fusion zone (Fig. 4), or in some cases in the centre of the sample, according to a specific path crossing the base metal and the fusion zone. In this last case, the fracture surface is only oxidised in the part crossing the base metal, which indicates that the crack initiates and progressively propagates within the base metal, the final catastrophic failure occurring in the fusion zone. In the samples buttered prior to welding, fracture takes place in the buttering layer after a low number of cycles (Fig. 10). Fig. 11 indicates the average number of cycles before failure of both the bulk material and the various welded samples. The bulk material shows a typical behaviour, the number of cycles before failure decreasing with the strain range. The welded materials show a much more complex behaviour: in most conditions, the number of cycles before failure increases with the strain range. This could be due to damage to the base material during welding, such as microcracking, which has not been detected by the microstructural analysis. The weld-repair with prior buttering with Invar does not show any significant difference with weld-repair without buttering at 750 • C. In contrast, it decreases drastically the number of cycles before failure at 920 • C. Note that only two fatigue samples were tested for each condition, which could explain the non-standard fatigue life behaviour observed on welded samples. Indeed, the possible presence of undetected microcracks in the samples induces wide lifetime scattering, which requires a more relevant statistical analysis, and then many more samples.

Discussion

Fatigue lifetime of weld-repaired components

The weld-repaired samples always exhibit a lower fatigue lifetime than the bulk non-welded material, both at 750 and 920 • C (Fig. 11). Consequently, it may be concluded that the sole verification of the absence of cracking over the investigated specimen, which is ensured by the deposition of a buttered layer prior to welding, does not guarantee acceptable fatigue strength.

In order to understand the failure mechanism during fatigue loading in heterogeneous welded samples better, a numerical simulation of strain-controlled fatigue test is performed using Abaqus software (Branza, 2005). Fig. 12 shows the typical stress distributions calculated in a welded sample, for a fixed strain range. The maximum stresses occur in the fusion zone. In addition, note that there is also, due to the mismatching effect, some stress concentration in the base metal, close to the interface with the welding zone and at the opposite side of the fusion zone. These regions strictly correspond to the zones where failure occurs in the samples.

However, the stress concentration due to the mis-matching between the welding zone and the base material (stiffness and yield strength differences) is not the only factor responsible for the strong decrease of the number of cycles before failure experimentally measured. Figs. 7 and8 show that, for the same strain, the stress amplitude is higher in the welded samples than in the bulk samples. Fig. 13 shows master S-N W öhler curves at 750 and 920 • C, plotting the stress amplitude versus the number of cycles to failure, including all experimental results obtained for bulk and welded materials. Note that the continuity between the data characteristic of the bulk and the welded material is rather good. This indicates that the high stress imposed on the welded specimens during straincontrolled fatigue loading could justify the lower number of cycles before failure assessed experimentally.

So the main detrimental effect resulting from welding is to enhance the stiffness of the samples, or of the structural parts in the case of industrial applications, which increases the stress amplitude when the material is subject to a fixed strain (induced by a thermal gradient for instance). A secondary effect is the generation of stress concentration due to the mis-matching between the welding zone and the base metal, which determines the location of fracture.

In the samples weld-repaired with prior buttering, the number of cycles before failure decreases compared to bulk samples or classically welded samples. In this case, the failure is systematically located in the buttering layer. However, for a given strain, the stress amplitude during the first cycles is in the same order of magnitude as that obtained for standard welding (Fig. 9). For the buttered specimens, a decrease of the stress amplitude is measured on the stress-strain loops after few cycles. This indicates a softening or more probably a damaging of the buttering layer, which in turn provokes its early cracking (Fig. 14). These results are in agreement with previous works, reporting the lower toughness of a softened HAZ confined between two harder zones (base metal and even-matching or over-matching welding zone) [START_REF] Angamuthu | Investigation of dynamic fracture toughness (JId) behaviour of strength mis-matched Q & T steel weldments using instrumented Charpy impact testing[END_REF]. This behaviour may result from an increase in the stress tri-axiality as the thickness of the soft layer decreases, thus preventing its plastic strain. This enhances the risk of cracking of this layer, in spite of the high ductility of the material [START_REF] Kim | Numerical analyses of strength mis-match effect on local stresses for ideally plastic materials[END_REF]. This also explains satisfactorily the establishment of stresses with similar amplitude, for a given strain, observed with or without buttering. In addition, the presence of a soft welding zone (under-matching) increases the crack-tip plastic deformation in this region and then favours the propagation of cracks [START_REF] Kim | Numerical analyses of strength mis-match effect on local stresses for ideally plastic materials[END_REF].

Weld-cracking

The mechanism of cracking of the base metal during welding is a typical solid-state cracking phenomenon due to the low ductility of the material. The propensity to cracking of the base metal depends on the stress level in this material. The two filling metals used show higher yield strength than the base material (over-matching), so they both promote the cracking of the base metal in isothermal conditions, as confirmed by isothermal fatigue test results. However, the problem is more complex during welding, because the strains and the temperatures are not uniform in the material. The filler material, which is deposited into the welding arc, is subject to a higher temperature than the base metal during its deposit and its cooling. The yield strength being lower when the temperature is higher, the difference between base metal and filler metal yield strength is then reduced. Consequently, the real over-matching effect during welding is lower than the over-matching in isothermal conditions. In addition, the nonuniform thermal strain established during welding creates a complex stress field compared to the uni-axial and isothermal fatigue loading. Despite these differences, it has been shown in a previous work that for a given over-matching, the higher residual stresses after welding occur in the fusion zone and in the base metal close to the interface, in the region where the thermal gradient is the highest (Branza et al., 2004b[START_REF] Branza | Influence of the welding processing on weld repair quality of SPF tools[END_REF] and where the cracks are experimentally observed in the welded samples without buttering. This unambiguously shows that the over-matching effect persists in welding conditions without buttering, with the filling materials used, despite the thermal gradient, so that the fatigue-cracking and the weld-cracking mechanisms are rather similar.

The similarities persist when the effect of both filling materials is compared. Alloy 2133Mn, which has the lower yield strength up to 900 • C (Fig. 15), produces a higher stress amplitude during strain-controlled fatigue tests of welded samples at 750 • C than alloy 6222Mo (Fig. 8). This could be due to an enhanced hardening effect in alloy 2133Mn, which is in agreement with the higher crack length observed after welding with this filling metal (Fig. 3).

Buttering samples prior to welding prevents weld-cracking, indicating a significant effect of the buttering on the magnitude of the stress field in the base material developed upon welding.

