Ouail Al Maghraoui
email: ouail.al-maghraoui@irt-systemx.frflore.vallet@irt-systemx.frjakob.puchinger@irt-systemx.fr

Flore Vallet

Jakob Puchinger

Bernard Yannou
email: bernard.yannou@centralesupelec.fr

Framing key concepts to design a human centered urban mobility system

Urban mobility poses some sustainability and design practice challenges. Mobility components such as vehicles, urban infrastructure, mobility services or other services delivered along a mobility experience are often designed separately. The performance of a global urban mobility system, at a city scale for instance, is therefore parceled out into components' ones that are not integrated from the perspective of a user who interacts with them in a door-to-door journey. This paper starts with relating different perspectives of urban mobility including, in crescendo, the human in the production of a global design solution. Through the examination of design and transport literature as well as practical examples, the paper highlights complexity factors of urban mobility that challenges engineering design. A second contribution is to identify relevant design objects aiming at providing a language for designing urban mobility.

INTRODUCTION

The worldwide proportion of people living in urban areas is expected to rise in the coming decades, reaching 67% by 2050 [START_REF] Van Audenhove | The Future of Urban Mobility 2.0: Imperatives to Shape Extended Mobility Ecosystems of Tomorrow[END_REF]. This growth gives rise to increasingly challenging situations for urban mobility: (1) People: traffic chaos, insecurity, traffic jam, decreasing quality of life and convenience. (2) Profit: overloaded infrastructures, insufficient public transport capacities, increasing motorization, limited parking places. (3) Planet: air pollution, CO2 emissions, noise, increasing ecological footprint. If urban areas are to become socially, environmentally and economically sustainable a holistic and integrated approach of urban mobility is needed. The social dimension covers safety, equality and fairness of both physical and economical accessibility to transportation. The environmental challenge is, for instance, on the use of non-fossil energy for vehicles, lowering the emissions of vehicles and infrastructures, promoting ecological behaviours, and eco-designing vehicles and infrastructure. Finally, the economic issue is about using resources efficiently, maximizing profit for the stakeholders (including taxes) and minimizing costs (e.g. investments in infrastructures) (UNHSP 2014). A sustainable urban mobility system satisfies current mobility needs of people in cities without compromising the ability of future generations to meet their own needs [START_REF] Kayal | Defining Sustainable Urban Mobility[END_REF]. One way to satisfy travelers' needs is to improve their door-to-door journey experience. However, the urban mobility system is still designed as an aggregation of products and services that are not operating in a user-centred harmony to offer a seamless mobility experience [START_REF] Preston | Integration for seamless Transport, UK[END_REF]. For example, in the Paris region, there are several transportation operators for different bus lines. At the exit of a train station, it is frequent to find the information about one operator but not the others. One of the reasons why such problems persist is that each line is designed separately from the others. The same logic applies on interchange between private car and public transportation (e.g. park and ride facility), or the need of different smartphone apps for one trip planning and monitoring (e.g. one for bus real time schedule and one for multimodal transfers). For that reason, governance is crucial for promoting more integrated transport policy in many senses: horizontally (between different agencies or sectors involved in policymaking), vertically (between different tiers of government), spatially (between geographically adjacent agencies), temporally (between policies with ICED17 different time horizons and/or implementation dates) and modally (between different systems and operators) [START_REF] Stead | Identifying Key Research Themes for Sustainable Urban Mobility[END_REF]. A holistic approach considering urban mobility as a system could be of value to formalize an integrated language of design for urban mobility. Considered as the ease to move from one destination to another or the demand generated from social and economic activities [START_REF] Hasan | Spatiotemporal Patterns of Urban Human Mobility[END_REF] [START_REF] Cascetta | Transportation systems analysis[END_REF], urban mobility takes different forms depending on the perspectives. This paper defines the Urban Mobility System (UMS) as all what makes possible for an individual to move from a point A to a point B in a defined geographical space (artefacts, information, people, and organizations). The complexity of the UMS poses challenges defining its design process, models, knowledge or expertise [START_REF] Sussman | New Approach to Transportation Planning for the 21st Century: Regional Strategic Transportation Planning as a Complex Large-Scale Integrated Open System[END_REF]. Examples of design for mobility are tackled in [START_REF] Hollauer | A Model to Describe Use Phase of Socio-Technical Sphere of Product-Service Systems[END_REF], [START_REF] Barbieri | Investigation on Optimal Mobility System using Axiomatic Design and Scoring Matrix: The "drive Ability" Experiment[END_REF], and [START_REF] Vidal | Impact indicators in transport infrastructures: A new market for engineering design researchers[END_REF] and they relate different sorts of difficulties for design practice. For example, the volume, the diversity and the asynchronicity of urban mobility components such as vehicles, infrastructure or information makes it non obvious to define an integrated design process of the whole UMS. Therefore, there is an opportunity to gain some insight in a global understanding of urban mobility with a design intent. The first section of the paper develops the complexity factors and dimensions that makes user-centred design of urban mobility a challenging issue. The second section proposes an integrative conceptualization based on three key concepts: mobility components, travel scenarios and travel-centred indicators. Finally, the paper discusses different perspectives of the proposed integration.

THE COMPLEXITY OF AN URBAN MOBILITY SYSTEM

A traveler interacts with many products and services while heading to some destination. Indeed, the traveler may be the user of a smartphone application to program the journey or check the schedule of the bus for example. He/she gets in the metro station and interacts with the tickets machine then boards the metro. He/she uses information panels at the station's exit. In this paper, the point of view is a user-centric one in order to include all products and services used during a whole journey. The complexity of the UMS encompasses different factors through several dimensions. The UMS contains a large amount of diverse and evolving stakeholders, physical components, information, and users, all interacting with each other in an urban context. Another form of complexity emerges from the diversity of combinations of use, the unstoppable usage (people moving 24/7), and a shared form of use. To represent this complexity, the technical aspect of urban mobility is first presented as the Technical UMS (TUMS). It is followed by the interactions with the TUMS through the traveler experience. Then the Socio-technical perspective of urban mobility is introduced. It is the extent to the market, where the TUMS is an offer and the travelers are generating the demand introduces.

Technical Urban Mobility System

The technical physical components of urban mobility are composed of roads, rail, fuel stations, train stations, bridges, energy and communication network, terminals and facilities etc. On the one hand, buses, cars, trains, trucks, boats, trams on the other hand. . Information and Communications Technology (ICT) (GPS, Internet of Things, mobile networks…) also play an important role in enabling the qualities of this technical system (safety, usefulness, fluidity…) (Kitchin 2013). However, the aggregation of infrastructure, vehicles and ICT is not sufficient to provide the whole picture of the technical urban mobility system. Indeed, trip-chaining [START_REF] Primerano | Defining and understanding trip chaining behaviour[END_REF]) connects the elements of infrastructure and vehicles. It brings out the necessity to have a global understanding taking multimodality into account and considering a global performance rather than that a single bus line, highway, or hub. For instance, bus lines that are feeding a regional rail line may operate with good performance indicators (e.g. schedule respect, good frequency) but if they arrive all at the same time at the train station they would cause a congestion and deteriorate the global performance indicators. The combined set of bus lines and regional train would then operate with a bad multi-modal efficiency indicator. Nevertheless, studying multi-modality is not sufficient to complete the whole technical view of urban mobility. Actually, there are other urban systems connected, such as households, industries, or workplaces [START_REF] Wegener | The future of mobility in cities: Challenges for urban modelling[END_REF]. The traveler interacts with TUMS and experiences mobility under different conditions, set as follows.

