Anis Hor
email: anis.hor@ensam.eu

Franck Morel

Jean-Lou Lebrun

Guénaël Germain

An experimental investigation of the behaviour of steels over large temperature and strain rate ranges

Keywords: rheology, compression tests, shear tests, large strain rate and temperature range, forging, machining, 42CrMo4 steel, 100Cr6 steel

During forging and machining manufacturing processes, the material is subject to large strains at high strain rates which provoke local heating and microstructural changes. Modelling of these phenomena requires precise knowledge of the stress-strain constitutive equations for a large range of strains, strain rates and temperatures. An experimental study of the rheology of both hyper and hypo-eutectoid steels (with different microstructrues) over a temperature range from 20 • C to 1000 • C and with strain rates from 10 -2 to 10 5 s -1 , has been undertaken. These tests were performed in compression on cylindrical specimens and in shear using hat-shaped specimens. Both a GLEEBLE 3500 thermo-mechanical testing machine and a Split-Hopkinson Pressure Bar apparatus were used. From these tests, three deformation domains have been identified as a function of the material behaviour and of the changes in the deformed microstructure. Each domain was characterized by its behaviour, including: the competition between hardening and softening, strain rate sensitivity on the flow stress and the softening phenomenon (i.e. recrystallisation or recovery, etc.). Finally, based on thermodynamical considerations, the conditions of thermo-plastic instability (i.e. shear bands, twinning, heterogeneities, etc.) and microstructural changes are highlighted using process maps of the dissipated power repartition.

Introduction

Recent work on thermo-mechanical manufacturing processes such as machining and forging is often based on simulation tools like the finite element method and is done with the aim of predicting the forming process. It can be noted that this work is often focused exclusively on the numerical problems related to the use of the finite element method in this context or the investigation of problems such as material separation, frictional contact, very large deformation, etc. Due to advances in these areas, the simulation of these manufacturing processes is more easily undertaken. However, other factors, such as frictional behaviour and stress-strain constitutive equations become the limiting factors in terms of undertaking accurate numerical simulations. Determining the stress-strain constitutive equations and the associated material parameters is extremely difficult due to the extreme character of the loads to which the material is subject during machining and / or forging operations.

The study of the thermo-mechanical loads seen by a material during forming has been the object of several investigations. For example, Changeux [START_REF] Changeux | Microstructural modifications and material response in machining of steels[END_REF] showed that during machining, the strain can attain values close to 5. M'Saoubi [START_REF] Msaoubi | A new method for cutting tool temperature measurement using ccd infrared technique: Influence of tool and coating[END_REF] estimated that during the machining of steels, the temperature in the secondary shear zone can be up to 800 • C. Calamaz [START_REF] Calamaz | A new material model for 2d numerical simulation of serrated chip formation when machining titanium alloy ta6v[END_REF] showed that the loading mode in machining results in a combination of shear stress and compressive stress. Finally, Montheillet [START_REF] Montheillet | Métallurgie en mise en forme[END_REF] placed the different manufacturing processes in a strain rate versus temperature diagram. He showed that during forging and machining, the material is submitted to strains with a strain rate ranging from 10 -2 to 10 5 s -1 (see Figure 1). Therefore, experimental rheology studies during machining and forging, require the use of tests that cover the complete strain rate range, going from the quasi-static to the dynamic range and the complete temperature range, going from ambient temperature to temperatures close to the melting temperature of the material. However, very little work has been published concerning the determination of the material parameters necessary to identify the stress-strain constitutive equations for the manufacturing processes modelling. The available studies are based on experimental data which is often very limited. In general a minimum number of tests are done in order to identify the chosen constitutive model without comparing its domain of validity with the loads applied to the material during machining [START_REF] Tounsi | From the basic mechanics of orthogonal metal cutting toward the identification of the constitutive equation[END_REF] or forging [START_REF] Mulyadi | Parameter optimisation in constitutive equations for hot forging[END_REF].

Studies considering the experimental behaviour over a large strain rate range and temperature range are rare. Based on data from the literature, Lurdos [START_REF] Lurdos | Loi de comportement et recristallisation dynamique: approches empirique et physique[END_REF] regrouped the experimental test conditions concerning the 316L stainless steel. He showed that much data is missing in the zone corresponding to semi-hot deformation (i.e. 0.3 ≤ T/T f ≤ 0.5) and there is a complete absence of data in the hot dynamic domain (i.e. domain 4 in Figure 1).

Processing maps offer a way of analyzing the constitutional behaviour of a material due to deformation. This approach can be used to highlight microstructural changes that occur during deformation. It is based on the evaluation of the complementary dissipated power (J) using the relation between strain rate (ε) and flow stress (σ) during the deformation process. The experimental data presented in this work are analyzed using this methodology.

Based on thermodynamical considerations, Prasad et al. [START_REF] Prasad | Processing maps : A status report[END_REF][START_REF] Prasad | Modeling of dynamic material behavior in hot deformation : forging of ti-6242[END_REF] proposed that the optimal thermo-mechanical conditions for deformation correspond to the maximum value of J. Indeed, Ziegler [START_REF] Ziegler | Some extremum principles in irreversible thermodynamics with applications to continuum mechanics[END_REF] has shown that the behaviour of the deformation process follows the principle of the maximum entropy production rate, which is equivalent to the principal of the minimum force or the minimum irreversible strain rate in a force or strain rate space. At a given temperature, the rate of the dissipated work is directly proportional to the internal entropy production rate, which, based on the irreversibility principle, is always positive:

P = σ ε = T dS dt ≥ 0 (1)
where σ is the effective stress, ε is the effective strain rate, T is the temperature and dS dt is the internal entropy production rate. The total entropy production rate is composed of two complementary parts. The first part, which is generally larger, is the conduction entropy which is due to heat conduction from the plastically deformed region to the colder regions of the specimen. The second part is due to the microstructural dissipation which reduces the flow stress required for plastic deformation. Ziegler [START_REF] Ziegler | Some extremum principles in irreversible thermodynamics with applications to continuum mechanics[END_REF] represented both contributions in terms of dissipative functions in the strain rate and force space, and showed that the instantaneous total power dissipated (σ, ε) is given by:

P = ˆε 0 σ d ε + ˆσ 0 ε dσ = G + J (2)
The first integral is the content dissipated power G, the second is the co-content or the complementary dissipated power J (see Figure 2-(a)). The shape of the σ = f (ε) curve gives an indication of the part of the dissipated power that is used for heat conduction and the part used for microstructural dissipation. Viscoplasticity is caused by this dissipation due to the microstructure.

A power law is often used to describe the relationship σ = f ( ε) for plastic deformation:

σ = K (T, ε, ε) εm(T,ε,ε) (3)
For fixed values of T and ε, it is possible to examine the quantities ( ∆J /∆G) and ( ∆J /∆P) for small variations of ε in order to obtain an indication of the change in dissipation mechanism. By assuming that the dependence of K and m on the strain rate ε is small, the instantaneous values of ∆J, ∆G and ∆P are given by the integrals:

∆J ≈ ´σ+△σ σ ε dσ ∆G ≈ ´ε 0 +∆ ε0 ε0 σ d ε ∆P ≈ K ( ε + ∆ε) m+1 (4)
Figure 2: Schematic representation of (a) the content (G) and co-content (J) dissipated power for a material with a behaviour given by a curve σ = f ( ε) (the total dissipated power is given by the rectangle) and (b) the maximum complementary dissipated power J max obtained for a sensitivity to the strain rate m = 1 [START_REF] Prasad | Modeling of dynamic material behavior in hot deformation : forging of ti-6242[END_REF].

Prasad [START_REF] Prasad | Processing maps : A status report[END_REF] defined the efficiency of the power dissipation (η) by a linear dissipater (m = 1) so that:

△J /△P ( △J /△P) linéaire = m /(m+1) 1 /2 = 2m m + 1 ≡ η (5)
This highlights the importance of the parameter m, which defines the relative importance of the division of the power between heat production and microstructural change. The value of m is between 0 and 1 for stable flow of a viscoplastic solid [START_REF] Ziegler | Some extremum principles in irreversible thermodynamics with applications to continuum mechanics[END_REF]. An increase in m results in an increase in microstructural dissipation. A negative value of m (commonly referred to negative strain rate sensitivity) is observed for dynamic ageing, where mobile dislocations are repeatedly locked and unlocked by the speed at which solute atoms are moved. In terms of solid mechanics, when m > 1 the term locking solids is discussed [START_REF] Prager | Viscoplasticity and locking solids[END_REF]. These two extremes represent unstable flow.

In this context, the present work consists of a complete experimental investigation of three steels: a) 42CrMo4 with a ferrito-perlitic microstructure, b) 42CrMo4 with a bainitic microstructure and c) 100Cr6 in the annealed state. The compressive and shear loading modes are studied.

