Amir Zidi
email: amir.zidi@univ-valenciennes.fr

Mourad Abed
email: mourad.abed@univ-valenciennes.fr

A Generalized Framework for Ontology-Based Information Retrieval Application to a public-transportation system

Keywords: Information Retrival (IR); Information retrieval, public-transportation ontology, semantic indexing, .entity retrieval. I

In this paper we present a generic framework for ontology-based information retrieval. We focus on the recognition of semantic information extracted from data sources and the mapping of this knowledge into ontology. In order to achieve more scalability, we propose an approach for semantic indexing based on entity retrieval model. In addition, we have used ontology of public transportation domain in order to validate these proposals. Finally, we evaluated our system using ontology mapping and real world data sources. Experiments show that our framework can provide meaningful search results.

INTRODUCTION AND BACKGROUND

The amount of content stored and shared on the Web and other document repositories is increasing fast and continuously. Consequently, the ability to access and select relevant information in these huge and heterogeneous masses of data remains a difficult task. However, most Information retrieval systems have limited abilities to exploit the conceptualizations involved in user needs and content meanings. This involves limitations such as the inability to describe relations between search terms.

In order to overcome these limitations, current Information Retrieval (IR) studies are focusing on relevant documents retrieval using additional knowledge. The main idea is to support a high-level of content and queries conceptual understanding. According to [START_REF] Egozi | Concept-Based Information Retrieval Using Explicit Semantic Analysis‖[END_REF], there are two main categories of conceptual-based information retrieval approaches. The first one concerns approaches that extract semantic meaning from documents and queries by analyzing the latent relationships between text words. The second category consists on approaches that, manually or automatically, construct taxonomy of semantic concepts and relations and map documents and queries onto them. Ontology, as a knowledge representation, is one of the most used technologies in the second category. The use of ontology in IR is an important parameter presented by [START_REF] Egozi | Concept-Based Information Retrieval Using Explicit Semantic Analysis‖[END_REF] to characterize ontology-based methods. The ontology may be used partially through a query expansion phase [START_REF] Carpineto | ‖A Survey of Automatic Query Expansion in Information Retrieval‖[END_REF]. It may also be advanced in both phases of indexing and retrieval. Several approaches exist in the literature such as [START_REF] Soner | An ontology-based retrieval system using semantic indexing‖[END_REF] and [START_REF] Fernández | Semantically enhanced Information Retrieval: An ontologybased approach‖[END_REF]. These approaches adopt an advanced use of ontology-based knowledge representation. They can be more efficient especially using domain-information extraction. However, they use specific language for semantic querying which is not easy to be used by the end-users. Formulating a query using such languages requires the knowledge of the domain ontology as well as the syntax of the language.

In this paper, we are focusing on adapting the keywordbased semantic retrieval system using domain ontology in three phases namely the knowledge phase, the indexing phase and the retrieval phase. We are trying to deal with three main issues of the semantic search and retrieval:  Scalability: it involves not only exploiting semantic metadata that are available in data sources but also managing huge amounts of information having a structured and unstructured content form [START_REF] Crescenzi | Automatic information extraction from large websites‖[END_REF]. In order to achieve more scalability, we propose a semantic indexing approach based on an entity retrieval model.

 Usability: In order to deal with usability issue, we adopt a keyword-based interface as it provides a comfortable and relaxed way to query about the enduser.

 Retrieval performance: we are trying to improve the retrieval performance by using a domain-specific information extraction, inference and rules.

The remainder of this paper is organized as follows: The next section presents the framework of ontology-based information retrieval. This section covers the general architecture and the main processes description including the use of public-transportation ontology, semantic indexing and querying. In order to validate the proposed Framework, the third section includes performed experiments which are based on real word data sources such as RATP open data 1 . The paper concludes with a summary and discussion of the outcomes of the presented work.

II. PROPOSAL OF FRAMEWORK

Our framework structure is mainly based on three processes: semantic knowledge representation, semantic indexing and semantic querying. The overall diagram of the framework is shown in Fig. 1.We describe the steps we take until the system becomes ready for semantic querying:

 Using the usable information from data source (web sites, data base ...) we populate the initial OWL files.

 We run the Reasoner over these files and obtain new OWL files containing the inferred information.

 We build indexes, using these inferred OWLs, which are used in semantic querying.

A. Semantic Knwoledge Representation

Ontology is considered as a key feature to represent semantic knowledge. RDF2 schema (RDFS3 ), which was built upon RDF, was used to develop ontology language. It extends RDF vocabulary with additional classes and properties such as rdfs:Class and rdfs:subClassOf [START_REF] Soner | An ontology-based retrieval system using semantic indexing‖[END_REF]. OWL4 further extends RDFS with additional features such as cardinality constraints, equality and disjoint classes, which enable users to better define their classes. In addition to that, OWL classes may be instantiated by adding new individuals. Generally, ontology design is based on the diagram presented in fig. 2. This is the diagram of entity types defined for RDF, RDFS and OWL. We can see that user's classes are defined and instantiated based on those entities.

In our work, an existing ontology is reused. It was developed by [START_REF] Kathia | Transportation ontology definition and application for the content personalization of user interfaces‖[END_REF] to facilitate information retrieval for transportation systems. To constitute our knowledge base, we use a wrapper-based method [START_REF] Crescenzi | Automatic information extraction from large websites‖[END_REF]. This latter has as input a data source (data base, www, document corpus). It analyzes and extracts data in order to populate the ontology with instances. The next step is inference. The main idea of this step is to expand knowledge base with new added instances using relations and rules defined by [START_REF] Kathia | Transportation ontology definition and application for the content personalization of user interfaces‖[END_REF]. An example for ontology population can be seen in fig. 2, where instances are extracted from RATP open data and Web Annuaire5 . In this example, we have two new instances (CONNECTION_POINT, OBSERVATOIRE-ASSAS (Paris)) and (SHELTER, Hôtel Istria Montparnasse). After the inference process, we obtain new Knowledge which is OBSERVATOIRE-ASSAS (Paris) is_encercled_by Hôtel Istria Montparnasse. Beyond the relations between classes, authors of the used ontology present a set of rules in order to offer better planning to passengers. As a result, we can have new knowledge about a trip from an origin to a destination. After this step, we obtain useful OWL files that will be indexed and used for the search.

B. Semantic Indexing

As our knowledge base is constituted of entities defined for RDF, RDFs and OWL, we designed an indexing system using entity retrieval model.

1) Entity retrieval model

A knowledge base, which is constituted of entities defined for RDF, is essentially a labeled and directed graph with the nodes being resources while the edges represent the properties [START_REF] Delbru | ‖ Searching Web Data: an Entity Retrieval and High-Performance Indexing Model‖[END_REF]. This graph is essentially a set of RDF Triple (N-Triples). An RDF Triple contains three components each of them is providing complementary pieces of information: subject (node), predicate (property) and object (node).

The subject identifies what object the triple is describing, the predicate defines the piece of data in the object we are giving a value to and the object is the actual value.

In this work, we adopted the Entity Attribute-Value model (EAV model) proposed by [START_REF] Delbru | ‖ Searching Web Data: an Entity Retrieval and High-Performance Indexing Model‖[END_REF]. Before describing our indexing system, we estimated useful to first introduce some basic definitions of EAV model. This later is based on a directly labeled graph G which covers the various types of data sources in particular RDF resources.

The graph G represents datasets, entities and their relationships: We illustrate an example of an RDF graph extracted from our knowledge base. We can see how dataset are divided into entities description (subgraph). We create our second index, RULES_INDEX, which contains all entities generated after the rule inferencing step. In this index, indexed documents are basically a set of journey pattern (Tab.2). This latter is composed of an entity set which may define a trip from origin to a destination. Taking the example of service journey pattern in which banks or post offices are available with the associated connection point. Note that RULES_INDEX is created for retrieval performance purpose.

Field

C. Semantic Querying

Once the semantic knowledge is represented and indexed, the next step is querying the EAV graph (e.g. RDF graph). In order to do that, we use SIREn 6 , an efficient semi-structured information retrieval for Lucene 7 . Three types of queries are supported:

 Full text: keyword-based query when the data structure is unknown. It allows the user to find all the relevant documents that contain all terms in the query using full-text search syntax.

 Structural: when the data structure is known, it produces precise search results using triple patterns to represent partial or complete triples. A triple pattern is a complete or partial representation of a triple <entity, attribute, value>.

 Semi-structural: combination of the two previous query types when the structure is partially known.

Full-text search is supported on any part of the triple, which means that the user can use the Keywordbased query syntax to describe his entity, attribute or value.

1) Search with SIREn

With SIREn, Querying RDF graph is commonly achieved using triple stores (i.e. RDF triple, EAV model). We developed a keyword-based interface as it provides a comfortable way to query about the end-user. Query results are achieved using a Boolean combination of attribute-value pairs based on the logical operator ˄, ˅ and ¬, this is called query algebra. In the following we present how we adapted the formal model of relational query algebra, which is used in SIREn and proposed by [START_REF] Delbru | ‖ Searching Web Data: an Entity Retrieval and High-Performance Indexing Model‖[END_REF] [8], to our work.

2) Query formulation

In this section, the field Dataset is denoted by d, the field Entity is denoted by e, the field Attribute is denoted by at and the field Value is denoted by v. Given a keyword selection condition c and a relation R, the keyword selection operator σ c (R) is defined as a set of relation instances {r|r∈R} for which the condition c is true. The condition c consists of testing if a given word denoted by k occurs in one of the field f of a relation R, which is denoted by f:k. we denoted the function of the test by W. More details about the function W can be found in [START_REF] Delbru | A Node Indexing Scheme for Web Entity Retrieval‖[END_REF]. For example if we test if the keyword k occurs in value label of a relation instance r (denoted by r.v):

σ v:k (R): {r|r∈R, k∈W(r.v)}
We denote by π f (R) the projection operator which allows extracting a specific column of field f from a relation R. The projection operator can be used to extract more than one column. For example π e,d (R) returns a relation with only two columns, dataset and entity. In the following, we present an example for a simple query formulation, in which, the user is searching for a Hotel Istria.

Q: Find all entities matching keywords Hotel and Istria.