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The user of the Technical Urban Mobility System

The TUMS is designed for different users who interact with its components, individually or collectively or, in different moments of the day, and with different itineraries. The EU-FP7 project METPEX (MEasurement Tool to determine the quality of Passenger EXperience), describes 'traveler experience' by decomposing the journey into different typical stages [START_REF] Woodcock | Deconstructing the Whole Journey Traveler Experience[END_REF]). The journey is decomposed into (1) the assessment of the need for mobility;(2) the planning stage (time, modes, routes…) and the gathering of the artefact needed during the journey (tickets/car paper, entertainment artefacts ...);

(3) the movement from the origin to the transport gateway/car; (4) the interaction with the transport service (payment, ingress…); (5) the traveling in the vehicle. An additional stage can take place: (6) the interchanges which includes finding the location of the next transport mean, schedule information, buying new tickets…). Finally, the egress from the service to destination concludes the journey. Along the journey, the user of the TUMS, the traveler, gives value to different aspects. [START_REF] Stradling | Performance, importance and user disgruntlement: A six-step method for measuring satisfaction with travel modes[END_REF], [START_REF] Woodcock | Measuring quality across the whole journey[END_REF], and [START_REF] Susilo | Exploring key determinants of travel satisfaction for multi-modal trips by different traveler groups[END_REF]) produced a set of values which are: price, journey and service speed, protection against weather while waiting and traveling, reliability (punctuality and regularity), availability (frequency and stop locations), physical environment, vehicle quality, cleanliness both at stations and on-board, quality on on-board, fellow travelers, seat availability, seat comfort, crowding both at stops and on board, station facilities, information accessibility, safety & security (at stops and on-board), tickets use and buying simplicity, connectivity (network wise and easy transfer). To analyse the journey experience, [START_REF] Susilo | Implementing a Behavioural Pilot Survey for the Stage-based Study of the whole Journey Traveler Experience[END_REF], based on an activity representation of travel as seen above, considers three variables for each activity. The first one is personal doing such as packing belongings, exit home, walk to station, or going across the road for 'leaving home' activity, for example. The second one is personal thinking like thinking over day's schedule, observing of people waiting, or wondering of the total time spent on waiting. The last one is personal feeling such as concern about hygiene in bus, noise of train arriving, or concern over weather. While moving from an origin to a destination, travelers collectively generate a demand on the TUMS which constitutes the offer of this mobility market.

Urban Mobility Market

The World Business Council for Sustainable has proposed a model of the UMS as being a set of three markets Development (WBCSD 2015) :(1) the travel market where spatial-temporal activity of people creates travel patterns, (2) the transport market where travel patterns meet -theoreticaltransport options in a transport patterns, (3) the traffic market, in which transport patterns are confronted with the actual supply of infrastructure and their associated traffic management systems, information systems, etc. The interactions between the TUMS elements produce both the demand for travel within a given area and the provision of transportation services to satisfy this demand [START_REF] Cascetta | Transportation systems analysis[END_REF]. There are many interactions between the components within the transportation system and between the activity system (the set of individual, social, and economic behaviours and interactions that give rise to travel demand) and transportation systems. For example, the level and spatial distribution of travel demand is defined, inter alia, by the location of both households and economic activities. On the other hand, the set of interactions generates feedback cycles. For example, travelers who individually choose the most efficient (fast and cheap) path within the available means, might collectively congest it and thereby deteriorate the global efficiency (transportation service performance). [START_REF] Hasan | Spatiotemporal Patterns of Urban Human Mobility[END_REF] gave more attention to the travel demand in its spatial dimension. Their approach covers all population displacement in the physical space, regardless of the duration and distance of travel, the means used, their causes and consequences. Mobility is represented as a map of passenger concentration in an urban space (mobility patterns). It gives insight of the most frequented places by profiles of people (e.g. using cars or public transport) or of all citizens together. Furthermore, these mobility patterns can be combined to actual city maps and explain the noticed concentrations. In summary, demand is characterized by attitudes and cultural background of travelers on one hand, and, by spatial distribution of social and economic activities on the other hand.

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Considering the supply and demand separately in representing the urban mobility has given some information about their interactions. However, this perspective does not take into account neither the effect of legal and political dimensions nor the role of individuals and institutions in operating urban mobility.

Socio-technical Urban Mobility

According to [START_REF] Auvinen | Future transport systems: Long-term visions and socio-technical transitions[END_REF], technological, social, economic, political, legal or environmental dimensions need to be considered in order to understand the complexity of urban mobility. They define the UMS as set of four main components: the infrastructure, the vehicles, the users and the governance. For example, from the environmental perspective, the infrastructure offering smart electricity grids and charging stations for cars and buses permits the development of emission-free and silent electrical fleets. Together with political support and standardization, this encourages responsible modal choice from users and finally generates a clean transport environment. For instance, people buying electric cars create a demand on charging stations and encourage the creation of new ones. However, if there is no charging station supply, people will not have the opportunity to buy electric cars. These loops permit the propagation of social values trough the global urban mobility. In a wider perspective, the UMS is a component of a larger urban system and interacts with e.g. energy systems and social structures. Hospitals and workplaces, for example, by the practice of telemedicine and teleworking, decrease the need for mobility, and consequently the transportation energy consumption. However, these new practices need involvement of people, commitment of companies, and the adequate technological and legal measures. For (Ottens et al. 2005), the main components of the UMS as a socio-technical system are: technical elements, social elements and actors. Technical elements include all physical components and the software to operate those, the actors are individuals or organizations that are directly running the system, and the social elements influence the functioning the UMS. Beyond the functional relations (e.g. bus providing information to bus station) and the physical relations (vehicles driving on roads), there are intentional and normative interactions. The intentional interactions are performed by actors where other elements are the object of their intention to take an action (e.g. a passenger has the intention to use a bike between metro station and work). The normative interactions represent rules for running a technical element or an actor, e.g. a public transport operator obliges passengers to have valid tickets. Thus, from the socio-technical perspective, the traveler is a part of the UMS and involved in its operation as a consumer and as an actor.

