text:
Ahmed Hefnawy

Taha Elhariri

Abdelaziz Bouras

Chantal Cherifi

Jeremy Robert

Sylvain Kubler

Kary Frӓmling

Lifecycle Management in the Smart City Context: Smart Parking Use-Case

Keywords: PLM, SLM, CL2M, SoS, Smart City

Lifecycle management enables enterprises to manage their products, services or product-service bundles. IoT and CPS have made products and services smarter by closing the loop of data across different phases of lifecycle. Similarly, CPS and IoT empower cities with real-time decision-making capabilities enabled by real-time data streams from heterogeneous objects. Yet, cities are smarter and more powerful when relevant data can be exchanged between different systems across different domains. From engineering perspective, smart city can be seen as a System of Systems (SoS) composed of interrelated/ interdependent smart systems and objects. To better integrate people, processes, and systems in the smart city ecosystem, this paper discusses the use of PLM/ SLM to better manage smart cities. Considering the differences between ordinary and smart service systems, this paper seeks better understanding of lifecycle aspects in the smart city context. As a proof-ofconcept, the proposed approach is applied to a smart parking use-case.

Introduction

Lifecycle Management is a concept [START_REF] Zhang | PLM components monitoring framework for SMEs based on a PLM maturity model and FAHP methodology[END_REF] that evolved in 1990s to improve several engineering aspects of an enterprise to manage its products across their lifecycles [START_REF] Zhang | Sustainability consideration within Product Lifecycle Management through Maturity Models Analysis[END_REF]. As per J. Li et al. [3], Product Lifecycle Management (PLM) is ideally used to manage the knowledge intensive process consisting mainly of market analysis, product design and process development, product manufacturing, distribution, product in use, post-sale service, and recycling. Despite what its name implies, PLM is not only about manufactured products; J. Stark [START_REF] Stark | Product Lifecycle Management: 21 century paradigm for product realization[END_REF] extends the definition of "product" to include services, package of services or a bundle of products and services. O. Isaksson et al. [START_REF] Isaksson | Development of Product-Service Systems: Challenges and Opportunities for the Manufacturing Firm[END_REF] also see "service" as part of the wider concept of "product".

The transformation from product-oriented to more service-oriented economies is part of a complete "servitization" revolution, with more than 70% of global workers engaged in service tasks [START_REF] Sassanelli | Towards a Lean Product Service Systems (PSS) Design: state of the art, opportunities and challenges[END_REF]. Therefore, traditional product-centric sectors evolve into service-centric sectors in order to meet the new challenges, with the aim to put customers and users at the center of their business models [START_REF] Freitag | An Approach to standardize a Service Life Cycle Management[END_REF]. Through servitization, companies seek unique selling proposition for their products, in which the physical artifact is extended by a surrounding provision of services, thus defining the concept of Product-Service System (PSS) [START_REF] Garetti | Life Cycle Simulation for the design of Product-Service Systems[END_REF]. J. Cassina et al. [START_REF] Cassina | Development of an Extended Product Lifecycle Management through Service Oriented Architecture[END_REF] call PSS as extended product, where the product is a complex result of tangible and intangible components.

The advancement of ICT and the evolve of Internet of Things (IoT) and Cyber Physical Systems (CPS) have made ordinary products smarter. D. Kiritsis [START_REF] Kiritsis | Closed-loop PLM for intelligent products in the era of the internet of things[END_REF] argues that smart products allow monitoring new parameters of the product and its environment along different phases of lifecycle. In this regard, the PROMISE, an EUfunded FP6 project, provides a mean to transfer critical information about a product back and forward across the lifecycle, and O-MI/ O-DF standards emerged out of this project [START_REF] Främling | Universal Messaging Standards for the IoT from a Lifecycle Management Perspective[END_REF]. Similarly, IoT and CPS have an enabling role in public services in the city environment, and can exist in many forms [START_REF] Hefnawy | Lifecycle Based Modeling of Smart City Ecosystem[END_REF]. The simplest form of CPS is the form of single objects, like sensors and actuators that collect data and execute commands respectively. CPS can also be in the form of smart systems that address domain-specific issues, like transportation, parking, energy, lightening, etc.

As it was proposed in previous research in [START_REF] Hefnawy | Lifecycle Based Modeling of Smart City Ecosystem[END_REF], [START_REF] Hefnawy | IoT for Smart City Services: Lifecycle Approach[END_REF] and [START_REF] Hefnawy | Integration of Smart City and Lifecycle Concepts for Enhanced large-scale Event Management[END_REF], and in line with ambitions of many cities and states around the world, there is a need for a more holistic vision of smart city as a complete ecosystem. This paper carries on the proposed lifecycle approach to ensure systematic involvement and seamless flow of information between different stakeholders of the smart city ecosystem. Nevertheless, this holistic vision of smart city implies interrelations and interdependence between multiple smart systems that in many cases are independently developed, operated and managed [START_REF] Lopes | Service Systems Engineering Applications[END_REF]. Hence this paper proposes a step further to extend lifecycle functionalities to smart cities, in order to exchange not only generated data but also system data, versions, variants and business processes. This research aims to understand some lifecycle aspects in the smart city context, considering some features like heterogeneity of data sources, interdependence between smart systems and integration between cyber and physical components.

The remainder of this paper consists of four sections. Section 2 includes related work of different types of lifecycle management. Section 3 projects lifecycle management aspects on smart city systems and explains the proposed meaning of different lifecycle components and functionalities in the smart city context. Section 4 demonstrates the lifecycle approach in a smart parking use-case. Section 5 includes discussion of this paper and the proposed future work.

