Parisa Dolati Neghabadi
email: parisa.dolatineghabadi@grenoble-inp.fr

Professor, Karine Evrard Samuel
email: karine.samuel@grenoble-inp.fr

Marie-Laure Espinouse
email: marie-laure.espinouse@g-scop.grenoble-inp.fr

CITY LOGISTICS: A REVIEW AND RESEARCH FRAMEWORK

Keywords: City Logistics, Review, Bibliometric Analysis, Clusters

City logistics is one of the significant branches of supply chain management, dealing with the logistics and transportation activities in urban areas. This research area has known an exponential growth of publications for the last five years. This paper presents a systematic literature review, which is the result of surveying more than 200 papers and research works published during the past five years. We identify the top contributing authors, countries and key research topics. The most common keywords used in city logistics literature are referred and allow us to propose six research clusters in which scholars could be encouraged to contribute.

Introduction

City logistics deals with the logistics and transportation activities in urban areas and is spotted as the last stage of supply chain management. By 2050, around 70% of the world's population will live in cities [START_REF] Lee | Urbanisation proceeding on unprecedented scale[END_REF]. In this manner, economic growth, improvement of urbanization environmental effects, resolving urban unemployment, safe urban freight transport and regional economy development will all be affected by city logistics. Thus, appropriate and optimal city logistics design and planning will have a great impact on social welfare and peace in society [START_REF] Rao | Location selection of city logistics centers under sustainability[END_REF]. In this stage, not only the interactions between freight forwarders exist but also the interactions between all others stakeholders: suppliers, retailers, consumers and local authorities need to be considered to approach the fully complex governance. City logistics includes plenty of subjects which require to be organized and then treated to achieve mobility, sustainability and livability within a city [START_REF] Taniguchi | Recent trends and innovations in modelling city logistics[END_REF].

To facilitate the study of city logistics field, it is fruitful to break it into several small subsets including different aspects of the problem to investigate each of which more deeply and precisely. In this study, various important aspects of city logistics are extracted from the literature and briefly discussed. Statistical analysis is done on the number of research works released since 2010. A classification based on keywords listed in articles is introduced to categorize related research activities on this field of research. More than 200 scientific works have been studied comprising conference and journal papers, books and book chapters, and Master's and PhD thesis. The aim of this paper is to present the different aspects of city logistics and its recent trends to provide a comprehensive view on this extensive research field.

In section 2, a review on city logistics literature is presented concerning its different definitions and contributions; section 3 explains our applied methodology to perform a systematic bibliometric analysis on city logistics by evaluating around 200 articles published since 2010; in section 4, we propose different classifications based on papers type, papers region profile and papers' keywords are presented. Regarding to these keywords, six established and emerging clusters are proposed which would encourage scholars to expand research on city logistics.

City logistics: significance and position

City logistics is one of the most argued concerns in most cities around the world regarding to recent issues linked to phenomena such as urbanization or the rise of citizen's expected welfare level [START_REF] Bozzo | Decision Support System for City Logistics: Literature Review, and Guidelines for an Ex-ante Model[END_REF]. Efficient city logistics creates more productive and attractive urban areas [START_REF] Taniguchi | Recent trends and innovations in modelling city logistics[END_REF]. Figure 1 City logistics is described as "The process for optimizing both logistics and transport activities done by private companies in urban areas while considering the traffic environment, traffic congestion and energy consumption within the framework of a market economy" [START_REF] Taniguchi | Modelling city logistics[END_REF].

The basic philosophy of city logistics is to propose a proper planning for goods distribution within a city [START_REF] Morfoulaki | Integrating city logistics into urban mobility policies[END_REF]. City logistics aims to optimally plan, manage and control the freight movements within a logistical network in a metropolitan area considering integration and coordination among involved stakeholders [START_REF] Amaral | City Logistics and Traffic Management: Modelling the Inner and Outer Urban Transport Flows in a Two-tiered System[END_REF]. The consequences of such planning are expected to reduce noise pollution, air pollution, congestion and carbon emission; it also implies measures and evaluation methods [START_REF] Browne | Light goods vehicles in urban areas[END_REF].

Two principal difficulties are addressed in comprehensive city logistics studies. The first difficulty is the high number of stakeholders and their heterogeneous aims [START_REF] Maggi | Simulating urban mobility and the role of public policies: the challenges of Agent Based Models[END_REF]. The main stakeholders are shippers, freight carriers, local authorities and residents (consumers) who preserve different objectives and perspectives towards their own goals [START_REF] Taniguchi | Emerging techniques for enhancing the practical application of city logistics models[END_REF]. The most conflicting interactions almost happen between private and public stakeholders [START_REF] Gonzalez-Feliu | Financing urban logistics projects. From public utility to public-private partnerships. Chapter in[END_REF]. Thus, it is essential to consider stakeholders' behavior in city logistics measures evaluation [START_REF] Tamagawa | Evaluating city logistics measures using a multi-agent model[END_REF]. The latter inconvenience is the lack of real data on city logistics. On one side, a wide number of involved stakeholders and their disinclination in sharing information and, on the other side, a lack of interest among the former local authorities to provide data as public stakeholders lead to real data shortage in city logistics context [START_REF] Russo | A model system for the ex-ante assessment of city logistics measures[END_REF].

Methodology

City logistics is a research area that crossbreeds several disciplines ranging from Social and Human Sciences to Engineering Sciences. In order to understand the current status of this large field, related research works are surveyed and analyzed in this paper. Our paper is a complementary study to literature reviews published in recent years (Ambrosini and Routhier, 2004;[START_REF] Anand | Relevance of city logistics modelling efforts: a review[END_REF][START_REF] Comi | Urban freight transport demand modelling: A state of the art[END_REF].

Methodology used to review and analyze the literature on city logistics is presented in Figure 2. We started by scanning definitions and finally, this work led to literature classification. This figure demonstrates that the number of research works in city logistics is exponentially increasing, and this growth is obvious since 2012 for the total number of published works, especially for conference papers. This shows the emergent nature of the research field.

Papers Region Profile

We considered the affiliation of first author to assign the research works to the countries.

Classification of articles based on region represents that each continent has its share, even low, in city logistics research field and implies that high number of articles have been published by European countries, especially France, Italy and Netherlands, Figure 4 (a andb).

This heterogeneity implies different city logistics environments in Europe comparing to other continents. In European countries, the most important issue is to deliver micro-size shipments into dense, historical centers within urban areas. So, contributions are mainly around improving the last mile deliveries through narrow streets and historical centers to preserve heritage sites and in parallel to attain sustainable and socially efficient delivery system. Local authorities in these countries are highly interested in designing appropriate restrictions and policies to limit the number of vehicle movements in congested areas [START_REF] Muñuzuri | Decision-making tools and procedures for City Logistics[END_REF]. dissimilarity within these keywords reveal the propagation of city logistics issues, its various aspects and also, the lack of consensus between authors. This led us to categorize them. The first step was to group keywords with common concepts into different categories. As some keywords could be assigned to more than one category, the most related one was selected for each of which. In this way, 490 keywords were sorted and the remaining keywords (which were cited only once) were discarded. Then, categories were organized into clusters.

Reassessing the proposed classes led to final clustering. Six clusters were introduced as follows: 1) Definition and Perimeters, 2) Policy, 3) Innovative Solutions, 4) Sustainability, 5) Methods and 6) Stakeholders. Each cluster is explained in the next sub-sections. It is necessary to note that all the used words within the categories are derived from keywords and that there is no new added word. Table 1 shows the number of articles that have at least one keyword in each cluster per year.

Definition and Perimeters

This cluster comprises all keywords used to define different aspects of city logistics issues.