However, fatigue tests at 750 and 920 • C show a similar stress amplitude for fatigue specimens either prepared with or without buttering, which is due to the stress tri-axiality in the thin buttering layer confined between two harder regions.

The effects of the isothermal fatigue loading and the thermo-mechanical cycles due to a multi-pass welding on the failure mechanisms of buttered samples are rather different. This could have two main reasons. First, the thermal gradient during welding lowers the mis-matching between the base material, the buttering alloy and the filling metal, as previously discussed. Secondly, the geometry as well as the loading are very different during fatigue test and multi-pass welding. In this last case, the geometry of the specimen progressively changes with the filling of the V-groove, whereas the induced mechanical loading is non-uniform and more "localised" (around the deposited pass). Then, we can assume the stress tri-axiality in the buttering zone during welding is lower than during a fatigue test. This allows the plastic flow of the buttering alloy and a beneficial release of the stress in the base metal.

Conclusion

The multi-pass weld-repair of a heat-resistant cast steel used for the manufacturing of SPF dies was carried out with two filler materials, with or without prior buttering of the base material surface. The non-buttered samples systematically crack after the tenth filling pass whereas the buttered samples remain free of cracks up to the deposition of the final fortieth pass. The strain-controlled fatigue lifetime of the heat-resistant cast steel is strongly reduced by the presence of a welding zone as a result of an increase of the stress amplitude due to the over-matching effect and to the presence of stress concentrations in the base material close to the interface with the welding zone.

In the case of the buttered samples (with Invar), the fatigue lifetime drastically decreases because of the occurrence of a tri-axial stress which inhibits the plastic strain of the soft buttering layer and enhances its cracking.

As a consequence, it is concluded that if the buttering technique efficiently prevents weld-cracking, it strongly limits the fatigue lifetime of the components. r e f e r e n c e s

Fig. 1 -

 1 Fig. 1 -Geometry of fatigue samples machined from weld-repaired blocks.

Fig. 2 -

 2 Fig. 2 -SEM micrographs of the base metal.

Fig. 3 -

 3 Fig. 3 -Effect of the type of filler metal and of the number of passes on the total crack length after welding.

Fig. 4 -

 4 Fig. 4 -Macrograph of the cracked weld-repaired sample, after a complete filling with 6222Mo, without buttering (a), and crack path across the primary carbides within the base metal (b).

Fig. 5 -

 5 Fig. 5 -Macrograph (a) and micrograph (b) of a weld-repair sample, after a complete filling with 6222Mo and a prior buttering with Invar.

Fig. 6 -

 6 Fig. 6 -Vickers micro-hardness profiles in the various zones of the weld-repaired sample cross-sections.

Fig. 7 -

 7 Fig. 7 -Cyclic behaviour of the base metal and of the welded samples, for a strain amplitude of 0.4%.

Fig. 8 -

 8 Fig. 8 -Stabilised stress amplitude during fatigue tests for various strain amplitudes and temperatures for the base material and the welded samples.

Fig. 9 -

 9 Fig. 9 -Comparison of the stabilised stress amplitude during fatigue tests on the base material, the welded samples with 6222Mo, and the welded samples with prior buttering with Invar.

Fig. 11 -

 11 Fig. 11 -Comparison of the number of cycles to failure for the various fatigue tests.

Fig. 12 -

 12 Fig. 12 -Axial stress field (in MPa) during fatigue test calculated by numerical simulation.

Fig. 13 -

 13 Fig. 13 -S-N W öhler curves showing the stabilised stress amplitude vs. the number of cycles to failure when the failure occurs in the base metal.

Fig. 14 -

 14 Fig. 14 -Evolution of the stress amplitude during fatigue tests of the samples welded with 6222Mo, with or without buttering with Invar.

Fig. 15 -

 15 Fig. 15 -Yield strength vs. temperature for the base and filling materials.

Table 1 -

 1 Chemical compositions (in wt.%) of the base material (BM), the filling metals (FM) and the buttering alloys (BA)

	El. wt.%	Fe	C	Cr	Ni	Mn	Si	P	S	Mo	W	Nb	V
	BM	Bal.	0.313	24.4	39.2	1.11	1.45	<0.001	0.0011	<0.02	< 0.002	0.328	0.038
	FM
	2133Mn 6222Mo	Bal. <1	0.1-0.17 0.03	20-22 32-34 3.5-5 22 63 0.8	≤0.6 0.3	0.005	0.005	9		1-1.5 3.5
	BA
	Safonte	48			52
	Invar	63			36	0.35	0.2
	modate weld-induced stresses, before carrying out the filling
	with a more resistant metal.

Table 2 -

 2 Mechanical properties of the base material

	Temperature ( • C)	Yield strength (MPa)	Ultimate tensile strength (MPa)	Elongation to rupture (%)
	20	220	335	2.0
	500	175	300	5.0
	920	77	140	35.0

Table 3 -

 3 Summary of welding parameters

	Filling	Buttering	Operation Current	Arc voltage	Welding	Filler wire	Number of	Time between
	material	alloy		(A)	(V)	speed (mm/s)	rate (mm/s)	weld passes	weld passes (min)
	2133Mn	None	Filling	63	24-25	2.7	-	10	5-10
	2133Mn	None	Filling	63	24-25	2.7	-	16	5-10
	2133Mn	None	Filling	63	24-25	2.7	-	40	5-10
	6222Mo	None	Filling	63	25-26	2.7	-	10	5-10
	6222Mo	None	Filling	63	25-26	2.7	-	16	5-10
	6222Mo	None	Filling	63	25-26	2.7	-	40	5-10
	6222Mo	Safonte	Buttering	75	23-25	2.7	-	18	5-10
			Filling	63	26	2.7	-	31	5-10
	6222Mo	Invar	Buttering	110	12	1.6	4-5	22	5-10
			Filling	63	26	2.7	-	31	5-10


Gilles Dusserre
email: gilles.dusserre@mines-albi.fr

Olivier Valentin

Fabien Nazaret

Thierry Cutard

Experimental and numerical investigation of the asymmetric primary creep of a fibre reinforced refractory concrete at 1200 • C

Keywords:

Tension/compression asymmetry is a key feature of the creep behaviour of ceramics at high temperature. This paper investigates the capability of Drucker-Prager creep model, based on a sensitivity of the equivalent creep stress to the pressure, to account for the creep behaviour of a refractory concrete subjected to compression and bending. A relevant calibration procedure is proposed. The analysis mainly focuses on the associativity of the creep flow. A profile analysis of a compressed sample allows concluding that the creep flow is dilatant and thus associated. A discussion on the neutral axis shift during bending creep is carried out with the model calibrated on the experiments.