The experimental campaign makes it possible to detect all of the macro and microscopic physical phenomena involved in the deformation process and to analyze the temperature and strain rate sensitivity of the materials investigated over a wide range of rheological parameters.

Materials and methods

Materials

This work is focused on two steels often used in machining and forging operations:

-The 42CrMo4 steel (AISI 4140) is a low alloy hypo-eutectoid steel (% C < 0.85) with chromium and molybdenum as main alloying elements. This steel is studied in two initial microstructural states. The first is the steel with an initial ferrito-perlitic microstructure is referred to as 42CrMo4-FP. The microstructure is composed of grains of ferritic and pearlitic (see Figure 3-(a)). The hardness is 248 HB. The second is the steel with an initial bainitic microstructure (see Figure 3-(b)) is referred to as 42CrMo4-BA. This material is harder (approximately 310 HB) than the ferrito-perlitic version. The chemical composition is very similar to that measured for the ferrito-perlitic microstructure.

-The 100Cr6 steel (AISI 52100) is a hyper-eutectoid, low chromium, steel (% C > 0.85). The chemical composition of this steel, determined by spark emission spectroscopy is given in Table 1. This steel is principally used for the manufacture of ring and ball bearings. Bars of the 100Cr6 material were used in the annealed state. Its microstructure consists of carbides dispersed in a ferritic matrix with a low carbon content (see Figure 3-(c)). The hardness is 272 HB. In the following this material will be referred to as 100Cr6-FP.

Experimental tools

In order to test the materials at temperatures ranging from ambient temperature to hot-forging temperatures and to cover the range of strain rates going from 10 -2 to 10 4 s -1 , two different experimental set-ups were used: -A GLEEBLE 3500 thermo-mechanical testing machine. The temperature measurement and control was done using a thermocouple welded to the specimen. The specimen was tested in a primary vacuum chamber. Mechanically, the GLEEBLE can achieve displacement speeds of 10 -4 to 2 ms -1 . The measurement and strain control was done using a quartz longitudinal extensometer placed on the jaws. Depending on the desired speed, two configurations can be used. For low speeds, the normal configuration was used, involving a hydraulic actuator capable of applying a compressive force to the specimen (see Figure 4). For high speeds (> 10 -2 ms -1 ) the hydraulic ram was decoupled in order to accelerate before hitting the specimen. This configuration results in a shock at the moment of impact between the two parts of the decoupled actuator.

-Split Hopkinson Pressure Bar (SHPB) apparatus. This technique based on elastic wave transmission between the bars and the specimen was largely used for dynamic tests. The apparatus used allows strain rates of the order of 10 3 s -1 in compression test. The projectile velocity can reach 25 ms -1 . However, it was not equipped with a heating system, therefore only tests at ambient temperature can be done.

Experimental tests

Two types of tests are analyzed in this study: compression test and shear test on hat shaped specimens. The elastic strain was deleted, taking into account the stiffness of the machine and the anvils.

For high strain rate and high temperature tests, the measured force-displacement curves were smoothed in order to reduce the effect of the shock and the thermal effects caused by the temperature regulation (i.e. the effect of the control system). Precautions were taken to ensure that the temperature and strain distribution within the specimens were uniform. The latter was judged uniform up to a value of 0.5 [START_REF] Hor | Simulation physique des conditions thermomécaniques de forgeage et d'usinage-caractérisation et modélisation de la rhéologie et de l'endommagement[END_REF]. Experimental techniques were also used to reduce the effects of friction (i.e. hot anvils, graphite sheets and nickel paste). The final shape of the specimens were verified to ensure that they remained globally cylindrical. The plastic strain is ε p = εσ E(T ) where E(T ) is the Young's modulus taking into account the machine stiffness evolution with temperature [START_REF] Hor | Simulation physique des conditions thermomécaniques de forgeage et d'usinage-caractérisation et modélisation de la rhéologie et de l'endommagement[END_REF][START_REF] Committee | [END_REF], and ε is the positive true strain. Finally, for tests at high speeds (ε greater than 1 s -1 ), since the strain rate is not uniform, average strain determined by ε = ε f /△t, where ε f is the total deformation and △t is the total test time. The instantaneous and the average strain rates were almost identical, except at the start of the test.

Furthermore, the compression tests described above highlighted the existence of several physical and microstructural changes in the tested steels. These phenomena are however affected by the loading mode. Moreover, the strain (up to 0.5) and the strain rate (up to 10 3 s -1 ) are lower than those encountered during machining for which macroscopic strain localization in the form of shear bands is often observed. Compression tests are generally homogeneous and cannot reproduce this type of localization. For these reasons, shear tests have been conducted on specimens referred to as "hat shaped specimens". They are often used to characterize the formation of adiabatic shear bands [START_REF] Longère | Adiabatic heat evaluation for dynamic plastic localization[END_REF][START_REF] Meyers | Evolution of microstructure and shear-band formation in alpha-hcp titanium[END_REF] and to identify constitutive equations for the machining simulation [START_REF] Changeux | Microstructural modifications and material response in machining of steels[END_REF][START_REF] Habak | A study of the influence of the metallurgical state on shear band and white layer generation in 100cr6 steel: Application to machining[END_REF].

The experimental set-up in the GLEEBLE testing machine and the Hopkinson bars are identical to those used for the compression tests, only the geometry of the specimen is different. The compression of this geometry results in a shear zone (see Figure 6). It starts at the fillet radius (between the two cylinders) and propagates towards the centre. The resulting shear produces a plastic instability (in particular for dynamic tests) close to that observed in the primary shear zone in machining. This shear zone is of very small width (0.1 mm max) which results in a higher equivalent strain rate, compared to the compression tests, with the same displacement rate. For example, for a displacement rate of 1 ms -1 , an equivalent strain rate of 10 4 s -1 can be obtained in shear, compared to 10 2 s -1 in compression. By using the parameters shown in Figure 6, the theoretical shear stress and strain are determined experimentally: This calculation assumes that W = 0.1 mm is the width of the shear zone. Several authors [START_REF] Changeux | Microstructural modifications and material response in machining of steels[END_REF][START_REF] Meyers | Evolution of microstructure and shear-band formation in alpha-hcp titanium[END_REF] have shown that it can vary depending on the imposed thermomechanical cycle (rate and temperature) and the material studied. In this work, the theoretical shear stress and strain have been verified using finite element calculations with the Johnson-Cook constitutive equation [START_REF] Johnson | A constitutive model and data for metals subjected to large strains, high strain rates and high temperatures[END_REF] identified for the 42CrMo4-BA steel. The theoretical values were very similar to the numerical ones, regardless of the thermo-mechanical conditions [START_REF] Hor | Simulation physique des conditions thermomécaniques de forgeage et d'usinage-caractérisation et modélisation de la rhéologie et de l'endommagement[END_REF].

τ theo = F πh d i +d e
The experimental results are presented in the (τ theo , γ p theo ) space. The equivalent strain rate is determined from the shear rate by ε = γtheo / √ 3. A large number of tests have been conducted on two materials: the 42CrMo4-BA steel and the 100Cr6-FP steel. The repeatability of these tests is generally low because of the complex specimen geometry especially at the inner and outer fillet radii which determines the shear zone (see Figure 6). Each test condition is repeated several times until at least three similar curves are obtained, the dimensions of each specimen being verified before testing.

Results

Global behaviour

The 0.2% offset yield stress has been determined from the experimental stress-strain curves in compression. The evolution of this parameter as a function of the temperature shows a change in slope when passing from one deformation domain to another for the three steels investigated In order to analyze the strain rate sensitivity on the compressive yield stress, the temperature dependence of compressive yield stress of the 42CrMo4-FP steel is compared for different strain rates (see Figure 7-(b)). shows that for strain rates between 10 -2 and 1 s -1 , the curves are almost identical. For all temperatures, an increase in the yield stress was observed for strain rates greater than 10 s -1 . This could be thought of as the limit between a quasi-static behaviour,

where the yield stress is effectively independent of the strain rate, and a dynamic behaviour. The temperature sensitivity on the yield stress was not affected by the strain rate. This simple analysis of the sensitivity of the compressive yield stress on two rheological parameters (i.e. temperature and strain rate) highlights the different types of behaviour of the three steels studied. That is, cold, semi-hot and hot deformation, in terms of temperature and quasi-static and dynamic behaviour, in terms of strain rate.

In the following, the material behaviour will be analyzed for both loading cases (compression and shear) in terms of strain rate and temperature sensitivity on the flow stress. The results for the compression tests are given in Figure 8 and in Figure 9 for the shear tests. From Figure 8, it can be seen that for a given strain rate (10 -2 s -1 ) an elevation of temperature translated the stress-plastic strain curves towards the bottom, for all three steels. A radical change could be observed in the shape of these curves. The slope of the curves can be both positive, indicating work hardening, or negative indicating softening. Therefore, as well as causing a decrease in the flow stress, the temperature often plays a role in softening the material. With respect to this competition between work hardening and softening, three deformation domains can be distinguished, as a function of the temperature.