We tried to distinguish between the close concepts to clarify their position and leverage in this issue. The result of this endeavor is depicted in Figure 5 which includes all the perimeters of city logistics. Since city logistics is considered as the last step of Supply Chain Management, keywords including supply chain frequently appear in city logistics literature. Thus, city logistics is presented as a subset for a supply chain. However, Freight Transport exists in different steps of a supply chain, and in the city logistics field, freight transport in urban area is considered. Logistics is a general concept which comprises several activities, but in city logistics domain, only logistics activities within cities are considered. Types and goals of logistics have been noticed as Logistics Land Use [START_REF] Diziain | How can we bring logistics back into cities? The case of Paris metropolitan area[END_REF], Green Logistics [START_REF] Jedliński | The position of green logistics in sustainable development of a smart green city[END_REF], and Intermodal Logistics [START_REF] Bandinelli | Advanced intermodal freight transportation system for sustainable city logistics: An application in fashion industry[END_REF]. Distribution in city logistics reflects freight distribution in urban area considering the existing constraints.

Delivery is a subset of distribution, and nowadays it is a notable issue in terms of pick-up and delivery spaces in the cities, mainly in European countries. Another important factor in city logistics refers to Urban Area that is mainly defined from size, character and desired quality of the studied area.

The flow of merchants within the city by the goal of distribution creates Urban Goods

Movement as an important issue in city logistics. In some optimization-based problem, urban goods movement is occasionally assumed equal to city logistics; however, city logistic is never limited to urban goods movement. If distribution is combined by delivery purpose, in this way it can be considered as a Last Mile Delivery issue, which has dramatically increased in recent years.

All these sub problems could be handled by different points of view, by Designing New Network or Facilities, providing new infrastructure such as Logistics Information Platform [START_REF] Liu | Research on Construction of Urban Logistics Information Platform[END_REF] and Geographical Information System [START_REF] Guerlain | Towards a collaborative Geographical Information System to support collective decision making for urban logistics initiative[END_REF] or Management approaches such as Freight Distribution Management [START_REF] Jiang | City Logistics: Freight Distribution Management with Time-Dependent Travel Times and Disruptive Events[END_REF] and Freight Demand Management [START_REF] Holguín-Veras | Sustainable urban freight systems and freight demand management[END_REF]. Polices are defined to organize a comprehensive framework for planning and developing efficient freight distribution systems in urban areas [START_REF] Stathopoulos | Urban freight policy innovation for Rome's LTZ: A stakeholder perspective. City distribution and urban freight transport[END_REF]. Policy is an overall word and can be broken to three different actions, as shown in Figure 6. In this area, governance refers to Urban Governance [START_REF] Schliwa | Sustainable city logistics-Making cargo cycles viable for urban freight transport[END_REF] and New Governance Model [START_REF] Marciani | How the URBeLOG Project Will Enable a New Governance Model for City Logistics in Italian Metropolitan Areas[END_REF], which is a topic rather underestimated in academic researches.

Planning is essential to achieve medium and long term objectives and is done for different aspects of city logistics such as City Logistics Planning (Nuzzolo andComi, 2014, Jiang and[START_REF] Jiang | Development of Urban Logistics in China[END_REF][START_REF] Awasthi | Collaboration partner selection for city logistics planning under municipal freight regulations[END_REF]Transport Planning (Betanzo-Quezada et al., 2013).

Measures are defined as actions that are taken to implement a plan or a decision. Measures are divided into two subsets regarding to their application. First, Regulations related to city logistics are defined by Local Authorities (considered as public sector) to exert restrictions in accessing to the urban area [START_REF] Ville | The limits of public policy intervention in urban logistics: Lessons from Vicenza (Italy)[END_REF]. Second subset of measures is Innovative Solutions which are widely applied and developed by private sectors, although their success highly depends on public sectors supports.

Because of interactions existing among different stakeholders (e.g. carriers, retailers, shippers, large or small transport operators and local authorities) whose interests are almost always conflicting, implementing policies is a difficult task. Several researches concentrate only on

Policy and endeavor to recognize its characteristics and influences on the whole parts of city logistics [START_REF] Marcucci | Urban Freight Modelling and Policy Analysis[END_REF][START_REF] Stathopoulos | Urban freight policy innovation for Rome's LTZ: A stakeholder perspective. City distribution and urban freight transport[END_REF][START_REF] Ville | The limits of public policy intervention in urban logistics: Lessons from Vicenza (Italy)[END_REF]. [START_REF] Morganti | The last food mile concept as a city logistics solution for perishable products. Enterprise Interoperability. Interoperability for Agility, Resilience and Plasticity of Collaborations Bppk[END_REF][START_REF] Diziain | How can we bring logistics back into cities? The case of Paris metropolitan area[END_REF] Consolidation [START_REF] Faure | How to anticipate the level of activity of a sustainable collaborative network: the case of urban freight delivery through logistics platforms[END_REF][START_REF] Van Duin | Towards E (lectric)-urban freight: first promising steps in the electronic vehicle revolution[END_REF][START_REF] Janjevic | Downscaling the consolidation of goods state of the art and transferability of micro-consolidation initiatives[END_REF][START_REF] Gonzalez-Feliu | A location-based accessibility analysis to estimate the suitability of urban consolidation facilities[END_REF][START_REF] Trentini | Developing urban logistics spaces: UCC and PLS in South-Western Europe[END_REF],

Pooling [START_REF] Ballot | Reducing transportation CO2 emissions through pooling of supply networks: perspectives from a case study in French retail chains[END_REF][START_REF] Gonzalez-Feliu | Costs and benefits of logistics pooling for urban freight distribution: scenario simulation and assessment for strategic decision support[END_REF][START_REF] Xu | Allocation of Transportation Cost & CO 2 Emission in Pooled Supply Chains Using Cooperative Game Theory[END_REF] and Hub [START_REF] Lange | Urban Retail Logistics-Research into the Bundled Urban Store Deliveries of the Future[END_REF] refer to Innovative Solutions based on collaboration. Other mostly argued solutions are the use of Light Goods Vehicle [START_REF] Browne | Light goods vehicles in urban areas[END_REF] [START_REF] Van Duin | Towards E (lectric)-urban freight: first promising steps in the electronic vehicle revolution[END_REF][START_REF] Amodeo | An ex-ante evaluation of last-mile freight distribution services for city logistics[END_REF][START_REF] Schöder | The Impact of E-Commerce Development on Urban Logistics Sustainability[END_REF][START_REF] Lebeau | Exploring the choice of battery electric vehicles in city logistics: A conjoint-based choice analysis[END_REF], Off Hour Deliveries [START_REF] Holguín-Veras | The New York City off-hour delivery project: lessons for city logistics[END_REF][START_REF] Santos | Exploring carriers' perception about city logistics initiatives 2[END_REF][START_REF] Verlinde | Who is in favor of off-hour deliveries to Brussels supermarkets? Applying Multi Actor Multi Criteria analysis (MAMCA) to measure stakeholder support[END_REF][START_REF] Jaller | Quantifying the impacts of sustainable city logistics measures in the Mexico City Metropolitan Area[END_REF], proposing manipulation deliveries during night and finally the use of Cycles [START_REF] Schliwa | Sustainable city logistics-Making cargo cycles viable for urban freight transport[END_REF][START_REF] Choubassi | An assessment of cargo cycles in varying urban contexts[END_REF] for last mile deliveries. Some other innovative solutions are proposed and their feasibility and applicability are discussed [START_REF] Samuel | Towards evaluation tools for last mile delivery projects: insights from france, 6 th METRANS International Urban Freight Conference[END_REF].

However [START_REF] Cattaruzza | Vehicle routing problems for city logistics[END_REF][START_REF] Heng | Multi-objective heterogeneous capacitated vehicle routing problem with time windows and simultaneous pickup and delivery for urban last mile logistics[END_REF][START_REF] Lim | Three-Dimensional Vehicle Routing Problem for Urban Last Mile Logistics: Problem Formulation and Computational Analysis[END_REF].