Introduction

Modelling the creep behaviour is a major issue for any application involving a long-term exposure of ceramic parts to high temperature. In this framework, accounting for the well-known asymmetry of ceramics creep behaviour is a key feature, from all points of view: experimental characterization, modelling and parameter identification. The tension/compression asymmetric creep of ceramics at high temperature is generally attributed to preferential damage [START_REF] Rosenfield | Estimating tensile creep data from flexure data[END_REF] or cavitation [START_REF] Lim | A finite element model for asymmetric creep behavior of ceramics[END_REF][START_REF] Wereszczak | Asymmetric tensile and compressive creep deformation of hot-isostatically-pressed Y2O3-doped -Si3N4[END_REF] in tension, or to the motion of solid particles in a glassy phase during creep deformation [START_REF] Dryden | On the creep in ceramics due to a viscous grain boundary phase[END_REF].

Asymmetric creep was evidenced from tensile and compressive tests for many materials, including ceramics [START_REF] Wereszczak | Asymmetric tensile and compressive creep deformation of hot-isostatically-pressed Y2O3-doped -Si3N4[END_REF][START_REF] Kottada | The high temperature tensile and compressive deformation characteristics of magnesia doped alumina[END_REF]. However tensile tests on brittle materials turn out to be delicate at high temperature, whereas bending tests (which provide tensile stresses in part of the specimen) are more easily carried out. In many studies, bending was thus preferred to tension in order to assess the tensile behaviour and complete compression data [START_REF] Rosenfield | Estimating tensile creep data from flexure data[END_REF][START_REF] Blond | Modelling of high temperature asymmetric creep behavior of ceramics[END_REF].

The experimental advantage of bending is counterbalanced by the complexity arising in the interpretation of the results. Indeed, a non-uniform elastic stress field is instantaneously built up and then progressively modified by the non-uniform inelastic strain [START_REF] Warshaw | Deformation behavior of polycrystalline aluminum oxide[END_REF][START_REF] Timoshenko | Non-stationary stress distribution in a ceramic bending beam during constant load[END_REF]. Primary creep still increases this non-stationary behaviour. The stress field change during asymmetric creep implies that the neutral axis in bending shifts from the specimen mid-plane [START_REF] Chuang | Estimation of power-law creep parameters from bend test data[END_REF]. The location of the neutral-axis can thus be used as an indicator of creep asymmetry [START_REF] Wiederhorn | Creep rupture of siliconized silicon carbide[END_REF][START_REF] Chen | Improved analysis for flexural creep with application to SiAlON ceramics[END_REF][START_REF] Bird | Creep behavior of a zirconium diboride-silicon carbide composite[END_REF]. More recently, asymmetric creep evidences were obtained from bending tests monitored by full-field measurement methods: 3D-Digital Image Correlation (3D-DIC) at room temperature after interrupted tests [START_REF] Dusserre | Applicability of image correlation techniques to characterise asymmetric refractory creep during bending tests[END_REF] or during the test at high temperature [START_REF] Leplay | Analysis of asymmetrical creep of a ceramic at 1350 • C by digital image correlation[END_REF] by a global approach to 2D-DIC. Both studies have shown that the neutral axis shift mainly occurs at the beginning of the creep.

In many studies [START_REF] Blond | Modelling of high temperature asymmetric creep behavior of ceramics[END_REF][START_REF] Chen | Improved analysis for flexural creep with application to SiAlON ceramics[END_REF][START_REF] Lee | Tensile creep in ceramics using four-point bending test[END_REF][START_REF] Lee | Estimating tensile creep rate of ceramics from flexure data[END_REF][START_REF] Peras | The flexural creep of refractory materials[END_REF][START_REF] Talty | Determination of tensile and compressive creep behaviour of ceramic materials from bend tests[END_REF] a steady-state creep law is enough to model the material behaviour. In the case of materials that exhibit a pronounced primary creep, this assumption no longer holds since the primary creep controls the stress field change. The analysis of the bending experiment is therefore much more complex. Moreover, the specimen may be subjected simultaneously to stationary creep on the upper and lower faces and to primary creep in the vicinity of the neutral axis, where the creep strain remains low. A previous work [START_REF] Dusserre | Applicability of image correlation techniques to characterise asymmetric refractory creep during bending tests[END_REF] has shown that the behaviour of the studied material necessitates taking the primary creep into account.

Understanding the non-uniform and non-stationary stress field involved in bending creep necessitates appropriate modelling tools. Many uniaxial creep laws exist that relate the strain rate to the applied stress. Some authors propose to explicitly account for the More generally, tri-axial stress fields usually consider that the strain rate tensor derives from pseudo-potential, function of the stress field, and is proportional to a scalar function that describes the uniaxial creep law. This scalar function depends on the stress field through an equivalent stress. In this framework, another way to implicitly account for the asymmetry is to build an equivalent stress that intrinsically includes this effect, for instance by introducing a dependency to the first stress invariant.

Asymmetry in ceramics is often related to damage occurring in the tensile area [START_REF] Leplay | Identification of asymmetric constitutive laws at high temperature based on Digital Image Correlation[END_REF]. However, plasticity models with yield surface dependent on the first stress invariant proved to be able to describe the behaviour of concretes under compression [START_REF] Grassl | Concrete in compression: a plasticity theory with a novel hardening law[END_REF]. In particular, Drucker-Prager [START_REF] Drucker | Soil mechanics and plastic analysis or limit design[END_REF][START_REF] Drucker | Concept of path independence and material stability for soils[END_REF] yield surface is often used to model the behaviour of confined concretes [START_REF] Jiang | Identification of material parameters for Drucker-Prager plasticity model for FRP confined circular concrete columns[END_REF][START_REF] Jiang | Application of drucker-Prager plasticity model for stress-Strain modeling of FRP confined concrete columns[END_REF][START_REF] Arslan | Sensitivity study of the Drucker-Prager modeling parameters in the prediction of the nonlinear response of reinforced concrete structures[END_REF], and less frequently the creep behaviour of refractories [START_REF] Valentin | Mechanical engineering creep simulation of ceramic materials with the Drucker-Prager model[END_REF][START_REF] Jin | Influence of creep on the thermomechanical behavior of a rh-snorkel[END_REF].