Q= π e,att,v (σ v:"Hotel" (R)) ∩π e,att,v (σ v :"Istria" (R)) Q=π e,att,v (σ v:"Hotel"˄ v:"Istria" (R))

Entity Attribute Value OBSERVATOIRE _ASSAS (Paris)
is_encercled_by Hôtel Istria Montparnasse Table 3. An example showing extracted query results using 3 columns

The proofs of used properties can be found in [START_REF] Maier | Theory of Relational Databases[END_REF].

III. EVALUATION PROCESS

A. Evaluation method

In order to evaluate the framework performance, we prepared a set of queries as the example shown in Table .3. We put the corresponding keyword query which was actually used in the evaluation. Then, we calculated the correct number of documents that should be retrieved, for each query. Finally, we run the queries and calculated the performance using evaluation metrics Precision, Recall and the F-Measure. Precision metric is the proportion of the related documents in the retrieved documents (true positives) to the total number of retrieved documents. Recall metric is the proportion of the retrieved related documents to the total number of related documents that should have been retrieved. F-Measure is used as it provides more robust evaluation criteria using Precision and recall together. They are calculated as follows:

B. Analysis of results

The obtained results (Tab.5) show that the exploitation of semantic fields shown fruit with high rate of precision and recall. With respect to the precision, scores show that the semantic search presents a high rate. This latter means that little unnecessary documents are provided by our framework and that the latter may be considered as "precise". Additional information (keyword) in Q3 produced a gain compared to Q2 and Q4 in terms of Recall. This gap is also explained by the lack of information about user query. As shown in (Tab.6), this gap can be reduced by separately indexing entities which are generated after the rule inferencing step. Finally, these results are confirmed by the F-measure. IV. CONCLUSION

In this paper, we presented a generic framework for ontology-based information retrieval system and its application in public-transportation domain. We tried to exploit the main advantages of semantic knowledge representation by using a domain-specific information extraction, inference and rules and also to take advantage of semantic indexing to enhance the retrieval performance.

The current implementation can be extended in many ways. We are planning to enrich indexed data by using more meaningful rules to better exploit underlying semantics in content being indexed. In addition, we will focus on a new aspect of a personalized search which integrates user's profile in the indexing phase. The main idea is to re-index contents after clustering user's profiles in order to get more relevant matching between well-defined resources and user queries.

Figure 1 .

 1 Figure 1.Over all framework diagram

Figure 2 .

 2 Figure 2.Example of ontology population

  V: set of nodes A: set of labelled edges V E : set of entity nodes V L : set of literal node L: set of labels composed of L V (set of node labels) and L A (set of edge labels) V E D : set of entity nodes which form a dataset D L E D : set of entity node labels which form a dataset D L V D : set of node labels which for a dataset D Graph G: is a graph over L and G=<V, A, λ > where λ: V→ L V is node labeling function. The set of labelled edges is defined as A⊆ {(e, α, v)| e∈V E , α∈ L A , v∈ V}. The components of edge a∈ A is denoted by source(a), label(a) and target (a) respectively A dataset provides information about an entity including its relationships with other entities and its attributes: Dataset D: a dataset over a graph G=<V, A, λ >is a tuple D=<V D , A D , L V D , λ> with V D ⊆V and A D ⊆A. A subgraph describing an entity can be extracted from a dataset; an entity description is defined as: Tuple <e, A e , V e >where e∈V E D the entity node, A e ⊆ {(e, α, v)| α∈ L A D , v∈V e }the set of labelled edges representing theattributes andV e ⊆VD the set of nodes representingvalues.

Figure 3 .

 3 Figure 3. RDF Graph

Q1

  Find the Hotel Istria. (query: -Hotel Istria") Q2 Find a trip from AEROPORT_CDG to PORT-ROYAL-Paris. (query: -Trip cdg Port-Royal") Q3 Find a trip from AEROPORT_CDG to PORT-ROYALwith a Hotel near to PORT-ROYAL-. (query: -trip cdg Port-Royal Hotel‖) Q4 Find a trip from AEROPORT_CDG to Hotel Istria (query: -trip cdg Hotel Istria")

Table 1 .

 1 An example of indexing Entity (POINT_ARRET_ROYAL)

	Field	Value
	Dataset	http://www.owlontologies.com/Ontology1256801179.owl#POIN
		T_ARRET_ROYAL
	Entity	POINT_ARRET_ROYAL
	Attribute station_name
	Value	PORT-ROYAL-Paris

Table 4 .

 4 Example of evaluation queriesBefore analyzing the results, we want to clarify the evaluation queries. Q1 is used to retrieve all entities matching keywords Hotel and Istria. Q2 is used to retrieve all entities matching keywords Trip, cdg and Port-Royal. Q3 is used to retrieve all entities matching keywords Trip, cdg and Hotel and Port-Royal. Q4 is used to retrieve all entities matching keywords Trip, cdg and Hotel Istria. By executing Q2, Q3 and Q4, user should access to all information about a trip from an origin to destination including entities, attributes and values.

Table 5 .

 5 Evaluation results (BASIC_INDEX)

	Queries	Precision	Recall (%)	F-measure
		(%)
	Q1	100	100	1
	Q2	75,0	100	0,857
	Q3	100	90,0	0,947
	Q4	100	63,0	0.777

Table 6 .

 6 Evaluation results (RULES_INDEX)

http://data.ratp.fr/ http://data.ratp.fr/

http://www.w3.org/RDF/

http://www.w3.org/2001/sw/wiki/RDFS

http://www.w3.org/OWL/

http://www.webannuaire.net/

http://siren.sindice.com/

http://lucene.apache.org/core/


Jérôme Lux
email: jerome.lux@univ-lr.fr

Elisangela Guzi De Moraes

Eric Maire

Jérôme Adrien

Lisa Biasetto

Gas permeability of Ti6Al4V foams prepared via gelcasting, experiments and modelling

Keywords:

The use of light components possessing tailored functional properties will represent one of the main issues for the development of new products. Open cells foams, where porosity is engineered in order to grant for lightness, high geometrical surface, interconnectivity and permeability, can be shaped using various techniques and possess regular or irregular structure. In this work, Ti6Al4V foams were produced using the gelcasting route, a technique usually employed for ceramics. Numerical simulations were carried out directly on 3D images of the foam's microstructure obtained by X-ray 3D µCT in order to predict the gas permeability properties of the produced samples. The obtained results were compared to experimental measurements of gas permeability. This approach could be useful for the prediction of Newtonian fluids permeability behavior of non-periodic and complex porous structures.

Introduction

Metallic foams offer a set of intriguing functional and structural properties in terms of high specific stiffness, impact and energy absorption. They can be used as filters and heat exchangers (open cells) or as thermal and acoustic insulators (close cells). Depending on their composition, they may find applications in different fields such as building (aluminum foams panels for facades and floors), automotive (sandwiches cores for lightweight vehicles, nickel-based cathode substrates for high efficiency batteries, titanium substrates for fuel cells) aeronautic and military (titanium foams for lightweight and high corrosion resistance components), biomedical (Ti6Al4V alloy foams for bone replacements) [START_REF] Pang | Dynamic characteristics of aluminium foams under impact crushing[END_REF][START_REF] Salimon | Potential applications for steel and titanium metal foams[END_REF][START_REF] Shu | Studies on palladium coated titanium foams cathode for Mg-H2O2 fuel cells[END_REF]. The use of metallic foams may represent a double advantage in terms of reduction of the use of raw materials and weight reduction of the final component (porosity is an empty phase). These two aspects make metallic foams a cutting edge material for circular economy European policies. However, wide scale applications are still limited by the high production costs, that make metallic foams often non-competitive in comparison to the polymer-based cheaper solutions.

Authors recently reported on the production of Ti6Al4V open-cell foams by a novel gelcasting route [START_REF] Biasetto | Ovalbumin as foaming agent for Ti6Al4V foams produced by gelcasting[END_REF]. The effect of processing parameters on foams morphology and composition were studied. Foams with a porosity ranging from 71 to 91 vol% and compression yield strength from 24 MPa up to 79 MPa were produced. The foams exhibited a double pore size distribution with large pores (530-730 µm) and smaller pores (79-124 µm), with large pores deriving from the air bubbles incorporated during the process and small pores deriving from empty spaces around the titanium spherical particles due to uncomplete sintering.

In the present work, optimized process parameters were used to produce Ti6Al4V foams possessing two different range of porosity and pores size. Their morphology was studied by means of X-ray micro-Computed Tomography (3D µCT). Their gas permeability was measured and compared to simulated values. As reported by other authors [START_REF] Dukhan | A two-permeability approach for assessing flow properties in metal foam[END_REF][START_REF] Rajamani | Extended Navier-Stokes Equations: Derivations and applications to Fluid Flow Problems[END_REF][START_REF] De Schampheleire | A discussion on the interpretation of the Darcy equation in case of open-cell metal foam based on numerical simulations[END_REF][START_REF] De Carvalho | Pore-scale numerical investigation of pressure drop behaviour across open-cell metal foams[END_REF][START_REF] Kumar | Investigation of fluid flow properties in open cell foams: Darcy and weak inertia regimes[END_REF][START_REF] Jobic | Predicting permeability tensors of foams using vector kinetic method[END_REF][START_REF] Lasseux | On the stationary macroscopic inertial effects for one phase flow in ordered and disordered porous media[END_REF][START_REF] Lasseux | On the developments of Darcy's law to include inertial and slip effects[END_REF][START_REF] Kumar | State-of-the-art of pressure drop in open-cell porous foams: review of experiments and correlations[END_REF], using the Forchheimer equation to calculate the Darcy permeability and the form drag coefficient can lead to significant errors when these coefficients are calculated on experimental data that cover different flow regimes. A complementary approach was used in this paper, which is to perform numerical simulations [START_REF] Diani | Numerical investigation of pressure drop and heat transfer through reconstructed metal foams and comparison against experiments[END_REF][START_REF] De Schampheleire | How to study thermal applications of open-cell metal foam: experiments and computational fluid dynamics[END_REF] directly on 3D images of the foam's microstructure obtained by X-ray 3D µCT. This is however particularly challenging in this kind of material featuring a non-periodic structure with a wide range of pore size.

