D Rittel

R Levin

H Maigre

The Influence of Mode-Mixity on Dynamic Failure Mode Transitions in Polycarbonate

The transition of a shear to opening type of failure mechanism has been reported for side impact experiments of notched or cracked plates (metallic alloys and Polycarbonate). The present paper addresses additional aspects of the phenomenon in relation to mode-mixity for actual fatigue cracks in Polycarbonate specimens. Two distinct experimental setups are used and systematically compared throughout the work: dominant mode II and dominant mode I loading. The experimental results show that the same characteristic failure mechanisms operate irrespective of the specimen geometry (loading mode) and crack-tip nature for a given impact velocity (evolution of the stress intensity factors).

Résumé. La transition des modes de rupture d'ouverture en cisaillement a été observée lors d'essais d'impact latéral de plaques entaillées ou fissurées (alliages métalliques et Polycarbonate). Cet article traite d'autres aspects de la transition en relation avec la mixité du chargement pour des fissures de fatigue dans du Polycarbonate. Nous utilisons et comparons systématiquement deux types d'expériences distinctes: chargement en mode 11 dominant ou en mode I dominant. Les résultats montrent que des mécanismes de ruine identiques opèrent dans les deux cas, en fonction de la vitesse d'impact (evolution des facteurs d'intensité des contraintes) indépendemment de la géometrié de l'éprouvette ou de la stature de la fissure.

INTRODUCTION

Dynamic crack initiation addresses the case of a crack subjected to transient loading. The issue is the determination of a criterion for the onset of crack initiation and if possible an estimate of its initial trajectory. Whereas the problem has been extensively investigated in its theoretical aspects for the linear elastic case [START_REF] Freund | Dynamic Fracture Mechanics[END_REF] a lot remains to be done at the experimental level to develop a sound criterion for dynamic crack initiation. Considering materials related aspects of dynamic fracture, Charpy tests have long been used to investigate the ductile to brittle transition for crack opening mode. A new kind of failure mode transition has been reported for side impact of notched (and cracked) plates. In these experiments, a projectile is shot parallel to the notch direction, inducing mixed mode loading conditions with a dominant mode II term. [START_REF] Kalthoff | Shadow optical analysis of dynamic fracture[END_REF] investigated maraging steel and reported that at lower impact velocities, fracture proceeds in a "brittle" way by forming an angle of about 70° with respect to the initial notch line. Increasing the impact velocity causes a shear band to develop and propagate for a short distance almost along the continuation of the original notch line. [START_REF] Lee | Fracture initiation due to asymmetric impact loading of an edge cracked plate[END_REF] studied this problem and provided an analytical solution for short times after impact such as to exclude wave reflections as a result of the specimen's finite size. They showed the development of a minor negative mode I component aside the dominant mode II. [START_REF] Ravi Chandar | On the failure mode transitions in polycarbonate under dynamic mixed mode loading[END_REF] worked on Polycarbonate and he reported the very same behavior also in terms of a brittle to ductile transition and fracture angle (66°). [START_REF] Zhou | Dynamically propagating shear bands in impact-loaded prenotched plates. I-Experimental investigations of temperature signatures and propagation speed[END_REF] carried out a detailed investigation of side impact in maraging steel and a titanium alloy. They report a shear to opening (with an angle of about 30°) fracture mode transition in the maraging steel. The titanium specimens fail by shear mechanism only. The cited works report about systematic shear band generation at the higher impact velocities with varying amounts of propagation and subsequent failure by opening mechanism at specific angles (either 70° or 30°). These works address side impact experiments only with the crack-tip fields determined by using optical methods (caustics or CGS). The present research addresses additional aspects of the problem with emphasis on the initiation process. We investigate actual cracks, grown experimentally as fatigue cracks and modeled numerically through contact and friction conditions. Since the problem is inherently related to mode-mixity, we systematically investigate and compare situations in which mode II is either the dominant (shear impact) or the minor mode [START_REF] Rittel | A new method for dynamic fracture toughness testing[END_REF]. Failure modes are characterized using optical and scanning electron microscopy. The results are discussed and compared such as to outline the role of mode mixity in this kind of experiments in particular and in dynamic crack initiation more generally.

EXPERIMENTAL AND NUMERICAL FRAMEWORK

Experimental

The experimental setup is of the "one point impact" type [START_REF] Giovanola | Investigation and application of the one-point-bend impact test[END_REF]. It comprises the impacting apparatus and the specific specimen. A 12.7 mm diameter instrumented (PH 17-4) steel bar is on one side brought in contact with the specimen and on its other side is impacted by a striker fired by a gas gun. Typical impact velocities range from 8 to 60 m/s. Strikers with length of 8 or 17 cm were used to produce impacts of varying duration (30 μs and 64 μs respectively). The striker induces a compressive pulse in the bar which is partly transmitted to the specimen and partly reflected as a tensile pulse. The pulses were collected by means of a digital oscilloscope (Nicollet 490 model) at a sampling rate of 5 MHz. In some experiments, we cemented single wire fracture gages in the vicinity of the crack-tip to signal the onset of crack-propagation. The experimental pulses were corrected for geometrical dispersion [START_REF] Lifshitz | Data processing in the split Hopkinson pressure bar tests[END_REF] using home made programs. The interfacial force P i and velocity V i pulses were determined from the incident and reflected pulses. Two kinds of specimens were used (Fig. 1): the rectangular plate, for shear impact ("mode II") experiments and the compact compression specimen (CCS) for "mode I" experiments [START_REF] Rittel | A new method for dynamic fracture toughness testing[END_REF]. Due to the asymmetrical loading, the crack-tip experiences mixed-mode conditions which were investigated by [START_REF] Maigre | Mixed-mode quantification for dynamic fracture initiation: application to the compact compression specimen[END_REF] for the CCS. Throughout the experiments, the specimen was brought in contact with the instrumented bar without special fixtures to support it so that fracture resulted from inertia only. The boundary conditions (applied force and displacement) on the specimen are monitored through the instrumented bar, and they can be used for subsequent numerical simulations. In each specimen, a 2-3 mm long fatigue crack was grown carefully. The investigated material was polycarbonate, supplied as 12.7 mm thick sheets. The mechanical properties of this material were assessed using quasi static tensile testing. An equivalent "transfer modulus" (to account for dynamic effects -see [START_REF] Rittel | On dynamic crack initiation in polycarbonate under mixed-mode loading[END_REF] was determined from longitudinal wave velocity measurements using ultrasonic technique. We also measured the quasi-static fracture toughness of the material using fatigue precracked CCS's (stroke control at a velocity of 0.2 mm/min). A representative value was found to be KIC = 3.6 MPa.m 1/2 .