Challenges of designing Urban Mobility

Urban mobility is a complex system where the users play an important role defining its dynamic and intervening in its performance. Starting from the technical dimension of the UMS, introducing the user, then a market view including both, finishing with a socio-technical integration of all UMS's components, several complexity factors have been identified (Table 1). Attempts have been made to tackle the complexity factors by some research works in the design community. Trying to model a bike sharing Product Service Systems (PSS) through use cases, [START_REF] Hollauer | A Model to Describe Use Phase of Socio-Technical Sphere of Product-Service Systems[END_REF] introduced: stakeholders, system's goals and stakeholders objectives, functions and sub-functions, infrastructure, hardware and software, interactions, and cycles. They faced challenges such as defining the relevant level of details, the expanse of stakeholders' integration in the system, or picking general key performance indicators (KPI) for the PSS. In order to compare different electric vehicle (EV) technologies, [START_REF] Barbieri | Investigation on Optimal Mobility System using Axiomatic Design and Scoring Matrix: The "drive Ability" Experiment[END_REF] used scoring matrix and axiomatic design. They defined multiple variables related to the technology used in the vehicle (e.g. feasibility, upgradability) and variables related to the users such as satisfaction and delighters. They were challenged by the qualitative nature of user's variables. For example, recruiting the appropriate sample (in size and nature) brought out questions such as: how likely would the respondents use an EV, how many, from which geographical area etc. [START_REF] Vidal | Impact indicators in transport infrastructures: A new market for engineering design researchers[END_REF] proposed to apply engineering design methods such as life cycle assessment, life cycle cost and risk analysis in order to develop transportation infrastructure sustainability KPIs. They faced an issue in defining the boundaries and therefore the interactions of the infrastructure with the other TUMS components.

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To summarize, issues of sampling, scaling, setting performance indicators, gathering and analysing qualitative data, involving stakeholders, and setting the boundaries of the system to design are not obvious when it comes to design a system at the scale of a city which is anchored in the urban life. In order to take into account these complexity consequences on design practice, a conceptual proposition to scope the design urban mobility system is proposed in next section. The Global performance of a transportation system depends on modal synchronicity (Primerano et al.

2008) Urban context

The functions of a transportation system depend on other urban systems [START_REF] Wegener | The future of mobility in cities: Challenges for urban modelling[END_REF])

User

Experience

The traveler interacts with a large number of different products and services along his travel (Woodcock, Osmond, et al.

2014) Mobility Value

The traveler values different aspects of mobility

KEY CONCEPTS FOR URBAN MOBILITY DESIGN

In order to help designers handle the complexity of the UMS and overcome the challenges, this section proposes a conceptualization that defines urban mobility design entities. A conceptualization is an implicit form of ontology. It is an abstract and a simplified view of the world, as a knowledge base, for some purpose [START_REF] Gruber | A translation approach to portable ontology specifications[END_REF]. The user-centred proposition aims to represent the UMS in order to bring out concepts that concerns the traveler experience. Therefore, the entities considered in the conceptualization are the components in direct interaction with the traveler, the travel experience representation, and the performance indicators from the perspective of the traveler. Based on User Experience design, the model proposed by (Ortíz Nicolás & Aurisicchio 2011) provides a relevant framework to organize the previous entities (Figure 1). The traveler is described by the attributes of physical condition, cognition, affection, motivation, expectation, and personality traits.

Technical Urban Mobility System components

The big picture of the TUMS can be obtained by identifying its components. Therefore, these components should be defined and modelled so that the boundaries of the TUMS are clear. Depending on which scale the TUMS is observed, some components appear or disappear (Figure 2). For example, the road and the parking appear as infrastructures where vehicles move or park, from a helicopter view. Zooming in, other artefacts such as ticket machines or information panels arise. Inside the bus, doors, seats, roll bars, or the line's map become identifiable artefacts. The travel information appear as well, in a smaller scale, on the traveler's smartphone. In the proposed conceptualization, different transportation means and stages are considered. Therefore, simple artefacts like a gear shifter or a smartphone charging spot belong to the TUMS's components. On the other hand, a dynamic schedule screen, as an integrated product (being connected to multiple servers and offering diverse information) also figures among the components. Moreover, it is to note that only components directly interacting with the traveler are considered. The layers behind the functioning of the UMS such as the servers, the underground energy network, or the cooling system of a train motor, are not taken into consideration.

Figure 2. Technical Urban Mobility System's components from different scales

The TUMS boundaries encompass components used to operate mobility activities (e.g. walking, driving, or buying a ticket) or to support travel stages. Therefore, are included: an umbrella protecting the traveler from weather while walking; a device to listen to music or a book used in a train or a bus for example. The directional signage is included as well, as long as it helps find the way while driving or walking. Based on section 2.1, Table 2 shows the decomposition of the TUMS into infrastructure and mobile components. The objective is to bring out the different natures of the technical entities of the TUMS, such that the representation of travel experience takes into account the different scales. This way, no matter the scale, the object of design is connected to the whole UMS and allows the designer to define its functions related to other TUMS components.

Travel Scenarios

The set of components of the TUMS are designed to meet travelers' need for urban mobility generated by their social and economic activities. Therefore, in order to understand how a traveler is using the TUMS for his /her mobility, the interactions between the two need to be modelled.

Imagery ©2016

, Map data ©2016 Google

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Figure 3 shows the observable entities of a travel experience. It includes the TUMS components, the activities of the traveler during the trip, the ones after and before the trip, and the situations that may happen around him/her depending on the urban context. Moreover, the traveler may interact with other fellow travelers, using the same TUMS components or a different one. Rainy day, the tram is delayed, crowd at the train station, bus drivers' strike, slippery ground For example, a traveler walks from home to the bus station (activity). It is a rainy day and the size of the bus shelter cannot cover all the passengers waiting for the bus (situation) (Table 3). The combination of the possible travel activities and situations, to perform a social/economic activity, generates the travel scenarios of the TUMS. It should be noted that a travel activity has, in turn different scales like the TUMS components. For instance, if 'interchange' is considered as an activity, then 'walking in the subway corridors' is one of its sub-activities. Moreover, depending on the TUMS component the activity is made on, the nature of the activity may vary between physical (e.g. walk to bus station), and sensorial/mental (e.g. reading signage).

In order to evaluate how the TUMS is responding to traveler's expectations, a system of indicators is needed, defined as follows in a traveler-centred way.

Traveler centred indicators

Based on the aspects reported in section 2.2 the indicators proposed include relevant aspects to describe how travelers appreciate the travel experience. This travel appreciation is translated here by 'travel value'.

(Table 4) sets a list of the values the TUMS can bring to the traveler. Getting to destination as fast as possible at the lowest price are the classical rational values travelers expect from a transportation system (Spickermann et al. 2013). So, bike sharing service, for example, is one cheap way to move in the city. Post pay pass allows one to pay, at the end of the month, only the trips made instead ICED17 of a fixed fare that might be higher than what have been used. E-tickets are a way to spare the public transportation travelers queues and wasting time at the travel gate buying tickets. Comfort, both physical and sensorial, is valued by travelers. A wheelchair lift is meant to spare the reduced mobility travelers the pain of pushing hard or asking for help to get in the bus. Shelters at tramway or bus stations protect travelers from rain or sun. Art at metro station and tunnels, whether it is music or paintings or photography, enhances the diversity of station's landscape.