Materials

Foams were prepared as reported in [START_REF] Biasetto | Ovalbumin as foaming agent for Ti6Al4V foams produced by gelcasting[END_REF]. Briefly, Ti6Al4V powder was added stepwise to the methylcellulose (MC) solutions containing 1 wt% of PEI, a cationic polyelectrolyte, upon continuous stirring with a laboratory mixer for 20 min. Afterwards, the 35 vol% concentrated slurry was aerated with a double shear mixer during 2 min followed by addition of 0.1 vol% of Tergitol™ TMN 10 and subsequently, 5 wt% of ovalbumin (amount calculated based on the metallic powder content) was added, then the suspension was vigorously stirred at different mixing rates (700 rpm or 1500 rpm) during 3 min in order to achieve foaming, and consequently to tailor the pore microstructure and porosity. Afterwards, the foams were poured in a Teflon mold and thermal gelling occurred in a dryer at 80 °C for 2 h. Sintering was conducted in two steps: pre-calcination at 650 °C (2h; 1 °C/min heating rate), to decompose the organic phase; final sintering at 1400 °C (4h, 2 °C/min heating rate) under 99.99% Argon flow and high vacuum. The gas permeability was subsequently measured, and some other samples were cut for the three-dimensional high-resolution X-ray adsorption tomography evaluation. In Table 1 an overview of the processing conditions and sample names is reported.

3D micro-computed tomography (3D µCT)

Acquisition

X-ray computed tomography was used at two different scales to digitize the morphology of the Ti6Al4V-700 and Ti6Al4V-1500 produced samples. The sample were first cut into parallelepipeds of approximate dimensions (5 × 5 × 20 mm) and scanned at low resolution (lateral voxel size = 7 µm). The tomograph used for this first acquisition is described in [START_REF] Buffiere | In situ experiments with X ray tomography: an attractive tool for experimental mechanics[END_REF]. It was operated at 100 kV and 220 µA. This allowed us to analyze the so-called "macro" porosity. 3D visualizations of reconstructed sub-volumes of the 1500 rpm and 700 rpm samples are shown in Fig. 1. This first low scale imaging was not sufficient to capture the smaller pores. The samples were then cut into smaller rods (1 × 1 × 10 mm) and scanned at higher resolution (voxel size = 0.7 µm) using a second tomograph described in [START_REF] Etiemble | 3D morphological analysis of copper foams as current collectors for Li-ion batteries by means of X-ray tomography[END_REF]. 3D visualizations of reconstructed sub-volume of the high-resolution images of the two samples are shown in Fig. 2.

Segmentation

Before the image can be analyzed, porous and solid phase must be identified. A simple threshold is applied on the high-resolution images of the two samples as well as the low-resolution image of the Ti6Al4V-700 sample. This gives a sufficiently good segmentation of the two phases. It is however not as simple for the low-resolution image of the Ti6Al4V-1500, because the smallest pores are not well resolved. The grey level of voxels containing both pore and metal is indeed a weighted average of the pore and metal grey levels (partial volume effect). In order to retrieve the majority of the small-scale porosity, the image was then smoothed using a Perona-Malik anisotropic diffusion filter [START_REF] Perona | Scale-space and edge detection using anisotropic diffusion[END_REF] followed by a sharpen filter to enhance the remaining small dark zones that correspond to small-scale porosity. A global threshold was applied and then isolated clusters of white voxels were deleted. The result of this process is shown in Fig. 3. The total porosity measured in the low-resolution image of the Ti6Al4V-1500 sample is 72.2%, which is very close to the experimental data of 70.9% (see Table 1).

Pore size distribution

A simple way to measure pore size distribution without prior segmentation of individual pores is to perform a granulometry by opening [START_REF] Serra | Image Analysis and Mathematical Morphology[END_REF][START_REF] Maire | Characterization of the morphology of cellular ceramics by 3D image processing of X-ray tomography[END_REF] on the pore space. An opening is a morphological operation that removes part of the considered set which is strictly smaller that the size of a chosen structuring element (generally a ball or a disk). Using a series of openings of increasing size, the pore size distribution can be deduced from the measures of the number of voxels deleted after each opening.

Pressure drop and flow regimes

In past few years, lots of studies have demonstrated the critical importance of determining the different flow regimes prior to evaluating the Darcy permeability and inertia correction [START_REF] Dukhan | A two-permeability approach for assessing flow properties in metal foam[END_REF][START_REF] De Schampheleire | A discussion on the interpretation of the Darcy equation in case of open-cell metal foam based on numerical simulations[END_REF][START_REF] Kumar | Investigation of fluid flow properties in open cell foams: Darcy and weak inertia regimes[END_REF][START_REF] Jobic | Predicting permeability tensors of foams using vector kinetic method[END_REF][START_REF] Lasseux | On the stationary macroscopic inertial effects for one phase flow in ordered and disordered porous media[END_REF][START_REF] Lasseux | On the developments of Darcy's law to include inertial and slip effects[END_REF][START_REF] Kumar | State-of-the-art of pressure drop in open-cell porous foams: review of experiments and correlations[END_REF]. Different flow regimes have been observed in the literature [START_REF] Jobic | Predicting permeability tensors of foams using vector kinetic method[END_REF][START_REF] Lasseux | On the stationary macroscopic inertial effects for one phase flow in ordered and disordered porous media[END_REF][START_REF] Bağcı | Flow regimes in packed beds of spheres from pre-Darcy to turbulent[END_REF]: Darcy, inertial (weak and strong) and turbulent. It has been shown that the macroscopic equations describing the flow in these different regimes are not the same. Note that the transition between different regimes can also exhibits a different behavior. That's why fitting Forchheimer equation to velocity vs pressure drop curve in order to determine Darcy permeability and inertial coefficient can lead to very different results depending on the range of flow velocity. In a remarkable paper, Lasseux et al. [START_REF] Lasseux | On the stationary macroscopic inertial effects for one phase flow in ordered and disordered porous media[END_REF] carried out a thorough numerical study of inertial effects in different laminar flow regimes, on ordered and disordered structures. Using the macroscopic flow equation derived from the volume averaging of Navier-Stokes equations, they find that:

-The inertial correction is always characterized by a cubic dependence on the velocity in the weak inertia regime but is generally insignificant except in the case of ordered structures with particular pressure gradient orientations. -The Forchheimer correction (e.g. a quadratic dependence on the velocity) is a robust approximation in a very large interval of Reynolds numbers, on disordered structure (note that this is not always the case on ordered structures). The correction is not purely quadratic, but of the form + av bv 2 , which explain why the perme- ability measured in inertia regime is not the same as the Darcy permeability.

In order to identify flow regimes, some authors [START_REF] Jobic | Predicting permeability tensors of foams using vector kinetic method[END_REF] use the Forchheimer number defined as:

= F ρv K β μ ( • ) o s D (1)
where K D is the Darcy permeability (m 2 ), β is the inertial coefficient ( - m ) 1 , μ the viscosity (Pa s), ρ is the fluid density -(kg m ) 3 and v s is the superficial velocity ( - m s 1 ) The problem is that both K D and β have to be known beforehand. In the case of porous media with large pore size distribution, it is also possible to define a Reynolds number using K D as a characteristic length [START_REF] Lasseux | On the stationary macroscopic inertial effects for one phase flow in ordered and disordered porous media[END_REF][START_REF] Bağcı | Flow regimes in packed beds of spheres from pre-Darcy to turbulent[END_REF]:

= Re ρv K μ K s D (2)
which can also be used to identify flow regimes. Some correlations between threshold Re K and porosity are given in [START_REF] Lasseux | On the stationary macroscopic inertial effects for one phase flow in ordered and disordered porous media[END_REF], but it is far from certain that these are also valid for other structures. The other way to identify flow regimes is simply to plot some meaningful data (pressure gradient, reduced pressure gradient P Lv Δ s , etc.) vs velocity or, if available, Re K [START_REF] Dukhan | A two-permeability approach for assessing flow properties in metal foam[END_REF][START_REF] Jobic | Predicting permeability tensors of foams using vector kinetic method[END_REF][START_REF] Bağcı | Flow regimes in packed beds of spheres from pre-Darcy to turbulent[END_REF]. This graphical method can however be error-prone due to dispersion of experimental data, small number of data points, etc.

In the Darcy regime and assuming a one-dimensional pressure gradient, we have the following macroscopic relation:

- ∂ ∂ = = P x P L μ K v Δ s D (3)
where P Δ is the pressure drop and L is the medium thickness along the macroscopic flow direction (m).

In the strong inertia regime, the Forchheimer equation can be written:

- ∂ ∂ = = + P x P L μ K v ρβv Δ F s s 2 (4)
where K F is the Forchheimer permeability (m 2 ), which is different from the Darcy permeability.

As stated previously, the Darcy permeability have to be estimated in Darcy regime using Eq. ( 3) and K F and β have to be estimated in the inertia regime using Eq. ( 4).

(4 can be modified to take into account compressible effects. If we assume that the gas is ideal, it gives, after integration of Eq. ( 4) between = x 0 and =

x L [START_REF] Innocentini | Influence of air compressibility on the permeability evaluation of refractory castables[END_REF]:

- = + P P P L μ K v ρβv 2 in out out F s s 2 2 2
(5)

where P in and P out are, respectively, the absolute fluid pressure at the entrance and at the exit of the medium. Another critical point in the determination of gas permeability of metallic foams is the measurement of the pressure drop P Δ through the sample thickness. In [START_REF] Bonnet | Flow laws in metal foams: compressibility and pore size effects[END_REF] compressibility effects were measured by collecting pressure profiles along the sample thickness (corresponding to the main flow axis). By this route it was confirmed that the macroscopic pressure drop is linear for incompressible fluid (water), whilst it is non-linear in the case of compressible fluids (air). It is clear that pressure measurements at the entrance and exit of fluid flow through the sample are affected by edges effects and differ from the pressure profiles measured along the sample thickness. However it was also observed that these edge effects can be minimized by reducing the sample thickness and the A flow over A sample ratio [START_REF] Innocentini | Influence of sample thickness and measurement set-up on the experimental evaluation of permeability of metallic foams[END_REF] without compromising the representativity of the sample.

