Jing Ye

Stéphane André

Laurent Farge

Nicolas Louvet

Stéphane Andre
email: stephane.andre@univ-lorraine.fr

Plastic instability studied experimentally on a semi-crystalline polymer through thermomechanical heat source identification: the flow stress concept revisited

Keywords: Thermomechanical Heat Source, Inverse methods, Plastic instability, Semi-Crystalline Polymers, Flow stress

Plastic instability studied experimentally on a semi-crystalline polymer through thermomechanical heat source identification: the flow stress concept revisited

Introduction

Plastic instabilities correspond to sudden phenomena associated to a strong transition of rheological regime. To quote the definition given by [START_REF] Coussot | Slow flows of yield stress fluids: yielding liquids or flowing solids?[END_REF], one could consider that they refer to "(i) a transition from a solid regime to a flowing regime which does not correspond to a true "liquid state," the flow in this regime may rather be seen as a succession of solid states during very large deformation; (ii) a strong tendency to localization of the yielded regions in some small region of the material while the rest of the material undergoes some deformation in its solid state". This description is in full agreement with experimental facts observed on semi-crystalline polymers over decades. But the physical phenomena related to the plastic instabilities are still to be fully understood and described, in particular to make any commercial Finite Element software that solves the mechanical stress/strain problem more predictive.

Scientific studies of the rheological behavior of solid materials involving plastic instabilities should normally favor a thermodynamic approach as they are always accompanied with dissipation. This is required especially because strong thermomechanical couplings occur, which result from a direct coupling between the dependent variables of the considered thermodynamic potential considered. With this aim in mind, many experimental studies have been conducted for years on a considerable number of different materials to measure calorimetric effects activated only by mechanical stress and strain variables [START_REF] Tauchert | Heat generated during torsional oscillations of polymethylmethacrylate tubes[END_REF][START_REF] Adams | Latent energy of deformation of bisphenol A Polycarbonate[END_REF][START_REF] Chrysochoos | An infrared image processing to analyze the calorific effects accompanying strain localization[END_REF][START_REF] Bhalla | Thermomechanics of slow stable crack growth: closing the loop between experiments and computational modeling[END_REF]. In the last 3 decades, this field of research has received much attention thanks to the widespread availability of Infra-Red (IR) imaging cameras. A lot of studies go no further than the analyzis of local temperature evolutions, to gain insight into the physical mechanisms at stake [START_REF] Maj | Analysis of plastic strain localization on the basis of strain and temperature fields[END_REF]. These studies remain qualitative, as temperature is not intrinsic to what happens in the bulk Representative Elementary Volume (REV) and thus proscribes any rheophysical understanding. The only way to advance in the direction of a real thermodynamically based study is to proceed to a reconstruction of the so-called Thermomechanical Heat Sources (THS).

In general, this requires a strong commitment to the field of mathematical inverse methods, as is seen for example in Aufray et al. (2000), although shortcut analyses are sometimes used [START_REF] Chrysochoos | An infrared image processing to analyze the calorific effects accompanying strain localization[END_REF][START_REF] Doudard | Identification of heat source from infrared thermography: Determination of 'self-heating' in a dual-phase steel by using a dog bone sample[END_REF][START_REF] Toussaint | Combining displacement, strain, temperature and heat source fields to investigate the thermomechanical response of an elastomeric specimen subjected to large deformations[END_REF][START_REF] Delpueyo | Heat source reconstruction from noisy temperature fields using an optimised derivative Gaussian filter[END_REF]. Many inverse methods have been put forward to reconstruct unknown source terms from the heat diffusion operator, either in the bulk of the material, at its boundaries, or corresponding to unknown initial conditions. Fewer methods are available when advection is considered along with diffusion [START_REF] Prigogine | The inverse source problem for the variable coefficients convection-diffusion equation[END_REF][START_REF] Maalej | Estimation of position and intensity of a pollutant source in channel flow using transmittance functions[END_REF]Souza and Roberty, 2012). This was necessary here as the development of plastic instabilities in polymers is responsible for strong kinematic effects. Everything presented in this paper is thus the result of a long process decomposed into three steps. A first paper [START_REF] Ye | Kinematic Study of Necking in a SemiCrystalline Polymer through 3D Digital Image correlation[END_REF] presents detailed results of the kinematic phenomenon through precise measurements made with 3-D stereo Digital Image Correlation (DIC). The plastic instability initiation, development and propagation are analyzed there through the local velocity or strain rate observables, expressed in either a Lagrangian or Eulerian framework. A second paper [START_REF] Ye | A numerical study of heat source reconstruction for the advection-diffusion operator: A conjugate gradient method stabilized with SVD[END_REF], is rather theoretical and gives all detailed information and bibliographic background concerning the mathematical method used to perform inverse reconstruction of the THS from IR recordings of temperature maps and velocity measurements. These were in fact synthetic data, produced (i) with an idealized model of the thermal transfer in a specimen for a well-known idealized mathematical form of THS and (ii) with idealized velocity profiles, but with the same characteristic profiles as those measured during real experiments [START_REF] Ye | Kinematic Study of Necking in a SemiCrystalline Polymer through 3D Digital Image correlation[END_REF]. Hence, the current report, which is the last paper in this series of three, aims to show the results that the inverse algorithm produces on real velocity and temperature data. Monitoring of the THS in both space and time variables can then be analyzed to identify different thermomechanical regimes in relation to the development of plastic instability. The paper is organized as follows:

In the next section, the experimental set-up used to produce the data is briefly described. In section "Thermodynamic Heat Transfer Modeling", we return to the thermodynamic foundations used in this study to derive the state laws, i.e. the behavior law of our material, which has both a rheological and a thermal component. The latter, through an entropy balance, naturally leads to the heat equation used for THS inversion. Our knowledge of the order of magnitudes of local velocities in plastic instabilities and of the associated dimensionless Peclet number suggests considering a reconstruction that accounts for the influence of both diffusion and advection. The Inverse Problem Formulation is then presented. In the "Thermomechanical observations from inversion" section, experimental results of reconstructed THS are presented.

These are discussed according to various possible options of the inversion process (linked to various possible model assumptions). Elements are provided that allow the reader to measure the difficulties inherent to this goal and the performances reached in this study. Lastly, the rheological implications that can be drawn from the THS measurements are discussed. The main finding of our work is that a flow stress can be calculated from the local THS measurements, as a function of the local strains and local strain rates. This "thermo-mechanically determined" flow stress is shown to correspond fairly well to the mechanical flow stress that can be reached from tensile-relaxation tests. This supports the simple idea that the THS associated with the plastic flow is essentially the sole contribution of heat dissipation.

Experimental setup

Experimental platform This is fully described in Andre et al. (2007) and [START_REF] Ye | Kinematic Study of Necking in a SemiCrystalline Polymer through 3D Digital Image correlation[END_REF]. It consists of a servo-hydraulic MTS ® 810 load frame surrounded by three metrological components (Fig. 1  A stereo 3-D DIC system (Aramis3D), which provides full field measurements of the displacement vector on the specimen and consequently full field strain tensor maps ( , , )

x y t  or longitudinal profiles. These are measured on the opposite face of the specimen with respect to the Video-extensometer real-time monitoring of the central strain. As will be seen later, the system provides the local velocity measurements necessary for this study.

 An IR camera (JadeIII-SC7000 from Cedip/Flir systems), which provides the temperature fields   ,, T x y t .

The DIC and IR equipments work alternatively rather than simultaneously, for two different tests on two different specimens. This can introduce a bias when comparing thermal and kinematic measurements and therefore makes it crucial to have the exact same ramp command

()  VE VE y0 y0
 realized by the set-up when tests are repeated to produce sets of DIC and IR movies.

