Ethical questions regarding the place of the child in ethnographic research at the children's home Julie Chapeau -EUSARF 2021

To introduce this communication about "Ethical questions regarding the place of the child in ethnographic research at the children's home", I would like to quote William Lillie from his classic book called An introduction to ethics:

« The question for ethics is not whether such an action was deliberately willed but whether the doer could have prevented it by taking thought about it [START_REF] Lillie | An Introduction to Ethics[END_REF] » With this in mind, how can we design an ethical framework adapted to ethnographic research at the children's home? [START_REF] Tillard | Enquêtes au domicile des familles : La recherche dans l'espace privé[END_REF]) How would we "be prepared to expect the unexpected" as Virginia Morrow recommends? [START_REF] Alderson | The ethics of research with children and young people : A practical handbook[END_REF]) What should we anticipate?

If ethics in research involving children has become a major concern for researchers and very well documented, there is still a lack of knowledge about home-based ethnographies involving children, especially in the field of child protection.

 As Ferguson shows, a lot of so-called ethnographic researches in the field use a repertoire of various techniques but practice ethnographies and particularly participant observations at home are "very rare in child and family social work" [START_REF] Ferguson | Researching Social Work Practice Close Up : Using Ethnographic and Mobile Methods to Understand Encounters between Social Workers, Children and Families[END_REF].  This scarcity can be explained by the difficulties to access to the fieldwork, but moreover by strong ethical issues related to this research design (even ethics committee's rejection for example).

Also, Ruch says that "Endeavouring to access more "direct" or "honest" accounts of what actually happens in encounters between social workers and vulnerable children, however, is fraught with ethical difficulties, not least determining how, and from whom, to obtain informed consent" [START_REF] Ruch | Helping Children Is a Human Process' : Researching the Challenges Social Workers Face in Communicating with Children[END_REF], quoted by (Ferguson, 2016, 157) Thus, is promoting an ethical approach in this research design an impossible challenge?

From one case examined in my study, I will identify issues arising and will make recommendations to adjust research ethics in France and provide ethical guidelines for this type of research.

My doctoral research is based on what [START_REF] Longhofer | An example of social work practice ethnography[END_REF] called in 2012 "practice ethnography". That means face-to-face interactions between social workers and service users (305) and more specifically practice-near ethnography in so far as my research considers the emotional and unconscious aspects of practices as suggest [START_REF] Froggett | Practice-near and practice-distant methods in human services research[END_REF]. I conducted participant observation of French social workers during one week per month from January to September 2019. I followed 12 social workers who visited 6 families and 22 children aged 0 to 14 years old at the children's home, the office and in neutral venues…

Looking at the scheme drew by Harry Shier, my research was clearly ON protected children.

Why did I choose this research design?

Because in the context of home-placement -which imposes a high frequency of home visits-, I wanted to highlight what really happens during the visits, how it impacts the adult/parents' relationship and the daily life, but also to unfold social worker's practices and especially their emotional labour. I was deliberately on the side of the "intervention", spending most of my time with the social workers. The relationship I established with the family I visited relied a lot on their own relationship with the practitioners. I was a complete stranger when the family was reluctant to the intervention or a facilitator who played with the child and doing so allowed the social worker to have a private conversation with the mom for example.

My choice was to let me carried by the intervention dynamic, to stay open to deliberate interactions with the parents or the children and to be attentive to what they said and expressed. I assumed an "immature methodology" defined by [START_REF] Gallacher | Methodological immaturity in childhood research ? Thinking throughparticipatory methods[END_REF] as an attitude that positions researchers as 'experimenting and creating' a world with others rather than reporting on things as they are in an observed world". [START_REF] Dennis | Ethnographic research in childhood institutions : Participations and entanglements[END_REF] In that configuration, I was entangled between the field of my research and the field of intervention, interfering in the family intimacy. Looking at the intervention dynamic through the lens of the social workers, my role was ambiguous for the children. Sticking to the social workers 'practices, I was at risk of habituation and of participating to the invisibilization of the children.

Research field ecosystem -Chapeau, 2021

To illustrate that, let's introduce the vignette presenting Sarah, an "ethically important moment" as described by [START_REF] Guillemin | Ethics, reflexivity, and "ethically important moments" in research[END_REF].

She was 4 years old during the participant observation. She is a brilliant little girl, playful and full of questions. She has been back home with her two younger sisters since a few months. An emergency placement was court-ordered after a serious household accident. She was placed in a residential care facility without her sisters for one year.

At the beginning of the interventions, Sarah was always angry especially at the adults around her. The intervention had a very positive impact on her behaviour week after week. She experienced less temper tantrums and agitation. But when I visited Sarah, she was much different than usual. One day, the social worker asked her why she was nervous in my presence. The mom replied: "Because she is afraid of Julie; she doesn't understand why she is here. She is scared of the white car, the one which brought her to the emergency shelter"

I didn't have the time to say something before the mom told Sarah: "You have to know; Julie is here because she is writing a book about children". The girl then was really curious about the book and her attitude toward myself was suddenly different. I took the time to answer her every single question. Doing that, I moved from the dangerous stranger to an interesting figure.

What did I miss in my research process?

The agitation of Sarah was a clear sign of changing consent, even though she was not able to express it clearly. Yet, I have respected ethical guidelines so far, the French ones at least.

In France, research ethics in human sciences are not regulated, including research involving children. Indeed, they have been excluded from the Jardé law in 2012 and implementing decrees so far. However, we can't say that investigations in human sciences are in the grey area of research ethics:

 The ministerial decree of 25 may 2016 requires every PhD student to undergo basic training on research ethics and scientific integrity.  More and more universities and research units have set up an ethics committee which is asked to render an advisory opinion on research projects. (no compulsory)  Researchers have to request an authorization from the CNIL (which is in charge of protection of personal data) to collect data.

Even more, PhD students are encouraged to complete their ethics self-assessment before starting the fieldwork, seize the ethics committee, create info sheets and ask participants to sign consent forms. Dissertation committees can evaluate PhD students on their ethical approach even if they are not required to.

… But insufficient attention is paid in this framework to children's rights and needs and ethical adjustments.

How did I develop and apply my ethical approach?

Procedural ethics. I started my PhD on 2017, just one year after the publication of the decree. I attended the research ethics training while starting the fieldwork and completed my ethics self-assessment. But I didn't have the chance to request the opinion of the ethics committee because of bad timing.