Gas permeability measurements

Permeation experiments were performed on one sample for each composition and were repeated three times, with argon flow at room temperature (20-24 °C) and atmospheric pressure (P atm = P out ≅ 1.025 bar), that were measured by a thermocouple fixed at the exit of the flow meter and a barometer, respectively. The setup for these measurements was previously described in [START_REF] Biasetto | Gas permeability of lanthanum oxycarbide targets for the SPES project[END_REF], a scheme of which is reported in Fig. 4.

The disk specimen under analysis was tightly fixed with rubber rings inside the sample holder that provided a useful flow diameter of 14 mm (A flow /A sample = 0.6, where A flow is the surface exposed to the fluid flow and A sample is the sample's surface area). As previously observed [START_REF] Innocentini | Influence of sample thickness and measurement set-up on the experimental evaluation of permeability of metallic foams[END_REF] the sample thickness as well as the A flow over A sample ratio may affect the measurement of permeability constants. Ratios far from 1 give rise to a stagnant space (where the fluid doesn't flow) so to an enlargement of the flow section. Decreasing the sample thickness can reduce this effect. Indeed, our experiments were performed at high A flow /A sample ratios and at low sample thickness ( ∼ 6 mm). So that we can estimate a reduced flow deformation and a consequent negligible effect on the measurement of the permeability constants.

Argon flow from a commercial cylinder (99.9% purity) upward through the disk and was controlled by a needle valve. Pressure drop across the disk ( P Δ sample ) as well as the pressure drop between the sample exit and the atmosphere ( P Δ exit ) were measured by a digital manometer (Greisinger electronic GmbH, Germanymodel GMH 3161-01 CE, range 0-25 mbar, resolution of 0.01 mbar). The pressure drop parameter used to fit the experimental data was calculated using Eq. ( 5 [START_REF] Biasetto | Gas permeability of lanthanum oxycarbide targets for the SPES project[END_REF]. The resulting volumetric argon flow rate (Q) across the disk was measured by a rotameter (CT Platon, France model NG GTF2BHS, 1-10 NL/min, resolution of 0.5 NL/min). Argon flow temperature was measured in the exit stream with help of a digital thermometer with a resolution of 0.1 °C. In order to ensure a reliable fitting analysis, at least 15 sets of pressure drop and flow rate were measured in steady-state conditions. The experiment was reproduced three times for each sample.

The collected data were fitted according to the least-square method using a parabolic model of the type = + y ax bx 2 . Using this procedure and Eq. ( 5), we have = y P L Δ / , = x v s , = a μ K / F and = b ρβ so that K F and β constants can be easily extrapolated from experimental data. Note that we use a one-dimensional Forchheimer equation here, because it was shown in [START_REF] Lasseux | On the stationary macroscopic inertial effects for one phase flow in ordered and disordered porous media[END_REF] that the Forchheimer correction tensor is diagonal for a disordered isotropic structure when estimated on a Representative Elementary Volume.

Numerical methods

3D µCT images show that the Ti6Al4V open-cell metallic foams feature a wide range of pore sizes. The low-resolution image is therefore a compromise between precision and representativity. Furthermore, we'll see later that we cannot be sure that the whole image is sufficiently large to be representative. As a consequence, the main numerical challenge is to be able to perform numerical computations on very large volumes. As it is not tractable to mesh such a large image [START_REF] Diani | Numerical investigation of pressure drop and heat transfer through reconstructed metal foams and comparison against experiments[END_REF], we use the implicit discretization of the image, i.e. we use voxels as elementary volumes. The main advantage of this approach is that we do not have to remesh the image, we just need to store the binary image and iterate through it to generate the linear systems at each iteration. The main drawback is that it is not easy to refine the mesh near pore/ solid interfaces or in small pores. We developed a homemade c++ code to solve the steady incompressible Navier-Stokes equations directly on binary images:

∇ = v • 0 (6) ∇ ⊗ =-∇ + v v v p μ •(ϱ ) Δ (7)
Equations are discretized using a staggered grid and a hybrid differencing scheme. We implemented SIMPLEC, SIMPLER and SIMPLER-R [START_REF] Li | An efficient SIMPLER-revised algorithm for incompressible flow with unstructured grids[END_REF] algorithms for pressure-velocity coupling. The SIMPLER-R algorithm was used for the Ti6Al4V-700 sample and for pressure drops over × 6 10 Pa/m 3 , where SIMPLEC and SIMPLER methods were not able to converge (see the reference book [START_REF] Versteeg | An Introduction to Computational Fluid Dynamics: The Finite Volume Method[END_REF] for further details about numerical methods).

To reproduce the experimental setup, a constant pressure is imposed at the inlet and at the outlet, while impervious boundary conditions are imposed on the other faces. No-slip condition is assumed at the solid/air interface.

Simulations are stopped when the relative maximum mass residual as well as the relative momentum components residuals are lower than

- 10 5 .
In a first series of numerical simulations, the Navier-Stokes equations are solved for different pressure gradients ranging from 10 3 to 10 5 Pa/m. In the same way as for experimental measurements, the superficial velocity v s is measured and the Forchheimer permeability K F as well as the Forchheimer inertial coefficient β are deduced from a parabolic fit using Eq. ( 4). In a second series of numerical simulations, we solve the steady Stokes equation (i.e. neglecting the inertial term

∇ ⊗ v v •(ϱ
) in Eq. ( 7) to compute directly the Darcy permeability K D . In this case, only one simulation is sufficient, since K D is a geometrical parameter and therefore does not depend on the pressure drop.

Results

Structural characterization

In Table 1 an overview of the main processing parameters and physical properties after sintering is reported.

Previous 2D morphological analysis [START_REF] Biasetto | Ovalbumin as foaming agent for Ti6Al4V foams produced by gelcasting[END_REF] have shown that the two samples have different cell size and porosity, as a consequence of the effectiveness of the gelcasting approach employed to stabilize the gas bubbles against the destabilization mechanisms. Thus, some bubbles may shrink and disappear whilst others may coalesce to form larger bubbles [START_REF] Sepulveda | Processing of cellular ceramics by foaming and in situ polymerisation of organic monomers[END_REF], due to a thinning process of the particle coating (lamella) surrounding the bubbles, and the consecutive increase of the mechanical shearing [START_REF] Mabille | Monodisperse fragmentation in emulsions: mechanisms and kinetics[END_REF][START_REF] Gonzenbach | Tailoring the microstructure of particle-stabilized wet foams[END_REF] (up to 1500 rpm); by this route, the cell sizes decreased [START_REF] Meille | Mechanical properties of porous ceramics in compression: on the transition between elastic, brittle, and cellular behavior[END_REF] from 732 to 532 µm, approximately. In addition, the gelation process can also lead to a significant local shrinkage of the wet foam favoring the rupture of the lamella around the bubbles, resulting in the interconnection of cells after drying and sintering. The total porosity of the final porous ceramic is linked to the amount of gas incorporated during the foaming process [START_REF] Sepulveda | Processing of cellular ceramics by foaming and in situ polymerisation of organic monomers[END_REF]. SEM images of the two samples show that the microstructure of the Ti6Al4V open-cell metallic foams presents an interconnected network of spherical cells, homogeneously distributed with apparently dense struts (Fig. 5). This is also clearly visible in the 3D µCT images (Figs. 6 and7). Using µCT, the big pores of the Ti6Al4V-1500 appears to be not always spherical, as opposed to the Ti6Al4V-700 sample, where the pores are also larger. No anisotropy could be noticed in the cellular structure. This was expected, given the process used, where pores are formed by air bubbles gelling and not by gas escape. In the high-resolution image (Fig. 7), we can distinguish a smaller porosity (d < 10 µm) in the cell walls where the Ti6Al4V powders are not in strict contact to each other due to partial sintering. To complete this qualitative analysis, the pore size distribution was measured on the low-resolution image of the two samples (Fig. 8). These results highlight the differences between the two samples. In the Ti6Al4V-700, the small pores in the cell walls represent about 5% of the total porosity, while 50% in the Ti6Al4V-1500 sample (pores size smaller than 50 µm approximately, tallest peak in Fig. 9). For a better precision, Fig. 9 shows the pore size distribution measured in the high-resolution image of the Ti6Al4V-1500 sample, for a pore size range between 2.8 and 50 µm. These results indicate that a more than 50% of the pores in the cell-wall are smaller than 17 µm. In Fig. 8, we can also notice that pores are also bigger in the Ti6Al4V-700 sample: maximum pore size is 2.4 mm vs 0.8 mm in the Ti6Al4V-1500 sample. The average pore size and standard deviation of the Ti6Al4V-700 and Ti6Al4V-1500 samples are ± 792.5 637.7 μm and ± 84.9 93 μm respec- tively (Fig. 7).

