Cécile Davoine

Gérald Portemont

Nicolas Horezan

Bertrand Langrand

Vincent Marcadon

Fabienne Popoff

Cellular materials made of stacked tubes: influence of the manufacturing process on the dynamic behavior of the constitutive material. Part I: microstructure

Cellular materials are very promising as lightweight aeronautical frames thanks to their superior specific mechanical properties such as impact resistance. However, because of the processing routes and heat treatments used in the production process, the material within the cell walls may behave differently from the bulk, and therefore the in situ mechanical properties are often unknown. The present work aims at investigating the link that exists between the processing of cellular architectures and the mechanical properties of their constitutive material. The cellular material studied is an Inconel® 600 tube stacking brazed together using a nickel-phosphorus alloy. The experimental works have been conducted in order to analyse the microstructures of Inconel® 600 tube stacking resulting from brazing and annealing heat treatment. In order to discuss the influence of the manufacturing process of tube stackings on the mechanical properties of their constitutive material, it has been proposed to perform uni-axial tensile tests on tubular specimens. Different material configurations (with or without heat treatments, with or without nickel-phosphorus coating) have been thus characterised and compared, involving both tensile tests from quasi-static to dynamic loads and electron back-scattered diffraction analyses (EBSD) performed on post-mortem specimens.

Introduction

High-temperature Ni-based brazing alloys was used for year in aerospace industry to bond aircraft structural parts, acoustical tail pipes, thrust reversers, turbines blades and seals. They provide good resistance to chemical corrosion and oxidation coupled with high strength at elevated temperatures. Vacuum brazing or controlled atmosphere brazing are the most common heating methods used. Typical temperatures of brazing are in the range of 890-1170°C. Alloys are available in electrochemical coating, powder, paste form, and melt spun foils or tape. A technic for fabricating a cellular structure closed from the honeycombs one consists of stacking and brazing hollow tubes in a compact arrangement. Such a structure made in metallic hollow tubes stacking is being considered for applications where multifunctional properties are required including structural properties. In the field of aeronautic industry, this kind of lightweight structure could provide solution for vibration damping, heat dissipation and storage, and crash energy absorption. The integration of metallic hollow tubes stacking as an engine component that will be routinely subjected to operating temperatures up to 700°C involved to achieve the brazing at high temperature in order to avoid the melting of the joints in use. As the behaviour of the material is governed by the amount of cold work, grain size and chemical composition, the purpose of this of this work is to study the impact of the brazing heat treatment on the microstructure of the tubes, and consequently, on their final mechanical properties.

Cellular material's processing route

Cellular materials of interest in the present work are brazed tube stackings. They are manufactured using tubes made of Inconel® 600 which are prior coated with a 50 µm thick Nickel-Phosphorus (NiP) alloy layer for the brazing. Inconel® 600 alloy is a standard nickel-based superalloy used for applications which require resistance to corrosion at high temperatures. Being not precipitation hardenable, this alloy can be hardened and strengthened only by cold work. Tubes, having external and internal diameters of 5 mm and 4 mm respectively, are stacked up regularly between two metallic skins to produce the desired structures (e.g. sandwiches with either a square stacking core or a hexagonal one as shown in Figure 1). Skins are made of Inconel® 600 too and their thickness equals 1 mm. A low-melting coating (nickel phosphorous alloy) deposited on the surface of each tubes by electroless plating was used for brazing. An electroless nickel bath which contains nickel sulfate as a Ni ion source and sodium hypophosphite as a reducing agent was used to coat a 50µm thick nickelphosphorous NiP layer on the surface of each tubes. Before immersion in the electroless NiP deposition bath, the tubes were etched in nitric acid to remove the native oxide layer. This bath provides NiP coating with a medium phosphorous content closed to the eutectic concentration, namely 19 at.% P. As the tubes are not hermetically sealed at the ends, the coating occurs in the inner surface of the tubes. The tubes were stacked in a graphite die lined with inconel® plates positioned at the top and at the bottom of the stacking in order to provide a sandwich structure. A close-packed arrangement of tubes is naturally created and provides "hexagonal stacking". A square stacking can also be obtained by tilting the mould during brazing. The bonding is achieved by applying a thermal treatment in a vacuum furnace. According to the Ni-P phase diagram, a medium phosphorous content alloy would be liquid above 1000 °C. As the liquid concentrates at the contact lines by capillarity, metallurgical joints are created between the tubes. In this study, a "soft" brazing heat treatment has been performed in order to avoid microstructural evolutions of the Inconel® 600 tubes and in particular grain growth. It consists in a progressive heating under vacuum (with a heating rate of 100°C/min) until 1000°C. Then the heating temperature is kept constant during 15mn. The heat treatment ends by a natural cooling down to room temperature. In order to provide a strong bond between the tubes, and between tubes and the sheet at the top and at the bottom of the sandwich structure, it is often recommended to increase the brazing duration to enhance the diffusion phenomenon and to improve the braze metal penetration into base metal. Some samples have been annealed under vacuum during 16 hours at 1050°C. A detailed scanning electron microscopy (SEM)-based microstructural and chemical characterization of the sandwich structure has been carried out to observe the effect of this annealing heat treatment. For these eight tubes wide and eight tubes high sandwich structures, the brazed joint width was around 1.5 mm. Braze joints was composed of a P-rich central area corresponding to the phase that remains liquid at the brazing temperature. During the cooling down to room temperature, the complete solidification of the remaining liquid phase results in the formation of the particular microstructure which consists of two distinct phases: a Ni-rich one embedded in an intermetallic matrix Ni 3 P. At the interface of the joints and the tubes, some Ni-rich grains resulting from a local depletion of phosphorus in the liquid phase appeared. The whole grains formed a layer, appearing in figure 2 (a), and whose thickness and composition depend on the heat treatment. Moreover, the grains in the tubes are visible thanks to electron channeling phenomenon. The electron channeling contrast imaging makes use of the strong dependence of the backscatter electron (BSE) signal on the orientation of the crystal lattice planes with respect to the incident electron beam due to the electron channeling mechanism.