The conclusions made based on the compression tests remain valid for the shear loading mode, in particular the existence of three deformation domains and similar temperature sensitivity on the flow stress (figure 9). Although the attained shear levels were very high ( 6), the saturation stress was observed only for certain temperatures and equivalent strain rates. The equivalent strain rate sensitivity on the flow stress is temperature dependent. This sensitivity is analyzed in the following for the three domains. The phenomenon between 200 and 400 • C observed for the two loading conditions (compression and shear) for the ferrito-pearlitic steels will be discussed later (see section 3.3 and Figure 25). Cold deformation domain (T/T f < 0.3)

In this domain, work hardening is the phenomenon observed at low strain rate, for the three steel studied. An increase in strain rate results in a slight increase in the stress level. The strain rate sensitivity on the stress is practically negligible in this strain rate interval (between 10 -2 and 10 2 s -1 ). However, the strain rate plays a role in terms of softening. The greater the strain rate, the lower the hardening. It even becomes negative for the 42CrMo4-BA steel and the 100Cr6-FP (see Figure 10). The softening phenomenon is a function of the strain rate in this deformation domain.

Softening caused by an increase in the strain rate can appear after a change in the deformation mechanism caused by the speed of the deformation process. It can also be caused by adiabatic heating effect. Indeed, during dynamic test, the plastic power is converted into heat which cannot be evacuated for this range of strain rate. We call this phenomenon self-heating. Note that softening also depends on the microstructure of the material. It is more intense for the 100Cr6-FP and the 42CrMo4-BA steels. At a strain rate of approximately 10 s -1 (depending on the material), the flow curves are of adiabatic nature. The plastic power conversion is responsible for heating the specimen and decreasing the flow strength. The flow stress of dynamic flow curves can be lower than the quasi-static curves after severe deformation. The fact that the heat conductivity of an austenitic structure is lower than that of a ferritic structure should also be taken into consideration. Comparison between the three steels studied.

To evaluate the level of self-heating, the evolution of the temperature was measured during a test in which the initial temperature was 20 • C and with a strain rate of ε = 10 2 s -1 . Figure 11 shows than an increase in the surface temperature of approximately 120 • C was measured during the test. It is very likely that the temperature in the centre of the specimen was higher. This is reinforced by the observed time difference between the rapid compression of the specimen and the appearance of a high surface temperature. A non-negligible period of time was necessary for the surface temperature to increase. A large part of this heat was also dissipated via conduction to the anvils used to compress the specimen or by radiation. This is consistent with the behaviour curves where softening occurs in the final part of the deformation. This phenomenon caused that the flow stress to be lower with respect to the quasi-static tests [START_REF] El-Magd | Characterization, modelling and simulation of deformation and fracture behaviour of the light-weight wrought alloys under high strain rate loading[END_REF]. In the literature, constitutive models are typically based on isothermal flow curves. Consequently adiabatic flow curves should be corrected to isothermal flow curves. The dependency of adiabatic heating should be included in the constitutive equations [START_REF] Meyer | A basic approach for strain rate dependent energy conversion including heat transfer effects: An experimental and numerical study[END_REF]. Self-heating is therefore a phenomenon that provokes a softening effect and reduces the level of work hardening for steel materials. However, it cannot be the only factor responsible for the softening phenomenon, given the comparison between the temperature sensitivity of the steel and the self-heating temperature (see figure 11-(a)). To quantify the influence of self-heating on the stress-strain curves for the 42CrMo4-BA steel, two axisymmetric finite element simulations were carried out. The first one assumes that the flow is isothermal and the second assumes adiabatic flow where the conversion coefficient of the plastic power into heat (Taylor-Quinney factor) is equal to 0.95. The Abaqus Explicit finite element code was used in conjunction with a Johnson-Cook constitutive equation [START_REF] Johnson | A constitutive model and data for metals subjected to large strains, high strain rates and high temperatures[END_REF]. The material parameters were identified in [START_REF] Hor | Simulation physique des conditions thermomécaniques de forgeage et d'usinage-caractérisation et modélisation de la rhéologie et de l'endommagement[END_REF]. Frictional effects were neglected in both cases. The conditions corresponding to a maximum increase in temperature (i.e. for a mean strain rate of 10 2 s -1 and a temperature of 20 • C) were chosen for investigation. This is because increasing the temperature or decreasing the strain rate results in a decrease of the plastic power and therefore causes the self-heating temperature to drop. For the adiabatic flow model, the temperature at the node corresponding to the thermocouples location is of the order of 135 • C for a mean strain of 0.5 (see figure 11-(b)). This result is almost identical to the experimental measurements obtained under the same conditions, but for the 42CrMo4-FP steel. Locally the temperature is higher. Figure 12 shows that, at the center of the specimen, the temperature can reach 180 • C(453 K). Finally the comparison between the isothermal and adiabatic flow curves and the experimental curve for an extreme case (20 • C and ε = 10 2 s -1 ) shows the phenomenon of softening due to the self-heating temperature increase for the experimental case and the adiabatic flow calculation (see figure 13). The trend was reversed when self-heating was neglected and the slope of the stressstrain curve became positive. However, the relative differences between the three curves remained low (below 7%) up to a strain of 0.4. The low conductivity of steel and the tests conducted with high strain rates definitely assist in absorbing a great deal of this error. The study of thermo-plastic stability was made for a range of strain rates between 10 -2 and 10 2 s -1 . Therefore, the effect of self-heating on the flow stress were neglected and the stress levels used to determine the processing maps corresponded to the levels obtained experimentally.

For the shear tests, the stress increased with the equivalent strain rate. For equivalent strain rates between 10 -1 and 10 3 s -1 , the increase was modest. From 10 3 s -1 , the rate sensitivity became important and the shape of the curves also changed for both steels (see Figure 14). These observations are consistent with the phenomena observed during compression tests. For the three materials studied, the strain rate sensitivity on the flow stress was low in the cold deformation domain. However, the strain rate range studied in the compression tests did not allow the observation of fast dynamic behaviour.

Concerning the shape of the quasi-static curves, the flow stress increased to a saturation state for strains of the order of 1.5. Beyond this value, the curves stabilized until the appearance of macroscopic damage (which was removed during the treatment of the curves). From the equivalent strain rate of 10 3 s -1 , the curves showed a local maximum and then decreased gradually. Damage was not observed in this case. This softening was due to self-heating. For high equivalent strain rates, the small shear zone, in which occured a large degree of deformation, underwent a rapid and localized increase in temperature due to plastic dissipation. Numerical simulation showed [START_REF] Changeux | Microstructural modifications and material response in machining of steels[END_REF] that this temperature increase can reach several hundred degrees Celsius, which explains the large decrease for high equivalent strain rates.

Semi-hot deformation domain (0.3 ≤ T/T f ≤ 0.5)

The phenomenon which dominates in this domain is softening. The slope of the stress-strain curves became negative (see Figure 15) for all strain rates and for the three steels. The strain rate did not have an influence on the softening but rather on the stress level. This structural softening was not provoked by self-heating but rather by a change in the deformation mechanisms.

The dislocation mobility was perturbed by thermally activated phenomena such as dynamic recrystallisation. These phenomena appeared even more clearly for the 42CrMO4-BA steel where oscillations in the stress-strain curve were observed (T = 600 • C and ε = 10 -1 s -1 ).These oscillations result from an interaction between interstially alloyed elements and dislocations, due to the diffusion of such elements and the pinning of dislocations. Obstacles have a narrow stress field and dislocations may climb. On the opposite, athermal indicates a long range stress field, which may not be overcome by a thermal activated process, no experimental results were obtained that confirm the appearance of this phenomenon in this temperature range. Finally, with respect to the first domain, the flow stress is more sensitive to the strain rate or the material is more viscous.