Sustainability

Environmental, social and economic considerations create a cluster around sustainability, which is one of the most significant issues among the researches in both city logistics and supply chain management. Competitive business environment, resource shortage and conflicting goals caused to appearance of several steady states to oblige agents pursuing these rules. City logistics could be established as a specialized discipline to face with sustainability issues encountered in urban freight transport [START_REF] Anand | GenCLOn: An ontology for city logistics[END_REF]. Sustainable city logistics almost addresses the problems caused by freight transportation within city centers [START_REF] Russo | City characteristics and urban goods movements: A way to environmental transportation system in a sustainable city[END_REF]. Sustainability concept has been reported in different sectors of city logistics such as Sustainable Urban Freight [START_REF] Holguín-Veras | Sustainable urban freight systems and freight demand management[END_REF], Sustainable mobility [START_REF] Trentini | Flow management of passengers and goods coexisting in the urban environment: Conceptual and operational points of view[END_REF], Sustainable Urban Logistics Plan [START_REF] Morfoulaki | Evaluation of specific policy measures to promote sustainable urban logistics in smallmedium sized cities: the case of Serres, Greece[END_REF]. Fuel Consumption (Koç, et al., 2015[START_REF] Koç | The impact of depot location, fleet composition and routing on emissions in city logistics[END_REF] and Polluting Emission [START_REF] Daniela | Reduced urban traffic and emissions within urban consolidation centre schemes: The case of Bristol[END_REF] are the environmental impact of city logistics. Social preoccupation mainly points out Quality of life [START_REF] Witkowski | The Role of Local Governments in the Development of City Logistics[END_REF], Congestion [START_REF] Alho | A state-of-the-art modeling framework to improve congestion by changing the configuration/enforcement of urban logistics loading/unloading bays[END_REF][START_REF] Jiang | City Logistics: Freight Distribution Management with Time-Dependent Travel Times and Disruptive Events[END_REF][START_REF] Jaller | Quantifying the impacts of sustainable city logistics measures in the Mexico City Metropolitan Area[END_REF] and concerns related to Historical Urban Area Preservation [START_REF] Pulawska | Ecological urban logistics in the historical centers of cities[END_REF].

Methods

There are a large number of techniques and approaches developed to solve city logistics problems by concentrating on different aspects of the problem. Indeed, they are almost different in nature. Some novel approaches which have been presented to cope with city logistics issues practically require a high infrastructural support to be launched [START_REF] Nowicka | Smart city logistics on cloud computing model[END_REF]; some others mostly try to only improve existing situations by defining and assessing different scenarios. There are also other solutions that have been already established, and for which researchers try to optimize their efficiency. Different types of methods are shown in [START_REF] Ballantyne | A comparative study of urban freight transport planning: addressing stakeholder needs[END_REF], Involvement [START_REF] Lebeau | Exploring the choice of battery electric vehicles in city logistics: A conjoint-based choice analysis[END_REF] and Engagement [START_REF] Marciani | Stakeholders' involvement and new governance models: The best practice case of Turin, Italy[END_REF]. Additionally, Trust among stakeholders is lately presented and discussed as a crucial factor (Daudi et al., 2012).

Conclusion

This paper presents a systematic literature review on city logistics and identifies main research trends to provide a comprehensive view on this extensive and emergent research field. The results are obtained by surveying more than 200 relevant articles since 2010. A bibliometric analysis is applied to extract important issues in city logistics. Three classifications are proposed based on papers type, papers region profile and papers keywords.

From the keywords analysis, we establish six clusters, which demonstrate the most important aspects of city logistics. These clusters will help those who are currently working on city logistics to appreciate the current status of the city logistics research field.

  illustrates the growing number of studies published in the city logistics field since 2010. Note that available published studies until end of May 2016 have been reviewed and taken into account for this study.

Figure 1 .

 1 Figure 1. Number of academic publications on city logistics

First

  , purpose and scope of the study are defined. In the second step, related keywords, which enable us to conceptualize the research and help to find relevant articles are identified. In our case, urban logistics, city logistics, urban freight transportation, freight transport, freight and logistics constitute the main used keywords for the search on the databases: Science Direct, Web of Science, Google Scholar, Wiley and Springer. More than 200 articles were found in the city logistics field since 2010 as the result of our research on the above-mentioned scientific platforms. The next step was to discover the scope of each paper in city logistics context through evaluating the article's title, abstract, keywords and full text. To focus purely on city logistics issues, less related articles were eliminated. At the end, 202 publications directly connected to the topic were left in our database built on an Excel file. Papers were classified based on predefined criteria in the last step. In our study, three different classifications based on papers type, papers region profile and paper's keywords are introduced. Detailed information about different classifications is explained in section 4.

Figure 2 .

 2 Figure 2. Applied framework for literature analysis (adapted from[START_REF] Gahm | Energy-efficient scheduling in manufacturing companies: A review and research framework[END_REF]

Figure 3 .

 3 Figure 3. Classification of different scientific works by origin and by year

  Figure 4.a Region profiles by continents

Figure 5 .

 5 Figure 5. Definition and Perimeters category extracted from keywords

Figure 6 .

 6 Figure 6. Policy category extracted from keywords

  Figure 7.

Figure 7 .

 7 Figure 7. Methods category extracted from keywords.

Table 1 .

 1 The number of articles in each cluster

	Clusters	Year (Articles N  )	2010 (8)	2011 (7)	2012 (23)	2013 (27)	2014 (34)	2015 (47)	2016 (56)
	Definition & Perimeters	7	6	20	22	27	38	49
	Policy		2	1	5	7	7	7	13
	Innovative solutions	2	0	9	8	7	20	16
	Sustainability	4	2	6	4	14	9	11
	Methods		2	4	15	19	22	30	37
	Stakeholders	1	2	2	3	2	2	6

  or Electrical Vehicle (Van

		Policy
	Governance	Planning	Measures
		Regulation	Innovative solutions
		Public		Private

  , solutions are almost the same all around the world. In France, a high percentage of research works insist on collaboration concepts to cope with city logistics issues. Italy has focused mainly on Policy definition and execution. In Netherland and United Kingdom the use of Electrical vehicle and Cycles are further discussed. In this context, United States and China mostly focus on mathematical approaches to optimize proposed solutions. The most argued issue in optimization context is Vehicle Routing Problem

  Two general divisions of stakeholders are Public and Private Stakeholder. Local Authority or Local Government are considered as public stakeholders (Lindholm and Behrends, 2012). Consumer, Supplier, Manufacturer, Logistics Service Provider and Retailer are grouped as Recently, stakeholder's analysis has absorbed attention of number of researchers. They mostly focused on Stakeholders' Needs

	Private Stakeholders.
	Optimization	Modelling
	Qualitative Methods	Methods	Evaluation
	Case Study	Data Collection

Acknowledgement

Support from Academic Research Communities (ARC8), Rhône-Alpes, France for funding this research is greatly acknowledged.


B Tanguy

C Bouchet

S Bugat

J Besson

Local approach to fracture based prediction of the ¢¡ ¤£ ¦¥ ¨ § and ¢¡ ¤© shifts due to irradiation for an A508 pressure vessel steel

Keywords: Irradiation embrittlement, Ductile to brittle transition, Charpy test, Fracture toughness

Nuclear pressure vessel steels are subjected to irradiation embrittlement which is monitored using Charpy tests. Reference index temperatures, such as the temperature for which the mean Charpy rupture energy is equal to 56 J (( 0) 21 23 ), are used as embrittlement indicators.

The safety integrity evaluation is performed assuming that the shift of the nil-ductility reference temperature 4 5( 76 98 7@ due to irradiation is equal to the shift of ( 0) 21 23 . A material model integrating a description of viscoplasticity, ductile damage and cleavage brittle fracture is used to simulate both the Charpy test and the fracture toughness test (CT geometry). The model is calibrated on the Charpy data obtained on an unirradiated A508 Cl.3 steel. It is then applied to irradiated materials assuming that irradiation affects solely hardening. Comparison with Charpy energy data for different amounts of irradiation shows that irradiation possibly also affects brittle fracture. The model is then applied to predict the fracture toughness shifts (A B( 0C ED GF IH P RQ GQ ) for different levels of irradiation.