Tensile tubular specimens

In order to analyse the influence of the manufacturing process on the mechanical properties of the Inconel® 600 base material, it is proposed to compare the mechanical behaviour of tensile tubular specimens under uni-axial tensile loading at different rates of strain for different material configurations. The machined specimen in Figure 4 has been developed to avoid plastic strain location and failure near the gripping areas and to make the analysis of the tube material behaviour possible under a wide range of average strains and strain rates. Tubular samples are made from 75mm length tubes with external and internal diameters equals to 6mm and 4mm respectively produced by TEXTPART using extrusion and cold-drawing. The external diameter in the center of the specimen corresponds to the external diameter of the tubes implemented in the cellular materials in Figures 1 and 2 ie 5mm. The table 2 gives the process steps of the different analysed specimens.

Figure 4: Dimensions of the tubular specimen for tensile tests (in mm)

Sample Machining NiP coating

Brazing heat treatment

Annealing Loading

Inco600M X Inco600BM X X Inco600HT X X X Inco600NiP X X X X Inco600HT-A X X X X Inco600NiP-A X X X X X
Table 2: Process steps of the tensile tubular specimens for EBSD Analysis.

This is particularly important that the tensile tubular specimens and the tubes within the cellular materials have a similar diffusion and penetration of brazing metal (Ni-P alloy layer) into the Inconel® 600 base material. The set of Inco600NiP specimens has been coated by 15µm Ni-P alloy layer. This thickness has been calibrated in order to produce a similar diffusion and penetration of the brazing metal into the base metal, compared to the actual manufacturing process. As the cellular materials are being subjected to possible dynamic loads (i.e., fan blade off application), the mechanical behaviour of the tube materials is also studied under different displacement rates to analyse possible strain rate effects.

Analysis of post-mortem specimens

Experimental procedure

In order to investigate the evolution of the microstructure resulting from dynamic load, Electron Backscattered Diffraction (EBSD) Spectroscopy analysis was carried out on post-mortem samples. All samples were cut mechanically on the longitudinal section of the tube, in a plane that passes through the tube axis. Then the cut samples were cold-mounted in epoxy and polished. The material configurations were analysed in order to study the effect of heat treatement and phosphorus on the mechanical properties of Inconel® 600 base material. The microstructure of low-speed loaded (6 mm.min -1 ) and high-speed loaded (1m.s -1 ) post-mortem samples were also compared (Table 1).

Material configuration

Load conditions

Name of the postmortem specimen

Inco600BM 6 mm.min -1 Inco600BM-6 1m.s -1 Inco600BM-1 Inco600HT 6 mm.min -1 Inco600HT-6 1m.s -1 Inco600 HT-1 Inco600NiP 6 mm.min -1 Inco600NiP-1 1m.s -1 Inco600NiP-6 Inco600HT-A 1m.s -1 Inco600HT-A-1 Inco600NiP-A 1m.s -1 Inco600NiP-A-1
Table 1: Material configurations and load conditions of the analysed post-mortem specimens Two areas of interest were compared: the fracture area (A), and a non-loaded area (B) (figure 5). EBSD patterns were collected from a Scanning Electron Microscope (SEM) Zeiss DSM 960 to map the crystallographic orientations distribution. The scanning was performed in order to put Transversal Direction (TD) parallel to the load direction, ie. the tubular specimen axis. The Normal Direction (ND) is normal to the specimen's surface. Two categories of Bravais unit cells are present: the wall in Inconel®600 and the Ni-rich grains in the brazing layer crystallized in a cubic unit cell (Inconel® 600 has a face-centered cubic structure as pure nickel), while Ni 3 P matrix of the brazing layer crystallizes in a quadratic system. A large area of 500 x 500 µm 2 with 0.5 µm steps at the center of the tube wall was investigated.

Microstructures resulting from dynamic load

The EBSD analysis in the loaded areas of all post-mortem tubular specimens revealed typical grain morphology of high-strained material illustrated in figure 6: the grains are elongated in the loading direction, ie. in the tubular specimen axis. The deformed area spreads on few mm 2 . The figure 6 shows the typical orientation mapping referring to TD and ND, and the corresponding pole figure for the Inco600BM-1 tubular specimen. Textures develop from the rotation of slip systems in grains during this plastic deformation. Comparing with that a non-loaded specimen (Figure 7), it is seen that there is a strong texture development in tube wall due to significant plastic deformation. This texture is observed both in TD and ND. The orientation mapping referring to TD shows that there are two preferential orientations of the cubic unit cell: major <111> in blue, and minor <100> in red as given by the corresponding key color. It means that, in the face-centred cubic cell, either the diagonal or the edge is parallel to TD. These two preferential orientations are also highlighted by the direct pole figures. In 001-axis and in 111-axis, the two pole figures show intensity peaks in the directions closed to TD. The orientation mapping referring to ND also reveals a texture. It appears that the grains are mostly in green: the <101> is preferentially parallel to ND. As the specimens have been cut in a plane that passes through the tube axis, and as ND is normal to the specimen surface, ND is a particular direction in the tube. EBSD analysis highlighted heterogeneity of grain size in the wall thickness in the Inco600 BM-1 specimen. The figure 7 shows the orientation mapping of a non-loaded Inco600BM tubular specimen in the central The corresponding pole figures show that no texture is present before loading. The grains appear twinned. The analysed areas, chosen to cross the wall thicknesses, highlight that the grains are finer at the inner surface of the tubular specimens. Grain size is dependent on processing. The initial tubes were manufactured through a mechanical cold-drawing process followed by annealing step. They are drawn through a mould using a guide bar to keep the tube straight and give the inner diameter after drawing. The outer diameter and wall-thickness of the tube are reduced and a

EBSD

Inco600 BM-1 Inco600 HT-A-1 significant plastic deformation occurs in material at the end of drawing. Grain size reduces quickly through the sequence of cold drawing and annealing. A heterogeneous plastic deformation during manufacturing process could explain the heterogeneity of grain size in the wall thickness of the tubular specimens. In general, material with a fine grain structure is preferred because it has better corrosion resistance and higher tensile, fatigue and impact strength. EBSD analysis highlighted no heterogeneity of grain size in the wall thickness in the Inco600 HT-A-1.