Cognitive comfort is necessary to travel, especially regarding information availability. Whether it is about the destination's location, or the schedule of public transport means, or the itinerary, or the real-time of the traffic jam etc. Google Maps, for example, is an integrated tool that provides the major information needed to move from a point A to a point B. Availability, both spatial and temporal, is an important value for travelers. Demand Responsive Transport (DRT) like Uber or taxis are vital when there is no public transport is available, by night for example. Safety and security are important values travelers expect from a mobility solution. A seat belt or a helmet are in this case reinforcing the feeling of safety while protecting against accidents. Finally, components improving travel activities are contributing to the improvement of the whole travel experience. A Wi-Fi hotspot in a train station or in a bus could, for instance, be valuable to travelers. Design research gained expertise upon dealing with artefacts, services, and even product-service systems. This paper proposes a way to consider design challenges that might be faced when dealing with urban mobility at the scale of a city. Several complexity factors have been identified. The diversity and the large number of UMS components and travelers make the understanding of the TUMS a difficult task. Thus, the influence of the users on the TUMS substantiate the relevance of including the traveler in modelling the UMS as a socio-technical system or as a market. The systemic modelling would allow to identify the existing links between UMS's entities of different natures such as stakeholders, a user, and a service. The contribution of the paper is a proposition of interrelated key concepts to set the scope of design for urban mobility (Figure 4): traveler, TUMS components, travel scenarios and travel-centred indicators. The identification of the TUMS components brings out the need to consider their diversity and their large number. Some components are used at the same time (e.g. vehicle on road), some others sequentially (e.g. bus station, bus, then train station). Moreover, one component is used continually by different users. Moreover, including social and economic activities of the travelers allows to enlarge the value proposition of transportation solutions and therefore connect the UMS to other urban systems. For instance, a famous GPS-based geographical navigation application includes a mobile marketing feature. Drivers are suggested shop and restaurants that are more likely to interest him/her depending on his/her driving patterns. Thirdly, the traveler-centred indicators evaluate how the traveler appreciates the experience of using the TUMS. For example, temporal availability of a tram line for a group of travelers during a time interval would be an alternative to the indicator of frequency and regularity. The proposed values are meant to describe if the TUMS components improve traveler experience (and how). Therefore, these need to be linked to the activities, situations and components. In conclusion, the preliminary step tackled in this paper was to provide engineering designers a holistic view of TUMS components. This was achieved through the hybridization of a UX conceptual model with acknowledged mobility concepts. Future work aim at developing an ontology that links components to travel scenarios and to the indicators system, in order to obtain a traveler-centred model of the UMS. The goal of the ontology is to capture the diversity of issues in urban mobility experiences in a need-seeker mode (as developed in [START_REF] Bekhradi | Usefulness Simulation of Design Concepts[END_REF]) for elderly falls for example). Several applicative situations are envisaged. An innovation department in the automotive sector may use the approach to define the coverage ability of new driverless shared solutions over a cluster of usage segments. In a context of market opening for competition, a public metro or train operator may analyse issues of mobility experiences to gain knowledge into determinants of modal choices over usage segments, or foster the design of new integrated offers. Developing innovative products and services that ensure well-being of travelers is indeed a strategic challenge for urban mobility practitioners (industrials, operators, start-ups etc.) that is guiding future developments of this research.

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(

Figure 1 .

 1 Figure 1. User/traveler experience analogy (Ortíz Nicolás & Aurisicchio 2011)

Figure

  Figure 3. The door to door travel experience Table 3. Travel scenarios concepts Concepts Definitions Examples Travel Activities Actions operated by the traveler from the origin to the destination Walking from home's door to the bus shelter, reading a book while waiting for the Metro, driving a car looking for some specific train station exit Social and economic activities Actions operated by the traveler after and before travel Shopping Having a work meeting Watching a movie Situations State of the surrounding of the traveler or events

Figure 4 .

 4 Figure 4. Key concepts for urban mobility designSome limitations have be reported. Firstly, the large number and the diversity of travelers poses the question of what profiles to consider in order to shape the travel scenarios. A classification of travel activities and situations can be useful to capture the variety of travel scenarios and travelers' profiles. This should be further investigated. Secondly, the travel scenarios describe the way the travelers can use the TUMS. However, the combination of situations and activities does not consider what the traveler thinks and feels, contrary to UX design models (Ortíz Nicolás & Aurisicchio 2011). The representation of traveler experience can be enhanced by linking thoughts and emotions to the activities and situations. Thirdly, the traveler-centred indicators evaluate how the traveler appreciates the experience of using the TUMS. For example, temporal availability of a tram line for a group of travelers during a time interval would be an alternative to the indicator of frequency and regularity. The proposed values are meant to describe if the TUMS components improve traveler experience (and how). Therefore, these need to be linked to the activities, situations and components. In conclusion, the preliminary step tackled in this paper was to provide engineering designers a holistic view of TUMS components. This was achieved through the hybridization of a UX conceptual model with acknowledged mobility concepts. Future work aim at developing an ontology that links components to travel scenarios and to the indicators system, in order to obtain a traveler-centred model of the UMS. The goal of the ontology is to capture the diversity of issues in urban mobility experiences in a need-seeker mode (as developed in[START_REF] Bekhradi | Usefulness Simulation of Design Concepts[END_REF]) for elderly falls for example). Several applicative situations are envisaged. An innovation department in the automotive sector may use the approach to define the coverage ability of new driverless shared solutions over a cluster of usage segments. In a context of market opening for competition, a public metro or train operator may analyse issues of mobility experiences to gain knowledge into determinants of modal choices over usage segments, or foster the design of new integrated offers. Developing innovative products and services that ensure well-being of travelers is indeed a strategic challenge for urban mobility practitioners (industrials, operators, start-ups etc.) that is guiding future developments of this research.

Table 1

 1 Complexity factors of the urban mobility system

	Domain	Perspective	Complexity Factors	References
	Technical Multi-modality

Table 2 .

 2 Components of Technical urban mobility system

	Components	Sub-components

Table 4 .

 4 Traveler centred urban mobility performance system

	Value

ACKNOWLEDGMENTS

This research work has been carried out in the framework of the Technological Research Institute SystemX, and therefore granted with public funds within the scope of the French Program "Investissements d'Avenir".