Fig. 1 also shows the geometry of the specimen, which is designed in such a way that the onset of plasticity takes place precisely at its center (REV of uniform cross-section). multiple physical investigations of the microstructure and also at the macroscopic scale [START_REF] Ye | Kinematic Study of Necking in a SemiCrystalline Polymer through 3D Digital Image correlation[END_REF][START_REF] Ye | Kinematic Study of Necking in a SemiCrystalline Polymer through 3D Digital Image correlation[END_REF][START_REF] Blaise | Advantages of a 3-parameter Reduced Constitutive Model for the Measurement of Polymers Elastic Modulus using Tensile Tests[END_REF].

Experimental conditions

Previous experiments of the same type aiming to measure THS were performed at low strain rates, with an inverse model based solely on the diffusion phenomenon (Andre et al., 2012). The metrological objective in that case is a hard challenge to achieve: at low strains, heat effects are of small magnitude (quasi non measurable for strain rates below 1 0.005 s  ). The signal-to-noise ratio of recorded temperatures is not very high and applying an inverse method to such data produces THS outputs of poor quality. In this paper we aimed to investigate higher strain rates. This provides thermomechanical effects of greater amplitudes, hence better THS outputs. However, there are drawbacks; the high local strain rates occurring with the onset of plasticity require both the development of an inverse model taking into account heat advection and the provision of velocity measurements as additional inputs of the inverse process. This was the original purpose of this research. Four imposed strain rates are considered for the experiments:

.

  VE 1 0 0 01 s  , .   VE 1 0 0 02 s  , .   VE 1 0 0 03 s  , .   VE 1 0 0 04 s  . To reach a maximum deformation of ( 0) 2    y y
in the most stretched central part of the specimen then requires between 50 and 200 seconds. Reaching higher strain rates is difficult because the controller PID settings of the actuator/feedback loop are not able to produce a perfectly defined strain rate ramp behavior. Higher strain rates also change the nature of the problem; different physical/rheological behavior of the material and simpler identification models such as an adiabatic assumption become relevant.

Thermodynamic Heat Transfer Modeling

Thermodynamic framework In the context of Thermodynamics of Irreversible Processes (T.I.P.), the rheological behavior of a REV of some material is obtained in a thermodynamically consistent way by writing state laws. These clearly express all the couplings between macroscopic and internal dual variables, the latter being in charge of describing the multiscale evolution of the microstructure and making the intrinsic heat dissipation calculation possible.

According to [START_REF] Callen | Thermodynamics and an Introduction to Thermostatics[END_REF] they are obtained by stating that the internal energy is a potential which contains all information about the system, even in out-of-equilibrium situations. Highly detailed information is available from previous research (Andre et al., 2012;[START_REF] Cunat | The DNLR approach and relaxation phenomena. Part I : Historical account and DNLR formalism[END_REF] and we reiterate only the basic outline and principles, which are sufficient to allow the reader to grasp the subject.

The volume-specific internal energy is given from Euler fundamental equation, summing its thermal (T,s variables), mechanical (, variables) and chemical ( k ,n k variables) contributions

 

, , :

j j j j u s z Ts A z        (1)
where a first but important step has already been made, which consists of substituting the classical   Az dual variables (see Prigogine, 1949;[START_REF] Callen | Thermodynamics and an Introduction to Thermostatics[END_REF] for affinities as generalized forces; [START_REF] Nowick | Anelastic relaxation in crystalline solids[END_REF][START_REF] Meixner | Thermodynamische theorie der elastichen relaxation[END_REF] for affinities as conjugates of normal internal variables). This is a classical procedure [START_REF] Kuiken | Thermodynamics of Irreversible Processes: Applications to Diffusion and Rheology[END_REF], but one can insist on the fact that, for the mechanical scientist, it allows realization of the scale transition. Various j  type processes are considered which involve the k  species in unknown chemical reactions. j

A is the affinity associated with process j representing a non-equilibrium generalized force, which allows the perturbed microstructure to proceed to an equilibrium recovery. Within this point of view, the state laws are straightforward [START_REF] Tisza | Generalized Thermodynamics[END_REF][START_REF] Cunat | The DNLR approach and relaxation phenomena. Part I : Historical account and DNLR formalism[END_REF] and lead to generalized matrix form, such as for example the following one when the Helmholtz potential is considered:

th mech tt th mech C Eb T sT E E b b b g z A                               (2- a b c     
In view of the 1-D tensile test, the mechanical tensor quantities  ,  , E are replaced by scalar ones and E corresponds to the rigidity or Young modulus. The upper bar stands for tensorial quantities linked to internal variables (the collection of j  type processes gathered in the z vector). , th mech bb stand for the coupling matrix between the macroscopic observable variables and the internal, non-observable variables. For this matrix form of state laws to be operational, the bz  products, which represent the non-elastic or dissipative component of the entropy and stress responses of eqs.2a, 2b, are further elucidated, but this aspect will not be detailed here. The interested reader can refer for example to [START_REF] Blaise | Advantages of a 3-parameter Reduced Constitutive Model for the Measurement of Polymers Elastic Modulus using Tensile Tests[END_REF]Andre et al., 2012) to learn more about this part of the theory. g stands for the coupling matrix between internal variables. In classical reversible thermodynamics, matrices b and g vanish and one recovers from (2) the state laws associated with the pure thermo-elastic material.

In view of reconstructing the THS responsible for the temperature field of the specimen, the entropy balance has to be written which, in accordance with T.I.P. treatises [START_REF] Kuiken | Thermodynamics of Irreversible Processes: Applications to Diffusion and Rheology[END_REF]Munster, 1966), gives equation ( 3):

1 dissipation exchanged internal production q i s s s s T       J  (3) with - 1 dissipation i s A z T 
: dissipation is the flux-force product of internal variables that reflect the non-equilibrium state; q J is the heat flux density composed in our case of both diffusive (Fourier's law) and advection components that is

  J V q k T cT      (k: thermal conductivity, cT  :
volume-specific enthalpy, V: local velocity).

Taking the negative of equation ( 2a) as left-hand term of eq.( 3) and rearranging terms gives the local heat equation

JJ THS th q q THS q cT ET T b z A z q              (4)
which states that the local enthalpy rate varies according to heat exchanges with the

surroundings q J ( 2 Wm  
) and the THS source term

THS q ( 3 Wm  
). This latter will be identified from the reconstruction process and can be further decomposed according to its three components:

 Reversible thermo-elastic effect ET   , remaining as long as the tensile test runs;

 Irreversible intrinsic heat dissipation 0 Az  D  Recoverable entropic couplings

Dimensional analysis and considerations concerning the inverse estimation of THS

Rewriting eq. ( 4) in a differential and usual form and reducing it to the 1-D case gives

2 _ 2 THS heat loss diffusion advection accumulation T T T c cV k q q t y y             (5)
where the _ heat loss q term figures the heat losses in the two transverse directions, as a result of the reduction in the dimension of the problem and non-adiabatic nature of the system. In practice, the time constant for such heat losses is very long compared to the experimental duration and the specimen is confined in an enclosure with reflecting walls. These will therefore be neglected in the treatment, keeping in mind a possible overestimation of the identified THS q . The second point to discuss concerning equation ( 5) lies in the necessity or not of accounting for both diffusion and advection for the THS reconstruction, especially when plastic instabilities occur. A preliminary answer is made possible by a precise measurement of local velocities during this phenomenon. These have been obtained using the DIC system and for tensile tests performed at four different imposed strain rates, as described in the Experimental conditions section. As an example (see Fig. 2), typical experimental velocity profiles show large differences with the deformation state (characterized through the 0  VE strain ) and the localization effect in the [-1;1] mm centered area (Lagrangian reference) starting at 0.12 y  when yielding starts. All these measurements are available (see [START_REF] Ye | Kinematic Study of Necking in a SemiCrystalline Polymer through 3D Digital Image correlation[END_REF]. They gave the numerical data used in the dimensional analysis presented below to estimate orders of magnitude of the different mechanisms of heat transfer to be compared. where k is the thermal conductivity, the characteristic length over which diffusion is considered, and T  the same considered difference. The material properties c  and k (or thermal diffusivity a ) have been assumed to be constant over the bulk material to obtain eq.( 5). Advection can be neglected with respect to diffusion when the ratio of these two remains small, i.e. the Peclet number

cV T V Pe 1 k T a    
 . The Peclet number is also the ratio of the characteristic time of diffusion to the time of propagation  Regarding velocities, it must be pointed out that the local velocity at the center of the specimen is always equal to 0 by symmetry. The maximum absolute velocity is always obtained symmetrically at the extremities of the specimen. The decrease in velocity magnitude can be seen as soon as necking proceeds ( 0 0.12 VE



). As a consequence, the Peclet number will vary along the tensile test.