The homogeneous grain size can be explained by grain growth of the finer grains localized at the inner part of the wall occurring during the annealing treatment. For Inconel® 600, grain growth occurs because of carbides coalescence above 980°C. At that temperature, the finely dispersed carbide particles in the alloy's microstructure, which inhibit grain growth, begin to coalesce. It means that the time and temperature of the annealing treatment (16 hours at 1050°C) is enough to dissolve the carbides completely and results in increased grain size. rate of local work hardening in inner wall of tubular sample is enough to recrystallize the microstructure during brazing step, and annealing treatment provide an homogeneous grain size into the wall of the tubular samples.

Local strain characterization using EBSD intragranular misorientation measurements

The state of residual strain in materials can be investigated by EBSD. The technique provides high spatial resolution, strain sensitive examination of plastic strain, enabling analyses that are otherwise difficult given the spatial / angular resolution and sampling limitations of XRD and TEM, respectively. The angular spreads into a grain were quantified. Figure 8a) gives the maps of grain orientation spread of some high-speed loaded tubular specimens (1m.s -1 ) in the loaded area: Inco600BM-1, Inco600NiP-1 and Inco600NiP-A-1. A grain was defined as a group of at least five neighboring indexed points having the same crystal orientation within a tolerated angular deviation of 5°. Intragranular misorientation measurements by EBSD capture strain manifested as a continuous change in local lattice orientation across grains of deformed materials, with the gradient of deformation a function of dislocation density and degree of dislocation order. Within each grain, the point with the lowest misorientation gradient (i.e. least deformed) is calculated. This is taken as a reference orientation, and the misorientations to all other points in the grain are calculated. The blue color indicates where grains have very little internal misorientation. The green, yellow, orange and red colors represent progressively increasing levels of internal misorientation. Grains are outlined with black boundaries. Such maps clearly show the most strained grains. The same colour scale, from 0 to 25° misorientation is used for the three analyses. It appears that three high-speed loaded tubular specimens are in the same state of residual strain. A majority of grains are colored in green. In the case of the specimen Inco600NiP-1, the grains colored in blue at the border of the wall belong to braze phases. As Ni 3 P is brittle, the braze layer breaks during loading, and few level of residual strain is observed. For the specimens Inco600BM-1 and Inco600NiP-1, the grains located at the inner surface

Fine grains

Coarse grains of the tubular samples appear in blue because of their small grain size. In fact, the plastic residual strain level is locally high as revealed by the Kernel Average Misorientation (KAM) maps given in figure 8b). The KAM is defined for a given point as the average misorientation of that point with all of its neighbors, which is calculated with the proviso that misorientations exceeding a tolerance value of 5° are excluded from averaging calculation. The magnitude of the KAM is affected by the scan step size, which was always set as 0.5 µm in the present paper. The figure 8b) show that the average misorientation in the grains located at the inner surface of the tubular samples of the specimens Inco600BM-1 and Inco600NiP-1 are higher than the rest of the wall. The maps of Kernel Average Misorientation and Grain Orientation Spread of the Inco600NiP-A-1 reveal a lower and homogeneous residual plastic strain, due to the annealing treatment.

Conclusion

A SEM analysis of tubular stacking has been conducted in order to explain the improvement of mechanical strength through annealing treatment (16h at 1050°C). During annealing, grain growth allowed to create a Ni-based junction between the tubes and to avoid premature joints failure. A systematic study of the microstructure of Inconel® 600 alloy loaded under dynamic condition has been thoroughly investigated as a function of the initial state of tubular samples with particular emphasis on local strain characterization using EBSD intragranular misorientation measurements.

Based on these studies, the follow conclusions are summarized:

-Mechanical properties of the tubes depend on their manufacturing process (residual work hardening); -A strong texture develop in tube wall due to significant plastic deformation; -Texture and residual strain level are not affect by the initial configuration or the load speed; -Because of its brittleness, the local strain in the braze layer is low; -Annealing treatment provides a homogeneous grain size and residual plastic strain into the thickness of the wall of the tubular samples.

Inco600BM-1 Inco600NiP-A-1 Inco600NiP-1 a) b)

Figure 1 :

 1 Figure 1: Arrangement of hollow tubes for fabricating cellular structure (a) square and (b) hexagonal stacking.

  SEM of a brazed joint between two neighbouring tubes; (a) without annealing treatment; (b) with annealing treatment.X-ray analyses were performed in order to have access to chemical distributions (figure3). The phosphorous distribution shows clearly the location of the Ni 3 P phase in the joint area. During heat treatment, the tubes are progressively enriched in phosphorus coming from the liquid phase. But our observation did not reveal a deep penetration of phosphorus element into the tubes walls, even with annealing treatment (figure3 (a)). Indeed, phosphorus diffuses up to its limit of solubility in nickel around 0.32% (atomic percent) at 1000°C according to the binary Ni-P diagram. As this solubility is very low, phosphorous appears clearly in the Ni 3 P phase where its concentration is around 33% (atomic percent). During brazing and annealing heat treatments, the diffusion of alloy elements of Inconel® 600 such as chromium and iron took place from the tubes to the joint area. Elemental distribution shows a heterogeneous composition of these two elements in the joint area. Indeed chromium and iron diffusion took place in the interfacial Ni-rich layer and decreased in the Ni 3 P phasis as highlighted by low chromium and iron contents in the Ni 3 P phase. According to the homogeneity of composition, especially in the case of annealing treatment, the diffusion of iron element into the interfacial Ni-rich layer seems to be faster than the one of chromium element. The comparison of the element distributions with and without annealing treatment (figure3(a) and (b)) shows that the annealing treatment allows grain growth of the interfacial Ni-rich layer and creates a Ni-based junction between the tubes. The interfacial Ni-rich layer appears thicker and richer in chromium content after annealing treatment. It creates a strong Ni-based junction which could avoid premature joints failure into the brittle Ni 3 P phase.

Figure 3 :

 3 Figure 3: Elemental distributions according to the X-ray mapping (Ni, Fe, Cr, P); (a) without and (b) with annealing treatment.

Figure 5 :

 5 Figure 5: Schematic illustrations of the analysed plans into a cut of a post-mortem sample: areas of interest in a section of a post-mortem sample: the fracture area (A), and a non-loaded area (B).