An important consequence of these models is that the assumption of creep incompressibility is not consistent with the hypothesis of normality [START_REF] Drucker | Concept of path independence and material stability for soils[END_REF][START_REF] Drucker | Relation of experiments to mathematical theories of plasticity[END_REF] if the equivalent creep stress depends on the first stress invariant [START_REF] Tsvelodub | The Stability Postulate and Its Applications in Creep Theory of Metallic Materials[END_REF]. However, according to Vermeer and de Borst [START_REF] Vermeer | Non-associated plasticity for soils concrete and rock[END_REF], for materials such as soils, rocks and concretes, experiments have disproved the hypothesis of normality. It is therefore mandatory to investigate the associativity of the creep flow. Theoretical developments describing the creep asymmetry as a consequence of an anisotropic kinematic hardening, have been applied to secondary [START_REF] Voyiadjis | Modeling of secondary creep behavior for anisotropic materials with different properties in tension and compression[END_REF] or primary creep [START_REF] Zolochevsky | Theory of creep deformation with kinematic hardening for materials with different properties in tension and compression[END_REF] of metallic alloys. In the present study, only monotonic behaviours will be considered, and the strain hardening will be assumed to be isotropic (as it is the case for soils [START_REF] Drucker | Concept of path independence and material stability for soils[END_REF]). Indeed the creep strain rate is very small and the Bauschinger effect, if any, is expected to become significant very late. However, it is noteworthy that the neutral axis shift toward the compressed area due to asymmetric creep implies that an amount of material initially under compression will later undergo tension, as described in Ref. [START_REF] Dusserre | Applicability of image correlation techniques to characterise asymmetric refractory creep during bending tests[END_REF]. This phenomenon will not be considered in this paper and bending tests will be considered as monotonic.

The present study aims at contributing to understand the stress field change and the neutral axis shift while bending creep, by combining both experimental and simulation approaches.

It is proposed in this paper to investigate the capability of a Drucker-Prager creep model to describe the asymmetric behaviour of a fibre reinforced refractory concrete assuming an isotropic hardening. An original calibration method is proposed to identify the model parameters from compression and bending data considering two extreme assumptions: in the first case, the flow is assumed to be associated (and thus dilatant), whereas the second one assumes a non-dilatant flow (and thus non-associated). The validity of these assumptions is discussed by comparing the experimental results to the numerical simulations.

Material and methods

Material

The material under study is a commercial refractory castable reinforced with 2 wt.% of FeCrAl hooked-end fibres (length 25 mm, diameter 0.7 mm). The concrete is constituted of bauxite aggregates (size between 0.2-5 mm), dispersed in commercial ultra low cement bauxite matrix (Table 1), composed of bauxite particles lower than 200 m in diameter, 8 wt.% of calcium alumina cement, fumed silica and ␣-alumina. The material is characterised by an open porosity of 10 vol.% and an apparent density of 2970 kg m -3 . A more detailed description of this material is available in [START_REF] Dusserre | Applicability of image correlation techniques to characterise asymmetric refractory creep during bending tests[END_REF].

Elasticity properties

The elasticity properties of the material were measured by impulse excitation of vibration method [START_REF]Standard Test Method for Dynamic Young's Modulus, Shear Modulus, and Poisson's Ratio by Impulse Excitation of Vibration[END_REF]. The Young's modulus of three 16 × 40 × 25 mm 3 samples was measured in bending configuration up to 1050 • C (the maximum temperature of the device) at 3 • C•min -1 . The dilatometric behaviour of the matrix is stable enough [START_REF] Marzagui | Characterisation of microstructural evolutions in refractory castables by in situ high temperature ESEM[END_REF] to consider that the modulus should not vary significantly up to 1200 • C, the temperature of the creep tests performed in this study. Moreover, the instantaneous strains and deflections will be subtracted from all experimental and numerical data considered for calibration purpose. The effect of elasticity is therefore expected to be only of the second order. A value of 45 GPa will be considered for the Young's modulus at 1200 • C, with a ±2 GPa dispersion related to the material heterogeneity. The shear modulus was measured at room temperature in a torsional configuration.

The value of the Poisson's ratio at room temperature, 0.1, has been assumed to stay constant with temperature. It is noteworthy that the Young's modulus of this kind of materials is the same in tension and compression [START_REF] Nazaret | Influence of the mechanical behaviour specificities of damaged refractory castables on the Young's modulus determination[END_REF]. The elastic part of the strain is therefore linear isotropic and symmetric, following the generalized Hooke's law.

Bending creep experiment

The bending data used in this study are those described in a previous paper [START_REF] Dusserre | Applicability of image correlation techniques to characterise asymmetric refractory creep during bending tests[END_REF] (the same batch of concrete was used) and provide the neutral axis location after bending creep at 1200 • C. This temperature is high enough to involve creep without macroscopic damage [START_REF] Dusserre | Applicability of image correlation techniques to characterise asymmetric refractory creep during bending tests[END_REF] and is just below the onset of an important shrinkage of the matrix [START_REF] Marzagui | Characterisation of microstructural evolutions in refractory castables by in situ high temperature ESEM[END_REF]. The bending specimens (35 × 40 × 165 mm3 ) were cast under vibrations and kept in the mould for 24 h at room temperature until the concrete sets. The specimen are then dried at 110 • C for 24 h and fired at 1200 • C for 30 h. Finally, the four faces of the specimen are ground to get plane and parallel bearing surfaces and obtain a relevant texture to perform 3D-DIC measurements.

Compressive creep experiment

The bending data are useful as they involve tension in the material. To complete these experiments, compression tests have been performed at 1200 • C with alumina plates using the same setup as depicted in [START_REF] Dusserre | Evaluation of viscous and elastic parameters of a borosilicate glass above transition temperature using optical instrumented squeeze tests[END_REF]. The specimens were 40 mm in height and 25 mm in diameter and were machined in a bending specimen with a diamond drill bit. The creep duration was 11 h for specimens under compressive stress levels of 3, 4, 5 and 6 MPa. Another test at 3 MPa for 25 h was also perform to assess the repeatability of the experiment and investigate the limit of the primary creep.

Friction plays a non-negligible role in the compression tests, as non-desired mechanism of dissipation, involving shear stresses and non-uniform loading. The coefficient of kinetic friction between a concrete pin with a 25 mm-radius spherical contact surface and an alumina plate with the same roughness as the compression plates was measured at room temperature. A mean value of 0.28 ± 0.03 was obtained with a normal load of 1 N and a linear sliding speed of 2 mm•s -1 . Another test at 1 m•s -1 has shown that the coefficient of friction is first of the same value, but quickly increases up to 1 and more, probably under the effect of temperature increase. Two assumptions will therefore be compared in the numerical analysis: a sliding contact (Coulomb model) with a coefficient of friction of 0.28, and a sticky contact.