From this information, Peclet numbers can be estimated for the different regimes, along with typical characteristic times, and the following analysis brings a physical understanding of what can be expected:

 At small strains ( 0.05  ), and according to the investigated strain rates, the experimental time at which to consider the influence of both mechanisms of transfer remains below 5s. During this time, the typical maximum length affected by diffusion is 1 at mm  . For this length, and the maximum velocity considered, the characteristic time for advection is of the order of 2s, which means a situation of Peclet approaching unity. At the same time, the THS is only made from the instantaneous thermo-elastic effect, which simultaneously affects all parts of the specimen and the dissipative phenomena have not really been triggered. The temperature gradients are then very small, reflecting the nearly homogeneous stress distribution ruled by the sole geometry of the specimen. In consequence, even if both advection and diffusion phenomena have nearly the same impact on heat transfer, this latter remains very weak because the temperature is nearly homogeneous in the specimen. On a REV scale, anywhere in the specimen and considering this short time range, the system appears adiabatic. In other words, the instantaneous thermo-elastic THS is directly responsible for an enthalpy change of the REV.

 When necking initiates ( 0 0.12  in the center of the specimen), the thermal effect and accompanying temperature gradients appear in the neck, but the velocity field has , which means that advection is much less efficient than diffusion.

 At the end of the test i.e. for high strains, the two necking shoulders appear as regions with high velocities/high strain rates and non-negligible temperature gradients (Fig. 9 in Andre, 2012). A rough estimate of the Peclet number can be made, for example for the strain state . Advection competes then locally with diffusion and has to be considered in the inversion process to produce good estimates of the THS in the shoulders.

This preliminary analysis leads to two important issues for future thermo-rheological investigations of this type:

 The THS reconstructed over the REV at the center of the specimen will always remain insensitive to advection. Due to symmetry, very low velocity magnitudes operate on flat temperature profiles in this area. If one aims only to produce a thermomechanical signal for the purpose of identifying the state laws of a given material (Eqs2), the best strategy is to identify the THS versus time at the center of the specimen in the simpler diffusion case.

 The THS reconstructed when the plastic instability propagates, requires taking into account the advective part. This is unavoidable when one aims to understand the local distribution of the THS with respect to strain rate distributions, a major objective and result in terms of this study.

Inverse Problem Formulation

This work, which is essentially of a mathematical order, was the object of a separate paper [START_REF] Ye | A numerical study of heat source reconstruction for the advection-diffusion operator: A conjugate gradient method stabilized with SVD[END_REF]. A short summary is given here so that the reader can gain insight into the kind of strategy used to build the inversion computational code. The solution to the direct problem (eq.5) was brought about by decomposition of both the temperature and heat source on the same spectral finite basis (dimension N) like for example,

      1 , N THS i i i q y t d t V y    (6)
where the i V 's are the selected spectral modes and the i d 's, the THS coefficients to be estimated as a function of time to reconstruct the heat source. Two algorithms have been considered which depends on the choice of a spectral basis: Algorithm A1 uses a very sophisticated spectral basis known as the Branch modes (see [START_REF] Videcoq | Heat source identification and on-line temperature control by a Branch Eigenmodes Reduced Model[END_REF] THS T M t q temperatures of the specimen:

            2 2 0 11 , , ; , , ; 22 f t LS THS f f THS f J q T M t T M t q dt T M t T M t q t          (7) Temperatures   ,;
THS T M t q are produced by the direct spectral model. The The best compromise between these different options was chosen for the inversion of the THS according to our good stabilized initial experimental conditions, the level of noise on the data, the physical insight obtained from the rough observables i.e. symmetry, the quality of the reconstruction (regularization process adapted to our signal to noise ratio). The inversion settings remained unchanged for all data inversion and were selected as those leading to the optimal choice from reference results obtained on synthetic noisy data. Results obtained with algorithm A1 will be presented first up to compare THS identifications on whole temperature profiles in order to highlight the differences obtained when neglecting or not the advection phenomenon. Results obtained with algorithm A2 on half profiles will be given starting from section Magnitude of the THS as a function of the strain rate and in the rest of the paper to discuss more quantitatively the physical implications resulting from THS estimations, on the rheological side.

Thermomechanical observations from inversion: results and discussion

Diffusive versus Advecto-Diffusive THS reconstructions The first objective of this study was to compare the influence of the two modes of heat transfer in real experimental conditions and for real data. Here we used the full experimental temperature profile with algorithm A1.

Fig. 3 shows reconstructed THS profiles in the Eulerian frame for the 4 different imposed strain rates VE 0  . The two curves in each plot correspond to the THS reconstruction performed with or without taking into account the advection and associated measured velocity profiles. The spectral approach adopted in the inversion process makes reconstructed THS profiles un-noisy as a result of the spectral approach: once the THS coefficients were identified from the temperature data, eq. ( 6) computed the linear combination with perfect bases functions. The reconstructions are given for the case of a true strain of 1.8, which corresponds to a high specimen deformation (around 600%) for which the neck shoulders had already propagated and the reconstruction biases were the greatest. They made it clear that neglecting or not advection made major differences, with an overestimation of the THS when merely diffusion was considered. These differences were marked at the specimen extremities, whereas it was clear that focusing on the area of pixels at the specimen center made it unnecessary to use a diffusion and advection model for inversion; reconstructions matched perfectly in both cases. This result legitimized a posteriori the thermomechanical results published in Andre (Andre et al., 2012) where the THS was only reconstructed for the central REV and with a pure diffusion model.

The existence of an extremity-oriented velocity flattened the temperature profiles: this additional mechanism of transfer favors temperature field homogeneity.

Drawbacks associated with the use of algorithm A1 are clear from the figures, showing THS profiles which lost the symmetry of the problem at high strain rates. Although more or less pronounced, this phenomenon was reproducible on all the experiments we carried out. It resulted from high sensitivity of the inversion process to all experimental imperfections, in the first rank of which is the lack of simultaneity of velocity and temperature profile acquisition.

(a)

. . All the strain was localized in this region and a null longitudinal strain was observed outside (see Fig. 5c in [START_REF] Ye | Kinematic Study of Necking in a SemiCrystalline Polymer through 3D Digital Image correlation[END_REF]. The reconstructed THS for all 4 strain rates clearly showed that the THS activity was localized in this region and increased with the strain rate. ADM or DM models gave roughly the same reconstruction as expected from the discussion of section 3.2. Slight discrepancies appeared logically where the velocity is the greatest ( 3 y mm 

). Outside of the   5;5 mm  window, no THS was associated with the null strain state. The temperature there is uniform i.e. there is no temperature gradient and identified THS were very weak. The curve was always wavy at its extremities, with large oscillations as a result of the influence of the first modes on the reconstructions.