Figure 6 :

 6 Figure 6: EBSD analysis of a Inco600 BM-1 and Inco600 HT-A-1 tubular specimens in the loaded area A; orientation mapping referring to (a) TD axis direction and (b) ND axis direction; the corresponding key color and the direct pole figure.

Figure 7 :

 7 Figure 7: EBSD analysis of a Inco600BM tubular specimen in a non-loaded Inco600BM tubular specimen in the central area: Orientation mapping referring to TD axis direction and key color; the corresponding pole figure.

Figure 8

 8 Figure 8: a) Maps of GOS (Grain Orientation Spread) of three high-speed loaded tubular specimens in the loaded area; b) Kernel Average Misorientation (KAM) maps.


Houssem Gaiech

Rachid El-Azouzi

Majed Haddad

Eitan Altman

Issam Mabrouki

Evolutionary Games for Multiple Access Control: From Egoism to Altruism ‡

This paper studies multiple access games with a large population of mobiles decomposed into several groups. Mobiles interfere with each others through many local interactions. We assume that each mobile (or player) cooperates with his group by taking into account the performance of his group. The degree of cooperation not only covers the fully noncooperative behavior and the fully cooperative behavior, but also the fully altruistic behavior. To do so, we make use of the evolutionary game theory which we extend to cover this kind of behavior. We define and characterize the equilibrium (called Evolutionary Stable Strategy) for these games and establish the optimal level of cooperation that maximizes the probability of successful transmission. We also study the game dynamics both in its classical form and in the presence of delay. Interestingly, we show that, in order to maximize the system performance, the mobiles should be less cooperative.

I. INTRODUCTION

Wireless networks are growing increasingly less structured, adopting many of the characteristics of ad-hoc networks. However, the dynamic interactions arising in these networks make it difficult to analyze and predict system performance, thereby inhibiting the development of wireless technologies. Game theory with several concepts, provides a powerful mathematical framework that can accommodate the preferences and requirements of various stakeholder in a given process. Recently, there has been a surge in research activities that employ game theory to model and analyze the performance of various networks, such as communication networks, neural networks, computer networks, social networks, biologically inspired networks, etc.

Evolutionary game theory has became a central tool for predicting and even designing evolution in many fields. Its origins come from biology where it was first introduced by [START_REF] Smith | Game theory and the evolution of fighting[END_REF], [START_REF] Smith | Evolution and the theory of Games[END_REF] to model conflicts among animals. It differs from classical game theory by (i) its focusing on the evolution dynamics of a fraction of members of the population that use a given strategy, and (ii) in the notion of Evolutionary Stable Strategy (ESS, [START_REF] Smith | Game theory and the evolution of fighting[END_REF]) which includes robustness against a deviation of a whole (possibly small) fraction of the population that may wish to deviate (this is in contrast with the standard Nash equilibrium concept that only incorporates robustness against deviation of a single user). Although ESS has been initially defined in the context of biological systems, more and more applications can be found studying multiple access protocols [START_REF] Srinivas Shakkottai | Evolutionary power control games in wireless networks[END_REF], [START_REF] Tembine | Mean field asymptotics of markov decision evolutionary games and teams[END_REF], multihoming [START_REF] Shakkottai | The Case for Non-Cooperative Multihoming of Users to Access Points in IEEE 802.11 WLANs[END_REF], delay tolerant networks [START_REF] El-Azouzi | Evolutionary forwarding games in delay tolerant networks: Equilibria, mechanism design and stochastic approximation[END_REF], evolution of transport protocols [START_REF] Altman | The evolution of transport protocols: An evolutionary game perspective[END_REF] and resources competition in the Internet [START_REF] Zheng | Evolutionary game and resources competition in the internet[END_REF].

In this paper, we consider a large population of sensors or relays that are deployed in a large area to detect the occurrence of a specific event. We assume that the population is decomposed into different groups. We consider an Aloha system in which mobiles interfere with each other through many local interactions (e.g., access points, throwboxes) where the collision can happen if more than one mobile transmit a packet in the same time slot. In particular, we consider that each node seeks to maximize some combination between its own performance and the performance of its group. We study the impact of cooperation in the context of multiple access control in many possible behaviors such as altruist behavior and fully non-cooperative behavior. In many problems, assumption about selfishness or rationality has been often questioned by economists and other sciences. Many research works shown that even in a simple game and controlled environment, individuals do not act selfishly. They are rather either altruistic or malicious. Several explanations have been considered for such behavior of players. Fairness reasons are argued by Fehr [START_REF] Fehr | A theory of fairness, competition and cooperation[END_REF] to consider the joint utility model, while reciprocity among agents are considered in [START_REF] Gintis | Moral sentiments and material interests[END_REF]. Cooperation among users, often referred to as altruism, are discussed in [START_REF] Levine | Modeling altruism and spitefulness in experiment[END_REF], [START_REF] Ledyard | Public goods: A survey of experimental research. princeton university press[END_REF], [START_REF] Chen | Altruism, selfishness, and spite in traffic routing[END_REF], [START_REF] Sharma | Stackelberg thresholds in network routing games or the value of altruism, games and economic behavior[END_REF]. Some of the models in [START_REF] Sobel | Interdependent preferences and reciprocity[END_REF] argues that the partial altruism mimics closely users' behavior often observed in practice.

In this paper, we present a new model for evolutionary games which takes into account both the altruism and selfishness of agents. Firstly, we begin by defining this new concept, driving it in several ways and exploring its major characteristics. The major focus of this paper is to study how the level of cooperation impacts the profile of population as well as the global performance of the system. Our theoretical results unveil some behaviors. More specifically, we show that when all users increase their level of cooperation, then the performance of the system is not necessary improved. In fact, for some scenarios, the performance of groups may lead to an improvement by adopting selfishness instead of altruism. This happened when the density of nodes is high. For low density, the degree of cooperation may indeed improve the performance of all groups.

The paper is structured as follows. We first formalize in the next section the system model. We then present, in Section III, the evolutionary game model that includes the cooperation aspect. In Section IV, we compute the expression of the ESS. We study in section V the replicator dynamics in the classical and delayed forms. In Section VI, we proceed to some optimization issues through the analysis of the probability of success. Section VII shows some numerical investigations on the equilibrium, the probability of success and the replicator dynamics. Section VIII concludes the paper.