I started working with the local authority representatives, practitioners and my supervisor on an ethical charter. This charter presents the research framework and design and ethical principles; terms and conditions about information on the research, consent, anonymity and professional secrecy, and researcher responsibility. Considering the specific issues related to the field of child protection, the discussion with the representatives was sometimes tough but really important. It made us realised our perceptual difference of child endangerment and the need or not to break professional secrecy [START_REF] Monceau | Enquêter ou intervenir ? Effets des recherches socio-cliniques[END_REF]. Doing so, I stepped up to be recognized as a researcher and not a practitioner and the responsibilities this imposes. [START_REF] Bruggeman | Conditions d'enquête et démarches méthodologiques de recherches « à domicile ». Le chercheur sur le terrain des familles[END_REF].

Alongside, I co-created the ethical material with the local authority and the child protection services involved in the research. Before visiting the family, the social workers had to transmit the info sheet and explain orally the goal of the research to obtain the prior consent of the parents. For the first visit, I was allowed to take time with the parents to introduce the research and ask for their consent, while social workers intervened with the children. I offered the possibility for parents to think about it for one month. I made clear that I won't intervene during the visits and I won't write notes at their home, even knowing that this would impact the quality of my data. I paid more attention to detail the anonymization process and my researcher's responsibility in the context of my observations. If they agreed and signed the consent form, I began the observation after explaining orally the research and my role to children presents considering their young age. If they were able to, I read to them the consent form and I allowed them to sign it if they wanted to. They could call me Julie, in contrary to the social workers who were named M.X Miss Y, and use the informal "you" to talk with me. I didn't want to appropriate "the least adult role" described by [START_REF] Mandell | The least-adult role in studying children. I Frances C Waksler, red : Studying the Social Worlds of Children : Sociological Readings[END_REF] and more recently [START_REF] Atkinson | Ethical complexities in participatory childhood research : Rethinking the 'least adult role[END_REF] [START_REF] Horton | A 'sense of failure' ? Everydayness and research ethics[END_REF]. But I wanted them to make a difference between me and the social workers.

Was my research really ethical, especially toward the children involved?

Respecting main ethical guidelines helped me to assess and anticipate most of the risks, particularly those incurred to the vulnerable children and if the potential research outcomes and potential benefits merit these risks. But I wasn't really prepared to deal with ethical issues arising in the fieldwork.

Yes, it is necessary to increase knowledge about home-based interventions and improve practices from that research results [START_REF] Deshayes | L'épaisseur de l'institution. L'enquête en protection de l'enfance, du face à face aux coulisses[END_REF][START_REF] Tillard | Enquêtes au domicile des familles : La recherche dans l'espace privé[END_REF]. But I don't totally agree with Ferguson when he says that "a researcher can be present at child protection encounters without having a negative impact on practice or service users" [START_REF] Ferguson | Researching Social Work Practice Close Up : Using Ethnographic and Mobile Methods to Understand Encounters between Social Workers, Children and Families[END_REF]. He obliterates the responsibility of the ethnographer. Sarah's story shows us that it's impossible to anticipate what happens for the children in our presence.

The most important in this situation is ongoing attention AND REFLEXIVITY. I cite "Being reflexive in an ethical sense means acknowledging and being sensitized to the microethical dimensions of research practice and in doing so, being alert to and prepared for ways of dealing with the ethical tensions that arise." Tensions can also have a positive impact on the data collection process and on the participants. Now, let's think about the story of Sarah a bit differently. The agitation of Sarah expressed something which go further than changing consent. Reacting to my presence, she tried to verbalize her need to be reassured about the reunification process, which has never been discussed before. After having presented my research, the social worker seized the opportunity to talk with her about the separation and the reasons for his intervention. My "situatedness "contributed to give Sarah voice and to obtain new data. This example illustrates that "most fascinating insights of my research emerged from children acting in unexpected ways", noted Gallacher and Gallagher (508).

Whereas "Ethics is so often seen as an extra rather separate set of tasks that researchers have to complete." reminds us Patricia Alderson, I would recommend to adapt procedural ethics in France and to reinforce the awareness of micro-ethics issues by:

• Translating, adapting and disseminating research ethics resources developed in others countries in France • Systematizing research projects involving children assessment by research ethics committees and ongoing monitoring by dissertation committees and supervisors • Developing specific training and adapting ethics self-assessment guides to the issues related to the participation of protected children [START_REF] Renold | Becoming Participant' : Problematizing `Informed Consent' in Participatory Research with Young People in Care[END_REF] • Increasing knowledge about research issues in the context of home-based ethnographies involving children.

  Toulouse, E.,Masseguin, C., Lafont, B., Mcgurk, G., Harbonn, A., Roberts, J.A., Granier, S., Dupeyron, A., & Bazin, J. (2018). French legal approach to clinical research. Anaesthesia, critical care & pain medicine, 37 6, 607-614 .


Bruno Levieil
email: bruno.levieil@ensta-bretagne.org

Cédric Doudard

David Thevenet

Florent Bridier

Anthony Ezanno

Sylvain Calloch

An original simplified method based on the use of an adjustable localization operator for low-cycle fatigue life predictions in the case of confined plasticity

This study aims to propose a methodology to rapidly obtain low-cycle fatigue life predictions. The main idea is to use a recently developed simplified method to estimate the stabilized stress-strain curve in confined plasticity in order to apply a fatigue criterion. This original simplified method is based on the use of an adjustable scale transition rule which has been demonstrated to be more robust to multiaxial loadings than classic plasticity correction rules. Different procedures are used to calibrate the localization operator and are compared to each other. The whole method is validated on a double notched specimen subjected either to compressive or tensile repeated loading. Kinematics and isotropic hardenings with threshold are used to model the partial mean stress relaxation that is observed experimentally. A modification of the Morrow fatigue criterion is proposed to enhance the prediction of the influence of the load ratio, represented by the local stabilized mean stress. The full design chain validation on notched specimens has shown a constant conservatism on the fatigue life predictions whatever the nominal load ratio and level. The results obtained with the simplified method are very similar to those obtained with finite elements.