Related Work

The term Lifecycle management has been mostly associated with "product", in Product Lifecycle Management (PLM) and "service", in Service Lifecycle Management (SLM). In addition, Lifecycle management is a very crucial aspect of Product-Service Systems (PSS), in which a manufacturing company sets its market proposition on extending the traditional functionality of its products by incorporating additional services for reaching new market competitive advantages [START_REF] Sassanelli | Towards a Lean Product Service Systems (PSS) Design: state of the art, opportunities and challenges[END_REF]. PLM. A product is defined as an output that results from a process. Products can be tangible or intangible, a thing or an idea, hardware or software, information or knowledge, a process or procedure, a service or function, or a concept or creation (ISO 9001:2000) [START_REF] Isaksson | Development of Product-Service Systems: Challenges and Opportunities for the Manufacturing Firm[END_REF]. PLM is commonly referred to as a strategic approach that incorporates the management of data associated with products of a particular type, and perhaps the versions and variants of that product type, as well as the business processes that surround it [START_REF] Främling | Universal Messaging Standards for the IoT from a Lifecycle Management Perspective[END_REF]. PLM has three main phases [START_REF] Zhang | Sustainability consideration within Product Lifecycle Management through Maturity Models Analysis[END_REF]: Beginning of Life (BOL), Middle of Life (MOL), and End of Life (EOL). BOL is the period in which product concept is generated, designed, and subsequently physically realized. MOL is the period when products are distributed, used, and maintained by customers or engineers. EOL is the period when products are recycled by manufacturers or disposed [3].

SLM. Service can be a product in itself, when it comes under the ISO definition of product. C. Sassanelli et al. [START_REF] Sassanelli | Towards a Lean Product Service Systems (PSS) Design: state of the art, opportunities and challenges[END_REF] define "service" as an activity done for others with an economic value and often done on a commercial basis. SLM is conceptually similar to PLM, however it manages the lifecycle of services instead of tangible products. SLM can be characterized by the same three main phases, like PLM: BOL, MOL, and EOL [START_REF] Mahut | Servicization of Product Lifecycle Management: towards Service Lifecycle Management[END_REF] [START_REF] Freitag | Requirements for a Service Lifecycle Management[END_REF]. BOL consists of service ideation, definition, design, test and service system implementation. MOL consists of service delivery and service system maintenance. EOL consists of service redesign and service system decommission.

PSS. C. Sassanelli et al. [START_REF] Sassanelli | Towards a Lean Product Service Systems (PSS) Design: state of the art, opportunities and challenges[END_REF] define "system" as a collection of elements including their relations. As part of the servitization trend, manufacturing companies extend their traditional products by incorporating additional services. This approach supports the development of service-oriented sectors, switching the emphasis from the "sale of products" to the "sale of use" and reshaping the same concept of customer values, from "possession" to "utilization" [START_REF] Sassanelli | Towards a Lean Product Service Systems (PSS) Design: state of the art, opportunities and challenges[END_REF] [START_REF] Schmidt | Customer-oriented Framework for Product-Service Systems". 7th Industrial Product-Service Systems Conference -PSS, industry transformation for sustainability and business[END_REF]. From business perspective, the traditional boundaries between manufacturing and services are blurring. Additionally, business models have dramatically changed. Manufacturers provide and guarantee functions/ solutions instead of products; ownership stays with manufacturers; hence, efficient use, maintenance and repair, in MOL, are becoming prevailing in the value chain [START_REF] Isaksson | Development of Product-Service Systems: Challenges and Opportunities for the Manufacturing Firm[END_REF].

CL2M. The objective of Closed-Loop Lifecycle Management is to allow information flow management to go beyond the customer, to close the product lifecycle information loops, and to enable seamless e-transformation of product lifecycle information to knowledge [START_REF] Kiritsis | Closed-loop PLM for intelligent products in the era of the internet of things[END_REF]. D. Kiritsis argues that a closed-loop PLM system allows all the actors who play a role during the lifecycle of a product (managers, designers, service and maintenance operators, recyclers, etc.) to track, manage and control product information at any phase of its lifecycle, at any time and any place [START_REF] Kiritsis | Closed-loop PLM for intelligent products in the era of the internet of things[END_REF]. As per K. Främling et al. [START_REF] Främling | Universal Messaging Standards for the IoT from a Lifecycle Management Perspective[END_REF], CL2M contribute to enhancing various aspects of PLM, particularly the following: (1) information manageability: to efficiently and properly process large amount of data; (2) information interoperability: to manage the many changes in data media and formats and to ensure information exchanges between any kind of products, users, and systems; (3) information visibility: to make data available for any system, anywhere, and at any time.

Smart City Lifecycle Management

Smart city is a composition of smart objects, smart systems, and smart services that focus on problems and issues that arise in service sectors, like transport, logistics, energy, waste management [START_REF] Cavalieri | Product-Service Systems Engineering: State of the art and research challenges[END_REF] [START_REF] Poncela | Smart Cities via Data Aggregation[END_REF]. Yet, smart city as a complete ecosystem goes beyond conventional product systems, service systems or PSS [START_REF] Jin | An Information Framework for Creating a Smart City through Internet of Things[END_REF] [START_REF] Sum | Service Systems Engineering: Framework & Systems Modeling[END_REF]; and hence, Smart City Lifecycle Management (SCLM) has some differences from ordinary PLM/ SLM, in terms of components, description and functionalities. As proposed in [START_REF] Hefnawy | Integration of Smart City and Lifecycle Concepts for Enhanced large-scale Event Management[END_REF] and shown in Fig. 1, Lifecycle Management can be used in the smart city context to manage data, versions, variants and business processes associated with heterogeneous, uniquely identified connected objects. Smart City as a SoS. Smart city is a SoS, where individual, heterogeneous, functional systems are linked together and organized in a hierarchy of subsystems to realize new features/ functionalities [START_REF] Lopes | Service Systems Engineering Applications[END_REF] [START_REF] Freitag | Requirements for a Service Lifecycle Management[END_REF]. For example, A. Medvedev et al. [START_REF] Jin | An Information Framework for Creating a Smart City through Internet of Things[END_REF], propose a smart waste collection system that enable dynamic scheduling and routing of waste trucks. The proposed system features data exchange between waste management, surveillance/ monitoring and transportation/ routing smart systems. Another example, J. Poncela, et al. [START_REF] Poncela | Smart Cities via Data Aggregation[END_REF] propose a CCTV camera video stream to feed to a video processing algorithm that extracts information such as numbers of cars/people/objects in a given street. Authors propose a middleware layer for selection and discovery of the appropriate data sources.