II. SYSTEM MODEL

Consider an Aloha system composed of a large population of mobiles (or sensors) operating in a low traffic condition. Mobiles are randomly deployed over a plane and each mobile may interact with a mobile in its group or with a mobile in another group. The channel is assumed to be ideal for transmission and all errors are only due to collision. A mobile decides to transmit a packet or not to transmit to a receiver when they are within transmission range of each other. Interference occurs as in the Aloha protocol: if more than one neighbor of a receiver transmit a packet at the same time, then there is a collision.

We assume, in particular, that we can ignore cases of interaction in which more than two sensors or relays transmit simultaneously causing interference to each others. An example where we may expect this to hold is when sensors are deployed in a large area to monitor the presence of some events, e.g., in Delay Tolerant Networks (DTNs) where the network is assumed to be sparse and the relay density is low. Under this setting, communication opportunities arise whenever two nodes are within mutual communication range because of the mobility pattern.

The size of each group of mobiles G i is denoted by α i with N i=1 α i = 1. Let µ be the probability that a mobile i has its receiver R(i) within its range. When a mobile i transmits to R(i), all mobiles within a circle of radius R centred at R(i) cause interference to the node i for its transmission to R(i). This means that more than one transmission within a distance R of the receiver in the same slot induce a collision and the loss of mobile i packet at R(i). Accordingly, each mobile has two actions: either to transmit T or to stay silent S. A mobile of group G i may use a mixed strategy p i = (p i , 1 -p i ) where p i is the probability to choose the action T . If a mobile transmits a packet, it incurs a transmission cost of ∆. The packet transmission is successful if the other users do not transmit (stay silent) in that given time slot. If a mobile transmits successfully a packet, it gets a reward of V . We suppose that the payoff V is greater than the cost of transmission, i.e., ∆ < V .

III. UTILITY FUNCTIONS

As already mentioned, we study a new aspect of evolutionary games for multiple access games where each mobile cooperates with other mobiles of his group in order to improve the performance of his group. Let β be the degree of cooperation. The utility of a tagged mobile choosing action a within group G i is a convex combination of the utility of his group and his own utility, namely

U i user (a, p -i ) = βU i group (a, p -i ) + (1 -β)U i self (a, p -i ) (1)
where U group is the utility of the group to which the tagged player belongs and U self is the individual utility of that player.

When the mobile plays T , resp. S, the utility of the group is given resp. by

U i group (T, p i , p -i ) = µ γ + (1 -γ) N j=1 α j (1 -p j ) U i group (S, p i , p -i ) = µ(1 -γ)α i p i with γ
being the probability that a mobile is alone in a given local interaction. Analogously, the selfish utility when the mobile chooses strategy T is

U i self (T, p i , p -i ) = µ (1 -△) -(1 -γ) N j=1 α j p j
while the selfish utility of user i when playing S is zero, namely

U i self (S, p i , p -i ) = 0
Combining the above results, the utility of a mobile of class i using strategy T is given by

U i user (T, p i , p -i ) = µ 1 -(1 -β)∆ -(1 -γ) N j=1 α j p j ,
while the utility of a mobile of class i when he plays S is

U i user (S, p i , p -i ) = µβ(1 -γ)α i p i IV. COMPUTING THE ESS
In evolutionary games, the most important concept of equilibrium is the ESS, which was introduced by [START_REF] Smith | The logic of animal conflict[END_REF] as a strategy that, if adopted by most members of a population, it is not invadable by mutant strategies in its suitably small neighbourhood. In our context, the definition of ESS is related to the robustness property inside each group. To be evolutionary stable, the strategy p * must be resistant against mutations in each group. There are two possible interpretations of ǫdeviations in this context:

1) A small deviation in the strategy by all members of a group. If the group G i plays according to strategy p * i , the ǫdeviation, where ǫ ∈ (0, 1), consist in a shift to the group's strategy pi

= ǫp i + (1 -ǫ)p * i ; 2)
The second is a deviation (possibly large) of a small number of individuals in a group G i , that means that a fraction ǫ of individuals in G i plays a different strategy p i . After mutation, the average of a non-mutant will be given

U i user (p * i , ǫp i + (1 -ǫ)p * i , p * -i ). Analogously, we can construct the average payoff of mutant U i user (p i , ǫp i + (1 -ǫ)p * i , p * -i ). A strategy p * = (p * 1 , p * 2 , . . . , p * N ) is an ESS if ∀ i and p i = p *
i , there exists some ǫ i ∈ (0, 1), which may depend on p i , such that for ǫ ∈ (0, ǫ i )

U i user (p * i , ǫp i + (1 -ǫ)p * i , p * -i ) > U i user (p i , ǫp i + (1 -ǫ)p * i , p * -i ) (2)
Equivalently, p * is an ESS if and only if it meets best reply conditions:

• Nash equilibrium condition:

U i user (p * i , (p * i , p * -i )) > U i user (p i , (p * i , p * -i ))
• Stability condition:

If p i = p * i , and U i user (p * i , (p * i , p * -i )) = U i user (p i , (p * i , p * -i )) then U i user (p i , (p i , p * -i )) < U i user (p * i , (p i , p * -i )) A.

Characterization of the equilibria

In this section, we provide the exact characterization of the equilibria induced by the game. We distinguish pure ESS equilibria and mixed ESS. Before studying the existence of ESS, we introduce some definitions needed in the sequel.

Definition 1.

• A fully mixed strategy p is the strategy when all actions for each group have to receive a positive probability, i.e., 0 < p i < 1 ∀i. • A mixer (pure) group i is the group that uses a mixed (pure) strategy 0 < p i < 1 (resp. p i ∈ {0, 1}). • An equilibrium with mixed and non mixed strategies is an equilibrium when there is at least a pure group and a mixer group.

Proposition 1 characterizes the condition on the existence of a fully mixed ESS. Proposition 1.