Introduction

Reactor pressure vessels (RPV) of commercial nuclear power plants are subjected to embrittlement due to the exposure to high energy neutrons from the core. The current way to determine the effects of the degradation by radiation on the Preprint submitted to Elsevier Science 25 May 2005 mechanical behavior of the RPV steels is the use of tensile and impact Charpy tests, from which reference index temperatures, such as the ductile to brittle transition (DBT) temperature, can be calculated. More conventionally in the French nuclear surveillance program, ) 21 ¢¡ , the temperature for which the mean Charpy energy is equal to 56J and its increase due to neutron irradiation are determined.

The safety integrity evaluation based on a fracture toughness lower bound curve

Effect of neutron irradiation on the Charpy transition curves and hardening properties

For a given fluence ( (n.cm¡ %

)) of neutrons (E¢ 1 MeV) and irradiation temperature, irradiation induced embrittlement is strongly dependent of the material chemical composition, and specially on Cu, Ni, P contents [START_REF] Haggag | Effects of irradiation temperature on embrittlement of nuclear pressure vessel steels[END_REF]. The volume fraction of copper is an important factor in the hardening-induced embrittlement due to the irradiation induced defects. This has led to consider these chemical elements in the empirical embrittlement correlation models for the prediction of the shift of the reference index temperatures [START_REF] Kirk | Updated embrittlement trend curve for reactor pressure vessels steels[END_REF][START_REF] Hiranuma | Mechanistic modeling of transition temperature shift of japonese RPV materials[END_REF]. On the physical point of view, irradiation produces fine scale microstructures [START_REF] Nikolaev | Radiation embrittlement of low-alloy steels[END_REF]. Their effects on the macroscopic behavior can be separated into two groups [START_REF] Nikolaev | Radiation embrittlement of low-alloy steels[END_REF][START_REF] Wagenhofer | Yield and toughness transition prediction for irradiated steels based on dislocation mechanics[END_REF]: (i) change of the plastic hardening, (ii) embrittlement, one of the most wellknown and well-described being phosphorous segregation at grain boundaries [START_REF] Nikolaev | Radiation embrittlement of low-alloy steels[END_REF]. Hardening mechanisms include matrix and age hardening. Matrix hardening is due to radiation-produced point defect clusters and dislocation loops, referred to as the matrix damage contribution [START_REF] Wagenhofer | Yield and toughness transition prediction for irradiated steels based on dislocation mechanics[END_REF]. Age hardening is an irradiation-enhanced formation of copper-rich precipitates. These two hardening mechanisms cause an increase of the yield strength whereas phosphorous segregation causes grain boundaries embrittlement without any increase of hardness and may be responsible for intergranular fracture [START_REF] Brillaud | In-service evaluation of french pressurized water reactor vessel steel[END_REF]. It should be underlined that the micromechanical models developed in this study do not consider the embrittlement of the grain boundaries due to phosphorous segregation. Intragranular cleavage is assumed to be the prevailing mechanism controlling brittle fracture. labeled irr. 1) the whole stress-strain curve of the unirradiated material is shifted to higher stress values by ¡ E , i.e. the strain hardening rate remains unchanged whereas the yield stress in increased [START_REF] Margolin | A new engineering method for prediction of the fracture toughness temperature dependence for RPV steels[END_REF][START_REF] Al Mundheri | Irradiation embrittlement of a low alloy steel interpreted in terms of a local approach of cleavage fracture[END_REF]. This case was also experimentally observed on ferritic alloys irradiated at 288a C by electron irradiation [START_REF] Farrell | Hardening of ferritic alloys at 288a C by electron irradiation[END_REF]. In the second case the yield stress is increased by ¡ E whereas the ultimate tensile stress (UTS) remains almost unaffected by irradiation (dotted curve labeled irr. 2), i.e. that the hardening rate is substantially lower than in the unirradiated conditions. This case is reported for steels (ferritic, martensitic) subjected to neutron irradiation at low temperatures ( ¢ ¤£ ¦¥ §¥ a C) [START_REF] Byun | Irradiation hardening behavior of polycrystalline metals after low temperature irradiation[END_REF] and its importance increases with decreasing irradiation temperature [START_REF] Odette | Cleavage fracture and irradiation embrittlement of fusion reactor alloys: mechanisms, multiscale models, toughness measurements and implications to structural integrity assessment[END_REF]. At high irradiation doses, it is also suggested that neutron irradiation has a similar effect as a plastic prestrain on strain hardening [START_REF] Byun | Plastic instabillity in polycrystalline metals after low temperature irradiation[END_REF].

The choice of the appropriate case is undoubtedly strongly dependent on the considered material: in pioneer studies on the interpretation of irradiation effects on low temperature fracture toughness, the second case was considered for an A533B steel [START_REF] Parks | Interpretation of irradiation effects on the fracture toughness of a pressure vessel steel in terms of crack tip stress analysis[END_REF] whereas it was the first case for an A508 Cl.3 steel in [START_REF] Al Mundheri | Irradiation embrittlement of a low alloy steel interpreted in terms of a local approach of cleavage fracture[END_REF]. The variation of the yield stress and the difference between the ultimate stress and the yield stress for an A508 Cl.3 steel from the French surveillance program [START_REF] Trouvain | Approches globale et locale de la rupture fragile d'un acier faiblement allié[END_REF][START_REF] Edf | Edf industry-nuclear generation division corporate laboratorie[END_REF] with a chemical composition very close to the unirradiated A508 Cl.3 material considered in this study [START_REF] Tanguy | Ductile to brittle transition of an A508 steel characterized by the Charpy impact test. Part-I: experimental results[END_REF] and their evolutions with irradiation fluence are reported on fig. 1b. The variation of the yield stress with fluence up to a maximum value of 6.0 ¨¥ $ © n/cm % is well represented by a power-law expression, ¡ E . A value of 0.51 is obtained for which is representative of the low doses regime where the work hardening behavior is not altered by irradiation [START_REF] Byun | Irradiation hardening behavior of polycrystalline metals after low temperature irradiation[END_REF][START_REF] Farrell | Deformation mode maps for tensile deformation of neutron-irradiated structural alloys[END_REF]. Data plotted on fig. 1b show that for fluence up to 6.10 $ © n/cm % (i.e. ¡ E ¥ MPa) the behavior of the investigated steel is closer to case irr-1 which however sligthly overestimates the actual UTS $

. Therefore the hypothesis of unaltered work hardening behavior will be the only one considered in the present study. This will lead to higher computed stresses. Considering both the impact and fracture toughness properties, it is well known that irradiation embrittlement induces an increase of the DBT temperature and therefore of the index ) 21 23 [2, 17, 19, 23, 28, 31]. As far as the upper shelf energy (USE) is concerned, most of the studies report a decrease with increasing irradiation level. This decrease being strongly dependent of the material considered for a given level of irradiation. Based on experimental evidences, linear correlations between the yield stress increase and the DBT temperature shift on the Charpy energy curves, ! #" ¡ E , have been proposed in the literature. In the French surveillance program, the shift of the index ) 21 23 ¦ is considered [19] whereas !$ $ 3 ¦ is used in the American regulations. Based on the experimental values given in [START_REF] Haggag | Effects of irradiation temperature on embrittlement of nuclear pressure vessel steels[END_REF][START_REF] Brillaud | In-service evaluation of french pressurized water reactor vessel steel[END_REF][START_REF] Edf | Edf industry-nuclear generation division corporate laboratorie[END_REF][START_REF] Sokolov | Comparison of irradiation-induced shifts of © ¡ ¤ and Charpy impact toughness for reactor pressure vessel steels[END_REF] where only A533B and A508 Cl.3 structural steels are considered linear correlations between ) 21 23 , !$ $ 3 and ¡ E for various RPV base metals were established:

) 21 23 ¥ &% ¥ ¥ ¡ E and !$ $ 3 ¥ '% £ )( 0 E (see

$

Rigorously, the variation of the ultimate stress is given by A 21 ( 43 U 65 87 )9 W A@ 4B DC ) $ showing that the hardening rate is slightly decrease with increasing level of irradiation. [START_REF] Edf | Edf industry-nuclear generation division corporate laboratorie[END_REF] [32] [START_REF] Haggag | Effects of irradiation temperature on embrittlement of nuclear pressure vessel steels[END_REF] ¢¡ ¤£ ¦¥ ¨ § © ! #" $ &% " (MPa) ' ( ) 0 1 fig. 2).