Numerical

The crack-tip fields are not monitored directly in our experiments. Consequently, the stress intensity factors are determined from a numerical (finite element) model of actual experiments (hybrid experimentalnumerical approach). We assumed two-dimensional plane strain deformations and linear elastic material behavior. We did not include thermo-mechanical couplings as in [START_REF] Zhou | Dynamically propagating shear bands in impact-loaded prenotched plates. I-Experimental investigations of temperature signatures and propagation speed[END_REF] since the emphasis here is on the analysis of the crack-tip fields prior to initiation without attempting to model shear band related phenomena. The crack tip is modeled using singular quarter point elements [START_REF] Barsoum | On the use of isoparametric finite elements in linear elastic fracture mechanics[END_REF] and the crack in all the analyses is kept stationary so that the stress intensity factors are a function of time only [START_REF] Freund | Dynamic Fracture Mechanics[END_REF]. Traction free conditions are assumed over all the specimen except at the specimen-bar interface. To simulate the experiment, the recorded force pulse is applied to this interface as boundary condition. Once the specimen-bar separation has occurred, measurement of the interfacial forces and velocities using the instrumented bar is not possible as the specimen is no longer loaded through the input bar. However, the crack-tip fields can be assessed for any duration by extending the calculation while applying traction free conditions. Fatigue cracks induce two specific conditions. The first is contact between the crack-flanks to avoid interpenetration following the negative mode I which develops for shear impact experiments [START_REF] Lee | Fracture initiation due to asymmetric impact loading of an edge cracked plate[END_REF]. Secondly, friction may develop between the crack faces which affects the mode II component of the crack-tip loading. Contact was modeled using special contact elements which apply a normal reaction when the gap between the two surfaces is closed (ANSYS, 1994). Friction was modeled as Coulomb friction and the coefficient of friction in all the analyses was arbitrarily fixed to u s = 0.4. The equations of equilibrium were solved using Newmark implicit scheme (Bathe, 1982). The mode I and mode II stress intensity factors were determined from the displacements "measured" at two control points in the vicinity of the crack-tip [START_REF] Bui | A new approach to the experimental determination of the dynamic stress intensity factor[END_REF].

RESULTS

Foreword

The framework described here is a joint experimental-numerical effort. A detailed report and discussion of our results is in preparation. In this paper, we will describe selected results of the work.

The influence of contact and friction

The influence of friction and contact on the stress intensity factors in our specimens was assessed by applying a gaussian pressure pulse (30μs wide -2x10 7 Pa at peak) to simulate transient loading characteristic of that applied in typical experiments. Fig. 2 shows the evolution of K I and K II (whose absolute values are meaningless) for a shear impact specimen to which we prescribed contact conditions with μ=0 (frictionless) and μ=0.4. This figure shows that contact does not allow for the negative mode I to develop while friction does not seem to affect crack opening (the slightly negative KI values are most likely related to the contact stiffness used in the numerical calculations). As the time increases (t norm > 10) a marked tendency for crack opening develops. On the other hand, the influence of friction is felt on the mode 11 in the sense that contact tends to restrain the shearing motion of the crack faces, but this is mostly evidenced at the later times (t norm > 10). It should be noted that the absence of a negative mode I component during a large initial time brings this experiment close to a "pure mode II" condition.

Shear experiments ("mode II")

A total of 16 specimens were tested. We observed that there is a range of velocities (the "lower impact velocities", e.g., below 25 m/s) for which the specimen fractures whereas in another range (the "higher impact velocities", e.g., 60 m/s) an adiabatic shear band is observed and the crack does not seem to grow (Rittel et al., 1997). Shear bands are clearly detected by visual examination only on specimens subjected to the high impact velocities. This result is similar to previously reported results on failure mode transitions (e.g., [START_REF] Ravi Chandar | On the failure mode transitions in polycarbonate under dynamic mixed mode loading[END_REF]. It can also be noted that regardless of the kind of the crack-tip, all the specimens which fractured (11 out of 16) did it at angles which did not exceed 40°. This value must be contrasted with the predicted value of 70° at which maximum tensile stress develops for mode II loading. Consequently, opening at such an angle suggests the joint operation of not only mode II but also mode I fracture mechanism as discussed later.

Compact compression experiments ("mode I")

A total of 8 specimens were tested. Here, fracture occurred in every case. However, similar to the shear impact experiments, the maximum kink angle does not exceed 40° regardless of the impact velocity.