SCLM phases. To allow evolutionary development of smart city, in most cases, smart city is composed of independently developed, operated and managed service systems. Therefore, SCLM has no clear phases similar to PLM/ SLM; instead, each component of the smart city has its own lifecycle; and, smart city components can be at different phases -BOL, MOL and EOL -in the same time. Therefore, the lifecycle of smart city is in fact a lifecycle of lifecycles [START_REF] Bkcase Editorial | The Guide to the Systems Engineering Body of Knowledge (SEBoK)[END_REF].

Bill of Materials(BOM).

BOM is a hierarchical structure showing the components that make up the end item [START_REF] Hefnawy | Integration of Smart City and Lifecycle Concepts for Enhanced large-scale Event Management[END_REF]. The end item in this case can be a smart city service system or a smart city SoS. In the smart city context, smart objects can be repurposed and reused [START_REF] Tao | Internet of Things and BOM-Based Life Cycle Assessment of Energy-Saving and Emission-Reduction of Products[END_REF]. As shown in the above examples, monitoring cameras were used for smart waste management as well as smart transportation, although the initial purpose is to be used in surveillance system. Therefore, BOM in the smart city context should allow for loose-coupling, modularity, composability, scalability interdependency and dynamic complexity [START_REF] Böhmann | Service Systems Engineering[END_REF] [START_REF] Silcher | A Service-Based Approach for Next-Generation Product Lifecycle Management[END_REF].

Product/ Service/ System data. The interdependence between different smart systems in the smart city context, as detailed in hierarchy structure of BOM, gives the right to interdependent systems to exchange product/ service/ system data that should be generated and used across lifecycle phases. The large scale of smart city will produce large amounts of data across lifecycle. Archiving and traceability requirements vary from one industry to another. Unique identification of every component is very crucial for traceability. Product/ Service/ System data can be in various states, including in-work, in-process, in-review, released, as-designed, asplanned, as-built, as-installed, as-maintained, and as operated [START_REF] Hefnawy | Integration of Smart City and Lifecycle Concepts for Enhanced large-scale Event Management[END_REF].

Ownership and Rights. Ownership in the smart city context is an important issue. In light of heterogeneity, repurposing and reusing of data sources, certain components can belong to multiple smart systems. Due to the dynamic complexity of smart city, rights may change during lifecycle. Rights include rights to access, create and modify data, and also rights to approve and promote.

Policies and Regulations. Smart city is subject to many policies and regulations related to the different utilities infrastructure, public services and applications. Cyber security, resiliency of ICT connectivity infrastructure and user data privacy are of absolute importance.

Versions, Variants and Options. During SCLM phases, smart city components can be modified or upgraded. Particularly software components in CPS. Hence, smart city components can have multiple versions, options, variants, releases and alternatives.

Processes. Processes include problem report process, engineering change process and enterprise notification process. For these processes, it's absolutely important to define actors and roles. In the smart city context, processes can include notifications, verifications and approvals between actors from different domains.

Smart Parking: Use-Case

This section carries on the use-case, presented in [START_REF] Hefnawy | IoT for Smart City Services: Lifecycle Approach[END_REF], for smart parking system. This is a hypothetical scenario that was built as a proof-of-concept. The proposed scenario was examined in collaboration with the on-going H2020 project named "bIoTope"0 F 1 to use the O-MI/ O-DF standards to exchange data between different nodes in the proposed smart parking system. Meanwhile, Aras Innovator® was used to examine some PLM functionalities in the proposed case. This paper will only focus on the lifecycle aspect of the smart parking system. As detailed in [START_REF] Hefnawy | IoT for Smart City Services: Lifecycle Approach[END_REF], the proposed smart parking system allocates parking spaces to users, based on the preferred entrance and eligibility to use allocated spots for people with disability. In this paper, we propose to use smart parking systems in FIFA World Cup 2022 stadia in Qatar. The main functions of the proposed system include: booking of parking spaces in-advance through online booking; parking space allocation as close as possible to entrance leading to the booked seat; fast track car entrance through gates that are equipped with plate number reader and only open to eligible cars; another plate number reading at each parking space to alert user in case of parking in a wrong space (not the allocated parking space). Fig. 2 presents the high-level illustration of the proposed smart parking system.

BOM.

To develop the BOM, we detailed a hierarchical structure of the components that make up the smart parking system. The smart parking system was structured in zones (1…n); each zone has one gate equipped with plate number reader; and certain number of parking spaces (1…j), each has its plate number reader. Fig. 3 shows a snapshot from BOM. Aras Innovator® was used for two purposes, first is to build and manage BOM; second is to export BOM in O-DF structure as XML file to build O-DF object tree. Fig. 3 is a screenshot from Aras Innovator®, showing the user interface for smart parking system; and Fig. 4 shows the implementation of the O-DF structure into the smart parking O-MI node which relies on the first reference implementation of O-MI/ O-DF standards. Hence, Aras Innovator® can be used as a master tool to manage all lifecycle aspects including BOM development and changes in case of new versions and/ or variants.

Versions, variants and options. As a step further in the smart parking use-case, this part discusses potential scenarios for new versions, variants and options. For this purpose, we denoted the smart parking system, as explained above, as version (V 1.0). After implementing (V 1.0) in a number of stadia that will host the world cup competitions, it was noticed that some users park in parking spaces different than their allocated ones, disregarding the red light alert. Problem Reports (PRs) were developed by parking zone administrators in three different stadia reporting the same problem. PRs were reviewed and verified by chief of staff in stadia. The manager of smart parking has approved PRs, as per the PR process shown in Fig. 5. As a response to the above mentioned PRs, an Engineering Change Request (ECR) was developed to overcome the violent parking, as per the ECR process shown in Fig. 6. The proposed solution was to add surveillance cameras to monitor violent parking cars in order to file cases against these cars. One stadium has rejected the ECR and decided to keep (V 1.0) smart parking system while dealing with the PR by increasing the number of security personnel who can immediately intervene and request violent cars to use their allocated parking spaces. The second stadium has approved the ECR through a fast track approval. Hence, the smart parking system evolved to version (V 2.0); accordingly, the BOM should be updated by adding 1 camera to each parking zone. The third stadium has approved the ECR and requested to add an option to connect the smart parking system with the traffic department system so that the applicable fine will go directly to the traffic department upon capturing the violent car. Hence, a new variant of smart parking will evolve (V 2.1). due to the relationship with other systems, the ECR in the last scenario should go through the Change Request Board (CRB) approval route that involve all relevant stakeholders.