The established ) 21 23 I ¦ ¡ E ¦ correlation will be considered as the reference experimental database to which the predictions developed in this study for Charpy tests will be compared. For fracture toughness tests, the level of 100 MPaP Q , which is also considered in the Master curve approach [33, 34] to define the reference temperature, B , will be used. The finite element (FE) simulations of fracture toughness tests for different levels of irradiation will be then used to determine the shift of the index C ED GF SR $ B B . These predicted shifts will be then compared to the shift of impact toughness ) 21 23 .

Modeling of the ductile to brittle transition

The behavior of the reference unirradiated A508 Cl.3 steel of this study is presented in the following. More details can be found in [START_REF] Tanguy | Ductile to brittle transition of a A508 steel characterized by the Charpy impact test. Part-II: modeling of the Charpy transition curve[END_REF][START_REF] Tanguy | Ductile to brittle transition of an A508 steel characterized by the Charpy impact test. Part-I: experimental results[END_REF][START_REF] Tanguy | Mechanical behaviour of A508 steel based on double nonlinear viscoplastic constitutive equation[END_REF]. The material model consists of three parts: (i) a temperature and strain rate dependent viscoplastic model describing the behavior of the undamaged material, (ii) a model for ductile tearing, (iii) a model for transgranular cleavage fracture.

Viscoplastic behavior and ductile failure

A modified Rousselier model [START_REF] Tanguy | An extension of the Rousselier model to viscoplastic temperature dependant materials[END_REF] which is able to handle strain rate and temperature dependence of the material behavior is used to represent ductile failure. In this model, as in the well-known GTN model [START_REF] Tvergaard | Material failure by void growth to coalescence[END_REF], a single damage parameter which represents the void volume fraction, I , is used to model ductile fracture.

The model is based on the implicit definition of an effective stress ( & ) which is a function of both the macroscopic Cauchy stress ( ) and of the porosity:

R ¡ ¦ ¨¨I & ¢ £ ( I ¤£ ¦¥ ¨ § © ¥ £ ¦ ¨¨I & ¨¨! R #" C $ &% ¥ (1)
where R ¡ is the von Mises equivalent stress and ' the trace of the stress tensor, while ¥ and £ (¥ are material parameters which need to be adjusted.

The viscoplastic yield function is written as:

) & ¨ ¦ 10 % (2)
where the flow stress as a function of the effective plastic strain, 0 is given by:

¦ 20 43 E ¦ ¢ 65 $ ¦ ¦ ¨¨ § © ¦ ¨7 $ ¦ 98 0 ¢ @5 % ¦ ¨¨ § © ¦ Ä7 % 8 0
(3) where the yield stress E ¦ , 5 $ ¦ and 7 $ ¦ are temperature dependent. The values of the parameters of eq. 3 can be found in [START_REF] Tanguy | Ductile to brittle transition of a A508 steel characterized by the Charpy impact test. Part-II: modeling of the Charpy transition curve[END_REF]. Viscoplastic flow occurs when

) CB
¥ . The equivalent plastic strain rate D 0 is given by the viscoplastic flow function D 0 F ¦ which is expressed as:

¨D0 ¨F ¨DE $ ¢ ¨DE % with D E GF IH & © F P RQ TS ¨3 £ (4)
The strain rates D E $ and D E % are each representative of a deformation mechanism:

(1) Peierls friction, (2) phonon drag. Deformation is controlled by the slowest mechanism. Here it is important to underline that the viscoplastic model is identified in the whole range of strain rates ( ¨¥ ¡ $ up to U ¥ §¥ §¥ s¡ $

), temperatures (¨¨W V ¥ up to ( ¥ §¥ a C ) and plastic strains (up to 1.0 using the Bridgman analysis) encountered during the Charpy test and the quasi-static fracture toughness test in the whole DBT range including fully ductile behavior.

The plastic strain rate tensor is given by the normality rule as:

D E X ¦ ¨¨I D 0 `Y ) Y ¦ ¨¨I D 0 aY & Y % (5)
The evolution of the damage variable is governed by mass conservation modified to account for nucleation of new voids [START_REF] Tvergaard | Material failure by void growth to coalescence[END_REF][START_REF] Chu | Void nucleation effects in biaxially stretched sheets[END_REF]:

D I ¦ ¨¨I cb ed gf ih p § ¦ D E X ¢ rq D 0 % (6)
In this expression, the first right handside term corresponds to void growth and the second one to strain controlled nucleation.

q is a material parameter representing the nucleation rate [START_REF] Zhang | The statistical modelling of brittle fracture in homogeneous and heterogeneous steel microstructures[END_REF]. All model parameters directly related to ductile fracture (i.e.

¥ , £ (¥ and q ) are assumed to be temperature independent.

Under quasi-static loading, isothermal conditions are assumed. Under rapid loading corresponding to adiabatic conditions, the temperature increases due to plastic deformation. In the following, the temperature evolution is written as:

X D ¢¡ D E X ¤£ ¦ ¨¨I ¥¡ D 0 & (7)
where X is the volume heat capacity and ¡ a constant factor. Note that X equals ¦ ¨¨I B X where B X is the heat capacity of the undamaged material so that B

X D ¡ D 0 & .
The constitutive equations lead to softening up to crack initiation and propagation so that a material scale length is required. In the following, this length, ¦ , is identified to the mesh size which must be adjusted. As the model parameters are numerous, some of them were directly obtained from the metallurgical analysis of the material. In the present case, the initial porosity is set equal to the Manganese sulfide (MnS) inclusions volume fraction (i.e. I B ¨% ¨ § £ ¨¥ ¡ $ ). Cross section examinations of notched bars and Charpy specimen, show that secondary voids are nucleated on cementite particles (Fe© C) at high levels of plastic strain. Nucleation starts for 0 ¥ &% £ [START_REF] Tanguy | Ductile to brittle transition of an A508 steel characterized by the Charpy impact test. Part-I: experimental results[END_REF]. The carbide volume fraction is obtained from the chemical composition of the material: 2.3%. This sets the maximum value of the nucleated porosity. Nucleation is assumed to end for 0 ¨% ¨which is the level of plastic deformation in a tensile bar close to the rupture surface. Consequently the nucleation rate parameter q (eq. 6) is set to 0.038 for ¥ &% £ ¢ 0 ¢ ¨% ët to ¥ '% ¥ otherwise. The other parameters were obtained fitting the model on experimental results from notch tensile bars (NT) with different notch radii at room temperature [START_REF] Tanguy | Ductile to brittle transition of a A508 steel characterized by the Charpy impact test. Part-II: modeling of the Charpy transition curve[END_REF]. The following model parameters were used: ¥ ¥ &% V , £ ¦¥ £ '% £ and ¦ ¨¥ §¥ Q .

Brittle failure

The description of brittle failure is derived from the Beremin model [START_REF] Beremin | A local criterion for cleavage fracture of a nuclear pressure vessel steel[END_REF] which assumes the well-accepted assumption that cleavage is controlled by the propagation of the weakest link among a pre-existing population of statistically size-distributed microcracks. The model uses the following definition of a local effective stress for brittle failure £ X :

£ X £ § © ¦ ¨0 if D 0 ¢ ¥ 3 0 ¢ 0 ¤ ¥ otherwise ( 8
)
where £ is the maximum principal stress of . The condition D 0 ¢ ¥ expresses the fact than failure can only occur when plastic deformation occurs. The § © ¦ ¨0

term has been proposed in [START_REF] Beremin | A local criterion for cleavage fracture of a nuclear pressure vessel steel[END_REF] to phenomenologically account for grain shape changes or crack blunting induced by plastic deformation. 0 ¤ is the critical strain over which cleavage can occur. This brittle failure model can be applied as a postprocessor of calculations including ductile tearing. In this case, care must be taken while computing the failure probability as ductile crack advance leads to unloading of the material left behind the crack front. Considering that each material point is subjected to a load history, ¦ ¡ 3 0 ¦ ¢ ( time) the probability of survival of each point at time is determined by the maximum load level in the time interval £ ¥ 3 ¤ ¦¥ .