Introduction

Fatigue life prediction of complex structures in low-cycle fatigue is still an industrial challenge. Indeed, the application of fatigue life criteria such as those proposed by Morrow [START_REF] Morrow | Low cycle fatigue behavior of quenched and tempered SAE 1045 steel[END_REF] or Smith-Watson-Topper [START_REF] Smith | A stress-strain function for the fatigue of metals[END_REF] requires the determination of the stabilized cyclic strain-stress curve. These values have to be determined at the critical point, around which confined plasticity frequently appears. This can be achieved with finite element analysis but the number of elements necessary to obtain a converged solution at the critical point may impede this solution for large structures. The number of cycles necessary to obtain a stabilized solution is also often a problem because of the plasticity that generally occurs during low-cycle fatigue and induces progressive slow meanstress relaxation.

In order to avoid long and costly finite element computations, rapid simplified methods have been developed. Their aim is to estimate the strain-stress histories during cyclic loadings only at the critical point. As early as 1961, Neuber [START_REF] Neuber | Theory of stress concentration for shear-strained prismatical bodies with arbitrary nonlinear stress-strain law[END_REF] proposed a rule for shear strained prismatic bodies with notches. Another approach has been then proposed by Molski-Glinka [START_REF] Molski | A method of elastic-plastic stress and strain calculation at a notch root[END_REF] called the equivalent strain energy density (ESED) method. Later on, Moftakhar [START_REF] Moftakhar | Calculation of elasto-plastic strains and stresses in notches under multiaxial loading[END_REF] suggested that Neuber's rule can be considered as an upper bound estimation whereas ESED as lower bound estimation of the actual elasto-plastic notch tip behavior. For the last 40 years, several authors have worked on the energetic interpretation of Neuber's rule [START_REF] Kujawski | On energy interpretations of the Neuber's rule[END_REF][START_REF] Kujawski | On deviatoric interpretation of Neuber's rule and the SWT parameter[END_REF][START_REF] Ye | Further investigation of Neuber's rule and the equivalent strain energy density (ESED) method[END_REF][START_REF] Desmorat | Fast estimation of localized plasticity and damage by energetic methods[END_REF][START_REF] Zappalorto | Neuber's rules and other solutions: Theoretical differences, formal analogies and energy interpretations[END_REF] and on its extension to different loading conditions such as multiaxial cases [START_REF] Moftakhar | Calculation of elasto-plastic strains and stresses in notches under multiaxial loading[END_REF][START_REF] Hoffmann | Generalized method for estimating multiaxial elasticplastic notch stresses and strains. Part 1: Theory[END_REF][START_REF] Barkey | Yield surface approach to the estimation of notch strains for proportional and nonproportional cyclic loading[END_REF][START_REF] Sethuraman | Evaluation of notch root elasto-plastic stress-strain state for general loadings using an elastic solution[END_REF][START_REF] Ye | A unified expression of elastic-plastic notch stress-strain calculation in bodies subjected to multiaxial cyclic loading[END_REF][START_REF] Darlet | Stress triaxiality and Lode angle along surfaces of elastoplastic structures[END_REF]. Nevertheless, it is established that the predictive capability of Neuber's rule or of ESED method remains dependent on the material constitutive behavior and the mode of loading [START_REF] Buczynski | An analysis of elasto-plastic strains and stresses in notched bodies subjected to cyclic non-proportional loading paths[END_REF][START_REF] Kilambi | Algorithm to estimate notch root strain in tension using numerical analysis[END_REF][START_REF] Lim | Application of local stress-strain approaches in the prediction of fatigue crack initiation life for cyclically non-stabilized and non-Masing steel[END_REF][START_REF] Salemi | A strain energy method for elastic-plastic analysis of notches under shear loading[END_REF][START_REF] Kilambi | Numerical evaluation of the original 'Neuber's rule' for pure out-of-plane shear loading[END_REF].

More recently, Herbland [START_REF] Herbland | A new technique for the fatigue life prediction in notched compoenents[END_REF] has proposed another method to solve the problem in the general multiaxial case. This method is based on the use of an adjustable scale transition rule. It implies the calibration of a localization operator, using the results of a finite element (FE) simulation of the first load. Darlet [START_REF] Darlet | Stress triaxiality and Lode angle along surfaces of elastoplastic structures[END_REF] has subsequently proposed a similar approach by using another identification method of the localization operator. The aim of this paper is to propose a full design chain integrating this new approach and to compare the predictions given by this chain with experimental results.

Experimental validation is accomplished on plate specimens with two notches, used to induce confined plasticity. In this study, Darlet's and Herbland's identification methods will thus be used and compared with one another as well as with FE predictions and experimental results. The discussion will cover key elements of strain-stress histories with focus on the chosen fatigue criterion, in particular in the case of compressive mean stress. This specific load case can occur under compressive loadings or when the structure presents compressive residual stresses [START_REF] Hanaki | Effect of compression residual stress on fatigue properties of stainless cast steel[END_REF]. It has been less studied than tensile loading cases that are more frequent in the literature [START_REF] Kamaya | Mean stress effect on fatigue strength of stainless steel[END_REF][START_REF] Ince | A modification of Morrow and Smith-Watson-Topper mean stress correction models[END_REF][START_REF] Paul | The effect of low cycle fatigue, ratcheting and mean stress relaxation on stress-strain response and microstructural development in a dual phase steel[END_REF][START_REF] Kujawski | A unified approach to mean stress effect on fatigue threshold conditions[END_REF].

The present paper is divided into five main sections. First, the fatigue tests conducted on notched specimens that will be used for the method validation are presented. Then, simplified methods for local stress/strain calculation are introduced with a focus on Darlet's and Herbland's propositions.

In the third part, the material behavior is identified. Its elastoplastic behavior is described using two alternative behavior laws identified using an original sequential identification procedure: a first law with classical non-linear hardenings as proposed by Armstrong & Frederick [START_REF] Armstrong | A mathematical representation of the multiaxial Bauschinger effect[END_REF] and Voce [START_REF] Voce | A practical strain-hardening function[END_REF]; and an original second law with threshold hardenings [START_REF] Chaboche | On some modifications of kinematic hardening to improve the description of ratchetting effects[END_REF].

The aim of using these two laws is to underline their mean stress relaxation prediction capability. Then, purely alternated strain controlled tests on axisymmetric specimens are performed to identify the low-cycle fatigue properties. A modification of the Morrow fatigue criterion is proposed to enhance the prediction of the mean stress effect, in the particular case of compressive mean stress. This experimental part is achieved with the introduction of the fatigue testing on notched specimens under tensile or compressive repeated loadings.