3.4 Fractographic analysis 3.4.1 Shear impact specimens The high velocity impact is characterized by a shear band which develops at the crack-tip but does not propagate significantly in our experiments. The shear band appears as a planar zone, 1-2 mm long, which spans across the specimen's width (Rittel et al., 1997). This band is best detected by tilting the specimen with respect to the incident light and carefully looking in the vicinity of the crack-tip. Optical microscopy at relatively low magnifications discloses minor extension of the fatigue crack within the shear band but this crack did not propagate much either and its tip appears to be quite blunted. Fractographic examination in the scanning electron microscope discloses a layered pattern which is characteristic of the band. This pattern is followed by a series of waves emanating concentrically from the shear band (Fig. 3a). The overall size of the band and the waves is of the order of the millimeter. The fracture surface topography in the waves appears to be featureless at low magnifications but inspection at higher magnifications reveals a characteristic fine scaled pattern of alternating patches of material over a uniform background. These patches can be attributed to the operation of a crazing mechanism sensitive to normal stresses. [START_REF] Kausch | Polymer Fracture[END_REF]. Further apart, the fracture surface is much rougher. Yet, higher magnifications show again a lack of distinctive features. The fracture surface typical of a lower velocity impact is shown in Fig. 3b. The fracture comprises two main features: arrays of linear markings which converge to the fatigue crack. These arrays delineate a seemingly featureless zone although patches of striations can somehow be discerned. Further examination at higher magnification clearly shows that the striations are made of alternating patches identical to those observed in the previous specimen. No shear band was observed as in the previous case.

Compact compression specimens

A typical fractograph of higher velocity impact in a compact compression specimen is shown in Fig. 4a. Fracture begins by a shear band with its characteristic pattern of parallel striations. Higher magnifications reveal the characteristic layered aspect of the fracture surface in the band. The band extends by a series of waves within which one can observe the alternating patches of material, both features similar to those observed in the side impact specimen. Farther away, the fracture surface is very rough, quite similar to the observations of the previous specimen. A typical fractograph of lower velocity impact in a compact compression specimen is shown in Fig. 4b. In the immediate vicinity of the fatigue crack, filaments are noticeable (similar to the previously mentioned linear features) and the general fracture surface topography seems once again rather dull at low magnifications, whereas at higher magnifications, craze markings are again observed. The fracture surface topography comprises rough features farther away from the initiation area. The fractographic observations can thus be summarized at this stage as follows. A high degree of similarity of the fractographic features can be noted between our "mode II" and "mode I" specimens. Specifically, at "lower" velocities shear bands are not observed. The fracture surface looks dull at low magnifications while at high magnifications a typical pattern of alternating patches (craze dominated fracture) of material is noted. For the higher velocity impacts at which a shear band is formed (with evidence of local heating), the latter extends by a series of wavy patterns emanating form its front. These patterns comprise the above mentioned alternating patterns of patches followed by very rough features. The key point here is that these characteristic features are observed regardless of the specimen and/or experiment in question.

DISCUSSION AND CONCLUSIONS

The investigation has been carried out along three complementary axes: mechanical (dynamic fracture tests)materials related (fracture mechanisms) -and numerical. A relatively large sample size of dominant mode II and dominant mode I specimens have been systematically tested. The experiments reported here all involve mixed-mode loading to some extent. Not only do these experiments confirm the previously reported trend for failure mode transition but they also shed additional light on its operation in various contexts. A first observation is that transition of failure modes is not related to the nature of the crack-tip (notch or fatigue crack). The selection of a failure mode is thus not sensitive to the level of negative mode I which develops in side-impact experiments. The important point in our "mode II" experiments is that the crack tip experiences initially almost "pure mode 11" conditions followed by mixed-mode loading at the later stages. Shear band initiation, propagation and final fracture are thus separable events in this experiment. The initiation phase of dynamic cracking with the occurrence (or lack) of adiabatic shear banding is the characteristic response of a material which possesses a sensitivity to two distinct failure modes, shearing and opening. As a remark, the same experiment carried out on a brittle polymer -characterized by a single failure mode (opening) -always yields fracture at about 70°. The occurrence of shear bands is neither restricted to a specific kind of experiment nor to the crack-tip geometry. This point was corroborated by the striking similarity of fractographic features between mode 11 and mode I specimens at matching impact velocities. Consequently, mixed-mode loading causes the joint operation of shear and opening fracture mechanisms. At high impact velocities, the shear failure mechanism is operative and it does not contribute significantly at lower impact velocities. This important cooperation of failure mechanisms should be accounted for when proposing a criterion for dynamic crack initiation.

Figure 1 :

 1 Figure 1: The specimens for dominant mode II and dominant mode I dynamic fracture testing

Figure 2 :

 2 Figure 2: Results of a numerical simulation. Mode I and II stress intensity factors for contact (w/o friction) conditions on the fatigue crack.

Figure 3 :

 3 Figure 3: SEM fractographs of (a) high and (b) low velocity impacted shear specimens ("mode 11")

Figure 4 :

 4 Figure 4: SEM fractographs of (a) high and (b) low velocity CC specimens ("mode I")

Acknowledgment

Support from Arc-en-Ciel cooperation program is gratefully acknowledged. D.R. wishes to acknowledge financial support from the Israel Science Foundation and from the Technion Fund for Research Promotion. References


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INTRODUCTION

The prediction of fatigue and failure of structures submitted to thermomechanical loadings is one of the major problems in mechanical engineering and has been the subject of major efforts in the last few decades. The main applications concern the nuclear and the aeronautical industries, and many results are essentially related to safety and lifetime predictions on simplified structures (see for example [START_REF] Chaboche | Thin wall thermal gradient: experimental study, F.E. analysis and fatigue life prediction[END_REF]). However, when mechanical behaviour of real structures submitted to non isothermal multiaxial loading is computed, methods do not integrate to a lifetime prediction [START_REF] Dambrine | Dimensionnement en fatigue et fluage des aubes de turbines[END_REF]. But in a design context one should be able to estimate by computation the response of any complex structure.