Geometrical representative elementary volume

If the small-scale images are clearly too small to be representative, an important question is to verify if low-resolution images contain a Geometric Representative Elementary Volume (GREV). A simple test is to measure the porosity in series of volumes of increasing sizes centered at different locations in the whole image. Figs. 10 and11 show the porosity measured in cubes centered at 10 random locations versus the ratio of the cube's volume over the image volume in the two samples. These results show that the porosity is not fully stabilized before reaching the whole volume. However, from volumes of about 30% of the total volume the variation of the measured porosity φ is less than × φ 0.025 tot , which is a criterion used in previous work [START_REF] De Carvalho | Pore-scale numerical investigation of pressure drop behaviour across open-cell metal foams[END_REF]. The porosity is however generally not sufficient to conclude about the representativity of a porous medium. In previous numerical studies [START_REF] De Carvalho | Pore-scale numerical investigation of pressure drop behaviour across open-cell metal foams[END_REF][START_REF] Brun | Microstructure and transport properties of cellular materials: representative volume element[END_REF], it was found that a computational volume containing about 3 to 4 mean pore diameter per side was sufficient to achieve a good representativity (both geometrically and physically). In our case, there are about 9 mean pores per side in the Ti6Al4V-700 image and 80 mean pores per side in the Ti6Al4V-1500 image. The pore size distributions of the two samples show however an important standard deviation and it could be more relevant to use the maximum pore diameter measured in the pore size distribution. In this case, the number of maximum pore per side is approximately 3 for the Ti6Al4V-700 image and 9 for the Ti6Al4V-1500 image. In view of these results, the image dimensions seem to be large enough to achieve a sufficient geometrical representativity, but only numerical simulations will tell if the volumes are sufficiently large to be representative for the studied phenomenon. (see Eq. ( 5)) as a function of fluid velocity for Ti6Al4V-700 and Ti6Al4V-1500 samples are reported in Fig. 12. In order to identify flow regimes, the reduced pressure gradient Lv ΔP s is plotted against the superficial velocity v s , as in [5,33] for the Ti6Al4V- 1500 sample (Fig. 13) and for the Ti6Al4V-700 sample (Fig. 14). Following the analysis in [START_REF] Dukhan | A two-permeability approach for assessing flow properties in metal foam[END_REF], it seems that the transition between Darcy and inertial regime occurs between 0.12 and 0.24 m s 1 for the Ti6Al4V-700 sample, so that a large majority of the data points are in the inertia regime. No clear threshold between weak and strong inertia can be detected here, due to precision of experimental data. For the Ti6Al4V-1500, it seems that the whole data range is in the inertia regime. The Forchheimer regime (i.e. linear part) seems to begin for velocity values greater than 0.6 m s 1 . Under that threshold velocity, we observe a transitional regime (possibly mixed with weakly inertial) between Darcy and Forchheimer regime.

Gas permeability: experimental results

Typical

In order to compute β and K F , we fit data points above 0.6 m s 1 to Eq. ( 5) for the Ti6Al4V-1500 sample and above 0.4 m s 1 for the Ti6Al4V-700 sample. The Darcy permeability K D is estimated using only the first point (approximately 0.12 m s 1 for both samples). Indeed, the reduced pressure gradient is constant vs velocity in the Darcy regime. All these values, as well as the relative difference

= - ε K K K | | D F D , are
reported in Table 2. Standard deviations are given in parenthesis. As expected from previous analysis, K D and K F are very close in the case of the Ti6Al4V-1500 sample because the whole data range lies in the inertia regime. It is therefore not possible to estimate K D with confidence in this case. The difference between K F and K D is more important in the case of Ti6Al4V-700 sample, which is expected since the Darcy permeability was estimated using a point at the end of the Darcy regime whereas K F was estimated only in the inertial regime.

In [START_REF] Lasseux | On the stationary macroscopic inertial effects for one phase flow in ordered and disordered porous media[END_REF], a correlation between porosity and the Reynolds number at the transition between weak and strong inertia regime (noted Re Kc ) was given for disordered structure:

= - Re p p 0.69 (1 ) Kc 1.49 0.36 ( 8
)
where p is the porosity.

Values of this crossover Reynolds number for each sample and corresponding velocity computed with Eq. ( 2) is given in Table 3 for porosity measured experimentally and on tomographic images. The results indicate that the transition velocity determined using the correlation corresponds well with the transition identified graphically. Even if the correlation was obtained for a different structure, it gives a rather good estimate of the threshold velocity between weak and strong inertia regimes.

At last, the measured values for Darcy and Forchheimer constants are in good agreement with literature data, which reports a behavior dependent on mixing rate: Ti6Al4V-1500 shows typical values of metallic foams, whilst Ti6Al4V-700 exhibits a behavior typical for granular or fibrous media [START_REF]Cellular Ceramics: Structure, Manufacturing, Properties and Applications[END_REF].

Numerical results

Mesh size

For the Ti6Al4V-700 sample, pores are big enough to downscale the image to a resolution of 14 µm/voxels without losing fine details. In this case the numerical simulations are performed on a sub-volume of size × × 384 384 384 voxels ( × × 5.38 5.38 5.38 mm 3 ). As we've seen previously, about 50% of the porosity of the Ti6Al4V-1500 sample consists of small pores of size smaller than 42 µm, i.e. 6 voxels at the original resolution of 7 µm/voxel. It is therefore not possible to downscale this image if we wish to keep the finer details of the Ti6Al4V-1500. The high-resolution image cannot be used as well, since its volume is far too small to be representative. The whole image's size is × × 1000 1000 1000 voxels at the original resolution. Without access to a high-performance computer cluster, it is not possible to perform numerical simulations on such large image, as it contains approximately × 720 10 6 finite volume cells. Numerical simulations are there- fore carried out in smaller computational volumes of × × 400 400 400 voxels ( × × 2.8 2.8 2.8 mm 3 ), which are sufficiently large to contains a few large pores (4-5 per side). The number of cells in these sub-volumes is then approximately 46 × 10 6 . Note that simulations were performed on small volumes at higher resolution (3.5 µm/voxel and 2.33 µm/ voxel) to verify that the results are independent of the mesh size. Relative differences from 5% to -15% for the Darcy permeability were observed for various subvolumes of the Ti6Al4V-1500 sample for a resolution of 2.33 µm/voxel (i.e. a × 3 upscaling).

Results

Results of simulations performed on the Ti6Al4V-700 as well as one of the corresponding experimental data are presented in Fig. 15. A very good agreement between the two approaches is achieved here. It is a good indication that the computational volume is sufficiently representative and that the resolution is well chosen. It is interesting to notice that the length of the computational volume is here only 2.25 times larger than the max pore size (2.4 mm) and approximately 6.7 times larger than the average pore size (792.5 μm). While we cannot be sure that the computational volume is a REV, these results indicate that a size of only a few maximum pore diameters is sufficient to achieve a good representativity. Values of Forchheimer permeability and coefficient estimated by curve fitting are = × - K 30.5 10 m ) and numerical K D is due to the fact that K D was estimated using only the first experimental point, which is not in pure Darcy regime. In this case, the numerical approach is better suited for the determination of the Darcy permeability.

Fig. 16 shows the evolution of pressure gradient versus measured superficial velocity for five different sub-volumes of the Ti6Al4V-1500 image, as well as the experimental results. An overview of the numerical results is presented in Table 4. Note that the absolute error on the average value of a property Z (here the permeability or Forchheimer coefficient) computed in n volumes of size V can be estimated by [START_REF] Kanit | Determination of the size of the representative volume element for random composites: statistical and numerical approach[END_REF]: Crossover Reynolds (from weak to strong inertia regime) calculated from experimental and image porosity, as well as corresponding velocity ranges from Eq. ( 8) and velocity estimated using experimental curves. It is clear that the curves obtained from numerical simulations show a noticeable deviation from the experimental ones. As a result, there is a significant discrepancy between the values of K D , K F and β estimated from numerical and experimental data. The relative difference is indeed greater than 70% for all parameters. Several factors can explain these differences: firstly, the resolution of the tomographic image of the Ti6Al4V-1500 sample is probably too low to discretize correctly the microstructure and particularly the smaller pores. With such a spatial resolution, it is likely that pores smaller than 7 µm are not detected and that the discretization effect reduces the size of pores between 7 and 14 µm, thus decreasing the apparent permeability. Note that pores which diameter is smaller than 14 µm represent 40% of the pores in the cell-wall (i.e. smaller than 50 µm), as shown in Fig. 9, which gives about 20% of the total porosity.

Secondly, the size of the computational volumes seems be smaller than the flow-dependent REV size, as it was observed in [START_REF] De Carvalho | Pore-scale numerical investigation of pressure drop behaviour across open-cell metal foams[END_REF][START_REF] Laschet | Effective permeability and thermal conductivity of open-cell metallic foams via homogenization on a microstructure model[END_REF]. While the values for Darcy permeability are all of the same order of magnitude, there is a noticeable variation for the Forchheimer coefficient. Note however that these values are strongly correlated to the porosity of each sub-volume. In previous studies [START_REF] De Carvalho | Pore-scale numerical investigation of pressure drop behaviour across open-cell metal foams[END_REF][START_REF] Brun | Microstructure and transport properties of cellular materials: representative volume element[END_REF], it was found that the REV must contains at least 3 average pore diameter, whereas in our case the side of each computational volume is 3.5 times larger than the max pore diameter (approximately 800 µm). The fact that this constraint is not sufficient to achieve a good representativity in the case of the Ti6Al4V-1500 sample can be due to the double pore size distribution observed previously. However, even if the computational volumes are smaller than the REV, K D and K F values are quite close for all volumes and it is likely that permeability values won't change that much with an increase of the computational volume.

At last, it is worth mentioning that it is not possible to reproduce exactly the experimental boundary conditions in numerical simulations. Indeed, what is measured in the experiments is an external pressure drop across the sample, while this the internal pressure drop which is fixed in the in numerical simulations. These two quantities may differ but it is difficult to estimate if this difference is meaningful or not. All the experimental and numerical results as well as the absolute value of the relative difference (always divided by the maximum value) are summarized in Table 5.

Figs. 17 and 18 show respectively the pressure and the velocity fields computed in the Ti6Al4V-1500 sample on a 2D cut-plane. These two fields show a behavior in agreement to other similar studies [START_REF] De Carvalho | Pore-scale numerical investigation of pressure drop behaviour across open-cell metal foams[END_REF]. The velocity field shows important variations with areas of high velocity in the narrow pores apertures and low-velocity areas in the wake of solid ligaments. In Fig. 19, we can observe the tortuous path of streamlines in the complex microstructure.