In the fourth part, Herbland's and Darlet's parameters are calculated using the material behavior law on a quasi-static finite element analysis. The predictions of the stabilized stress-strain curves obtained with the simplified methods are thus compared to those obtained using a reference non-linear finite element cyclic analysis rather than to compare results to other simplified methods such as Neuber or ESED. Indeed, the results obtained with finite elements analysis are considered the most accurate. The same comparative work is accomplished employing the stabilized parameters used for the fatigue life calculation. The last part of this paper aims at validating the fatigue life prediction for the notched specimens. A multiaxial equivalent formulation of the scalar parameters used in the fatigue life criterion is proposed. Fatigue life prediction can then be obtained. A comparison is made with the experimental data obtained on the notched specimens under tensile repeated loadings or compressive repeated loadings.

Case study

Fatigue tests were conducted on ferritic high-strength steel specimens under two different load ratios. The geometry of these notched specimens is indicated in Fig. 1. These specimens were machined with an electrical discharge machine. A slight electro-polishing was performed at the notch surface to reduce all residual stresses. Initial stress measurements were achieved using an X-Ray diffraction device. All initial residual stresses were lower than 5% of the material elastic limit.

A nominal stress-controlled loading was applied at the end of the notched specimens in the → e 1 direction. (1)

The notched specimens were tested in fatigue for repeated nominal stress loads. Compressive (R σ nom =-∞) and tensile (R σ nom = 0) nominal loads were applied. Fatigue tests were conducted at 2 Hz. Crack initiation was detected with the alternating current potential drop technique [START_REF] Matelect | Practical aspects of the ACPD technique[END_REF] calibrated for a 0.8 mm crack length. Two distinct Wöhler curves, shown in Fig. 2, were obtained indicating that, as expected, the nominal stress ratio has an important effect as compressive tests showed longer lifespan. It is worth noting that, although the nominal stress ratio is controlled, the local stress ratio can only be evaluated numerically and evolves during the test until strain and stresses are stabilized.

Simplified methods and fast prediction of stabilized stressstrain curves in a located plastic zone

When confined plasticity appears around notches or any other geometric singularity, a specific methodology has to be employed to predict the stabilized stress-strain curve, necessary for the fatigue criterion application. Cyclic finite element analysis is a common solution, but the computation time necessary to obtain an accurate prediction of the stabilized stress-strain curve can be costly due to the local plastic behavior that induces slow cyclic stress relaxation. This is an additional motivation to the development of simplified methods to estimate the stabilized stress-strain curve only at the critical point. Since Neuber's rule [START_REF] Neuber | Theory of stress concentration for shear-strained prismatical bodies with arbitrary nonlinear stress-strain law[END_REF], many authors have proposed different simplified approaches [START_REF] Molski | A method of elastic-plastic stress and strain calculation at a notch root[END_REF][START_REF] Moftakhar | Calculation of elasto-plastic strains and stresses in notches under multiaxial loading[END_REF][START_REF] Ellyin | An energy-based fatigue failure criterion[END_REF]. In recent years, Herbland has proposed a multiaxial approach that has shown better correlation under cyclic multiaxial loadings with numerical finite element analysis predictions than Neuber's method [START_REF] Herbland | Une méthode de correction élastoplastique pour le calcul en fatigue des zones de concentration de contraintes sous chargement cyclique multiaxial non proportionnel[END_REF][START_REF] Chouman | Elastic-viscoplastic notch correction methods[END_REF]. Herbland's method is based on the use of a relationship between the stress tensor at the critical point, σ̲ loc , and the stress tensor at the same point if an elastic behavior is assumed, inspired by localization laws of multi-scale approaches [START_REF] Eshelby | The determination of the elastic field of an ellipsoidal inclusion, and related problems[END_REF]. Fig. 3 illustrates the analogy between the confined plasticity problem and the localization problem written for an elasto-plastic inclusion in an elastic matrix [START_REF] Kröner | Zur plastischen verformung des vielkristalls[END_REF][START_REF] Berveiller | An extension of the self-consistent scheme to plastically-flowing polycrystals[END_REF]. For this problem the stress tensor in the inclusion, σ̲ I , can be related to the stress tensor applied in the matrix, Σ̲ ,b y

=- σL ̲̇ Σ ̲ ̲ ̲ : ̲ IE p I ∊ ∊ (2)
where ̲ p I ∊ ∊ is the plastic strain tensor in the inclusion. The dot operator over a letter refers to its time derivative. L̲ ̲ E is Eshelby's fourth-order localization operator, which depends on the shear modulus, G, and the Poisson's ratio of the material, ν,

=- LGβ J 21 ̲ ̲ () ̲ ̲ E (3) with = - - β ν ν 24 5 15 1
() () , (4)
given by the Eshelby analysis [START_REF] Eshelby | The determination of the elastic field of an ellipsoidal inclusion, and related problems[END_REF], and J̲ ̲ the deviatoric projector ex- pressed as For the confined plasticity problem, Eq. ( 2) can be replaced by

= ⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ -- -- -- ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ J 23 13
=- σK σ L ̲̇ ̲ ̲ ̲̇ ̲ ̲ : ̲ loc T ν nom p loc ∊ ∊ (6)
where K ̲ ̲ T ν is the fourth-order tensor of the stress concentration factors calibrated on any elastic increment of the finite element simulation, ̲ p loc ∊ ∊ is the plastic strain tensor at the critical point. As this problem is different from Eshelby's problem, the localization operator L̲ ̲ differs. It depends on the material, the geometry and the loading conditions. However, some simplifications can be performed to reduce the number of terms to be identified using the free edge condition and the symmetries at the critical point [START_REF] Herbland | A new technique for the fatigue life prediction in notched compoenents[END_REF]. Assuming → e () 2 is the normal di- rection to the free surface, tensors can be simplified to

= ⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ = ⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ = ⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ ⎟
∊ ∊ ∊ ∊ ∊ ∊ ∊ ∊ (7)
Herbland proposed to identity the localization operator (denoted L̲ ̲ H in this case) on the numerical elasto-plastic simulation of the first quarter cycle. Once the localization and the stress concentration operators are identified the cyclic stress-strain curves are simulated analytically for a few tens of cycles until the critical point behavior is stabilized. The analytical calculation is immediate which means that the numerical cost is only the cost of simulating the first quarter cycle. In order to obtain the Wöhler curve, only the highest load has to be simulated and the different values of the localization operators are obtained thanks to the different increments of the calculation.