Rodrigue Desmorat

Fabrice Gatuingt

Frédéric Ragueneau

Thermodynamics framework for robust computations with induced anisotropic damage

Many anisotropic damage models have been proposed for different materials, including concrete. The main drawback of the corresponding analyses is that a large number of material parameters is often introduced, leading to identification difficulties but also to models complexity and associated numerical difficulties. It is also sometimes difficult to ensure the continuity of the stresses if the quasi-unilateral effect of microcracks closure and the dissymmetry tension/compression are represented. One considers here an anisotropic damage model with a restricted number of material parameters (5 including the Young's modulus and Poisson's ratio of the initially isotropic material) and built in the thermodynamics framework. The large dissymmetry tension/compression response of concrete is due to the loading induced damage anisotropy. A non standard thermodynamics framework is used with damage states represented by a symmetric second order tensor and with a damage rate governed by the positive part of the strain tensor. The proof of the positivity of the intrinsic dissipation is given for any damage law ensuring (anisotropic) damage increase -in terms of positive principal values of the damage rate tensor. This extends then to induced anisotropy the isotropic case property of a positive damage rate. Altogether with the fact that the thermodynamics potential can be continuously differentiated, the considered anisotropic damage model allows for robust Finite Element implementation. Both space and time regularizations are used and applied to quasi-static and dynamic cases. Examples on concrete and reinforced concrete structures are given, with the consideration of either nonlocal Mazars criterion or of Mazars criterion regularized with viscous damage.

INTRODUCTION

In standard thermodynamics framework [START_REF] Halphen | Sur les matériaux standards généralisés[END_REF] both the state and the evolution laws derive from potentials, Gibbs free enthalpy ρψ ⋆ written in terms of stress σ σ σ in the present work for the first one, a dissipation or pseudo-dissipation potential for the second one. The elasticity law and the strain energy release rate density Y Y Y , variable associated with D D D, are respectively gained as the derivative of the state potential with respect to the elastic strain tensor ǫ ǫ ǫ e and with respect to the damage tensor D D D. In the standard framework of thermodynamics a pseudodissipation potential quadratic function of Y Y Y is most often considered, the damage law taking the form

Ḋ D D = λ J J J : Y Y Y (1)
with λ a positive multiplier and J J J a positive fourth order tensor, eventually nonlinear function of the thermodynamics variables, so that the dissipation D = Y Y Y : Ḋ D D = λ Y Y Y : J J J : Y Y Y due to the degradation mechanisms remains positive for any kind of loading. The choice of damage laws of the form (1) is very restrictive, for instance concerning the possibility to model induced anisotropy directly driven by the strains or the stresses. But, as one will illustrate on numerical examples, to ensure the positivity of the intrinsic dissipation also has good numerical properties.

NON STANDARD ANISOTROPIC FRAME-

WORK A general form for the strain energy coupled with anisotropic damage has been proposed by Ladevèze [START_REF] Ladevèze | TComputational damage mechanics approach for laminates: identification and comparison with experimental results, chapter Damage Mechanics in Engineering Materials[END_REF][START_REF] Lemaitre | Engineering Damage Mechanics : Ductile, Creep, Fatigue and Brittle Failures[END_REF], as ρψ ⋆ =a 1 tr (H H Hσ σ σH H Hσ σ σ) + a 2 σ σ σ : σ σ σ + a 3 g(D H )(tr σ σ σ) 2 + a 4 (tr σ σ σ) 2

(2) with H H H = (1 1 1 -D D D) -1/2 a symmetric tensor and D H = 1 3 tr D D D, where g(D H ) is a positive increasing function of D H , 1/(1 -ηD H ) for example, and where the a i (a 1 ≥ 0, a 3 ≥ 0) as well as η > 0 are material parameters. The first term of Eq. ( 2) can take different -non equivalent -forms [START_REF] Ladevèze | TComputational damage mechanics approach for laminates: identification and comparison with experimental results, chapter Damage Mechanics in Engineering Materials[END_REF][START_REF] Papa | Anisotropic damage model for the multi-axial static and fatigue behaviour of plain concrete[END_REF][START_REF] Lemaitre | Engineering Damage Mechanics : Ductile, Creep, Fatigue and Brittle Failures[END_REF], a 1 tr H H Hσ σ σ D H H Hσ σ σ D

(3) or a 1 tr (H H Hσ σ σ + H H Hσ σ σ + ) + σ σ σ -: σ σ σ -

or

a 1 tr H H Hσ σ σ D + H H Hσ σ σ D + + σ σ σ D -: σ σ σ D - (5)
allowing in the last case for the mechanical representation of the quasi-unilateral effect of microdefects closure, with σ σ σ + (resp. σ σ σ D + ) a special positive part [START_REF] Ladevèze | TComputational damage mechanics approach for laminates: identification and comparison with experimental results, chapter Damage Mechanics in Engineering Materials[END_REF][START_REF] Desmorat | Quasi-unilateral conditions in anisotropic elasticity[END_REF] built from the eigenvalues and the eigenvectors of H H Hσ σ σ (resp. of H H Hσ σ σ D ) and with . -the negative part in terms of principal components of a tensor. Two examples of strain energy densities are a first one based on a splitting deviatoric / hydrostatic quantities (with E and ν the Young's modulus and Poisson's ratio of the undamaged material),

ρψ ⋆ = 1 + ν 2E tr H H Hσ σ σ D H H Hσ σ σ D + 1 -2ν 6E (tr σ σ σ) 2 1 -ηD H (6)
a second one based on the feature of a constant ν/E ratio,

ρψ ⋆ = 1 2E tr (H H Hσ σ σH H Hσ σ σ) + ν 2E σ σ σ : σ σ σ -(tr σ σ σ) 2 (7)
Concerning the evolution laws, many non standard damage laws can be formulated for induced anisotropy [START_REF] Mazars | The unilateral behavior of damage concrete[END_REF][START_REF] Dragon | Modélisation de l'endommagement par mésofissuration : comportement unilatéral et anisotropie induite[END_REF][START_REF] Lemaitre | Anisotropic damage law of evolution[END_REF][START_REF] Billardon | Creep damage behaviour of a copper alloy on a large temperature range[END_REF], with second order tensorial damage rate proportional • to the positive part of the strain tensor ǫ ǫ ǫ + or to ǫ ǫ ǫ α + with α a damage exponent, • to the absolute value of the plastic strain tensor ǫ ǫ ǫ p , to the positive part of ǫ ǫ ǫ p , or to any linear combination α |ǫ ǫ ǫ p | + (1 -α) ǫ ǫ ǫ p + , • to a power 2s of the stress tensor,

• to a linear combination α σ σ σ 2 + + (1 -α) σ σ σ 2 -(eventually at the power s) where to take α = 1 will lead to the modeling of the unilateral damage effect of no damage growth in compression, and to take 1/2 < α < 1 will lead to the modeling of the quasi-unilateral damage effect of a damage growth in compression smaller than in tension.

Induced damage anisotropy governed by the positive extensions is adapted to quasi-brittle materials as concrete. The other expressions will allow to generalize to induced anisotropy Lemaitre's damage law Ḋ = (Y /S) s ṗ of a damage rate governed by the accumulated plastic strain rate ṗ and enhanced by the strain energy Y = 1 2 ǫ ǫ ǫ e : E E E : ǫ ǫ ǫ e , with E E E Hooke's tensor. As possible generalization, one has

Ḋ D D = Y S s α |ǫ ǫ ǫ p | + (1 -α) ǫ ǫ ǫ p + (8) Ḋ D D = α σ σ σ 2 + + (1 -α) σ σ σ 2 - 2ES s ṗ (9)
where E denotes the Young's modulus and S and s the damage parameters. For more details on ductile damage, refer to [START_REF] Lemaitre | Engineering Damage Mechanics : Ductile, Creep, Fatigue and Brittle Failures[END_REF].