Below, we only consider the THS reconstructed using the full advection-diffusive model (ADM). This is of course due to the thermoelastic effect which will induce negative THS. This short time non uniformity is simply the result of the non homogenous cross-section along the specimen length. Very rapidly then, the thermomechanical effects produce a raise in temperature which can be seen to be localized first in the central region and further spreads out on the full specimen.  At short times, the THS is negative, with a wide extension over the specimen (Figs. 6a andb). It only corresponds to the thermoelastic effect, which is instantaneous and completely reversible. No heat is dissipated (D term of THS q ) nor involved in entropy processes coupled to internal variables. Of course it is non uniform as it obeys the strain-rate distribution which is also non uniform due to the non constant cross-section along the tensile-specimen axis (even before localization takes place).

 The THS activity very quickly becomes positive, simultaneously with the yield stress crossing, which implies a plastic activity. It is then concentrated in a very small area around the specimen center (the necked part, which is approximately shown between the dotted lines of ) shows the profiles given in Fig. 3a, demonstrating a ratio of THS activity of nearly 2 between the average peak levels and the curve's well level in the central position. . In the Lagrangian frame, and as time proceeds, material points having initially been within the camera's field of view exit from it, which explains the lack of information in the left end right upper corners of the Figures. The first observation is that there is a clear agreement between the two observables, which undoubtedly means a simple and direct correspondence between strain rates and THS as necking proceeds. The second consideration is related to clear differences in quality between these two maps, which underline the fact that inversion of temperature data as measured with the best quality cameras available to date appeared unable to compete with data measured in a direct process. However, obtaining DIC measurements in our conditions, where the specimen has a finite width of 2.5 mm and an extension of nearly 30 mm is not a straightforward task, as explanations are given in [START_REF] Ye | Kinematic Study of Necking in a SemiCrystalline Polymer through 3D Digital Image correlation[END_REF]. Despite that, a third conclusion can be drawn, related to the physical behavior of necking in semi-crystalline polymers. The Natural Draw Ratio was defined in the polymer community as the highest draw ratio obtained before the neck began to travel over the whole length of the sample. We showed recently [START_REF] Ye | Kinematic Study of Necking in a SemiCrystalline Polymer through 3D Digital Image correlation[END_REF] that indeed it corresponds to an intrinsic property of the material when measured through full field kinematic information: the strain value at which the central region with the highest magnitude of strain rate splits into two shoulders. The measurement supplies 0 1.2 NDR . It is interesting to note that the Natural Draw Ratio is recovered at the exact same strain on the THS signature, where the THS central region with the maximum activity also splits into two parts (Figs. 7a andb). Once again, this indicates more quantitatively that the THS and strain rate fields are fully correlated in the plastic regime. to be difficult to produce, due to a rather complex experimental protocol and an inversion method applied to data of relatively small signal magnitudes. As explained before in the "Inverse Problem Formulation" section, obtaining more precise quantitative conclusions led us to prefer algorithm A2

We first show the THS evolution as a function of time or strain in the middle of the specimen ( 0 y  ). This corresponds to the curves obtained from the cutting plane P of Fig. 6a. where additional heat effects come into play with the hardening regime. 4), the thermoelastic effect is more intense with proportional strain rates. The hardening is accompanied by more intense heat effects with strain rates. A clear overshoot is now obtained at the yield stress and the onset of plastic instability.

The oscillations behind this point again show a small experimental artefact. For strain rates greater than 1 0.01 s  , a problem of reactivity of the feedback loop to ensure a constant strain rate VE 0  from the real-time local measurements of the Video-extensometer exists. The best PID settings of the feedback loop cannot avoid very small strain oscillations due to the actuator displacement, which result in large amplitudes of the THS signal due to the contribution of the large thermoelastic effects. This problem increases when the strain rate level is raised, but as it corresponds to an instantaneous and reversible effect, it does not affect the average behavior.

We do not intend to further investigate here a well-defined phenomenon such as thermoelasticity, but one could imagine correction of the THS curve oscillations from the recorded effective strain rate signal in relation to thermoelastic THS y q ET    . Fig. 9 THS versus strain at the neck and specimen center for different strain rates.

An alternative to investigate trends accompanying the THS magnitude and strain rates relies on the joint analysis of both observable profiles along the specimen axis as shown in

Rheological investigation: a thermomechanical-based flow stress estimation

From all the local data (THS and strain rate measurements), we attempted to obtain a quantitative insight into the general behavior of the THS versus strain rate and associated rheological outcomes. For this, we relied on the conclusions of a previous study (Andre et al., 2012), which made use of the thermodynamic model of section 3 and showed in all likelihood that the thermoelastic and entropic components of the THS balance one another out in the steady state regime , i.e. when necking is established and plasticity governs deformation. The consequence was that the measured THS roughly corresponded to the irreversible dissipated power: THS A z  D

. The question to attempt to answer is: does the THS or intrinsic power D behave as proportional to  as a result of a solid friction type of behavior, as has been widely discussed when considering extensional flows of melt polymer systems? Or alternatively can it be rather related to 2

 as a result of a fluid-like behavior and viscous effect? In this section, the local true strain rate is denoted simply by  , and refers to precise measurements obtained by DIC.   , single-peak regime, just before neck propagation;

Measurements made in

 Curve C : 1.4 plateau   , initiation of the double-peak regime;

 Curves D and E :

1.6

  and 1.8 respectively, full double-peak regime.

The solid line curves correspond to the linear regression of each data set (prime notation). The THS is measured directly from Fig. 9, with appropriate interpolation of the curves, or from profiles such as those shown for example in Fig. 10, where averaging of a series of profiles taken before and after a given  is necessary because of the noisy character of the reconstructed THS. The associated error bars were evaluated by accounting for the discrepancies observed on repeated experiments. The first comment is that the THS or dissipated power not only depends on  but also on the strain level. In a recent paper reporting on the full scanning of a specimen undergoing tension during synchrotron Small Angle X-ray Scattering real-time experiments [START_REF] Farge | Anisotropy development during necking of HDPE studied at the microscale with in situ continuous 1D SAXS scans[END_REF], it was shown that the local strain univocally governs the microstructural state. It is then normal to observe an intrinsic dissipation which directly depends on the underlying microstructural state. Regarding the strain rate dependency and accounting for a significant measurement uncertainty, the global trend seems to be a linear dependency.

Measurements made in the specimen shoulders

To go further, we can take advantage of our local measurements and add additional points to obtain another range of strain rates. This is possible when measuring the maximum THS recorded in the double-peak regime (for the imposed central strain measurement 1.8   ). In the neck shoulders where THS is at its maximum, DIC gives a local strain measurement of only 1.3   , which falls between the two selected strain states of plots B and C. Nevertheless, within that regime, the associated local strain rates jump towards higher values, giving additional interesting points. For the four tensile tests performed, at We now show that the flow stress values obtained through THS measurements can be absolutely related to the stress relaxation decrease when drawing is stopped and a constant strain dwell is imposed. This way of measuring the flow stress is the natural consequence of the physical meaning of flow stress: the stress necessary to sustain further plastic deformation at a particular strain. For this, we refer to data from true stress-true strain curves obtained in a previous study (Blaise et al., 2012) for a Tensile-Relaxation test performed at a strain rate of 1 0 0.005

VE s   
with some values also being given for a strain rate of 1 0 0.02

VE s   
. The tensile stage is stopped at various levels of an imposed strain  (Video-extensometry measurements at neck center). From this set of curves (Fig. 12), we measured the flow stress . In Table 1 (line 3) some specific values of ).