For the shear tests, for all equivalent strain rates, the shear stress-strain curves are characterised by a local maximum followed by a decrease in the flow stress towards a saturation stress. The rate of this softening, increased with increasing equivalent strain rate. This is a result of the effect of the very localised self-heating in the shear zone of the specimen. The curves in Figure 16-(a) show the limits of the compression tests which only highlighted part of the phenomena accompanying plastic deformation. By limiting these stress-strain curves to a shear strain of 2 (which is equivalent to a strain of 0.5 in compression) it can be seen that the curves are almost Comparison between the three steels studied.

identical to those obtained in compression with a local maximum followed by a decrease in the slope which is identical for all strain rates. With these results, it can be concluded that depending on the level of deformation, the equivalent strain rate sensitivity on the flow stress changes, in particular, at saturation, this sensitivity completely changes sign. Finally, this phenomenon is not observed in compression, not only because the strain and the strain rates are lower, but also because the test is more homogeneous and the self-heating is less intense and less localized. Hot deformation domain (T/T f > 0.5)

The austenization temperature is between 800 and 900 • C. At these temperatures the material starts to transform from a body centred cubic (BCC) structure to a face centred cubic (FCC) structure. In this temperature interval, the shape of the stress-strain curves changes completely (see Figure 17). The stress is more sensitive to the strain rate than in the cold and semi-hot deformation domains. The material behaves in a more viscous manner. Like in the cold domain, the strain rate has an influence on the competition between work hardening and softening. The effect is opposite to the one observed in the first temperature interval. That is, the higher the strain rate, the lower the level of softening and the higher the work hardening. Also, softening that occurs at lower strain rates is different to that observed in the cold and semi-hot deformation domains. The curves drop before achieving a saturation stress at a strain of between 0.3 and 0.4 for the three steel studied.

This saturation stress was never achieved in the two other deformation domains. Comparison between the three steels studied.

For the shear tests, the trends are identical to those encountered in compression even for higher strain rates. The equivalent strain rate sensitivity on the stress is limited to a significant increase with strain rate increments. There is no softening phenomena resulting from a local self-heating (Figure 16-(b)). For these temperatures, the flow stress is so low that the power dissipation causing self-heating remains low regardless of the equivalent strain rate and the level of strain.

Montheillet [START_REF] Montheillet | Métallurgie en mise en forme[END_REF] showed that, for materials with low stacking fault energy, it is possible to reach the saturation stress for strains inferior to 1. For the case of materials with high stacking fault energy, the stationary regime is obtained for strains of approximately 50. In this work, it has been confirmed that following austenitization the material behaviour changes from that of a material with high stacking fault energy (Figure 18 To better understand this transition, tests have been done close to the austenitization temperature (see Figure 19). At 700 and 750 • C, the shape of the curve is similar to the behaviour seen in the semi-hot deformation domain. Continuous softening is observed up until the end of the curve.

From 800 • C onwards, softening converges to a saturation state. The change in phase then occurs at a temperature between 750 and 800 • C for the 42CrMo4-FP steel.

Processing Maps

Efficiency maps

Efficiency maps represent the variation of the efficiency (equation ( 5)) as a function of temperature and strain rate, in the shape of an iso-efficiency contour plot. The parameter η can also be written as η = J /J max . The variation of the efficiency η as a function of the thermo-mechanical conditions therefore represents the evolution of the complementary dissipated power J for this temperature and strain rate range. Several metallurgical mechanisms can contribute to the cocontent dissipation during hot deformation. Often, for materials with complex microstructures or for alloys with two phases, these microstructural phenomena occur simultaneously and / or are interactive. The value of J is the global result of these interactions. The metallurgical phenomena such as dynamic recovery (DRV) and dynamic recrystallization (DRX), internal fracture (i.e. the formation of cavities or the opening of micro-cracks), dissolution or growth of particles or phase, phase transformation or strain-induced precipitation, contributing to the change in power dissipation [START_REF] Prasad | Modeling of dynamic material behavior in hot deformation : forging of ti-6242[END_REF]. In the case of hot forming, the most favorable conditions are those corresponding to the highest values of J (where η is the highest). In the maps shown in Figure 21, the parameter η is presented in terms of a percentage. For the 42CrMo4-FP steel (see Figure 21-(a)), high values of η (up to 40%) are observed in the temperature range close to the phase change temperature. A slight increase in this parameter is also observed close to a temperature of 400 • C. It is at this temperature that the phenomenon of softening of the flow stress starts. An important variation of this parameter as a function of the strain rate is also detected for a temperature of approximately 1000 • C. This variation can be explained by the phenomenon of recrystallization which is sensitive to the strain rate. For the 42CrMo4-BA steel, the observations are almost the same. An increase in the dissipation power, close to the austenitization temperature is also seen. However, unlike the previous case, no increase in the efficiency η is observed between the temperatures range of 200 to 400 • C (see Figure 21-(b)). For the 100Cr6-FP steel the same peeks in dissipation power have also been observed (see Figure 21-(c)).

Instability maps

This map is based on the instability criterion derived from extreme value principles of irreversible thermodynamics applied to large deformation continuum mechanics [START_REF] Ziegler | Some extremum principles in irreversible thermodynamics with applications to continuum mechanics[END_REF]. As per the first map, the principle of dividing the dissipation power is also applied so that flow instabilities caused by microstructural changes are revealed by this map. The criterion is given by the dimensionless parameter ξ for unstable flow:

ξ (ε) = ∂ ln [ m /(m+1)] ∂ ln ε + m ≦ 0 (7)
The temperature and strain rate regimes where ξ is negative implies unstable plastic deformation. Instabilities typically appear in the form of adiabatic shear bands or strain localization in the microstructure. The robustness of this criterion has been largely validated by microstructural observations [START_REF] Prasad | Processing maps : A status report[END_REF][START_REF] Prasad | Modeling of dynamic material behavior in hot deformation : forging of ti-6242[END_REF]. This approach is often used for temperature ranges and very low strain rates, included in the hot deformation and quasi-static domains. In this work, an attempt is made to apply the complementary power dissipation method over a wide range of thermo-mechanical conditions. For the 42CrMo4-FP steel, the hot deformation domain is generally stable, regardless of strain rate. Instabilities appear for the cold deformation domain and more precisely between 200 and 400 • C (see Figure 22-(a)). These instabilities could be the source of a flow stress at 400 • C that is higher than the one determined at 200 • C.

Based on Ziegler criterion, two zones, stable and unstable depending on the sign of ξ, must be observed. Chiba et al. [START_REF] Chiba | Construction of processing map for biomedical co-28cr-6mo-0.16n alloy by studying its hot deformation behavior using compression tests[END_REF] justified the existence of a metastable zone observed for values of ξ between -1 and 0 by microstructural observations. This zone is generally in between stable and unstable flow. For the 42CrMo4-BA steel, it is observed in cold flow plastic conditions (see Figure

22-(b))

. A metastable state is observed for slightly higher temperatures, the metastability being probably caused by stress softening. The deformation of the 100Cr6-FP steel is more stable than the two previous materials. Indeed, the zones of instability and metastability are reduced.

Finally, according to Prasad et al. [START_REF] Prasad | Modeling of dynamic material behavior in hot deformation : forging of ti-6242[END_REF], these instabilities are often observed during cold deformation due to strain localization and the appearance of shear bands. This result confirms previous findings on the influence of increasing strain rates on the microstructure. Indeed, it has been noted that this increase has an effect on the material principally for cold deformation, by the appearance and localization of shear bands due to self-heating and that for hot deformation, it causes only an increase in flow stress.

Microstructural changes

To extract additional information concerning microstructural changes, metallurgical analyzes of the deformed specimens were done. Hardness measurements and SEM observations were conducted at the center of the specimen section after testing. These examinations also help to distinguish the three domains of deformation:

Figure 23: Hardness of the cylindrical specimens after testing at different temperatures and at a constant strain rate of (10 -2 s -1 ) and for a constant strain of 0.6. Comparison between the three steels studied.

-For cold deformation: Hardness remains constant in this domain and is higher than the initial hardness for the three steels (see Figure 23). This is due to strain hardening. The microstructure does not change.

-For semi-hot deformation: The hardness gradually decreases and above 600 • C becomes lower than the initial hardness of the material (the ratio of hardness/initial hardness is less than 1) except for the 42CrMo4-BA steel. There is more softening than work hardening in this case. This result is confirmed by the stress-strain curves where the flow stress at the end of the deformation is less than the initial yield strength.

-For hot deformation: The austenitization temperature of these steels is between 750 and 800 • C (see Figure 20). Above this temperature, the material begins to transform from a BCC structure to FCC structure. In this temperature range, the deformed microstructure is not the same, it is partially or completely transformed. This explains the change in shape of the curves. The hardness (normalized by the initial hardness) increases to 2 (see Figure 23).

The material is completely transformed during deformation and the final microstructure, recovered after cooling, is martensite.

Metallurgic observations, of the 42CrMo4-FP steel for example, confirm these findings (see Figure 24). Indeed, for temperatures below 800 • C, the microstructure is still ferrito-pearlitic and above 800 • C, it is transformed into martensite. It should be noted that the average cooling rate in the Gleeble is between 30 and 50 • C/s. Microscopically, no change in grain size was observed during softening, although a decrease in the hardness of the deformed specimens was often noticed when the curve exhibits softening behaviour.