There are essentially three major difficulties in attaining this objective : the constitutive law which must be validated in a wide range of temperatures, its efficient implementation in a finite element standard code and the fatigue criterion itself. To be compatible with an industrial structural computation, the constitutive law should permit computer simulations in a relatively small calculation time of elastoviscoplastic structures subjected to complex thermomechanical loadings and the fatigue criterion should predict both isothermal and anisothermal LCF failure in a large range of temperatures and multiaxial loadings. Therefore we should use a simple robust criterion at a global macroscopical level to predict low cycle thermomechanical fatigue failure structures. This complete approach has recently been illustrated [START_REF] Charkaluk | On a failure criterion for thermomechanical multiaxial low cycle fatigue criterion and its industrial application[END_REF] for cast iron exhaust manifolds in an industrial context. As a continuation of that paper, we present here a detailed discussion on the determination of a robust fatigue criterion in a thermomechanical context. After an interpretation of the tests results using classical fatigue criteria, such as Manson-Coffin and Ostergren, it is shown that the dissipated energy per cycle can be considered as a good damage indicator.

An energetic approach in thermomechanical fatigue for silicon molybdenum cast iron E. Charkaluk 1 and A. Constantinescu 2 1 P.S.A. Peugeot-Citroën, Direction de la Recherche et de l'Innovation, Chemin de la Malmaison, 91570 Bievres, France 2 L.M.S, Laboratoire de Mécanique des Solides (CNRS UMR 7649), Ecole Polytechnique, 91128 Palaiseau, France

The purpose of this paper is to define a low cycle fatigue criterion in order to predict the failure of engineering structures. The major problem in defining a predictive fatigue criterion is that it should be applicable for structures submitted to complex multiaxial thermo-mechanical loadings but should be identifiable from simple experiments on specimens. After a short critical review of the principal criteria used in low cycle fatigue it will be shown that the dissipated energy per cycle permits a correlation of isothermal and anisothermal results obtained on silicon molybdenum cast iron in the case of specimens and also on structures. The energetic approach has been appraised beginning with the pioneering paper of Feltner and Morrow [START_REF] Feltner | Micro-plastic strain hysteresis energy as a criterion for fatigue fracture[END_REF] who asserted that : "…to start with a specimen in one piece and then after the application of a finite number of load cycles find it to be in two pieces, requires a conversion of energy. The energy necessary to cause fracture is collected in small amounts during the course of the cyclic loading and is observable in terms of strain hysteresis". This approach has been justified by experiments. For example, Luong and Dang Van [START_REF] Luong | Infrared thermographic evaluation of fatigue limit in metals[END_REF] have shown using infrared thermography measurements, that the rate of energy dissipation can be correlated with the fatigue limit of the material and that it can be correlated with classical damage indicators such as plastic strain.

Keywords

The energetic approach has been heavily criticized as the global dissipated energy per cycle cannot yet be split into dissipated heat and damage accumulation in a convincing manner. Being conscious of this drawback, we shall show in that a precise estimate of the dissipated energy can be conducted as a first approximation of a fatigue criterion unifying LCF and TMF experiments. For structures and loads where it is impossible to obtain closed form solutions, the dissipated energy per cycle in TMF experiments will be obtained from numerical computations. Our results show that the method predicts the lifetimes as well as the damage area (macroscopic cracks).

TESTS AND SPECIMENS

The silicon molybdenum cast iron

The studied material is a silicon molybdenum spheroidal graphite cast iron (SiMo SG cast iron with a composition presented in Table 1) employed in the automotive industry for the production of exhaust manifolds which are subject to temperatures ranging between -20°C and 800°C with heating rates of 10-20°C/s.

The metallographic structure of this material exhibits a ferritic matrix with pearlitic zones especially around grain boundaries and graphite nodules homogeneously dispatched in the matrix (see Figure 1). At high temperatures, pearlitic zones grow around grain boundaries with molybdenum precipitates (see Figure 2).

In isothermal tension-compression tests, the material displays mainly kinematic hardening with stable strain-stress loops being obtained after 20 cycles. Its cyclic behaviour can be considered to be saturated after a very short time, as discussed later in detail.

Low cycle fatigue tests (LCF)

Fully reversed isothermal low cycle fatigue tests (R ⑀ = -1) were carried out in a temperature range of 200-700°C. The specimens have a gauge section of 7 ϫ 20 millimeters (diameter ϫ length). The conditions for each temperature and the number of tests for each conditions are given in Table 2.

Thermal fatigue tests (TMF)

The thermal fatigue tests were conducted on a specific test machine. The specimen was clamped between two cantilever beams and heated by the Joule effect, applying a continuous current at its ends. The temperature was controlled by a spot welded thermocouple in the middle of the specimen. A precise scheme of the machine and the measurement points are presented in Figure 3.

The specimen had a central useful part of dimensions: 12 ϫ 40 mm (diameter ϫ length). The temperature distribution in the specimen varies between 40 and 700°C in the middle and within 40 and 200°C in the ends. The temperature in the ends has been maintained low by continuous water circulation in the hollow heads. In each section, the temperature distribution can be considered constant as long as no major crack appears. Thermal FEM computations showed less than 15°C of radial difference between the center and the surface at the maximum 700°C level. At the maximum temperature, during the dwell time, the on-off control of the temperature regulation induces temperature variations of ±15°C. FEM analysis has shown that only 5% of the strain variations on the total cumulated strain was due to the dwell time. Therefore these variations will be neglected in the following.

The maximum stress in the specimen is partly related to the axial rigidity of the machine, i.e. the sum of the axial rigidity of the pillars and the bending rigidity of the cantilever beams. This rigidity depends on the distance between the two pillars and has been considered as a test parameter (see Table 3). The machine acts as a spring clamped in parallel with the specimen at its ends. Therefore the axial stress in the specimen can be estimated precisely by a strain gauge on the pillars. As an example, axial stresses were of ≈30 MPa in the dwell period at maximum temperature 700°C and of ≈200-400 MPa at room temperature, depending on the machine rigidity.