Conclusions

Ti6Al4V interconnected foams were produced using the gelcasting route where porosity and pore size were controlled by means of the mixer rotation speed during foaming of ovalbumine. The two used rotation speeds used gave rise to two sets of samples, Ti6Al4V-700 and Ti6Al4V-1500, with measured porosities of ∼81% and 79% respectively. Low resolution and high resolution micro-tomography analyses confirmed the presence of similar level of porosities where a double pores size distribution was observed. The average pore size being larger 2 ). The pore space is colored by the norm of the velocity field.

for Ti6Al4V-700 than for Ti6Al4V-1500. For this last set of samples, high resolution images were necessary to obtain useful information concerning smaller pores size distribution. Flow regime were first identified using plots of the reduced pressure gradient vs superficial velocity. In the case of Ti6Al4V-1500 sample the inertial effect became dominant on the overall pressure drop for fluid velocities higher than 0.6 m/s. This can be attributed to the presence of pores of small size (⩽ 50 μm). In the case of the Ti6Al4V-700 foam, the inertial regime begins for velocities higher than 0.24 m/s approximately. These transition velocity values are very close to those calculated using correlations given in [START_REF] Lasseux | On the stationary macroscopic inertial effects for one phase flow in ordered and disordered porous media[END_REF] for estimated porosities. Then, Darcy permeability and Forchheimer coefficient values were derived from interpolation of gas permeability data by Darcy and Forchheimer equations. Simulations performed using the Navier-Stokes equation on large volumes (46 × 10 6 finite volume elements) in the low-resolution tomography show very good agreement with the experimental results for the Ti6Al4V-700 foams whilst a clear deviation was observed for the Ti6Al4V-1500 samples. This last result can be attributed to the double pore size distribution of the sample which make the choice of the image's resolution quite difficult. It is indeed not possible at the moment to have both a sufficient spatial resolution to discretize the fine scale details and a sufficient volume size for representativity. However, the difference between experimental and numerical results in the case of the Ti6Al4V-1500 foam is still less than one order of magnitude, which is not so bad, in view of the discretization and representativity issue. A possible solution could be to use a multiscale approach (using upscaling techniques like volume averaging [START_REF] Whitaker | The Forchheimer equation: a theoretical development[END_REF] or periodic homogenization [START_REF] Sanchez-Palancia | Non-homogeneous media and vibration theory[END_REF]) to deal with the double porosity. The numerical approach coupled with micro-tomographic imaging is however an invaluable tool to study the relation between morphology and transport properties of metallic foams. As such, it could be used for process optimization of these materials.

Fig. 1 .

 1 Fig. 1. (a) 3D visualization of the low-resolution image of the Ti6Al4V-700 sample (5.4 × × 5.4 5.4 mm 3 , resolution 14 µm/voxelsafter binning of the initial reconstruction) and (b) of Ti6Al4V-1500 sample (2.8 × × 2.8 2.8 mm 3 , resolution 7 µm/voxelsno binning). Solid matrix in light grey.

Fig. 2 .

 2 Fig. 2. (a) 3D visualization of the Ti6Al4V-700 sample (0.54 × × 0.54 0.54 mm 3 , resolution 1.4 µm/voxels after binning) and (b) of the high-resolution image of Ti6Al4V-1500 sample (0.47 × × 0.58 0.63 mm 3 , resolution 1.4 µm/voxels after binning). Solid matrix in light grey.

Fig. 3 .

 3 Fig. 3. Slice (1000 × 1000 pixels) of the low-resolution grey level image (a) and corresponding slice in the binary image (b).

Fig. 4 .

 4 Fig. 4. Schematic drawing of the experimental setup used for the measurement of permeability.

  experimental data of the measure of the pressure gradient defined as = -

Fig. 5 .

 5 Fig. 5. FEG-SEM pictures of foams processed at 700 rpm (left) and 1500 rpm (right).

Fig. 6 .

 6 Fig. 6. (a) 3D binary representation of the Ti6Al4V-700 sample, (5.4 × × 5.4 5.4 mm 3 , resolution 14 µm/voxelsafter binning of the initial reconstruction) and (b) of the low-resolution image of the Ti6Al4V-1500 sample (2.8 × × 2.8 2.8 mm 3 , resolution 7 µm/voxelsno binning. Solid matrix in white, porosity in black.

Fig. 7 .Fig. 8 .Fig. 9 .

 789 Fig. 7. (a) 3D binary representation of the high-resolution image of the Ti6Al4V-700 sample, 1.4 µm/voxel (after binning) and (b) of the Ti6Al4V-1500 sample, 7 µm/ voxel (after binning). Solid matrix in white, porosity in black.

Fig. 10 .

 10 Fig. 10. Evolution of porosity in ten cubes centered randomly in the whole lowresolution image of the Ti6Al4V-700 sample versus the ratio of the cube volume over the image volume.

Fig. 11 .

 11 Fig. 11. Evolution of porosity in ten cubes centered randomly in the whole lowresolution image of the Ti6Al4V-1500 sample versus the ratio of the cube volume over the image volume.

  the value of the Darcy permeability computed using Stokes flow simulation is = in the experimental case, K F and β are estimated in the inertia regime only while K D is determined using Stokes flow simulation. The difference observed between experimental (21.6 × -

Fig. 12 .

 12 Fig. 12. Sample curves for pressure gradient versus fluid velocity for Ti6Al4V-700 and Ti6Al4V-1500 and their quadratic fittings.

Fig. 13 .

 13 Fig. 13. Evolution of the reduced pressure gradient with superficial velocity for 3 experimental results on the Ti6Al4V-1500 sample.

Fig. 14 .

 14 Fig. 14. Evolution of the reduced pressure drop with superficial velocity for 3 experimental results on the Ti6Al4V-1500 sample.

Fig. 15 .

 15 Fig. 15. Pressure gradient versus superficial velocity for the Ti6Al4V-700 sample.

Fig. 16 .

 16 Fig. 16. Pressure gradient versus superficial velocity for the Ti6Al4V-1500 sample.

F 10 *

 10 Experimental K D was not estimated in pure Darcy regime and is therefore under estimated.

Fig. 17 .

 17 Fig. 17. Pressure field in a slice of the Ti6Al4V-700 image ( × mm 6.1 6.12 ). Flow direction from left to right.

Fig. 18 .

 18 Fig. 18. Velocity field in a slice of the Ti6Al4V-700 image ( × mm 6.1 6.12 ). The pore space is colored by the norm of the velocity field.

Table 1

 1 Overview of main processing parameters and physical properties of Ti6Al4V-700 and Ti6AL4V-1500 samples.

	Sample	Mixing	T sint	Density [g/cm 3 ] Porosity	Porosity
		Rate	[°C]		(Experimental)	(Image
		[rpm]			[vol.%]	analysis)
						[vol.%]
	Ti6Al4V-	700	1400	0.79 ± 0.08	81.3 ± 0.5	83.4
	700
	Ti6Al4V-	1500	1400	1.25 ± 0.01	70.9 ± 0.9	72.2
	1500

Table 2

 2 Flow parameters extrapolated from experimental results (standard deviation in parenthesis).

	Sample	-K [10 D 10 2 m ]	-K [10 F 10 2 m ]	-β [10 m ] 4 1	ε
	Ti6Al4V-700	21.6 (1.3)	27.2 (2.9)	0.53 (0.032)	25.8%
	Ti6Al4V-1500	1.55 (0.03)	1.54 (0.14)	12.1 (0.36)	0.38%

Table 3

 3

Table 4

 4 Overview of the numerical results for the Ti6Al4V-1500 sample.

	Sub-volume	Porosity K D from Stokes	K F from curve	β from curve
			simulations	fitting	fitting
			× -10 11 (m ) 2	× -10 11 (m ) 2	× 10 4 ( -m ) 1
	VOL1	71.8%	2.70	2.85	55.1
	VOL2	74.5%	4.05	3.77	17.4
	VOL3	74.2%	5.18	4.49	6.52
	VOL4	69.6%	2.30	2.54	95.3
	VOL5	73.7%	3.67	4.12	61.3
	Average	72.8%	± 3.58 0.39	± 3.56 0.31	± 47.1 28.1
	Standard	1.65%	0.86	0.68	12.6
	Deviation

Table 5

 5 Summary of numerical and experimental results as well as their relative difference.

	Sample	Parameters	Numerical Experimental Relative
							difference
	Ti6Al4V-700	K	D	× -10 10 (m 2 ) 27.7	21.6 *	22%
		K		× -10

Acknowledgment

Authors wish to thank Dr. S. Corradetti and Dr. S. Carturan, Laboratori Nazionali di Legnaro, INFN-Italy for kind support in gas permeability measurements and heat treatments of foams.

Funding sources

This work was funded by Università di Padova project # BIAS_SID_16_01.

Data availability

The raw/processed data required to reproduce these findings can be shared on demand.

Author contributions

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript.

Declarations of interest

None.


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From sustainability to sobriety

Do not think of yourself more highly than you ought, but rather think of yourself with sober judgment 1 Saint-Paul

Considering the current ecological crisis, 2 the concept of sustainable development, or 'sustainability' appears to have failed to meet the goals laid out by its authors at the 1992 Earth Summit of Rio de Janeiro. 3 Sustainable development was originally perceived as the torchbearer of a new project, 4 a new hope of protecting humanity's general interest, 5 a 'magic formula meant to reconcile free trade, economic growth and environment'. 6 As it turns out, the contradiction inherent in sustainable development has never been wholly resolved: 'In Rio, some countries talked about the environment only, whereas others talked about development only'. 7 Today, a sense of scepticism, of wariness towards a 'superficial ecology,' 8 reflects the failure of sustainable development to fulfill its original mission of preserving the environment, having ultimately become a simple foil for 'soft' economic growth, also known as 'green growth'. In other words, development has prevailed over sustainability. Multiple technologies and 'low-tech innovations' 9 have been developed to curb the adverse effects of economic growth, but they never genuinely challenge the imperative of growth. 10 This raises the question of what kind of economy and by extension what kind of 1 SAINT PAUL, Epistle to the Romans, AD 57, 12--3 (NIV translation), 'Non plus sapere quam opertet sapere, sed sapere ad sobrietatem'. 2 Already diagnosed in the 1970s (WHITE Lynn Townsend, 'The Historical Roots of Our Ecological Crisis ', Science, vol. 155, n°3767, 10 mars 1967, pp.1203--1207), the crisis is an increasingly serious concern, fuelled by alarming recent reports on global warming (Intergovernmental Panel on Climate Change (IPCC), Special Report on Global Warming of 1.5°C, 8 Oct. 2018) and biodiversity collapse (DUROUSSEAU Michel, « Le constat : la biodiversité en crise », RJE, 2008, hors--série, pp. 11--19).