1) For

γ < 1-1-(1-β)∆
(β+N ) min αi then there exists a unique fully mixed ESS p * = (p * i ) i=1,. . . ,N , where

p * i = 1 -(1 -β)∆ α i (β + N )(1 -γ) 2) For 1 -1-(1-β)∆ (β+N ) min αi ≤ γ ≤ 1 -1-(1-β)∆
1+β max αi , then there exists a unique ESS with mixed and non mixed strategies.

3) For γ > 1 -1-(1-β)∆
1+β max αi , then there exists a fully pure ESS where all groups play pure strategy T . Proof: 1) From the definition of ESS in (2), we have ∀i ∈ {1, . . . , N },

(p i -p * i ) U i user (T, ǫp i + (1 -ǫ)p * i , p * -i ) -U i user (S, ǫp i + (1 -ǫ)p * i , p * -i ) < 0 Thus (p i -p * i ) 1 -(1 -β)∆ -(1 -γ) N j=1 α j p * j -(1 -γ)βα i p * i +ǫ(p i -p * i ) 2 -(1 -γ)(1 + β)α i < 0
The mixed Nash equilibrium is obtained when the first term of the previous inequality is strictly negative. While (p i -p * i ) can be positive or negative for p * i / ∈ {0, 1}, the following equation holds

1 -(1 -β)∆ -(1 -γ) N j=1 α j p * j -(1 -γ)βα i p * i = 0
By summing this equation from 1 to N, we get

N (1 -(1 -β)∆) -N (1 -γ) N j=1 α j p * j -(1 -γ)β N i=1 α i p * i = 0
and so

N j=1 α j p * j = N (1 -(1 -β)∆) (1 -γ)(β + N )
Finally, we deduce

p * i = 1 -(1 -β)∆ α i (1 -γ)(β + N ) Hence p * is fully mixed Nash equilibrium if γ < 1 -1-(1-β)∆
(β+N ) min αi . Furthermore, since -(1 -γ)(1 + β)α i ) < 0, the stability condition is always satisfied which implies that p * is an ESS. This complete the proof of (1).

2) Without loss of generality, we assume that the sizes of groups are ordered as follows:

α 1 ≤ α 2 ≤ . . . ≤ α N .
We assume that for a given value of γ we have n T groups playing strategy T and N -n T groups play mixed strategy. Hence the profile of population becomes (T, . . . , T, p nT +1 , . . . , p N ). For mixed group i (i ∈ {n T + 1, . . . , N }), we have the following relation

1 -(1 -β)∆ -(1 -γ) nT j=1 α j -(1 -γ) • N j=nT +1 α j p j -(1 -γ)βα i p i = 0
Thus, a strategy of a mixer group is given by

p * i = 1 -(1 -β)∆ -(1 -γ)α T (1 -γ)α i (β + N -n T )
where α T = nT j=1 α j . For pure groups playing T, the following inequality holds: ∀i ∈ {1, . . . , n T }

1 -(1 -β)∆ -(1 -γ)[α T + α i (β + N -n T )] ≥ 0 which completes the proof of 2).
3) Assume that ∀i ∈ {1, . . . , N }, the group i transmit all the time (p * i = 1). The Nash equilibrium conditions become : ∀i ∈ {1, . . . , N },

1 -(1 -β)∆ -(1 -γ)(1 + βα i ) > 0 ⇒ γ > 1 - 1 -(1 -β)∆ 1 + βα i (3)
and the the proof is complete.

The previous proposition claims that an increased network density results in more transmission which leads to more collision. It also states that, in order to avoid collision between mobiles belonging to the same group, the cooperation degree tends to decrease the probability of transmission within the same group.

Proposition 2. At the Nash equilibrium, there is no group playing pure strategy S.

Proof: Let p * be the ESS. By contradiction, suppose that there exists a group k playing the strategy S at ESS, i.e., p * k = 0. The Nash equilibrium condition for group k becomes

p k 1 -(1 -β)∆ -(1 -γ) N j=1,j =k α j p * j < 0 ⇒ 1 -(1 -β)∆ -(1 -γ) N j=1,j =k α j p * j < 0
If all groups use pure strategy S at ESS, the last condition becomes

1 -(1 -β)∆ < 0
But this contradicts our assumptions on β and ∆. Hence there exists at least a group l playing strategy p * l such that p * l ∈]0, 1]. The Nash equilibrium condition is expressed as following:

(p l -p * l ) 1 -(1 -β)∆ -(1 -γ) • N j=1,j =k α j p * j -(1 -γ)βα l p * l < 0 Thus 1 -(1 -β)∆ -(1 -γ) N j=1,j =k α j p * j ≥ (1 -γ)βα l p * l ( 4
)
Combining conditions ( 3), ( 4) and p * l > 0, we get (1-γ)βα l < 0 which is a contradiction. This completes the proof.

V. REPLICATOR DYNAMICS

Evolutionary games study not only equilibrium behavior but also the dynamics of competition. We introduce the replicator dynamics which describe the evolution in groups of the various strategies. Replicator dynamic is one of the most studied dynamics in evolutionary game theory. In this dynamic, the frequency of a given strategy in the population grows at a rate equal to the difference between the expected utility of that strategy and the average utility of group i. Hence, successful strategies are more likely to spread over the population. In this paper, we study the replicator dynamics for the case of two groups. The general case of N groups will be handled in a future work.

A. Replicator dynamics without delay

The proportion of mobiles in the a group i programmed to play strategy T , denoted p i , evolves according to the replicator dynamic equation given by:

ṗi (t) = p i (t)[U i user (T, p i (t), p -i (t)) -Ū i user (p i (t), p -i (t))] (5)
where

Ū i user (p i (t), p -i (t)) = p i (t)U i user (T, p i (t), p -i (t)) + (1 -p i (t))U i user (S, p i (t), p -i (t)) Thus ṗi (t) = p i (t)(1 -p i (t)) 1 -(1 -β)∆ -(1 -γ)• 2 j=1 α j p 2 (t) -(1 -γ)βα i p i (t) (6)
By expressing Equation ( 6) for i = 1, 2, we obtain a system of two non-linear ordinary differential equations (ODEs) in [START_REF] El-Azouzi | Evolutionary forwarding games in delay tolerant networks: Equilibria, mechanism design and stochastic approximation[END_REF]. There are several stationary points in which at least one group playing a pure strategy and a unique interior stationary point

p * = (p * 1 , p * 2 ) with 0 < p * i < 1.
The interior stationary point corresponds to the fully mixed ESS given by Proposition 1 and it is the only stationary point at which all mixed strategies coexist. Assuming that the state space is the unit square and that p * exists, the dynamic properties of this equilibrium point are brought out in the next theorem.