RESULTS

Microscopic observations of fracture surfaces

As an introduction to the fatigue results some macroscopic observations on fractured specimens for LCF and thermal fatigue tests will be presented and compared with similar aspects observed on steels [START_REF] Solomon | Effects of frequency and environment on fatigue crack growth in A286 at 1100F. Fatigue at elevated temperatures[END_REF][START_REF] Gell | Mechanisms of high-temperature fatigue. Fatigue at elevated temperatures[END_REF].

A summary of these observations is that LCF test conditions produce transgranular fatigue crack growth with a final ductile fracture (see Figure 4) and that TMF tests produce intergranular fatigue crack growth with a final ductile fracture (see Figure 5). The inspected fractographic surfaces correspond to 700°C for the LCF tests and to a 1 minute dwell time for the TMF tests.

For LCF tests, the frequency was 3-12 cycles per minute (cpm) corresponding to strain rates of ≈10 -3 s -1 and strain ranges ≈±0.25-±1%.

For TMF tests, the frequency range was 0.06-0.3 cpm and corresponding to thermal strain rates ≈10 -4 s -1 . It is important to notice that the frequency is mainly determined by the dwell time, as the total cycle is approximately divided into 30 seconds of heating, 2 minutes of cooling and 1-15 minutes of dwell time.

It is now obvious that the transgranular and intergranular fatigue crack growth for LCF and TMF tests respectively can also be correlated with the notion of frequency [START_REF] Solomon | Effects of frequency and environment on fatigue crack growth in A286 at 1100F. Fatigue at elevated temperatures[END_REF][START_REF] Gell | Mechanisms of high-temperature fatigue. Fatigue at elevated temperatures[END_REF]. Low frequency TMF tests lead to an intergranular fatigue crack growth characteristic of creep damage whereas high frequency LCF tests lead to transgranular fatigue crack growth. A complete conclusion could only be drawn after performing LCF tests at the same frequency.

Figure 5

Intergranular fracture surface of a thermal fatigue specimen tested between 40 and 700°C with a mean strain rate of 10 -4 s -1 .

Similar results have been reported on steels by Solomon et al. [START_REF] Solomon | Effects of frequency and environment on fatigue crack growth in A286 at 1100F. Fatigue at elevated temperatures[END_REF] (for A286 steel ) and Gell et al. [START_REF] Gell | Mechanisms of high-temperature fatigue. Fatigue at elevated temperatures[END_REF] (for A304 steel). They reported an intergranular fracture surface at low frequencies (< 0.1 cpm in their case), and a transgranular fracture surface at high frequencies (> 5 cpm). In both cases, the frequencies on cast iron are of the same order as that for steels. More than that, TMF tests show that a dwell time with a tensile stress increased the chances for intergranular failure which was also one of the conclusions in [START_REF] Gell | Mechanisms of high-temperature fatigue. Fatigue at elevated temperatures[END_REF].

The observation of fracture surfaces shows that apparently two microscopic mechanisms are driving the failure in different groups of testing conditions. However, in the sequel, we shall focus on the search for a unique global macroscopic quantity characterizing fatigue damage. Therefore the test results will be interpreted from the point of view of classical fatigue criteria, pointing out the difficulties due to the anisothermal and multiaxial context.

Plastic strain range influence

As a first approximation let us interpret the results using the classical Manson-Coffin fatigue criterion [START_REF] Manson | Behaviour of materials under conditions of thermal stresses[END_REF][START_REF] Coffin | A study of the effects of cyclic thermal stresses on a ductile metal[END_REF], based on the cumulated plastic strain:

⌬⑀ p .N f ␤ = c (1)
where ⌬⑀ p is the plastic strain range, N f is the number of cycles to failure and ␤ and c are material constants. The LCF tests results are reported in Figure 6 and Table 4.

For 200, 600 and 700°C the results can be placed in a scattered band with a factor of ±2 on the number of cycles to failure corresponding to the standard deviation.

The lifetime decreased by a significant factor of 10 between 350 and 400°C. This phenomenon might be explained by dynamic strain aging of cast iron at this temperatures already reported for static loadings by Bastid [START_REF] Bastid | Comportement thermomécanique des fontes à graphite sphéroïdal pour collecteurs d'échappement[END_REF]. To our knowledge, no similar observations have been reported for cast iron for cyclic loading. Further experiments and microscopic observations should be therefore carried out to explain this phenomenon.

As a first conclusion, we can state that the cumulated plastic strain for cast iron experiments has not been influenced by temperature, except for the important drop in lifetime at 350 and 400°C due to the brittleness. These results are in opposition with the observations made on different steels. A significant influence of temperature has been reported by Coffin [START_REF] Coffin | Fatigue at high temperatures. Fatigue at elevated temperatures[END_REF] on five different steels (A409, A304, A316, A348, A286) and Taira [START_REF] Taira | Relationship between thermal fatigue and low cycle fatigue at elevated temperature. Fatigue at elevated temperatures[END_REF] on a 0.16% carbon steel.

The results in [START_REF] Coffin | Fatigue at high temperatures. Fatigue at elevated temperatures[END_REF] show that if the maximum temperature of the cycle T max is smaller than 500°C, damage is controlled by dislocation gliding and the value of the exponent ␤ is close to 0.5 whereas if T max is greater than 500°C, damage is controlled by diffusion phenomena inducing creep and ␤ is close to 1.

Coffin also notices [START_REF] Coffin | Fatigue at high temperatures. Fatigue at elevated temperatures[END_REF] the convergence of the fatigue curves for "very low" cycle fatigue tests (less than 100 cycles). This was not observed for cast iron and probably more tests are needed before drawing a definite conclusion on this item.