future we want, at odds with the fatalistic view that we are 'destined to produce not only goods, but also new servitudes '. 11 While these 'technological revolutions' may be desirable, some authors see such projects as mirages in that they suggest that energy transition is possible without drastically reducing consumption: 12 'the most reckless ones essentially never raise the question of natural resources and raw materials: where will rare earths, plastics, steel, aluminium, copper, gold, silver, construction materials, etc., come from? The most flagrant example of this is certainly the construction of a car factory by the company Tesla, whose annual production of half a million of battery-operated electric cars would absorb the current global production of lithium-ion.' 13 Paradoxically, it appears that reducing the reliance of some resources causes increased dependence on others, as in the case of the 'dematerialised' economy, which accounted for 7 per cent of the world's electricity consumption as of 2017 according to Greenpeace. 14 Sobriety -from the Latin sobrietas -in the sense of caution, moderation, temperance or frugality, as opposed to 'sustainability' understood as green growth as the only possible horizon, should help us conceive a different approach to environmental protection. This involves no longer only limiting the consumption of natural resources, but seriously considering not using them all. 15 Generally, sobriety should be conceived functionally, meaning that it is liable to change behaviours. Drawing on such an approach, the introduction of an 'environmental ceiling' and a 'social foundation' has been suggested to work towards a 'resilient' development, 16 tackling a wide range of issues, including environmental inequalities.17 Is the idea of sobriety a new concept that may allow us to resolve the contradictions inherent in the sustainable development concept? Can it help change behaviours in law? 18 Beyond the concept itself, can it be implemented by a variety of economic actors, especially firms? To answer these questions, this paper will first show that sobriety is actually an old concept that is currently experiencing a renewal (I) and then examine illustrations of it in various branches of French law (II).

l'environnement et le développement, Thèse, 2014, p.474 : 'The spirit of capitalism produces unprecedented material wealth but it is also destroying the arts of life that Keynes cherished so. The arts of life are among the desirable goods that men must seek to lead a good life.' 11 ZAFRANI Avishag, « Quelle économie pour quel avenir ? Hans Jonas et la responsabilité », Arch. phil., 2016/3, t.79, p.192: 'Technological power not only creates a new field of possibles for human agency, it also introduces new necessities, new needs, that always come before the question of whether they are desirable is posed. As a result, following economic impulses, we are destined to produce not only goods, but also new servitudes.' Humanity", Oxfam Discussion Papers, 2012 [online] https://www.oxfam.org/sites/www.oxfam.org/files/dp--a--safe--and--just--space--for--humanity--130212--en.pdf

I.

Sobriety -an old concept experiencing a renewal

Although sobriety is not defined by French law, legislators have used the terms sobriety and sober in the field of the ecological transition since 2012, with the adoption of a 2015-2020 national strategy for ecological transition towards a sustainable development, designed to 'bring about the emergence of a new model of society, combining economic, ecological and human progress'. 19 The word sobriety appears in that document, as well as in various legislations pertaining to environmental protection. While it may appear recent, the concept is not new.

The idea of sobriety was already in use in Ancient times, when the Greek wrote of temperance (sophrosyne), as opposed to hubris, meaning immoderation, excessive behaviour considered as a vice, as a moral failing. Temperance was one of four cardinal virtues in human action, alongside prudence or wisdom, fortitude or courage and justice. Hence, the Greek adage pan metron calls for 'moderation in all things'. The Romans conveyed the same sense of temperance and moderation with the maxim Aut facio hoc moderate, et possum, aut facio hoc immoderate, et non possum, which translates as 'either I do this with moderation, and then I can, or I do this in an immoderate manner, and then I cannot do it'. Sobriety was also perceived in terms of health; the Latin adage medicorum nutrix est intemperantia, attributed to Publilius Syrus, contends that intemperance is the physician's provider. Likewise, in 1558, the Italian Luigi Cornaro wrote: 'The man, also, who is of a bad constitution, may, by dint of reason and a regular and sober life, live to a great age and in good health'. 20 Sobriety is also present in the Christian religion, in which the indulgence controversy was a major juncture. Indulgences were a lucrative trade for the Church, which promised a reduced number of years in purgatory in exchange for sums of money, a practice denounced by Martin Luther in his Ninety-Five Theses (1517), 21 one of the triggers of the Protestant Reform in the early sixteenth century, followed by Calvinism -both stressing the importance of asceticism in everyday life.

During the Enlightenment and the French Revolution, 22 , the neighbouring term frugality was perceived as a means to ensure democracy. To Montesquieu, equality and frugality were inseparable: 'The love of equality, in a democracy, limits ambition to the sole desire, to the sole happiness, of doing greater services to our country than the rest of our 21 LUTHER M., Disputatio pro declaratione virtutis indulgentiarum, Wittemberg, 1517. The following theses are particularly important: '27. They preach only human doctrines who say that as soon as the money clinks into the money chest, the soul flies out of Purgatory. 28. It is certain that when money clinks in the money chest, greed and avarice increases; but when the church intercedes, the result is in the hands of God alone. 45. Christians are to be taught that he who sees a needy man and passes him by, yet gives his money for indulgences, does not buy papal indulgences but God's wrath. 46. Christians are to be taught that, unless they have more than they need, they must reserve enough for their family needs and by no means squander it on indulgences. 47. Christians are to be taught that they buying of indulgences is a matter of free choice, not commanded. 48. Christians are to be taught that the Pope, in granting indulgences, needs and thus desires their devout prayer more than their money. 56. The true treasures of the Church, out of which the Pope distributes indulgences, are not sufficiently discussed or known among the people of Christ. 57. That indulgences are not temporal treasures is certainly clear, for many indulgence sellers do not distribute them freely but only gather them.' 22 OUTREPONT Ch.--L D', Discours sur la frugalité, prononcé au temple national de Bruxelles, le 10 ventôse de l'an troisième., Ed. impr. De G. Huyghe, 1795; see also MONTESQUIEU, De l'Esprit des lois, éd. Nourse, 1772, T. 1, Livre V chapitre VI --Comment les lois doivent entretenir la frugalité dans la démocratie, pp. 56--57.

fellow-citizens', whereas 'The love of frugality limits the desire of having, to the study of procuring, necessaries to our family, and superfluities to our country. Riches give a power which a citizen cannot use for himself, for then he would be no longer equal. They likewise procure pleasures which he ought not to enjoy, because these would be also repugnant to the equality.' Montesquieu argues that in order for individuals to aspire to this love of equality and frugality, they need to enjoy it, and thus laws must establish equality and maintain frugality: 'It is not sufficient, in a well-regulated democracy, that the divisions of land be equal; they ought also to be small (…). As equality of fortunes supports frugality, so the latter maintains the former. These things, though in themselves different, are of such a nature as to be unable to subsist separately: they reciprocally act upon each other: if one withdraws itself from a democracy, the other surely follows it.' 23 While Montesquieu thought of the absence of equality and frugality as a threat to democracy itself, 24 the same could be said today of the absence of sobriety.

With the industrial revolution and the rise of capitalism came a historically unprecedented shift towards an immoderate accumulation of goods. Then, in the 'glorious' post-war years, 25 the first challenges to a system based on exponential growth and its nefarious consequences were voiced. In the wake of the events of May 1968 and of the mobilization of part of French society, the Club of Rome published a report entitled The Limits to Growth (1972), which pondered the limits of infinite growth in a finite world. 26 In 1987, the Brundtland report highlighted the idea of 'sustainable development', which would be picked up at the 1992 Earth Summit, understood as development that meets present needs without compromising the ability of future generations to meet theirs.

In French law, since 2005, the Charter for the Environment, which was literally grafted onto (adossée) the Constitution of 4 October 1958, indicates in its preamble that 'Biological diversity, the fulfilment of the person and the progress of human societies are affected by certain types of consumption or production and by excessive exploitation of natural resources'. The phrase 'excessive exploitation' conveys the idea of immoderation and of a necessary return to sobriety to ensure a sustainable development. Likewise, Article 191 of the Treaty on the Functioning of the European Union (TFEU) states that the EU's environmental policy shall contribute to the pursuit of objectives such as the 'prudent and rational utilisation of natural resources'. The idea of sobriety is thus present in these texts even though the actual term is not used. At any rate, the objective of sustainable development inscribed in the Charter for the Environment involves the concept of sobriety. This calls for establishing what role the latter plays in French law -which legal materials and texts translate Montesquieu illustrates his argument with a reference to Roman soldiers who asked for a bigger portion of conquered land than was necessary to feed a man (Plutarch, Moral works, Lives of the ancient Kings and Commanders). 24 Ibid. To Montesquieu, the 'spirit of commerce' itself does not bring the inconveniences of inequality; 'excessive wealth destroys the spirit of commerce'. 'In order to support this spirit, commerce should be carried on by the principal citizens: this should be their sole aim and study; this the chief object of the laws: and these very laws, by dividing the estates of individuals in proportion to the increase of commerce, should set every poor citizen so far at his ease, as to be able to work like the rest; and every wealthy citizen in such a mediocrity, as to be obliged to take some pains either in preserving or acquiring a fortune.' 25 The French phrase Trente Glorieuses, referring to the country's years of unprecedented growth and prosperity between 1946 and the 1974 oil shock, was coined by the economist Jean Fourastié. FOURASTIÉ Jean, Les Trente Glorieuses : ou La révolution invisible de 1946 à 1975, Paris, Pluriel, 2011 (first published 1979). 26 DELAUNAY J., « Halte à la croissance ? », Enquête sur le Club de Rome et Rapport sur les limites de la croissance par MEADOWS Donella H., MEADOWS Dennis L., RANDERS Jorgens, BEHRENS III William W du Massachusetts Institute of Technology, trad. de l'anglais par DELAUNAY Jacques, Coll. Ecologie, Fayard, 1972 the concept of sobriety?

II. Sobriety in French law

Based on the objective of the pursuit of a 'prudent and rational utilisation of natural resources' laid out by Article 191 of the TFEU, two types of sobriety can be identified. The first is quantitative sobriety -using natural resources less or prudently -(A); the second is qualitative sobriety -using natural resources better or rationally -(B). It should be noted, however, that sobriety does not only pertain to natural resources; it may also relate to the use of new technologies liable to affect human health or the environment. These two types of sobriety may thus exist cumulatively in several areas of law, such as the sectors of energy, health, consumption, telecommunications, food and public procurement. Depending on the area in question, they may concern the State, local authorities and/or firms.