Theorem 1. The interior stationary point p * is globally asymptotically stable in the replicator dynamics.

Proof: The proof is based on a linearization of the system of non linear ODEs around p * . We introduce a small perturbation around p * defined by x i (t) = p i (t) -p * i for i = 1, 2. Keeping only linear terms in x i , we obtain the following linearized replicator dynamics:

ẋi (t) ≈ ρ i 1 -(1 -β)∆ -(1 -γ) 2 j=1 α j (x j (t) + p * j ) -(1 -γ)βα i (x i (t) + p * i ) (7)
with ρ i = p * i (1 -p * i ) and p * is the interior stationary point of the ODE system. Equation [START_REF] Altman | The evolution of transport protocols: An evolutionary game perspective[END_REF] becomes

ẋi (t) ≈ -(1 -γ)ρ i 2 j=1 α j x j (t) + βα i x i (t) (8)
This linearized system is of the form Ẋ(t) = AX(t) where

A = -(1 -γ) ρ 1 0 0 ρ 2 α(1 + β) 1 -α α (1 -α)(1 + β)
We note that the previous system is asymptotically stable if the eigenvalues of the matrix A has negative real parts. In order to investigate the eigenvalues of the matrix A, we express the following characteristic polynomial of A:

χ A = det(λI 2 -A) = λ 2 -tr(A)λ + det(A)
Hence, the determinant and the trace of the matrix B are given resp. by

det(A) = (1 -γ) 2 ρ 1 ρ 2 α(1 -α) (1 + β) 2 -1 = (1 -γ) 2 ρ 1 ρ 2 α(1 -α)β(β + 2)
and

tr(A) = -(1 -γ)(1 + β) ρ 1 α + ρ 2 (1 -α)
The discriminant of this polynomial is:

D = tr(A) 2 -4 • det(A).
Let λ 1 and λ 2 be two eigenvalues of A. Thus λ 1 + λ 2 = tr(A) and λ 1 λ 2 = det(A). Since det(A) ≥ 0 and tra(A) ≤ 0, it easy to check that the real parts of λ 1 and λ 2 are negative. Hence, the interior fixed point p * is asymptotically stable in the replicator dynamics.

B. Replicator Dynamics with delay

In the classical replicator dynamics, the fitness of strategy a at time t has an instantaneous impact on the rate of growth of the population size that uses it. A more realistic alternative model for replicator dynamic would be to introduce some delay: a mobile belonging to group i perceives the fitness about his group utility after a given delay τ . Hence, the group utility acquired at time t will impact the rate of growth τ time later. Under this assumption, the replicator dynamics equation for the group i is given by:

ṗi (t) = p i (t)(1 -p i (t)) 1 -(1 -β)∆ -(1 -γ)• N j=1 α j (βp j (t -τ ) + (1 -β)p j (t)) -(1 -γ)βα i p i (t -τ ) (9)
Similarly to the non-delayed case, we proceed to the linearization of the replicator dynamics equations by introducing a small perturbation around the interior equilibrium p * i defined by x i (t) = p i (t) -p * i . We get the following ODEs system:

ẋi (t) ≈ -(1 -γ)ρ i N j=1 α j βx j (t -τ ) + (1 -β)x j (t) +βα i x i (t -τ ) (10)
The Laplace transform of the system (10) is given by:

[λ + (1 -γ)ρ i α i βe -τ λ ]X i + (1 -γ)ρ i (1 -β + βe -τ λ ) • N j=1 α j X j = 0
For the case of two groups, the characteristic equation of the ODEs system is given by:

λ 2 + λ(1 -γ)(1 -β + 2βe -2τ λ )(αρ 1 + (1 -α)ρ 2 ) + (1 -γ) 2 ρ 1 ρ 2 α(1 -α) 2β(1 -β)e -τ λ + 3β 2 e -2τ λ = 0 (11)
The zero solution of the linearized system above is asymptotically stable if and only if all solutions of the corresponding characteristic equation [START_REF] Levine | Modeling altruism and spitefulness in experiment[END_REF] have negative real parts. The form of this equation was studied in [START_REF] Hara | Stability region for systems of differentialdifference equations[END_REF]. The mixed intermediate ESS is an asymptotically stable state in the time-delayed replicator dynamics if and only if τ < τ 0 = min( π 2|λ+| , π 2|λ-| ), with λ + and λ -the roots of the non-delayed characteristic equation (τ = 0). Remember that, according to the proof of Theorem 1, the eigenvalues of the differential system have negatives real parts.

VI. OPTIMIZATION ISSUES

According to the structure of the ESS, we try to evaluate the performance of the global system in order to derive the optimal degree of cooperation β. The performance of the system can be presented by the measure of the probability of success in a given local interaction for a mobile randomly selected from all mobiles of all groups. This probability of success is expressed as follows

P succ = γ N j=i α i p * i + (1 -γ) N i=1 N j=1 α i α j p * i (1 -p * j ) +p * j (1 -p * i )
Let us now study the expression of the probability of success depending on the structure of the game model considered.

A. Fully Mixed ESS

In the fully mixed ESS, the equilibrium is given ∀i ∈ {1, . . . , N } by

p * i = 1 -(1 -β)∆ α i (β + N )(1 -γ)
This gives a probability of success

P succ = N (1 -(1 -β)∆) (1 -γ)(β + N ) 2 • 2(β + N (1 -β)∆) -γ(β + N )
Having this expression, we calculate the level of cooperation β that maximizes the P succ . We find that P succ is maximized for

β * = (4∆ -γ -2)N 4N ∆ + γ -2 B. Pure-Mixed ESS
We note that in this structure, there are n T groups using pure strategy T at the equilibrium and the other N -n T groups using mixed strategies. Then, the probability of success is expressed by

P succ = (α T + N j=nT +1 α i p * i ) • 2 -γ -2(1 -γ)(α T + N j=nT +1 α i p * i )
with

p * i = 1 -(1 -β)∆ -(1 -γ)α T (1 -γ)α i (β + N -n T )
Finally

P succ = βα T (1 -γ) + (N -n T )(1 -(1 -β)∆) (β + N -n T ) 2 • β(1 -2α T ) + N -n T + 2(N -n T )(1 -β)∆ + β -(N -n T ) (1 -γ)
We notice here, that P succ (β) depends on both n T and α T . These two variables are step functions of β. The optimal value β * which maximizes P succ for each value of γ will be computed through an iterative algorithm.