The Enterprise Change Notice (ECN) is a process by which changes are implemented within the smart parking system, as shown in Fig. 7. The change, in case of (V 2.0) and the variant (V 2.1) is the addition of cameras, as new parts, to all parking zones. The ECN process is used to take the new parts from preliminary lifecycle state to a released lifecycle state. The relevant PRs and ECR can be attached to the ECN for tracking and reporting. Aras Innovator® was used to manage PRs, ECR and ECN processes and update the BOM accordingly.

Discussion and Future Work

As lifecycle management has enabled large enterprises to better manage their products, services and product-service bundles; similarly, lifecycle management can enable city operators to better manage public services and supporting infrastructure. The wide spread of IoT technologies and CPS systems in the city environment closes lifecycle data/ information loops across different phases and between heterogeneous objects/ systems. From engineering perspective, smart city as a service system has some features like: heterogeneity and loose-coupling of data sources; complexity of systems and composability of parts; customer oriented and service based systems. These features require some distinctions between lifecycle management in the smart city context, and PLM and SLM.

The vision of applying lifecycle management in the smart city domain(s) is to better integrate people, processes, and systems; and assure information consistency, traceability, and long-term archiving. To achieve such a holistic vision of complete smart city ecosystem, there is a need for two types of data to be exchanged. First, data collected from heterogeneous data sources that can be used in different domains. Second, system data that include BOM, versions, variants, stats and other lifecycle related data. The conducted proof-of-concept has shown good level of integration between lifecycle management tool and the O-MI node tool to exchange the two types of data. Future work will include expanding the use-case to ensure exchange of the two types of data between different systems in the smart city. Another required effort is to build general smart city BOM that includes as much as possible categories and parts that compose a smart city.

Fig. 1 .

 1 Fig. 1. Smart City: High-Level Conceptual Model [14].

Fig. 2 .

 2 Fig. 2. Smart Parking: High-level Illustration.

Fig. 3 .

 3 Fig. 3. BOM: Screenshot from Aras Innovator®.

Fig. 4 .

 4 Fig. 4. O-DF Object Tree.

Fig. 5 .

 5 Fig. 5. PR process.

Fig. 6 .

 6 Fig. 6. ECR Workflow Process.

Fig. 7 .

 7 Fig. 7. ECN Workflow Process.

http://biotope.cs.hut.fi/


Xingjun Wang
email: xingjun.wang@gmail.com

Olivier Hubert

Song He

Frederick-Sorel Mballa-Mballa

Ying Huang

Reversible magneto-mechanical modeling of heterogeneous media

Keywords: Ferromagnetic materials, DP steels, magneto-mechanical modeling, localization

Localization and homogenization of the magneto-mechanical fields in heterogeneous media are discussed. The magneto-elastic modeling for magnetic polycrystals is extended from the description of the single crystal modeling. This model is then applied to the prediction of the behavior of dual-phase steels. Each phase can be considered as a sphere embedded in a homogeneous equivalent medium. The magnetic model used for each phase is an explicit single crystalline model representative of the behaviour of the corresponding phase. Homogenization rules allow the simulation of the average medium. Experiments are carried out and compared to the numerical results.

Introduction

Non destructive evaluation (NDE) is used in the industries of steel since magnetic behavior demonstrates a good sensitivity to phase fraction of different nature, applied stress, residual stresses and plasticity. Many researchers have devoted themselves to developing higher performance modeling in order to implement inverse identification, ie predict the thermo-metallurgico-mechanic state of a material from a simple magnetic measurement.

On line NDE is particularly an important point, in order to optimize on line parameters of a forming or elaborating process. Models should take on the other hand account for the multiphased state of real materials and for the coupling with mechanical stress. This coupling is characterized by the influence of stress on the magnetic susceptibility and by magnetostriction. The magnetostriction µ ε consists in a dilatation or a contraction depending on the applied field H !!" level and direction [START_REF] Térmolet De Lacheisserie | Magnetostriction-Theory and Applications of Magnetoelasticity[END_REF]. This deformation is sensitive to any change of the material microstructure or applied stress [START_REF] Cullity | Introduction to magnetic materials[END_REF][START_REF] Daniel | [END_REF]. The effect of mechanical stress σ on magnetization M !!" of a material (change in magnetic susceptibility) is a second manifestation of the magneto-mechanical coupling. In case of uniaxial stress, the effect of tensile or compressive stress is usually not symmetric [START_REF] Cullity | Introduction to magnetic materials[END_REF]; it depends on material composition, magnetic field and stress levels. Multiaxiality of stress can on the other hand play a significant role [4]. At higher stress levels, plastic straining occurs, leading to a sharp degradation of the magnetic properties (strong decrease of susceptibility, increase of hysteresis losses) [5] and a change in the magnetostriction amplitude [6]. Classical magnetoelasticity is able to explain these phenomena assuming that the driving force associated to plasticity is the long range internal stresses [6,7].

The modeling of magneto-mechanical phenomena is usually done using an energetic approach where variables are written at an appropriate scale. On the one hand, several macroscopic models have been proposed. The applicability of these approaches is limited to a short range of loadings, isotropic materials and rarely accounted for multiaxial stress. These models do not take account of the microstructure of the material; magneto-mechanical coupling is consequently partially described. The development of micromagnetic or fully multiscale magneto-elastic models is a promising issue [8,9].

Because the number of degrees of freedom and interactions is increasing quickly with the number of magnetic moments, these simulations are always used for small size systems. When a complex heterogeneous material is considered, these models lead to dissuasive computation times. Inverse identification is consequently still not reachable.