The next question will be wether one automatically can ensure the positvity of the dissipation with the simple feature of a positive (tensorial) damage rate Ḋ D D.

POSITIVITY OF THE INTRINSIC DISSIPA-

TION Previous elastic energy densities can be continuously differentiated as

dρψ ⋆ = ǫ ǫ ǫ e : dσ σ σ + Y Y Y : dD D D (10)
or

dρψ ⋆ = [2a 1 (H H Hσ σ σH H H) + 2a 2 σ σ σ + 2a 3 g(D H ) tr σ σ σ 1 1 1 +2a 4 tr σ σ σ 1 1 1] : dσ σ σ + 2a 1 (σ σ σH H Hσ σ σ) : dH H H + 1 3 a 3 g ′ (D H )(tr σ σ σ) 2 tr dD D D (11)
leading to a dissipation due to damage mechanisms expressed as

D = Y Y Y : Ḋ D D = 2a 1 (σ σ σH H Hσ σ σ) : Ḣ H H + 1 3 a 3 g ′ (D H )(tr σ σ σ) 2 tr Ḋ D D (
D D = 3 J=1 ( Ḋ D D) J is positive so that 1 3 a 3 g ′ (D H )(tr σ σ σ) 2 tr Ḋ D D ≥ 0. It is important to precise that the eigenvalues ( Ḋ D D) J of Ḋ D D
are not the derivatives ḊJ of the eigenvalues of D D D (except in the particular case where D D D and Ḋ D D have the same principal directions), the positivity of the eigenvalues ( Ḋ D D) J nevertheless implies the increase of the eigenvalues of D D D.

Concerning the term 2a 1 (s s sH H Hs s s) : Ḣ H H, note that the expression H H H = (1 1 1 -D D D) -1/2 rewritten in terms of principal components

H J = 1 √ 1 -D J (13)
gives positive increasing eigenvalues H J of tensor H H H which is then also positive and increasing during any damage process. The positivity of the symmetric matrix (s s sH H Hs s s) is gained by seeking the sign of its eigenvalues, denoted χ, solution of (s s sH H Hs s s) g = χ g, with g the corresponding eigenvectors. The eigenvalues χ are equivalently solution of

(H H Hs s s) 2 g = χH H H g (14)
with obviously (H H Hs s s) 2 a positive matrix. These eigenvalues take the form

χ = g T (H H Hs s s) 2 g g T H H H g (15)
and, as ratio of positive terms, are positive. Last, the tensorial product of two symmetric positive tensors (s s sH H Hs s s) and Ḣ H H being positive, one can conclude to the positivity of the intrinsic dissipation D for any damaging loading, monotonic or not, uniaxial or multiaxial, proportional or non proportional... at the simple condition extended here to anisotropic damage that the damage rate Ḋ D D must remain a positive tensor. Considering conversely the set of states represented by deviatoric tensors σ σ σ = σ σ σ D , the dissipation reduced to 2a 1 (s s sH H Hs s s) : Ḣ H H ≥ 0 ∀ (s s sH H Hs s s) ≥ 0 0 0 leads to Ḣ H H ≥ 0 0 0, therefore to the fact that Ḋ D D ≥ 0 0 0 is a necessary and sufficient condition for the positivity of the dissipation due to damage

D = Y Y Y : Ḋ D D.
4 ANISOTROPIC DAMAGE MODEL FOR CON-CRETE An anisotropic damage model has been proposed for concrete in previous non standard thermodynamics framework [START_REF] Desmorat | 5è journéees du Regroupement Francophone pour la Recherche et la Formation sur le Béton (RF2B)[END_REF][START_REF] Desmorat | Nonlocal anisotropic damage model and related computational aspects for quasi-brittle materials[END_REF] introducing a single damage variable, a second order tensor, as the representation of the damage state due to microcracking. Mainly due to induced anisotropy, the dissymmetric response of concrete in tension and in compression is obtained with a low number of material parameters: 2 for elasticity, 1 as damage threshold, 2 for damage evolution. Mazars strain damage criterion [START_REF] Mazars | Application de la mécanique de l'endommagement au comportement non linéaire et à la rupture du béton de structure[END_REF] is used in this initial model, with the advantage of simplicity for instance at the numerical level.

Induced anisotropic damage model

For concrete, the microcracks due to tension are mainly orthogonal to the loading direction, when the microcracks due to compression are mainly parallel to the loading direction. The damage state has then to be represented by a tensorial variable D D D either a fourth rank tensor or a second rank tensor. The use of a second order damage tensor is more convenient for practical applications (as well as for the material parameters identification) and this is the choice made here. The damage anisotropy induced by either tension or compression is simply modeled by the consideration of damage evolution laws ensuring a damage rate proportional to the positive part of the strain tensor, i.e. a damage governed by the principal extensions [START_REF] Mazars | The unilateral behavior of damage concrete[END_REF].

The full set of constitutive equations reads

• Elasticity,

ǫ ǫ ǫ = 1 + ν E σ σ σ - ν E tr σ σ σ 1 1 1 or ǫ ǫ ǫ = E E E -1 : σ σ σ (16) • Effective stress, σ σ σ = (1 1 1 -D D D) -1/2 σ σ σ D (1 1 1 -D D D) -1/2 D + 1 3 tr σ σ σ + 1 -tr D D D + tr σ σ σ -1 1 1 (17)
• Damage criterion f = ǫ -κ(tr D D D), so that the condition f < 0 -→ elastic loading or unloading, f = 0, ḟ = 0 -→ damage growth, where ǫ = ǫ ǫ ǫ + : ǫ ǫ ǫ + is Mazars equivalent strain and where κ -1 (ǫ) = aA arctan ǫ a -arctan κ 0 a (18) introducing κ 0 as damage threshold, A and a as damage parameters.

• Induced damage anisotropy governed by the positive extensions,

Ḋ D D = λ ǫ ǫ ǫ 2 + (19)
The damage multiplier λ is determined from the consistency condition f = 0, ḟ = 0.