Conclusion

The experimental study reported here concerned the calorimetric effects occurring under the mechanical drawing of semi-crystalline polymers. This was shown to be a lengthy and tedious process. To obtain THS measurements from mechanical tests is tedious. It involves a good metrology, associated with complicated mathematical treatments such as inversion, and lastly a consistent and efficient thermodynamic model to account for couplings between rheological, thermal and microstructural behavior. Additionally, even for polymers in the glass transition regime, for which irreversible processes are in evidence, the strong thermomechanical couplings like those studied here, where the thermal effects are produced by the material itself, are very limited in magnitude. Strain rates above 1 0.01 s  were necessary in this study to induce measurable quantities, but were soon limited in their upper bound by the machine capacities and interest of using a non-adiabatic model for inversion. Future efforts in the direction of measuring calorimetric effects in specimens undergoing mechanical solicitation should therefore be considerably rethought.

Despite that point, THS measurements have been proved to be fully in phase with pure kinematical effects; namely THS behaves exactly like strain-rates. From that consideration, and trying to go further into the rheological consequences of that observation, we have shown that THS versus strain rate dependency leads to measurement of the flow stress associated with neck propagation, at least when THS can be considered as representative of the dissipated power i.e. internal entropy generation. The good comparison obtained with flow stress measured alternatively using simple tensile-relaxation tests, validates the rheological findings we made. It suggests that additional efforts from the community could reach calorimetric measurements of such a type in a more precise and direct way.

  ):  A homemade Video-extensometry system allows the axial or longitudinal true strain in situ and in real-time in the middle of our specimen   0 y  . Thanks to this external signal, the tensile machine is controlled to perform tensile tests at constant imposed strain rates ()  VE VE y0 y0  . As a consequence, most of the results presented below are indexed on the true strain VE 0  .

Fig. 1

 1 Fig. 1 Sketch of the experiment and geometry of the specimen (left)

  for the affinity-extent of the reaction   , jj

  thT b z requiring a model that describes the behavior of the internal processes.

Fig. 2

 2 Fig. 2 Velocity profiles versus material (lagrangian) coordinates for different strains 0  VE obtained for a test at a strain rate of -1 0 0.02 s  VE .



  of the three different variables ,, Va has to be estimated:  The thermal diffusivity coefficient is a material parameter. It has been precisely measured by a flash experiment developed in our laboratory: The characteristic diffusion length is estimated as the minimum (metrological) length allowing for a good thermal gradient resolution from IR images. It has been found empirically in various controlled thermal situations that 4 pixels always allow a good computation of thermal gradients. The resolution of 1 130 px m  in our optical configuration makes the estimation of min 550 m   .

.

  dropped to very low values due to localization. The investigated local strain rates of .These values are perfectly consistent with the velocities of the neck frontiers for this strain level, (see Fig. A-1 in Supplementary material). The diffusion time a t is then of the order of 1s and the local Peclet number ranges from

.

  The absolute velocities are of the order of ./ V 0 5 mm s  (see Fig. A-2 in Supplementary material) which makes a Peclet number of the order of Pe V a 1 

  . It allows considering any kind of boundary conditions. It was specifically used to reconstruct the whole experimental THS profiles on a [-L,L] segment along the y-axis of the specimen. Algorithm A2 uses the more classical Fourier modes. Even if the two algorithms produce close results[START_REF] Ye | A numerical study of heat source reconstruction for the advection-diffusion operator: A conjugate gradient method stabilized with SVD[END_REF], this latter was proved to give slightly more precise results, taking advantage of the obvious physical symmetry of necking which leads to consider only half-profiles of temperature data, with the nice consequence of producing an exact fixed null flux -a null temperature gradient-boundary condition. The cosine family offers then a perfect tool to respect this boundary condition. Additionally, the temperature profile can be averaged over its two mirror-sides and this favors a better signal-to-noise ratio, a determinant parameter in difficult inverse problems. The inversion runs then over temperature profile data along the [0,L] y-axis length of the specimen.For both algorithms, an optimization process has been designed to minimize a least-square criterion

  arguments of the temperature variable shall not disturb the reader: the objective is well and truly to identify the decomposition coefficients   i dt. The number N of decomposition coefficients i.e. the dimension of the selected spectral basis can be chosen classically in such a way that it produces a good regularization of the problem, ensuring stability and uniqueness of the solution.Because solving this problem requires an estimation of the descent gradient large size problem, the method of Lagrange multipliers or of the adjoint state is used in algorithm A2. It is known commonly as the Conjugate Gradient Method. It produces an iterative algorithm[START_REF] Ye | A numerical study of heat source reconstruction for the advection-diffusion operator: A conjugate gradient method stabilized with SVD[END_REF]) which converges to the exact solution but, traditionally in inverse problems, may lack stable convergence. To overcome this technical issue, a special feature has been developed during this study. It consists of applying a Truncated Singular Value Decomposition on the Gram matrix of LS J  . Such treatment which can be considered as a filtering process on high frequency singular vectors allows the representation ofLS J with the minimum degree of freedom and then the production of a set of descending directions having the best impact on the optimization process. The performance of the estimation can be assessed through the quality achieved in the residuals and has been extensively studied on theoretical cases[START_REF] Ye | A numerical study of heat source reconstruction for the advection-diffusion operator: A conjugate gradient method stabilized with SVD[END_REF]. The algorithms that were developed allow the user to choose between:-two different spectral bases (A1 and A2) used for the temperature/THS decomposition, -two different boundary conditions in the y-direction (Neuman or Robin type) leading to two different strategies; identification is performed from the whole longitudinal profile or half of it, taking advantage of the symmetry of the problem and the resulting good consequence of averaging the signal on both sides, -two different initial conditions (real temperature profiles on the specimen or ideally considered as set to a constant "room temperature"),-two different inverse "engines" (with or without a priori information and different regularization techniques)

Fig. 3

 3 Fig. 3 Reconstructed THS profiles for the Diffusion Model (DM -Dotted lines) and Advection Diffusion Model (ADM -Solid lines) at a true strain 0 1.8 VE for different imposed tensile

Fig. 4 .

 4 Fig. 4 Reconstructed THS profiles for the Diffusion Model (DM-Dotted lines) and Advection Diffusion Model (ADM-Solid lines) at a true strain 0 0.4 VE  for different imposed tensile

Fig. 5

 5 Fig. 5 Temperature map   , T y t in the Eulerian frame for

Fig.6b corresponds

  Fig.6bcorresponds to a material domain of 4mm length, i.e. included in the initial material domain of uniform cross section). The THS magnitude is no longer due solely to the contribution of thermoelasticity, which is still ongoing at the same magnitude, but to additional

Fig

  Fig. 6a THS map

Fig

  Fig. 7 Strain rate map

Fig

  Fig.8shows the

Fig. 8

 8 Fig. 8 THS (Left axis) and True Stress (Right axis) versus strain at the specimen's center. Reproducibility is as found in 4 repeated experiments and reconstructions at

FigFig. 10

 10 Fig.10 for

  the specimen neck center In Fig.11 the local measurements of THS versus  obtained in the specimen neck center ( 0 y  ) are shown, for five different strain states (Curves A,B,C,D,E). The curves correspond to the following cases:

  difference between the stress reached when the tensile stage was stopped at a given value of  , and the asymptotic stress , r   reached for indefinite relaxation, also known as the relaxed stress. Both were obtained by an identification process performed on the whole curve due to the model detailed in Blaise et al., (2016) (See Fig. D in Supplementary material). The confidence bounds on , flow TR   were estimated classically, by computation of the sensitivity coefficients and variance-covariance matrix. They range from 0

Fig. 12

 12 Fig. 12 Tensile tests followed by relaxations initiated at different  strains.

Fig. 13

 13 Fig.13plots all available points (star dots) obtained in this manner. A fairly good correspondence between both flow stress estimations can then be observed. This original result makes it clear that the flow stress needed to make the material flow in a plastic regime (assimilated to neck propagation) is responsible for the internal heat dissipation. It would be interesting now to observe this rheological evidence on other systems (a purely amorphous material for example).