As part of Nondestructive testing (NDT), a fast accurate modeling to estimate the magneto-elastic behaviour has become a crucial element to be taken into account. Our objective is to combine the accuracy of micromagnetic modeling in order to describe complex magneto-mechanical phenomena, to the simplicity and speed of macroscopic approaches. Quantitative inverse identification should be reached using few experiments.

The paper is divided into four parts. In the first part, localization and homogenization rules are discussed, which allow to define residual (internal) stress tensor and local demagnetizing field. The second part presents the single-crystalline magnetomechanical model. In the third part, specific numerical implementation for the magnetic field is detailed: a method using Taylor series for heterogeneous field is proposed. In the last part, the proposed model is used to simulate a dual-phase steel-polycrystalline and multiphased medium-, where heterogeneity is associated to the anisotropic behavior of the single crystal and to the nature of the crystal (ie ferrite / martensite). The modeling considers homogeneous magnetic field and strain within the single crystal. Hysteretic phenomena are not considered. The proposed modeling is associated to anhysteretic magnetic field strengthening and can be developed for ferri-or ferromagnetic materials.

Localization and homogenization of the magneto-mechanical fields in

Heterogeneous media

The representative volume element (RVE) is seen as an assembly of volume of interest (VOI). VOI is for example a grain inside a polycrystal or an assembly of grains. The problem will be solved (calculation of magnetic, mechanical and coupled quantities) at the VOI scale. The different VOIs have to communicate with each others. Localization (expression of local loadings as function of global loadings) and homogenization (averaging) operations can be made.

Localization and homogenization of the magnetic fields in heterogeneous media

The aim of this step consists in deriving the local magnetic field I H !!" from the external loading ( magnetic field H, and stress ),

H !! "
postulating a given form for the function h:

( , )

I H h H = ∑ !! " !! " (1)
This equation is usually written (in electrotechnical engineering) in the form

I d H H H = + !! " !! " !! " (2)
where the local perturbation of the macroscopic magnetic field is taken into account through the demagnetising field

d H !!"
. The general form of the localization law can be written:

( )

I I H H M M -= - !! " !! " !!" !!" !! " !! " !!" !!" !! " !! " !!" !!" !! " !! " !!" !!" L (3)
with M !!" the mean magnetization in the material, I M !!" the magnetization of the VOI. L is a 2nd order localization operator, depending on the magnetization, on the stress state, and on the shape choosen for the inclusion I. In the case of stress independent linear isotropic magnetic behavior, and spherical inclusions, the tensor L reduces to scalar value N usually called demagnetizing factor so that:

1 3 2 N = + χ (4)
with the equivalent medium susceptibility. We get 1 ( ) 3 2

I I H H M M = + - + χ !! " !! " !!" !!" (5)
This approach demonstrated for linear medium has been extended with success to anisotropic non-linear magnetic behavior [9] using:

M H = χ (6)
Formulation 5 becomes self-consistent. Its numerical implementation remains a challenge. A new algorithm is proposed further in the text, avoiding a step by step calculation.

Localization and homogenization of the mechanical fields in heterogeneous media

The aim of this step consists in deriving the local stress σ from the external loading (magnetic field H, and stress

), postulating a particular form for the function g in:

( , )

I g H σ = Σ !! " (7)
The function g is here deduced from a self-consistent approach. Each inclusion is considered imbedded in a homogeneous equivalent medium equivalent so that the problem can be linked to the solution of the Eshelby inclusion problem [10].

Consider an unloaded homogeneous infinite medium of stiffness tensor # . A VOI is submitted to a free magnetostriction strain I µ ε . This strain is the strain that would act in the VOI if no stress was exerted by the surrounding medium. The actual strain I ε in the VOI can be linked to the free strain using the Eshelby tensor E S [10]:

:

E I I = µ ε ε S (8)
The fourth order tensor E S only depends on the stiffness tensor # and the shape chosen for the VOI. It is calculated from the Green's functions and FDO [9]. The stress in the VOI I ε is related to the elastic strain of the VOI thanks to the Hooke's law:

:

e I I I σ ε = # (9)
Considering the strain decomposition, this decomposition is acceptable only when deformation levels are low (<10 -2 ), we get:

e I I I µ ε ε ε = + ( 10
)
The stress can be expressed as function of the free strain:

: ( )

I I I I µ σ ε ε = - # (11)
Considering an applied stress over the VOI, the stress at the grain scale g σ is derived from the implicit equation ( 13).

:

t g E µ µ ε =< Β > (12)
: :( )

acc I I E µ µ σ ε = Β Σ + - # (13) With 1 0 1 1 ( ) ( :(( ) )
)

acc E I S - - - = + -Ι # # # . 0
# is the stiffness tensor of the effective media [11]. Since a self-consistent scheme has been chosen, 0 # refers to the self-consistent stiffness tensor [9]. B is the so-called stress concentration tensor.

Multidomain magneto-mechanical modeling of single-crystals

The prediction of the influence of heterogeneity on magnetic behavior supposes the introduction of the complete mechanical loading into a magnetoelastic modeling. As already mentioned, the few practically implemented models describing the effect of stress on magnetic behavior are restricted to uniaxial mechanical loadings (tension or compression) [12]. A first approach to build a simple magneto-elastic model consists in the definition of magneto-elastic constitutive laws including the multiaxiality of stress at the local scale. A multidomain modeling following this requirement is briefly presented.