Model

The creep model used in this study is a built-in function of the commercial software ABAQUS/Standard and detailed in the user manual. A brief description is given here in order to clearly state the assumptions made. This section also provides a description of the numerical simulation of the experiments performed with ABAQUS/Standard. In this model, the strain tensor is additively decomposed into an elastic component and a creep component.

Uniaxial primary creep law

As explained previously in the introduction, the processing of bending data necessitates considering the primary creep to properly compute the stress field in the specimen during the whole test even if a stationary creep arises. It is also important for compression tests since friction at the contact with the plates induces a triaxial state of stress that imply that the stationary creep, if any, is not reached simultaneously in the whole specimen. The creep law Table 2 Values of A1, m and n identified in the first one-dimensional case.

A1 (MPa -n s -(m + 1) ) m n
2.57 10 -5 -0.593 0.716 considered here is a usual 3 parameters creep law with strain hardening, Eq. ( 1), including a Norton-type stress contribution, usually used for refractories creep modelling [START_REF] Jin | Compressive creep testing of refractories at elevated loads-Device, material law and evaluation techniques[END_REF].

εcr = A( ) n (m + 1) ε cr m 1 m+1
(1)

Drucker-Prager primary creep law

Equivalent creep stress

The tension/compression asymmetric creep behaviour of the refractory concrete is modelled using a Drucker-Prager creep law. The Drucker-Prager creep is an extension of the Drucker-Prager plasticity criterion [START_REF] Drucker | Soil mechanics and plastic analysis or limit design[END_REF][START_REF] Drucker | Concept of path independence and material stability for soils[END_REF] that assumes that the creep intensity depends not only on Mises equivalent stress q (function of the second stress deviator invariant) but also on pressure p (function of the first stress invariant). The tension/compression asymmetry thus derives from a pressure effect and is not an effect of the third invariant. The advantage of this model is that the creep law is the same in tension and compression, allowing a reduced number of parameters. The model introduces an equivalent stress cr , Eq. ( 2), function of q and p. Eq. ( 2) is the expression of the equivalent stress when the model is calibrated from uniaxial compression data. Indeed, for a compressive uniaxial state of stress ( < 0), Eq. ( 2) reduces to ¯ cr = -, whereas for a tensile uniaxial state of stress ( > 0), it

leads to ¯ cr = 1+ 1 3 tan 1- 1 3 tan ˇ>
. For a given absolute value of the uniaxial stress , the equivalent stress is therefore higher in tension than in compression implying a higher creep rate in tension.

¯ cr = q -p tan 1 -1 3 tan ˇ(2)
This model implies that any state of stress with the same equivalent stress induces the same creep intensity. In the meridional stress plane (p-q plane), the states of stress with the same equivalent stress are defined by a straight line passing through the actual (p,q) point and making an angle ␤ with the p-axis. ␤ is the material angle of friction. In the deviatoric stress plane, the equivalent-creep surface thus defined is the Mises circle. It is noteworthy that if q ≤ p • tan ˇ (and assuming 1 -1

Creep flow

The creep flow derives from a hyperbolic pseudo-potential G cr , Eq. (3), defined by three parameters: the dilation angle at high confining pressure, , the eccentricity, ε, that defines the curvature of the potential, and the initial yield stress, ¯ 0 , that must be set high enough in the present case to prevent plasticity and activate only a creep behaviour. This pseudo-potential is hyperbolic in the p-q plane and corresponds to the Mises circle in the deviatoric stress plane, and limits the model to isotropic hardening.

G cr = ε ¯ 0 tan 2 + q 2 -p tan (3)
This potential form is continuous and smooth, especially at the intersection with the p-axis, and asymptotically approaches the equivalent-creep surface. In the present case, only bending and compressive data are considered. Therefore the potential will be chosen as close as possible to the equivalent-creep surface by setting arbitrarily the eccentricity parameter to the minimum value allowable, 10 -6 . In such conditions, except in the vicinity of the paxis, the potential is very close to a straight line passing through the actual (p,q) point and making an angle with the p-axis, and the potential is mainly defined by the value of the dilation angle.

As discussed in the introduction, the dilatancy of the flow is a key feature of the creep behaviour when the first stress invariant is introduced in the equivalent creep stress. Two assumptions are considered in the paper. First, the assumption of creep incompressibility is formulated, leading to a non-dilatant flow with = 0 (the potential only depends on q). Then the hypothesis of normality will be assessed by setting the dilation angle to the friction angle value, = ˇ. It is noteworthy that even with the minimum value of the eccentricity, the flow is not exactly associated, especially in the vicinity of the p-axis.

The uniaxial creep law, Eq. ( 1), is translated to the triaxial case, Eq. ( 4), and relates the equivalent creep strain rate, ˙ε cr , to the equivalent creep stress, ¯ cr , and the equivalent creep strain, εcr , in order to account for the non-linear primary creep through an isotropic strain hardening.

˙ε cr = A ¯ cr n (m + 1) εcr m 1 m+1 (4)
From Eq. ( 4), the scalar equivalent creep strain rate, ˙ε cr , can be computed at each time step, and contributes to calculate the creep strain rate tensor = ε cr , calculated from Eq. ( 5), where f cr is defined in such a way that = ε cr is work conjugate to = , Eq. ( 6), as well as ˙ε cr is work conjugate to ¯ cr (cf. ABAQUS Theory Manual for further details on the numerical integration of these equations by the backward Euler method, selected through the activation of geometrically nonlin- ear analyses even if only small strains are involved in the problems under consideration).

= cr

ε = ˙εcr f cr ∂G cr ∂ = (5)
f cr = 1 ¯ cr = : ∂G cr ∂ = (6)

Finite element computations

The numerical simulations of the compression creep tests were carried out with axi-symmetric elements and a symmetry plane was defined at the mid-height of the specimen, as depicted in Fig. 1a. The compression plate was schematised as a discrete rigid body and a load is applied on it. The 12.5 × 20 mm 2 rectangular geometry was meshed with 170 linear quadrilateral elements by considering a single bias in both directions in order to ensure a finer element size in the vicinity of the edge of the specimen, in contact with the compression plates. The length of the element edge thus ranges between 0.4 and 2 mm (the simulations provide similar results using a twice finer mesh).