Finally, a phenomenon is observed for the 42CrMo4-FP and the 100Cr6-FP steels, in which the flow stress in compression or shear at 400 • C is higher than at the temperature of 200 • C. This phenomenon disappears for the bainitic microstructure of the 42CrMo4 steel. In order to observe changes in the shape of the stress-strain curves for intermediate temperatures in this range, a more detailed study was carried out for the 100Cr6-FP steel (see Figure 25). For temperatures between 20 and 350 • C, the stress increased with temperature. The maximum stress gradually decreased and the characteristic "bump" shape of the stress-strain curve for this steel progressively disappeared.

The hardening that follows this local maximum was also slightly increased. From a temperature of T = 350 • C, the local stress maximum disappeared, the hardening rate decreased and the evolution of stress with temperature decreased with increasing temperatures. This study was complemented by microscopic observations and an analysis of the hardness (see Figure 25). The microstructure does not exhibit significant changes: carbides have same size. For temperatures up to 350 • C, the hardness increased and then decreased, which validates the findings concerning work hardening.

T < 800 °C T ≥ 800 °C

Figure 24: The microstructure of deformed specimens at different temperatures, at a constant strain rate of (10 -2 s -1 ) and for a constant strain of 0.6. For the 42CrMo4-FP steel.

This phenomenon may be identical to instabilities encountered in the literature [START_REF] Ranc | Some aspects of portevin-le chatelier plastic instabilities investigated by infrared pyrometry[END_REF][START_REF] Ranc | Experimental study by pyrometry of portevin-le chatelier plastic instabilities -type a to type b transition[END_REF]. For the phenomenon of ageing dynamics, this inversion of the sensitivity of the stress as a function of temperature, can be observed. This phenomenon usually appears in the ferritic phase of steels alloyed with manganese for temperatures between 20 and 400 • C. It corresponds to the formation of manganese carbides and nitrides following the association of carbon and nitrogen atoms, which are in solid solution in the ferrite, with manganese atoms. This transformation can be enhanced by plastic deformation and is very sensitive to the deformation temperature. Wagner [START_REF] Wagner | Measurement of sensitivity to dynamic strain aging in c-mn steels by internal friction experiments[END_REF] showed that a ferrito-pearlitic steel A48 (with 0.769 % Mn, 0.198 % C and 0.0083 % N) has a higher tensile yield stress at 200 and 300 • C compared to 20 • C. These phenomena occur at the atomic scale and require special means of observations such as a TEM to observe the precipitates formed during deformation and internal friction tests (mechanical spectroscopy) to quantify the distribution of the atoms in solid solution [START_REF] Wagner | Measurement of sensitivity to dynamic strain aging in c-mn steels by internal friction experiments[END_REF]. For these reasons, very few microscopic investigations of ageing dynamics have been undertaken. Finally, the inversion of the strain rate sensitivity on the stress for the same steel but with a different microstructure (bainite) confirms that the phenomenon explaining this inversion is ageing dynamics and appears only in the ferritic phase.

Discussion

This investigation of the deformation behaviour of three steels over a large range of strain rates and temperatures, for two loading conditions (i.e. compression and shear), has highlighted the existence of three deformation domains (cold, semi-hot and hot deformation) that can be defined by certain temperature ranges. The shape of the stress-strain curves and the evolution of the flow stress as a function of the temperature are very different in each domain. As well as the influence on the flow stress, the strain rate plays a role on phenomena accompanying plastic deformation such as work hardening and softening. Comparison between the three steels shows much similarity in terms of thermomechanical behaviour (see Figure 26). Indeed, despite their compositional and microstructural variations, temperature and strain rate sensitivity is almost identical. The observed phenomena in each deformation domains are very similar, which is encouraging from a modelling point of view. The development of a global constitutive law that can be identified over such a large range of rheological parameters appears to be possible. However, small differences have also been observed in terms of the shape of the stress-strain curves at low temperature and low strain rate, especially for the 100Cr6 steel, where local maximum appears at the start of the plastic domain, followed by an increase in the flow stress. The globular microstructure is perhaps the origin of this local maximum that subsequently disappears when temperature is increased.

Finally, this rich experimental database offers the possibility to propose and adjust more sophisticated constitutive models that can account for certain phenomena essential to the improvement of numerical simulations of manufacturing processes. Indeed, Calamaz [START_REF] Calamaz | A new material model for 2d numerical simulation of serrated chip formation when machining titanium alloy ta6v[END_REF] showed that the inclusion of thermal softening in a modified Johnson-Cook model [START_REF] Johnson | A constitutive model and data for metals subjected to large strains, high strain rates and high temperatures[END_REF] results in a more accurate chip form for the simulation of the machining of TA6V for different cutting conditions. Lurdos [START_REF] Lurdos | Loi de comportement et recristallisation dynamique: approches empirique et physique[END_REF] modified the Voce law [START_REF] Voce | The relationship between stress and strain for homogeneous deformation[END_REF] in order to propose a more general thermo-visco-plastic model that takes into account certain phenomena observed over a wide range of rheological parameters such as the local maximum observed for dynamic tests on a 316L stainless steel. These studies and the experimental work conducted by here show that the three main phenomena observed (the effect of the temperature, the effect of the strain rate and strain hardening-softening) cannot be decoupled.

Strong coupling in the modeling of these effects is therefore necessary. Multiplicative constitutive equations such as the Johnson-Cook model, which does not include coupling, cannot be applied to manufacturing processes such as machining where the material is subject to a very wide range of temperatures and strain rates. A comparison between the different classes of constitutive models: empirical models (coupled and uncoupled), semi-empirical models and physical models, applied to the numerical simulation of orthogonal cutting and to plane strain tests is presented in [START_REF] Hor | Simulation physique des conditions thermomécaniques de forgeage et d'usinage-caractérisation et modélisation de la rhéologie et de l'endommagement[END_REF].

In summary, the dissipation and instability maps show that the three steels investigated can be used for hot forming processes. This is the domain in which their power dissipation is a maximum and they have greatest stability, especially for the 100Cr6-FP steel where the stability zone is large, which confirms the good forgeability of this steel for the manufacturing of bearings. The approach based on the complementary power dissipation is often used to characterize the dynamic behaviour during deformation process. Despite the thermodynamic considerations, this approach is very phenomenological. The processing maps do not fully estimate the microstructural changes.

Montheillet et al. [START_REF] Montheillet | Modeling of dynamic material behavior : a critical evaluation of the dissipator power co-content approach[END_REF] have discussed the short-comings of this approach. They showed that these maps do not enable estimation of the conditions for dynamic recrystallization. In terms of the present investigation, even though the saturation stress was not used, the dissipation and instability maps can highlight certain phenomena but no definitive conclusions can be made about their existence. Alone they are insufficient to establish favourable conditions for deformation.

Additional information must be provided by microstructural observations. The determination of these maps also requires the implementation of several tests at different temperatures and very low strain rates. Interpolation is often used to complete the required database. This can lead to undetection of a number of phenomena occurring at specific temperatures or strain rates. In terms of the present study, despite the extensive database with broad range of rheological parameters, interpolation using Bezier polynomials was necessary to determine the maps.

One of the major problems of studies concerning rheology in machining and forging is the lack of experimental data. This penalizes numerical analysts who use phenomenological and very simple models (such as the Johnson Cook model [START_REF] Johnson | A constitutive model and data for metals subjected to large strains, high strain rates and high temperatures[END_REF]) [START_REF] Barge | Numerical modelling of orthogonal cutting: influence of numerical parameters[END_REF], the identification of which requires only a few tests [START_REF] Tounsi | From the basic mechanics of orthogonal metal cutting toward the identification of the constitutive equation[END_REF][START_REF] Zemzemi | Identification of a friction model at tool/chip/workpiece interfaces in dry machining of aisi4142 treated steels[END_REF] and that have a limited range of validity, which is more restrictive than the range of the rheological parameters observed in machining or forging operations. Also, it is possible to find in the literature different values of the same model parameters for identical materials [START_REF] Huang | Force modelling in shallow cuts with large negative rake angle and large nose radius toolsapplication to hard turning[END_REF][START_REF] Guo | Dynamic material behavior modeling using internal state variable plasticity and its application in hard machining simulations[END_REF]. Contradictions in the experimental results can also be found depending on the different experimental devices used to cover a wide range of rheological parameters.

Conclusion

In this work, an experimental study of the rheology of two major families of steels with different microstructures was conducted over a wide range of temperatures and strain rates, characteristic of forging and machining forming processes. Three deformation domains have been distinguished depending on the behaviour and the changes in microstructure (both hardness and phase changes). Following this study, each domain was characterized by its thermomechanical behaviour (i.e. the competition between hardening and softening, the strain rate sensitivity on the stress, softening (recrystallization or recovery ),etc.). Finally, based on thermodynamical considerations, the conditions of thermomechanical instabilities and microstructural changes are highlighted by mapping the distribution of dissipated power.