An effective failure stress § £ X is then defined as:

§ £ X ¦ ¡ Q f G© © B R £ X ¦ ¡ ¦ (9)
The failure probability, ¥ , is obtained by computing the Weibull stress, :

! " $# &% § (' £ X 0) 21 1 B 43 $ 65 ' ¥ ¨¨ § © 87 @9 ! !A CB ' ED (10)
where the volume integral is taken over the whole volume of the specimen. 1 B is a reference volume which can be arbitrarily fixed. F is the Weibull modulus which describes the scatter of the distribution, GA is a scaling parameter which can be interpreted as a measure of the mean cleavage strength of a reference volume 1 B .

Model parameters ( HA , F , , 0 ¤ ) must be adjusted.

The original model is adapted to account for the temperature dependence of the model parameters [START_REF] Tanguy | Ductile to brittle transition of a A508 steel characterized by the Charpy impact test. Part-II: modeling of the Charpy transition curve[END_REF]. The rupture probability of a volume element is then no longer represented by £ X but by I ¦ £ X !A ' where both HA and F may be temperature dependent. The load history integrating stress variations but also temperature changes is represented by:

§I ¦ ¡ Q f G© © B R I ¦ ¢ ¦ (11)
Finally the failure probability is given by:

¥ ¦ ¡ ¨¨ § © ¦ 0P ¦ ¡ with P ¦ ¡ # % §I ¦ ¢ ) 21 1 B % (12)
The parameters of the model for brittle fracture have been identified using NT bars

tested at low temperature ( ¢ ¨¨£ ¥ a C): F ¨ § '% , U , 0 ¤ ¥ and !A £ V £ £ MPa for 1 B ¨¥ §¥ Q © .

Ductile to brittle transition

The ductile to brittle transition is modeled by simulating ductile tearing and by post-processing the results in order to obtain the brittle failure probability. There is therefore no specific model for the transition. This methodology has shown to be able to predict mean values as well as scatter of impact and quasi-static Charpy energies from low temperatures up to ¨ ¥ a C (including the 56 Joules level of impact Charpy energy) using the cleavage model parameters identified at low temperatures ( ¢ ¨¨£ ¥ a C), i.e. with constant values for these parameters.

However it is necessary to introduce a temperature dependent (A parameter above a threshold temperature ( ¨ ¥ a C) to model the sharp upturn of the Charpy transition curve and to describe the upper part of the curve [START_REF] Tanguy | Ductile to brittle transition of a A508 steel characterized by the Charpy impact test. Part-II: modeling of the Charpy transition curve[END_REF] (Fig. 5). This apparent effect of temperature on the cleavage mechanism which maybe combines the effects of large plastic strain, temperature-induced inhomogeneities of local stress-strain fields [START_REF] Lefevre | A modified beremin model to simulate the warm pre-stress effect[END_REF] and a possible change of the critical defects population is not within the scope of this paper and is under investigation.

Simulation of the ductile to brittle transition for irradiated materials

All the simulations presented here are conducted using the implicit finite element code ZeBuLon [START_REF] Besson | Large Scale Object-Oriented Finite Element Code Design[END_REF]. The FE meshes used to simulate the Charpy and the Compact tension specimens are shown on fig. 3. Details can be found in [START_REF] Tanguy | Ductile to brittle transition of a A508 steel characterized by the Charpy impact test. Part-II: modeling of the Charpy transition curve[END_REF]. For both specimens only $ $ is meshed due to symmetries. The Charpy specimen, striker and anvil sizes and geometries are those of the AFNOR standard [START_REF] Afnor | Essai de flexion par choc sur éprouvette Charpy. Partie 1 : méthode d'essai[END_REF]. For the CT specimens an initial crack ratio B ¢¡ ¥ &% £ £ has been considered. Full 3D computations have been carried out for both geometries.

Using the previously described models and the hypothesis that the effect of neutron irradiation on DBT temperature shifts can be quantitatively assessed in terms of the effects of irradiation on the flow properties of the material, it becomes possible to model the effect of irradiation on the ductile to brittle transition and to quantitatively forecast the Charpy energy or fracture toughness versus temperature curves. Obviously the increase of the flow stress has two consequences: (i) stresses in the material increase causing earlier brittle failure, (ii) the macroscopic load on the Charpy specimen increases causing an increase of the dissipated energy in the ductile regime so that the USE also increases. The USE is computed by propagating the ductile crack through the whole Charpy specimen. It is here assumed that the ductile and cleavage micromechanisms are not affected by irradiation. Based on literature results this assumption is discussed in the following sections.

Effect of irradiation on ductile damage mechanisms

Studies of ductile damage mechanisms sensibility to irradiation are seldom in the published literature. The possible effect of irradiation on ductile damage has to be considered through the nucleation, growth and coalescence processes.

Nucleation. At a nano-scale level, numerous experiments on RPV steels, using techniques as small angle neutron scattering (SANS) [START_REF] Böhmert | Correlation between irradiationinduced changes of microstructural parameters and mechanical properties of RPV steels[END_REF] or atom probe analysis [START_REF] Miller | Understanding pressure vessel steels: an atom probe perspective[END_REF], have reported the creation of irradiation induced defects.

Considering their mean size (few nm) it is unlikely they can act as potential nuclei for ductile damage. This assumption seems to be confirmed by the micrographs of fracture surfaces of irradiated Charpy specimen tested in the upper-shelf domain [START_REF] Gurovich | Intergranular and intragranular phosphorus segregation in Russian pressure vessel steels due to neutron irradiation[END_REF][START_REF] Kuleshova | Comparison of microstructural features of radiation embrittlement of VVER-440 and VVER-1000 reactor pressure vessel steels[END_REF]. On these micrographs the mean size of the ductile voids appears to be of the order of few microns. Assuming that these voids have nucleated around nanosize precipitates would imply a void growth ratio of about 200 which is unrealistic. On the other hand the observed dimple size is characteristic of voids nucleated around carbides in such steels. Results concerning the irradiation effect on carbides are more ambiguous and seem to be strongly dependent on the chemical composition of the considered steel. In the case of French ferritic pressure vessel steels, it has been shown that no change in composition and size of M % C or M© C carbides can be attributed to the irradiation process [START_REF] Auger | Synthesis of atom probe experiments on irradiation-induced solute segregation in French ferritic pressure vessel steels[END_REF]. The last population of particles to be considered is the MnS inclusions. Previous studies on A503 Cl.3 steels have shown that the void nucleation from such inclusions occurs for a very low deformation ( £ %) [START_REF] Beremin | Influence de la triaxialité des contraintes sur la rupture par déchirement ductile et la rupture fragile par clivage d'un acier doux[END_REF] so that it can be considered as voids present in the virgin material. Therefore it is unlikely that irradiation will affect the void nucleation process of MnS.

Based on these results, it is considered that the nature of defects involved in the nucleation process are unchanged by irradiation. Lacking of studies devoted to the correlation between the carbides void nucleation rate and irradiation has led to keep the void nucleation law established for the unirradiated material unchanged (eq. 6). However it should be kept in mind that an increase of the void nucleation rate may be due to irradiation as it was considered in [START_REF] Margolin | Radiation embrittlement modelling for reactor pressure vessels steels: II. Ductile fracture toughness prediction[END_REF] for vanadium alloyed RPV steels. In this study [START_REF] Margolin | Radiation embrittlement modelling for reactor pressure vessels steels: II. Ductile fracture toughness prediction[END_REF] it was considered that the segregation of phosphorous on carbide and non-metallic inclusion interfaces reduce the critical void nucleation strain. So that both the nucleation rate and total nucleated volume fraction are increased. Growth. At a continuum scale changes of hardening properties may have an influence of the growth process of existing cavities issued from large inclusions.