More recently, Darlet et al. [START_REF] Darlet | Stress triaxiality and Lode angle along surfaces of elastoplastic structures[END_REF] have proposed another methodology to identify this localization operator thanks to the analogy with a hemispheric inclusion at the surface of a matrix (Fig. 3). The localization operator can then be written as

Sauzay numerically estimated = Z 1.79 and = zZ 0.79/ in the case of a flat surface [START_REF] Sauzay | Effets de surface et d'anisotropie en fatigue multiaxiale[END_REF]. For the confined plasticity problem, Darlet proposed to consider = Z 1.79 as for the reference problem but a value of z which depends of the material, the geometry and the loading conditions. To determine the expression of this parameter, it is worth noting that the localization equation implies a direction move in the stress space for σ̲̇ loc , when plasticity appears except in specific cases, i.e., that σ̲̇ loc does not remain proportional to Kσ ̲ ̲ : ̲̇ [START_REF] Desmorat | Fast estimation of localized plasticity and damage by energetic methods[END_REF] withK ̲ ̲ T 0.5 the stress concentration tensor using a 0. ∊ ∊ ∊ ∊ , Eq. ( 6) can be simplified as follows

= LK σ ̲ ̲ ̲ ̲ ̲ : ̲̇ p loc T ν nom ∊ ∊ (10)
(i.e., σ̲̇ loc is negligible with respect to L̲ ̲ ̲ p loc ∊ ∊ ). Using the normality rule in the previous equation (i.e., the plastic flow is normal to the von Mises yield surface), it can be shown that [START_REF] Hoffmann | Generalized method for estimating multiaxial elasticplastic notch stresses and strains. Part 1: Theory[END_REF] Using Eq. ( 9), Eq. ( 11) can be expressed as [START_REF] Barkey | Yield surface approach to the estimation of notch strains for proportional and nonproportional cyclic loading[END_REF] Considering that the nominal loading direction is → e 1 and the normal direction to the free surface is → e 2 , Eq. ( 12) implies ,3311 [START_REF] Sethuraman | Evaluation of notch root elasto-plastic stress-strain state for general loadings using an elastic solution[END_REF] To satisfy this equation, the value of z can be finally be evaluated as As shear stresses and strains can be neglected in this study, Eq. ( 6) can also be reduces to the following system of equations with three parameters in the localization operator

∝ LJσ K σ ̲ ̲ : ̲ ̲ : ̲ ̲ ̲ : ̲̇ loc T νn o m
∝ LJK σ K σ ̲ ̲ : ̲ ̲ : ̲ ̲ : ̲ ̲ ̲ : ̲̇ T nom T ν nom 0.5
⎛ ⎝ ⎜ ⎜ ⎜ -+ - -+ - ⎞ ⎠ ⎟ ⎟ ⎟ ∝ ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ( ) ( ) ( ) ( )
= -+ - -+ - z KK
Consequently, to identify the localization operator in Darlet's proposition, two purely elastic FE analysis are needed, one considering the material Poisson's coefficient ν and one considering that the Poisson's coefficient is equal to 0.5 to represent the incompressible plastic behavior. The localization operator is the same for every load levels unlike in Herbland's method. In both method, a numerical model is necessary to identify the localization operator and the material elasto-plastic behavior has also to be determined for the analytical cyclic calculation.

Material properties

Material elasto-plastic behavior

In order to identify the material elasto-plastic behavior, a straincontrolled tensile test is performed with reversed compressive loadings (until the compressive elastic limit was reached) at different chosen plastic strain levels. The objective of this so-called "load-unload" test is to observe the evolution with plastic strain, of the size and the center of the elastic domain, and thus to characterize respectively the material isotropic and kinematic hardenings [START_REF] Lemaitre | Mechanics of Solid Materials[END_REF]. The test procedure, illustrated in Fig. 4 is the following:

(1) The specimen is first loaded in tension. The initial Young's modulus is evaluated using a linear regression between two stress values, corresponding approximately to 10 and 40% of σ r , where σ r is an Fig. 5. Load-unload test results.

Table 1

Material parameters for behavior Law 1 (without thresholds). 6) When the stress-strain curve crosses the compressive offset line (orange dashed line in Fig. 4), the compressive elastic limit for the chosen plastic strain p ∊ ∊ is detected. The specimen load is reversed again. ( 7) Steps 2-6 are repeated for each desired plastic strain level. The Young's modulus is recalculated for each loading or unloading.

Parameter σ C / 1 y γ 1 σ C / 2 y σ C / 3 y γ 3 σ X/ 3 lim, y b 1 σ Q / 1 y σ R /
Parameter σ C / 1 y γ 1 σ C / 2 y σ C / 3 y γ 3 σ X/ 3 lim, y b 1 σ Q / 1 y σ R /
During each unloading, the specimen load varies from the tensile elastic limit to the compressive elastic limit at a given plastic strain.

The load-unload test is plotted in Fig. 5. For confidentiality matter, stresses are normalized by the elastic limit of the material σ y and strains by = σE / .

yy ∊ ∊

This experimental elasto-plastic behavior can be modeled using a mixed model proposed by Chaboche [START_REF] Chaboche | Constitutive equations for cyclic plasticity and cyclic viscoplasticity[END_REF]. It combines various isotropic (R i ) and kinematic hardenings (X ̲ i ). The elastic domain is defined by [START_REF] Buczynski | An analysis of elasto-plastic strains and stresses in notched bodies subjected to cyclic non-proportional loading paths[END_REF] where J (•) 2 refers to the von Mises equivalent stress. Chaboche proposed a threshold kinematic hardening [START_REF] Chaboche | On some modifications of kinematic hardening to improve the description of ratchetting effects[END_REF] expressed as

∑∑ =- - + = ( ) fJ σ X R σ 0 ( ̲ ̲ ) i i y 2
=-〈 - 〉 XC γ X JX Xp 2 3 1 ̲̇ ̲ ̇( ̲ ) ̲ i p i lim i i i i , 2 ∊ ∊ ( 17
)
where X lim i , is the threshold value,

+ Cγ X (/ ) ii l i m i 2 3 ,
  is the saturation value in the uniaxial case and γ i rules the kinetic of the kinematic hardening. When plasticity initiates, < J XX ( ̲ )

il i m i 2,
, the hardening is linear, as in the Prager model [START_REF] Prager | Recent developments in the mathematical theory of plasticity[END_REF]. When the hardening reaches the threshold value X lim i , , it describes a non-linear kinematic hardening as in the Armstrong-Frederick model [START_REF] Armstrong | A mathematical representation of the multiaxial Bauschinger effect[END_REF].