C. Fully Pure ESS

When all groups play pure strategy T at the equilibrium, i.e. p * i = 1 ∀i ∈ {1, . . . , N }

P succ = N i=1 α i p * i 2 -γ -2(1 -γ) N i=1 α i p * i = γ

VII. NUMERICAL RESULTS

A. Impact of the transmission cost on the equilibrium

We first investigate the case where two groups compete to access to the medium with α 1 = α = 0.2 and α 2 = 0.8. In a sparse environment (corresponding to a high value of γ), an anonymous mobile of group G i is more likely to be alone when transmitting to a destination. This suggests that he will play the strategy T all the time (p * = 1). However, in a dense environment (low values of γ), he is more likely to be in competition with another mobile while transmitting to the destination. In this situation, the strategy played by the mobile i will depend on the cost of transmission ∆. In Figure 1, we consider a low transmission cost (∆ = 0.2). We found that the mobile gives less interest to the effect of collision as the cost of transmitting is very low. In fact, loosing a packet does not affect the mobile's utility compared to what he would earn if the transmission is successful. This fact justifies the aggressive behavior of the mobile. However, when the transmission cost ∆ is high, the equilibrium structure differs. In Figure 2 consider a higher cost (∆ = 0.9). In this case, the mobile becomes more cautious and take into account the effect of collision since it degrades his utility. Thereby, he lowers his level of transmission.

B. Impact of the cooperation on the equilibrium

In Figure 3, we keep a high level of transmission cost (∆ = 0.9) and we change the behaviour of the mobiles. We change the degree of cooperation to pass from a nearly egoistic behavior with β = 0.1 to a nearly altruistic behaviour with β = 0.9. We notice that by increasing the degree of cooperation, users have more incentive to use strategy T . This suggests that increasing the degree of cooperation among users induces a coordination pattern in which users have more incentive to use strategy T .

C. Impact of the cooperation and the transmission cost on the probability of success

In this section, we investigate the evolution of the P succ according to γ. Intuitively, we can expect that when the mobiles are fully cooperative inside groups, this leads to a better system performance. However, we will show that we obtain a different result. We consider the same system as previously: two interacting groups, the smaller with proportion α = 0.2 and the bigger with proportion 1 -α = 0.8. We start with low cost of transmission (∆ = 0.2). In this situation, as shown in Figure 4, the more the mobiles cooperate the more the probability of success increases. We found that the full altruistic behaviour is the unique optimal solution up to a value of γ ≃ 0.45. Beyond this value of γ, P succ = γ and becomes, thus, independent of β. Hence, all levels of cooperation give the same performance of the system . However, when the cost of transmission becomes high (∆ = 0.9), we notice, through Figure 5, that the P succ takes different values according to the level of cooperation β and we remark that the fully altruistic behavior is no more the optimal solution. In fact, for low values of γ, the level of cooperation that optimizes the performance of the system is unique. The level of cooperation β opt is a decreasing function of γ, which confirms the analytical result. The uniqueness of the level of β opt remains until a value of γ ≃ 0.68 beyond which several levels of cooperation give the same system performance. Hence, we deduce a counter-intuitive result. We would expect that the fully altruistic behavior is always the best decision that should be adopted to maximize the system performance. However, we found that the mobiles have to be, often, less cooperative. In Figure 6, we represent the margin between the performance of the system when adopting a fully altruistic behavior and this performance when behaving somewhat selfishly but optimally.

D. Impact of the delay on the stability of the replicator dynamics

The presence of delay in the replicator dynamic equations does not influence its convergence to the ESS. However, it has an impact on its stability. We investigate this fact through the following numerical example. We consider N = 2, α = 0.4, ∆ = 0.7, γ = 0.2 and β = 0.75. This example corresponds to a fully mixed ESS (see Proposition 1). In Figure 7, we observe that the replicator dynamics converge to the ESS and remain stable, which confirm the Theorem 1. However, for τ = 4 in Figure 8, we obtain the stability but the convergence is much slower. The boundary of stability of the replicator dynamics is τ 0 ≈ 7.5. In Figure 9, this boundary increases and we observe, that the replicator dynamics oscillate and become no longer stable.

VIII. CONCLUSION

In this paper, we have presented a new model of Medium Access Control problem through evolutionary game theory which takes in to account pairwise interactions. Our contribution was to include and investigate the aspect of cooperation between agents of the same group. We have studied the equilibrium of the system through the notion of Evolutionary Stable Strategies and study the effect of transmission cost and cooperation level. We have found that the mobiles tend to transmit less when the energy cost is high, whereas they may profit by the cooperation aspect to rise their transmission levels. Thereafter, we have evaluated the performance of the system in terms of the probability of success. We have studied the stability of replicator dynamics in the classical and delayed form. In a future work, we plan to study the general model with specific distributions.

Fig. 1 .Fig. 2 .

 12 Fig. 1. Evolution of the ESS as a function of γ for ∆ = 0.2 and β = 0.1.

Fig. 3 .Fig. 4 .

 34 Fig.3. Evolution of ESS as a function of γ for ∆ = 0.9 and β = 0.9.

Fig. 5 .Fig. 6 .

 56 Fig. 5. Variation of Psucc with different levels of β for ∆ = 0.9.

2 Fig. 7 .

 27 Fig. 7. Stability of the replicator dynamics for τ = 0 .

2 Fig. 8 .

 28 Fig. 8. Stability of the replicator dynamics for τ = 4 .

2 Fig. 9 .

 29 Fig. 9. Instability of the replicator dynamics for τ = 12 .

‡ This work has been partially supported by the European Commission within the framework of the CONGAS project FP7-ICT-2001-8-317672