The use of a damage criterion function f written in terms of strains instead of stresses allows for a simple implementation in a Finite Element computer code [START_REF] Desmorat | Nonlocal anisotropic damage model and related computational aspects for quasi-brittle materials[END_REF]. Even if Euler backward scheme is used, there is no need of an iterative process. Note that at the final stage of numerical implementation the elasticity law needs to be inverted. This can be done in a closed form as:

σ σ σ =(1 1 1 -D D D) 1/2 σ σ σ (1 1 1 -D D D) 1/2 - (1 1 1 -D D D) : σ σ σ 3 -tr D D D (1 1 1 -D D D) + 1 3 [(1 -tr D D D) tr σ σ σ + + tr σ σ σ -]1 1 1 (20)

Extension to nonlocal

Classical mesh dependency occurs when using previous local damage model in a Finite Element code. A nonlocal regularization can be used to gain the mesh independency. One just has to replace local Mazars strain ǫ in the damage criterion f by nonlocal Mazars strain ǫnl and to consider as nonlocal criterion,

f = ǫnl -κ (21)
where nonlocal equivalent strain can be defined using an integral form with W a nonlocal weight function [START_REF] Pijaudier-Cabot | Nonlocal damage theory[END_REF],

ǫnl =ǫ nl (x) = 1 V r Ω W(x -s) ǫ(s) ds V r =V r (x) = Ω W(x -s)ds (22)
or using a second gradient form [START_REF] Aifantis | The physics of plastic deformation[END_REF][START_REF] Peerlings | Gradient-enhanced damage model for quasibrittle materials[END_REF],

ǫnl -c ∇ 2 ǫnl = ǫ (23)
Both the integral form (through W) and the gradient form (through c) introduce a characteristic length l c . Even if induced anisotropy is considered next, the introduction of a single (isotropic) internal length will prove sufficient for practical applications.

VISCOUS REGULARIZATION FOR IMPACT

A regularization possibility for fast dynamics and impact consists in taking into account the strain rate effect on the dynamic response of concrete. For instance, introduce a characteristic time which, altogether with the consideration of the laws of dynamics, will indirectly defines a characteristic length. In the present case of elasticity coupled with damage this can simply be done by introducing a viscosity law ǫ v = ǫ v ( Ḋ) in Mazars criterion. The damage evolution occurs not anymore at f = 0 but at f = ǫ v > 0. A classical law for isotropic damage is Norton-Perzyna power law, ǫ v = k Ḋ1/n , with k and n the viscosity parameters. It leads to an unbounded damage rate often too high at high strain rates.

It is possible to bound the damage rate, for instance by the maximum rate Ḋ∞ = 1/τ c material dependent equal to the inverse of the characteristic time τ c [START_REF] Allix | Delay-damage modelling for fracture prediction of laminated composites under dynamic loading[END_REF][START_REF] Ladevèze | TComputational damage mechanics approach for laminates: identification and comparison with experimental results, chapter Damage Mechanics in Engineering Materials[END_REF]. To gain this property, these authors rewrite the criterion surface as f = g(ǫ) -D (with g = κ -1 ) and define the viscosity law as

f = D v > 0 D v = - 1 b ln Ḋ∞ - Ḋ Ḋ∞ (24)
from which derives the delay-damage law, saturating at high strain rates,

Ḋ = Ḋ∞ [1 -exp (-b(g(ǫ) -D))] (25)
The viscosity parameters, material dependent, are then Ḋ∞ and b. This regularization is defined locally (i.e. at a structure Gauss point) and is well adapted for dynamics computations. It can be extended to the case of induced anisotropic damage by setting [START_REF] Desmorat | Anisotropic 3d delaydamage model for concrete[END_REF]:

tr Ḋ D D = Ḋ∞ 1 -exp -b κ -1 (ǫ) -tr D D D (26)
The full set of constitutive equations now reads

• Elasticity,

ǫ ǫ ǫ = 1 + ν E σ σ σ - ν E tr σ σ σ 1 1 1 or ǫ ǫ ǫ = E E E -1 : σ σ σ (27)
• Effective stress,

σ σ σ = (1 1 1 -D D D) -1/2 σ σ σ D (1 1 1 -D D D) -1/2 D + 1 3 tr σ σ σ + 1 -tr D D D + tr σ σ σ -1 1 1 (28) • Damage criterion (local) f = κ -1 (ǫ) -tr D D D, us- ing the viscous regularization (24), f ≤ 0 -→ elastic loading or unloading f > 0 with f = - 1 b ln Ḋ∞ -tr Ḋ D D Ḋ∞ -→ damage growth (29)
with Ḋ∞ and b the delay-damage parameters.

• Induced damage anisotropy governed by the positive extensions,

Ḋ D D = λ ǫ ǫ ǫ 2 + (30)
The damage multiplier λ is determined from the damage criterion expression for f > 0 (Eq. 29).

The delay-damage law ( 25) is recovered from previous equations. Again, the numerical implementation in a Finite Elements computer code is quite simple as -once again -it does not need an iterative process [START_REF] Desmorat | Anisotropic 3d delaydamage model for concrete[END_REF].

STRUCTURAL EXAMPLES

The local and nonlocal integral anisotropic damage model has been implemented in CEA CAST3M Finite Element code. The anisotropic delay-damage model has been implemented in the explicit code LS-Dyna. The numerical scheme for the time integration is for both cases Euler's backward scheme but solved explicitly [START_REF] Desmorat | Anisotropic 3d delaydamage model for concrete[END_REF][START_REF] Desmorat | Nonlocal anisotropic damage model and related computational aspects for quasi-brittle materials[END_REF]. Finite Element examples on 3D structures are given next to illustrate the model capabilities.

Plain concrete mixed-mode fracture

The double edge notched specimen tested by Nooru-Mohamed (Nooru-Mohamed, 1992) is analysed using the implementation of the model developed in the previous section. The specimen is a symmetric 200 mm× 200 mm mortar square with two horizontal notches, 30 mm long and 5 mm thick.

The rotation of the external boundary of the plate is restricted around the vertical axis. The concrete specimen is first loaded by an increasing shear (lateral) force F (t) applied on the lateral surface. During the application of the shear force, the vertical displacement of the upper surface is totally free. In a second time, a vertical displacement U(t) is applied up to failure at constant F = F M ax , the higher F M ax the more curved the crack path.

The case study is here carried out for a lateral load F M ax = 10 kN. The FE discretization of the specimen is made by the use of four node tetrahedron elements with one integration point. In order to perform the computations in 3D at reasonable cost, a FE mesh with a 5 mm width is used when the real width of the specimen is 50 mm. The model parameters used for the simulation are: E = 42000 MPa, ν = 0.2, κ 0 = 5 10 -5 , A = 5 10 3 , a = 2.93 10 -4 . The nonlocal length used in the integral weight function is l c = 2 mm, small value indeed justified by the fact that the material is a mortar with very small constituents [START_REF] Bazant | Measurement of characteristic length of nonlocal continuum[END_REF][START_REF] Ragueneau | Mechanical behavior related to continuum damage mechanics for concrete[END_REF]. Three meshes are used: a coarse mesh with a total of 4936 elements, a medium mesh with 11294 elements, and a fine mesh with 14766 elements. The characteristic length corresponds then close to the notch to 2 elements of the coarse mesh, to 8 elements of the medium mesh, to 14 elements of the fine mesh.