The numerical simulations of the bending creep tests were performed with plane strain elements and a symmetry plane was defined in the mid-length of the specimen, as depicted in Fig. 1b. The support and loading points are 12 mm diameter arcs of circle schematised as analytical surfaces. The former is fixed and a load of 200 N is applied on the latter. The results reported in Section 4 for the calibration of the model were obtained by meshing the 40 × 80 mm 2 rectangular geometry with 800 linear 2 mm edge square elements. Indeed, a sensitivity analysis to the mesh size (performed a priori with arbitrary parameters values and a posteriori with the optimal set of parameters) showed that the strain field at the symmetry plane is not sensitive to the element size: after a creep step of 25 min, the discrepancy between the minimal strain obtained with a mesh size of 2 mm and 0.5 mm is about 2%, and lower than 1% for the maximal strains. The strain field being linear, the neutral axis location is therefore insensitive to the mesh size.

Table 3

Values of A2 and ˇidentified from bending creep full-field measurements.

Assumption ( • ) A2 (MPa -n s -(m + 1) ) ˇ(• )
Non-dilatant flow 0 2.57 10 -5 56.5 Associated flow 42.9 1.80 10 -5 42.9

In contrast, the bending creep results reported in Section 5 and computed with the optimal set of parameters, were obtained with a mesh size of 0.5 mm. Indeed, the specimen deflection turns out to be sensitive to the mesh size, because of the material deformation at the contact with the support and loading points: a discrepancy of 13% is obtained considering element sizes of 0.5 and 2 mm (the error is estimated at 15% for 2 mm and 1.4% for 0.5 mm, assuming a quadratic fit of the deflection computed with an element size of 0.5, 1, 1.5 and 2 mm). Moreover the computation time is not so important here since only one computation is performed.

In all cases, the load was applied on the reference point that controls the motion of the rigid body and the analytical surface, i.e. respectively the compression plate and the upper loading point. The contact between the specimen and the rigid bodies assumes either sliding with a coefficient of friction of 0.28 or a sticky contact.

Results of model calibration

The methodology involved to identify the parameters of the Drucker-Prager creep law includes three steps. The compression data are first processed as one-dimensional creep to identify the parameters of the creep law, m and n, and to assess a first value A 1 of parameter A. The angle of friction ˇof the material is then identified from kinematic full-field measurements performed during interrupted bending creep tests, to account for the shift of the neutral axis location. Finally, the compression tests are simulated to take friction with the plate into account and identify the final value of A.

One-dimensional calibration

A first set of parameters was identified using a one-dimensional model consisting in a numerical integration of Eq. (1) using the trapezoidal rule. The experimental stress is imposed as boundary condition. A Newton algorithm was used to identify the values of parameters m and n and a first value A 1 of parameter A, considering the four compressive tests at 1200 • C under nominal stress levels of 3, 4, 5 and 6 MPa.

Fig. 2 compares the experimental data and the results of this model obtained with the optimum set of parameters reported in Table 2. Considering the unavoidable scattering inherent to the characterisation of heterogeneous materials, a very proper agreement is observed over the whole duration of the tests irrespective of the experiment under consideration, with a maximum gap lower than 9% for the 4 MPa curve and lower than 6% for the other curves. The form of the creep law, Eq. ( 1), is thus able to properly account for the hardening occurring during the primary creep under compression.

Identification of the angle of friction

Ťhe angle of friction ˇis the parameter that controls the sensitivity of the equivalent creep stress to the pressure, and thus controls the creep asymmetry, assessed in this study through the neutral axis location measured in the interrupted bending experiment reported in [START_REF] Dusserre | Applicability of image correlation techniques to characterise asymmetric refractory creep during bending tests[END_REF]. As the equivalent creep stress depends on ˇ, the value of this parameter may be calibrated together with the parameter A to ensure that the creep strain rate remains the same when changing ˇ(and thus still matches the experimental creep curves). The parameter m of the creep law can keep the same value identified in the unidirectional approach, since ˇdoes not affect the isotropic hardening. It was also verified that the last parameter of the creep law, n, could be kept constant to still properly represent the effect of stress. Indeed considering pure compression tests under several loads, the equivalent creep stress is proportional to the Mises equivalent stress (independent on the material parameters), and the effect of changing ˇcan be compensated only by adjusting A.

Therefore ˇis identified together with a second value A 2 of parameter A. This calibration is performed based upon matching the longitudinal strain field computed along the symmetry line (Fig. 1b) to the experimental strain field reported in [START_REF] Dusserre | Applicability of image correlation techniques to characterise asymmetric refractory creep during bending tests[END_REF]. Several simulations of the bending creep experiment were performed varying the values of A and ˇin order to build the response surface of the model, considering both assumptions of non-dilatant flow (Fig. 3a) and associated flow (Fig. 3b).

For each simulation, the longitudinal component of the total strain along the symmetry line (cf. Fig. 1b) was extracted and fitted using a 6-th degree polynomial in order to interpolate the data at the experimental values, and to compute the gap with the experimental strain field using the least square method. The coordinate where the longitudinal component of the total strain is zero was also interpolated using the polynomial fit to compute the location of the simulated neutral axis.

These results are reported in Fig. 3 and show that an optimal value can be found for each assumption, as evidenced by the contour plot under the surfaces. The value A 2 of parameter A and of čorresponding to the optimisation considering both assumptions are summarized in Table 3. This identification shows that the value of ˇis significantly affected by the assumption under consideration. A very high value of 56.5 • is found assuming a non-dilatant flow.

Fig. 4 compares the experimental strain field to the results of both calibrations assuming a non-dilatant flow or an associated flow. A first observation shows that each assumption satisfactorily reproduces the strain field in the areas where the strain is large, i.e. far away from the neutral axis. In the vicinity of the neutral axis, the assumption of associated flow seems closer to the experimental curve but the experimental data may not be accurate enough in this low-strain area to argue the validity of this assumption.

In this approach, it has been assumed that the location of the neutral axis didn't change between the interruption of the creep test (at 1200 • C) and the full-field measurement performed at room temperature (the elastic springback and relaxation occurring after unloading and during cooling are not accounted for). This assumption will be discussed further considering the numerical results (Section 5.2).

Therefore A 2 is not expected to be the actual value of A because the simulation doesn't account for the specimen deformation occurring between the interruption of the creep test and the fullfield measurement. The experimental and simulated strain fields may thus not be the same, and only the location of the neutral axis is assumed to be the same. y/h Longitudinal total strain, LE11 Experiment first interruption [START_REF] Dusserre | Applicability of image correlation techniques to characterise asymmetric refractory creep during bending tests[END_REF] Experiment second interruption [START_REF] Dusserre | Applicability of image correlation techniques to characterise asymmetric refractory creep during bending tests[END_REF] Simulation first interruption Simulation second interruption identified in this step is not the final one, the value of ˇis the one that implies the right neutral axis location and thus controls the creep asymmetry. A similar conclusion can be formulated considering the non-dilatant assumption.