In the case of TMF tests the amplitude of plastic strain, ⌬⑀ p , lacks a precise mechanical interpretation. On one hand the specimen is subject to a temperature gradient and therefore the value of ⌬⑀ p depends of the spatial point where the measurement takes place. On the other hand the load cycle is anisothermal, the yield limit is temperature dependent and a strain amplitude would encompass different temperatures with different yield limits and damage influences.

Mean stress influence

In order to refine the previous approach including the effect of a mean stress in LCF tests, Smith et al. [START_REF] Smith | A stress-strain function for the fatigue of metals[END_REF], Jaske [START_REF] Jaske | Thermal-mechanical, low cycle fatigue of AISI 1010 steel. Thermal fatigue of materials and components[END_REF] and Ostergren [START_REF] Ostergren | A damage function and associated failure equations for predicting hold time and frequency effects in elevated tempera[END_REF] included stress in the damage indicator of the lifetime prediction. This change should also take into account the influence of the strain dwells as shown in [START_REF] Xi | High temperature fatigue-creep behaviour of single crystal SRR99 Nickel base superalloys: Part II -Fatigue-creep life behaviour[END_REF].

With (1) as the lifetime prediction equation, Smith-Topper-Watson (STW) proposed:

√ max .⌬⑀.E (2)
as a damage indicator. A similar indicator: ⌬⑀ --. max (3) 2 was introduced by Jaske [START_REF] Jaske | Thermal-mechanical, low cycle fatigue of AISI 1010 steel. Thermal fatigue of materials and components[END_REF]. In these relations ⌬⑀, max and E are respectively the total mechanical strain range, the maximum stress of the cycle and Young's modulus.

The lifetime equation proposed by Ostergren [START_REF] Ostergren | A damage function and associated failure equations for predicting hold time and frequency effects in elevated tempera[END_REF] is:

T .⌬⑀ p .N f ␤ . ␤(k-1) = C (4)
where denotes the frequency, T is the tensile stress and ␤ and k are material constants. A simplest form of this expression is used in our case where no hold time is realised during LCF tests. The modified Ostergren relation is of the following form:

T .⌬⑀ p .N f ␤ Figure 6
Manson-Coffin results obtained at different temperatures with the SiMo SG cast iron with the test conditions corresponding to Table 2. The LCF tests results analysed with the Smith-Topper-Watson and the modified Ostergreen relation respectively are plotted in Figures 7 and8. One can observe that the predicted lifetime for the same imposed strain amplitude in LCF tests decreases as a function of increasing temperature using both criteria. The Jaske criterion leads to a smaller dispersion band, due to the variation of Young's modulus with temperature (≈150 GPa at 20°C and ≈80 GPa at 800°C). As a conclusion one can state that the net tensile hysteretic energy permits the smallest scatter of the LCF test results. Unfortunately, the proposed damage indicators do not have a precise physical interpretation and are not intrinsic and therefore not appropriate for lifetime prediction in an anisothermal context.

At this point, in spite of the criticism several remarks can be made. Jaske's criterion represents a direct rough estimation of the dissipated energy per cycle. In an isothermal context, one can easily correlate the amplitude of plastic strain of the Manson-Coffin law with the dissipated energy per cycle. For perfect plasticity, this result is straightforward. For a simple viscoplastic equation, this relation has already been proven by Lemaitre [START_REF] Lemaitre | Evaluation of dissipation and damage in metals submitted to dynamic loading[END_REF].

One can conclude on the importance of the underlying assumptions of Ostergren's criterion:

• only the tensile part of the energy is damaging, as LCF phenomena are driven by growth of crack type defects to a critical size; • damage does not increase unless a critical threshold stress limit is exceeded. Experience show that this limit can often be neglected; • damage is dependent on test frequency. These remarks show that the previous approaches use damage indicators, whose physical interpretation is not too far from a dissipated energy. As a consequence, a precise computation of the dissipated energy could integrate the influence of the preceding phenomena.

In the next section we shall analyse the tests using a energy criterion computed using an adequate elastoviscoplastic constitutive law.

DISCUSSION

The preceding fatigue criteria were completely phenomenological and assumed that the response of a structure to cyclic loading is also cyclic in a strict sense. It is well known [START_REF] Lemaitre | Mécanique des matériaux solides[END_REF] that structures and materials exhibit stress hardening or softening to imposed load, depending on a series of factors like the nature of the material, the temperature or the initial state. Starting from this observation, we shall follow the analyses of Skelton [START_REF] Skelton | Energy criterion for high temperature low cycle fatigue failure[END_REF][START_REF] Skelton | Cyclic hardening, softening, and crack growth during high temperature fatigue[END_REF] in order to define a low cycle fatigue criterion for cast iron structures based on dissipated energy.

Cyclic behaviour and accumulated energy

The SiMo SG cast iron presents a short period of hardening at 200, 350, 400 and 600°C (except at ±1% at 600°C) and one of softening at 700°C. Both are followed by a long plateau, as presented for two temperatures in Figure 9.

An interpretation of this behaviour from the point of view of low cycle fatigue has been given by Skelton in [START_REF] Skelton | Energy criterion for high temperature low cycle fatigue failure[END_REF]. He proposed the identification of three different phases defined respectively by the saturation, tangent and final points, denoted respectively by subscripts : sat , tan , f (see Figure 10): • The hardening/softening phase [0 to N sat ] corresponds to the stabilization of the plastic properties of the material. This phase took for cast iron less than 20 cycles at all temperatures. • The stabilized phase [N sat to N tan ] represented by a flat plateau with small property variations. For cast iron this phase is really flat for the tests at 600°C and presents limited softening for the tests at 700°C (see Figure 9). • a final failure phase [N tan to N f ], where the load drops quickly and failure is inevitable. Conventionally N f is defined by a 5% or 10% load drop.

Reaching the saturation of the plastic cyclic behaviour implies a stabilized cyclic behaviour of the structure.