To conclude, it is hoped that this comprehensive investigation of the rheology of steels over a wide range of deformation, temperature and strain rate will allow the development of more elaborate constitutive laws that take into account certain phenomena active in forming processes such as machining and forging. This will subsequently improve the numerical simulation of such process by introducing more precise stress-strain constitutive equations, in which the model parameters can be identified in a deterministic manner.

Figure 1 :

 1 Figure 1: Imposed strain rate (in s -1 ) and temperature ranges in forging and machining operations and different domains of mechanical behaviour.

Figure 3 :

 3 Figure 3: The initial microstructures of the three steels studied: (a) 42CrMo4-FP, (b) 42CrMo4-BA and (c) 100Cr6-FP.

Figure 4 :

 4 Figure 4: The three configurations used for compression (and shear) tests done using a GLEEBLE machine and a Split Hopkinson Pressure Bars apparatus.

Figure 5 :

 5 Figure 5: (a) Applied thermal-mechanical cycle, (b) the different test conditions investigated.

2 2 =0. 1

 21 and, γ theo = tan α = △l W (6) Where d i and d e are the interior diameter of the hollow part and the exterior diameter of the solid part of the specimen respectively. h = 2 mm is the length of the shear zone, W = d i -d e mm is the theoretical width of this zone and △l is the displacement measured by a longitudinal extensometer.

Figure 6 :

 6 Figure 6: Theoretical shear zone.

(see Figure 7 -

 7 (a)). For the cold deformation domain (T/T f < 0.3), the influence of temperature on the yield strength is small. However, during semi-hot deformation (i.e. 0.3 ≤ T/T f ≤ 0.5) this material parameter becomes very sensitive to the temperature. Finally, in the hot deformation range (i.e. T/T f > 0.5) the yield strength is very low and is almost independent of temperature.

Figure 7 :

 7 Figure 7: Evolution of the compressive yield stress as a function of the temperature (a) for the three steels investigated, and (b) for the 42CrMo4-FP steel, at different strain rates.

Figure 8 :

 8 Figure 8: Evolution of the compressive stress as a function of plastic strain for the three steels investigated at different temperatures and at a strain rate of 10 -2 s -1 .

Figure 9 :

 9 Figure 9: Evolution of the shear stress as a function of the plastic shear strain for (a) the 42CrMo4-BA steel and (B) the 100Cr6-FP steel, at different temperatures and an equivalent strain rate of 10 2 s -1 .

Figure 10 :

 10 Figure 10: Evolution of the compressive stress as a function of the plastic strain at 20 • C for different strain rates.

Figure 11 :

 11 Figure 11: Increase of the temperature during deformation due to self-heating for a compression test at T = 20 • C and ε = 10 2 s -1 . Comparison betwen (a) The experimental measurement the 42CrMo4-FP steel and (b) The FE prediction for the 42CrMo4-BA steel.

Figure 12 :

 12 Figure 12: Comparison between equivalent strain, temperature and equivalent stress maps obtained by FE calculation with Abaqus Explicit for (a) adiabatic flow and (b) isothermal flow.

Figure 13 :

 13 Figure 13: Comparison between the stress-strain curves for the case of adiabatic flow (with a Taylor-Quinney factor = 0.95) and isothermal flow for the 42CrMo4-BA steel at T = 20°C and ε = 10 2 s -1 .

Figure 14 :

 14 Figure 14: Evolution of the shear stress as a function of the plastic shear strain at 20 • C for different equivalent strain rates, for steels (a) 42CrMo4-BA and (b) 100Cr6-FP.

Figure 15 :

 15 Figure 15: Evolution of the compressive stress as a function of the plastic strain at 600 • C for different strain rates.

Figure 16 :

 16 Figure 16: Evolution of the shear stress as a function of the plastic shear strain at (a) 600 • C and (b) 1000 • C for different equivalent strain rates for the steel 42CrMo4-BA.

Figure 17 :

 17 Figure 17: Evolution of the compressive stress as a function of the plastic strain at 1000 • C for different strain rates.

  -(b)) to one with low stacking fault energy (Figure 18-(a)).

Figure 18 :

 18 Figure 18: Characteristic stress-strain curves for a metal with (a) low stacking fault energy and (b) high stacking fault energy [4].

Figure 19 :

 19 Figure 19: Evolution of the compressive stress as a function of plastic strain for a constant strain rate of (10 -2 s -1 ) and for different temperatures close to the phase change temperature for the 42CrMo4-FP steel.

Figure 20

 20 Figure 20 shows thermal expansion curves, for the three steels, measured during heating of the specimens to 1000 • C. A brutal change can be seen, which corresponds to the phase transformation 21

Figure 20 :

 20 Figure 20: Evolution of the thermal dilation as a function of the temperature during the heating step.

Figure 21 :

 21 Figure 21: Efficiency maps (in %) for the three steels studied: (a) 42CrMo4-FP, (b) 42CrMo4-BA and (c) 100Cr6-FP.

Figure 22 :

 22 Figure 22: Unstable flow maps for the three steels investigated: (a) 42CrMo4-FP, (b) 42CrMo4-BA and (c) 100Cr6-FP.

Figure 25 :

 25 Figure 25: Behaviour curves for the 100Cr6 steel loaded in compression for a strain rate of ε = 10 -2 s -1 and for temperatures ranging from 20 to 500 • C and the hardness of the deformed specimens.

Figure 26 :

 26 Figure 26: Comparison of the shape of the stress-plastic strain curves for three steels studied in the three deformation domains.

Table 1 :

 1 Chemical composition of the three steels studied.

	Element	C	Cr	Mn	Si	Cu	Ni	Mo	S
	42CrMo4-FP (wt%) 0.40 1.10 0.85 0.23 0.24 0.10 0.20 0.06
	42CrMo4-BA (wt%) 0.43 1.16 0.72 0.175 0.2 0.21 0.17 0.04
	100Cr6-FP (wt%)	1.03 1.41 0.33 0.24 0.14 0.13 0.02 0.01


* Miemczyk
email: jmiemczyk@audencia.com

J & Johnsen
email: tjohnsen@audencia.com

Sustainability in purchasing and supply: defining the territory

The purpose of this paper is to provide a review of definitions for sustainability in the area of purchasing and supply. In order to do this a literature review is used to define the term sustainability at different levels of supply including the: 1) purchasing, procurement and sourcing 2) supply chain and 3) network levels. Furthermore, measures used in recent studies are also presented in order to demonstrate the typical scope of studies.

Introduction

This differentiation in levels of analysis is important, for example Seuring and Muller (2008 pg 460) show "supplier management (particularly addressing issues at the supplier-buyer interface)" and "Supply chain management (issues across all companies involved in the supply chain)". Moving further from traditional conceptualizations we also include the network level of analysis. There are a number of reasons why networks are relevant here. As Roome (2001 p.) aptly states "we increasingly see ourselves part of a network society living in a network age", "concerns about complex environmental and social consequences of industrial activity have provoked the need for more frequent and meaningful engagement between companies and stakeholders" and "networks have an identified role in innovation for environmental management and sustainable development", as well as "knowledge suggests that ecosystems are based on organisms connected through complex networks of energy and material flows". Alongside these definitions of sustainability we also provide a review of the principal measures used to gauge the degree of sustainability at these different levels of analysis. These measures include those used in empirical and conceptual frameworks as well as those typically used in practitioner models. In summary we ask: what are the definitions and measures used to describe sustainability at different levels of analysis in the area of purchasing and supply?

Method

This paper is a literature review. One of the difficulties in defining sustainability at different levels in a supply chain is that authors often describe their research as addressing supply chains when in fact the level of analysis is rather more restricted for example to a buyer or supplier perspective. Hence we are not limiting this review to empirical works (as these are rather limited across all levels), but to include other conceptual, theoretical papers as well. At the same time the review does not consider analytical modeling approaches, though there are significant research studies in this area we are more concerned with what firms are actually doing. The research asks what are the definitions and measures of sustainability at different levels in the supply chain. Papers are analyzed regarding the conceptual and empirical levels of analysis. At the same time we search for explicit and implicit measures used to detect differences in elements of sustainability at these different levels of SCM.

For the initial search of relevant papers the authors used the well established databases Emerald and Ebsco. For this first part of the study, journal article titles were searched. Using the following search terms. We used a number of criteria for the selection of articles: relevance, contains definitions and measures, recent (we decided an arbitrary cut off point at year 2000). As this is a continuing study, many of the limitations stem from the methodology. Hence, from this initial review we intend to extend the review further to provide an exhaustive list (attempting to achieve saturation in the results).