Original formulation

The multidomain modeling is a two-scale reversible modeling allowing the prediction of the magneto-mechanical behavior of isotropic polycrystals. It comes from a simplification (which is simplified in four points: simplified microstructure, homogeneous field and stress, uniaxial, stress, no minimization of energy.) of the so-called multiscale model [9]. A six magnetic domains configuration is considered associated to the six easy axes of cubic symmetry for materials that exhibit a positive magnetocrystalline constant (figure 1a). Each domain family α is defined by a magnetization vector a M !!" so that

a s M M = !!"
, and by a magnetostriction tensor a µ ε (15) (γ i parameters figure the direction cosines of magnetization; λ 100 and λ 111 are the two magnetostrictive constants). This single crystal is considered as submitted to a magnetic field H !! "

and/or stress σ. Uniform strain and field hypotheses are used over the crystal and domain walls contribution to the total energy is neglected [9]. The energy of a magnetic domain a W is the sum of the magnetostatic energy H a W , the magneto crystalline energy K a W and of the magnetoelastic energy a W σ (16) (K 1 is the magneto crystalline constant of the material). The stress tensor is supposed uniaxial; magnetic field and stress are applied along a same direction c n " defined by angles c φ and c θ of the spherical frame (figure 1b). This direction is restricted to the standard triangle defined by crystallographical directions < 100 >, < 110 > and < 111 >: cubic symmetry means that at any loading direction is corresponding a direction in this triangle. The resolution of the problem (ie calculation of the mean magnetization and deformation) requires to evaluate the direction of magnetization and the volumetric fraction of each domain family α.

1 2 3 [ , , ] t a s M M γ γ γ = !!" (14)
( ) 3 3 1 ( ) 2 3 1 ( ) 3 a µ λ γ λ γ γ λ γ γ ε λ γγ λ γ λ γ γ λ γ γ λ γ γ λ γ   -       = -       -     ( 15) (1)
(2)

(3) (4)
(5) .

(6) y x z φ c θ c <100> <110> <111> <001> <010> {Η,σ} (a) (b)
:

(( ) ( ) ( ) ) H a a a a K a W HM W W K σ µ µ σ ε γ γ γ γ γ γ = - = - = + + !! " !!" (16)
The volumetric fraction a f of a domain is calculated thanks to statistical Boltzmann formula (18); s A is an adjusting parameter corresponding to the inverse of entropy following Boltzmann formalism.

A simple Taylor expansion of volumetric fraction at low field is showing that s A is proportional to the initial susceptibility 0 χ of the magnetization curve. We get:

0 2 2 0 3 . .(cos sin ) s s c c A M χ µ φ θ = .
[cos( ) sin( ),sin( ) sin( ),cos( )]

t c c c c c c n φ θ φ θ θ = " (17) exp( . ) exp(
. )

s a a s a a A W f A W - = ∑ - (18)
The magnetization direction of a domain is defined by angles a φ and a θ of the spherical frame (19).

These angles are usually estimated thanks to a minimization of the potential energy (20). In the present study, the restriction to standard triangle allows an analytical minimization, so that a constitutive law for the angles of each domain is obtained as function of magnetic field, stress, and loading direction parameters. As an example, equations (21) give the constitutive laws of the two spherical angles for domain α=1(Cf. figure 1a).

( , , , , , ) 0 ( , , , , , ) 0

a c c a a a a c c a a a dW H d dW H d σ φ θ φ θ φ σ φ θ φ θ θ = = (20) 0 1 1 1 1 0 1 1 0 0 0 111 1 0 1 1 0 0 3 tan( sin(2 )) 2 ( , , ) 2 3 cos(2 ) 3 ( ) tan( sin(2( ))) 2 2 2 ( , , ) 2 3 cos(2( )) 2 s c c c s c s c c c s c M H a H M H K M H a H M H K µ φ λ σ φ φ σ φ µ λ σ φ π π µ θ λ σ θ θ σ θ π µ λ σ θ + = + + - + - = + + - (21)
Average magnetization and magnetostriction ( 22) are projected along the loading axis c n "

leading to the behavior of the single crystal ( , )

M H σ and // ( , )

H µ ε σ (23). a a a a a a M f M f µ µ ε ε = = ∑ ∑ !!" !!" (22)
// . . .

t c c c M M n n n µ µ ε ε = = !!" " " " (23)

Extension to loading out of the standard triangle

Multi-domain single-crystal model can be employed only when the loading is applied in the standard triangle. Thus, the local loading (Magnetic field and stress) need a linear transformation in order to mach the standard triangle. A permutation matrix is used to make the load to be adapted for a symmetries cubic crystal.

Since the standard triangle is a so-called spherical triangle defined between <100>, <110> and <111> crystallographic directions, 48 triangles are required to map the whole surface of the sphere. .

Let consider

i c c i n P n = " " (24)
48 permutation matrix can consequently be defined, allowing to define a loading direction inside 1 T triangle whatever the loading direction. 1 P is the permutation matrix from triangle 1 T to triangle 1 T

( 1 c c n n = " "
); 2 P is the permutation matrix from triangle 2 T to triangle 1 T ; etc…, leading to:

1 1 1 2 2 1 1 0 0 0 1 0 0 0 1 0 1 0 1 0 0 0 0 1 p P I p P → →     = = =           = =       (25)
The permutation must be applied to magnetic field and uniaxial stress direction and so on, so that: . .

i i i i H P H P σ σ = = !! " !! " !" !" (26)
The constitutive law is applied then Magnetic and magnetostrictive quantities are calculated using multidomain modeling. Results must be submitted to a reversal permutation in order to get the magnetization and magnetostriction tensor in the appropriate frame. .

i i i i i M P M P P µ µ ε ε - - = = !!" !!" (27)

Extension to multiaxial stress

The second step is to calculate stress used for rotation. We change the multiaxial stresses σ into uniaxial magneto-mechanical equivalent stresses according to the direction of the magnetic loading i c n "

. We use for that purpose an equivalent stress recently defined in [13]. In the frame (

, , h t t " " " ) associated to the magnetic field direction h " and two orthogonal directions associated to the matter symmetries. We note k the product s m A λ as a material dependent parameter: for negative magneto crystalline anisotropy materials. 0 χ denotes the initial anhysteretic susceptibility of the material, 0 µ is the permeability of vacuum and s M is the saturation magnetization of the material. This definition of equivalent stress finally requires to know some relatively usual material parameters and no supplementary adjusting parameter. This expression can be degraded in a more simple expression (deviatoric equivalent stress (31)) applicable to moderate loadings and/or relatively low magnetostrictive materials. [START_REF] Daniel | [END_REF] 2

1 1 2 2 3 3 exp exp 2 2 2 ln 3 2 t t t eq k k t St t St hS h k σ       +             = -       " " !" !" " " (28)
t i i eq c c n Sn σ = " " (31)
The same value of equivalent stress is considered to be applied along c n "

. We remark that permutation operation can be made on the stress tensor, so we have:

1 . i i c c n P n - = " " (32) 1 1 3 . 2 t eq t i i c c n P SP n σ - - = " " ( 33
) ' 3 . 2 t eq c c n S n σ = " " (34)
The same operation can be applied to the generalized formulation of equivalent stress.