After Skelton [START_REF] Skelton | Energy criterion for high temperature low cycle fatigue failure[END_REF] the saturation is reached at N sat for hardening and at N tan for a softening material behaviour. Let us denote by W sat and respectively W tan the accumulated dissipated energies at these points and by W f the accumulated dissipated energy at N f . The measured values the dissipated energies W sat , W tan and W f , for cast iron in LCF test at different temperatures, are displayed in Table 5. One can notice that the accumulated energy is very small for the test at temperatures where hardening behavior occurs, i.e. 200, 350, 400 and 600°C. This is directly related to the short saturation phase for cast iron. Typically less than 20 cycles passed before N sat , for softening or hardening material behaviour, compared to the several hundreds of cycles in the stabilized phase, N tan -N sat . As W sat and W tan are computed at two different points for hardening or softening behaviour, this also implies that the cumulated energy will have different values. In the case of A316L steel [START_REF] Skelton | Energy criterion for high temperature low cycle fatigue failure[END_REF][START_REF] Skelton | Cyclic hardening, softening, and crack growth during high temperature fatigue[END_REF], Skelton reported similar values of cumulated energy for both types of behaviour. The values are 10-100 times greater in comparison with these obtained for cast iron (see Table 6). This comparison might seem astonishing if one oversees the hundreds of cycles to the saturation point, N sat , observed for steels which easily explain both differences.

It is important to remark, that during stabilized phase, cracks appear in the specimens. For steels their size develops from 0.004 to 0.02 mm at N sat to 0.1-1.5 mm at N tan or N f as stated by Skelton et al. [START_REF] Skelton | Energy criteria and cumulative damage during fatigue crack growth[END_REF]. Even if no similar observations have been made for cast iron, one can assume that the behaviour should be analogous. It is also well accepted that for the softening behaviour one cannot consider N tan as a saturation indicator, due to the development of cracks. Accordingly, it seems natural to consider N sat as an indicator of the saturation of the cyclic material behaviour. This is also justified by the observation that: W f , the cumulated dissipated energy at N f is well approximated by: N f ϫ ⌬w sat , where ⌬w sat is the dissipated energy per cycle at N sat (see Table 5).

In conclusion, we shall consider that N sat is the stabilized point of the cyclic material behaviour and that ⌬w sat , the dissipated energy per cycle at saturation is a good candidate to characterize the fatigue loading of the material during the cycles. The stabilized phase is then the part where fatigue damage evolves.

Low cycle fatigue criterion

The conclusion drawn by Skelton et al. [START_REF] Skelton | Energy criteria and cumulative damage during fatigue crack growth[END_REF] from observations was a predictive fatigue crack growth criterion as a function of the critical cumulated energy W sat representative of the cyclic behaviour. The crack length observed at N sat for steels in the case of softening or hardening (0.004-0.02 mm) are of the same order of magnitude as the process zone at a crack tip (0.005-0.02 mm). Therefore Skelton et al. chose the representative point of the material behaviour for softened materials at the end of rapid softening and not the tangent point where the crack depth is considerably greater than the process zone size. This permits one to express the crack growth rate da/dN as a function of the cumulated energy to saturation W sat , the elastic modulus E and the external load expressed through the stress intensity factor ⌬K as follows:

This relation is of great interest as it introduces the stabi-

da dN = ⌬K 2 ⋅ (1 -) 2 ⋅ ⋅ E ⋅ W sat
lized behaviour of the material as an important parameter for lifetime prediction. It has, however, a drawback as fatigue design cannot be based on a critical crack approach in some industrial applications. For example, in the case of exhaust manifolds, no macroscopic cracks should form during the lifetime and this can be fixed as a design objective. Monitoring crack initiation, propagation, and critical length is a very tedious, if not impossible, task. However, one can remark that crack growth can be separated into two stages: an initial one where the crack is confined in a small volume and does not influence the macroscopic behaviour of the structure and a final one where the crack length directly influences the load distribution in the structure. The transition point between these two stages is represented by N f . The design criterion will now require that structures do not enter the final stage during their lifetime.

In conclusion, summarizing all the arguments, one can assume that the stabilized behaviour of the cast iron is well represented by ⌬w sat , the dissipated energy per cycle at N sat , and that the lifetime is characterized by N f .

For the LCF tests ⌬w sat was determined from the experimental strain -stress curve at the N sat point. For TMF tests on specimen or structure ⌬w was computed using finite elements assuming elastoviscoplastic material behaviour. The energy was computed for a stabilized cycle on the structure. The number of cycles to failure N f has been ascertained during TMF tests conducted on specimens or exhaust manifolds at the initiation of macroscopic cracks. These results interpreted in terms of predicted versus experimental lifetime are represented in Figure11 and show a good correlation both on specimens and structures. At this point is interesting to remark in Table 5 that the accumulated energy to failure, computed as W f = N f ϫ ⌬w sat has not a constant value but is a function of the loading level. As Halford has already pointed out, in an exhaustive review of results obtained for steels [START_REF] Halford | The energy required for fatigue[END_REF], the accumulated energy to failure, W f , is a monotonic increasing function of the number of cycles to failure and this is a consequence of the loading dependance of W f . This is a direct proof that not all dissipated energy is damaging. In order to determine precisely the damaging part of the dissipated energy more studies should be performed. The dissipated energy per cycle did, however, permit the construction of a material objective TMF criterion.

CONCLUSION

The purpose of this paper was to define a low cycle thermomechanical fatigue criterion in order to predict failure of engineering structures. Based on LCF test results is has been shown that classical criteria are not sufficient to predict lifetime of complex multiaxial structures. A series of notions like frequency or plastic strain range are not intrinsic from a thermomechanical point of view and, therefore, cannot be employed for characterizing fatigue for complex load paths.

The dissipated energy per cycle does not exhibit these drawbacks and permitted us to explain at the same time isothermal and anisothermal results obtained on cast-iron specimens and structures.