The isotropic hardening is adapted from the Voce model [START_REF] Voce | A practical strain-hardening function[END_REF] to obtain a similar behavior

=-〈 - 〉 Rb Q R R p ̇(| | ) ii i i l i m i , (18)
where R lim i , is the threshold value,

+ Q R il i m i
, is the saturation value and b i rules the kinetic of the isotropic hardening.

The equivalent cumulative plasticity rate is defined as

= p d 2 3 dd ̲ : ̲ . pp ∊ ∊∊ ∊ (19)
The plasticity flow is calculated using the normality rule given by

= - -∑ p σX σX J dd 3 2 ̲ ( ̲ ̲ ) ( ̲ ̲ ) , p D i 2 ∊ ∊ ( 20
)
with σ̲ D the deviatoric stress tensor.

To identify the hardening parameters, the threshold hardening values R lim i , and X lim i , are set at zero initially. The model identified is thus a mixed model using the Voce isotropic hardening model and the Armstrong-Frederick kinematic hardening model. This behavior law is called "Law 1". Besides an easier description of the identification procedure, the introduction of this first law will later help to point out the advantages of the second behavior law that uses threshold hardenings ("Law 2") to improve the description of the cyclic behavior.

A sequential identification in which each parameter is identified separately is proposed here. The complete identification of the first behavior law is described in the following:

(1) First, the yield strength σ y and the elastic modulus E are de- termined.

(2) The tensile behavior is identified using as many kinematic hardenings as necessary. Here, one non-linear kinematic hardening X ̲ 1 (parameters C 1 and γ 1 ) and one linear kinematic hardening X ̲ 2 (parameters C 2 and = γ 0 2

) are used. (3) Then, isotropic hardening is identified with as many isotropic hardenings as necessary. In this study, only one isotropic hardening R 1 was used, using parameters b 1 and Q 1 . However, the description of the tensile curve is then modified by the introduction of isotropic hardening and needs to be corrected. This procedure can be set on this material because the isotropic hardening is negative (the size of the elastic domain decreases with plastic strain), leading to a negative value of the Q 1 parameter and thus to a positive value of C 3 parameter. Nevertheless, in the case of a po- sitive isotropic hardening, this procedure has to be adjusted as it would lead to a negative C 3 parameter. In that case, steps 2-4 are applied replacing isotropic hardenings by kinematic hardenings and vice-versa.

As previously stated, a second behavior law is identified. In this law, X lim i , and R lim i , values are still set at zero except for the isotropic and kinematic hardenings identified in steps 3 and 4 in the previous sequential identification procedure, (X ̲ 3 andR ) 1 . To conserve the hardenings mutual compensation, the non-zero threshold values are set equal, i.e., = XR lim lim ,3 ,1 . The threshold value is optimized to suppress any step effect in the mean stress relaxation curve, visible in Fig. 12 and described in the next section.

Identified parameters for both behavior laws are given in Tables 1 and2. Parameters γ 2 , X lim,1 and X lim,2 are equal to zero. C i , X lim i , , Q i , and R lim i , are divided by the elastic limit σ y for confidentiality matters. Both identified laws are presented in Fig. 6 showing that the measured tensile behavior as well as the hardening evolutions can be correctly described by the two laws. Although the kinetic of the hardenings is slightly modified by the presence of an initial linear behavior in Law 2, the monotonic tensile curve is not influenced by the thresholds parameters as the hardenings still compensate each other. The main interest induced by the presence of thresholds can be underlined by the cyclic behavior study presented in the next section.

Material cyclic properties

Uniaxial purely alternated strain-controlled tests ( =-R 1)

∊ ∊ were Fig. 15. Comparison between experimental gauge measurements and numerical predictions using the two behavior laws of stabilized strain range in 11 direction.

Fig. 16. Numerical predictions of the local stress ratios using the two behavior laws.

carried out on axisymmetric specimen to calibrate the Manson-Coffin-Basquin (MCB) criterion [START_REF] Manson | Fatigue: A complex subject-Some simple approximations[END_REF][START_REF] Coffin | A study of the effect of cyclic thermal stresses on a ductile metal[END_REF] given by

=+= + εε ε σ E Nε N ∆ 2 ∆ 2 ∆ 2 22 (. ) (.
). Three other load ratios were considered to study their effect on the fatigue life:

-Repeated strain-imposed tests = ∊ R 0 () , -Alternated tests with positive mean strain =- ∊ R 0.2 () , -Alternated tests with compressive mean strain =- ∊ R 5 () .
The strain ratio influence is shown on the stabilized stress-strain curves (Fig. 8). Partial mean stress relaxation can be observed for the lowest strain amplitudes whereas a complete mean stress relaxation is observed for the highest strain amplitude levels, whatever the load ratio.

The consequence of the strain ratio on the fatigue life is shown in Fig. 9. Compressive load ratios corresponding to <-R 1 ε do not affect the fatigue life whereas tensile load ratios with >-R 1 ε do reduce the fatigue life. This observation can be explained if both main effects of the strain ratio are considered. The first effect is the modification of the mean stress itself, which enhances the fatigue life if mean stress is compressive or reduces it otherwise. The second effect is the increasing of the maximum absolute plastic strain, which increases the damage and thus reduces the fatigue life. In the case of a tensile load ratio, both effects decrease the fatigue life, whereas they can compensate each other, in the case of negative mean stress as illustrated in Fig. 10. Generally, the load ratio effect is taken into account introducing the mean stress σ m influence or the maximum stress σ max influence in MCB criterion. In the first case, this corresponds to the Morrow criterion [1] that is formulated as

= - + ε σσ E Nε N ∆ 2 22 (• ) (• ). f m r b f r c ' ' (22)
In the second case, this corresponds to the Smith-Watson-Topper (SWT) [START_REF] Smith | A stress-strain function for the fatigue of metals[END_REF] criterion that is expressed as