Single crystal modeling

For a single iron crystal, it is possible to subdivide the volume in six domain families associated with the six easy magnetization directions. Each domain family is defined by a specific magnetization and magnetostriction strain tensor. Considered loadings are 3D multiaxial stress and/or magnetic field.

Orientation of magnetization of each domain defined analytically avoiding energetic minimization: use of equivalent axial loading inside the standard triangle. Volume fraction of domains is calculated thanks to energy consideration [14].

Multidomain magneto-mechanical modeling of polycrystals

This method is applied to calculate the average behavior of polycrystalline medium.

Orientation Distribution Function -ODF

We consider in a first approach an isotropic texture represented by an orientation distribution function at 546 discrete grains oriented to pave properly the entire sphere surface. Euler angles are as: The following figure (Fig. 2) shows the associated pole figures. Each grain orientation is defined by a matrix Q providing the transition of local frame to macroscopic frame (x, y, z). We can thus define a local orientation of the load. We perform the following operations:

1) Transition of vector fields from the macroscopic frame to the local frame: localization.

2) Transition of vector fields from the local frame (sphere) to the standard triangle by appropriate permutation.

3) Application of constitutive law; Calculation of the local magnetization and magnetostriction.

4) Passage of the magnetization vector of the local landmark macroscopically.

5) Calculation of mean fields and magnetizations of the 546 directions.

The first method uses the implicit nature of the formulation of field location.

Assuming the magnetic field is known to each grain I H , one can use the following operations to determine the magnetization field in each grain I M and the mean fields M .

The problem is proposed as following: for a given external magnetic field H , we can define the magnetization field I M and the magnetic field I H for each grain, there is two relations between I M and I H : on one part, once I M is known, I H can be determined using Eq. ( 5), on the other hand, I H can be known if the process above is used, then I M can be calculated. For simplify the notation, the relation is denoted:

( ),

I I M F H = (35)
Where F is a function that depends the transition matrices Q and can be determined explicitly.

But in practice, it is difficult to give the analytical formula for F. By consequence, So, iteration will continue until the difference between these two iterations is considered sufficiently small. Combining these two equations above (Eq. ( 35) and Eq. ( 5)), we can get

( 1) ( ) ( ) 1 ( ( ) ) , 3 2 n n n I I M F H M M + = + - + χ (36)
n n I I M G M + =
Where the function can be determined by F, H and ; and we go back to a problem of fixed point. As we mentioned above, the convergence property of the algorithm depends on the quality of function G.

In fact, G is needed to be contractive, i.e.

I I I I G M G M M M - - is calculated.
For reason of computational complexity, three grains are considered to simply the case. As we do not know the state of the equilibrium of the magnetization field I M , many cases (>8000) have been tested in order to cover all of the possible configurations of I M .

We notice that the values of * * ( ( ) ( ) )/ ( )

I I I I G M G M M M - -
are superior to 1 in many cases, so that the convergence of algorithm is not ensured. Some implemented tests have confirmed our fears: the algorithm does not provide convergence except for the case where only two grains are considered.

The usual proposition to avoid this problem is to use an incremental procedure. We consider two steps of magnetic field denoted ( ) n H and ( 1) n H + . The local magnetic behavior at step (n+1) is given by macrocropic field at the same step and demagnetizing field calculated at the previous step (n) following equation (5). We get:

( 1)

) 3 2 n n n n I I n H H M M x + + = + - + !! " !! " !!" !!" ( ( 1) ( ) ( ) ( ) 1 (
) 37
Average condition is still verified

( 1) ( 1) n n I H H + + =< > !! " !! " (38)
We propose in the next subsection a new formulation allowing a true calculation of the demagnetizing field.

Algorithms

The Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point. The Taylor series can be defined in the following manner.

n n n a x n a f a x a f a x a f a x a f a f x f ) ( ! ) ( ... ) ( ! 3 ) ( ) ( ! 2 ) ( ' ' ) ( ! 1 ) ( ' ) ( ) ( 0 ) ( 3 ) 3 ( 2 - = + - + - + - + = ∑ ∞ = ( 39
)
where n! denotes the factorial of n and (n) denotes the th n derivative of evaluated at the point .

Assuming the first and second right terms of equation ( 39) are much bigger than the other terms, equation (40

) is obtained '( ) ( ) ( ) ( ) 1! f a f x f a x a = + - (40)
On the other hand, the localization law is considered (corresponding to the 3 projections (k) of equation):

1 1 ( ( ) ) 3 2 i i i i j j i H H M M k + = + - + χ ( 41
)
where j is the index of grains, i is the number of loop for localization (not corresponding to magnetic field step).

i M is the average magnetization for all the grains at iteration i so that

i i j M M =< >
and i χ is the mean secant susceptibility given by /

i i i M H = χ !!" !! " Convergence is achieved when 1 ( ) 0 i i j j k H H + - →
The magnetization j M is on the other hand function of the magnetic field j H thanks to local constitutive behavior F so that:

( ) ( ) ( ) ( )
1 1 i i i i j j j j M F H k or M F H k + + = = (42)
Using equation (41), the constitutive law gives: Table 2. Chemical composition of DP steel used for the study.

The behaviour of martensitic phase is still unknown. Tab. 2 gives the chemical composition of the DP steel used for the study. The carbon content is close to 0.15wt%. Figure 3 gives the typical microstructure observed for the material as quenched. The volume fraction of martensite has been evaluated to 40%. Assuming that carbon content of ferrite is about 0.02wt%, the carbon content in the martensite phase can be so evaluated to 0.4wt%.