Figure 1 shows the anisotropic damage patterns computed for the different meshes (at the figure top is the cracking pattern experimentally observed [START_REF] Nooru-Mohamed | Mixed-mode fracture of concrete: An experimental approach[END_REF]). The left column corresponds to the D 22 damage field, the right one to the D 11 damage field, both in nonlocal computations. They exhibit the now classical convergence and mesh independence of the results obtained with a nonlocal model, here in case of anisotropic damage. The application of the shear load up to F M ax yields localized damage at the notch tip. The structural failure is then due to the application of the vertical displacement U(t) with mainly mode I cracks represented here in the Continuum Damage Mechanics framework by large D 22 values. The damage patterns computed corresponds to the crack patterns observed with the rotation of two main cracks represented. The cracks are not perfectly symmetric with respect to the center of the specimen due to the application of the experimental boundary conditions.

Reinforced concrete structure

The objectives of this section are to evaluate the robustness and the ability of the anisotropic damage model to deal with a reinforced concrete element subject to flexion. The structure is a reinforced square cross section beam, subject to three point bend loading. Figure 2 shows geometric features for concrete and steel. During loading, multiple loading paths are encountered in different parts of the beam: tension on the lower part, compression on the upper part, shear near the edge and along the reinforcing bars. The corresponding different features of the constitutive equations are activated at the same time and the occurrence during loading of several competitive cracks usually makes difficult the global convergence scheme. For these reasons, this case-study was part of the international MECA benchmark, launched by E.D.F. to compare and discriminate different 3D constitutive models for concrete [START_REF] Delaplace | Modèles de fissuration de béton -projet MECA[END_REF].

For concrete, the material parameters used in the following computations are those of previous section. For steel, elasto-plasticity with linear hardening is considered and the material parameters are imposed by the benchmark: Young's modulus E = 200000 MPa, Poisson ratio ν = 0.3, yield stress of 480 MPa, plastic modulus of 20000 MPa. For the computation, a 3D specimen has been meshed with 2 elements in the thickness for a total of 600 eight node parallelepipedic elements. Accounting for the different symmetries of the problem, only one reinforcing steel bar is modelled. The mean dimension of the finite element size is 50 mm.

The monotonic loading is applied up to failure. Two computations with two different characteristic lengths are performed in order to appreciate the effect of the nonlocal length (Gaussian weight function, l c = 150 The choice l c = 150 mm is the more physical as it corresponds here to a characteristic length equal to 3 or 4 times the maximum aggregate size [START_REF] Bazant | Measurement of characteristic length of nonlocal continuum[END_REF].

As one can see in the D 11 -damage maps given in Figure 3, due to the reinforcing steel bar implying a flexural rupture the effect of the characteristic length is important on the cracking pattern as it can be seen from damage maps. The choice l c = 150 mm is the more appropriate here, as expected. The computation represents quite well the multiple cracks propagation for the characteristic length l c = 150 mm.

3D dynamics tension test

In order to get tensile results at very high strain rates, tensile tests by scabbing were developed [START_REF] Klepaczko | An experimental method for dynamic tensile testing of concrete by spalling[END_REF][START_REF] Schuler | Spall experiments for the measurement of the tensile strength and fracture energy of concrete at high strain rates[END_REF]. Figure 4 shows the principle of the test. The setup consists of a striker (launched at the velocity V), an input bar and the tested specimen. The input bar of [START_REF] Klepaczko | An experimental method for dynamic tensile testing of concrete by spalling[END_REF] experiment has a diameter of 40 mm for a one meter length, while the concrete sample has the same diameter for a length of 120 mm. After the impact of the striker, an incident wave propagates in the input bar. One part of the wave is transmitted into the specimen and another one is reflected at the bar/specimen interface. The transmitted compression wave is reflected at the free end and becomes a tensile wave. This leads to fracture in the spall plane.

Three Finite Element meshes have been used. A coarse mesh is made of 24000 underintegrated 8nodes brick elements, a medium mesh of 48000 elements and a fine mesh of 96000 elements. At time t = 0 the mesh boundaries are free and the experimental pressure wave is applied on the right face of the specimen. The simulation of the test must make it possible to find the rupture of the sample experimentally observed, i.e. a single main rupture crack at the distance X = 65.8 mm of the impacted face.

Figure 5 shows the damage field D 11 associated with the axial axe e 1 . When the material is subjected to compression the imposed strain is not sufficient to damage the material. To the opposite, when the state of tension becomes sufficiently large after the compressive wave reflection on the free surface, one obtains a damage D 11 close to 1 in a cross section. This cross section, which represents the location in which cracking will be initiated, is located at a distance X ranging between 64 and 69 mm from the impacted face. This measure is in good agreement with experimental one.

In order to illustrate the mesh independency (due

Impact on a reinforced concrete slab

In order to evaluate the ability of the anisotropic damage model to describe the concrete behavior in a case rather complex but representative of an industrial application, a test in which a projectile impacts a concrete slab has been carried out. The projectile is a cylinder representative of a Cessna engine (masse=200 kg, velocity=83,3 m/s, cross section=1 m 2 ) with an elastic behavior.

Figure 6 shows the finite element mesh used for the simulations on a 4 meters width and 0.5 meters thick slab. The slab is meshed with 24000 3D underintegrated elements and the reinforcements are repre- nomenon. In our simulation, damages D 11 and D 22 are quite large exhibiting a shear rupture of the concrete slab with the apparition of a punch cone as experimentally observed in cases of thin slabs. In the same time, the damage D 33 remains small and does not exhibit scabbing.

CONCLUSION

To conclude, a non standard thermodynamics framework for induced anisotropic damage guaranties the positivity of the intrinsic dissipation. A proof is given and the scalar feature Ḋ ≥ 0 of a positive damage rate for isotropic modeling is extended to anisotropy as the simple feature Ḋ D D ≥ 0 of a positive damage rate tensor. The convexity of the state potential with respect to the damage variable is not necessary.

An anisotropic damage model is described in previous non standard framework, with a quite low numbers of material parameters (5 including the elasticity parameters for the initial plus an internal length for the nonlocal model or plus 2 viscosity paramteers for the delay-damage model). The good mathematical properties of the model (differentiability of the state potential, positivity of the dissipation due to anisotropic damage even for complex non proportional loading) prove to be efficient for numerical computations of 3D structures, in both quasi-static and dynamic cases.

Figure 1 :Figure 2 :

 12 Figure 1: Damage maps for Nooru-Mohamed test at U = 3.5 10 -3 mm-(a) left column: D 22 fields, (b) right column: D 11 fields

Figure 3 :

 3 Figure 3: Mesh and D 11 damage field obtained at the begining of steels yielding (left : l c = 150 mm, right: l c = 250 mm -left and right correspond to two different computations)

Figure 4 :

 4 Figure 4: Principle of the dynamic tension test Stricker Input bar Sample

Figure

  Figure 6: Finite Element mesh

Figure 7

 7 Figure 7 shows the damages D 11 , D 22 and D 33 into the slab. One can notice that due to the symmetry condition, the damage D 11 and D 22 have a similar pattern. The damage D 33 represents the cracks in the slab thickness and is representative of the scabbing phe-

Figure

  Figure 7: Damage map