Cell computations [START_REF] Faleskog | Cell model for nonlinear fracture analysis-I : Micromechanics calibration[END_REF] have shown that when the hardening rate remains unchanged, as it is almost the case for the steel considered in this study, there is little effect on void growth kinetic so that the model parameters are kept constant. Coalescence. As the hardening rate of the studied material is not strongly affected by irradiation, at a first approximation it was considered that the localization kinetic which controls the coalescence process is not irradiation dependent. Nevertheless it is reminded that in case of high irradiation doses dislocation channel deformation develops and promotes strain localization. Moreover additional irradiation-induced precipitates between large voids issued from inclusions, as sulfides, are likely to change the coalescence process as described in [START_REF] Faleskog | Micromechanics of coalescence-I. Synergistic effects of elasticity, plastic yielding and multi-size-scale voids[END_REF].

Effect of irradiation on transgranular cleavage mechanisms

In the case of cleavage fracture, it has been often considered that at low temperatures the critical cleavage stress is not affected by irradiation [START_REF] Al Mundheri | Irradiation embrittlement of a low alloy steel interpreted in terms of a local approach of cleavage fracture[END_REF][START_REF] Parks | Interpretation of irradiation effects on the fracture toughness of a pressure vessel steel in terms of crack tip stress analysis[END_REF] provided the fundamental mechanism remains identical, i.e. stress propagationcontrolled transgranular cleavage. It is often argued in the literature that for ferritic steels the local critical cleavage stress, ¤ , is controlled by the coarsescale trigger particle microstructure and by the microarrest toughness, F ¤ , of the ferrite matrix [START_REF] Hahn | The influence of microstructure on brittle fracture toughness[END_REF] and that fine scale irradiation induced features are not expected to have an influence on ¤ and F ¤ . Therefore the mean cleavage fracture stress will be first considered as independent on irradiation. Considering the cleavage model presented in section 3.2 it means that all the model parameters are unaffected by irradiation.

Effect of irradiation on the temperature dependence of the yield stress

In order to simulate the whole DBT curve, FE modeling of Charpy-V notch and CT specimens have been carried out in the temperature ranges [¨¨U ¥ a C £ ¢ ¨¥ §¥ a C] and [¨¨¥ §¥ a C £ ¢ ¨¥ §¥ a C], respectively. The increase of yield stress due to irradiation in the whole temperature range investigated is then needed. A first approach is to consider that the hardening observed at one temperature is the same in the whole temperature range [START_REF] Kirk | Shift in toughness transition temperature due to irradiation: B vs !$ $ 3 , a comparison and rationalization of differences[END_REF][START_REF] Al Mundheri | Irradiation embrittlement of a low alloy steel interpreted in terms of a local approach of cleavage fracture[END_REF]. This relies on the fact that the microstructures produced by both matrix-and age-hardening provide only long-range barriers to dislocation motion whereas the temperature dependence of the flow properties is controlled by the lattice spacing so that the thermal part of the yield stress should be weakly affected by irradiation. However experimental data published in the literature show a slight modification of the yield stress temperature dependence with irradiation especially at low temperatures [START_REF] Edf | Edf industry-nuclear generation division corporate laboratorie[END_REF][START_REF] Hunter | Fracture and tensile behavior of neutronirradiated A533-B pressure vessel steel[END_REF][START_REF] Gurovich | Assessment of relative contributions from different mechanisms to radiation embrittlement of reactor pressure vessel steels[END_REF]. Using these experimental data the yield stress can be expressed as a function of the temperature and of the fluence as

E ¦ 3 E ¦ ¥ 3 ¢ ¡ E ¦ 3 ( ¥ §¥ 98 ¡ ¦ ( 13
)
where ¦ is given by:

¦ ¨¢ ¥ &% ( § © 9 ¨" $ B B ¨¥ &% ¥ ¨¥ § © 9 ¨" % B B (14)
with " $ £ '% ¨, " % ( % ¥ and B £ § ( % ¨£ a C.

Simulation of the irradiation-induced shifts of the Charpy DBT curve

In a previous study [START_REF] Bouchet | Prediction of the effects of neutron irradiation on the Charpy ductile to brittle transition curve of an A508 pressure vessel steel[END_REF], the Charpy transition curve was simulated for different values of 0 E , i.e. for different levels of irradiation, equal to 45 (low irradiation level), 88 (mean irradiation level) and 150 MPa (high irradiation level). For irradiation levels higher than those reported on fig. 1b and up to ¡ E ¨£ ¥ MPa, it is assumed that the saturation in irradiation hardening is not reached and that the deformation mode remains unchanged [START_REF] Byun | Irradiation hardening behavior of polycrystalline metals after low temperature irradiation[END_REF][START_REF] Farrell | Tensile properties of ferritic/martensitic steels irradiated in HFIR, and comparison with spallation irradiation data[END_REF] (i.e. no dislocation channel deformation). Results for the shifts obtained on the reference temperatures, ) 21 23 and !$ $ 3 are reported on Fig. 4.

As the trends observed for both temperature indexes are similar, comments will focus on the ) 21 23 index (Fig. 4a). For the three values of ¡ E , it is shown that ) 21 23 is always underestimated assuming that the temperature dependent value of !A ¦ is unaffected by irradiation (dotted line in fig. 4a). Assuming that (A is a constant equal to the low temperature value for the unirradiated material (2925 MPa) gives a conservative estimation of ) 21 23 (dashed line in fig. 4a). Up to a value of about ¡ E # § MPa, the model gives an estimation which corresponds to the upper band of the experimental data. To obtain the mean temperature shift given in fig. 2b, the parameter GA has to be assumed to be also affected by irradiation (full line in fig. 4a). The parameter for the irradiated material is then expressed as:

¢ ¤£ ¥£ A !A ¦
¢ §¦ . The shift ¨¦ depends on the level of irradiation, it is calculated in order to represent the experimental correlation between ) 21 23 and ¡ E (Fig. 2b) and is given in fig. 5 for the three values of ¡ E . It is shown in fig. 4b, that the prediction for $ $ 3 is in reasonable agreement with experimental bounds up to the maximum ¡ E considered in this study (150 MPa) using the same A ¦ ¢ ¨¦ function. (i) constant value (low temperature value for the unirradiated material), (ii) temperature dependent C A (unirradiated material), (iii) temperature and irradiation dependent C A (Fig. 5). Thin lines recall the experimental range. corresponding to a 50% cleavage failure probability for the considered ¡ E values are reported on fig. 6. On this figure the predicted Charpy energies corresponding to a ductile crack propagation of 0.2 and 0.5 mm are also drawn. The simulation results show that the energy level of 56J (drawn with an arrow on each graph) corresponds to a ductile tearing between 0.2 and 0.5 mm. It is therefore necessary

¡ © B B

The ability of the model to predict fracture toughness for the unirradiated material using the !A ¦ relation obtained from Charpy data was shown in a previous study [START_REF] Tanguy | Comparison of predicted transition temperature shifts between static fracture toughness and Charpy-V impact properties due to irradiation for an A508 pressure vessel steel[END_REF] and is reported on fig. 7a. As in this study no tests have been performed on irradiated A508 Cl.3 steel, data from literature on similar A508 Cl.3 steels [START_REF] Trouvain | Approches globale et locale de la rupture fragile d'un acier faiblement allié[END_REF][START_REF] Al Mundheri | Irradiation embrittlement of a low alloy steel interpreted in terms of a local approach of cleavage fracture[END_REF] were used for comparison with the model predictions. The effect of irradiation on the shifts of the reference temperature, C £ ¥¤ R $ B B is reported on fig. 7b, c andd. For ¡ E U £ and 88 MPa, a good agreement is obtained between the model predictions and available experimental data (see fig. 7 b andc). The fracture toughness levels corresponding to a ductile tearing of 0.2 and 0.5 mm are also shown on fig. 7. It appears that for a level of 100 MPaP Q from which the shift of the toughness curve is obtained, ductile damage has not started to develop, the hypotheses concerning the irradiation effect on ductile damage do not have an influence on the predicted C £ ¥¤ R $ B B . The C £ ¥¤ R $ B B shift values obtained considering the 50% failure probability prediction are reported on each graph and a comparison with ) 21 23 is made in tab. 1. The predicted C £ ¥¤ R $ B B values are slightly higher than the shifts ) 21 23 , i.e. ¡ £¢ 6 8 7@ . The difference decreases with increasing ¡ E . Nevertheless as the predicted C £ ¥¤ R $ B B shifts are more important than the ) 21 23 shifts ( ¡ £¢ 6 8 7@ ), the conservatism of the ASME © £ ¥¤ curve has to be checked for the irradiated material. The ASME © £ ¥¤ curve is plotted on each graph of fig. 7 (thick line) using the ) 21 23 values given in tab. 1 and the 6 8 7@ value of the unirradiated material ( A F Y aY 6 98 7@ ¨£ § a C). It is shown that in the investigated range The ductile to brittle transition characterized by the Charpy impact test and quasi-static fracture toughness test has been modeled in the case of a RPV steel using constitutive equations for viscoplasticity, ductile tearing and brittle failure. The whole Charpy and quasi-static fracture toughness transition curve of the base unirradiated material can be modeled provided the parameter (A of the Beremin model is considered to be an increasing function of temperature. This apparent effect of temperature on the cleavage mechanism is considered to be a combination of the effect of large plastic strain, temperature-induced inhomogeneity of local stress-strain fields and a possible change of the critical defects population. Moreover it should be reminded that the cleavage model considered in this study does not consider the microcracks nucleation step of the cleavage process (see e.g. [START_REF] Bordet | A new statistical local criterion for cleavage fracture in steel. part i: model presentation[END_REF]).