Fig

  Fig. 13 Experimental values of () flow MPa   obtained from THS measurements and from

  that depends on the local strain state: values given in the legend of Fig.11 and reportedin Table1(line 2).For a given microstructural state, we have then access to the local instantaneous stress producing the plastic deformation. This stress can in no way be assimilated to the yield stress as it is much less sensitive to strain rates in that range and almost equal to 33[START_REF] Ahmad | Flow stress of high density polyethylene and nylon 66 at high rates of strain[END_REF]. This thermomechanical approach made it clear that the macroscopic mechanical yield stress should not be mixed up with the thermomechanical flow stress. Where the yield stress corresponds to the critical stress necessary for the necking development of such polymers, the flow stress is directly connected to the heat dissipation produced locally by flow.

	, flow THS
	
											yield	MPa
	obtained for example for the tensile test at		0 VE		0.01 s	1 	, see Fig.8. This value is very
	commonly obtained on HDPE (Strain  0.3	1.2	1.3	1.4	1.6	1.8
		, flow THS 	() MPa			18.3	26.5	29.7	34.8	50.5	76
			(from fig.11)
		flow 	,TR () MPa			17.5	28.1	30.3 ♦	40	54.2	78
								0 VE		1 0.01 ,0.02 ,0.03 ,0.04 1 1 s s s   	s	1 	, local strain rates in the shoulders
	reach values of about				1 0.025 ,0.045 ,0.061 ,0.067 1 1 s s s   	s	1 	respectively. The results are
	reported in the same Fig.11 with curves F (filled squares) and F' (dashed line = Linear model).
	The linear regression of the data (F') clearly gives a line with a slope that falls between those
	obtained for	 	1.2	and	 	1.4	. Additionally, the two points corresponding to the highest 
	values still fall in the interval defined by the two linear models obtained for	 	1.2	and



(From Fig.12 and Model identification)

Table 1

 1 Experimental values of ()

flow MPa   obtained from THS measurements and form Tensile-Relaxation tests for different strain levels  . ( ♦ = value obtained by linear interpolation between strains 1.2 and 1.4 )

 

. This tends to validate the extrapolation of the linear behavior. Additionally, we checked that a quadratic dependency (another possible physical behavior) failed to encompass a global trend of these data (see Fig. C in Supplementary material).

Fig. 11 THS versus strain rate as measured locally for 6 different strain states.

Rheological implications: THS measurements as a direct measurement of the flow stress

All these data correspond to strains and strain rates associated with the necking development in a steady-state regime, beyond the onset of plastic instability but in the plastic regime of deformation, either in the primary stage ( single peak regime) or in the secondary stage (double-peak regime) when the plastic instability propagates in the specimen shoulders.

The linear dependency of dissipative THS q as function of the strain rate leads us to write   , the strain rate really imposed in the middle of the specimen is around 75% of the ideal target


Patrick Maillé
email: patrick.maille@imt.fr

Bruno Tuffin
email: bruno.tuffin@inria.fr

A Network Reliability Game

In an ad hoc network, accessing a point depends on the participation of other, intermediate, nodes. Each node behaving selfishly, we end up with a non-cooperative game where each node incurs a cost for providing a reliable connection but whose success depends not only on its own reliability investment but also on the investment of nodes which can be on a path to the access point. Our purpose here is to formally define and analyze such a game: existence of an equilibrium output, comparison with the optimal cooperative case, etc.

Introduction: problem definition

We consider a topology with nodes wishing to be connected to a network access point. If from a node there is no direct connection to the access point, the connection has to be routed through other nodes. The feasibility/success/quality you experience as a node therefore depends on the quality you provide yourself, but also on the participation of peers to the network.

This type of problem has many applications. Typical, but non-exhaustive, ones are:

• An ad hoc network [START_REF] Toh | Wireless ATM and Ad-Hoc Networks: Protocols and Architectures[END_REF], which is a network without any fixed infrastructure where nodes serve as relays for their neighbors. Such an organization presents the advantages of being decentralized, and of incurring no deployment cost since relying on the collaboration and willingness to participate of nodes. There are many practical applications of those networks: Army tactical ad how networks for a fast deployment and operation in a war context when no fix/wireless network exists, smart phone ad hoc networks not requiring the traditional wireless carriers, vehicular ad hoc networks for autonomous vehicles, sensor networks, etc. Investment/participation of nodes is a key issue for the success in these applications.

1

• Security/reliability issues: nodes in a network can be subject to (stochastic) failures reducing the path possibilities to the target. There may also be attacks on some nodes, rendering paths to the target through infected nodes not practicable. Investing on security procedures limits the infection risks but incurs a cost; this trade-off and the impact on the whole system in the case of selfish nodes requires attention.

The natural modeling framework is that of non-cooperative game theory [START_REF] Fudenberg | Game Theory[END_REF][START_REF] Osborne | A Course in Game theory[END_REF]. In the literature, security and related investment games have been studied quite a lot; see for instance [START_REF] Maillé | Interplay between security providers, consumers, and attackers: a weighted congestion game approach[END_REF][START_REF] Maillé | Of threats and costs: a game-theoretic approach to security risk management[END_REF] and the references therein. Security and free riding have been studied in [START_REF] Varian | System reliability and free riding[END_REF]. The closest paper to ours is [START_REF] Jiang | How bad are selfish investments in network security[END_REF], where a general framework for security investment is provided, with functional interactions between nodes. Our model fits this framework in general, but we focus here on something more specific, and our utility functions are different, hence the results in [START_REF] Jiang | How bad are selfish investments in network security[END_REF] need to be adapted.

Our contribution is the following: we design a graph model with nodes as players and links corresponding to possible direct connections between nodes. Each node is subject to failures and has to set up its reliability for a given cost. On the other hand, each node is interested in the existence (in probability) of a path to the target access node, meaning that its service quality also depends on the others' choices. Our specificity in this game-theoretic context is therefore the use of the so-called static reliability analysis which computes the probably of such a path. Numerous methods exist for this NP-hard problem, efficient Monte Carlo simulation being applied in the case of large networks [START_REF] Cancela | Rare events analysis by Monte Carlo techniques in static models[END_REF][START_REF] Rubino | Network reliability evaluation[END_REF]. We show how this problem can be analyzed, with specific results.

The remaining of the paper is organized as follows. Section 2 describes the formal mathematical model we are going to analyze and the goals of the paper. Section 3 presents two illustrative examples, helpful to grasp the stakes and difficulties at hand, as well as some results on the Price of Anarchy, measuring the loss of efficiency due to selfishness. The general results on the output of the game are presented in Section 4, and Section 5 concludes the paper by giving directions of future research.

Model

Graph model

We consider an undirected connected graph G = (N , L) where N = {0, 1, . . . , n} is the set of nodes, and L is the set of links/possible connections between nodes. Node numbered 0 is the access node. Links are assumed to always work, but nodes 1 to n are subject to (independent) failures, due to vulnerabilities, attacks, etc.. The access node is assumed perfect. Let r i be the probability to be up for Node i, and q i = 1-r i . A configuration of the graph is given by vector X = (X 1 , . . . , X n ) where X i = 1 if i is up and 0 if it is down. Retaining only the set N of "up" nodes, we obtain a random partial graph G = (N , L).