Table 2: Journals used in the review

There have been a number of studies that have taken the purchasing level of analysis [START_REF] Bowen | The role of supply management capabilities in green supply[END_REF][START_REF] Grankvist | The impact of environmental information on professional purchasers' choice of products[END_REF][START_REF] Green | Purchasing and environmental management: interactions, policies and opportunities[END_REF], 1998;[START_REF] Min | Green Purchasing Strategies: Trends and Implications[END_REF][START_REF] Min | Green purchasing practices of US firms[END_REF]. While [START_REF] Walker | Drivers and barriers to environmental supply chain management practices: Lessons from the public and private sectors[END_REF] describe green supply chain practices, the evidence is collected from a purchasing perspective, whereby "In this context green supply chain management practices are understood as supply management activities that attempt to improve the environmental performance of purchased inputs, or of the suppliers that provide them" (pg 75). Sustainable sourcing is also a term often used by practitioners but has not become a mainstream academic construct. Variants on this include ethical sourcing (Blowfield 2000), and research in this area tends to focus on guidelines and codes of practice for managing relationships (trading) with suppliers along social and environmental dimensions, in particular using quality control (audit) type systems (Hamprecht et al 2005). Other implications include organizational integration issues to support new sourcing priorities (Koplin et al 2006). The following table provides examples of definitions of sustainability at the purchasing, procurement or sourcing level, typically involving the purchasing department or the dyad (buyer-supplier). an environmentally conscious purchasing initiative that tries to ensure that purchased products or materials meet environmental objectives set by the purchasing firm, such as reducing the sources of wastages, promoting recycling, reuse, resource reduction, and substitution of materials (Carter et al., 1998;[START_REF] Min | Green purchasing practices of US firms[END_REF]Zsidisin and Siferd, 2001).

2010

El Tayeb et al

Sustainable supply chain management

Few studies have attempted to address sustainability across the whole supply chain [START_REF] Seuring | Core issues in sustainable supply chain management -a Delphi study[END_REF]. Work that claims to address the supply chain level is fairly diverse in scope [START_REF] Beamon | Designing the Green Supply Chain[END_REF][START_REF] Gavaghan | The Greening of the Supply Chain[END_REF][START_REF] Handfield | Green' value chain practices in the furniture industry[END_REF][START_REF] Lambert | Life-Cycle Chain Analysis, Including Recycling[END_REF][START_REF] Murphy | Green logistics: comparative views of environmental progressives, moderates and conservatives[END_REF][START_REF] Preuss | Rhetoric and reality of corporate greening: a view from the supply chain management function[END_REF][START_REF] Rao | Do green supply chains lead to competitiveness and economic performance[END_REF][START_REF] Van Hoek | From Reversed Logistics to Green Supply Chains[END_REF][START_REF] Walker | Drivers and barriers to environmental supply chain management practices: Lessons from the public and private sectors[END_REF][START_REF] Walton | The Green Supply Chain: Integrating Suppliers into Environmental Management Processes[END_REF][START_REF] Zhu | Green supply chain management in China: pressures, practices and performance[END_REF][START_REF] Zhu | Confirmation of a measurement model for green supply chain management practices implementation[END_REF]. Furthermore, some researchers claim to address the supply chain, while the analytical level is actually limited to the firm [START_REF] Matos | Integrating sustainable development in the supply chain: The case of life cycle assessment in oil and gas and agricultural biotechnology[END_REF]. Other terms used includes 'green' as a focus for sustainability [START_REF] Darnall | Environmental management systems and green supply chain management: complements for sustainability?[END_REF]. As an example, defining green supply chain management Walker et al (2008 pg 69) view associated practices as "reducing packaging and waste, assessing vendors on their environmental performance, developing more eco-friendly products and reducing carbon emissions associated with transport of goods". The supply chain level implies a greater scope of management spanning boundaries up and downstream of operations including issues of external integration (Andersen & Skjoett-Larson 2009). Green SCM practices include internal environmental management, external green SCM, investment recovery, and eco-design or design for environment practices.

2004

Zhu and Sarkis Defines GSCM as 'integrating environmental thinking into supply chain management, including product design, material sourcing and selection, manufacturing processes, delivery of the final product to the consumer as well as end-of-life management of the product after its useful life'.

Sustainability at the network level

At this level we cast the net wider from the companies involved in the supply chain to also include the broader network of organizations. Typically this is a lens used in stakeholder studies, often drawing on network theory [START_REF] Cumming | Engaging stakeholders in corporate accountability programmes: A crosssectoral analysis of UK and transnational experience[END_REF][START_REF] Delmas | Stakeholders and competitive advantage: the case of ISO14001[END_REF][START_REF] Fineman | Green stakeholders: industry interpretations and response[END_REF][START_REF] Matos | Integrating sustainable development in the supply chain: The case of life cycle assessment in oil and gas and agricultural biotechnology[END_REF][START_REF] Peter | Risk management in sustainable supply chain management (SSCM): lessons learnt from the case of GMO-free soybeans[END_REF]. The advantage of adopting this level of analysis is that the stakeholders in sustainable development are many and varied, including consumers, businesses and government, NGOs, shareholders, activists, competitors, suppliers and individual managers. Achieving sustainability therefore involves multiple inter-connected actors, who are likely to have different ambitions and objectives, and the friction amongst these network actors may be critical [START_REF] Araujo | Path dependence, agency and technological evolution[END_REF]. It is clear from the literature review that studies looking at sustainability at the network level are far more rare. We find very few studies really take this wider view, but instead often use network and supply chain synonymously. An example of the wider network view includes industrial symbiosis involving the use of one firm's residual resources and byproducts as inputs (supplies) for another [START_REF] Bansal | PUSHING BACK AND PEERING SIDEWAYS: ANALYZING THE SUSTAINABILITY OF INDUSTRIAL SYMBIOSIS[END_REF]. Many articles attempt to take this view in their explanation but clear definitions are largely absent [START_REF] Neto | Designing and evaluating sustainable logistics networks[END_REF][START_REF] Mehalik | Sustainable Network Design: A Commercial Fabric Case Study[END_REF][START_REF] Roome | Conceptualizing and studying the contribution of networks in environmental management and sustainable development[END_REF][START_REF] Vachon | Linking supply chain strength to sustainable development: a country-level analysis[END_REF][START_REF] Von Malmborg | Stimulating learning and innovation in networks for regional sustainable development: the role of local authorities[END_REF][START_REF] Wheeler | Creating Sustainable Local Enterprise Networks[END_REF][START_REF] Young | Sustainable Supply Network Management[END_REF].

Measures of sustainability

In this part of the paper we present measures that have been used to gauge the level of sustainability in the areas of purchasing and supply. As in the previous section divide this into the three level of analysis in order to understand how sustainability has been measured

Sustainable purchasing, procurement / sourcing measures

Looking at corporate social responsibility from a purchasing perspective, Carter (2005) used four dimensions of CSR related to diversity, environment, human right philanthropy and safety. Measures for the environmental dimension included using LCA, design for disassembly, design for recycling and reuse, waste reduction commitment, reducing packaging and so forth.

Table 5: Measures of sustainability at the purchasing, procurement or sourcing level

Examples of Measures used Study SM ethical responsibility (based on the definitions of Carroll, 1991, andCarter andJennings, 2004)  1-I am very knowledgeable of my firm's ethical policies.  2 -I am in compliance with my firm's ethical policies.

Eltantawy et al

 Uses a life-cycle analysis to evaluate the environmental friendliness of products and packaging  Has a formal MWBE (minority/women-owned enterprise) supplier purchase programme  Participates in the design of products for recycling or reuse  Ensures the safe incoming movement of product to our facilities  Purchases from MWBE suppliers  Volunteers at local charities [START_REF] Walker | Sustainable procurement in the United Kingdom public sector[END_REF] See also Carter and Jennings, 2004  Asks suppliers to commit to waste reduction goals  Purchases from small suppliers  Visits suppliers' plants to ensure that they are not using sweatshop labour  Participates in the design of products for disassembly  Asks suppliers to pay a "living wage" greater than a country's or region's minimum wage  Donates to philanthropic organisations  Ensures that suppliers' locations are operated in a safe manner  Ensures that suppliers comply with child labour laws  Purchases from local suppliers  Reduces packaging material Green purchasing (9 item measures;  GP1 Provides design specifications to suppliers that include environmental requirements for purchased items  GP2 Requires its suppliers to develop and maintain an EMS  GP3 Requires its suppliers to have a certified EMS such as ISO 14001  GP4 Uses a questionnaire to collect information about its suppliers' environmental aspects, activities and/or management systems  GP5 Makes sure that its purchased products must contain green attributes such as recycled or reusable items  GP6 Makes sure that its purchased products must not contain environmentally undesirable items such as lead or other hazardous or toxic materials  GP7 Evaluates its suppliers based on specific environmental criteria  GP8 Evaluates the environmental aspects of its second-tier suppliers  GP9 Makes sure that its suppliers meet its environmental objectives Plus Regulations 7 item measures, Customer pressures 9 item measures ; Social responsibility 9 item measures Expected business benefits 9 item measures ElTayeb et al 2010



Sustainable supply chain measures

In measuring green supply chain management Darnell et al (2008) used OECD established metrics covering assessment of the environmental performance of suppliers, requiring suppliers to establish environmental practices and tracking the cost of waste throughout the supply chain, as well as, if facilities informed buyers of ways to reduce their environmental impacts. For these measures it can be seen that sustainability in a supply chain context is taken up primarily through collaboration or cooperation with suppliers to achieve various outcomes and monitoring of suppliers. It is the monitoring part that seems well developed, such as checking compliance to regulations and rules and voluntary standards such as ISO14001. -Developing a mutual understanding of responsibilities regarding environmental performance.