Modeling results of DP steels

Several simulations were carried out using MATLAB 2009 for a dual phase steel. In order to get a reasonable computation time, a fewer number of grains was chosen. On the other hand, the chosen grains should show obvious isotropic transverse symmetry. Computations have been made using a 546 grains RVE of a dual phase steel polycrystal [10].

The experiment results presented herein are performed thanks to a magnetic measurement benchmark where two coils are wounded around the sample. A DP600 from Arcelormittal has been considered for the study in order to underline the effect of the thermo-mechanical history on the magnetic behaviour of these steels. Samples for all experiments consists of 200 mm long and 12.5 mm wide bands cut by electro erosion machining in order to avoid residual stresses that have a strong influence on the magneto mechanical behavior.

Comparison between experimental and numerical results

shows that the multiscale model seems to describe correctly the effect of an applied stress. For the positive magnetic field, a DP steel with lower martensite content has higher magnetic value, and the modeling reproduces it. The result shows that it is more difficult to magnetize the DP steels which contain high volume fractions of martensite Experimental observations (Fig. 6b) show that the magnetostriction strain first increases with the applied field and then decreases until a saturation point is reached. Experimental results illustrate that the DP steel with lower martensite contents has a higher magnetostriction strain, which is correctly reproduced by the numerical result (Fig. 6a).

Conclusion

In this paper, a model for the reversible magneto-mechanical behavior of magnetic material, accounting for magnetization and microstructure has been presented. It is based on a description of the magneto-mechanical coupling at several scales (domain, single crystal, polycrystal). The modeling strategy allows a fast prediction of the magneto-mechanic behaviour of a dual phase microstructure. This estimation requires the knowledge of the behaviour and the volume fraction of the both phases. Numerical results have been compared with experimental results with very good agreement.

Fig. 1 .

 1 Fig. 1. (a) Initial domain structure of single crystal considered for the multidomain modeling; (b) Standard triangle and parameters for magneto-mechanical loading axis.

1 2 3 [

 3 , , ] [cos( )sin( ),sin( )sin( ),cos( )]

  to the same behavior that a loading along i c n " due to cubic symmetry of the single crystal. i P is the corresponding permutation matrix verifying:

  is the deviatoric tensor associated to . Using the hypothesis of uniform stress, we get

Fig. 2 .

 2 Fig. 2. Pole figure associated with the distribution function of orientations 546 discrete grains uniformly distributed in space

  be determined analytically. Thus, an iterative process is proposed to determine their values. Assuming that ( ) n M is known at the Nth iteration, noted( )

  Satisfying at least for X and Y in the vicinity of the solution I M .G is a non-analytical function; it is difficult to verify this contracting condition theoretically. For this reason, we use the numerical verification. One state of I M is chosen and a little bit of disturbing is

Fig. 3 .Fig. 4 .

 34 Fig. 3. Dual phase steel microstructure: distribution of martensite in white, and ferrite in dark. 1) Behavior and modeling of the single crystal of martensite: A C38 carbon steel has been considered for this step (wt% C=0.38%). It has been submitted to various heat treatment (quenching) in order to transform the initial microstructure (fig. 4a) into a microstructure close to the microstructure of the second phase of the dual phase steel. Hardness measurements are used as indicator of the quality of the new microstructure. Figure 4b shows the microstructure obtained after the heat treatment that has been selected. As expected we have a quite homogeneous microstructure, composed of martensite needles, strained ferrite and bainite.

Fig. 5 .Fig. 6 .

 56 Fig. 5. Magnetization curve for different volume fractions of martensite: (a) numerical results; (b) experimental results

TABLE 1

 1

		Euler angles of isotropic polycrystal of 546 grains
	Euler angles	Domain	Number of
				values
	φ	1	[0; 2π]	13
	cosψ	[0; π]	7

φ [0; 2π] 6

Acknowledgments

This research was carried out in the frame of the project DPS-MMOD "Dual-Phases Steel Magnetism Modelling" Grant No. MATETPRO 08 -322447, 2008., funded by the ANR French research agency." This work is supported by Department of Education of Guizhou Province Foundation (Qian Teaching and Research R&D [2012] -Grant No. 213), China.

( )

When the loop step turns to infinite, the magnetization should be a stable value. The magnetization of 1 th i + loop step equals to th i loop step magnetization.

From equation ( 44) and (43) the magnetization for grain is obtained:

( )

Because M and χ are function of M j , H being a constant for a given loop, we get the following self-consistent relation:

We consider now a Taylor expansion of G function:

where (n) indicates the n th magnetic field step. It is assumed that G function is sufficiently linear between two magnetic field step and so two magnetization steps. Equation ( 47) is rewritten:

The matrix inversion increases considerably the computation time if 546 orientations are considered.

Please note that a calculation field takes about 20 minutes with a laptop and a trade algorithm coded in MATLAB. The taylor expansion is applied to the "local" behavior in the study.

Application to dual-phase steel

Dual-Phase steels are produced thanks to a complex forming process that leads to a microstructure exhibiting two phases (fig. 3). The morphology and the distribution of the martensite islands in the ferritic matrix are the keys of the dual phases steel behaviour. This microstructure is obtained by quenching of a low carbon steel (wt% C<0.3% typically) from austeno-ferritic domain of the phase diagram.

Dual-phase ferrite-martensite steel

The magneto-mechanical behaviour of a dual phase steel is deeply linked to its microstructure and especially to the behaviour of its constituents. An accurate modeling of the mean medium supposes consequently to have an accurate modeling of each phase separately. The behaviour of ferrite is supposed very close to the behaviour of pure iron, which was the subject of many works. The measurements presented hereafter have been carried out using an Armco pure iron sample.

The main shortcomings of the model are first that magnetic and mechanical fields are considered homogeneous within a single crystal; second that domain walls are not taken into account. Future works will consist in introducing residual stress effects such as plasticity or thermal stress, and dynamic phenomena.