Node i is interested in the probability R i that in G there is a path connecting i to 0. This type of problem has been extensively studied in the literature [START_REF] Cancela | Rare events analysis by Monte Carlo techniques in static models[END_REF][START_REF] Rubino | Network reliability evaluation[END_REF]; basically,

R i = x∈{0,1} n Ψ i (x)P[X = x] = x∈{0,1} n Ψ i (x) n j=1 (r j x j + q j (1 -x j ))
where Ψ i (x) is 0 if there no path in the subgraph G corresponding to configuration x (a vector (x 1 , ..., x n ) where x i = 1 means Node i is up), and 1 if there is at least one such path. It is easy to compute analytically for small topologies, as a finite sum, but it is in general an NP-hard problem for which approximation methods, among which efficient simulation algorithms, must be used for large topologies. See [START_REF] Cancela | Rare events analysis by Monte Carlo techniques in static models[END_REF][START_REF] Rubino | Network reliability evaluation[END_REF] for a general description of the methodologies. Since it is not the purpose of the present paper, we assume that how to get a sufficiently accurate estimation is known.

Utility functions

We assume that Node i has a valuation f i (R i ) for the service, depending on the probability to get a connection to the access point. Each function f i is assumed increasing: the larger the quality/reliability, the more it is appreciated, and f i is also assumed concave, a usual and reasonable assumption to express that getting a given reliability amount when you are at a low level is seen more valuable than when you are at a high level. Node i is also assumed to experience a cost c i (r i ) to maintain a level r i of reliability/security/quality of its own equipment. Function c i is assumed increasing too, and strictly convex, with c i (0) = 0 and lim r→1 c i (r) = ∞, again standard assumptions.

The utility u i of Node i is then

u i (r 1 , . . . , r n ) = f i (R i ) -c i (r i )
where we recall that R i is a function of r = (r 1 , . . . , r n ).

We end up with a non-cooperative game where each node i plays with the level of investment resulting in (equivalent) reliability r i , but his/her strategy impacts the valuation of other nodes.

Questions to be answered

From the definition of this problem, here are the questions we aim at answering:

• What is the outcome of the game? Does a Nash equilibrium exist? If yes, is it unique?

• How does the output compare with a cooperative situation? This is evaluated by computing the Price of Anarchy (PoA) measuring how the efficiency of a system degrades due to selfish behavior of its players as the ratio of optimal social utility divided by the worst social utility at a Nash equilibrium (the social utility being the sum of utilities over all players).

Examples

A simple example with non-trivial interactions among nodes

Consider the topology described in Figure 1 where each node i ∈ {1, . . . , 4} wants to be connected to node 0. We let c i (x) = α x 1-x and f i (x) = x, which verifies the 0 2 1 3 4

Figure 1: A simple topology assumptions listed in the previous section. Since Node i is on all paths to Node 0, reliability R i can be expressed as R i = r i R i where R i is the probability that there is a path from Node i to Node 0 when i is operational, and R i does not depend on r i .

To compute the best response of Node i, we differentiate u i = r i R i -α r i 1-r i with respect to r i . It gives the optimal value

r i = 1 -α/R i + (1) with x + = max(x, 0).
For Nodes 2 and 3 at distance 1 of node 0, hence with R i = 1, we then get r i = 0 if α ≥ 1 (that is, the investment cost is too high and the node prefers not to participate), and 1 -√ α otherwise. Let us look at Nodes 1 and 4 now. Observe that we have

R 1 = r 2 + (1 -r 2 )r 3 r 4 and R 4 = r 3 +(1-r 3 )r 2 r 1 .
The utility of one node depends on the reliability choice of the other, therefore we have a game between nodes 1 and 4. Looking for the solution, we try to solve the system of equations

∂u 1 ∂r 1 = r 2 + (1 -r 2 )r 3 r 4 - α (1 -r 1 ) 2 = 0 ∂u 4 ∂r 4 = r 3 + (1 -r 3 )r 2 r 1 - α (1 -r 4 ) 2 = 0 α r 2 = r 3 r 1 = r 4 0.3
0.4522774425 0.2330400670 0.1 0.6837722340 0.6517825785 0.01 0.9 0.8990317600 0.001 0.9683772234 0.9683460395 0.0001 0.99 0.9899990048 Table 1: Some numerical results in terms of α for the example of Section 3.1

giving

r 1 = 1 - α r 2 + (1 -r 2 )r 3 r 4 r 4 = 1 - α r 3 + (1 -r 3 )r 2 r 1 .
Solving it is intractable in general, but we can easily get numerical results. Examples of solutions in terms of α are given in Table 1. The lower the cost, the higher the reliability.

For this example with homogeneous utility valuations and costs, it is interesting to see that all r i are converging to 1 as α → 0: the lower the investment cost, the better it is to participate because there is a gain at no cost.

But we can say more when α tends to 0:

r 1 = r 4 = 1 - √ α/ (1 - √ α)(1 + r 1 √ α) = 1 - √ α(1 + o(1))
where the o(•) is for α → 0. This yields r 1 = r 2 + o( √ α). Hence the r i are asymptotically equivalent, even in relative terms if we consider the difference with respect to 1 since 1 -

r 1 = √ α + o( √ α) = 1 -r 2 + o(1 -r 2 ), i.e., 1-r 1 1-r 2 = 1 + o(1)
. Finally, we can look at what would happen if the nodes were collaborating, that is, if they were trying to maximize the social utility defined as

4 i=1 u i = r 2 + r 3 + r 1 (r 2 + (1 -r 2 )r 3 r 4 ) + r 4 (r 3 + (1 -r 3 )r 2 r 1 ) -α 4 i=1 r i 1 -r i .
Getting the global optimum for (r 1 , . . . , r 4 ) ∈ [0, 1] 4 seems intractable, but we can easily get numerical values again. Table 2 displays for the same values of α as in Table 1 the maximum value of 4 i=1 u i , the corresponding r i s, and the Price of Anarchy, that is that optimal value divided by the value of 4 i=1 u i at equilibrium. Note that the PoA is necessarily larger than 1. It is interesting to note that the Price of Anarchy is close to one. It also seems to converge to 1 as α → 0. In other words, selfishness of nodes does lead to negligible losses with respect to the cooperative case, particularly as α is close to 0. The Nash equilibrium is easy to compute, since the actions of Nodes 2 to n only affect themselves and not Node 1. For Node 1, from (1) the Nash equilibrium (NE) strategy is simply

r 1 = 1 - √ α + := x NE .
Then Nodes 2 to n all see R i = r 1 , and therefore they all set, again from (1)

r 2 = • • • = r n = 1 - α [1 - √ α] + + := y NE .
Now, let us investigate the globally optimal choices. By symmetry, Nodes 2 to n should set the same reliability level y, and if we denote by x the reliability of Node 1, the objective function is

n i=1 u i = x -α x 1 -x + (n -1) xy -α y 1 -y , (2)
which gives y = 1 -α x + , and can be plugged into (2) to compute the optimal x (we do this numerically to avoid cumbersome algebra). But we can already notice that the Price of Anarchy is unbounded in general: if we take α = 0.49, at equilibrium Node 1 selects r i = 1 -√ α = 0.3 which leads all the other nodes to select r i = 0, so the social utility is u NE 1 = 0.09. On the other hand, the optimal social utility is at least the one we would reach with r 1 = 0.9 and r 2 = r 3 = ... = r n = 0.3, which equals -3.57 + 0.06n, hence the Price of Anarchy is at least 2 3 n -40, which increases linearly with n.

To treat our example for a more general set of parameters, Figure 3 shows the equilibrium and globally optimal strategies of Node 1, and of Nodes 2 to n. The corresponding costs are computed according to [START_REF] Fudenberg | Game Theory[END_REF], and are compared through the Price of Anarchy, plotted in Figure 4. As in the previous example, we observe that the Price of Anarchy tends to 1 when α tends to 0 (if the cost of reliability is null, all nodes should select r i close to 1, both at equilibrium and at the socially optimal outcome).