= ⎡ ⎣ ⎢ ⎢ + ⎤ ⎦ ⎥ ⎥ + ε σ E Nσ ε N σ ∆ 2 22 1 (• ) (• ) . f r b ff r bc max 2 ' '' 2 (23)
Both criteria cannot take into account the experimentally observed asymmetric effect of the load ratio (Fig. 9). A modified Morrow criterion (MM) is thus proposed considering the positive part of the mean stress, noted using Macaulay brackets 〈 〉 σ m . The modified Morrow cri- terion is thus defined by

= -〈 〉 + ε σσ E Nε N ∆ 2 22 (• ) (• ). f m r b f r c ' ' (24)
The comparison of experimental versus predicted fatigue lifespans for the four criteria (MCB, Morrow, SWT and MM) using the experimental stabilized values is shown in Fig. 11. Predictions are improved with the MM criterion, which is also the only one to predict conservative fatigue life whatever the load ratio.

The last validation step of the fatigue criteria consists in using the predicted stabilized stress-strain curves, instead of the experimental ones. The predicted mean stress is thus compared to the experimental mean stresses in Fig. 12. It shows that, with the behavior Law 1, a complete mean stress relaxation is predicted for strain amplitudes higher than ∊ 0.7 y . It leads to an underestimation of the stabilized mean stress which implies non-conservative fatigue life predictions. With the behavior Law 2, a partial mean stress relaxation is obtained. Mean stress absolute values are uniformly overestimated which could be due to a long term softening of the material that is not taken into account in the model. Fig. 13 shows the influence of the behavior law on the fatigue lifespans obtained using the modified Morrow criterion with the predicted stabilized stress-strain curves. Predictions are more conservative using Law 2 than using Law 1 that can underestimate the stabilized mean stress. The conservatism factor can be as high as three for the highest lifespan of cyclic test with positive mean strain. This corresponds to applied strain amplitudes lower than ∊ 0.7 y where the mean stress is slightly overestimated, whatever the chosen behavior law. For applied strain amplitudes higher than ∊ 0.7 y or for cyclic tests with negative mean strain, lifespans are correctly predicted.

Prediction of the stabilised local behavior

Finite element modeling and validation

One fourth of the double-notched specimen was meshed using 8-node hexahedra with linear integration. The in-plane mesh was extruded to obtain the 3D model as shown in Fig. 14 (10 elements are used through the thickness). Experimental gauge measurements at the notch tip were accomplished using KYOWA® KFG-02-120-C1-11 strain gauges. Correlation between strain range measurements and FE [START_REF] Paul | The effect of low cycle fatigue, ratcheting and mean stress relaxation on stress-strain response and microstructural development in a dual phase steel[END_REF] Fig. 16 shows that, whatever the nominal load ratio and level, the predicted local stress ratio with behavior Law 1 is close to -1, indicating that the mean stress is totally relaxed. It implies thus that, according to Law 1, the nominal load ratio has no effect on the local load ratio and thus on the fatigue life, which is in contradiction with the obtained experimental results (Fig. 2). On the contrary, using behavior Law 2, the predicted local stress ratio varies from -5t o-1 for repeated compressive nominal loadings and from -0.2 to -1 for repeated tensile nominal loadings.

Simplified methods validation

Cyclic behavior can then be obtained for both localization operators. Stabilized curves at = R 0 σ nom , obtained using both simplified methods (HSM and DSM) are plotted in Fig. 17 ). The shape of the stabilized curves obtained using simplified methods is similar to those obtained with FE, especially in the principal direction. For both load ratios and behavior laws, the shape and the size of the stabilized curves are correctly described although the maximum and minimum strain are underestimated by simplified methods, in comparison with FE curves. Fortunately, the only strain parameter used in the fatigue criteria is the strain amplitude which is similar between the different methods.

6. Fatigue life prediction using stabilized stress-strain curves

Multiaxial formulation of fatigue parameters

In order to obtain fatigue lifespan for the notched specimens using the modified Morrow (MM) criterion, a multiaxial formulation using equivalent scalar parameters has to be employed as expressed in the following relationship

= -〈 〉 + σσ E Nε N 2 22 Δ (• ) (• ). eq f m eq r b f r c ' ' ∊ ∊ (26)
Some formulations have been proposed based on von Mises formulations of the strain amplitude and maximum stress [START_REF] Chen | Low cycle fatigue life prediction of 63Sn-37Pb solder under proportional and non-proportional loading[END_REF]. Here, an equivalent strain amplitude parameter is used, written as

= + + ν ∆∆ 12 2 3 ∆∆ Δ ̲ : ̲ ̲ : ̲ . eq ee pp 2 ∊ ∊ ∊ ∊∊ ∊ ∊ ∊∊ ∊ (27)
It can be linked to a critical plane approach under an uniaxial strain but unlike such approaches, it can be easily implemented in a finite element code, as it is not necessary to determine a critical plane.

In order to preserve the mean stress sign, the von Mises equivalent stress cannot be employed directly as it would lead to positive equivalent mean stress under compressive loadings. For this reason, the first invariant of the mean stress is chosen to apply the MM criterion

= σ tr σ ( ̲ ) m eq m (28)
A comparison of the stabilized values of both equivalent parameters used in the proposed fatigue criterion is shown in Fig. 18. It appears that, whatever the behavior law, parameters obtained using simplified methods are very similar to FE. It is important to note that the equivalent mean stress is always totally relaxed with Law 1 whatever the load ratio or level, whereas Law 2 predicts an effect of the load ratio and load level.

The highest difference in the predicted equivalent mean stress, between simplified method results and FE results, reaches 0.18σ y . It comes from the difference in the mean stress prediction in the → e 3 direction as indicated by the stabilized curves in Fig. 10d). However, this variance does not induce a significative change in fatigue life prediction as will be shown later. The equivalent strain amplitude is less sensitive to this difference because of the quadratic norm employed in Eq. ( 27).

Fatigue life prediction

The fatigue life predictions obtained using the MM criterion are plotted in Fig. 19. The first observation is that all criteria and methods are conservative. Predictions obtained with both simplified methods are very similar to those obtained using finite elements, showing that the differences observed on the stress-strain curves and on the stabilized equivalent parameters are negligible, with respect to the global conservatism of the whole design process. This conclusion is in accord with [START_REF] Visvanatha | Influence of strain estimation methods on life predictions using the local strain approach[END_REF].