Calibration of A from compressive creep

The compressive creep experiments are simulated in ABAQUS/Standard in order to impose proper boundary conditions at the contact between the sample and the plates. The parameters m, n, and ˇare set to their value identified previously.

The final value of A is identified in such a way that the simulation matches the compressive creep experiments. For each assumption (dilatant flow with sliding contact, associated flow with sliding contact and associated flow with sticky contact), four Newton iterations are performed to obtain the values reported in Table 4 and provide the same creep curves when compared to experiment (Fig. 6), i.e. a gap lower than 2% with the unidirectional calibration. It is noteworthy that the comparison to the experimental compressive creep curves provides approximately the same results, depending on the residual of the identification, irrespective of the assumption of unidirectional compression, non-dilatant or Fig. 9. Neutral axis shift during a bending creep test computed considering the locus of zero stress (S11 = 0) and the locus of zero total strain (LE11 = 0). associated flow, which highlights that the compression tests are not sufficient to characterize the creep behaviour of the concrete.

Analysis of the sample profile

In order to further investigate the dilatancy assumption, as well as the actual tribological behaviour during the compressive creep experiments, a simulated sample profile will be compared to an experimental one. The reference experimental sample profile is the average of four linear scans performed along the axis of the sample subjected to a compression stress of 3 MPa for 25 h (and thus not used for calibration) with an extended field confocal microscope (AltiSurf520, Altimet). Fig. 7 compares the change in radius along the sample axis obtained from simulation and measured on the compression specimen subjected to 3 MPa for 25 h at 1200 • C. The experimental profile was measured using the extended field confocal microscope, and was used together with a measurement of the maximal specimen diameter with a micrometer to assess the change in radius. This value is therefore less accurate than the profile itself and an uncertainty on the vertical position of the experimental curve must be considered in Fig. 7. However, the assumption of a nondilatant flow provides an average change in radius significantly lower than the experimental one. In addition, the very good agreement (highlighted by the translated experimental curve) between the experimental profile and the simulated profile demonstrates that both assumptions of sticky contact and associated flow are very realistic, and are the only ones (among the tested assumptions) that allow to accurately account for the compressive creep curves, the neutral axis location and the sample curvature.

Discussion

In this section, the model results are compared to experiment in order to assess its limits. Finally the discussion will focus on the neutral axis shift during bending creep.

Comparison of simulated and experimental bending behaviours

The creep bending experiment was simulated using the optimized set of parameters, i.e. corresponding to the calibration assuming an associated flow with sticky contact for compression (Table 4). For bending simulation, the sticky contact assumption provides inconsistent results compared to the experimental strain field, and a sliding contact with friction (coefficient of friction of 0.28) was therefore assumed, as it was also assumed to identify the value of ˇ(Section 4.2). The difference between the contact conditions in both cases, compression and bending, is not surprising considering the different geometries in contact, as well as the different contact pressures and tangential stresses. Moreover the bending loading points are not fastened on their support and may be free to sligthly rotate. Fig. 8a compares the longitudinal total strain field computed with the aforementioned model during interrupted bending creep to the experimental results reported in [START_REF] Dusserre | Applicability of image correlation techniques to characterise asymmetric refractory creep during bending tests[END_REF]. As expected from the discussion of the influence of ˇ, all curves intersect each other at a strain of zero, and the simulation provides higher strain values due to the high value of A necessary to fit the compression data.

Fig. 8b compares the bending deflection computed considering a continuous bending creep and an interrupted bending creep to experimental data obtained from two different specimens during continuous and interrupted creep. The first rough comparison of simulation and experiment confirms the strain overestimation shown in Fig. 8a since the computed deflection is overall 20% higher than the experimental one. Considering the available experimental curves, it seems that this cannot result from the material heterogeneity alone. Moreover, the deflection decrease during the unloading, relaxation and cooling steps (partly plotted only for the interrupted experiment) is widely underestimated by the simulation of the unloading step. During the loading steps after test interruption, the experiment shows a significant deflection increase, and the curve tends toward a creep curve parallel to the continuous experiment. The simulation doesn't account for this behaviour and the computed creep curve tends toward the continuous computed behaviour. It is noteworthy that at least part of this behaviour may result from the setup deformation during loading and unloading steps, and also from a compaction of the material at the contact with the support and loading points.

However the assumption of isotropic hardening is here invalidated by the loading behaviour after a creep-unloading sequence. Indeed, the isotropic hardening remains constant during unloading while a kinematic hardening would change. It could indicate that the main mechanism involved during primary creep is related to the motion of hard particles in a glassy matrix, more consistent with a kinematic hardening than an isotropic one. This mechanism is also consistent with the dilatant behaviour of the material if the hard particles are not spherical as it is the case in this material. Nevertheless, under monotonic conditions, the model will be able to properly describe the material behaviour and will be used in Section 5.2 to investigate the neutral axis shift while bending creep.

Analysis of the neutral axis shift during bending creep

The simulation of bending creep allows analysing the shift of the neutral axis. Fig. 9 compares the neutral axis shift during creep (loading not considered) depending on the chosen criterion: locus of zero-stress (S11 = 0) or locus of zero-total strain (LE11 = 0). Due to the very low elastic strain compared to total strain, the locus of zero-creep strain is very similar to the locus of zero-total strain. The neutral axis at the end of creep is thus very closed to its location after unloading, as assumed in Section 4.2 for the identification Longitudinal total strain at y/h = 0.272 S 11 LE 11 of ˇ. Irrespective of the neutral axis definition, the neutral axis location rapidly stabilizes whereas the deflection curve is still nonlinear. In contrast, Leplay et al. [START_REF] Leplay | Analysis of asymmetrical creep of a ceramic at 1350 • C by digital image correlation[END_REF] report a significant neutral axis shift together with a perfectly linear bending creep, showing that the asymmetric creep alone doesn't imply that the global bending behaviour is non-linear. It is noteworthy that the final neutral axis location ( y/h = 0.09) reported in [START_REF] Leplay | Analysis of asymmetrical creep of a ceramic at 1350 • C by digital image correlation[END_REF] is much lower than the value found in this study ( y/h = 0.27) considering the locus of zero-total strain. Fig. 10a plots the total strain computed at the mid-plane (y/h = 0) and on the lower (y/h = -0.5) and upper (y/h = 0.5) faces of the specimen, as well as in the vicinity of the final neutral axis location (y/h = 0.272), during creep (loading not considered). The creep strain follows the same evolution at each point. This figure illustrates that the total and creep strains fields remain almost linear in the height of the specimen all along the creep.