Analytical results

While the previous section was dealing with instructive examples, we provide here general results towards the analysis of Nash equilibria and the resulting outcome.

A first result, observed on the examples in the previous section, is about the nodes directly connected to the access point (Node 0): Proposition 1. For nodes at distance 1 from Node 0, the decision does not depend on other nodes. Proof. It comes from the fact that for such nodes, R i = 1 because there is a direct link between i and 0.

We now interested in the Nash equilibria of the game. Since Nash equilibria are fixed-points of best-response correspondences, we first provide results on bestresponses.

Proposition 2. Given the assumptions of concavity of f i and strict convexity of c i , Node i has a unique best response in [0, 1).

Proof. Note that R i is a linear function of r i (actually R i = r i R i with R i the reliability of the random subgraph where both nodes 0 and i are perfect, and R i does not depend on r i ). As a consequence u i (r 1 , . . . , r n ) is strictly concave in r i , hence the result. Proposition 3. The game has at least one Nash Equilibrium.

Proof. It is a direct consequence of Rosen's theorem with concave utility functions, with action sets on a compact, here [0, 1] for each player. This is similar to what was shown in [START_REF] Jiang | How bad are selfish investments in network security[END_REF], something expected. Regarding the equilibrium uniqueness, we can apply existing results to express a sufficient condition, but that condition is difficult to verify in practice: Proposition 4. Let U (r) be the Jacobian matrix U (x) = ( ∂u i (r) ∂r j ∂r i ) 1≤i,j≤n . Let U t (r) be the transpose matrix of U (r). If the symmetric matrix U (r) + U t (r) is negative definite (that is, for all r ∈ [0, 1] n we have y t (U (r) + U t (r))y < 0 ∀y = 0) the Nash equilibrium is unique.

Proof. It is a sufficient condition of Rosen's uniqueness condition [START_REF] Rosen | Existence and uniqueness of equilibrium points for concave n-person games[END_REF].

Conclusions

In this paper, we have introduced a specific model for the interactions among participants (nodes) in an ad-hoc network in terms of reliability investments, when all nodes want a reliable access to a given point. When nodes are selfish, the non-cooperative game can be difficult to analyze, but we have proved that, under reasonable conditions, it always has an equilibrium. We have highlighted the contrast with globally optimized decisions, for which the of nodes close to the sink are higher than in the non-cooperative setting, due to the positive externality they create. We have shown that the loss of efficiency due to user selfishness, measured through the Price of Anarchy, can be arbitrarily large in general, but is in practice small for many settings (in particular, when reliability is cheap). This suggests some careful consideration is needed before deciding whether some coordination should be enforced.

Figure 2 :

 2 Figure 2: A specific topology with n nodes

= 2

 2 Opt., n 10 Opt., n = 100 Opt., n = 1000

Figure 3 :

 3 Figure 3: Equilibrium and globally optimal strategies for Node 1 (left), and for the other nodes (right), for the instance of Figure 2.

Figure 4 :

 4 Figure 4: Price of Anarchy for the instance of Figure 2.

α

  Optimum (r 1 = r 4 , r 2 = r 3 )

			4 i=1 u i	PoA
	0.3	(0.3445638958, 0.5265329288) 0.4924406891	1.06014
	0.1	(0.6842015722, 0.7383790483) 1.670241284	1.021059
	0.01	(0.9027739526, 0.90999676)	3.208617313	1.000922
	0.001	(0.9687904071, 0.969673)	3.747335781	1.0000352
	0.0001	(0.9900470095, 0.9900)	3.920006440 1.000000122

Table 2 :

 2 Social Optimum and Price of Anarchy in terms of α for the example of Section 3.13.2 A simple example with unbounded Price of AnarchyWe introduce here another simple example, where the interactions among nodes is not really a game (only one node affects the others), but the example illustrates what can happen in terms of the Price of Anarchy.Let us consider the topology of Figure2, with only Node 1 directly connected to the target node, and Nodes 2 to n only connected to Node 1.


Jonin Pierre-Yves

Noël Audrey

Gabriel Besson

Serge Belliard

Barillot Christian

Retrieval Practice Based On Recognition Memory: Testing the Retrieval Effort Hypothesis

Background Aim

Can recognition memory support a retrieval practice effect?

£ The finding that taking memory tests improves long-term memory and overcomes repeated studying is called retrieval practice effect or « testing effect »(1,2). While it has been much replicated within recall paradigms, a mechanistic account is still lacking. One way to move forward is to test predictions derived from current accounts £ The « Retrieval Effort Hypothesis » states that controlled (effortful) retrieval (e.g. recall) supports more elaborative and integrative processing than passive restudying, thus increasing the available retrieval cues (3,4)

£ Since recognition memory involves much less controlled retrieval than recall, repeated recognition should not yield a retrieval practice effect, especially if familiarity alone supports recognition

Discussion

u Experiment 1 shows that the retrieval practice effect can be observed when retrieval is based on recognition memory rather than recall. Thus, learning does occur during recognition testing u Importantly, both experiments show that the benefits of memory retrieval based on recognition memory are immune to negative side effects like extra false alarms u When retrieval is constrained to fast and automatic processes (around 320 ms.), thus being mostly familiarity-based, the generation of elaborative retrieval cues and / or effortful (controlled) processing are quite unlikely. Even there, extensive restudying does not outreach retrieval practice. Repeated automatic retrieval yields similar learning levels than extensive restudying, up to a 6 months delay u Familiarity-based recognition memory can support a retrieval practice effect, and resists to a 6 months delay similarly to restudying, thus challenging a core prediction of the « Retrieval Effort Hypothesis »

£«uu 1 uu

 1 Between-subjects design, N = 76 £ Recognition memory= typical Old/New task £ Manipulation of the intervening tasks: -2 successive study-test trials= « Study-Test » group -2 successive study trials= « Study » group - 2 successive test trials= « Test » group £ Main outcome: Performance at final test (25 min. delay) £ Matching for Age, Education, FSIQ, Verbal Memory Before final test, study duration was on average 11 minutes in the « Study-Test » group, 7.4 minutes in the « Study » group and only 6Still, « Study-Test » & « Test » conditions yielded better long-term memory (A,B), without increase in False Alarms (C), and « Test » condition led to better 25 minutes -retention (D) Retention (delta_Hits) Effect Size; 95%IC: -1.194, [-2.025; -0.382] Experiment 1 provides the first evidence for a retrieval practice effect based on recognition memory However, a contribution of controlled recollective processes cannot be ruled out, which is addressed in experiment 2 subjects design, N = 30 £ Manipulation of the learning schedules: -1, 2 or 3 repetitions of study trials= « Study » group -1, 2 or 3 repetitions of test trials= « Test » group £ Main outcomes: Performance at short-and long-term final tests £ Probing familiarity-based recognition memory: The «Speed and Accuracy Boosting procedure » (SAB) is a speeded Old/New memory test providing a direct estimate of familiarity-based recognition memory (5). à Use of the SAB procedure for all test phases £ Matching for Age, Education, FSIQ, Verbal Memory Subjects in the « Study » group spent twice as much time studying AND had up to three times more opportunities to encode the stimuli u Similar minimal reaction times (minRTs) were achieved in both groups, well below 400ms, strongly constraining responses to familiarity-based recognition memory(5) u Repeated retrieval was therefore based on automatic & fast processing, rather than slow, effortful, recollection Still, repeated testing proved as beneficial as restudying for short-(A) and long-term(B) retention u This did not came with an extra false alarms cost (C&D) Time spent studying does not drive learning efficiency. Instead, Experiment 2 provides unique evidence that learning occurs through repeated familiarity-based retrieval, i.e. even when retrieval is automatic