A series of questions related to the dissipated energy remain open:

• how can the damaging part of the dissipated energy be estimated? • is there a critical, material intrinsic, value characterizing failure or crack initiation energy? • can one characterize the part of creep and plasticity in the dissipated energy?

  : thermomechanical fatigue, silicon molybdenum, cast iron ␤ = Manson-Coffin and Ostergren relations exponent E = Young modulus ⌬⑀ = total mechanical strain range ⌬⑀ p = plastic strain range k = Ostergren relation frequency exponent N sat = number of cycles to saturation N tan = number of cycles to tangent point N f = number of cycles to failure = frequency max = maximum stress of a cycle T = maximum tensile stress of a cycle ⌬w = dissipated energy density ⌬w sat = experimental dissipated energy density at saturation W sat = cumulated energy to saturation W tan = cumulated energy to tangent point W f = cumulated energy to fracture

Figure 1 Figure 2

 12 Figure 1 Metallographic structure of the SiMo SG cast iron (ferritic matrix, pearlitic area, carbon nodules) at room temperature

Figure 3

 3 Figure 3 Thermal fatigue tests. An electric current is applied between both extremities of the specimen which is heated by Joule effect.

Figure 4

 4 Figure 4Transgranular fracture surface of a low cycle fatigue specimen tested at 700°C with a strain rate of 10 -3 s -1 .

Figure 8

 8 Figure 8Net tensile hysteretic energy obtained with LCF tests results on SiMo SG cast iron with the test conditions corresponding to Table2. All values are in J/mm 3 .

Figure 7

 7 Figure 7 STW function for the SiMo SG cast iron at different temperatures with the test conditions corresponding to Table2. All values are in J/mm 3 .

Figure 9

 9 Figure 9 Load versus number of cycles for a LCF test on SiMo SG cast iron at 600°C (+) and 700°C (o) (De = ±0,25%).

Figure 10

 10 Figure 10 Cyclic behaviour of the material for isothermal LCF tests. Three particular points can be pointed out : N sat , N tan and N f .

Table 1

 1 SiMo SG cast iron composition

	Element	C	Si	Mo	Mg	Ni
	Weight	3.47	3.86	0.63	0.22	0.56

Table 2

 2 Experimental

	±0.25%	±0.5%	±1%

conditions for LCF tests and number of tests for each condition Strain range Temperature (°C) Strain rate (s -1 )

Table 3

 3 Experimental conditions for thermal fatigue tests on cast-iron corresponding to the machine in Figure3

	temperature	clamp rigidity	dwell time	heating rate
	range (°C)	(N/mm)	(s)	(°C s -1 )
	40-700	227,000	60	20
	40-700	227,000	900	20
	40-700	183,000	60	20
	40-700	183,000	900	20

Table 4

 4 Exponent of the Manson-Coffin law at different temperatures

	Temperature	200°C	350°C	400°C	600°C	700°C
	␤	0.58	0.53		0.74	0.56

Table 5

 5 Accumulated energies obtained on SiMo SG cast iron at different temperatures. W sat is taken at N sat for hardened materials and at N tan for softened materials. All values are in J/mm 3 and ⌬w sat is the dissipated energy per cycle at saturation

	Specimen	±⌬⑀ (%)	W sat	W tan	W f	N f .⌬w sat
	200°C	0.25	2.8		7.4	6.8
	200°C	0.5	1.4		3.4	3.5
	200°C	0.5	0.03		3.6	3.6
	200°C	1	0.05		0.3	0.6
	200°C	1	0.04		0.4	0.7
	350°C	0.35	0.07		0.6	0.6
	350°C	0.35	0.1		0.6	0.6
	350°C	0.5	0.02		0.2	0.2
	350°C	0.5	0.05		0.3	0.3
	350°C	0.5	0.07		0.3	0.3
	350°C	1	0.1		0.1	0.1
	400°C	0.25	0.03		0.1	0.2
	400°C	0.25	0.02		0.2	0.3
	400°C	0.25	0.003		0.2	0.3
	400°C	0.5	0.02		0.1	0.1
	400°C	0.5	0.09		0.2	0.3
	600°C	0.25	0.001		1.8	1.3
	600°C	0.25	0.002		1.9	2.5
	600°C	0.25	0.001		1.6	1.2
	600°C	0.25	0.002		1.1	1.1
	600°C	0.5	0.02		0.9	0.9
	600°C	0.5	0.007		0.8	1.2
	600°C	0.5	0.02		1	1
	600°C	1		0.09	0.7	0.7
	600°C	1		0.1	0.5	0.5
	700°C	0.25		0.5	1	1.1
	700°C	0.25		0.6	1.2	1.3
	700°C	0.25		0.6	1	1.2
	700°C	0.5		0.1	0.4	0.4
	700°C	0.5		0.2	0.5	0.5
	700°C	1		0.2	0.4	0.4
	700°C	1		0.3	0.4	0.4

Table 6

 6 Cumulated energies obtained by Skelton[START_REF] Skelton | Energy criterion for high temperature low cycle fatigue failure[END_REF][START_REF] Skelton | Cyclic hardening, softening, and crack growth during high temperature fatigue[END_REF] on different materials

	Values in J/mm 3	W sat mean	W tan mean	W f mean
	A316L	0.69 ± 0.23		1.04 ± 0.18
	9Cr1Mo	0.94 ± 0.07	3.20 ± 0.29
	Nimonic 100		0.48 ± 0.20

Figure 11 Comparison between experimental and predicted lifetime for LCF and thermal tests and for exhaust manifolds.

ACKNOWLEDGMENTS

The authors would like to thank Ky Dang Van and André Bignonnet for fruitful discussion during this work. This work was funded by PSA (France) and their supported is gratefully acknowledged.