-Working together to reduce environmental impact of our activities.

-Conducting joint planning to anticipate and resolve environmental-related problems.

-Making joint decisions about ways to reduce overall environmental impact of our products  Environmental collaboration with customers: (5 items)  Environmental monitoring of suppliers: (4 items)

-Providing suppliers with written environmental requirements.

-Sending environmental questionnaires to suppliers in order to monitor their compliance.

-Requiring that suppliers have an implemented environmental management system (e.g. ISO 14000). -Asking suppliers to commit to waste reduction goals.  Environmental monitoring by customers: (5 items)  Environmental technologies: (5 items) [START_REF] Vachon | Green project partnership in the supply chain: the case of the package printing industry[END_REF][START_REF] Vachon | Linking supply chain strength to sustainable development: a country-level analysis[END_REF] Environmental performance measures: 42 measures of GSCM across all possible dimensions [START_REF] Aref | Performance measurement for green supply chain management[END_REF]  SGO1 Environmentally sound product (importance over the last three years)  SGO2 Environmentally sound product (current importance)  SGO3 Improving your environmental performance (for the next three years [START_REF] Hong | Implementation of strategic green orientation in supply chain: An empirical study of manufacturing firms[END_REF] External GSCM practices  Providing design specification to suppliers that include environmental requirements for purchased item  Cooperation with suppliers for environmental objectives  Environmental audit for suppliers' internal management  Suppliers' ISO14000 certification  Second-tier supplier environmentally friendly practice evaluation  Cooperation with customer for eco-design  Cooperation with customers for cleaner production  Cooperation with customers for green packaging [START_REF] Zhu | Relationships between operational practices and performance among early adopters of green supply chain management practices in Chinese manufacturing enterprises[END_REF]

Sustainable network measures

These measures are rather more scarce but can relate to industry level measures of environmental practices (Delmas and Toffel 2004) as well as managerial attitudes to broader stakeholders e.g. reactive, defensive, accommodative and proactive (Henriques and Sadorsky 1999). However, our review failed to find measures at a network level that show the degree of engagement within sustainable network (including the supply chain but also extending beyond typical supply chain companies, and out to other stakeholders). As this area is rather new, we might expect that methods used to study this phenomenon may be more restricted to case studies where measures could be embedded within narrative analysis. Thus further exploration of articles may reveal measures of how companies engage in sustainability at the network level.

Conclusions

The review reveals there is a great deal of confusion in the definitions and measures used, often cutting across levels of analysis. Hence our contribution is to show how definitions vary across different levels of analysis. As a general observation we can see that definitions at the purchasing (function or buying company) level tend to share many of the same elements as at the supply chain level, but the majority of the purchasing level definitions focus on the impact on purchased inputs. Definitions at the supply chain level tend to increase scope and focuses on the practices that cut across the supply chain. However it is the area of measures that provides the most usefulness, as this operationalises the definitions that have been given and shows how sustainability is actually detected. As such supply chain sustainability appears to be manifest through collaboration with and monitoring of suppliers (up to a 2 nd tier in the case of monitoring). We also develop a research agenda that addresses sustainable supply chain management at each different level and suggests measures for each of these levels of research.

Limitations

This working paper reports on on-going research into the definition and measurement of Sustainable purchasing and supply. As such this paper reports on current progress in the project and not final outcomes. The main area to be further developed is the method, whereby an extension of search terms may reveal further definitions and measures of sustainability at various levels of analysis.

  in green supply chains involves activities of transferring or disseminating green knowledge from green manufacturing firms to their partners with a view to developing new capabilities for effective actions.2008Cheng et al Sustainable supply chain management as the management of material and information flows as well as cooperation among companies along the supply chain while taking goals from all three dimensions of sustainable development, i.e. economic, environmental and social, and stakeholder requirements into account 2008 Seuring et al. SSCM as the strategic, transparent integration and achievement of an organization's social, systemic coordination of key interorganizational business processes for improving the environmental, and economic goals in the long-term economic performance of the individual company and its supply chains. 2008 Carter & Rogers GSCM is designed to incorporate environmental considerations into decision making at each stage of an organization's materials management and logistics functions through post-consumer disposal 2008 Zhu et al GSC encompasses a broad range of practices from green purchasing to integrated supply chains flowing from suppliers, to manufacturers, to customers, and to the reverse supply chain, which is "closing the loop" 2008 Lee one that performs well on both traditional measures of profit and loss as well as on an expanded conceptualization of performance that includes social and natural dimensions 2009 Pagell and Wu The sustainable supply chain discourse thus differs from mainstream supply chain management, as it involves the recognition of stakeholders within and beyond the 2010 Hall and Matos supply chain.

Table 1 :

 1 Search terms used and results

					Number
	Term	Term	Emerald Ebsco	Selected
	Green	supply	14	43	20
	Green	purchasing	8	10	7
	Sustainable	supply	15	55	27
	Sustainable	purchasing	1	2	2
	Sustainability	supply	4	20	13
	Sustainability	purchasing	0	2	1
	Sustainable	Procurement	6	10	10
	Sustainability	Procurement	0	2	2
	Green	Procurement	0	13	5
	Sustainable	Network	-	45	10
		Totals	48	157	97

Table 3 :

 3 Definitions of sustainability at the purchasing, procurement or sourcing level

		Year	Authors
	Definition
	Socially responsible organizational buying is that which attempts to take into	1994	Drumwright
	account the public consequences of organizational buying or bring about positive		(exception
	social change through organizational buying behavior		included)
	Green supply refers to the way in which innovations in supply chain management	1996	Green et al.
	and industrial purchasing may be considered in the context of the environment

"supply management activities that attempt to improve the environmental performance of purchased inputs, or of the suppliers that provide them" (pg 75)

2008

Walker et al social sustainability: A product or system that meets the performance requirements and expectations of customer stakeholders without causing harm to the wellbeing of society and its members across different time periods.

2008

Lindgren et al managing the optimal flow of high-quality, value-for-money materials, components or services from a suitable set of innovative suppliers in a fair, consistent, and reasonable manner that meets or exceeds societal norms, even though not legally required. 2009 Eltantawy et al Sustainable procurement (SP) is procurement that is consistent with the principles of sustainable development, such as ensuring a strong, healthy and just society, living within environmental limits, and promoting good governance.' 2009 Walker and Brammer

Table 4 :

 4 Definitions of sustainability at the supply chain level

		Year	Authors
	Definition
	Green Supply Chain Management (GSCM)= Green Purchasing + Green	2005	Aref et al
	Manufacturing & Materials Management + Green Distribution + Marketing + Reverse
	Logistics
	Environmental supply chain dynamics (ESCD) are a phenomenon where	2000	Hall
	environmental innovations diffuse from a customer firm to a supplier firm, with
	environmental innovation defined as being either a product, process, technology or
	technique developed to reduce environmental impacts

  Environmental performance measures: 42 measures of GSCM across all possible dimensions

		Aref et al 2005
	 Product based	Preuss 2009
	 Process based
	 Support and Indirect measures
	 Buyer GSC practices:	Lee 2008
	 Buyer1 Incorporate environmental considerations in selecting their supplies and suppliers.
	 Buyer2 Request us to have an environmental management system (e.g. ISO 14001).
	 Buyer3 Have interest in greening the supply chain.
	 Buyer4 Provide us with environmental training, education, or technical assistance.
	 1 Holding environmental awareness seminars for Suppliers	Rao 2009
	 2. Guiding/helping suppliers to establish their own environmental programs .
	 3 Bringing suppliers together
	 4 Informing suppliers about the benefits of environment friendly technologies
	 5. Urging suppliers to take environmental actions
	 6. Choice of suppliers by environmental criteria
	 7. Arranging funds to help suppliers for their environment programs
	 8. Sending company auditors to appraise environmental compliance of suppliers

Table 6 :

 6 Measures of sustainability at the supply chain level

	Study
	Examples of Measures used
	 Environmental collaboration with suppliers (5items)
	-Achieving environmental goals collectively.