In contrast, the stress field change due to creep at the lower and upper faces (Fig. 10b) results in a bilinear profile shape due to asymmetric creep, with a lower slope in the tensile area than in the compressive one, and a tensile area larger than the compressive one. When the creep flow stabilizes, the maximal tensile stress begins to decrease because of a fast creep related to a high equivalent stress; the mean stress in the tensile area therefore tends to decrease, what is counterbalanced by a second shift of the neutral axis (between 300 and 500 s in Fig. 9).

As shown in [START_REF] Dusserre | Applicability of image correlation techniques to characterise asymmetric refractory creep during bending tests[END_REF] using an analytical model of stationary creep, Fig. 11 confirms that an amount of material, initially under compression during the loading step, is finally subjected to tensile stress, as the result of the neutral axis shift. At y/h = 0.272, the transition from a compressive to a tensile state of stress leads the total strain, initially negative, to tend toward positive values during the creep. This phenomenon may explain why the model overes-timates the creep strain in bending, as this test is not monotonic anywhere in the specimen. The assumption of isotropic hardening would therefore imply some inconstancy in the modelling results.

Conclusions

This paper reports an analysis of experimental data obtained from simple monotonic compression and bending creep tests at 1200 • C through the Drucker-Prager creep model. The aim of the study is to assess the capability of the model to properly account for the tension/compression asymmetry evidenced in the creep behaviour of the fibre-reinforced refractory concrete under study. The analysis focuses on the primary creep of the material through an isotropic strain hardening law, assumed a priori.

A multi-steps method is proposed to calibrate this model, using kinematic full-field measurements performed on interrupted bending creep specimens. A calibration method involving only usual tests at high temperature, instead of complex triaxial tests, is of prior interest for technical applications and in order to characterize the creep behaviour over a large temperature range.

The model proved to be able to properly describe the compressive behaviour, together with accounting for the location of the neutral axis of bending specimen, shifted from the mid-plane after creep. It was possible irrespective of the assumption formulated on the dilatancy of the creep flow. However the geometrical characterisation of a compression sample shows that the flow is dilatant, and that the flow pseudo-potential must be built in such a way that the hypothesis of normality holds true (here, dilatancy and friction angles are equal). However, the model is not able to simultaneously reproduce compressive and bending creep curves (the simulated bending deflection is 20% higher than the experimental one and the simulated strain field is twice the experimental one), because the assumption of isotropic hardening is probably not relevant.

Despite this partial agreement with experimental results, showing that an assumption of kinematic hardening would better describe the behaviour during unloading and reloading, the model allows analysing the neutral axis shift during creep bending. This analysis shows that the neutral axis very quickly stabilizes toward a given location. These results show that simulation is unavoidable to catch the complexity of the stress field change in bending creep, and further build a more relevant model.

  asymmetry by combining two different behaviours depending on the sign of the stress[START_REF] Lim | A finite element model for asymmetric creep behavior of ceramics[END_REF][START_REF] Blond | Modelling of high temperature asymmetric creep behavior of ceramics[END_REF].

Fig. 1 .

 1 Fig. 1. Meshes, boundary conditions and load application points for compressive (a) and bending (b) creep simulations.

Fig. 2 .Fig. 3 .

 23 Fig. 2. Comparison of experiment (o) with the creep model (-) calibrated assuming unidirectional compression.

Fig. 4 .

 4 Fig.4. Comparison of the longitudinal total strain field in the bending specimen computed at the end of creep test and measured after unloading assuming a non-dilatant or an associated flow.

Fig. 5 .

 5 Fig. 5. Neutral axis location ( y/h) as a function of ˇand A assuming an associated flow.

Fig. 6 .

 6 Fig.[START_REF] Blond | Modelling of high temperature asymmetric creep behavior of ceramics[END_REF]. Comparison of experimental (circles and squares) and simulated (lines) compressive creep behaviours assuming (a) a non-dilatant flow and a sliding contact between the specimen and the plates, (b) an associated flow and a sliding contact between the specimen and the plates and (c) an associated flow and sticky contact between the specimen and the plates.

Fig. 7 .

 7 Fig.7. Anamorphic view of the sample profile after compressive creep under 3 MPa for 25 h at 1200 • C: experimental data and simulated profiles assuming a non-dilatant flow and a sliding contact, an associated flow and a sliding contact or an associated flow and a sticky contact (the light grey experimental curve is a translation of the raw experimental one in dark grey and highlights that the curvature is very close to the simulation assuming an associated flow and a sticky contact).

Fig. 5

 5 completes the results of Fig.3aand plots the neutral axis location (parameterized by the ratio between the neutral axis shift y and the specimen height h) computed assuming an associated flow, as a function of parameters ˇand A. The figure shows that the neutral axis location is a one-to-one function of ˇranging between 30 and 45 • , irrespective of the A value. This result validates the methodology of calibration by showing that even if the value of A 2

Fig. 8 .

 8 Fig. 8. Comparison of simulated and experimental bending creep: (a) longitudinal total strain fields after the first and the second test interruption and (b) time-deflection curves for the interrupted test (used for full-field measurement) and for a continuous test.

Fig. 10 .

 10 Fig.10. Change of (a) longitudinal total strain LE11 and (b) of longitudinal stress S11 during a bending creep test at several locations along the specimen thickness (y/h = -0.5, 0, 0.272 and 0.5).

Fig. 11 .

 11 Fig. 11. Variation of longitudinal stress and creep strain at y/h = 0.272.

Table 1

 1 Composition of the refractory castable.

	Component	Composition (wt.%)
	Al2O3	85
	SiO2	10
	CaO	1.1
	Fe2O3	1
	Water requirement	4.2-5.2

Table 4

 4 Values of A identified from compressive creep experiments.

	Flow assumption	Contact	( • )	A (MPa -n s -(m + 1) )	ˇ(• )
	Non-dilatant flow	Sliding with friction	0	3.18 10 -5	56.5
	Associated flow	Sliding with friction	42.9	3.01 10 -5	42.9
	Associated flow	Sticky	42.9	3.54 10 -5	42.9

tan ˇ> 0), then ¯ cr ≤ 0, what means that the material do not creep. The states of stress involving a high hydrostatic pressure, for which the model would predict no creep, do not fall into the scope of the present model, only identified in bending and uniaxial compression.

Acknowledgement

The authors greatly acknowledge the contributions of Dr Christine Boher, Mrs Sabine Leroux and Mr Fabrice Rossi to the experimental results.