A. QUANTITATIVE SOBRIETY

The objective of 'prudent utilisation' calls for the quantitative limitation of the use or harvesting of resources and of emissions resulting from the production of goods and services into the water, the air and the ground that may affect living beings. This objective is now relevant in several areas. The Law no. 2015-992 on Energy Transition for Green Growth provides for the reduction of energy consumption. The Law amended the Energy Code, whose Art. L100-1 1° now holds that: 'Energy policy: 1° Favours the emergence of a competitive, job-rich economy by mobilising all industrials sectors, including that of green growth, defined as an environment-friendly mode of economic development, both sober and efficient in terms of energy and resource consumption, socially inclusive, boosting the potential for innovation and ensuring the competitiveness of firms.'

To apply this article, the State and local authorities have been asked to enlist firms, non-profits and citizens in working towards energy sobriety, meaning reducing energy needs, which involves improving energy efficiency and adopting new measures (energy classification for products, energy performance diagnosis, 27 European energy labelling 28 for lightbulbs and household appliances such as washing machines, dishwashers and fridges).

In France, the so-called 'green growth' law of 17 August 2015 has further prioritised waste prevention, which consists in reducing the quantity and noxiousness of waste through measures impacting both production methods and consumption 29 (Extended Producer Responsibility) 30 , combating excessive packaging by having producers contribute for each 27 For instance, mentioning the energy efficiency class of a property is mandatory when it is being sold or rented.

Article *R134--5--1, French code of construction and housing. consumer sales unit. 31 Sobriety does not only concern the harvesting and consumption of natural resources. Service activities may indeed also cause harmful effects and pollution. Sobriety has for instance been promoted in the telecommunications sector, where reduced public exposure to electromagnetic fields is being pursued. For instance, it has become mandatory for facilities offering a Wi-Fi access to the public to indicate this on a sticker placed at the entrance. 32 The law no. 2015-136 of 9 February 2015 on sobriety, transparency, information and communication on exposure to electromagnetic waves, also known as Abeille law, provides for the French frequency agency to place a map of existing relay masts at the public's disposal at municipal level. Lastly, it remains mandatory to inform customers about the specific absorption rate (SAR) 33 of each wireless telecommunications equipment and of about the precautions to take in using them. 34 Fighting food waste has also been a recent concern for legislators, who have introduced an obligation for the State, public establishments and local authorities to implement measures to reduce food waste in catering. 35 A new offence has also been introduced in the French environmental code, providing for penalties for the removal of still edible foods from the market. 36 Lastly, where quantitative sobriety is concerned, Article L101-2 of the French urban planning code provides that public authorities in matters of urban planning shall act in compliance with goals such as 'controlled urban development' and with an 'economical use of natural spaces'.

B. QUALITATIVE SOBRIETY

For producers of goods and services, reducing the amount of natural resources used is necessary but not enough in itself. Curbing the harvest of natural resources may be praiseworthy, but their rational utilisation is also crucial. Below, I present several legislative illustrations of this in the areas of food, consumption and public procurement.

31 Ibid, Appendix 1, on consumer sales units: 'Consumer Sales Unit': a conditioned product unit that a consumer can buy separately. For the application of the present specifications, each "POS" and shipping packaging is considered as an independent unit, equivalent to a CSU. The CSU may be composed of different elements from different materials.' 32 Article 184 of law no. 2010--788 of 12 July 2010 (Grenelle II), as amended by law no. 2015--136 of 9 February 2015 on sobriety, transparency, information and communication on exposure to electromagnetic waves provides that in order to ensure the sobriety of public exposure to electromagnetic fields: 1° Instruction for use of radio--electric terminal equipment shall include clear information on practical indications for activating and de--activating wireless internet access; 2° Equipment emitting electromagnetic fields exceeding a threshold set by decree shall not be installed on private premises used for housing purposes without clear and legible information to occupants as to the presence of radiation, and when appropriate, recommendations for minimising exposure to radiation; 3° Establishments offering Wi--Fi access to the public shall indicate this clearly by affixing a sticker at their entrance. See also Article L32--1--II--9° of the French postal and electronic communications code: 'Within the framework of their respective powers, the Minister in charge of electronic communications and the regulatory authority for postal and electronic communications shall, in objective and transparent conditions, take reasonable and proportionate measures to reach the following goals: (…) 9° The sobriety of public exposure to electromagnetic fields'. 33 French postal and electronic communications code, art. R9: '"Specific absorption rate" shall be understood as the rate at which the energy produced by an equipment is absorbed per mass of body tissue, expressed in watts per kilogram (W/ kg), measured over the whole body, or on one of its parts'. 34 French postal and electronic communications code, Art. R20--10.

Concerning food, a bill (no. 3280, submitted to the National Assembly on 25 November 2015 and still under discussion as 19 February 2018) aims at promoting the reliance on local and sustainable foods. The bill's main article provides for the creation of a new article L 230-5-1 in the French rural and maritime fishing code, which reads: 'In a manner consistent with the food policy objectives defined in Article L. 1 of the present code, by 1 January 2020, the State, local authorities and public facilities shall in the meals served in their catering operations include 40 per cent of sustainable food products, i.e., produced under labels certifying quality and origin or featuring prestige-enhancing indications, defined by Article L. 640-2 of the rural and maritime fishing code, coming from short supply chains or meeting sustainable development criteria, including the seasonality of produce. 20 per cent of the products served shall result from organic farming.' Farming businesses will undoubtedly have to change their practices to be able to meet the demands of sustainable food for catering in facilities owned by the State, local authorities or public institutions. It is also conceivable that catering facilities owned by private parties could spontaneously take similar measures, or be subject to similar regulation in the future.

Concerning consumption in a broader sense, a legislation aimed at combatting planned obsolescence has recently been adopted. Indeed, sobriety calls for not systematically replacing a faulty product, but instead repairing it and providing for the availability of the parts needed to repair it. Therefore, developing a non-repairable product is now punishable by law: planned obsolescence has been defined, prohibited and made an offence under Article L441-2 of the consumer code.

Planned obsolescence has been defined as the recourse to techniques aimed at deliberately reducing the lifespan of a product so as to increase its replacement rate.

Combatting unplanned obsolescence, however, poses a bigger challenge. Indeed, an increase in product replacement rates may be simply due to technological innovation, making it obsolete and encouraging replacement (this is particularly visible in the field of new technologies). 37 This is also the case when the product becomes passé because of its design or fashion (cars, clothes), which calls for educating the consumer to prefer products that are less sensitive to trends.

To support the fight against planned obsolescence, a law has provided for mandatory pre-contractual consumer information as to the period during which spare parts indispensable for the use of the purchased good are available. 38 In the same spirit, in the automobile sector, decree no. 2016-703 of 30 May 2016 relating to the use of spare parts from the circular economy, allows the consumer to be informed, when their vehicle is undergoing maintenance or repairs, about the possibility of opting for spare parts from the circular economy instead of new ones. These parts may for instance come from licensed centres for the treatment of endof-life vehicles (ELV), and be reconditioned for reuse in a 'standard exchange'. This applies for instance to optical parts, removable body parts, brakes and steering components. 39 This means saving non-renewable resources, energy and raw materials for the environment, and 37 See Thierry LIBAERT, Ed. Les petits matins, Institut Veblen, 2017. The author for instance calls for displaying the life spans of products. 38 Law no. 2014--344 of 17 March 2014 on consumer protection. See. Art. L111--3 of the consumer code.

giving consumers access to more affordable spare parts. 40

*** Conclusion

Although the concept of sobriety appears in various legislative texts, there is no single sobriety principle, only elements towards a definition that suggest -as per the Greek adage pan metron (moderation in all things) -that sobriety consists in the quantitative and qualitative monitoring of the use and consumption of resources and materials for economic activity.

This tentative definition is obviously open to criticism. It might be argued that it is only a material definition, as the formulation only pertains to economic production -'having' and 'doing'. In the broader sense, we should also include the dimension of 'being' inherent in the underlying philosophical and historical roots of the objectives of the production of goods and services -this is what is at stake in the ecological transition. 41 In that sense, the plan to amend Article 1835 of the civil code to mention 'the raison d'être that the company intends to embrace in pursuing its activity' 42 is a first step towards behavioural sobriety. *

19

  French Ministry of ecology, sustainable development, Stratégie nationale de transition écologique vers un développement durable 2015--2020, adopted by the Council of Ministers on 4 February 2015, p.127: 'Ecological transition: evolution towards a new economic and social model, a model of sustainable development that renews our ways of consuming, producing, working, living together to face major environmental challenges; those of climate change, resource scarcity, accelerated biodiversity loss and multiplying environmental health risks.' 20 CORNARO L., Discourses on the Sober Life. How to Live 100 Years, 1558.

23

  MONTESQUIEU, The Spirit of Laws, Cambridge University Press, 1999, (1777), edited by A.M. Cohler, B.C. Miller and H.S. Stone, Volume 1, Book V, Chapter VI, "In what manner the laws must maintain frugality in a democracy".

  28 Art. L 122--11, French consumer code, established by Order no. 2016--301 of 14 March 2016 : 'When advertising materials, regardless of their support, present products subject to European energy labelling and indicate their retail price, they must include a mention of the energy efficiency class of these products, which is to be as visible, legible and intelligible as the indication of their retail price.'; Directive 2010/30/EU of 19 May 2010 on the indication by labelling and standard product information of the consumption of energy and other resources by energy--related products. Directive 2010/30/UE has been replaced with a European Council regulation (26 June 2017) for clearer energy labelling, replacing the A+, A++ and A+++ classes with a simpler A to G scale. 29 Art. L 541--1 and L541--1--1, French environmental code.

Art. L 541--1--I--3° and L541--10--II, French environmental code.

French environmental code, Art. L541--15--3.

See articles L541--15--5 and L541--15--6--III of the environmental code.

The provisions of this decree issued pursuant to Law no. 2015--992 of 17 August 2015 are listed in Articles R 121--26 to R121--29 of the consumer code.

This article received support from the Maison Interuniversitaire des Sciences de l'Homme d'Alsace (MISHA) and the Excellence Initiative of the University of Strasbourg.