To complete and conclude this study, Fig. 20 shows the comparison between the experimental data and the predicted fatigue lifespans with the MM fatigue criterion and Herbland simplified method. The factor of security appears to be dependent of the load ratio with Law 1 but not with Law 2. A slight dependence on the load level can be observed with Law 2, which may be explained by a potential long-term softening of the material that cannot be taken into account by the behavior law. The global conservatism is not linked to the proposed simplified approach as results obtained with finite element analysis are identical. The first order factor in the fatigue life criterion is the strain amplitude which suggests that the prediction obtained with both methods overestimate the real strain amplitude. However, the strain amplitude values predicted with the behavior law (whether it is with the simplified method or with finite element analysis) are very close to the strain gauges measurements shown in Fig. 15. The main hypothesis of the important conservatism factor is, in the author's opinion, the strain amplitude gradient around the notch. Indeed, the volume that undergoes the maximum strain amplitude differs in the case of the criterion calibration specimens and the notched specimen. To take this effect into account, an effective volume approach could be used [START_REF] Härkegård | Assessment of methods for prediction of notch and size effects at the fatigue limit based on test data by Böhm and Magin[END_REF].

Conclusion

In this study, a full design chain to predict fatigue life of structures in confined plasticity cases was presented. This method is sum-up in Fig. 21. The main conclusions on the different steps of the proposed design chain are:

-The mean stress cannot be predicted correctly with classic nonlinear hardenings as proposed by Armstrong & Frederick or Voce. Threshold kinematic hardenings, as proposed by Chaboche, can be used to enhance the mean stress prediction. In this paper, a similar non-linear isotropic hardening with threshold is proposed to permit a sequential identification of the behavior law using a particular load-unload test. -Simplified methods based on the use of an adjustable scale transition rule can be used to predict the stabilized cyclic stress-strain curve. Darlet's and Herbland's localization operators were compared in this study and have shown similar capability. Fatigue life predictions obtained using both approaches are also similar to those obtained using FE showing that the use of the simplified method only reduces the computation time but not the fatigue life prediction accuracy. -As Darlet's operator only requires elastic calculations, it is easier to implement than Herbland's approach, especially if the behavior law is not already implemented in the finite element code. However, validation of the approach on multiaxial and non-proportional loads could also help to choose one or the other approach. Comparison between these two approaches and other already existing methods will be of interest for future work. -The method could be enhanced to reduce the conservatism. The material volume that undergo the same load is different between the calibration samples and the notched specimen. This could explain part of the conservatism because of the local approach. The use of an effective volume approach could help to reduce the conservatism factor [START_REF] Härkegård | Assessment of methods for prediction of notch and size effects at the fatigue limit based on test data by Böhm and Magin[END_REF]. The over-estimation of the mean stress could also be a reason of the conservatism. -The particular case of compressive loads was studied. Tensile mean stress reduces the fatigue life whereas compressive mean stress does not affect it. A modified version of the Morrow criterion is proposed to account for this effect.

Fig. 1 .

 1 Fig. 1. Specimen geometry (units in mm).

Fig. 2 .

 2 Fig.2. Fatigue curves of the notched specimen for two nominal load ratios.

Fig. 3 .

 3 Fig. 3. Analogy between Kröner's approach and the confined plasticity problem.

Fig. 4 .

 4 Fig. 4. Load-unload test control procedure.

Fig. 6 .

 6 Fig.6. Stress and hardening evolution with strain for the two behavior laws compared to the experimental data.

Fig. 7 .Fig. 8 .

 78 Fig. 7. Identification of MCB parameters for uniaxial cyclic strain-controlled tests =-R 1 ∊ ∊

Fig. 9 .

 9 Fig. 9. Fatigue life curves for different load ratios compared to the Manson-Coffin-Basquin criterion.

Fig. 10 .

 10 Fig. 10. Experimental stabilized stress/strain curves for three load ratios at two strain amplitudes: Δε/2 = 0.69ε y (a) and Δε/2 = 0.87ε y (b).

Fig. 11 .

 11 Fig. 11. Experimental vs. predicted lifespans for uniaxial cyclic strain-controlled tests under four load ratios for four fatigue criteria: MCB (a), Morrow (b), SWT (c), MM (d). Stabilized parameters used for the predictions are measured experimentally.

Fig. 12 .

 12 Fig. 12. Comparison of experimental versus predicted mean stress with a classic Chaboche model (a) and with threshold hardenings (b).

Fig. 13 .

 13 Fig. 13. Experimental vs. predicted lifespans for uniaxial cyclic strain-controlled tests under four load ratios for the modified Morrow criterion. Stabilized parameters used for the predictions are obtained using Law 1 (11.a). and Law 2 (11.b).

Fig. 14 .

 14 Fig. 14. Part mesh with boundary conditions.

( 4 )

 4 To compensate this influence, a third kinematic hardening (X ̲ )

  amplitude, σ f ' and b (resp. ε f ' and c) are parameters of the elastic (resp. plastic) regression in a log-log diagram, shown in Fig. 7 [16].

Fig. 17 .

 17 Fig. 17. Stabilized curves in direction 11 and 33 for the two behavior laws at = R 0 σ nom

Fig. 18 .

 18 Fig. 18. Stabilized parameters for the two behavior laws and the two load ratios ( = R 0 σ nom

Fig. 19 .

 19 Fig. 19. Comparison of fatigue lifespans predicted using the modified Morrow criterion with FE and SM with experimental data.

Fig. 20 .

 20 Fig. 20. Fatigue life prediction using HSM and the modified Morrow criterion for the two load ratios ( = R 0 σ nom

  and compared to FE. Curves are plotted for three load levels: the highest and lowest bounds used for the fatigue testing (

Fig. 21 .

 21 Fig. 21. Full fatigue life design chain diagram.

13 13 23 13 13 13 23 000 000 000 000 000 000 100 010 001

		//	/
		//	/
	̲ ̲	//	/
			(5)

in Voigt notation.

Table 2 Material

 2

	lim,	1	y

parameters for behavior Law 2 (with thresholds).

Acknowledgements

The financial support of DGA/MRIS (Mission for Scientific Research and Innovation) is gratefully acknowledged.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.