¤ ¦¥ ¨ § / MPa ¤ © / C ¤ © "! $# % '& "& / C ¤ (© ) "! $# % '& & 10 2¤ © / C 45

Simulation of the USE decrease due to irradiation.

The irradiation-induced USE decrease reported in many experimental studies cannot be reproduced considering only the increase of the hardening properties.

Following experimental results obtained on an A508 steel, the work hardening rate was considered to be unaffected by irradiation, at least in the fluence range considered here. Preliminary FE simulations of the Charpy test, not reported here, considering a decreasing hardening rate with irradiation (i.e. hypothesis irr. 2 on fig. 1a), show that the enhanced localization induced by a lower hardening rate is not able to compensate for the increase of the maximum load level induced by the increase of the yield strength. As the Charpy energy is given by the area under the load-deflection curve, the deflection being the same when the behavior is fully ductile, the amount of energy is more important. This results in a higher upper shelf level. It should be also noted that fully ductile failure is obtained on irradiated materials at higher testing temperatures. This will induce a decrease of the USE if irradiation hardening is not considered. However for high levels of irradiation where up to a 50% decrease of the USE is reported [START_REF] Gurovich | Intergranular and intragranular phosphorus segregation in Russian pressure vessel steels due to neutron irradiation[END_REF], this temperature effect will not be sufficient to explain such decrease. In that case a possible change of nucleation rate may also be involved in the observed decrease of the USE level. Assuming that only a mode I ductile fracture is prevalent, a larger nucleation rate will lead to an acceleration of the ductile damage, so that the ductile crack will propagate faster resulting in a lower energy for a given temperature. Numerically it has been shown in case of strain controlled nucleation, a decrease of the critical nucleation strain favors shear-type fracture (cup-cone or slant fracture) [START_REF] Saje | Void nucleation effects on shear localization in porous plastic solids[END_REF]. Consequently it may be inferred that the change of nucleation rate due to increasing irradiation would lead to larger shear lips on the fracture surface. This change of macroscopic fracture behavior may be linked to the decrease of the USE. Fractographic investigations of irradiated specimens have to be carried out to investigate this hypothesis. Simulation of the embrittlement due to irradiation. Irradiation hardens the material causing an increase of the stresses thus causing earlier brittle failure. Considering hypothesis irr. 1, this effect cannot account quantitatively for the whole experimental reference temperature shift ) 21 23 ¦ . It is therefore necessary to consider that irradiation also affects GA and consequently the mean macroscopic cleavage stress. Irradiation effect is then equivalent to "a cooling of the material", the HA ¦ curve being shifted toward higher temperatures. The micromechanical mechanism needs however to be identified. A first hypothesis which is actually under investigation is the interaction between the irradiationinduced plastic deformation and population of potential nuclei for cleavage.

Using the value of HA ¦ ¢ §¦ determined from Charpy data, the shifts of fracture toughness index temperature for different level of hardening corresponding to different level of irradiation were found to be slightly higher than the corresponding shifts obtained from Charpy energy curves. These fracture toughness predictions were found to be in agreement with literature data. The difference decreases with increasing hardening. However it has been checked that the ASME © £ ¥¤ curve with ¡ £¢ 6 98 7@ ) 21 23 remains a lower bound for the CT(1T) fracture toughness data. Effect of the irradiation-induced hardening evolution on the predicted DBT shifts and USE. Assuming hypothesis irr. 2 would have led for a given deflection to smaller stresses and consequently to lower cleavage failure probability. This implies that using HA determined for the unirradiated material would have led to a larger underestimation of the DBT shifts. Consequently the conclusion that the mean cleavage stress (i.e. GA ) is affected by irradiation is valid whatever the chosen hypothesis. This also holds for the actual plastic behavior which is bounded by hypotheses irr. 1 and irr. 2. The computed ¦ shown on fig. 5 would have been larger. Concerning the USE prediction, using hypothesis irr. 2 would not have led to a decrease but to smaller increase. Consequently the conclusion that nucleation may be affected by irradiation is valid.

Fig. 1 .

 1 Fig. 1. a) Two principal hypotheses concerning the evolution of the flow stress after neutron irradiation. b) Evolution of A DC FE and UTSG HC IE as a function of the irradiation flux for an A508 Cl.3 steel (base metal) (( QP SR 'T SP VU XW Y `Y ba C) [21].

Fig. 2 .

 2 Fig. 2. Evolution of temperature indexes, A ( $ $ 3 and A B( ) 21 23 as a function of A DC E for various RPV base metals.

Fig. 3 .

 3 Fig. 3. Finite element meshes used for the analysis. a) Charpy specimen b) CT specimen.

Fig. 4 .

 4 Fig. 4. Prediction of A B( ) 21 23 and A ( $ $ 3 . Three hypotheses are used for the parameter C A :

Fig. 5 .

 5 Fig. 5. Variation of C A as a function of temperature for the unirradiated material and for irradiation levels corresponding to an increase of the yield stress equal to 45, 88 and 150 MPa. Arrow indicates the temperature shift, A B( ¦ , caused by irradiation.

  D n/cmE[23]

Fig. 7 .

 7 Fig. 7. Comparison of the unirradiated and irradiated predicted fracture toughness curve bounds (V Y U 65 #Y XW S Y XW 7 WS %) of as a function of temperature for different irradiation-induced increases of the yield stress, A DC FE . a) unirradiated material, b) A DC FE U Q TS MPa, c) A DC IE U U WU MPa, d) A DC IE U 5 XS Y MPa. The calculated temperature shifts at 100 MPaY referring to the unirradiated state are reported on each graph. The ASME a £ ¥¤ curves are represented with thick line (4 5( A F Y aY 6 98 7@ U G ¢b c a C). Fracture toughness levels corresponding to a ductile tearing of 0.2 and 0.5 mm are drawn with thin dashed lines on each graph.

Table 1

 1 Comparison between A B( ) 21 23 and predicted A ( C £ ¥¤ R $ B B shifts

		30	49	+19
	88	56	73	+17
	150	95	104	+9

for ¡ E , the model predictions are always less conservative than the ASME curve which remains a lower bound.

7 Discussion and conclusions Simulation of the DBT curves for the unirradiated material.

Acknowledgments

Financial support from Direction de la Sureté des Installations nucléaires (DSIN) and Electricité de France (EdF) is acknowledged.

to take into account ductile damage to numerically estimate the shift of the index ) 21 ¢¡ .

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, model ¥ §¦ ¡ ©£

) "0 1 ) "0 1 32 0 1 54 0 60 1 54 87 0

) "0 1 ) "0 1 32 0 1 54 0 60 1 54 87 0